Jimmy Teng - A Bayesian Theory of Games

A Bayesian Theory of Games

A Bayesian Theory of Games

Iterative conjectures and determination of equilibrium

J IMMY T ENG

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Published in 2014 by Chartridge Books Oxford

ISBN print: 978-1-909287-76-1

ISBN digital (pdf): 978-1-909287-77-8

ISBN digital book (epub): 978-1-909287-78-5

ISBN digital book (mobi): 978-1-909287-79-2

J. Teng 2014

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Contents

Preface

This book introduces a new games theory equilibrium concept and solution algorithm that provide a unified treatment for broad categories of games that are presently solved using the different equilibrium concepts of Nash equilibrium, sub-game perfect equilibrium, Bayesian Nash equilibrium and perfect Bayesian equilibrium.

The new method achieves consistency in equilibrium results that current games theory at times fails to, such as those between Perfect Bayesian Equilibrium and backward induction (sub-game Perfect Equilibrium). The new equilibrium concept is Bayesian equilibrium by iterative conjectures (BEIC) and its associated algorithm is the Bayesian iterative conjecture algorithm. BEIC requires players to make predictions on the strategies of other players using the Bayesian iterative conjecture algorithm. The Bayesian iterative conjectures algorithm makes predictions starting from first order uninformative predictive distribution functions (or conjectures) and keeps updating with the Bayesian statistical decision theoretic and game theoretic reasoning until a convergence of conjectures is achieved. Information known by the players such as the reaction functions are thereby incorporated into the higher order conjectures and help to determine the convergent conjectures and the associated equilibrium.

In a BEIC, conjectures are consistent with the equilibrium or equilibriums they support and so rationality is achieved for actions, strategies and conjectures and (statistical) decision rule.

The BEIC approach is capable of analyzing a larger set of games than current games theory, including games with noisy inaccurate observations and games with multiple-sided incomplete information games. On the other hand, for the set of games analyzed by the current games theory, it generates smaller numbers of equilibriums and normally achieves uniqueness in equilibrium. It treats games with complete and perfect information as special cases of games with incomplete information and noisy observations, whereby the variance of the prior distribution function on type and the variance of the observation noise term tend to zero. Consequently, there is the issue of indeterminacy in statistical inference and decision-making in these games as the equilibrium solution depends on which variance tends to zero first. It therefore identifies equilibriums in these games that have so far eluded current treatments.

Acknowledgments

I thank D. Banks, J. Bono, P. Carolyn, M. Clyde, J. Duan, I. Horstmann, P.Y. Lai, M. Lavine, J. Mintz, R. Nau, M. Osborne, J. Roberts, D. Schoch, R. Winkler, R. Wolpert, F.Y. Chiou, and G. Xia for their comments.

I thank the students of my 2005, 2008 and 2009 games theory classes (at the Department and Graduate Institute of Political Science in the National Taiwan University in Taipei, Taiwan) for their enthusiasm in learning, and interesting questions raised in class.

I thank the participants of my three-day games theory workshop (at the Graduate Institute of Political Science in the National Sun Yat Sen University in Kaohsiung, Taiwan) for their questions.