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23
An approach to bounded rationality
 In Advances in Neural Information Processing Systems 19 (Proc. of NIPS 2006
, 2007
"... A central question in game theory and artificial intelligence is how a rational agent should behave in a complex environment, given that it cannot perform unbounded computations. We study strategic aspects of this question by formulating a simple model of a game with additional costs (computational ..."
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A central question in game theory and artificial intelligence is how a rational agent should behave in a complex environment, given that it cannot perform unbounded computations. We study strategic aspects of this question by formulating a simple model of a game with additional costs (computational or otherwise) for each strategy. First we connect this to zerosum games, proving a counterintuitive generalization of the classic minmax theorem to zerosum games with the addition of strategy costs. We then show that potential games with strategy costs remain potential games. Both zerosum and potential games with strategy costs maintain a very appealing property: simple learning dynamics converge to equilibrium. 1 The Approach and Basic Model How should an intelligent agent play a complicated game like chess, given that it does not have unlimited time to think? This question reflects one fundamental aspect of “bounded rationality, ” a term coined by Herbert Simon [1]. However, bounded rationality has proven to be a slippery concept to formalize (prior work has focused largely on finite automata playing simple repeated games such
Modeling the Economic Interaction of Agents with Diverse Abilities to Recognize Equilibrium Patterns
"... We model differences among agents in their ability to recognize temporal patterns of prices. Using the concept of DeBruijn sequences in two dynamic models of markets, we demonstrate the existence of equilibria in which prices fluctuate in a pattern that is independent of the fundamentals and that ca ..."
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We model differences among agents in their ability to recognize temporal patterns of prices. Using the concept of DeBruijn sequences in two dynamic models of markets, we demonstrate the existence of equilibria in which prices fluctuate in a pattern that is independent of the fundamentals and that can be recognized only by the more competent agents.
Cooperation, Repetition, and Automata
 Cooperation: Game Theoretic Approaches, NATO ASI Series F
, 1995
"... This talk studies the implications of bounding the complexity of players' strategies in long term interactions. The complexity of a strategy is measured by the size of the minimal automaton that can implement it. A finite automaton has a finite number of states and an initial state. It prescribes t ..."
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Cited by 9 (0 self)
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This talk studies the implications of bounding the complexity of players' strategies in long term interactions. The complexity of a strategy is measured by the size of the minimal automaton that can implement it. A finite automaton has a finite number of states and an initial state. It prescribes the action to be taken as a function of the current state and its next state is a function of its current state and the actions of the other players. The size of an automaton is its number of states. The results study the equilibrium payoffs per stage of the repeated games when players' strategies are restricted to those implementable by automata of bounded size. The first talk will concentrate maily on the 0sum case and address the following topics/questions. 1 What is the relation between the bounds of the automata sizes and the quantitative advantage of the player with the larger bound. (Theorems 1 and 3 of the enclosed paper) 2 What is the duration (number of repetition) needed for an unrestricted player to exploit fully his advantage over a player with bound automata (Conjecture 2 of the enclosed paper including a positive solution of its second part). 3 The existence of a deterministic periodic sequence (with period n) which is asymptotically random for every automata of size o(n= log n) (Proposition 2 of the enclosed paper). 1
Niche strategies: the Prisoner’s Dilemma computer tournaments revisited
 JOURNAL OF EVOLUTIONARY ECONOMICS
, 1989
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Bounded Rationality in Repeated Games and Mechanism Design for Agents in Computational Settings
, 2000
"... In Part I, we study bounded rationality in repeated twoperson zerosum games. First we investigate infinitely repeated games in which both players are restricted to pure strategies that can be executed on a finite automaton. In particular, we provide an upper bound on the number of states that Play ..."
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In Part I, we study bounded rationality in repeated twoperson zerosum games. First we investigate infinitely repeated games in which both players are restricted to pure strategies that can be executed on a finite automaton. In particular, we provide an upper bound on the number of states that Player 2 needs to defeat Player 1 when Player 1 is restricted to simple cycles of length m. Next we argue that the finite automaton approach to bounded rationality is not satisfactory. As an alternative, we propose limiting the number of strategies available to the players. We provide a thorough study of finitely repeatedly zero sum games in which Player 1 is restricted to mixing over a fixed number of pure strategies while Player 2 is unrestricted. We describe an optimal set of pure strategies for Player 1 and a method for describing these strategies such that any strategy from this set can be efficiently executed given its description. We develop upper and lower bounds on the value of these games and discuss how the value is related to the strategic entropy function defined by Neyman and Okada (1999). Finally, we show that an approximately optimal set can be produced in time which is linear in the size of the set. This set achieves a total expected payoff that is within an additive constant of the optimal.
On Finite Strategy Sets for Finitely Repeated ZeroSum Games
 Games and Economic Behavior
, 2003
"... We study nitely repeated twoperson zerosum games in which Player 1 is restricted to mixing over a xed number of pure strategies while Player 2 is unrestricted. We describe an optimal set of pure strategies for Player 1 along with an optimal mixed strategy. We show that the entropy of this mix ..."
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We study nitely repeated twoperson zerosum games in which Player 1 is restricted to mixing over a xed number of pure strategies while Player 2 is unrestricted. We describe an optimal set of pure strategies for Player 1 along with an optimal mixed strategy. We show that the entropy of this mixed strategy appears as a factor in an exact formula for the value of the game and thus is seen to have a direct numerical eect on the game's value. We develop upper and lower bounds on the value of these games that are within an additive constant and discuss how our results are related to the work of Neyman and Okada on strategic entropy (Neyman and Okada, 1999, Games Econ. Behavior 29, 191223). Finally, we use these results to bound the value of repeated games in which one of the players uses a computer with a bounded memory and is further restricted to using a constant amount of time at each stage. Journal of Economic Literature Classi cation Number: C72 Key Words: Bounded rationality, entropy, repeated games, nite automata This is a preprint of an article that appears in Games and Economic Behavior 43 (2003) 107136. Page numbering and gure placement may dier.
Online Concealed Correlation and Bounded Rationality ∗
, 2005
"... Correlation of players ’ actions may evolve in the common course of play of a repeated game with perfect monitoring (“online correlation”), and we study the concealment of such correlation from a boundedly rational player. We show that “strong ” players, i.e., players whose strategic complexity is l ..."
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Correlation of players ’ actions may evolve in the common course of play of a repeated game with perfect monitoring (“online correlation”), and we study the concealment of such correlation from a boundedly rational player. We show that “strong ” players, i.e., players whose strategic complexity is less stringently bounded, can orchestrate the online correlation of the actions of “weak ” players, where this correlation is concealed from an opponent of “intermediate ” strength. The feasibility of such “online concealed correlation ” is reflected in the individually rational payoff of the opponent and in the equilibrium payoffs of the repeated game. This result enables the derivation of a folk theorem that characterizes the set of equilibrium payoffs in a class of repeated games with boundedly rational players and a mechanism designer who sends public signals.
Repeated games with public signals and bounded recall
, 2006
"... This paper studies repeated games with public signals, symmetric bounded recall and pure strategies. Examples of equilibria for such games are provided and the convergence of the set of equilibrium payoffs is studied as the size of the recall increases. Convergence to the set of equilibria of the in ..."
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This paper studies repeated games with public signals, symmetric bounded recall and pure strategies. Examples of equilibria for such games are provided and the convergence of the set of equilibrium payoffs is studied as the size of the recall increases. Convergence to the set of equilibria of the infinitely repeated game does not hold in general but for particular signals and games. The difference between private and public strategies is relevant and the corresponding sets of equilibria behave differently. Key words: folk theorem, de Bruijn sequence, imperfect monitoring, uniform equilibrium.
Edgar Allan Poe’s Riddle: Framing Effects in Repeated Matching Pennies Games*
, 2009
"... Framing effects have a significant influence on the finitely repeated matching pennies game. The combination of being labelled "a guesser", and having the objective of matching the opponent’s action, appears to be advantageous. We find that being a player who aims to match the opponent’s action is a ..."
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Framing effects have a significant influence on the finitely repeated matching pennies game. The combination of being labelled "a guesser", and having the objective of matching the opponent’s action, appears to be advantageous. We find that being a player who aims to match the opponent’s action is advantageous irrespective of whether the player moves first or second. We examine alternative explanations for our results and relate them to Edgar Allan Poe’s "The Purloined Letter". We propose a behavioral model which generates the observed asymmetry in the players’ performance. JEL Classification: C91, C72
Complexity and Mixed Strategy Equilibria ∗
"... Unpredictable behavior is central for optimal play in many strategic situations because a predictable pattern leaves a player vulnerable to exploitation. A theory of unpredictable behavior is presented in the context of repeated twoperson zerosum games in which the stage games have no pure strateg ..."
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Unpredictable behavior is central for optimal play in many strategic situations because a predictable pattern leaves a player vulnerable to exploitation. A theory of unpredictable behavior is presented in the context of repeated twoperson zerosum games in which the stage games have no pure strategy equilibrium. Computational complexity considerations are introduced to restrict players ’ strategy sets. The use of Kolmogorov complexity allows us to obtain a sufficient condition for equilibrium existence. The resulting theory has implications for the empirical literature that tests the equilibrium hypothesis in a similar context. In particular, the failure of some tests for randomness does not justify rejection of equilibrium play.