# Discrete, Bounded Reasoning in Games

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1 Discrete, Bounded Reasoning in Games Level-k Thinking and Cognitive Hierarchies Joe Corliss Graduate Group in Applied Mathematics Department of Mathematics University of California, Davis June 12, 2015 Joe Corliss (UC Davis) Discrete, Bounded Reasoning June 12, / 21

2 References [1] Nagel, Rosemarie Unraveling in Guessing Games: An Experimental Study. American Economic Review 85 (5): [2] Camerer, Colin F., Teck-Hua Ho, and Juin-Kuan Chong A Cognitive Hierarchy Model of Games. Quarterly Journal of Economics 119 (3): [3] Arad, Ayala and Ariel Rubinstein The Money Request Game: A Level-k Reasoning Study. American Economic Review 102 (7): Joe Corliss (UC Davis) Discrete, Bounded Reasoning June 12, / 21

3 Outline 1. Motivating example 2. Level-k reasoning: theory and experiments 2.1 The Beauty Contest [1] Money Request Game [3] 3. Cognitive hierarchies [2] Joe Corliss (UC Davis) Discrete, Bounded Reasoning June 12, / 21

4 Example: Guess 2/3 of the Average Each player picks a real number in the interval [0, 100]. The winner(s) are those whose chosen number(s) are closest to 2/3 of the average of all chosen numbers. The unique pure NE: all players choose 0. In fact, IEWDA leaves only the action 0. Traditional game theory says that all players are perfectly rational, so we should play 0, right? Let s find out. Joe Corliss (UC Davis) Discrete, Bounded Reasoning June 12, / 21

5 Example: Guess 2/3 of the Average What s the problem? In using IEWDA we assume that all other players also use IEWDA. Apparently this is a poor assumption in practice. Indeed, in 2005 the Danish newspaper Politiken played this game with 19,196 of its readers, with a prize of 5,000 Danish kroner. The mean was Google-translated Politiken headline: Q: How can we model behavior outside equilibrium? If players don t reason completely, how much do they reason? Is there a distribution of amounts of reasoning? Joe Corliss (UC Davis) Discrete, Bounded Reasoning June 12, / 21

6 Level-k Reasoning [1], [3] Level-k reasoning assumes that players form beliefs over opponents actions in discrete steps. Moreover, every player in a game has a type describing the amount of reasoning he does. A 0-step or level-0 player (L0) forms no beliefs over opponents actions. His action depends on the model, and could be: 1. Best-responds to the rules of the game, ignoring opponents; or 2. Plays (uniformly) randomly; or 3. Chooses a salient or intuitive action, a.k.a. a focal point [Schelling, 1960] For k 1, a k-step or level-k player (Lk) best-responds to the belief that all opponents are level-(k 1). Joe Corliss (UC Davis) Discrete, Bounded Reasoning June 12, / 21

7 Modeling a Game [3] To develop a level-k model of a game, we need to determine: 1. What an L0 player would do, hopefully unambiguously 2. The distribution of player types The L0 action recursively determines the actions for each Lk, k 1. For (2.), we typically collect some data and choose a distribution of types to fit the data. This suggests there is not a universal distribution of types. That is, a player s level of reasoning is determined in part by the game. Joe Corliss (UC Davis) Discrete, Bounded Reasoning June 12, / 21

8 The Keynesian Beauty Contest [1] The general version of Guess 2/3 of the Average. Do players exhibit level-k reasoning? The game: Each player chooses a real number in the interval [0, 100]. Let 0 p < 1 be a parameter. The winner(s) are those whose chosen number(s) are closest to p times the mean of all chosen numbers. The winner(s) split a fixed prize; all others receive nothing. Game-theoretic solutions Unique NE: all players choose 0 IEWDA maximal reduction: {0} Joe Corliss (UC Davis) Discrete, Bounded Reasoning June 12, / 21

9 The Keynesian Beauty Contest [1] The level-k model: L0 player: chooses 50. Interpretation: chooses uniformly randomly over [0, 100], or 50 is a focal point Lk player, k 1: chooses 50p k, the best response to L(k 1) 50p k 0 as k (recover game theoretic solution) Keynes predicted that most players are Lk for some k 3, and that higher k is rare [2]. Q: Does player behavior fit into these reasoning categories? If so, is Keynes prediction correct? Joe Corliss (UC Davis) Discrete, Bounded Reasoning June 12, / 21

10 The Keynesian Beauty Contest [1] Experiment: p = 2/3. 66 German students. Small cash prizes. About 50% fall near step 1 or 2. All higher steps account for less than 10%. Supports Keynes idea. Q: The model fits behavior, but does it explain behavior? Q: How to account for choices > 50? Mistakes? IEWDA? Joe Corliss (UC Davis) Discrete, Bounded Reasoning June 12, / 21

11 The Money Request Game [3] Arad and Rubinstein design a simple game that naturally triggers level-k reasoning. Goal: observe unobstructed, unambiguous level-k reasoning. The game: You and another player are playing a game in which each player requests an amount of money. The amount must be (an integer) between 11 and 20 shekels. Each player will receive the amount he requests. A player will receive an additional amount of 20 shekels if he asks for exactly one shekel less than the other player. What amount of money would you request? [3] Joe Corliss (UC Davis) Discrete, Bounded Reasoning June 12, / 21

12 The Money Request Game [3] Nice properties of this game for level-k reasoning (compare to Beauty Contest): 1. Unambiguous level-0 action: Best responses are clear and simple. 3. Best responses are robust to a wide range of beliefs; e.g., 18 is a best response if > 52.5% of players choose Iterative reasoning is natural; game is defined in terms of best responses. No other clear decision method*: no pure NE, no dominated strategies. *So observed actions are very likely to be the result of iterative reasoning and not some other decision process. Joe Corliss (UC Davis) Discrete, Bounded Reasoning June 12, / 21

13 The Money Request Game [3] Experiment: The game was played by 108 undergraduate economics students at Tel Aviv University, who had not studied game theory before. After choosing their actions, the subjects were asked to give written explanations of their choices. The subjects submissions were randomly paired together, and the payoffs distributed. Results on the next slide. Joe Corliss (UC Davis) Discrete, Bounded Reasoning June 12, / 21

14 The Money Request Game [3] Observations: Subjects behavior is not near Nash equilibrium. This is statistically significant (chi-square test, p < ). The choice 20 was usually explained as a safety strategy. Almost all the choices were explained in terms of level-k reasoning. The choices were explained as guesses, rather than 4 9 reasoning steps. The model with the best statistical fit consists of the player types L0 L3, and a random/error type ( 20%! Perhaps L0?) Joe Corliss (UC Davis) Discrete, Bounded Reasoning June 12, / 21

15 The Money Request Game [3] Conclusions According to the level-k model, subjects did not use more than three steps of reasoning in choosing their actions. Past studies of level-k reasoning reached the same conclusion. However, past games had obstacles to level-k reasoning. This game was designed to remove all such obstacles. Q: Why is higher-level reasoning invariably absent in level-k models? A modification of level-k, called cognitive hierarchies, could explain this. Joe Corliss (UC Davis) Discrete, Bounded Reasoning June 12, / 21

16 Cognitive Hierarchies [2] Players are grouped into types as before. The level-0 type is unchanged. Let f (k) be the true distribution of types in a game. NEW: For k 1, a level-k player re-normalizes the actual frequencies to form his beliefs over the other types: f (q) f k (q) := k 1 q {0,..., k 1} i=0 f (i), ( I m the smartest player! ) A level-k player, k 1, best-responds given his beliefs over the other types. Joe Corliss (UC Davis) Discrete, Bounded Reasoning June 12, / 21

17 Cognitive Hierarchies [2] Compare: Level-k 1. Beliefs do not converge. 2. Beliefs are less accurate as k grows. 3. High-level players act differently. 4. Endless perceived benefits from thinking. Could reason forever. Cognitive Hierarchies 1. Beliefs f k converge to f as k. 2. Beliefs are more accurate as k grows. Level- is omniscient. 3. High-level players act similarly. 4. Diminishing benefits from thinking. Reasoning will terminate. Joe Corliss (UC Davis) Discrete, Bounded Reasoning June 12, / 21

18 Cognitive Hierarchies [2] How to determine f? If you have no theory, then you have to use data to estimate the values f (0),..., f (k) via maximum likelihood estimation. But we want a theory. What properties should f have? A discrete distribution. Decreasing term-to-term ratios; in fact, we desire the very specific property that f (k) f (k 1) 1 k One distribution very conveniently has exactly this property: Joe Corliss (UC Davis) Discrete, Bounded Reasoning June 12, / 21

19 Cognitive Hierarchies [2] Poisson. We assume that the types are Poisson distributed with mean/variance τ > 0, which depends on the player population and the game. f (k) := e τ τ k, k 0 k! Mode: τ if τ / Z, or τ and τ 1 if τ Z. Determine τ by fitting to data. E.g., minimize the (absolute) difference between the predicted and actual mean of chosen numbers [2]. Usually, 1 < τ < 2. Joe Corliss (UC Davis) Discrete, Bounded Reasoning June 12, / 21

20 Cognitive Hierarchies [2], [3] Revisit the game: In yellow are the CH-Poisson predictions, where τ = 2.36 gives a best fit. [3] claims all types 3 choose 17 I think this is wrong. Types 3 and 4 choose 17; I think types 5 choose 16. Q: Is CH better? Do we learn more? Joe Corliss (UC Davis) Discrete, Bounded Reasoning June 12, / 21

21 This high-level player will now answer your questions. Joe Corliss (UC Davis) Discrete, Bounded Reasoning June 12, / 21

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