On the one hand, a new subject called algorithmic game theory has emerged that is concerned with the study of the algorithmic theory of finite games with multiple players. Hence, all players apart from player 0 win, but player 0 cannot improve her payoff since no clause vertex is visited. For computational purposes, we assume that these probabilities are rational numbers. Finally, we claim that α satisfies φ. Hence, a vertex of the form C, X i is visited infinitely often. I am indebted to my primary supervisor Erich Grädel for giving me the opportunity to pursue these studies, for introducing me to the scientific community and for giving me advice just when I needed it.
These smaller games are usually obtained from the original game by restricting to a subarena. G , v 0 has a positional subgame-perfect equilibrium with payoff 1, 21 ; 3. Author: Michael Ummels Publisher: Amsterdam : Amsterdam University Press, 2010. The next lemma shows that, conversely, we can turn every strategy profile that fulfils this property into a Nash equilibrium. Realizable and unrealizable specifications of reactive systems. One main result is that the constrained existence of a Nash equilibrium becomes undecidable in this setting.
The basis of this work is a comprehensive complexitytheoretic analysis of the standard game-theoretic solution concepts in the context of stochastic games over a finite state space. All lower bounds also hold for the corresponding problems for subgame-perfect equilibria, and the upper bounds for games with Rabin objectives also hold for games with Streett-Rabin objectives. The aim of this work is to bring together algorithmic game theory with the games that are used in verification by extending the algorithmic theory of stochastic two-player zero-sum games to incorporate multiple players, whose objectives are not necessarily conflicting. This impossibility result is accompanied by several positive results, including efficient algorithms for natural special cases. As in the previous chapter, all games in this chapter are finite. Now, assume that σ is not a Nash ,τ equilibrium.
G , C 1 has a subgame-perfect equilibrium with payoff 1, 0 ; 3. In 1992 the symposium was organized in Grimentz, Switzerland, under the supervision of an international scientific committee and with the help of a local organizing committee based at University of Geneva. At first glance, it seems that this game does not have a Nash equilibrium; if the 16 1. Let F be the Borel σ-algebra over A , and let P be a probability measure on F. Whether the latter problem is decidable in polynomial time is a long-standing open 50 2. The proofs are virtually identical to the proofs of Theorems 5.
Consider the game G , v 0 constructed in the proof of Theorem 4. The question arises whether we can also guarantee the existence of a positional Nash equilibrium in such games. An immediate corollary to this result is that there exists a fixed formula of stochastic game logic Baier et al. Modalities for model checking: branching time logic strikes bsack. For example, the philosophers could proceed in rounds: in each round, only one philosopher eats and all others think see below. Consider any player i whose objective is violated. However, almost pure strategies, which require randomisation only for finitely many histories, do suffice for this purpose.
In Essays in Game Theory and Mathematical Economics in Honor of Oskar Morgenstern, pp. Stochastic games provide a versatile model for reactive systems that are a'ected by random events. The resulting set of vertices may not be strongly connected any more and fewer objectives may be satisfied; hence, the procedure has to be called recursively. In the two-player zero-sum variant, these games occur naturally when one wants to build a controller for a system interacting with a probabilistic environment Baier et al. From a computer science point of view, the mere existence of an object is not sufficient; we also want to compute it. The basis of this work is a comprehensive complexity theoretic analysis of the standard game-theoretic solution concepts in the context of stochastic games over a finite state space. If they go to different concerts, then each of them has an incentive to go to the respective other concert since their main concern is to enjoy a concert together.
An exponential lower bound for the parity game strategy improvement algorithm as we know it. To obtain meaningful results, we assume that all transition probabilities in G as well as the thresholds x and y are rational numbers with numerator and denominator given in binary and that all objectives are ω-regular. Hence, σ induces a Nash equilibrium of G1 , v 1 with payoff 1,. However, if for every possible initial vertex there exists an ε- optimal strategy, then there also exists a globally ε- optimal strategy. It follows from Theorem 2. Of course, it also follows from Theorem 2.
To establish the reduction, we need to show that φ is satisfiable if and only if there exists a play of G that is won by each player. In Chapter 4, we studied the different decision problems associated to a solution concept and a strategy type in their full generality. We call the resulting decision problem the qualitative decision problem. A substantial part of this book is based on joint work with Dominik Wojtczak. Recursive Markov decision processes and recursive stochastic games. Obviously, player 0 wins almost surely in this strategy profile.
Finally, in a perfect-information stochastic two-player zero-sum game S2G , there are only two players, player 0 and player 1, who have opposing objectives: one player wants to fulfil her objective, while the other one wants to prevent her from doing so. As for ε-equilibria and k, t -robust equilibria, we do not know whether our results carry over to this equilibrium notion. Nash and subgame-perfect equilibria, and prove their existence for subclasses of stochastic games. What is a good protocol in such a situation? The corresponding statements about subgame-perfect equilibria are only true for deterministic games Theorems 3. There are edges from a clause C j to each vertex C j , L such that L occurs in C j and from there to C j mod m +1 , and there is an edge from each vertex of the form C, ¬X to . All vertices are controlled by player 0. Formal Aspects of Computing, vol.
Note that a positional strategy profile that is globally optimal is also residually optimal. The proof of this claim uses the monotone class theorem and resembles the proof of the corresponding claim in the proof of Lemma 3. This dissertation advances the algorithmic theory of stochastic games to incorporate multiple players, whose objectives are not necessarily conflicting. Hence, if one philosopher eats, then his two neighbours cannot eat at the same time. Similarly, subgame-perfect equilibria correspond to pairs of residually optimal strategies. Games and Economic Behavior, vol. The claim now follows from Lemma 3.