Optimal strategy in games with chance nodes

In this paper, games with chance nodes are analysed. The evaluation of these game trees uses the expectiminimax algorithm. We present pruning techniques involving random effects. The gamma-pruning aims at increasing the efficiency of expectiminimax (analogously to alpha-beta pruning and the classical minimax). Some interesting properties of these games are shown: for instance, a game without draw can be fair. A fair game may not be fair any more if it is played iteratively. To handle these phenomena, the use of additional indicators, such as the minimal guaranteed outcome value, is suggested

A Decision Analysis Approach To Solving the Signaling Game

Decision analysis has traditionally been applied to choices under uncertainty involving a single decision maker. Game theory has been applied to solving games of strategic interaction between two or more players. Building upon recent work of van Binsbergen and Marx (2007. Exploring relations between decision analysis and game theory. Decision Anal. 4(1) 32–40.), this paper defines a modified decision-theoretic approach to solving games of strategic interaction between two players. Using this method, the choices of the two players are modeled with separate decision trees comprised entirely of chance nodes. Optimal policies are reflected in the probabilities in the decision trees of each player. In many cases, the optimal strategy for each player can be obtained by rolling back the opponent’s decision tree. Results are demonstrated for the multi-stage signaling game, which is difficult to model using decision nodes to represent strategies,as in the approach of van Binsbergen and Marx

Quasi-Perfect Stackelberg Equilibrium

Equilibrium refinements are important in extensive-form (i.e., tree-form) games, where they amend weaknesses of the Nash equilibrium concept by requiring sequential rationality and other beneficial properties. One of the most attractive refinement concepts is quasi-perfect equilibrium. While quasi-perfection has been studied in extensive-form games, it is poorly understood in Stackelberg settings---that is, settings where a leader can commit to a strategy---which are important for modeling, for example, security games. In this paper, we introduce the axiomatic definition of quasi-perfect Stackelberg equilibrium. We develop a broad class of game perturbation schemes that lead to them in the limit. Our class of perturbation schemes strictly generalizes prior perturbation schemes introduced for the computation of (non-Stackelberg) quasi-perfect equilibria. Based on our perturbation schemes, we develop a branch-and-bound algorithm for computing a quasi-perfect Stackelberg equilibrium. It leverages a perturbed variant of the linear program for computing a Stackelberg extensive-form correlated equilibrium. Experiments show that our algorithm can be used to find an approximate quasi-perfect Stackelberg equilibrium in games with thousands of nodes

Synthesising Strategy Improvement and Recursive Algorithms for Solving 2.5 Player Parity Games

2.5 player parity games combine the challenges posed by 2.5 player reachability games and the qualitative analysis of parity games. These two types of problems are best approached with different types of algorithms: strategy improvement algorithms for 2.5 player reachability games and recursive algorithms for the qualitative analysis of parity games. We present a method that - in contrast to existing techniques - tackles both aspects with the best suited approach and works exclusively on the 2.5 player game itself. The resulting technique is powerful enough to handle games with several million states