Cognitive Ability in a Team-Play Beauty Contest Game: An Experiment Proposal THESIS

Transcription

1 Cognitive Ability in a Team-Play Beauty Contest Game: An Experiment Proposal THESIS Presented in Partial Fulfillment of the Requirements for the Degree Master of Mathematical Sciences in the Graduate School of The Ohio State University By Daniel Lee Kashner Graduate Program in Mathematics The Ohio State University 2014 Master's Examination Committee: Rodica Costin, Advisor Paul J. Healy

2 Copyright by Daniel Lee Kashner 2014

3 Abstract The p-beauty Contest game is introduced using concepts from game theory. Previous experimental results are reviewed, with an emphasis on how player behavior can be analyzed according to the level-k model. An experiment is proposed to test how players use information about their opponents cognitive abilities to predict the level of strategic sophistication at which they will play. ii

4 Acknowledgments The author wishes to thank Professors Rodica Costin and Paul J. Healy for advising on this thesis. The author also wishes to thank Professors John Kagel and James Peck for their guidance. iii

5 Vita Cookeville High School B.A., University of Memphis Field of Study Major Field: Mathematics iv

6 Table of Contents Abstract... ii Acknowledgments... iii Vita... iv List of Figures... vi Chapter 1: Introducing the p-beauty Contest... 1 Chapter 2: Experimental Results: A Review of the Literature Chapter 3: An Experiment Proposal References v

7 List of Figures Figure 1: Matching Pennies Figure 2: The Prisoners' Dilemma vi

8 Chapter 1: Introducing the p-beauty Contest Keynes compared financial investment/speculation to a beauty contest run by some newspapers in which readers are shown dozens of faces and asked to rank them according to attractiveness 1. The pictures were ranked according to the submitted votes. The winner of the contest was the reader whose individual ranking best matched the calculated official ranking. A strategic reader then, rather than voting according to his own preferences for physical attractiveness, might try to anticipate which faces would be considered most beautiful according to general opinion. Some might even try to anticipate what the general opinion about the general opinion about the faces might be, and so on. Defining the Game A p-beauty contest is a game in which players write down a real number between 0 and 100 (inclusive), along with their name. After all numbers are submitted, they are 1 Nagel (1995). 1

9 averaged. The winner is the player whose guess is closest to a pre-established number p times the average of the submitted numbers. The number p is often chosen to be p=2/3, and the game is sometimes called the Guess Two-Thirds the Average game. The winner receives a fixed prize, and the prize is split in case of a tie. This game is often used to illustrate common game theoretic concepts. We can analyze any strategic interaction involving multiple agents (players) mathematically as a game. In these strategic situations, an individual s payoff depends not only on his own decision over several options, but also on the decision of the other players in the game. Mathematically, a game is defined as G := (N,S,u) by specifying the following elements: A finite set of players, N ={1,, i,,n} For each player i, a nonempty set of actions/strategies,, with S = being called the set of strategy profiles For each player i, a utility (or payoff) function, with representing a preference relationship over the elements of S, in that for simply means that player i (weakly) prefers outcome to outcome we want to emphasize player i s role in a strategy profile, we can write. If as where. For the p-beauty contest with n players and, we have for all Further, we define the average, and a set of winning players (who share the prize) by. Then for 2

10 all players, if where #W denotes the number of elements of W, and T is the size of the prize to be split, and 0 if i is not in the set W. Nash Equilibrium A fundamental concept in game theory is the Nash equilibrium. A Nash Equilibrium (NE) of a game (N,S,u) is defined to be a strategy profile with the property that for every player,. In words, a strategy profile is a Nash Equilibrium if, given the fact that the other players are playing their strategy in the profile, no player can increase his payoff by deviating from his NE strategy. Equivalently, the Nash Equilibrium can be defined in terms of a best-response function. Given any define to be the (set-valued) bestresponse function of player i. So, a Nash Equilibrium can be defined as a strategy profile for which. In words, in a Nash Equilibrium, all players are mutually best-responding to their opponents. For the p-beauty contest, it can be shown that the (unique) Nash equilibrium (for p < 1) is for all players to choose 0 as their number. Assuming symmetry (for simplicity), the proof is as follows. Given the same set of strategies, all players will choose the same number in equilibrium. If a player i s opponents are all choosing a number x, then the average of the other guesses will be x. So, player i s best response is to get closer than his opponents to the winning value of px. But, since it is assumed that 3

11 the opponents are also best-responding, it must be that x = px. We are given that. So, x must be 0. Rationalizable Strategies A Nash Equilibrium can be seen as a steady state, perhaps as the result of players being familiar with their opponents strategies from analysis of the history of previous games 2. However, in a one-shot game like the p-beauty contest, the question of how a Nash Equilibrium might result is an important one. If presented with the game for the first time, how might players realize the Nash Equilibrium simply through their reasoning prior to selecting a strategy? In the background of the Nash Equilibrium concept, it is implicitly assumed players know their opponents strategies to which they can best respond. In reality, players may not know their opponents strategies with certainty. Instead, they form beliefs about their opponents strategies. Mathematically formulated, a belief player i holds about his opponents is a probability distribution over, denoted :=, called the simplex over. The intuitive interpretation is that is the probability that player i assigns to the collection of opponents strategies. Then, the best response function can include best responses to beliefs, where responses are chosen to maximize the expected value of the utility function, given the probability distribution. That is 2 See Osborne (2000), Chapter 12, Rationalizability. 4

12 is the strategy that maximizes. A player who chooses action is called rational if there exists a belief such that. The question of whether or not a player s opponents are rational is important for the player to form beliefs about them. If there is no belief that makes a strategy an element of, then we say that is strictly dominated. Intuitively, this means that if I am rational, I will not play a dominated strategy. Further, if I know that my opponent is rational, I will only choose actions from any actions for which the probability is not zero) for for which the support (those only includes opponent strategies that are undominated. In other words, if I know my opponent is rational, I will expect him to best respond to his beliefs, so I will not place positive probability on the chance of his playing any actions that are dominated. If I assume that my opponent is also rational, then the set of beliefs I can hold is a subset of the beliefs I could originally hold (without the assumption that my opponent is rational), so the set of actions I can play and still be rational is a subset of my original set of rationalizable actions. If I further assume that I am rational, I know my opponent is rational, I know that my opponent knows that I am rational, and on and on ; dominated strategies are eliminated iteratively from each of the players set of rationalizable strategies. If this process continues under the assumption of common knowledge of rationality (that everyone knows that everyone knows that knows that every player is rational), this Iterative Elimination of Dominated Strategies will result in a subset of rationalizable strategy profiles, of which any Nash Equilibria will be a subset. 5

13 In the p-beauty contest, it is easy to see that any strategy which includes guessing a number above 100p is weakly dominated by guessing 100p. That is, for p = 2/3, even if all of my opponents choose 100 as their number, making the average 100, I cannot improve my payoff by guessing any number higher than. Therefore, if I know my opponents are rational, I know that they will not guess any number higher than 100p, so now I know that p(100p) weakly dominates any number above it. Similarly, if I know that all of my opponents know that all of their opponents are rational, then they will not guess a number higher than 100, which means 100 weakly dominates any number above it, and so on As before, everyone guessing 0 remains the only rationalizable strategy profile, assuming common knowledge of rationality. 3 As will be shown below in the section which reviews experimental results 4, it may be more common for a player to reason toward the Nash Equilibrium of the p-beauty contest through a different method. Rather than first imagining all possible opponent actions, then eliminating those strategies that are dominated according to the extent of assumed knowledge of rationality at that round of elimination, a player may instead first form a belief about opponent actions and then consider best responding to it, iteratively imagining in each round that his opponents will hold beliefs identical to his own in the previous round. That is, a player may (through some means) come up with the initial 3 It should be noted that Iterated Elimination of Dominated Strategies (IEDS) usually applies for strict dominance. That is, if, then a strictly dominates b. Here, the inequality is weak, and elimination of weakly dominated strategies may present some problems, since the order in which strategies are eliminated may affect which set of outcomes survive all rounds of elimination. However, a game is dominance solvable if all players are indifferent between all outcomes that survive the iterative procedure in which all the weakly dominated actions of each player are eliminated at each stage [Osborne and Rubinstein, A Course in Game Theory (1994), p. 63], and this is the case for the p-beauty contest. 4 See Nagel (1995), Bosch-Domenech, et al. (2002), Breitmoser (2012), Burchardi and Penczynski (2012), Georganas, et al. (2012), etc. 6

14 belief that all opponents will guess numbers according to some distribution, often assumed to be uniform distribution over the interval [0,100]. A rational player will then best respond according to the expected value of that distribution. A player who knows his opponents are rational will then best respond to the best response to the expected value of a uniform distribution, and so on. In other words, a player who at first attributes no rationality to his opponents may at first assume they will guess randomly, resulting in a uniform distribution over [0,100], to which he should best-respond by guessing 50p. Upon attributing rationality to his opponents, the act of updating his beliefs about his opponents leads him to best-respond by guessing 50, and so on, as before. Again, this results in the Nash Equilibrium guess of 0. A Brief History of Game Theory It is useful to explore the elements of game theory before focusing on the p- beauty contest. The first major figure in game theory is the mathematician John von Neumann ( ). 5 He was a child prodigy in mathematics and published his first scientific paper when he was 19 years old. He earned his PhD from the University of Budapest in 1926 and taught in Berlin, Hamburg, and at Princeton University. There, he collaborated with economist Oskar Morgenstern to write the book that established the field of game theory: Theory of games and economic behavior, published in Osborne (2000), Chapter 1. 7

15 The book argues that any competitive game could be modeled mathematically by a simple structure, called the normal form of a game (as outlined in the previous section): There is a set of players, each player has a set of strategies, each player has a payoff function from the Cartesian product of these strategy sets into the real numbers, and each player must choose his strategy independently of the other players. 6 Von Neumann and Morgenstern focused on two-player zero-sum games, defined as games where the utilities for each player for a given outcome sum to zero. This can be thought of as one player being forced to pay the other player a certain dollar amount for each outcome, so that if player 1 s utility is positive five, player 2 s utility must be negative five. John Forbes Nash Jr. is the second major figure in game theory. Nash did not limit his attention to two-player zero-sum games. Also, whereas the Brouwer fixed-point theorem had been used previously, Nash used the Kakutani fixed-point theorem to look at strategy profiles that were mutually optimal. 7 For his contributions in game theory, Nash was awarded the Nobel prize in economics in 1994, which he shared with John C. Harsanyi and Reinhard Selten. 8 Nash s equilibrium concept is one of the most fundamental concepts in game theory. Game theory can be seen as an extension of the theory of rational choice to multiple players rather than a single individual. The theory of rational choice simply states that the action chosen by a decision-maker is at least as good, according to her 6 Myerson (1999), p Myerson (1999) 8 Osborne (2000), Chapter 1 8

16 preferences, as every other available action. 9 The structure of an individual choice problem is exactly the same as a game, except that the set of players N is a set with one element (a singleton set). Along with the theory of rational choice, game theory has been used as a mathematical foundation for the social sciences, and it has been used across disciplines, from economics to political science. 10 Some Two-Player Game Examples 11 Two player games are often represented in a game matrix. The rows denote player one s set of strategies, and the columns denote player two s set of strategies. Each cell then represents a strategy profile that results from player one playing the strategy associated with that row and player two playing the strategy associated with that column. Within each cell, two numbers are listed, separated by a comma. The first (left) number reveals the utility player one gets from that cell s strategy profile, and the second (right) number reveals the utility that player two gets. Example 1: Matching pennies Matching pennies is two-player zero-sum game in which each player chooses to show either the Head or Tail of a penny. If both players show the same side (both Heads 9 Osborne (2000), p Myerson (1999) 11 Taken from Osborne (2000) 9

17 or both Tails), player two must give player one a dollar. If both show a different side, then player one must give player two a dollar. As described above, the game can be represented by the following game matrix. Head Tail Head 1,-1-1,1 Tail -1,1 1,-1 Figure 1: Matching Pennies Example 2: The Prisoners Dilemma One of the most famous two-player games is called The Prisoners Dilemma. Two suspects of a serious crime are questioned separately by the police. The police tell each suspect that they have enough evidence to convict the suspect of a minor crime, but they need a confession to get a conviction for the serious crime. They offer each suspect a deal: if the suspect confesses but his partner remains quiet, that suspect will go free and his partner will spend ten years in prison. If both suspects confess, they will each do five years in prison. However, if neither suspect confesses, they will both be convicted of the minor crime and pay a heavy fine. Obviously each suspect prefers going free to paying a 10

18 fine to doing five years to doing ten years. So, the utility function will assign utility 3, 2, 1, and 0 to each outcome, respectively. The game can be represented by the following matrix. Quiet Confess Quiet 2,2 0,3 Confess 3,0 1,1 Figure 2: The Prisoners' Dilemma To illustrate the concepts of dominance and Nash Equilibrium, we will adopt the perspective of player 1. Player 1 does not know what player 2 will do. However, he can consider each possible action that player 2 might take and determine, in turn, what his best response to that action would be. Suppose player 2 will keep Quiet. If player 1 keeps Quiet, he gains utility 2, but if he Confesses, he gains utility 3. So, it is better for him to Confess if player 2 keeps Quiet. Now, suppose player 2 will Confess. If player 1 keeps Quiet, he gains utility 0. If he Confesses, he gains utility 1. So, it is better for him to Confess if player 2 Confesses. Notice, that it is better for player 1 to Confess no matter what player 2 does. By our previously developed terminology, Confessing 11

19 dominates keeping Quiet, regardless of what player 2 does. Since the game is symmetric, each player will play their dominant strategy, and the resulting strategy profile will be (Confess, Confess). Since Confessing is a dominant strategy, it is certainly a bestresponse (element of the best response function developed earlier). Therefore, (Confess, Confess) is a Nash Equilibrium. Note that (Quiet, Quiet) would be the outcome most preferred by each player. However, the way the game is set up (by not allowing players to form binding agreements with each other), the less efficient Nash Equilibrium outcome (Confess, Confess) is the expected result. 12

20 Chapter 2: Experimental Results: A Review of the Literature The Nash equilibrium for the p-beauty contest is never actually observed in experiments. The apparent simplicity of the game and the somewhat systematic departure from Nash equilibrium makes this an interesting game in which to explore alternative equilibrium concepts and issues related to higher order beliefs. The main assumption behind the analysis of the experimental data is that one can determine the level of strategic sophistication or depth of reasoning that the player engaged in from his guess. Nagel describes the thought processes of players with varying degrees of strategic sophistication as follows: A person is strategic of degree n if he chooses the number So, a guess of 50 may be anticipated (by more sophisticated players) from a player who is strategic of degree 0 (either as the expected value from guessing randomly over a symmetric distribution, or as a salient point that draws the player s focus to the midpoint of the interval.) A degree n player best responds to a degree n-1 player, with higher values of n indicating both a higher degree of sophistication and the belief that one s opponents are more strategic. This designation for a player s level of sophistication is sometimes called the player s k-level, and the model used to describe 12 Nagel (1995) 13

21 players actions in games where this kind of behavior may exist is often called the levelk model. Nagel conducted experiments in which the game was repeated, and players were informed of results from the previous round. The p-value was different in each experimental treatment, including one with p > 1 (which adds another Nash equilibrium in which all players guess 100). Compared to first-round play, guesses approached the Nash equilibrium after a few rounds in all treatments. Nagel tried to analyze not only players level of sophistication based on first-round guesses, but also the rate of learning by how well each player used the results of the previous round to adjust their guess in the next round. By observing the values of the winning guesses, one key question that is raised is which guesses are optimal for the actual distribution of players guesses. In other words, since players are not in equilibrium, how many iterations of reasoning will yield the highest expected payoff? This is important because the Nash equilibrium guess of 0 is not a best response to the other players actual guesses, according to the data. Stahl and Wilson explore this fact by looking at different types of players who are sophisticated enough to recognize the Nash equilibrium of the game, distinguished by the beliefs they hold about their opponents. Stahl and Wilson develop a model in which players can differ in two different ways that lead to non-equilibrium play: (1) they may hold different prior beliefs about their opponents, and (2) they may differ in their ability to best respond to these beliefs Stahl and Wilson (1995) 14

22 Unlike Nagel (and later Costa-Gomes and Crawford 14 ), Stahl and Wilson allow a level 2 player s priors to be a convex combination of level 0 (whose guesses are still assumed to follow a uniform distribution) and level 1 players (as opposed to the assumption that all level-n players believe all of their opponents to be level n-1 players.) They also notice the tendency for players to either make 0, 1, or 2 iterations of reasoning about their opponents, or else recognize that guesses that follow continued reasoning approach the Nash equilibrium in the limit. (This simplification seems valuable, since it is unlikely a player will undergo 5 or 6 levels of reasoning, developing fourth- or fifth-order beliefs about his opponents, without recognizing the Nash equilibrium of the game and expecting (at least some of) his opponents to recognize it as well.) The Nash types then are distinguished by whether they expect some or all of their opponents to recognize (and play) the Nash equilibrium of the game. A naive Nash type assumes all of his opponents will join him in playing the Nash equilibrium (in games with unique symmetric NE). A worldly type believes some players are naive Nash types, while others are level 0, 1, or 2 players. Stahl and Wilson also posit the existence of a third Nash type, but find no evidence for it in the data. A rational expectations type is aware that some opponents are just as unboundedly rational as he is, and responds to a mixture of (potentially) all types, including other rational expectations types. Stahl and Wilson s model includes two belief parameters one indicating the proportion of his opponents a player believes to be level 1 (who believe all other 14 Costa-Gomes and Crawford (2006) 15

23 opponents are behaving randomly, i.e. all others are level 0) and one indicating the proportion of his opponents a player believes to be naive Nash types as well as one precision parameter that allows variation in a player s response to his beliefs (similar to the QRE model). Since the parameters for types with degenerate beliefs (that are not a convex combination over multiple types, e.g. L0, L1, and naive Nash types) must be exactly 0 or 1, a wide range of values are left for the non-degenerate L2 and worldly types. Together with the precision response parameter, certain type descriptions allow room for data-fitting, and Stahl and Wilson address the problem of the worldly type becoming a residual category. They apply their model to data from twelve symmetric 3x3 games. While these games are not similar to the p-beauty contest, the paper is important for the general level-k/cognitive hierarchy literature. With so many potential types of opponents and parameters describing their beliefs and the precision with which they respond to those beliefs, one might wonder if there is a simpler model (perhaps with a single summary statistic of the average number of iterations that players reason through) for a player or analyst to adopt. Camerer et al. develop the cognitive hierarchy model and find that players do 1.5 steps of reasoning on average 15. They look for a distribution of thinking steps among all players and find that the Poisson distribution fits their data well. They use the beauty contest example to motivate their discussion of the cognitive hierarchy theory, but they use the model to examine other games, including coordination games and market entry games. 15 Camerer et al.(2004) 16

24 While the assumption that the levels of players follow a Poisson distribution seems more valuable to the analyst in describing results than it would be to a player trying to model his opponents, the Camerer paper is important because it mentions so many interesting issues raised by the cognitive hierarchy theory. They suggest, for example, that a player s level could be endogenous and perhaps reflect a cost-benefit analysis on the part of a player, where this cost is measured in cognitive effort, resulting in more steps of reasoning when stakes are higher. They also raise concerns regarding experiment protocol, where questions asked of the players about what they think their opponents will do can also increase the number of levels. Camerer et al. talk about the economic value of a normative theory describing players behavior and define it as the winnings that could be expected by a player responding according to the theory compared with the average winnings that players actually earn. Since equilibrium requires mutual consistency between players beliefs and the actual choices of their opponents, if a non-equilibrium theory of play (that decreases the gap between beliefs and actual choices) adds value, that value can be used as a measure of degree of disequilibrium present in the data. They show that the Poisson-CH model of play adds more value than someone playing equilibrium in the games that they examined. Georganas et al. raise many other interesting issues regarding the cognitive hierarchy theory 16. They seek to examine whether a player s level of reasoning corresponds to some personal characteristic, and if so, whether an individual s level will 16 Georganas et al. (2012) 17

25 remain the same across different kinds of games (perhaps at least relative to other players). They develop an undercutting game, where a player will win (and cause his opponent to lose) a certain amount if he can guess exactly one number less than his opponent s guess. The set-up is designed to encourage level-k reasoning by drawing attention to opponent strategies. While they find cross-game stability of a player s level for the undercutting games, they do not find evidence for it in two-player guessing games first used by Costa-Gomes and Crawford (similar to a two-player p-beauty contest, but with different p-values and intervals for each player). They draw a distinction between the level-k model (which the analyst might use after the experiment to describe his data) and the level-k heuristic (which a player might use to think about his opponents possible behaviors). In my opinion, although an as if model might accurately describe data no matter how well it actually matches the thought processes of experiment participants, the value (and accuracy) of the level-k model will likely increase with the proportion of players in which the level-k heuristic is triggered. Georganas et al. find that the level-k heuristic may be triggered more often in beauty contests, simple matrix games, and their undercutting game, but may not be as common in common-value auctions, global games, or endogenous-timing investment games. They also note that framing effects or training that focus players on iteratively calculating best responses might trigger the heuristic. 18

26 Issues with Modeling Inconsistent Higher Order Beliefs Strzalecki creates a very general model (able to incorporate existing models) by developing the notion of a cognitive type space that consists of a 3-tuple for each player that includes his type, his cognitive ability (level), and his beliefs (a distribution over all other types) 17. In this model, type is distinguished by ability and beliefs, so that for example, a level 2 who believes all opponents are level 1 can be distinguished from a level 2 who believes half opponents are level 1 and half are level 0. While this model exceeds the Stahl-Wilson model in generality (by not restricting bounded types to levels 0, 1, and 2), it does not incorporate any precision term, and so cannot account for players who make mistakes in responding to their beliefs. Also, since the belief parameter can vary for individuals with the same level, but the level could be inferred from the belief parameter (as one level higher than the highest level on which the player places positive probability), it is not obvious to me what is gained by having three parameters. In theory, a player with a relatively high level might put zero weight on the levels immediately before him, but unless a precision parameter (which might be a function of a player s level, with higher levels better responding than lower levels) is incorporated, then there is little practical difference between this player and someone whose type is a few levels lower, since they are both best responding to the same beliefs. The other difficulty of any model is in dealing with different Nash types, as defined before according to the Stahl-Wilson taxonomy of types, including Naive Nash 17 Strzalecki (2009) 19

27 as well as Worldly and Rational Expectiations (although these last two are sometimes both simply referred to as Sophisticated.) 18 Often modeled as level, this notation might be appropriate, since with a continuous interval over which to guess in a p-beauty contest, no finite number of iterations will lead a player to guess the NE of 0. However, an interesting question is how many steps of thinking a player must actually go through before taking their inductive step in realizing the NE through some sort of epiphany. Issues with p-beauty Contests While the p-beauty contest is often used to look at higher order beliefs because the mathematics involved in coming up with a guess is simple and intuitive, it is easy to overlook a few subtleties. Nagel mentions in a footnote that a player should guess in a way that accounts for the effect of his own guess on the average. Thus, if he believes the average of the other players guesses is µ, he should guess to solve:. Note that for large n, this effect becomes smaller. Several authors also note that for n = 2, guessing 0 is a (weakly) dominant strategy for p < 1. One problem with using p-beauty contests to determine a player s type or k-level is that a single guessed number is the only piece of information the analyst has about the player. For p = 2/3, does a guess of 22 indicate a level 2 player with degenerate beliefs? 18 Georganas et el. (2012) 20

28 Is it a worldly player who tried to estimate the proportions of naive Nash and all bounded types and decided it was the best response to the distribution of opponents? Or, is it a Level 0 (L0) player whose favorite number is 22 and who already decided he was going to guess 22 before the experimenter even got to the part of the instructions about calculating 2/3 of the average? (In this regard, eliciting comments that will be judged in case of a tie, like in Bosch-Domènech et al., may be illuminating. 19 ) Breitmoser points out that it is not necessary to try to guess the winning number exactly. 20 Depending on the beliefs a player holds about his opponents, this might not even be optimal. A level 1 player, defined by his belief that his opponents are playing randomly, should choose his guess so as to maximize the region over which his guess will be closest to the winning value (of p times the average) rather than try to guess the winning value exactly, based on the expected distribution of the n-1 variables. In this respect, it seems important to clarify whether level 0 players (who in some instances may only exist as an anchor for level 1 beliefs) are actually playing according to a uniform distribution, or picking an action that seems most salient, or perhaps something else. In the p-beauty contest, it may be necessary to vary experiment protocols in order to determine this. 19 Bosh-Domenech et al. (2002) 20 Breitmoser (2012) 21

29 Further Questions The value of the cognitive hierarchy model (as has been described in this section) seems contingent upon whether players actually think about their opponents as performing limited steps of reasoning (i.e. whether players actually iteratively best respond). In some games, people deviate from equilibrium in ways that appear to simply be mistakes and would be hard to justify according to inconsistent beliefs about their opponents. However, the p-beauty contest seems to be one such game where players may actually reason in steps. It is unclear whether an individual s characteristics are correlated with his steps of reasoning in a game (and whether this will remain constant across games). If so, it would be interesting to know which characteristics are most revealing. Georganas et al. test for what might be called empathy and cognitive reflection in players. I might label the ability to develop accurate beliefs about one s opponents strategic empathy, and the ability to best respond to those beliefs precision. Camerer mentions limits in working memory as a possible reason for bounded steps of reasoning. 21 The author would be interested in more exploration about the value of incorporating into a model a cognitive cost function, unique to the individual, which may constrain the cognitive effort it becomes profitable for an individual to exert in certain situations. While training may affect this cost in the long term, in the short term, 21 Camerer et al. (2004) 22

30 this could account for play being affected by something like the stakes at risk in the game or the presence of non-monetary payoffs. Even if no individual characteristic, such as a player s k-level, exists across games, there may still be some benefit in exploring Camerer s idea of the economic value of a normative theory. If I know how players in aggregate actually behave, I can determine an optimal strategy to maximize my expected winnings. Even though the cognitive hierarchy theory is mostly applicable to one-shot games or first-round games (before equilibrium play emerges), there are many real-world situations best described by this type of game. 23

31 Chapter 3: An Experiment Proposal This new experiment is proposed to explore some issues related to observed behavior in previous p-beauty contest experiments conducted by some researchers. A key question that the proposed experiment aims to answer is whether players who have access to information about their opponents cognitive abilities might use that information to form beliefs about how their opponents will play. Another question is whether there are specific kinds of cognitive ability related to an opponent s expected level of strategic sophistication. Specifically, is a player s iterative reasoning ability independent of his theory of mind ability? If so, how might relative weakness in one ability relative to the other be perceived by his opponents to affect his expected play? The question about whether players respond to information about their opponent s level of strategic sophistication by playing differently against different types of opponents has already been explored in Agranov et al. (2011) and Georganas et al. (2012). The other questions might be unique to this paper. By relaxing the assumption of common knowledge of rationality (as defined in the first section), most models explain observed behavior as resulting from either inconsistent beliefs (about an opponent s rationality) or inconsistent rationality (on the 24

32 part of the player making errors in responding to his beliefs). In the p-beauty contest, most belief-based models (those that rely on the first explanation) propose that a player reasons in steps (through iterated elimination of dominated strategies, or through iterated best responding to the expected value of the distribution of anticipated opponents actions). Most belief-based models exploring the p-beauty contest suggest that players reason in steps. They either suppose that the highest possible average is 100, so any guess above (100) is dominated. Repeating, is eliminated during the iteration. Alternatively, players hold initial beliefs about hypothetical nonstrategic opponents, and best respond to the expected value of the distribution of these opponents guesses (for k iterations). As k increases without bound, only 0 remains undominated (respectively, a best response). For most players, however, the number of steps a player reasons through falls short of recognizing the Nash equilibrium guess of 0. The question of why this happens remains unanswered. Cognitive Levels Does a player who does not guess 0 lack the cognitive ability to recognize the equilibrium play, does he believe others will lack the cognitive ability to do so, or are the two somehow intertwined? In Agranov, et al (2012), the authors note the distinction between a player s objective cognitive level (representing the number of steps of reasoning that a player is capable of in the game) versus his observed cognitive level 25

33 (the level at which the player chooses to play, which might be lower than his objective cognitive level if he is best responding to opponents he believes will have low observed cognitive levels) 22. Most papers on the subject classify players according to their observed cognitive level (or possibly by their objective cognitive level, if that could somehow be inferred). Previous studies have examined correlation between how close a player s guess is to 0 and his cognitive ability. Assuming deviations from equilibrium can be explained by inconsistent beliefs (rather than inconsistent action, i.e. cognitive errors or mistakes in responding to beliefs), another criterion by which to classify players might be the accuracy of their beliefs. Since the Nash equilibrium is never observed, this might be the best predictor of a player s payoff. Other criteria could be related to what might be called the sophistication of a player s beliefs: how well developed are beliefs about opponents? Do they involve beliefs about opponent beliefs, or just about opponent actions? Do they allow for heterogeneity of opponents? Heterogeneity of beliefs among a single cognitive level of opponents? Etc. For now, however, let s assume that the most valuable criterion is a player s objective cognitive level (that may need to be inferred from his observed cognitive level). A remaining question is how to characterize players according to the data. In other words, how exactly does an experimenter infer a player s objective cognitive level from his observed cognitive level? In fact, how is the observed cognitive level observed from the data available to the experimenter? Even if players are assumed to behave according to level-k thinking, in most experiments the experimenter is expected to 22 Agranov et al. (2012) 26

34 determine a player s cognitive level from a single piece of information: the number he guessed. So, does a guess of 22 indicate a Level 2 player who believed everyone else would guess 33 (the closest integer to 2/3 of 50, the expected value of a uniform distribution over [0,100])? Or, does it indicate a sophisticated player who attempted to estimate the proportions of all bounded types, Naive Nash types, and other sophisticated types (like herself) accounting for the fact that even within the subset of players performing the same number of steps of reasoning, (assuming no mistakes in responding to beliefs) there still might be heterogeneity in the beliefs they hold about the Level 0 action distribution or about the distribution of the proportions of types below them (as well as the beliefs those lower types might hold) and decided that 22 would be the guess closest to the winning value? Or, does it indicate a Level 0 player whose favorite number is 22 and who had already decided he was going to guess 22 before the experimenter even got to the part of the instructions about 2/3 of something? Identifying the Players One possible solution to the problem of how to categorize players according to their capacity for step-level reasoning in one-shot games (that is, games that will not be repeated) is through eliciting comments. Building on the innovative introduction of 27

35 team chat into experiments by Cooper and Kagel (2005) 23, a forthcoming paper by Burchardi and Penczynski (2012) 24 makes an important methodological contribution. Players are placed in teams of two to play a p-beauty contest. Each individual is told the rules and asked to write to his partner a suggested decision of what number to play, along with an explanation for how he chose that number. The suggestions are then simultaneously shown to the partners, after which each individual chooses a final decision to play. With probability ½, an individual s final decision is chosen to represent his team s action. Thus, since their payoff might depend on their partner s final decision, players are incentivized to explain their reasoning well enough in their comments to persuade their partners to choose the same final decision that they do. The simultaneous exchange of comments also helps reduce learning effects. Since according to the level-k model, players are classified according to the number of reasoning steps they make, the comments are then judged by this criterion. A lower bound on the number of iterations (of best responding or elimination of dominated strategies) that are clearly indicated in the comments, as well as an upper bound on the number of steps that one might infer the player took, based on the comment, are both coded and reported. So, using the authors example, a comment like, I presume everybody else will play 33, so let us play only explicitly demonstrates one step of best responding to a hypothetical opponent s action, but a previous step (of assuming everybody else will play 33 because they assume their opponents random guesses will 23 Cooper and Kagel (2005) 24 Burchardi and Penczynski (2012) 25 Burchardi and Penczynski (2012), p

36 have an expected value of 50) might be inferred, so the comment would be coded as indicating a player with a lower bound of L1 and an upper bound of L2. 26 Given this new way of interpreting a player s cognitive level in a p-beauty contest, several questions still remain. Do players in a p-beauty contest actually reason in steps? Do they model their opponents according to level-k? Is there any connection between the hypothetical opponents they respond to in their minds and their actual opponents? How does available information about their actual opponents affect their beliefs about their opponents? Modeling Opponents: Understanding Beliefs An example may help illustrate the issues mentioned above. Some players may be capable of recognizing that the Nash equilibrium of the p-beauty contest is for every player to guess 0. However, not all of them who recognize this actually guess 0. The Naive (or Stubborn) 27 Nash type guesses 0, but the sophisticated (or Worldly) type tries to estimate the distribution of other types (including Naive Nash types) and guesses accordingly. (Since the Nash equilibrium is never the observed outcome of these 26 The authors do not assume that players beliefs about L0 opponents (or the actual actions of non-strategic L0 players) will necessarily be represented by a uniform distribution, and in fact, their data show that this assumption might not be accurate, with the perceived salience of numbers having an effect. From my own experience, I ve had more than a few people immediately respond with a guess of Sixty-six! when presented with this problem, and I imagine the salience of certain numbers for non-strategic types might also be recognized by some more reflective types. 27 Descriptive names for types according to level-k models vary by author. Most of my choices come from Stahl and Wilson (1995); however, Sophisticated can include two types according to the Stahl/Wilson taxonomy: worldly and rational expectations. Stubborn Nash might be a term unique to Georganas, et al. (2010). 29

37 experiments, the sophisticated type is more likely to win the game, depending on the accuracy of his beliefs about this distribution.) So, why does the Naive Nash type guess 0? It could be that he has well-developed (but highly inaccurate) beliefs about his opponents, and he believes they will all reason the same way that he did, and they will all guess If so, the Naive Nash type is the only type to place full support on his own type (for the distribution with which he models his opponents). Compare this with the beliefs that Level 1 players hold about their opponents actions that they will be completely non-strategic. If Level 1 players lack strategic empathy or are overly pessimistic, then perhaps Naive Nash players lack strategic sympathy and are overly optimistic. However, it could be that the Naive Nash s beliefs are not very well developed. Perhaps the hypothetical opponents in the mind of a Naive Nash player do nothing more than help generate a geometric sequence 29 that converges to 0. Perhaps there exists no connection between these hypothetical opponents and the other players in the game. If Level 0 players are simply playing a guessing game, without any awareness of the strategic element involved in the situation, then perhaps the Naive Nash type is simply 28 With enough players, and choices allowed to be any real number, this is the only rational explanation for guessing 0 exactly. 29, where X is distributed according to the player s L0 beliefs for iterated best responding, or where guesses are chosen from the interval [a,b] for iterated elimination of dominated strategies. 30

38 solving a math problem, equally unaware of how his payoff depends not only on his choosing the right answer for his own action, but on the actions of other players. Assuming for a moment that there is a connection between the opponents a player best responds to in his thinking process before choosing his action in a one-shot game and the actual other players in the game. Exactly how might information about his opponents affect a player s beliefs about them? Ideally, if I want to win the game, all I need to do is know exactly what every other player will guess! Short of knowing all of the guesses with certainty, is there any information (personal characteristics) about the individual I m playing against that would help me predict the number of reasoning steps he might take in choosing his number? In other words, if players do indeed model their opponents according to level-k for the p-beauty contest, but there is no opportunity to update beliefs about actual opponents in a one-shot game, then would players make use of information about their opponents to update their beliefs before guessing? And, if so, what information would they consider the most valuable in attempting to type their opponents? The proposed experiment will address these questions from the point of view of a potential Nash type (unbounded objective cognitive level, but nothing assumed about the sophistication of his beliefs or what his observed cognitive level will be), aware of the 30 A third possibility is that there are non-monetary payoffs involved for this player guessing the right answer. Based on some conversations with colleagues in the Math department, the term stubborn Nash might actually be more appropriate than Naive. 31 Also note that the observed cognitive level is equal to the objective cognitive level for the Naïve Nash type, but is lower than the objective cognitive level for the Worldly type. So, even though the Worldly type s beliefs could be described as more sophisticated than the Naïve Nash type s, the Worldly type would be classified as a lower level, based only on the number he guessed, under some methods of analyzing the data. 31

39 Nash equilibrium guess, but possibly also aware that everyone else may not be aware of this. Adopting the perspective of a player, if I am presented with information about my opponent that indicates that he will likely fail to recognize the NE (that his cognitive level is bounded), then I should not guess 0 myself (and be labeled naive ). Rather, I should do my best to predict what he might guess (and be labeled sophisticated ) and respond to it effectively, thus increasing my chance at winning. As a player in the game, how would I make such a judgment about my opponent s strategic ability? I might be interested in personal characteristics related to his overall intelligence, possibly revealed by information about his SAT score, or major and GPA. Or, perhaps I might be interested in his capacity for cognitive reflection. I think the popularity of the p-beauty contest is in some ways due to its deceivingly simple nature (at first blush) that unravels with devious complexity the more attention one devotes to choosing a number. So, the ability (and willingness) to devote extended effort to considering the complexities of a problem that at first seems simple might be revealed by something like the Cognitive Reflection Test. 32 I might also be interested in his theory of mind abilities whether he is likely to attribute a capacity for reasoning to his opponents, or whether their actions are simply draws from a uniform distribution to him 32 This test was used in Georganas et al. (2012). The test includes three trick questions in which the answers that may come immediately to mind are incorrect, and the correct answers require further reflection on the question. (1) A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost? cents (2) If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets? minutes (3) In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake? days 32