We study the computational complexity of computing or approximating a quasi-proper equilibrium for a given finite extensive form game of perfect recall. We show that the task of computing a symbolic quasi-proper equilibrium is \(\mathrm {PPAD}\)-complete for two-player games. For the case of zero-sum games we obtain a polynomial time algorithm based on Linear Programming. For general n-player games we show that computing an approximation of a quasi-proper equilibrium is \(\mathrm {FIXP}_a\)-complete. Towards our results for two-player games we devise a new perturbation of the strategy space of an extensive form game which in particular gives a new proof of existence of quasi-proper equilibria for general n-player games.

We study the computational complexity of proper equilibrium in finite games and prove the following results. First, for two-player games in strategic form we show that the task of simply verifying the proper equilibrium conditions of a given pure Nash equilibrium is NP-complete. Next, for n -player games in strategic form we show that the task of computing an approximation of a proper equilibrium is FIXPa-complete. Finally, for n -player polymatrix games we show that the task of computing a symbolic proper equilibrium is PPAD-complete.

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.

The Stackelberg equilibrium is a solution concept that describes optimal strategies to commit to: Player 1 ( the leader ) first commits to a strategy that is publicly announced, then Player 2 ( the follower ) plays a best response to the leader’s choice. We study Stackelberg equilibria in finite sequential (i.e., extensive-form) games and provide new exact algorithms, approximate algorithms, and hardness results for finding equilibria for several classes of such two-player games. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Gambles in casinos are usually set up so that the casino makes a profit in expectation -- as long as gamblers play honestly. However, some gamblers are able to cheat, reducing the casino’s profit. How should the casino address this? A common strategy is to selectively kick gamblers out, possibly even without being sure that they were cheating. In this paper, we address the following question: Based solely on a gambler’s track record,when is it optimal for the casino to kick the gambler out? Because cheaters will adapt to the casino’s policy, this is a game-theoretic question. Specifically, we model the problem as a Bayesian game in which the casino is a Stackelberg leader that can commit to a (possibly randomized) policy for when to kick gamblers out, and we provide efficient algorithms for computing the optimal policy. Besides being potentially useful to casinos, we imagine that similar techniques could be useful for addressing related problems -- for example, illegal trades in financial markets.

We provide the first pivoting-type algorithm that computes an exact proper equilibrium of a bimatrix game. This is achieved by using Lemke's algorithm to solve a linear complementarity problem (LCP) of polynomial size. This also proves that computing a simple refinement of proper equilibria for bimatrix game is PPAD-complete. The algorithm also computes a witness in the form of a parameterized strategy that is an epsilon-proper equilibrium for any given sufficiently small epsilon, allowing polynomial-time verification of the properties of the refined equilibrium. The same technique can be applied to matrix games (two-player zero-sum), thereby computing a parameterized epsilon-proper strategy in polynomial time using linear programming.

Starting from Zermelo’s classical formal treatment of chess, we trace through history the analysis of two-player win/lose/draw games with perfect information and potentially infinite play. Such chess-like games have appeared in many different research communities, and methods for solving them, such as retrograde analysis, have been rediscovered independently. We then revisit Washburn’s deterministic graphical games (DGGs), a natural generalization of chess-like games to arbitrary zero-sum payoffs. We study the complexity of solving DGGs and obtain an almost-linear time comparison-based algorithm for finding optimal strategies in such games. The existence of a linear time comparison-based algorithm remains an open problem.