Experimetrics: A Survey

This paper presents a new, probabilistic model of learning in games which investigates the often stated intuition that common knowledge of strategic intent may arise from repeated interaction. The model is set in the usual repeated game framework, but the two key assumptions are framed in terms of the likelihood of beliefs and actions conditional on the history of play. The first assumption formalizes the basic intuition of the learning approach; the second, the indeterminacy that inspired resort to learning models in the first place. Together the assumptions imply that, almost surely, play will remain almost always within one of the stage game's minimal inclusive sets. In important classes of games, including those with strategic complementarities, potential functions, and bandwagon effects, all such sets are singleton Nash.

Level-k analysis of experimental centipede games

Abstract. The aim of this research was to develop new learning models which integrating between learning strategies and learning medias to the atomic structure subject. The research was done in the first state secondary vocational school of Talawi. The research population consisted of four classes. All of the population were used as the research samples, so there were four experiment classes. They were first class of experiment was taught by using cooperative learning model was integrated with computer animation media; the second class of experiment was taught by using game learning model was integrated with computer animation media; the third class of experiment was taught by using game learning model was integrated with card media and the fourth class of experiment was taught by using game learning model was integrated with card media. The result of this research showed that learning strategies had influence to the student learning result significantly. The learning media had influence to the student learning result significantly. There were interaction between learning strategy with learning media. The most optimum of learning model to increase student learning result was game learning model was integrated with card media. Keywords : cooperative, game learning, animation media

Sponsored search auctions (SSAs) have attracted a lot of research attention in recent years and different equilibrium concepts have been studied in order to understand advertisers' bidding strategies. However, the assumption that advertisers are perfectly rational in these studies is unrealistic in the real world. In this work, we apply the quantal response equilibrium (QRE), which is powerful in modeling bounded rationality, to SSAs. Due to high computational complexity, existing methods for QRE computation have very poor scalability for SSAs. Through exploiting the structures of QRE for SSAs, this paper presents an efficient homotopy-based algorithm to compute the QRE for large-size SSAs, which features the following two novelties: 1) we represent the SSAs as an Action Graph Game (AGG) which can compute the expected utilities in polynomial time; 2) we further significantly reduce redundant calculations by leveraging the underlying relations between advertisers' utilities. We also develop an estimator to infer parameters of SSAs and fit the QRE model into a dataset from a commercial search engine. Our experimental results indicate that the algorithm can significantly improve the scalability of QRE computation for SSAs and the QRE model can describe the real-world bidding behaviors in a very accurate manner.

The quantal response equilibrium (QRE) notion of Richard D. McKelvey and Thomas R. Palfrey (1995) has recently attracted considerable attention, due in part to its widely documented ability to rationalize observed behavior in games played by experimental subjects. However, even with strong a priori restrictions on unobservables, QRE imposes no falsifiable restrictions: it can rationalize any distribution of behavior in any normal form game. After demonstrating this, we discuss several approaches to testing QRE under additional maintained assumptions. (JEL C72, D84)

In this paper, we design and implement an experiment aimed at testing the level-k model of auctions. We begin by asking which (simple) environments can best disentangle the level-k model from its leading rival, Bayes-Nash equilibrium. We find two environments that are particularly suited to this purpose: an all-pay auction with uniformly distributed values, and a first-price auction with the possibility of cancelled bids. We then implement both of these environments in a virtual laboratory in order to see which theory can best explain observed bidding behaviour. We find that, when plausibly calibrated, the level-k model substantially under-predicts the observed bids and is clearly out-performed by equilibrium. Moreover, attempting to fit the level-k model to the observed data results in implausibly high estimated levels, which in turn bear no relation to the levels inferred from a game known to trigger level-k reasoning. Finally, subjects almost never appeal to iterated reasoning when asked to explain how they bid. Overall, these findings suggest that, despite its notable success in predicting behaviour in other strategic settings, the level-k model (and its close cousin cognitive hierarchy) cannot explain behaviour in auctions.

In this paper, we design and implement an experiment aimed at testing the level-k model of auctions. We begin by asking which (simple) environments can best disentangle the level-k model from its leading rival, Bayes-Nash equilibrium. We find two environments that are particularly suited to this purpose: an all-pay auction with uniformly distributed values, and a first-price auction with the possibility of cancelled bids. We then implement both of these environments in a virtual laboratory in order to see which theory can best explain observed bidding behaviour. We find that, when plausibly calibrated, the level-k model substantially under-predicts the observed bids and is clearly out-performed by equilibrium. Moreover, attempting to fit the level-k model to the observed data results in implausibly high estimated levels, which in turn bear no relation to the levels inferred from a game known to trigger level-k reasoning. Finally, subjects almost never appeal to iterated reasoning when asked to explain how they bid. Overall, these findings suggest that, despite its notable success in predicting behaviour in other strategic settings, the level-k model (and its close cousin cognitive hierarchy) cannot explain behaviour in auctions.

This article reviews individual models of learning in games. We show that the experience-weighted attraction (EWA) learning nests different forms of reinforcement and belief learning, and that belief learning is mathematically equivalent to generalized reinforcement, where even unchosen strategies are reinforced. Many studies consisting of thousands of observations suggest that the EWA model predicts behaviour out-of-sample better than its special cases. We also describe a generalization of EWA learning to investigate anticipation by some players that others are learning. This generalized framework links equilibrium and learning models, and improves predictive performance when players are experienced and sophisticated.KeywordsBelief learningCurse of knowledgeEquilibriumExperience-weighted attraction (EWA) learningExtensive-form gamesFictitious playForgone payoffsIndividual learning in gamesIndividual models of learningMaximum likelihoodMixed-strategy equilibriumNoiseOverconfidencePopulation models of learningQuantal response equilibriumReinforcement learningSignallingSocial calibrationSophisticated playersJEL ClassificationsC9

This article reviews individual models of learning in games. We show that the experience-weighted attraction (EWA) learning nests different forms of reinforcement and belief learning, and that belief learning is mathematically equivalent to generalized reinforcement, where even unchosen strategies are reinforced. Many studies consisting of thousands of observations suggest that the EWA model predicts behaviour out-of-sample better than its special cases. We also describe a generalization of EWA learning to investigate anticipation by some players that others are learning. This generalized framework links equilibrium and learning models, and improves predictive performance when players are experienced and sophisticated.

By using data from a voluntary contribution mechanism experiment with heterogeneous endowments and asymmetric information, we estimate a quantal response equilibrium (QRE) model to assess the relative importance of efficiency concerns versus noise in accounting for subjects overcontribution in public good games. In the benchmark specification, homogeneous agents, overcontribution is mainly explained by error and noise in behavior. Results change when we consider a more general QRE specification with cross-subject heterogeneity in concerns for (group) efficiency. In this case, we find that the majority of the subjects make contributions that are compatible with the hypothesis of preference for (group) efficiency. A likelihood-ratio test confirms the superiority of the more general specification of the QRE model over alternative specifications.

This chapter lays out the general theory of quantal response equilibrium (QRE) for normal-form games. It starts with the reduced-form approach to QR, based on the direct specification of “regular” quantal or smoothed best-response functions required to satisfy four intuitive axioms of stochastic choice. A simple asymmetric matching pennies game is used to illustrate these ideas and show that QRE imposes strong restrictions on the data, even without parametric assumptions on the quantal response functions. Particular attention is given to the logit QRE, since it is the most commonly used approach taken when QRE is applied to experimental or other data. The discussion includes the topological and limiting properties of logit QRE and connections with refinement concepts. QRE is also related to several other equilibrium models of imperfectly rational behavior in games, including a game-theoretic equilibrium version of Luce's (1959) model of individual choice, Rosenthal's (1989) linear response model, and Van Damme's (1987) control cost model; these connections are explained in the chapter.

Power utility allocates defense resources to prevent unscheduled load shedding due to transmission line failure caused by the malicious physical attacks. Game theory explains the interaction between the defender and the attacker, overcoming the shortage of unilateral vulnerability analysis. Different from previous researches typically assuming the players are rational and the total defense resources are fixed, this paper investigates the bounded rationality and allows variable total defense resources aiming for a higher level of practicability. Stochastic response to strategies of the bounded rational attacker is described by the Quantal Response Equilibrium (QRE) model. A two-layer defense resources allocation optimization framework is established to obtain both optimal total defense resources and optimal resource distribution, and we design a combined power-of-two and dichotomy (CPTD) algorithm for solution. We also explore the multiple properties of bounded rational behaviors in the attack-defense game. Besides, a unified resources allocation framework of bilateral players is established, which advantages in both cost and reward calculations. To the knowledge of the authors, this work is more general and more practical than previous relevant publications. The experimental studies on the IEEE 14-bus system and the IEEE 118-bus system empirically justify the improvements by our modeling and solution approaches.

Quantal Response Equilibrium and Overbidding in Private-Value Auctions

Rationalizable partition-confirmed equilibrium with heterogeneous beliefs

Models of learning in games based on imitation have provided fundamental insights as the relevance of risk-dominance equilibria or perfectly competitive outcomes. It has also been shown, however, that the introduction of nontrivial memory in those models fundamentally alters the results. This paper further considers the effect of asymmetric memory length in the population. We focus on two classical results and find that, while asymmetric memory crucially affects equilibrium selection in coordination games, it reinforces the stability of perfectly competitive outcomes in oligopoly games. The latter result is generalized to aggregative games and to finite population ESS in symmetric games.