Strategies For Two-player Games Research Articles

Logical time control of concurrent DES

The synthesis of controllers for reactive systems can be done by computing winning strategies in two-player games. Timed (game) Automata are an appropriate formalism to model real-time embedded systems but are not easy to use for controller synthesis for two reasons: i) timed models require the knowledge of the precise timings of the system; for example, if an action must occur in the future, the deadline of this occurrence must be known, ii) in practice, the dense state space makes the computation of the controller often impossible for complex systems. This paper introduces an extension of untimed game automata with logical time. The new semantics introduces two new types of uncontrollable actions: delayed actions which are possibly avoidable, and ineluctable actions which will eventually happen if nothing is done to abort it. The controller synthesis problem is adapted to this new semantics. This paper focuses specifically on the reachability and safety objectives and gives algorithms to generate a controller. The paper then extends these results to Game Petri Nets which can express concurrent timed behaviors and where an avoidable transition can lose its avoidability by the elapsing of time. The usefulness of this new model is illustrated by a real device driver synthesis example.

Approximating maxmin strategies in imperfect recall games using A-loss recall property

Extensive-form games with imperfect recall are an important model of dynamic games where the players are allowed to forget previously known information. Often, imperfect recall games result from an abstraction algorithm that simplifies a large game with perfect recall. Solving imperfect recall games is known to be a hard problem, and thus it is useful to search for a subclass of imperfect recall games which offers sufficient memory savings while being efficiently solvable. The abstraction process can then be guided to result in a game from this class. We focus on a subclass of imperfect recall games called A-loss recall games. First, we provide a complete picture of the complexity of solving imperfect recall and A-loss recall games. We show that the A-loss recall property allows us to compute a best response in polynomial time (computing a best response is NP-hard in imperfect recall games). This allows us to create a practical algorithm for approximating maxmin strategies in two-player games where the maximizing player has imperfect recall and the minimizing player has A-loss recall. This algorithm is capable of solving some games with up to 5⋅109 states in approximately 1 hour. Finally, we demonstrate that the use of imperfect recall abstraction can reduce the size of the strategy representation to as low as 0.03% of the size of the strategy representation in the original perfect recall game without sacrificing the quality of the maxmin strategy obtained by solving this abstraction.

Strategy logic

We introduce strategy logic, a logic that treats strategies in two-player games as explicit first-order objects. The explicit treatment of strategies allows us to specify properties of nonzero-sum games in a simple and natural way. We show that the one-alternation fragment of strategy logic is strong enough to express the existence of Nash equilibria and secure equilibria, and subsumes other logics that were introduced to reason about games, such as ATL, ATL ∗, and game logic. We show that strategy logic is decidable, by constructing tree automata that recognize sets of strategies. While for the general logic, our decision procedure is nonelementary, for the simple fragment that is used above we show that the complexity is polynomial in the size of the game graph and optimal in the size of the formula (ranging from polynomial to 2EXPTIME depending on the form of the formula).

The Evolution of Fuzzy Rules as Strategies in Two-Player Games

This paper describes simulations using fuzzy that show how Nash equilibrium behavior can be achieved by boundedly rational agents in two-player games with infinite strategy spaces. That is, we show how agents using simple rules of thumb can achieve near-equilibrium outcomes without any overt computation of the equilibrium. This is accomplished by using a genetic algorithm to approximate repeated play. Two games of differing complexities, both with analytic solutions, are examined: a repeated linear-demand Cournot game and a contestable rent game. When fuzzy used only the most recent information, the games we examined converged to outcomes similar to their respective Cournot-Nash equilibrium outcomes. When fuzzy remembered play from the more distant past, we found that the games converged more slowly, if at all. How to model players' strategies has been an important issue in the study of repeated games. The problem has been reasonably well addressed for the case in which strategy choices in the stage game are finite because the history at any point in time consists only of a series of choices from a finite set of stage game strategies. For example, Miller (1996) and Linster (1992, 1994) used finite automata to encode strategies in the repeated prisoner's dilemma. These finite automata, or Moore machines, can capture the idea of bounded rationality, but they can only be used when strategy choices are finite in number. The purpose of this paper is to illustrate the use of fuzzy strategies in two familiar two-player games with continuous strategy spaces. We chose games for which analytic solutions are available so as to enable comparisons between equilibrium outcomes of our fuzzy models and Cournot-Nash, competitive, and collusive outcomes. These simulations based on fuzzy strategies can be used in models for which analytic solutions do not exist or are computationally very difficult to obtain. A rule described by a small number of parameters can express only limited rationality within the context of a complex model, but these might be good to describe human behavior in a variety of situations. Consider as an example the task of running an automobile's heating and cooling system prior to the advent of climate control. A heater temperature/air conditioner lever position is continuously and minutely variable. No doubt a complicated optimal nonlinear response function could be specified in which temperature readings are taken and minute adjustments are made to the lever position. Instead, we hypothesize that as few as three simple might be enough to