Timeability of Extensive-Form Games

Extensive-form games constitute the standard representation scheme for games with a temporal component. But do all extensive-form games correspond to protocols that we can implement in the real world? We often rule out games with imperfect recall, which prescribe that an agent forget something that she knew before. In this paper, we show that even some games with perfect recall can be problematic to implement. Specifically, we show that if the agents have a sense of time passing (say, access to a clock), then some extensive-form games can no longer be implemented; no matter how we attempt to time the game, some information will leak to the agents that they are not supposed to have. We say such a game is not exactly timeable. We provide easy-to-check necessary and sufficient conditions for a game to be exactly timeable. Most of the technical depth of the paper concerns how to approximately time games, which we show can always be done, though it may require large amounts of time. Specifically, we show that some games require time proportional to the power tower of height proportional to the number of players, which in practice would make them untimeable. We hope to convince the reader that timeability should be a standard assumption, just as perfect recall is today. Besides the conceptual contribution to game theory, we show that timeability has implications for onion routing protocols.

[1]  Peter Bro Miltersen,et al.  Fast algorithms for finding proper strategies in game trees , 2008, SODA '08.

[2]  H. W. Kuhn,et al.  11. Extensive Games and the Problem of Information , 1953 .

[3]  Ariel Rubinstein,et al.  On the Interpretation of Decision Problems with Imperfect Recall , 1996, TARK.

[4]  B. Stengel,et al.  Efficient Computation of Behavior Strategies , 1996 .

[5]  Kevin Waugh,et al.  A Practical Use of Imperfect Recall , 2009, SARA.

[6]  Sergiu Hart,et al.  The Forgetful Passenger , 1997 .

[7]  Philip Wolfe,et al.  Contributions to the theory of games , 1953 .

[8]  Sergiu Hart,et al.  The Absent-Minded Driver , 1996, TARK.

[9]  J. Weibull On self-enforcement in extensive-form games , 1992 .

[10]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[11]  Sune K. Jakobsen A Numbers-on-Foreheads Game , 2015, MFCS.

[12]  Tim Roughgarden,et al.  Algorithmic Game Theory , 2007 .


[14]  Kristoffer Arnsfelt Hansen,et al.  Finding Equilibria in Games of No Chance , 2007, COCOON.

[15]  Clifford Stein,et al.  Introduction to Algorithms, 2nd edition. , 2001 .

[16]  Thomas H. Cormen,et al.  Introduction to algorithms [2nd ed.] , 2001 .

[17]  Peter Bro Miltersen,et al.  Computing a quasi-perfect equilibrium of a two-player game , 2010 .

[18]  Nick Mathewson,et al.  Tor: The Second-Generation Onion Router , 2004, USENIX Security Symposium.

[19]  Bernhard von Stengel,et al.  Extensive-Form Correlated Equilibrium: Definition and Computational Complexity , 2008, Math. Oper. Res..

[20]  Tuomas Sandholm,et al.  Extensive-form game abstraction with bounds , 2014, EC.

[21]  J. Fabris,et al.  Gravity: An Introduction to Einstein's General Relativity , 2004 .