Buffett was born in Omaha, Nebraska. He developed an interest in business and investing in his youth, eventually entering the Wharton School of the University of Pennsylvania in 1947 before transferring to and graduating from the University of Nebraska at 19. He went on to graduate from Columbia Business School, where he molded his investment philosophy around the concept of value investing pioneered by Benjamin Graham. He attended New York Institute of Finance to focus his economics background and soon after began various business partnerships, including one with Graham. He created Buffett Partnership, Ltd in 1956 and his firm eventually acquired a textile manufacturing firm called Berkshire Hathaway, assuming its name to create a diversified holding company. In 1978, Charlie Munger joined Buffett as vice-chairman.[6][7]

Buffett has been the chairman and largest shareholder of Berkshire Hathaway since 1970.[8] He has been referred to as the "Oracle" or "Sage" of Omaha by global media.[9][10] He is noted for his adherence to value investing, and his personal frugality despite his immense wealth.[11] Research published at the University of Oxford characterizes Buffett's investment methodology as falling within "founder centrism", defined by a deference to managers with a founder's mindset, an ethical disposition towards the shareholder collective, and an intense focus on exponential value creation. Essentially, Buffett's concentrated investments shelter managers from the short-term pressures of the market.[12]

Buffett is a notable philanthropist, having pledged to give away 99 percent[13] of his fortune to philanthropic causes, primarily via the Bill & Melinda Gates Foundation. He founded The Giving Pledge in 2009 with Bill Gates, whereby billionaires pledge to give away at least half of their fortunes.[14]

Buffett was elected to the American Philosophical Society in 2009.[15]

Early life and education Buffett was born in 1930 in Omaha, Nebraska, the second of three children and the only son of Leila (née Stahl) and Congressman Howard Buffett.[16] He began his education at Rose Hill Elementary School. In 1942, his father was elected to the first of four terms in the United States Congress, and after moving with his family to Washington, D.C., Warren finished elementary school, attended Alice Deal Junior High School and graduated from Woodrow Wilson High School in 1947, where his senior yearbook picture reads: "likes math; a future stockbroker."[17] After finishing high school and finding success with his side entrepreneurial and investment ventures, Buffett wanted to skip college to go directly into business but was overruled by his father.[18][19]

Buffett displayed an interest in business and investing at a young age. He was inspired by a book he borrowed from the Omaha public library at age seven, One Thousand Ways to Make $1000.[20] Much of Buffett's early childhood years were enlivened with entrepreneurial ventures. In one of his first business ventures, Buffett sold chewing gum, Coca-Cola bottles, and weekly magazines door to door. He worked in his grandfather's grocery store. While still in high school, he made money delivering newspapers, selling golf balls and stamps, and detailing cars, among other means. On his first income tax return in 1944, Buffett took a $35 deduction for the use of his bicycle and watch on his paper route.[21] In 1945, as a high school sophomore, Buffett and a friend spent $25 to purchase a used pinball machine, which they placed in the local barber shop. Within months, they owned several machines in three different barber shops across Omaha. They sold the business later in the year for $1,200 to a war veteran.[22]

Investor Benjamin Graham influenced young Buffett Buffett's interest in the stock market and investing dated to schoolboy days he spent in the customers' lounge of a regional stock brokerage near his father's own brokerage office. On a trip to New York City at age ten, he made a point to visit the New York Stock Exchange. At 11, he bought three shares of Cities Service Preferred for himself, and three for his sister Doris Buffett (who also became a philanthropist).[23][24][25] At 15, Warren made more than $175 monthly delivering Washington Post newspapers. In high school, he invested in a business owned by his father and bought a 40-acre farm worked by a tenant farmer. He bought the land when he was 14 years old with $1,200 of his savings. By the time he finished college, Buffett had accumulated $9,800 in savings (about $107,000 today).[22][26]

In 1947, Buffett entered the Wharton School of the University of Pennsylvania. He would have preferred to focus on his business ventures, but his father pressured him to enroll.[22] Warren studied there for two years and joined the Alpha Sigma Phi fraternity.[27] He then transferred to the University of Nebraska where at 19, he graduated with a Bachelor of Science in Business Administration. After being rejected by Harvard Business School, Buffett enrolled at Columbia Business School of Columbia University upon learning that Benjamin Graham taught there. He earned a Master of Science in Economics from Columbia in 1951. After graduating, Buffett attended the New York Institute of Finance.[28]

The basic ideas of investing are to look at stocks as business, use the market's fluctuations to your advantage, and seek a margin of safety. That's what Ben Graham taught us. A hundred years from now they will still be the cornerstones of investing.[29][30][31]

— Warren Buffett Investment career Further information on Warren Buffett's time at Berkshire Hathaway: List of assets owned by Berkshire Hathaway Early business career Buffett worked from 1951 to 1954 at Buffett-Falk & Co. as an investment salesman; from 1954 to 1956 at Graham-Newman Corp. as a securities analyst; from 1956 to 1969 at Buffett Partnership, Ltd. as a general partner; and from 1970 as Chairman and CEO of Berkshire Hathaway Inc.

In 1951,[32] Buffett discovered that Graham was on the board of GEICO insurance. Taking a train to Washington, D.C. on a Saturday, he knocked on the door of GEICO's headquarters until a janitor admitted him. There he met Lorimer Davidson, GEICO's vice president, and the two discussed the insurance business for hours. Davidson would eventually become Buffett's lifelong friend and a lasting influence,[33] and would later recall that he found Buffett to be an "extraordinary man" after only fifteen minutes. Buffett wanted to work on Wall Street but both his father and Ben Graham urged him not to. He offered to work for Graham for free, but Graham refused.[34]

Buffett returned to Omaha and worked as a stockbroker while taking a Dale Carnegie public speaking course.[35] Using what he learned, he felt confident enough to teach an "Investment Principles" night class at the University of Nebraska-Omaha. The average age of his students was more than twice his own. During this time he also purchased a Sinclair gas station as a side investment but it was unsuccessful.[36]

In 1952,[37] Buffett married Susan Thompson at Dundee Presbyterian Church. The next year they had their first child, Susan Alice. In 1954, Buffett accepted a job at Benjamin Graham's partnership. His starting salary was $12,000 a year (about $116,000 today).[26] There he worked closely with Walter Schloss. Graham was a tough boss. He was adamant that stocks provide a wide margin of safety after weighing the trade-off between their price and their intrinsic value. That same year the Buffetts had their second child, Howard Graham. In 1956, Benjamin Graham retired and closed his partnership. At this time Buffett's personal savings were over $174,000 (about $1.66 million today)[26] and he started Buffett Partnership Ltd.

Buffett's home in Omaha, Nebraska In 1957, Buffett operated three partnerships. He purchased a five-bedroom stucco house in Omaha, where he still lives, for $31,500.[38][39] In 1958 the Buffetts' third child, Peter Andrew, was born. Buffett operated five partnerships that year. In 1959, the company grew to six partnerships and Buffett met future partner Charlie Munger. By 1960, Buffett operated seven partnerships. He asked one of his partners, a doctor, to find ten other doctors willing to invest $10,000 each in his partnership. Eventually, eleven agreed, and Buffett pooled their money with a mere $100 original investment of his own.

In 1961, Buffett revealed that 35% of the partnership's assets were invested in the Sanborn Map Company. He explained that Sanborn stock sold for only $45 per share in 1958, but the company's investment portfolio was worth $65 per share. This meant that Sanborn's map business was being valued at "minus $20." Buffett eventually purchased 23% of the company's outstanding shares as an activist investor, obtaining a seat for himself on the Board of Directors, and allied with other dissatisfied shareholders to control 44% of the shares. To avoid a proxy fight, the Board offered to repurchase shares at fair value, paying with a portion of its investment portfolio. 77% of the outstanding shares were turned in.[40][41] Buffett had obtained a 50% return on investment in just two years.[42]

Assuming Berkshire In 1962, Buffett became a millionaire because of his partnerships, which in January 1962 had an excess of $7,178,500, of which over $1,025,000 belonged to Buffett. He merged these partnerships into one. Buffett invested in and eventually took control of a textile manufacturing firm, Berkshire Hathaway. He began buying shares in Berkshire from Seabury Stanton, the owner, whom he later fired. Buffett's partnerships began purchasing shares at $7.60 per share. In 1965, when Buffett's partnerships began purchasing Berkshire aggressively, they paid $14.86 per share while the company had working capital of $19 per share. This did not include the value of fixed assets (factory and equipment). Buffett took control of Berkshire Hathaway at a board meeting and named a new president, Ken Chace, to run the company. In 1966, Buffett closed the partnership to new money. He later claimed that the textile business had been his worst trade.[43] He then moved the business into the insurance sector, and, in 1985, the last of the mills that had been the core business of Berkshire Hathaway was sold.

In a second letter, Buffett announced his first investment in a private business — Hochschild, Kohn and Co, a privately owned Baltimore department store. In 1967, Berkshire paid out its first and only dividend of 10 cents.[44] In 1969, Buffett liquidated the partnership and transferred their assets to his partners including shares of Berkshire Hathaway. In 1970, Buffett began writing his now-famous annual letters to shareholders. He lived solely on his salary of $50,000 per year and his outside investment income.

In 1973, Berkshire began to acquire stock in the Washington Post Company. Buffett became close friends with Katharine Graham, who controlled the company and its flagship newspaper and joined its board. In 1974, the SEC opened a formal investigation into Buffett and Berkshire's acquisition of Wesco Financial, due to possible conflict of interest. No charges were brought. In 1977, Berkshire indirectly purchased the Buffalo Evening News for $32.5 million. Antitrust charges started, instigated by its rival, the Buffalo Courier-Express. Both papers lost money until the Courier-Express folded in 1982.

In 1979, Berkshire began to acquire stock in ABC. Capital Cities announced a $3.5 billion purchase of ABC on March 18, 1985, surprising the media industry, as ABC was four times bigger than Capital Cities at the time. Buffett helped finance the deal in return for a 25% stake in the combined company.[45] The newly merged company, known as Capital Cities/ABC (or CapCities/ABC), was forced to sell some stations due to U.S. Federal Communications Commission ownership rules. The two companies also owned several radio stations in the same markets.[46]

In 1987, Berkshire Hathaway purchased a 12% stake in Salomon Inc., making it the largest shareholder and Buffett a director. In 1990, a scandal involving John Gutfreund (former CEO of Salomon Brothers) surfaced. A rogue trader, Paul Mozer, was submitting bids in excess of what was allowed by Treasury rules. When this was brought to Gutfreund's attention, he did not immediately suspend the rogue trader. Gutfreund left the company in August 1991.[47] Buffett became Chairman of Salomon until the crisis passed.[48]

In 1988, Buffett began buying The Coca-Cola Company stock, eventually purchasing up to 7% of the company for $1.02 billion.[49] It would turn out to be one of Berkshire's most lucrative investments and one which it still holds.[50]

As a billionaire Buffett became a billionaire when Berkshire Hathaway began selling class A shares on May 29, 1990, with the market closing at $7,175 a share.[51] In 1998 he acquired General Re (Gen Re) as a subsidiary in a deal that presented difficulties—according to the Rational Walk investment website, "underwriting standards proved to be inadequate," while a "problematic derivatives book" was resolved after numerous years and a significant loss.[52] Gen Re later provided reinsurance after Buffett became involved with Maurice R. Greenberg at AIG in 2002.[53]

Buffett meets with President Barack Obama at the White House in July 2011 During a 2005 investigation of an accounting fraud case involving AIG, Gen Re executives became implicated. On March 15, 2005, the AIG board forced Greenberg to resign from his post as chairman and CEO after New York state regulators claimed that AIG had engaged in questionable transactions and improper accounting.[54] On February 9, 2006, AIG agreed to pay a $1.6 billion fine.[55] In 2010, the U.S. government agreed to a $92 million settlement with Gen Re, allowing the Berkshire Hathaway subsidiary to avoid prosecution in the AIG case. Gen Re also made a commitment to implement "corporate governance concessions," which required Berkshire Hathaway's Chief Financial Officer to attend General Re's audit committee meetings and mandated the appointment of an independent director.[52]

In 2002, Buffett entered in $11 billion worth of forward contracts to deliver U.S. dollars against other currencies. By April 2006, his total gain on these contracts was over $2 billion. In 2006, Buffett announced in June that he gradually would give away 85% of his Berkshire holdings to five foundations in annual gifts of stock, starting in July 2006—the largest contribution would go to the Bill and Melinda Gates Foundation.[56]

In 2007, in a letter to shareholders, Buffett announced that he was looking for a younger successor, or perhaps successors, to run his investment business.[57]

2007–08 financial crisis Buffett ran into criticism during the subprime mortgage crisis of 2007 and 2008, part of the Great Recession starting in 2007, that he had allocated capital too early resulting in suboptimal deals.[58] "Buy American. I am." he wrote for an opinion piece published in the New York Times in 2008.[59] Buffett called the downturn in the financial sector that started in 2007 "poetic justice".[60] Buffett's Berkshire Hathaway suffered a 77% drop in earnings during Q3 2008 and several of his later deals suffered large mark-to-market losses.[61]

Berkshire Hathaway acquired 10% perpetual preferred stock of Goldman Sachs.[62] Some of Buffett's put options (European exercise at expiry only) that he wrote (sold) were running at around $6.73 billion mark-to-market losses as of late 2008.[63] The scale of the potential loss prompted the SEC to demand that Berkshire produce, "a more robust disclosure" of factors used to value the contracts.[63] Buffett also helped Dow Chemical pay for its $18.8 billion takeover of Rohm & Haas. He thus became the single largest shareholder in the enlarged group with his Berkshire Hathaway, which provided $3 billion, underlining his instrumental role during the crisis in debt and equity markets.[64]

In 2008, Buffett became the richest person in the world, with a total net worth estimated at $62 billion[65] by Forbes and at $58 billion[66] by Yahoo, overtaking Bill Gates, who had been number one on the Forbes list for 13 consecutive years.[67] In 2009, Gates regained the top position on the Forbes list, with Buffett shifted to second place. Both of the men's values dropped, to $40 billion and $37 billion respectively—according to Forbes, Buffett lost $25 billion over a 12-month period during 2008/2009.[68]

In October 2008, the media reported that Buffett had agreed to buy General Electric (GE) preferred stock.[69] The operation included special incentives: He received an option to buy three billion shares of GE stock, at $22.25, over the five years following the agreement, and Buffett also received a 10% dividend (callable within three years). In February 2009, Buffett sold some Procter & Gamble Co. and Johnson & Johnson shares from his personal portfolio.[70]

In addition to suggestions of mistiming, the wisdom in keeping some of Berkshire's major holdings, including The Coca-Cola Company, which in 1998 peaked at $86, raised questions. Buffett discussed the difficulties of knowing when to sell in the company's 2004 annual report:

That may seem easy to do when one looks through an always-clean, rear-view mirror. Unfortunately, however, it's the windshield through which investors must peer, and that glass is invariably fogged.[71]

In March 2009, Buffett said in a cable television interview that the economy had "fallen off a cliff ... Not only has the economy slowed down a lot, but people have really changed their habits like I haven't seen". Additionally, Buffett feared that inflation levels that occurred in the 1970s—which led to years of painful stagflation—might re-emerge.[72][73]

A capitalized Berkshire On August 14, 2014, the price of Berkshire Hathaway's shares hit $200,000 a share for the first time, capitalizing the company at $328 billion. While Buffett had given away much of his stock to charities by this time, he still held 321,000 shares worth $64.2 billion. On August 20, 2014, Berkshire Hathaway was fined $896,000 for failing to report December 9, 2013, purchase of shares in USG Corporation as required.[74]

In 2009, Buffett invested $2.6 billion as a part of Swiss Re's campaign to raise equity capital.[75][76] Berkshire Hathaway already owned a 3% stake, with rights to own more than 20%.[77] Also in 2009, Buffett acquired Burlington Northern Santa Fe Corp. for $34 billion in cash and stock. Alice Schroeder, author of Snowball, said that a key reason for the purchase was to diversify Berkshire Hathaway from the financial industry.[78] Measured by market capitalization in the Financial Times Global 500, Berkshire Hathaway was the eighteenth largest corporation in the world as of June 2009.[79]

In 2009, Buffett divested his failed investment in ConocoPhillips, saying to his Berkshire investors,

I bought a large amount of ConocoPhillips stock when oil and gas prices were near their peak. I in no way anticipated the dramatic fall in energy prices that occurred in the last half of the year. I still believe the odds are good that oil sells far higher in the future than the current $40–$50 price. But so far I have been dead wrong. Even if prices should rise, moreover, the terrible timing of my purchase has cost Berkshire several billion dollars.[80]

The merger with the Burlington Northern Santa Fe Railway (BNSF) closed upon BNSF shareholder approval during Q1 of 2010. This deal was valued at approximately $44 billion (with $10 billion of outstanding BNSF debt) and represented an increase of the previously existing stake of 22%.[81][82] In June 2010, Buffett defended the credit-rating agencies for their role in the US financial crisis, claiming:

Very, very few people could appreciate the bubble. That's the nature of bubbles – they're mass delusions.[83]

On March 18, 2011, Goldman Sachs was given Federal Reserve approval to buy back Berkshire's preferred stock in Goldman. Buffett had been reluctant to give up the stock, which averaged $1.4 million in dividends per day,[84][85] saying:

I'm going to be the Osama bin Laden of capitalism. I'm on my way to an unknown destination in Asia where I'm going to look for a cave. If the U.S. Armed forces can't find Osama bin Laden in 10 years, let Goldman Sachs try to find me.[86]

In November 2011, it was announced that over the course of the previous eight months, Buffett had bought 64 million shares of International Business Machine Corp (IBM) stock, worth around $11 billion. This unanticipated investment raised his stake in the company to around 5.5 percent—the largest stake in IBM alongside that of State Street Global Advisors. Buffett had said on numerous prior occasions that he would not invest in technology because he did not fully understand it, so the move came as a surprise to many investors and observers. During the interview, in which he revealed the investment to the public, Buffett stated that he was impressed by the company's ability to retain corporate clients and said, "I don't know of any large company that really has been as specific on what they intend to do and how they intend to do it as IBM."[87]

In May 2012, Buffett's acquisition of Media General, consisting of 63 newspapers in the south-eastern U.S., was announced.[88] The company was the second news print purchase made by Buffett in one year.[89]

Interim publisher James W. Hopson announced on July 18, 2013, that the Press of Atlantic City would be sold to Buffett's BH Media Group by ABARTA, a private holding company based in Pittsburgh, U.S. At the Berkshire shareholders meeting in May 2013, Buffett explained that he did not expect to "move the needle" at Berkshire with newspaper acquisitions, but he anticipates an annual return of 10 percent. The Press of Atlantic City became Berkshire's 30th daily newspaper, following other purchases such as Virginia, U.S.' Roanoke Times and The Tulsa World in Oklahoma, U.S.[90]

During a presentation to Georgetown University students in Washington, D.C. in late September 2013, Buffett compared the U.S. Federal Reserve to a hedge fund and stated that the bank is generating "$80 billion or $90 billion a year probably" in revenue for the U.S. government. Buffett also advocated further on the issue of wealth equality in society:

We have learned to turn out lots of goods and services, but we haven't learned as well how to have everybody share in the bounty. The obligation of a society as prosperous as ours is to figure out how nobody gets left too far behind.[91]

After the difficulties of the economic crisis, Buffett managed to bring its company back to its pre-recession standards: in Q2 2014, Berkshire Hathaway made $6.4 billion in net profit, the most it had ever made in a three-month period.[92]

COVID-19 pandemic In a June 2021 interview with CNBC, Buffet said that the economic impact of the COVID-19 pandemic has increased economic inequality and bemoaned that most people are unaware that "hundreds of thousands or millions" of small businesses have been negatively impacted. He also stated that the markets and the economy will likely be unpredictible well into the post-pandemic recovery period, even with the Biden administration and the United States Federal Reserve having a plan in place. He said the unpredictability and the effects of COVID-19 are far from over.[93]

Investment philosophy Warren Buffett's writings include his annual reports and various articles. Buffett is recognized by communicators[94] as a great story-teller, as evidenced by his annual letters to shareholders. He has warned about the pernicious effects of inflation:[95]

The arithmetic makes it plain that inflation is a far more devastating tax than anything that has been enacted by our legislatures. The inflation tax has a fantastic ability to simply consume capital. It makes no difference to a widow with her savings in a 5 percent passbook account whether she pays 100 percent income tax on her interest income during a period of zero inflation, or pays no income taxes during years of 5 percent inflation.

— Buffett, Fortune (1977) In his article, "The Superinvestors of Graham-and-Doddsville", Buffett rebutted the academic efficient-market hypothesis, that beating the S&P 500 was "pure chance", by highlighting the results achieved by a number of students of the Graham and Dodd value investing school of thought. In addition to himself, Buffett named Walter J. Schloss, Tom Knapp, Ed Anderson (Tweedy, Browne LLC), William J. Ruane (Sequoia Fund), Charlie Munger (Buffett's partner at Berkshire), Rick Guerin (Pacific Partners Ltd.), and Stan Perlmeter (Perlmeter Investments).[96] In his November 1999 Fortune article, he warned of investors' unrealistic expectations:[97]

Let me summarize what I've been saying about the stock market: I think it's very hard to come up with a persuasive case that equities will over the next 17 years perform anything like—anything like—they've performed in the past 17. If I had to pick the most probable return, from appreciation and dividends combined, that investors in aggregate—repeat, aggregate—would earn in a world of constant interest rates, 2% inflation, and those ever hurtful frictional costs, it would be 6%!

— Buffett, Fortune (1999) Index funds vis-à-vis active management Buffett has been a supporter of index funds for people who are either not interested in managing their own money or don't have the time. Buffett is skeptical that active management can outperform the market in the long run, and has advised both individual and institutional investors to move their money to low-cost index funds that track broad, diversified stock market indices. Buffett said in one of his letters to shareholders that "when trillions of dollars are managed by Wall Streeters charging high fees, it will usually be the managers who reap outsized profits, not the clients."[98] In 2007, Buffett made a bet with numerous managers that a simple S&P 500 index fund will outperform hedge funds that charge exorbitant fees. By 2017, the index fund was outperforming every hedge fund that made the bet against Buffett.[98]

Personal life

Buffett with Gary Green in 2010 In 1949, Buffett was infatuated with a young woman whose boyfriend had a ukulele. In an attempt to compete, he bought one of the instruments and has been playing it ever since. Though the attempt was unsuccessful, his music interest was a key part of his becoming a part of Susan Thompson's life and led to their marriage. Buffett often plays the instrument at stockholder meetings and other opportunities. His love of the instrument led to the commissioning of two custom Dairy Queen ukuleles by Dave Talsma, one of which was auctioned for charity.[99]

Buffett married Susan Buffett (born Thompson) in 1952. They had three children, Susie, Howard and Peter. The couple began living separately in 1977, although they remained married until Susan Buffett's death in July 2004. Their daughter, Susie, lives in Omaha, is a national board member of Girls, Inc., and does charitable work through the Susan A. Buffett Foundation.[100]

In 2006, on his 76th birthday, Buffett married his longtime companion, Astrid Menks, who was then 60 years old—she had lived with him since his wife's departure to San Francisco in 1977.[101][102] Susan had arranged for the two to meet before she left Omaha to pursue her singing career. All three were close and Christmas cards to friends were signed "Warren, Susie and Astrid".[103] Susan briefly discussed this relationship in an interview on the Charlie Rose Show shortly before her death, in a rare glimpse into Buffett's personal life.[104]

Buffett disowned his son Peter's adopted daughter, Nicole, in 2006 after she participated in the Jamie Johnson documentary The One Percent about the growing economic inequality between the wealthy and the average citizen in the United States. Although his first wife referred to Nicole as one of her "adored grandchildren",[105] Buffett wrote her a letter stating, "I have not emotionally or legally adopted you as a grandchild, nor have the rest of my family adopted you as a niece or a cousin."[106][107][108]

His 2006 annual salary was about $100,000, which is small compared to senior executive remuneration in comparable companies.[109] In 2008, he earned a total compensation of $175,000, which included a base salary of just $100,000.[110] He continued to live in the same house in the central Dundee neighborhood of Omaha that he bought in 1958 for $31,500, a fraction of today's value. He also owns a $4 million house in Laguna Beach, California.[111] In 1989, after spending nearly $6.7 million of Berkshire's funds on a private jet, Buffett named it "The Indefensible". This act was a break from his past condemnation of extravagant purchases by other CEOs and his history of using more public transportation.[112]

Bridge is such a sensational game that I wouldn't mind being in jail if I had three cellmates who were decent players and who were willing to keep the game going twenty-four hours a day.

—Buffett on bridge[113] Buffett is an avid bridge player, which he plays with fellow fan Gates[114]—he allegedly spends 12 hours a week playing the game.[115] In 2006, he sponsored a bridge match for the Buffett Cup. Modeled on the Ryder Cup in golf—held immediately before it in the same city—the teams are chosen by invitation, with a female team and five male teams provided by each country.[116]

He is a dedicated, lifelong follower of Nebraska football, and attends as many games as his schedule permits. He supported the hire of Bo Pelini, following the 2007 season, stating, "It was getting kind of desperate around here".[117] He watched the 2009 game against Oklahoma from the Nebraska sideline, after being named an honorary assistant coach.[118]

Buffett worked with Christopher Webber on an animated series called "Secret Millionaires Club" with chief Andy Heyward of DiC Entertainment. The series features Buffett and Munger and teaches children healthy financial habits.[119][120]

Buffett was raised as a Presbyterian, but has since described himself as agnostic.[121] In December 2006, it was reported that Buffett did not carry a mobile phone, did not have a computer at his desk, and drove his own automobile,[122] a Cadillac DTS.[123] In contrast to that, at the 2018 Berkshire Hathaway's shareholder meeting, he stated he uses Google as his preferred search engine.[124] In 2013 he had an old Nokia flip phone and had sent one email in his entire life.[125] In February 2020, Buffett revealed in a CNBC interview that he had traded in his flip phone for an iPhone 11.[126] Buffett reads five newspapers every day, beginning with the Omaha World Herald, which his company acquired in 2011.

Buffett's speeches are known for mixing business discussions with humor. Each year, Buffett presides over Berkshire Hathaway's annual shareholder meeting in the Qwest Center in Omaha, Nebraska, an event drawing over 20,000 visitors from both the United States and abroad, giving it the nickname "Woodstock of Capitalism". Berkshire's annual reports and letters to shareholders, prepared by Buffett, frequently receive coverage by the financial media. Buffett's writings are known for containing quotations from sources as varied as the Bible and Mae West,[127] as well as advice in a folksy, Midwestern style and numerous jokes.

In April 2017, Buffett (an avid Coca-Cola drinker and shareholder in the company) agreed to have his likeness placed on Cherry Coke products in China. Buffett was not compensated for this advertisement.[128][129]

Health On April 11, 2012, Buffett was diagnosed with stage I prostate cancer during a routine test.[130] He announced he would begin two months of daily radiation treatment from mid-July. In a letter to shareholders, Buffett said he felt "great – as if I were in my normal excellent health – and my energy level is 100 percent."[130] On September 15, 2012, Buffett announced that he had completed the full 44-day radiation treatment cycle, saying "it's a great day for me" and "I am so glad to say that's over."[131]

Wealth and philanthropy

Buffett, Kathy Ireland and Bill Gates at the 2015 Berkshire Hathaway shareholders meeting In 2008, Buffett was ranked by Forbes as the richest person in the world with an estimated net worth of approximately $62 billion.[132] In 2009, after donating billions of dollars to charity, he was ranked as the second richest man in the United States with a net worth of $37 billion[133][134] with only Bill Gates ranked higher than Buffett. His net worth had risen to $58.5 billion as of September 2013.[135]

In 1999, Buffett was named the top money manager of the Twentieth Century in a survey by the Carson Group, ahead of Peter Lynch and John Templeton.[136] In 2007, he was listed among Time's 100 Most Influential People in the world.[137] In 2011, President Barack Obama awarded him the Presidential Medal of Freedom.[138] Buffett, along with Bill Gates, was named the most influential global thinker in Foreign Policy's 2010 report.[139]

Buffett has written several times of his belief that, in a market economy, the rich earn outsized rewards for their talents.[140] His children will not inherit a significant proportion of his wealth. He once commented, "I want to give my kids just enough so that they would feel that they could do anything, but not so much that they would feel like doing nothing".[141]

Buffett had long stated his intention to give away his fortune to charity, and in June 2006, he announced a new plan to give 83% of it to the Bill & Melinda Gates Foundation (BMGF).[142] He pledged about the equivalent of 10 million Berkshire Hathaway Class B shares to the Bill & Melinda Gates Foundation (worth approximately $30.7 billion as of June 23, 2006),[143] making it the largest charitable donation in history, and Buffett one of the leaders of philanthrocapitalism.[144] The foundation will receive 5% of the total each July, beginning in 2006. The pledge is conditional upon three requirements:

Bill or Melinda Gates must be alive and active in BMGF BMGF must continue to qualify as a charity Each year BMGF must give away an amount equal to the prior year's Berkshire gift plus the additional 5% of net assets as required of all US foundations Buffett joined the Gates Foundation's board, but did not plan to be actively involved in the foundation's investments.[145][146] Buffett announced his resignation as a trustee of the Gates Foundation on June 23, 2021.[147]

This represented a significant shift from Buffett's previous statements, to the effect that most of his fortune would pass to his Buffett Foundation.[148] The bulk of the estate of his wife, valued at $2.6 billion, went there when she died in 2004.[149] He also pledged $50 million to the Nuclear Threat Initiative, in Washington, where he began serving as an adviser in 2002.[150]

In 2006, he auctioned his 2001 Lincoln Town Car[151] on eBay to raise money for Girls, Inc.[152] In 2007, he auctioned a luncheon with himself that raised a final bid of $650,100 for the Glide Foundation.[153] Later auctions raised $2.1 million[154][155] $1.7 million[156] and $3.5 million. The winners traditionally dine with Buffett at New York's Smith and Wollensky steak house. The restaurant donates at least $10,000 to Glide each year to host the meal.[157]

In 2009, Ralph Nader wrote the book Only the Super Rich Can Save Us, a novel about "a movement of billionaires led by Warren Buffett and featuring, among others, Ted Turner, George Soros and Barry Diller, who use their fortunes to clean up America." On C-SPAN BookTV, Nader said Buffett invited him to breakfast after the book came out and was "quite intrigued by the book." He also told Nader of his plan to get "billionaires all over the world to donate 50% of their estate to charity or good works."[158] On December 9, 2010, Buffett, Bill Gates, and Facebook CEO Mark Zuckerberg signed a promise they called the "Gates-Buffett Giving Pledge", in which they promise to donate to charity at least half of their wealth, and invite other wealthy people to follow suit.[14][159] In 2018, after making almost $3.4 billion donations,[160] Buffett was ranked 3rd in the Forbes' List of Billionaire 2018.[161]

Warren Buffett continues to help fund and support his family's individual foundations which include Susan Buffett's Susan Thompson Buffett Foundation, Susan Alice Buffett's Sherwood Foundation, Howard Graham Buffett's Howard G. Buffett Foundation, and Peter Buffett's NoVo Foundation.[162][163] Warren Buffett was also supportive of his sister Doris Buffett's Letters Foundation and Learning By Giving Foundation.[164][165]

Political and public policy views

Buffett and President Obama in the Oval Office, July 14, 2010 In addition to political contributions over the years, Buffett endorsed and made campaign contributions to Barack Obama's presidential campaign. On July 2, 2008, Buffett attended a $28,500 per plate fundraiser for Obama's campaign in Chicago.[166] Buffett intimated that John McCain's views on social justice were so far from his own that McCain would need a "lobotomy" for Buffett to change his endorsement.[167] During the second 2008 U.S. presidential debate, McCain and Obama, after being asked first by presidential debate mediator Tom Brokaw, both mentioned Buffett as a possible future Secretary of the Treasury.[168] Later, in the third and final presidential debate, Obama mentioned Buffett as a potential economic advisor.[169] Buffett was also a financial advisor to Republican candidate Arnold Schwarzenegger during the 2003 California gubernatorial election.[170]

On December 16, 2015, Buffett endorsed Democratic candidate Hillary Clinton for president.[171] On August 1, 2016, Buffett challenged Donald Trump to release his tax returns.[172][173] On October 10, 2016, after a reference to him in the second presidential debate, Buffett released his own tax return.[174][175] He said he had paid $1.85 million in federal income taxes in 2015 on an adjusted gross income of $11.6 million, meaning he had an effective federal income tax rate of around 16 percent. Buffett also said he had made more than $2.8 billion worth of donations last year.[175] In response to Trump saying he was unable to release his tax information due to being under audit, Buffett said, "I have been audited by the IRS multiple times and am currently being audited. I have no problem in releasing my tax information while under audit. Neither would Mr. Trump — at least he would have no legal problem."[175]

Buffett has said he would judge President Donald Trump by his results on national safety, economic growth and economic participation when deciding if he would vote for him in the 2020 presidential election.[176][177]

Health care Buffett described the health care reform under President Barack Obama as insufficient to deal with the costs of health care in the US, though he supports its aim of expanding health insurance coverage.[178] Buffett compared health care costs to a tapeworm, saying that they compromise US economic competitiveness by increasing manufacturing costs.[178] Buffett thinks health care costs should head towards 13 to 14% of GDP.[179] Buffett said "If you want the very best, I mean if you want to spend a million dollars to prolong your life 3 months in a coma or something then the US is probably the best", but he also said that other countries spend much less and receive much more in health care value (visits, hospital beds, doctors and nurses per capita).[180]

Buffett faults the incentives in the United States medical industry, that payers reimburse doctors for procedures (fee-for-service) leading to unnecessary care (overutilization), instead of paying for results.[181] He cited Atul Gawande's 2009 article in the New Yorker[182] as a useful consideration of US health care, with its documentation of unwarranted variation in Medicare expenditures between McAllen, Texas and El Paso, Texas.[181] Buffett raised the problem of lobbying by the medical industry, saying that they are very focused on maintaining their income.[183]

Curbing population growth Buffett has been reported to have concerns about unchecked population growth. In 2009, he met with several other billionaires to discuss healthcare, education and slowing population growth. Called "The Good Club" by an insider, the billionaires had given away $45 billion to philanthropic causes and included well known names such as Oprah Winfrey, Michael Bloomberg and David Rockefeller, Jr..[184] The meeting has drawn criticism from some right-wing alarmists, with some fringe elements believing the group to be a part of a secret sterilization society.[185]

Buffett is a long time supporter of family planning. The Buffett Foundation has given over $1.5 billion to abortion research to include $427 million to Planned Parenthood.[186]

Taxes See also: Buffett Rule File:President Obama Speaks on the Buffett Rule.webm President Obama announcing the "Buffett Rule" Buffett stated that he only paid 19 percent of his income for 2006 ($48.1 million) in total federal taxes (due to their source as dividends and capital gains), while his employees paid 33 percent of theirs, despite making much less money.[187] "How can this be fair?" Buffett asked, regarding how little he pays in taxes compared to his employees. "How can this be right?" He also added, "There's class warfare, all right, but it's my class, the rich class, that's making war, and we're winning."[188][189] After Donald Trump accused him of taking "massive deductions," Buffett countered, "I have copies of all 72 of my returns and none uses a carryforward."[190]

Buffett favors the inheritance tax, saying that repealing it would be like "choosing the 2020 Olympic team by picking the eldest sons of the gold-medal winners in the 2000 Olympics".[191] In 2007, Buffett testified before the Senate and urged them to preserve the estate tax so as to avoid a plutocracy.[192] Some critics argued that Buffett (through Berkshire Hathaway) has a personal interest in the continuation of the estate tax, since Berkshire Hathaway benefited from the estate tax in past business dealings and had developed and marketed insurance policies to protect policy holders against future estate tax payments.[193]

Buffett believes government should not be in the business of gambling, or legalizing casinos, calling it a tax on ignorance.[194]

Dollar and gold The trade deficit induced Buffett to enter the foreign currency market for the first time in 2002. He substantially reduced his stake in 2005 as changing interest rates increased the costs of holding currency contracts. Buffett remained bearish on the dollar, stating that he was looking to acquire companies with substantial foreign revenues. Buffett has been critical of gold as an investment, with his critique being based primarily on its non-productive nature. In a 1998 address at Harvard, Buffett said:

It gets dug out of the ground in Africa, or someplace. Then we melt it down, dig another hole, bury it again and pay people to stand around guarding it. It has no utility. Anyone watching from Mars would be scratching their head.

In 1977, about stocks, gold, farmland and inflation, he stated:

Stocks are probably still the best of all the poor alternatives in an era of inflation – at least they are if you buy in at appropriate prices.[195]

China Buffett invested in PetroChina Company Limited and in a rare move, posted a commentary[196] on Berkshire Hathaway's website stating why he would not divest over its connection with the Sudanese civil war that caused Harvard to divest. He sold this stake soon afterwards, sparing him the billions of dollars he would have lost had he held on to the company in the midst of the steep drop in oil prices beginning in the summer of 2008.

In October 2008, Buffett invested $230 million for 10% of battery maker BYD Company (SEHK: 1211), which runs a subsidiary of electric automobile manufacturer BYD Auto. In less than one year, the investment reaped over 500% return.[197]

In May 2018, BYD's shares had a substantial fall with a total net investment loss of $9 billion. This was Buffett's worst investment in China.[198]

Tobacco During the RJR Nabisco, Inc. hostile takeover fight in 1987, Buffett was quoted as telling John Gutfreund:[199]

I'll tell you why I like the cigarette business. It costs a penny to make. Sell it for a dollar. It's addictive. And there's fantastic brand loyalty.

— Buffett, quoted in Barbarians at the Gate: The Fall of RJR Nabisco Speaking at Berkshire Hathaway Inc.'s 1994 annual meeting, Buffett said investments in tobacco are:[200]

fraught with questions that relate to societal attitudes and those of the present administration. I would not like to have a significant percentage of my net worth invested in tobacco businesses. The economy of the business may be fine, but that doesn't mean it has a bright future.

— Buffett, Berkshire Hathaway annual meeting Coal In 2007, Buffett's PacifiCorp, a subsidiary of his MidAmerican Energy Company, canceled six proposed coal-fired power plants. These included Utah's Intermountain Power Project Unit 3, Jim Bridger Unit 5, and four proposed plants previously included in PacifiCorp's Integrated Resource Plan. The cancellations came in the wake of pressure from regulators and citizen groups.[201]

Renewable energy Native American tribes and salmon fishermen sought to win support from Buffett for a proposal to remove four hydroelectric dams from the Klamath River owned by PacifiCorp which is a Berkshire Hathaway company. David Sokol responded on Buffett's behalf, stating that the FERC would decide the question.[202][203]

Expensing of stock options He has been a strong proponent of stock option expensing on corporate income statements. At the 2004 annual meeting, he lambasted a bill before the United States Congress that would consider only some company-issued stock options compensation as an expense, likening the bill to one that was almost passed by the Indiana House of Representatives to change the value of Pi from 3.14159 to 3.2 through legislative fiat.[204]

When a company gives something of value to its employees in return for their services, it is clearly a compensation expense. And if expenses don't belong in the earnings statement, where in the world do they belong?[205]

Technology In May 2012, Buffett said he had avoided buying stock in new social media companies such as Facebook and Google because it is hard to estimate future value. He also stated that initial public offering (IPO) of stock are almost always bad investments. Investors should be looking to companies that will have good value in ten years.[206]

Bitcoin and cryptocurrencies In an interview with CNBC in January 2018, Buffett said that the recent craze over Bitcoin and other cryptocurrencies won't end well, adding that "when it happens or how or anything else, I don't know." But he said he would not take a short position on bitcoin futures.[207]

In terms of cryptocurrencies, generally, I can say with almost certainty that they will come to a bad ending.

Formula investing is a method of investing that rigidly follows a prescribed theory or formula to determine investment policy. Formula investing can be related to how an investor handles asset allocation, invests in funds or securities, or decides when and how much money to invest.

KEY TAKEAWAYS

• With formula investing, a market participant follows a structured plan that determines factors such as asset allocation, types of securities invested in, or the amount and frequency of investments.

• Some examples of common styles of formula investing include dollar-cost averaging, dividend reinvesting and ladders.

• Formula investing is appealing to market participants who find active investing stressful or overwhelming; formula investing is structured and consistent.

• The downside to formula investing is that it doesn't leave much room for an investor to make changes to adjust to unforeseen market or economic changes.

Understanding Formula Investing

Formula investing takes most of the discretionary decision-making out of the investment process, which can reduce stress for investors and help them automate their strategies; investors simply follow the rules or formula and invest accordingly. A drawback of using formula investing is the inability to adapt to changing market conditions. For instance, during a period of extreme volatility, an investor may achieve better results by making a discretionary adjustment to their investment strategy.

An investor must make sure that the formula fits with his or her risk tolerance, time horizon and liquidity requirements for it to be effective. Dollar-cost averaging, dividend reinvesting and ladders are examples of simple formula investing strategies.

Formula investing may simplify the investment process for inexperienced investors or those who lack the time to actively manage their accounts; however, the risk is that a formula investor can't react fast enough to changes in the market or the economy.

Formula Investing Strategies

• Dollar-Cost Averaging: This strategy involves buying a fixed dollar amount of an investment on a set schedule, regardless of how the investment performs. For example, a market participant invests $1,000 in a particular mutual fund on the first day of the month, every month for a year, ultimately investing $12,000. Dollar-cost averaging helps to build a portfolio in a piecemeal fashion, adding small amounts of money over a consistent time frame.

• Dividend Reinvesting: Investors may set up a dividend reinvestment plan (DRIP) to reinvest dividends to purchase additional stock. This strategy has the advantage of compounding wealth, providing the company pays consistent dividends. For example, an investor holds $10,000 in stock that pays an annual yield of 5%. After a year, the investor reinvests the $500 dividend and now has stock holdings of $10,500. After two years, the investor reinvests the $525 dividend and has holdings of $11,025. The compounding effect continues as long as the investor keeps reinvesting dividends. This example assumes the share price stayed unchanged over the two-year period.

• Ladders: Investors use this strategy for fixed-income investments, such as bonds. Investors purchase a portfolio of bonds with different maturity dates. By staggering the maturity dates, the short-term bonds offset the volatility of the long-term bonds. Cash received from maturing bonds is then used to buy additional bonds to keep the defined structure.

Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum game and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by the 1944 book Theory of Games and Economic Behavior, co-written with Oskar Morgenstern, which considered cooperative games of several players. The second edition of this book provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty.

Game theory was developed extensively in the 1950s by many scholars. It was explicitly applied to evolution in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been widely recognized as an important tool in many fields. As of 2014, with the Nobel Memorial Prize in Economic Sciences going to game theorist Jean Tirole, eleven game theorists have won the economics Nobel Prize. John Maynard Smith was awarded the Crafoord Prize for his application of evolutionary game theory.

Discussions on the mathematics of games began long before the rise of modern mathematical game theory. Cardano's work on games of chance in Liber de ludo aleae (Book on Games of Chance), which was written around 1564 but published posthumously in 1663, formulated some of the field's basic ideas. In the 1650s, Pascal and Huygens developed the concept of expectation on reasoning about the structure of games of chance, and Huygens published his gambling calculus in De ratiociniis in ludo aleæ (On Reasoning in Games of Chance) in 1657.

In 1713, a letter attributed to Charles Waldegrave analyzed a game called "le Her". He was an active Jacobite and uncle to James Waldegrave, a British diplomat.[2][3] In this letter, Waldegrave provided a minimax mixed strategy solution to a two-person version of the card game le Her, and the problem is now known as Waldegrave problem. In his 1838 Recherches sur les principes mathématiques de la théorie des richesses (Researches into the Mathematical Principles of the Theory of Wealth), Antoine Augustin Cournot considered a duopoly and presented a solution that is the Nash equilibrium of the game.

In 1913, Ernst Zermelo published Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels (On an Application of Set Theory to the Theory of the Game of Chess), which proved that the optimal chess strategy is strictly determined. This paved the way for more general theorems.[4]

In 1938, the Danish mathematical economist Frederik Zeuthen proved that the mathematical model had a winning strategy by using Brouwer's fixed point theorem.[5] In his 1938 book Applications aux Jeux de Hasard and earlier notes, Émile Borel proved a minimax theorem for two-person zero-sum matrix games only when the pay-off matrix is symmetric and provided a solution to a non-trivial infinite game (known in English as Blotto game). Borel conjectured the non-existence of mixed-strategy equilibria in finite two-person zero-sum games, a conjecture that was proved false by von Neumann.

Birth and early developments

John von Neumann Game theory did not really exist as a unique field until John von Neumann published the paper On the Theory of Games of Strategy in 1928.[6][7] Von Neumann's original proof used Brouwer's fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by his 1944 book Theory of Games and Economic Behavior co-authored with Oskar Morgenstern.[8] The second edition of this book provided an axiomatic theory of utility, which reincarnated Daniel Bernoulli's old theory of utility (of money) as an independent discipline. Von Neumann's work in game theory culminated in this 1944 book. This foundational work contains the method for finding mutually consistent solutions for two-person zero-sum games. Subsequent work focused primarily on cooperative game theory, which analyzes optimal strategies for groups of individuals, presuming that they can enforce agreements between them about proper strategies.[9]

John Nash In 1950, the first mathematical discussion of the prisoner's dilemma appeared, and an experiment was undertaken by notable mathematicians Merrill M. Flood and Melvin Dresher, as part of the RAND Corporation's investigations into game theory. RAND pursued the studies because of possible applications to global nuclear strategy.[10] Around this same time, John Nash developed a criterion for mutual consistency of players' strategies known as the Nash equilibrium, applicable to a wider variety of games than the criterion proposed by von Neumann and Morgenstern. Nash proved that every finite n-player, non-zero-sum (not just two-player zero-sum) non-cooperative game has what is now known as a Nash equilibrium in mixed strategies.

Game theory experienced a flurry of activity in the 1950s, during which the concepts of the core, the extensive form game, fictitious play, repeated games, and the Shapley value were developed. The 1950s also saw the first applications of game theory to philosophy and political science.

Prize-winning achievements In 1965, Reinhard Selten introduced his solution concept of subgame perfect equilibria, which further refined the Nash equilibrium. Later he would introduce trembling hand perfection as well. In 1994 Nash, Selten and Harsanyi became Economics Nobel Laureates for their contributions to economic game theory.

In the 1970s, game theory was extensively applied in biology, largely as a result of the work of John Maynard Smith and his evolutionarily stable strategy. In addition, the concepts of correlated equilibrium, trembling hand perfection, and common knowledge[a] were introduced and analyzed.

In 2005, game theorists Thomas Schelling and Robert Aumann followed Nash, Selten, and Harsanyi as Nobel Laureates. Schelling worked on dynamic models, early examples of evolutionary game theory. Aumann contributed more to the equilibrium school, introducing equilibrium coarsening and correlated equilibria, and developing an extensive formal analysis of the assumption of common knowledge and of its consequences.

In 2007, Leonid Hurwicz, Eric Maskin, and Roger Myerson were awarded the Nobel Prize in Economics "for having laid the foundations of mechanism design theory". Myerson's contributions include the notion of proper equilibrium, and an important graduate text: Game Theory, Analysis of Conflict.[1] Hurwicz introduced and formalized the concept of incentive compatibility.

In 2012, Alvin E. Roth and Lloyd S. Shapley were awarded the Nobel Prize in Economics "for the theory of stable allocations and the practice of market design". In 2014, the Nobel went to game theorist Jean Tirole.

Game types See also: List of games in game theory Cooperative / non-cooperative Main articles: Cooperative game theory and Non-cooperative game A game is cooperative if the players are able to form binding commitments externally enforced (e.g. through contract law). A game is non-cooperative if players cannot form alliances or if all agreements need to be self-enforcing (e.g. through credible threats).[11]

Cooperative games are often analyzed through the framework of cooperative game theory, which focuses on predicting which coalitions will form, the joint actions that groups take, and the resulting collective payoffs. It is opposed to the traditional non-cooperative game theory which focuses on predicting individual players' actions and payoffs and analyzing Nash equilibria.[12][13] The focus on individual payoff can result in a phenomenon known as Tragedy of the Commons, where resources are used to a collectively inefficient level. The lack of formal negotiation leads to the deterioration of public goods through over-use and under provision that stems from private incentives.[14]

Cooperative game theory provides a high-level approach as it describes only the structure, strategies, and payoffs of coalitions, whereas non-cooperative game theory also looks at how bargaining procedures will affect the distribution of payoffs within each coalition. As non-cooperative game theory is more general, cooperative games can be analyzed through the approach of non-cooperative game theory (the converse does not hold) provided that sufficient assumptions are made to encompass all the possible strategies available to players due to the possibility of external enforcement of cooperation. While using a single theory may be desirable, in many instances insufficient information is available to accurately model the formal procedures available during the strategic bargaining process, or the resulting model would be too complex to offer a practical tool in the real world. In such cases, cooperative game theory provides a simplified approach that allows analysis of the game at large without having to make any assumption about bargaining powers.

Symmetric / asymmetric E F E 1, 2 0, 0 F 0, 0 1, 2 An asymmetric game Main article: Symmetric game A symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. That is, if the identities of the players can be changed without changing the payoff to the strategies, then a game is symmetric. Many of the commonly studied 2×2 games are symmetric. The standard representations of chicken, the prisoner's dilemma, and the stag hunt are all symmetric games. Some[who?] scholars would consider certain asymmetric games as examples of these games as well. However, the most common payoffs for each of these games are symmetric.

The most commonly studied asymmetric games are games where there are not identical strategy sets for both players. For instance, the ultimatum game and similarly the dictator game have different strategies for each player. It is possible, however, for a game to have identical strategies for both players, yet be asymmetric. For example, the game pictured in this section's graphic is asymmetric despite having identical strategy sets for both players.

Zero-sum / non-zero-sum A B A –1, 1 3, –3 B 0, 0 –2, 2 A zero-sum game Main article: Zero-sum game Zero-sum games are a special case of constant-sum games in which choices by players can neither increase nor decrease the available resources. In zero-sum games, the total benefit goes to all players in a game, for every combination of strategies, always adds to zero (more informally, a player benefits only at the equal expense of others).[15] Poker exemplifies a zero-sum game (ignoring the possibility of the house's cut), because one wins exactly the amount one's opponents lose. Other zero-sum games include matching pennies and most classical board games including Go and chess.

Many games studied by game theorists (including the famed prisoner's dilemma) are non-zero-sum games, because the outcome has net results greater or less than zero. Informally, in non-zero-sum games, a gain by one player does not necessarily correspond with a loss by another.

Constant-sum games correspond to activities like theft and gambling, but not to the fundamental economic situation in which there are potential gains from trade. It is possible to transform any game into a (possibly asymmetric) zero-sum game by adding a dummy player (often called "the board") whose losses compensate the players' net winnings.

Simultaneous / sequential Main articles: Simultaneous game and Sequential game Simultaneous games are games where both players move simultaneously, or instead the later players are unaware of the earlier players' actions (making them effectively simultaneous). Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions. This need not be perfect information about every action of earlier players; it might be very little knowledge. For instance, a player may know that an earlier player did not perform one particular action, while they do not know which of the other available actions the first player actually performed.

The difference between simultaneous and sequential games is captured in the different representations discussed above. Often, normal form is used to represent simultaneous games, while extensive form is used to represent sequential ones. The transformation of extensive to normal form is one way, meaning that multiple extensive form games correspond to the same normal form. Consequently, notions of equilibrium for simultaneous games are insufficient for reasoning about sequential games; see subgame perfection.

In short, the differences between sequential and simultaneous games are as follows:

Sequential Simultaneous Normally denoted by Decision trees Payoff matrices Prior knowledge of opponent's move? Yes No Time axis? Yes No Also known as Extensive-form game Extensive game Strategy game Strategic game Cournot Competition The Cournot competition model involves players choosing quantity of a homogenous product to produce independently and simultaneously, where marginal cost can be different for each firm and the firm's payoff is profit. The production costs are public information and the firm aims to find their profit-maximising quantity based on what they believe the other firm will produce and behave like monopolies. In this game firms want to produce at the monopoly quantity but there is a high incentive to deviate and produce more, which decreases the market-clearing price.[16] For example, firms may be tempted to deviate from the monopoly quantity if there is a low monopoly quantity and high price, with the aim of increasing production to maximise profit.[16] However this option does not provide the highest payoff, as a firm's ability to maximise profits depends on its market share and the elasticity of the market demand.[17] The Cournot equilibrium is reached when each firm operates on their reaction function with no incentive to deviate, as they have the best response based on the other firms output.[16] Within the game, firms reach the Nash equilibrium when the Cournot equilibrium is achieved.

Equilibrium for Cournot quantity competition Bertrand Competition Main article: Bertrand competition The Bertrand competition, assumes homogenous products and a constant marginal cost and players choose the prices.[16] The equilibrium of price competition is where the price is equal to marginal costs, assuming complete information about the competitors' costs. Therefore, the firms have an incentive to deviate from the equilibrium because a homogenous product with a lower price will gain all of the market share, known as a cost advantage.[18]

Perfect information and imperfect information Main article: Perfect information

A game of imperfect information (the dotted line represents ignorance on the part of player 2, formally called an information set) An important subset of sequential games consists of games of perfect information. A game is one of perfect information if all players, at every move in the game, know the moves previously made by all other players. In reality, this can be applied to firms and consumers having information about price and quality of all the available goods in a market.[19] An imperfect information game is played when the players do not know all moves already made by the opponent such as a simultaneous move game.[16] Most games studied in game theory are imperfect-information games.[citation needed] Examples of perfect-information games include tic-tac-toe, checkers, infinite chess, and Go.[20][21][22][23]

Many card games are games of imperfect information, such as poker and bridge.[24] Perfect information is often confused with complete information, which is a similar concept.[citation needed] Complete information requires that every player know the strategies and payoffs available to the other players but not necessarily the actions taken, whereas perfect information is knowledge of all aspects of the game and players.[25] Games of incomplete information can be reduced, however, to games of imperfect information by introducing "moves by nature".[26]

Bayesian game Main article: Bayesian game For one of the assumptions behind the concept of Nash equilibrium, every player has right beliefs about the actions of the other players. In game theory, there are many situations where participants do not fully understand the characteristics of their opponents. Negotiators may be unaware of their opponent's valuation of the object of negotiation, companies may be unaware of their opponent's cost functions, combatants may be unaware of their opponent's strengths, and jurors may be unaware of their colleague's interpretation of the evidence at trial. In some cases, participants may know the character of their opponent well, but may not know how well their opponent knows his or her own character.[27]

Bayesian game means a strategic game with incomplete information. For a strategic game, decision makers are players, and every player has a group of actions. A core part of the imperfect information specification is the set of states. Every state completely describes a collection of characteristics relevant to the player such as their preferences and details about them. There must be a state for every set of features that some player believes may exist.[28]

example of bayesian game For example, where Player 1 is unsure whether Player 2 would rather date her or get away from her, while Player 2 understands Player 1's preferences as before. To be specific, supposing that Player 1 believes that Player 2 wants to date her under a probability of 1/2 and get away from her under a probability of 1/2 (this evaluation comes from Player 1's experience probably: she faces players who want to date her half of the time in such a case and players who want to avoid her half of the time). Due to the probability involved, the analysis of this situation requires to understand the player's preference for the draw, even though people are only interested in pure strategic equilibrium.

Combinatorial games Games in which the difficulty of finding an optimal strategy stems from the multiplicity of possible moves are called combinatorial games. Examples include chess and go. Games that involve imperfect information may also have a strong combinatorial character, for instance backgammon. There is no unified theory addressing combinatorial elements in games. There are, however, mathematical tools that can solve particular problems and answer general questions.[29]

Games of perfect information have been studied in combinatorial game theory, which has developed novel representations, e.g. surreal numbers, as well as combinatorial and algebraic (and sometimes non-constructive) proof methods to solve games of certain types, including "loopy" games that may result in infinitely long sequences of moves. These methods address games with higher combinatorial complexity than those usually considered in traditional (or "economic") game theory.[30][31] A typical game that has been solved this way is Hex. A related field of study, drawing from computational complexity theory, is game complexity, which is concerned with estimating the computational difficulty of finding optimal strategies.[32]

Research in artificial intelligence has addressed both perfect and imperfect information games that have very complex combinatorial structures (like chess, go, or backgammon) for which no provable optimal strategies have been found. The practical solutions involve computational heuristics, like alpha–beta pruning or use of artificial neural networks trained by reinforcement learning, which make games more tractable in computing practice.[29][33]

Infinitely long games Main article: Determinacy Games, as studied by economists and real-world game players, are generally finished in finitely many moves. Pure mathematicians are not so constrained, and set theorists in particular study games that last for infinitely many moves, with the winner (or other payoff) not known until after all those moves are completed.

The focus of attention is usually not so much on the best way to play such a game, but whether one player has a winning strategy. (It can be proven, using the axiom of choice, that there are games – even with perfect information and where the only outcomes are "win" or "lose" – for which neither player has a winning strategy.) The existence of such strategies, for cleverly designed games, has important consequences in descriptive set theory.

Discrete and continuous games Much of game theory is concerned with finite, discrete games that have a finite number of players, moves, events, outcomes, etc. Many concepts can be extended, however. Continuous games allow players to choose a strategy from a continuous strategy set. For instance, Cournot competition is typically modeled with players' strategies being any non-negative quantities, including fractional quantities.

Differential games Differential games such as the continuous pursuit and evasion game are continuous games where the evolution of the players' state variables is governed by differential equations. The problem of finding an optimal strategy in a differential game is closely related to the optimal control theory. In particular, there are two types of strategies: the open-loop strategies are found using the Pontryagin maximum principle while the closed-loop strategies are found using Bellman's Dynamic Programming method.

A particular case of differential games are the games with a random time horizon.[34] In such games, the terminal time is a random variable with a given probability distribution function. Therefore, the players maximize the mathematical expectation of the cost function. It was shown that the modified optimization problem can be reformulated as a discounted differential game over an infinite time interval.

Evolutionary game theory Evolutionary game theory studies players who adjust their strategies over time according to rules that are not necessarily rational or farsighted.[35] In general, the evolution of strategies over time according to such rules is modeled as a Markov chain with a state variable such as the current strategy profile or how the game has been played in the recent past. Such rules may feature imitation, optimization, or survival of the fittest.

In biology, such models can represent evolution, in which offspring adopt their parents' strategies and parents who play more successful strategies (i.e. corresponding to higher payoffs) have a greater number of offspring. In the social sciences, such models typically represent strategic adjustment by players who play a game many times within their lifetime and, consciously or unconsciously, occasionally adjust their strategies.[36]

Stochastic outcomes (and relation to other fields) Individual decision problems with stochastic outcomes are sometimes considered "one-player games". These situations are not considered game theoretical by some authors.[by whom?] They may be modeled using similar tools within the related disciplines of decision theory, operations research, and areas of artificial intelligence, particularly AI planning (with uncertainty) and multi-agent system. Although these fields may have different motivators, the mathematics involved are substantially the same, e.g. using Markov decision processes (MDP).[37]

Stochastic outcomes can also be modeled in terms of game theory by adding a randomly acting player who makes "chance moves" ("moves by nature").[38] This player is not typically considered a third player in what is otherwise a two-player game, but merely serves to provide a roll of the dice where required by the game.

For some problems, different approaches to modeling stochastic outcomes may lead to different solutions. For example, the difference in approach between MDPs and the minimax solution is that the latter considers the worst-case over a set of adversarial moves, rather than reasoning in expectation about these moves given a fixed probability distribution. The minimax approach may be advantageous where stochastic models of uncertainty are not available, but may also be overestimating extremely unlikely (but costly) events, dramatically swaying the strategy in such scenarios if it is assumed that an adversary can force such an event to happen.[39] (See Black swan theory for more discussion on this kind of modeling issue, particularly as it relates to predicting and limiting losses in investment banking.)

General models that include all elements of stochastic outcomes, adversaries, and partial or noisy observability (of moves by other players) have also been studied. The "gold standard" is considered to be partially observable stochastic game (POSG), but few realistic problems are computationally feasible in POSG representation.[39]

Metagames These are games the play of which is the development of the rules for another game, the target or subject game. Metagames seek to maximize the utility value of the rule set developed. The theory of metagames is related to mechanism design theory.

The term metagame analysis is also used to refer to a practical approach developed by Nigel Howard.[40] whereby a situation is framed as a strategic game in which stakeholders try to realize their objectives by means of the options available to them. Subsequent developments have led to the formulation of confrontation analysis.

Pooling games These are games prevailing over all forms of society. Pooling games are repeated plays with changing payoff table in general over an experienced path, and their equilibrium strategies usually take a form of evolutionary social convention and economic convention. Pooling game theory emerges to formally recognize the interaction between optimal choice in one play and the emergence of forthcoming payoff table update path, identify the invariance existence and robustness, and predict variance over time. The theory is based upon topological transformation classification of payoff table update over time to predict variance and invariance, and is also within the jurisdiction of the computational law of reachable optimality for ordered system.[41]

Mean field game theory Mean field game theory is the study of strategic decision making in very large populations of small interacting agents. This class of problems was considered in the economics literature by Boyan Jovanovic and Robert W. Rosenthal, in the engineering literature by Peter E. Caines, and by mathematician Pierre-Louis Lions and Jean-Michel Lasry.

Representation of games The games studied in game theory are well-defined mathematical objects. To be fully defined, a game must specify the following elements: the players of the game, the information and actions available to each player at each decision point, and the payoffs for each outcome. (Eric Rasmusen refers to these four "essential elements" by the acronym "PAPI".)[42][43][44][45] A game theorist typically uses these elements, along with a solution concept of their choosing, to deduce a set of equilibrium strategies for each player such that, when these strategies are employed, no player can profit by unilaterally deviating from their strategy. These equilibrium strategies determine an equilibrium to the game—a stable state in which either one outcome occurs or a set of outcomes occur with known probability.

Most cooperative games are presented in the characteristic function form, while the extensive and the normal forms are used to define noncooperative games.

Extensive form Main article: Extensive form game

An extensive form game The extensive form can be used to formalize games with a time sequencing of moves. Games here are played on trees (as pictured here). Here each vertex (or node) represents a point of choice for a player. The player is specified by a number listed by the vertex. The lines out of the vertex represent a possible action for that player. The payoffs are specified at the bottom of the tree. The extensive form can be viewed as a multi-player generalization of a decision tree.[46] To solve any extensive form game, backward induction must be used. It involves working backward up the game tree to determine what a rational player would do at the last vertex of the tree, what the player with the previous move would do given that the player with the last move is rational, and so on until the first vertex of the tree is reached.[47]

The game pictured consists of two players. The way this particular game is structured (i.e., with sequential decision making and perfect information), Player 1 "moves" first by choosing either F or U (fair or unfair). Next in the sequence, Player 2, who has now seen Player 1's move, chooses to play either A or R. Once Player 2 has made their choice, the game is considered finished and each player gets their respective payoff. Suppose that Player 1 chooses U and then Player 2 chooses A: Player 1 then gets a payoff of "eight" (which in real-world terms can be interpreted in many ways, the simplest of which is in terms of money but could mean things such as eight days of vacation or eight countries conquered or even eight more opportunities to play the same game against other players) and Player 2 gets a payoff of "two".

The extensive form can also capture simultaneous-move games and games with imperfect information. To represent it, either a dotted line connects different vertices to represent them as being part of the same information set (i.e. the players do not know at which point they are), or a closed line is drawn around them. (See example in the imperfect information section.)

Normal form Player 2 chooses Left Player 2 chooses Right Player 1 chooses Up 4, 3 –1, –1 Player 1 chooses Down 0, 0 3, 4 Normal form or payoff matrix of a 2-player, 2-strategy game Main article: Normal-form game The normal (or strategic form) game is usually represented by a matrix which shows the players, strategies, and payoffs (see the example to the right). More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions. In the accompanying example there are two players; one chooses the row and the other chooses the column. Each player has two strategies, which are specified by the number of rows and the number of columns. The payoffs are provided in the interior. The first number is the payoff received by the row player (Player 1 in our example); the second is the payoff for the column player (Player 2 in our example). Suppose that Player 1 plays Up and that Player 2 plays Left. Then Player 1 gets a payoff of 4, and Player 2 gets 3.

When a game is presented in normal form, it is presumed that each player acts simultaneously or, at least, without knowing the actions of the other. If players have some information about the choices of other players, the game is usually presented in extensive form.

Every extensive-form game has an equivalent normal-form game, however, the transformation to normal form may result in an exponential blowup in the size of the representation, making it computationally impractical.[48]

Characteristic function form Main article: Cooperative game theory In games that possess removable utility, separate rewards are not given; rather, the characteristic function decides the payoff of each unity. The idea is that the unity that is 'empty', so to speak, does not receive a reward at all.

The origin of this form is to be found in John von Neumann and Oskar Morgenstern's book; when looking at these instances, they guessed that when a union {\displaystyle \mathbf {C} }\mathbf {C} appears, it works against the fraction {\displaystyle \left({\frac {\mathbf {N} }{\mathbf {C} }}\right)}\left({\frac {\mathbf {N} }{\mathbf {C} }}\right) as if two individuals were playing a normal game. The balanced payoff of C is a basic function. Although there are differing examples that help determine coalitional amounts from normal games, not all appear that in their function form can be derived from such.

Formally, a characteristic function is seen as: (N,v), where N represents the group of people and {\displaystyle v:2^{N}\to \mathbf {R} }v:2^{N}\to \mathbf {R} is a normal utility.

Such characteristic functions have expanded to describe games where there is no removable utility.

Alternative game representations See also: Succinct game Alternative game representation forms exist and are used for some subclasses of games or adjusted to the needs of interdisciplinary research.[49] In addition to classical game representations, some of the alternative representations also encode time related aspects.

General and applied uses As a method of applied mathematics, game theory has been used to study a wide variety of human and animal behaviors. It was initially developed in economics to understand a large collection of economic behaviors, including behaviors of firms, markets, and consumers. The first use of game-theoretic analysis was by Antoine Augustin Cournot in 1838 with his solution of the Cournot duopoly. The use of game theory in the social sciences has expanded, and game theory has been applied to political, sociological, and psychological behaviors as well.[63]

Although pre-twentieth-century naturalists such as Charles Darwin made game-theoretic kinds of statements, the use of game-theoretic analysis in biology began with Ronald Fisher's studies of animal behavior during the 1930s. This work predates the name "game theory", but it shares many important features with this field. The developments in economics were later applied to biology largely by John Maynard Smith in his 1982 book Evolution and the Theory of Games.[64]

In addition to being used to describe, predict, and explain behavior, game theory has also been used to develop theories of ethical or normative behavior and to prescribe such behavior.[65] In economics and philosophy, scholars have applied game theory to help in the understanding of good or proper behavior. Game-theoretic arguments of this type can be found as far back as Plato.[66] An alternative version of game theory, called chemical game theory, represents the player's choices as metaphorical chemical reactant molecules called "knowlecules".[67] Chemical game theory then calculates the outcomes as equilibrium solutions to a system of chemical reactions.

Description and modeling

A four-stage centipede game The primary use of game theory is to describe and model how human populations behave.[citation needed] Some[who?] scholars believe that by finding the equilibria of games they can predict how actual human populations will behave when confronted with situations analogous to the game being studied. This particular view of game theory has been criticized. It is argued that the assumptions made by game theorists are often violated when applied to real-world situations. Game theorists usually assume players act rationally, but in practice, human behavior often deviates from this model. Game theorists respond by comparing their assumptions to those used in physics. Thus while their assumptions do not always hold, they can treat game theory as a reasonable scientific ideal akin to the models used by physicists. However, empirical work has shown that in some classic games, such as the centipede game, guess 2/3 of the average game, and the dictator game, people regularly do not play Nash equilibria. There is an ongoing debate regarding the importance of these experiments and whether the analysis of the experiments fully captures all aspects of the relevant situation.[b]

Some game theorists, following the work of John Maynard Smith and George R. Price, have turned to evolutionary game theory in order to resolve these issues. These models presume either no rationality or bounded rationality on the part of players. Despite the name, evolutionary game theory does not necessarily presume natural selection in the biological sense. Evolutionary game theory includes both biological as well as cultural evolution and also models of individual learning (for example, fictitious play dynamics).

Prescriptive or normative analysis Cooperate Defect Cooperate -1, -1 -10, 0 Defect 0, -10 -5, -5 The prisoner's dilemma Some scholars see game theory not as a predictive tool for the behavior of human beings, but as a suggestion for how people ought to behave. Since a strategy, corresponding to a Nash equilibrium of a game constitutes one's best response to the actions of the other players – provided they are in (the same) Nash equilibrium – playing a strategy that is part of a Nash equilibrium seems appropriate. This normative use of game theory has also come under criticism.[citation needed]

Economics and business Game theory is a major method used in mathematical economics and business for modeling competing behaviors of interacting agents.[c][69][70][71] Applications include a wide array of economic phenomena and approaches, such as auctions, bargaining, mergers and acquisitions pricing,[72] fair division, duopolies, oligopolies, social network formation, agent-based computational economics,[73][74] general equilibrium, mechanism design,[75][76][77][78][79] and voting systems;[80] and across such broad areas as experimental economics,[81][82][83][84][85] behavioral economics,[86][87][88][89][90][91] information economics,[42][43][44][45] industrial organization,[92][93][94][95] and political economy.[96][97][98][99]

This research usually focuses on particular sets of strategies known as "solution concepts" or "equilibria". A common assumption is that players act rationally. In non-cooperative games, the most famous of these is the Nash equilibrium. A set of strategies is a Nash equilibrium if each represents a best response to the other strategies. If all the players are playing the strategies in a Nash equilibrium, they have no unilateral incentive to deviate, since their strategy is the best they can do given what others are doing.[100][101]

The payoffs of the game are generally taken to represent the utility of individual players.

A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of a particular economic situation. One or more solution concepts are chosen, and the author demonstrates which strategy sets in the presented game are equilibria of the appropriate type. Economists and business professors suggest two primary uses (noted above): descriptive and prescriptive.[65]

The Chartered Institute of Procurement & Supply (CIPS) promotes knowledge and use of game theory within the context of business procurement.[102] CIPS and TWS Partners have conducted a series of surveys designed to explore the understanding, awareness and application of game theory among procurement professionals. Some of the main findings in their third annual survey (2019) include:

application of game theory to procurement activity has increased – at the time it was at 19% across all survey respondents 65% of participants predict that use of game theory applications will grow 70% of respondents say that they have "only a basic or a below basic understanding" of game theory 20% of participants had undertaken on-the-job training in game theory 50% of respondents said that new or improved software solutions were desirable 90% of respondents said that they do not have the software they need for their work.[103] Project management Sensible decision-making is critical for the success of projects. In project management, game theory is used to model the decision-making process of players, such as investors, project managers, contractors, sub-contractors, governments and customers. Quite often, these players have competing interests, and sometimes their interests are directly detrimental to other players, making project management scenarios well-suited to be modeled by game theory.

Piraveenan (2019)[104] in his review provides several examples where game theory is used to model project management scenarios. For instance, an investor typically has several investment options, and each option will likely result in a different project, and thus one of the investment options has to be chosen before the project charter can be produced. Similarly, any large project involving subcontractors, for instance, a construction project, has a complex interplay between the main contractor (the project manager) and subcontractors, or among the subcontractors themselves, which typically has several decision points. For example, if there is an ambiguity in the contract between the contractor and subcontractor, each must decide how hard to push their case without jeopardizing the whole project, and thus their own stake in it. Similarly, when projects from competing organizations are launched, the marketing personnel have to decide what is the best timing and strategy to market the project, or its resultant product or service, so that it can gain maximum traction in the face of competition. In each of these scenarios, the required decisions depend on the decisions of other players who, in some way, have competing interests to the interests of the decision-maker, and thus can ideally be modeled using game theory.

Piraveenan[104] summarises that two-player games are predominantly used to model project management scenarios, and based on the identity of these players, five distinct types of games are used in project management.

Government-sector–private-sector games (games that model public–private partnerships) Contractor–contractor games Contractor–subcontractor games Subcontractor–subcontractor games Games involving other players In terms of types of games, both cooperative as well as non-cooperative, normal-form as well as extensive-form, and zero-sum as well as non-zero-sum are used to model various project management scenarios.

Political science The application of game theory to political science is focused in the overlapping areas of fair division, political economy, public choice, war bargaining, positive political theory, and social choice theory. In each of these areas, researchers have developed game-theoretic models in which the players are often voters, states, special interest groups, and politicians.

Early examples of game theory applied to political science are provided by Anthony Downs. In his 1957 book An Economic Theory of Democracy,[105] he applies the Hotelling firm location model to the political process. In the Downsian model, political candidates commit to ideologies on a one-dimensional policy space. Downs first shows how the political candidates will converge to the ideology preferred by the median voter if voters are fully informed, but then argues that voters choose to remain rationally ignorant which allows for candidate divergence. Game theory was applied in 1962 to the Cuban Missile Crisis during the presidency of John F. Kennedy.[106]

It has also been proposed that game theory explains the stability of any form of political government. Taking the simplest case of a monarchy, for example, the king, being only one person, does not and cannot maintain his authority by personally exercising physical control over all or even any significant number of his subjects. Sovereign control is instead explained by the recognition by each citizen that all other citizens expect each other to view the king (or other established government) as the person whose orders will be followed. Coordinating communication among citizens to replace the sovereign is effectively barred, since conspiracy to replace the sovereign is generally punishable as a crime. Thus, in a process that can be modeled by variants of the prisoner's dilemma, during periods of stability no citizen will find it rational to move to replace the sovereign, even if all the citizens know they would be better off if they were all to act collectively.[107]

A game-theoretic explanation for democratic peace is that public and open debate in democracies sends clear and reliable information regarding their intentions to other states. In contrast, it is difficult to know the intentions of nondemocratic leaders, what effect concessions will have, and if promises will be kept. Thus there will be mistrust and unwillingness to make concessions if at least one of the parties in a dispute is a non-democracy.[108]

However, game theory predicts that two countries may still go to war even if their leaders are cognizant of the costs of fighting. War may result from asymmetric information; two countries may have incentives to mis-represent the amount of military resources they have on hand, rendering them unable to settle disputes agreeably without resorting to fighting. Moreover, war may arise because of commitment problems: if two countries wish to settle a dispute via peaceful means, but each wishes to go back on the terms of that settlement, they may have no choice but to resort to warfare. Finally, war may result from issue indivisibilities.[109]

Game theory could also help predict a nation's responses when there is a new rule or law to be applied to that nation. One example is Peter John Wood's (2013) research looking into what nations could do to help reduce climate change. Wood thought this could be accomplished by making treaties with other nations to reduce greenhouse gas emissions. However, he concluded that this idea could not work because it would create a prisoner's dilemma for the nations.[110]

Biology Hawk Dove Hawk 20, 20 80, 40 Dove 40, 80 60, 60 The hawk-dove game Main article: Evolutionary game theory Unlike those in economics, the payoffs for games in biology are often interpreted as corresponding to fitness. In addition, the focus has been less on equilibria that correspond to a notion of rationality and more on ones that would be maintained by evolutionary forces. The best-known equilibrium in biology is known as the evolutionarily stable strategy (ESS), first introduced in (Maynard Smith & Price 1973). Although its initial motivation did not involve any of the mental requirements of the Nash equilibrium, every ESS is a Nash equilibrium.

In biology, game theory has been used as a model to understand many different phenomena. It was first used to explain the evolution (and stability) of the approximate 1:1 sex ratios. (Fisher 1930) suggested that the 1:1 sex ratios are a result of evolutionary forces acting on individuals who could be seen as trying to maximize their number of grandchildren.

Additionally, biologists have used evolutionary game theory and the ESS to explain the emergence of animal communication.[111] The analysis of signaling games and other communication games has provided insight into the evolution of communication among animals. For example, the mobbing behavior of many species, in which a large number of prey animals attack a larger predator, seems to be an example of spontaneous emergent organization. Ants have also been shown to exhibit feed-forward behavior akin to fashion (see Paul Ormerod's Butterfly Economics).

Biologists have used the game of chicken to analyze fighting behavior and territoriality.[112]

According to Maynard Smith, in the preface to Evolution and the Theory of Games, "paradoxically, it has turned out that game theory is more readily applied to biology than to the field of economic behaviour for which it was originally designed". Evolutionary game theory has been used to explain many seemingly incongruous phenomena in nature.[113]

One such phenomenon is known as biological altruism. This is a situation in which an organism appears to act in a way that benefits other organisms and is detrimental to itself. This is distinct from traditional notions of altruism because such actions are not conscious, but appear to be evolutionary adaptations to increase overall fitness. Examples can be found in species ranging from vampire bats that regurgitate blood they have obtained from a night's hunting and give it to group members who have failed to feed, to worker bees that care for the queen bee for their entire lives and never mate, to vervet monkeys that warn group members of a predator's approach, even when it endangers that individual's chance of survival.[114] All of these actions increase the overall fitness of a group, but occur at a cost to the individual.

Evolutionary game theory explains this altruism with the idea of kin selection. Altruists discriminate between the individuals they help and favor relatives. Hamilton's rule explains the evolutionary rationale behind this selection with the equation c < b × r, where the cost c to the altruist must be less than the benefit b to the recipient multiplied by the coefficient of relatedness r. The more closely related two organisms are causes the incidences of altruism to increase because they share many of the same alleles. This means that the altruistic individual, by ensuring that the alleles of its close relative are passed on through survival of its offspring, can forgo the option of having offspring itself because the same number of alleles are passed on. For example, helping a sibling (in diploid animals) has a coefficient of 1⁄2, because (on average) an individual shares half of the alleles in its sibling's offspring. Ensuring that enough of a sibling's offspring survive to adulthood precludes the necessity of the altruistic individual producing offspring.[114] The coefficient values depend heavily on the scope of the playing field; for example if the choice of whom to favor includes all genetic living things, not just all relatives, we assume the discrepancy between all humans only accounts for approximately 1% of the diversity in the playing field, a coefficient that was 1⁄2 in the smaller field becomes 0.995. Similarly if it is considered that information other than that of a genetic nature (e.g. epigenetics, religion, science, etc.) persisted through time the playing field becomes larger still, and the discrepancies smaller.

Computer science and logic Game theory has come to play an increasingly important role in logic and in computer science. Several logical theories have a basis in game semantics. In addition, computer scientists have used games to model interactive computations. Also, game theory provides a theoretical basis to the field of multi-agent systems.[115]

Separately, game theory has played a role in online algorithms; in particular, the k-server problem, which has in the past been referred to as games with moving costs and request-answer games.[116] Yao's principle is a game-theoretic technique for proving lower bounds on the computational complexity of randomized algorithms, especially online algorithms.

The emergence of the Internet has motivated the development of algorithms for finding equilibria in games, markets, computational auctions, peer-to-peer systems, and security and information markets. Algorithmic game theory[117] and within it algorithmic mechanism design[118] combine computational algorithm design and analysis of complex systems with economic theory.[119][120][121]

Philosophy Stag Hare Stag 3, 3 0, 2 Hare 2, 0 2, 2 Stag hunt Game theory has been put to several uses in philosophy. Responding to two papers by W.V.O. Quine (1960, 1967), Lewis (1969) used game theory to develop a philosophical account of convention. In so doing, he provided the first analysis of common knowledge and employed it in analyzing play in coordination games. In addition, he first suggested that one can understand meaning in terms of signaling games. This later suggestion has been pursued by several philosophers since Lewis.[122][123] Following Lewis (1969) game-theoretic account of conventions, Edna Ullmann-Margalit (1977) and Bicchieri (2006) have developed theories of social norms that define them as Nash equilibria that result from transforming a mixed-motive game into a coordination game.[124][125]

Game theory has also challenged philosophers to think in terms of interactive epistemology: what it means for a collective to have common beliefs or knowledge, and what are the consequences of this knowledge for the social outcomes resulting from the interactions of agents. Philosophers who have worked in this area include Bicchieri (1989, 1993),[126][127] Skyrms (1990),[128] and Stalnaker (1999).[129]

In ethics, some (most notably David Gauthier, Gregory Kavka, and Jean Hampton)[who?] authors have attempted to pursue Thomas Hobbes' project of deriving morality from self-interest. Since games like the prisoner's dilemma present an apparent conflict between morality and self-interest, explaining why cooperation is required by self-interest is an important component of this project. This general strategy is a component of the general social contract view in political philosophy (for examples, see Gauthier (1986) and Kavka (1986)).[d]

Other authors have attempted to use evolutionary game theory in order to explain the emergence of human attitudes about morality and corresponding animal behaviors. These authors look at several games including the prisoner's dilemma, stag hunt, and the Nash bargaining game as providing an explanation for the emergence of attitudes about morality (see, e.g., Skyrms (1996, 2004) and Sober and Wilson (1998)).

Retail and consumer product pricing Game theory applications are used heavily in the pricing strategies of retail and consumer markets, particularly for the sale of inelastic goods. With retailers constantly competing against one another for consumer market share, it has become a fairly common practice for retailers to discount certain goods, intermittently, in the hopes of increasing foot-traffic in brick and mortar locations (websites visits for e-commerce retailers) or increasing sales of ancillary or complimentary products.[130]

Black Friday, a popular shopping holiday in the US, is when many retailers focus on optimal pricing strategies to capture the holiday shopping market. In the Black Friday scenario, retailers using game theory applications typically ask "what is the dominant competitor's reaction to me?"[131] In such a scenario, the game has two players: the retailer, and the consumer. The retailer is focused on an optimal pricing strategy, while the consumer is focused on the best deal. In this closed system, there often is no dominant strategy as both players have alternative options. That is, retailers can find a different customer, and consumers can shop at a different retailer.[131] Given the market competition that day, however, the dominant strategy for retailers lies in outperforming competitors. The open system assumes multiple retailers selling similar goods, and a finite number of consumers demanding the goods at an optimal price. A blog by a Cornell University professor provided an example of such a strategy, when Amazon priced a Samsung TV $100 below retail value, effectively undercutting competitors. Amazon made up part of the difference by increasing the price of HDMI cables, as it has been found that consumers are less price discriminatory when it comes to the sale of secondary items.[131]

Retail markets continue to evolve strategies and applications of game theory when it comes to pricing consumer goods. The key insights found between simulations in a controlled environment and real-world retail experiences show that the applications of such strategies are more complex, as each retailer has to find an optimal balance between pricing, supplier relations, brand image, and the potential to cannibalize the sale of more profitable items.[132]

Epidemiology Since the decision to take a vaccine for a particular disease is often made by individuals, who may consider a range of factors and parameters in making this decision (such as the incidence and prevalence of the disease, perceived and real risks associated with contracting the disease, mortality rate, perceived and real risks associated with vaccination, and financial cost of vaccination), game theory has been used to model and predict vaccination uptake in a society.[133][134]