In game theory, "guess 2/3 of the average" is a game that explores how a player’s strategic reasoning process takes into account the mental process of others in the game.[1]
In this game, players simultaneously select a real number between 0 and 100, inclusive. The winner of the game is the player(s) who select a number closest to 2/3 of the average of numbers chosen by all players.[2]
History
Distribution of the 2898 answers to 1983 tie breaker Jeux et Stratégie contest.
Alain Ledoux is the founding father of the "guess 2/3 of the average" game. In 1981, Ledoux used this game as a tie breaker in his French magazine Jeux et Stratégie. He asked about 4,000 readers, who reached the same number of points in previous puzzles, to state an integer between 1 and 1,000,000,000. The winner was the one who guessed closest to 2/3 of the average guess.[3] Rosemarie Nagel (1995) revealed the potential of guessing games of that kind: They are able to disclose participants' "depth of reasoning."[4]
In his influential book, Keynes compared the determination of prices in a stock market to that of a beauty contest. The competitors had to pick out the six prettiest faces from 100 photos, and the winner is the competitor whose choices best matches the average preferences of all the competitors. Keynes observed that "It is not a case of choosing those that, to the best of one’s judgment, are really the prettiest, nor even those that average opinion genuinely thinks the prettiest. 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 practice the fourth, fifth and higher degrees."[5]
Due to the analogy to Keynes's comparison of newspaper beauty contests and stock market investments[6] the guessing game is also known as the Keynesian beauty contest.[7] Rosemarie Nagel's experimental beauty contest became a famous game in experimental economics. The forgotten inventor of this game was unearthed in 2009 during an online beauty contest experiment with chess players provided by the University of Kassel:[8] Alain Ledoux, together with over 6,000 other chess players, participated in that experiment which looked familiar to him.[9][10]
Intuitively, guessing any number higher than 2/3 of what you expect others to guess on average cannot be part of a Nash equilibrium. The highest possible average that would occur if everyone guessed 100 is 66+2/3. Therefore, choosing a number that lies above 66+2/3 is strictly dominated for every player. These guesses can thus be eliminated. Once these strategies are eliminated for every player, 66+2/3 becomes the new highest possible average (that is, if everyone chooses 66+2/3). Therefore, any guess above 44+4/9 is weakly dominated for every player since no player will guess above 66+2/3, and 2/3 of 66+2/3 is 44+4/9. This process will continue as this logic is continually applied, If the same group of people play the game consistently, with each step, the highest possible logical answer keeps getting smaller, the average will move close to 0, all other numbers above 0 have been eliminated. If all players understand this logic and select 0, the game reaches its Nash equilibrium, which also happens to be the Pareto optimal solution.[11] At this state, every player has chosen to play the best response strategy for themselves, given what everyone else is choosing.
However, this degeneration does not occur in quite the same way if choices are restricted to, for example, the integers between 0 and 100. In this case, all integers except 0 and 1 vanish; it becomes advantageous to select 0 if you expect that at least 1/4 of all players will do so, and select 1 otherwise. (In this way, it is a lopsided version of the so-called "consensus game", where one wins by being in the majority.)
Rationality versus common knowledge of rationality
This game illustrates the difference between the perfect rationality of an actor and the common knowledge of the rationality of all players. To achieve its Nash equilibrium of 0, this game requires all players to be perfectly rational, rationality to be common knowledge, and all players to expect everyone else to behave accordingly.[12] Common knowledge means that every player has the same information, and they also know that everyone else knows that, and that everyone else knows that everyone else knows that, and so on, infinitely.[13] Common knowledge of rationality of all players is the reason why the winning guess is 0.
Economic game theorists have modelled this relationship between rationality and the common knowledge of rationality through K-level reasoning. K stands for the number of times a cycle of reasoning is repeated. A Level-k model usually assumes that k-level 0 agents would approach the game naively and make choices distributed uniformly over the range [0, 100]. In accordance with cognitive hierarchy theory, level 1 players select the best responses to level 0 choices, while level 2 players select the best responses to level 1 choices.[14] Level 1 players would assume that everyone else was playing at level 0, responding to an assumed average of 50 in relation to naive play, and thus their guess would be 33 (2/3 of 50). At k-level 2, a player would play more sophisticatedly and assume that all other players are playing at k-level 1, so they would choose 22 (2/3 of 33).[15] Players are presumptively aware of the probability distributions of selections at each higher level. It would take approximately 21 k-levels to reach 0, the Nash equilibrium of the game.
The guessing game depends on three elements: (1) the subject's perceptions of the level 0 would play; (2) the subject's expectations about the cognitive level of other players; and (3) the number of in-game reasoning steps that the subject is capable of completing.[16] Evidence suggest that most people play at k-levels 0 to 3,[17] so you would just have to think one step ahead of that to have a higher chance at winning the game. Therefore, being aware of this logic allow players to adjust their strategy. This means that perfectly rational players playing in such a game should not guess 0 unless they know that the other players are rational as well, and that all players' rationality is common knowledge. If a rational player reasonably believes that other players will not follow the chain of elimination described above, it would be rational for him/her to guess a number above 0 as their best response.
In reality, we can assume that most players are not perfectly rational, and do not have common knowledge of each other's rationality.[18] As a result, they will also expect others to have a bounded rationality and thus guess a number higher than 0.
Experimental results
This game is a common demonstration in game theory classes. It reveals the significant heterogeneity of behaviour.[17] It is unlikely that many people will play rationally according to the Nash Equilibrium. This is because the game has no strictly dominant strategy, so it requires players to consider what others will do. For Nash Equilibrium to be played, players would need to assume both that everyone else is rational and that there is common knowledge of rationality. However, this is a strong assumption.
Experiments demonstrate that many people make mistakes and do not assume common knowledge of rationality. It has been demonstrated that even economics graduate students do not guess 0.[4] When performed among ordinary people it is usually found that the winner's guess is much higher than 0: the winning value was found to be 33 in a large online competition organized by the Danish newspaper Politiken. 19,196 people participated and the prize was 5000 Danish kroner.[19]
The mean number chosen when playing the "guess 2/3 of the average" game four consecutive rounds
Grosskopf and Nagel’s investigation also revealed that most players do not choose 0 the first time they play this game. Instead, they realise that 0 is the Nash Equilibrium after some repetitions.[20] A study by Nagel reported an average initial choice of around 36. This corresponds to approximately two levels of k-level reasoning.[21]
Kocher and Sutter compared the behaviours between individual and groups in playing this type of game. They observed that while both subjects applied roughly the same level of reasoning, groups learned faster. This demonstrated that repetition enabled a group of individuals to observe others’ behaviour in previous games and correspondingly choose a number that increases their chances of winning the game.[22]
Sbriglia’s investigation also revealed that non-winners often try to imitate winners’ understanding of the game’s structure. Accordingly, other players adopt strategies which are best responses to the imitators’ behaviour instead of to the average level of rationality. This accelerates the attainment of the game’s Nash equilibrium.[12]
Real-life examples of K-level reasoning
K-level reasoning can be useful in several social and competitive interactions. For example, deciding when to sell or buy stock in the stock market before too many others do it and decrease your profitability.[1] Philosophers and psychologists observe this as an ability to consider other’s mental states to predict their actions.[23]
Another example of K-level reasoning is when stock traders evaluate stocks based on the value that others place on those stocks. Their goal is to foresee changes in valuation ahead of the general public.[2] Their choice is also likely influenced by other individuals’ choices, especially if those choices have been successful previously. This demonstrates the importance of social learning in arriving at the equilibrium of any decision-making.[12] Empirical studies show, shrewd traders like hedge fund managers frequently benefit from the cognitive biases of ordinary investors, "Second level thinking" is essential for active investors to achieve superior returns.[24]
Howard Marks, co-founder of one of the largest hedge funds in distressed securities, has given the example that when a company reports good news about future profits, first-level retail investors will buy its shares based on that good news alone. However, a second-level thinker with more sophistication would argue that if everyone only buys in response to good news, then the good news actually becomes bad news because it overvalues the stock's price, making it a bad choice.[24]
Similarly, during penalty kicks in soccer, both the shooter and goalie simultaneously decide whether to go left or right depending on what they expect the other person to do. Goalies tend to memorise the behavioural patterns of their opponents, but penalty shooters know that and will act accordingly. In each example, individuals will weigh their own understanding of the best response against how well they think others understand the situation (i.e., how rational they are).[25]
↑ Ledoux, Alain (1981). "Concours résultats complets. Les victimes se sont plu à jouer le 14 d'atout" [Competition results complete. The victims were pleased to play the trump 14]. Jeux & Stratégie (in French). 2 (10): 10–11.
↑ Schou, Astrid (22 September 2005). "Gæt-et-tal konkurrence afslører at vi er irrationelle". Politiken (in Danish). Retrieved 29 August 2017. Includes a histogram of the guesses. Note that some of the players guessed close to 100. A large number of players guessed 33.3 (i.e. 2/3 of 50), indicating an assumption that players would guess randomly. A smaller but significant number of players guessed 22.2 (i.e. 2/3 of 33.3), indicating a second iteration of this theory based on an assumption that players would guess 33.3. The final number of 33 was slightly below this peak, implying that on average each player iterated their assumption 1.07 times.
Game theory is the study of mathematical models of strategic interactions among rational agents. It has applications in many fields of social science, used extensively in economics as well as in logic, systems science and computer science. Traditional game theory addressed two-person zero-sum games, in which a participant's gains or losses are exactly balanced by the losses and gains of the other participant. In the 21st century, game theory applies to a wider range of behavioral relations, and it is now an umbrella term for the science of logical decision making in humans, animals, as well as computers.
In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equilibrium strategies of the other players, and no one has anything to gain by changing only one's own strategy. The principle of Nash equilibrium dates back to the time of Cournot, who in 1838 applied it to competing firms choosing outputs.
In economics and game theory, a participant is considered to have superrationality if they have perfect rationality but assume that all other players are superrational too and that a superrational individual will always come up with the same strategy as any other superrational thinker when facing the same problem. Applying this definition, a superrational player playing against a superrational opponent in a prisoner's dilemma will cooperate while a rationally self-interested player would defect.
In game theory, the centipede game, first introduced by Robert Rosenthal in 1981, is an extensive form game in which two players take turns choosing either to take a slightly larger share of an increasing pot, or to pass the pot to the other player. The payoffs are arranged so that if one passes the pot to one's opponent and the opponent takes the pot on the next round, one receives slightly less than if one had taken the pot on this round, but after an additional switch the potential payoff will be higher. Therefore, although at each round a player has an incentive to take the pot, it would be better for them to wait. Although the traditional centipede game had a limit of 100 rounds, any game with this structure but a different number of rounds is called a centipede game.
Matching pennies is a non-cooperative game studied in game theory. It is played between two players, Even and Odd. Each player has a penny and must secretly turn the penny to heads or tails. The players then reveal their choices simultaneously. If the pennies match, then Even wins and keeps both pennies. If the pennies do not match, then Odd wins and keeps both pennies.
In game theory, a player's strategy is any of the options which they choose in a setting where the optimal outcome depends not only on their own actions but on the actions of others. The discipline mainly concerns the action of a player in a game affecting the behavior or actions of other players. Some examples of "games" include chess, bridge, poker, monopoly, diplomacy or battleship. A player's strategy will determine the action which the player will take at any stage of the game. In studying game theory, economists enlist a more rational lens in analyzing decisions rather than the psychological or sociological perspectives taken when analyzing relationships between decisions of two or more parties in different disciplines.
In game theory, a Bayesian game is a strategic decision-making model which assumes players have incomplete information. Players hold private information relevant to the game, meaning that the payoffs are not common knowledge. Bayesian games model the outcome of player interactions using aspects of Bayesian probability. They are notable because they allowed, for the first time in game theory, for the specification of the solutions to games with incomplete information.
The public goods game is a standard of experimental economics. In the basic game, subjects secretly choose how many of their private tokens to put into a public pot. The tokens in this pot are multiplied by a factor and this "public good" payoff is evenly divided among players. Each subject also keeps the tokens they do not contribute.
Backward induction is the process of determining a sequence of optimal choices by employing reasoning backward from the end of a problem or situation to its beginning, choice by choice. It involves examining the last point at which a decision is to be made and identifying the most optimal choice of action at that point. Using this information, one can then determine what to do at the second-to-last point of decision. This process continues backward until the best action for every possible point along the sequence is determined. Backward induction was first utilized in 1875 by Arthur Cayley, who discovered the method while attempting to solve the Secretary problem.
The El Farol bar problem is a problem in game theory. Every Thursday night, a fixed population want to go have fun at the El Farol Bar, unless it's too crowded.
In game theory, strategic dominance occurs when one strategy is better than another strategy for one player, no matter how that player's opponents may play. Many simple games can be solved using dominance. The opposite, intransitivity, occurs in games where one strategy may be better or worse than another strategy for one player, depending on how the player's opponents may play.
In game theory, a focal point is a solution that people tend to choose by default in the absence of communication in order to avoid coordination failure. The concept was introduced by the American economist Thomas Schelling in his book The Strategy of Conflict (1960). Schelling states that "(p)eople can often concert their intentions or expectations with others if each knows that the other is trying to do the same" in a cooperative situation, so their action would converge on a focal point which has some kind of prominence compared with the environment. However, the conspicuousness of the focal point depends on time, place and people themselves. It may not be a definite solution.
A Keynesian beauty contest describes a beauty contest where judges are rewarded for selecting the most popular faces among all judges, rather than those they may personally find the most attractive. This idea is often applied in financial markets, whereby investors could profit more by buying whichever stocks they think other investors will buy, rather than the stocks that have fundamentally the best value. Because when other people buy a stock, they bid up the price, allowing an earlier investor to cash out with a profit, regardless of whether the price increases are supported by its fundamentals.
In game theory, folk theorems are a class of theorems describing an abundance of Nash equilibrium payoff profiles in repeated games. The original Folk Theorem concerned the payoffs of all the Nash equilibria of an infinitely repeated game. This result was called the Folk Theorem because it was widely known among game theorists in the 1950s, even though no one had published it. Friedman's (1971) Theorem concerns the payoffs of certain subgame-perfect Nash equilibria (SPE) of an infinitely repeated game, and so strengthens the original Folk Theorem by using a stronger equilibrium concept: subgame-perfect Nash equilibria rather than Nash equilibria.
Quantal response equilibrium (QRE) is a solution concept in game theory. First introduced by Richard McKelvey and Thomas Palfrey, it provides an equilibrium notion with bounded rationality. QRE is not an equilibrium refinement, and it can give significantly different results from Nash equilibrium. QRE is only defined for games with discrete strategies, although there are continuous-strategy analogues.
In game theory, an epsilon-equilibrium, or near-Nash equilibrium, is a strategy profile that approximately satisfies the condition of Nash equilibrium. In a Nash equilibrium, no player has an incentive to change his behavior. In an approximate Nash equilibrium, this requirement is weakened to allow the possibility that a player may have a small incentive to do something different. This may still be considered an adequate solution concept, assuming for example status quo bias. This solution concept may be preferred to Nash equilibrium due to being easier to compute, or alternatively due to the possibility that in games of more than 2 players, the probabilities involved in an exact Nash equilibrium need not be rational numbers.
In game theory, the traveler's dilemma is a non-zero-sum game in which each player proposes a payoff. The lower of the two proposals wins; the lowball player receives the lowball payoff plus a small bonus, and the highball player receives the same lowball payoff, minus a small penalty. Surprisingly, the Nash equilibrium is for both players to aggressively lowball. The traveler's dilemma is notable in that naive play appears to outperform the Nash equilibrium; this apparent paradox also appears in the centipede game and the finitely-iterated prisoner's dilemma.
Cognitive hierarchy theory (CHT) is a behavioral model originating in behavioral economics and game theory that attempts to describe human thought processes in strategic games. CHT aims to improve upon the accuracy of predictions made by standard analytic methods, which can deviate considerably from actual experimental outcomes.
Behavioral game theory seeks to examine how people's strategic decision-making behavior is shaped by social preferences, social utility and other psychological factors. Behavioral game theory analyzes interactive strategic decisions and behavior using the methods of game theory, experimental economics, and experimental psychology. Experiments include testing deviations from typical simplifications of economic theory such as the independence axiom and neglect of altruism, fairness, and framing effects. As a research program, the subject is a development of the last three decades.
Hervé Moulin is a French mathematician who is the Donald J. Robertson Chair of Economics at the Adam Smith Business School at the University of Glasgow. He is known for his research contributions in mathematical economics, in particular in the fields of mechanism design, social choice, game theory and fair division. He has written five books and over 100 peer-reviewed articles.
This page is based on this Wikipedia article Text is available under the CC BY-SA 4.0 license; additional terms may apply. Images, videos and audio are available under their respective licenses.
