Introducing Game Theory: A Graphic Guide (Graphic Guides)

by Ivan Pastine

Game theory is a set of tools used to help analyze situations where an individual’s best course of action depends on what others do or are expected to do. Game theory allows us to understand how people act in situations where they are interconnected.

In economics, the decisions of firms are affected by their expectations of a competitor’s choice of product, price and advertising. In political science, a candidate’s policy platform is influenced by the policy announcements of their rival. In biology, animals must compete for scarce resources, but can be hurt if they are too aggressive with the wrong rival. In computer science, networked computers compete for bandwidth. In sociology, public displays of non-conformist attitudes are influenced by others’ behaviour, which is shaped by social culture.

Game theory is useful whenever there is strategic interaction, whenever how well you do depends on the actions of others as well as your own choices.

Game theory is the study of strategic interaction. Strategic interaction is also the key element of most board games, which is where it gets its name. Your decision affects the other player’s actions and vice versa. Much of the jargon of game theory is borrowed directly from games. The decision makers are called players. Players make a move when they make a decision.

In human interaction, for instance, it’s not just our decisions, but also our expressions, our tone of voice and our body language that influence others. People bring different histories and points of view to their dealings with others. This infinite variety can create very complex situations that are difficult to analyze. We can circumvent this complexity by creating simplistic structures, called models.

backward induction: you can figure out your opponent’s response to your possible actions and take that into consideration before making your own move.

aim is to improve our understanding of interactions between people, companies, countries, animals, etc., when the actual problems are too complex to fully understand.

A good model is simple enough to allow us to fully understand the incentives motivating players. At the same time, it must capture important elements of reality, which involves creative insight and judgement to determine which elements are most relevant.

English economist John Maynard Keynes (1883–1946) likens investment in financial markets to a newspaper competition in which readers have to choose the “prettiest face”; the readers who choose the most frequently chosen face win. ‘It is not a case of choosing those which, to the best of one’s judgement, are really the prettiest, nor even those which average opinion genuinely thinks the prettiest… we devote our intelligences to anticipating what average opinion expects the average opinion to be.’ At first glance, Keynes’ Beauty Contest has very little to do with financial markets: there are no prices, and there are no buyers or sellers. But they have one crucial feature in common. Success in financial markets depends on being one step ahead of the crowd. If you can predict the behaviour of the average investor, you can make a killing. Likewise in Keynes’ Beauty Contest, if you can predict the average choice of newspaper readers, you can win the contest. I think brunettes are prettier, but most people prefer blondes, so I think the blonde will be the most popular.

First, there was the crash in stock prices in October 1987. The late 1990s saw a spectacular rise and fall in technology stocks. The irrational exuberance shifted to real estate, leading up to the peak in August 2006, followed by a crash that helped cause the global financial crisis. Even former chairman of the Federal Reserve Alan Greenspan apologised: “Those of us who have looked to the self-interest of lending institutions to protect shareholders’ equity — myself especially — are in a state of shocked disbelief.” Many other economists who were ardent supporters of the efficient market hypothesis (EMH) have also been surprised by recent history but there is one man who would not have been “shocked”: John Maynard Keynes. The efficient market hypothesis (EMH), alternatively known as the efficient market theory, is a hypothesis that states that share prices reflect all information and consistent alpha (risk adjusted excess returns) generation is impossible. According to the EMH, stocks always trade at their fair value on exchanges, making it impossible for investors to purchase undervalued stocks or sell stocks for inflated prices. Therefore, it should be impossible to outperform the overall market through expert stock selection or market timing, and the only way an investor can obtain higher returns is by purchasing riskier investments. Although it is a cornerstone of modern financial theory, the EMH is highly controversial and often disputed. Believers argue it is pointless to search for undervalued stocks or to try to predict trends in the market through either fundamental or technical analysis. Theoretically, neither technical nor fundamental analysis can produce risk-adjusted excess returns (alpha) consistently, and only inside information can result in outsized risk-adjusted returns. While academics point to a large body of evidence in support of EMH, an equal amount of dissension also exists. For example, investors such as Warren Buffett have consistently beaten the market over long periods, which by definition is impossible according to the EMH. Detractors of the EMH also point to events such as the 1987 stock market crash, when the Dow Jones Industrial Average (DJIA) fell by over 20 percent in a single day, as evidence that stock prices can seriously deviate from their fair values. Special Considerations Proponents of the Efficient Market Hypothesis conclude that, because of the randomness of the market, investors could do better by investing in a low-cost, passive portfolio. Data compiled by Morningstar Inc., in its June 2019 Active/Passive Barometer study, supports the EMH. Morningstar compared active managers’ returns in all categories against a composite made of related index funds and exchange-traded funds (ETFs). The study found that over a 10 year period beginning June 2009, only 23% of active managers were able to outperform their passive peers. Better success rates were found in foreign equity funds and bond funds. Lower success rates were found in US large cap funds. In general, investors have fared better by investing in low-cost index funds or ETFs. Keynes is remembered for his view that governments should spend money in recessions to regain full employment, an argument made famous in The General Theory of Employment, Interest, and Money (1936). Few, however, realise that Keynes was a true forerunner of behavioural finance. Had more people, including Greenspan, studied the chapter of The General Theory on financial markets, the crisis might have been avoided. Keynes thought markets had been more “efficient” at the beginning of the 20th century, when managers owned most of the shares in a company and knew what it was worth. As shares became more widely dispersed, “the element of real knowledge in the valuation of investments by those who own them or contemplate purchasing them . . . seriously declined”. By the time of The General Theory, Keynes had concluded that markets had gone crazy. “Day-to-day fluctuations in the profits of existing investments, which are obviously of an ephemeral and non-significant character, tend to have an altogether excessive, and even an absurd, influence on the market.” To buttress his point, he noted the fact that shares of ice companies were higher in summer months when sales are higher. This fact is surprising because in an efficient market, stock prices reflect the long-run value of a company, and do not rise in good seasons. Recent academic studies show this pattern is still true. Keynes was also sceptical that professional money managers would perform the role of the “smart money” that EMH defenders rely upon to keep markets efficient. Rather, he thought they were more likely to ride a wave of irrational exuberance than to fight it. One reason is that it is risky to be a contrarian. “Worldly wisdom teaches that it is better for reputation to fail conventionally than to succeed unconventionally.” Instead, Keynes thought that professional money managers were playing an intricate guessing game. He likened it to a common newspaper game “in which the competitors have to pick out the six prettiest faces from 100 photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole: so that each competitor has to pick, not those faces that he himself finds prettiest, but those that he thinks likeliest to catch the fancy of the other competitors, all of whom are looking at the problem from the same point of view . . . We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practise the fourth, fifth, and higher degrees.” I believe Keynes’s beauty-contest analogy remains an apt description of how financial markets work, as well as of the key role played by behavioural factors. To understand his analogy, try out this puzzle that Tim Harford recently posed on my behalf to FT readers:

In 1997 the American behavioural economist Richard Thaler (b. 1945) ran an experiment in the Financial Times a Guessing Game which was a version of Keynes’ Beauty Contest.

Guess a number from zero to 100, with the goal of making your guess as close as possible to two-thirds of the average guess of all those participating in the contest. To help you think about this puzzle, suppose there are three players who guessed 20, 30 and 40 respectively. The average guess would be 30, two-thirds of which is 20, so the person who guessed 20 would win. If you did not enter the contest, you might consider what your guess might have been. Now that you have thought, consider what I will call a zero-level thinker. He says: “I don’t know. This seems like a maths problem. I will just pick a number at random.” Lots of people guessing a number between zero and 100 at random will produce an average guess of 50. How about a first-level thinker? She says: “The rest of these players don’t like to think much, they will probably pick a number at random, averaging 50, so I should guess 33, two-thirds of 50.” A second-level thinker will say: “Most players will be first-level thinkers and think that other players are a bit dim, so they will guess 33. Therefore I will guess 22.” A third-level thinker: “Most players will discern how the game works and will figure that most people will guess 33. As a result they will guess 22, so I will guess 15.” Of course, there is no convenient place to get off this train of thinking. Do you want to change your guess? Here is another question: what is the Nash equilibrium for this scenario? Named for John Nash, the mathematician and subject of the film A Beautiful Mind who sadly was recently killed in a car crash, the Nash equilibrium in this game is a number that if everyone guessed it, no one would want to change their guess. The only Nash equilibrium in this game is zero. To see why, suppose everyone guessed three. Then the average guess would be three and you would want to guess two-thirds of that, or two. But if everyone guessed two you would want to guess 1.33, and so forth. If, and only if, all participants guessed zero would no one want to change his or her guess. Formally, this game is identical to Keynes’s beauty contest: you have to guess what other people are thinking that other people are thinking. In economics, the “number guessing game” is commonly referred to as the “beauty contest”. Thanks to the FT, this is the second time I have run this experiment on a large scale [see panel]. In 1997 we offered two business-class tickets to North America. Now, in these days of austerity, entrants were offered what I have been assured is a posh travel bag. Personally, I am also throwing in an autographed copy of my recent book Misbehaving, on which this essay is based. How have things changed? Well, one finding will comfort tradition-bound economists. When the prize was two business-class tickets we had 1,382 contestants. With only a travel bag on offer, entrants dropped to 583. Economic theory is redeemed! Even with the smaller number of entrants, the results were nearly identical. In 1997 the average guess was 18.9, meaning the winning guess was 13. This time the average guess was 17.3, leading to a winning guess of 12. The distribution of guesses also looks like the one from 1997. Many contestants were able to figure out the Nash equilibrium and guessed zero or one, thinking everyone else would be as clever as they were. A large number also guessed 22, showing second-level thinking. Just as last time, there was an assortment of pranksters who guessed 99 or 100, trying to skew the results. Keynes’s beauty-contest analogy remains an apt description of what money managers do. Many investors call themselves “value managers”, meaning they try to buy stocks that are cheap. Others call themselves “growth managers”, meaning they try to buy stocks that will grow quickly. But of course no one is seeking to buy stocks that are expensive or stocks of companies that will shrink. So what these managers are really trying to do is buy stocks that will go up in value — or, in other words, stocks that they think other investors will later decide should be worth more. Buying a stock that the market does not fully appreciate today is fine, as long as the rest of the market comes around to your point of view sooner rather than later. Remember another of Keynes’s famous lines: “In the long run we are all dead.” The typical long run for a portfolio manager is no more than a few years; often just a few months! So to beat the market a money manager has to have a theory about how other investors will change their minds. In other words, their approach has to be behavioural. The Thaler challenge Reading through the submissions was to meet a wonderful slice of FT readers: from the witty to the saboteurs, from accountants to students, writes Caroline Daniel. Many were pleased with their own logic; most were men. Inevitably, there were readers who suggested 42 on the grounds that in The Hitchhiker’s Guide to the Galaxy “42 is the answer to the ultimate question of life”. Another reader joked: “33, the age of Jesus when crucified by the Romans.” Some readers chose numbers for personal reasons: “65 means both financial freedom and also the freedom to do as I will.” Another picked 56 as it was “the number I was issued when I became a special constable” in the Met. One chose a number based on “how many kilometres I ran in the park while thinking about Keynes and his idea of burying piles of money there”. There were instances of sabotage. One entrant confessed he chose 99 “in an evil attempt to render the statisticians’ attempts at guessing a correct answer useless”. A round-robin group all nominated 100, ominously declaring, “All of your base belong to us.” A sorry few did not even bother to explain their logic. “The answer is 10, because it isn’t nine or 11.” One of my favourite responses came from Benjamin Mueller of Keble College, Oxford. He noted that the winning answer in 1997 was 13. “Schooled as I am in neoclassical economics, I also assume that incentives matter. In 1997 the prize was two business-class tickets for a flight from London to New York. This makes it likely that the contest attracted a higher calibre of participants who thought harder about the puzzle than when the reward is a bag. The education effect is cancelled out by the diminished worth of prize. My guess, therefore, is 13.” He came close to winning — as did many readers (and I should add that the Dom Reilly bag is a very splendid one). I want to thank all those who participated. Several people identified 12, the winning number, but Richard Thaler picked Anatoly Lebedev, executive director, commodities electronic trading, at Goldman Sachs for his logic. Lebedev added this excellent warning: “If the competition was checked by a computer, there would be a ‘hacker’ solution of submitting a billion times one same number from fake accounts and then calling two-thirds of the number from a real account.” Saboteurs, watch out!

iterative reasoning forever – a process of reasoning that involves repetition of the same process, taking the result from one round as a starting point for the next. Game theorists solve the Guessing Game in a similar fashion using iterative elimination of dominated strategies.

Human behaviour is probably better approximated by bounded rationality. That is, human rationality is limited by the tractability of the decision problem (how easy it is to manage), the cognitive limitations of our minds, the time available in which to make the decision, and how important the decision is to us.

The Guessing Game and Keynes’ Beauty Contest can explain the interesting fact that in financial markets we observe bubbles – excessively inflated prices – even if all participants are rational. This is because of a lack of common knowledge of rationality.