Interactions among agents can be conveniently described by game trees. In order to analyze a game, it is important to derive optimal (or equilibrium) strategies for the different players. The standard approach to finding such strategies in games with imperfect information is, in general, computationally intractable. The approach is to generate the normal form of the game (the matrix containing the payoff for each strategy combination), and then solve a linear program (LP) or a linear complementarity problem (LCP). The size of the normal form, however, is typically exponential in the size of the game tree, thus making this method impractical in all but the simplest cases. This paper describes a new representation of strategies which results in a practical linear formulation of the problem of two-player games with perfect recall (i.e., games where players never forget anything, which is a standard assumption). Standard LP or LCP solvers can then be applied to find optimal randomized strategies. The resulting algorithms are, in general, exponentially better than the standard ones, both in terms of time and in terms of space.

In this paper we consider the following game: players must alternately color the lowest numbered uncolored vertex of a given graph G = ({1, 2, ..., n), E) with a color, taken from a given set C, such that never two adjacent vertices are colored with the same color. In one variant, the first player which is unable to move, loses the game. In another variant, player 1 wins the game if and only if the game ends with all vertices colored. We show that for both variants, the problem to determine whether there is a winning strategy for player 1 is PSPACE-complete for any C with [C I _ 3, but the problems are solvable in O(n + ea(e, n)), and O(n + e) time, respectively, if [C I = 2. We also give polynomial time algorithms for the problems with certain restrictions on the graphs and orderings of the vertices. We give some partial results for the versions, where no order for coloring the vertices is specified.

Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we begin with general background in combinatorial game theory, which analyzes ideal play in perfect-information games. Then we survey results about the complexity of determining ideal play in these games, and the related problems of solving puzzles, in terms of both polynomial-time algorithms and computational intractability results. Our review of background and survey of algorithmic results are by no means complete, but should serve as a useful primer. 1

We initiate an investigation of probabilistically checkable debate systems (PCDS's), a natural generalization of probabilistically checkable proof systems. A PCDS for a language L consists of a probabilistic polynomial-time verifier V and a debate between player 1, who claims that the input x is in L, and player 0, who claims that the input x is not in L. We show that there is a PCDS for L in which V flips O(log n) random coins and reads O(1) bits of the debate if and only if L is in PSPACE. This characterization of PSPACE is used to show that certain PSPACE-hard functions are as hard to approximate closely as they are to compute exactly. 1 Introduction Suppose that two candidates, B and C, agree to a debate format. Voter V is too busy to catch more than a very small number of bits of the debate. How does V decide which of B or C won the debate? In this paper, we show that if These results first appeared in our Technical Memorandum [8]. They were presented in preliminary form at ...

. A probabilistically checkable debate system (PCDS) for a language L consists of a probabilistic polynomial-time verifier V and a debate between Player 1, who claims that the input x is in L, and Player 0, who claims that the input x is not in L. It is known that there is a PCDS for L in which V flips O(logn) coins and reads O(1) bits of the debate if and only if L is in PSPACE ([Condon et al., Proc. 25th ACM Symposium on Theory of Computing, 1993, pp. 304--315]). In this paper, we restrict attention to RPCDS's, which are PCDS's in which Player 0 follows a very simple strategy: On each turn, Player 0 chooses uniformly at random from the set of legal moves. We prove the following result. Theorem: L has an RPCDS in which the verifier flips O(logn) coins and reads O(1) bits of the debate if and only if L is in PSPACE. This new characterization of PSPACE is used to show that certain stochastic PSPACE-hard functions are as hard to approximate closely as they are to compute exactly. Exam...

We describe new results in parameterized complexity theory. In particular, we prove a number of concrete hardness results for W [P], the top level of the hardness hierarchy introduced by Downey and Fellows in a series of earlier papers. We also study the parameterized complexity of analogues of P SP ACE via certain natural problems concerning k-move games. Finally, we examine several aspects of the structural complexity of W [P] and related classes. For instance, we show that W [P] can be characterized in terms of the DT IME(2 o(n)) and NP. 1

Abstract. Aim: To present a systematic development of part of the theory of combinatorial games from the ground up. Approach: Computational complexity. Combinatorial games are completely determined; the questions of interest are efficiencies of strategies. Methodology: Divide and conquer. Ascend from Nim to chess in small strides at a gradient that’s not too steep. Presentation: Informal; examples of games sampled from various strategic viewing points along scenic mountain trails illustrate the theory. 1.

Abstract Studying the precise nature of the complexity of games enables gamesters to attain a deeper understanding of the difficulties involved in certain new and old open game problems, which is a key to their solution. For algorithmicians, such studies provide new interesting algorithmic challenges. Substantiations of these claims are illustrated on hand of many sample games, leading to a definition of the tractability, polynomiality and efficiency of subsets of games. In particular, there are tractable games that are not polynomial, polynomial games that are not efficient. We also define and explore the nature of the subclasses PlayGames and MathGames.

. We propose and analyze a 2-parameter family of 2-player games on two heaps of tokens, and present a strategy based on a class of sequences. The strategy looks easy, but it is actually hard. A class of exotic numeration systems is then used, which enables us to decide whether the family has an efficient strategy or not. We introduce yet another class of sequences and demonstrate its equivalence with the class of sequences defined for the strategy of our games. Keywords: heap games, numeration systems, sequences 1. Example Given a 2-player game played on two heaps (piles) of finitely many tokens. There are two types of moves: (I) Take any positive number of tokens from one heap, possibly the entire heap. (II) Take from both heaps, k from one and l from the other, with, say, k l. Then the move is constrained by the condition 0 ! k l ! 2k + 2, which is equivalent to 0 l \Gamma k ! k +2; k ? 0. The player making the last move (after which both heaps are empty) wins, and the opponent ...

We propose the \Competing Salesmen Problem" (CSP), a 2-player competitive version of the classical Traveling Salesman Problem. This problem arises when considering two competing salesmen instead of just one. The concern for a shortest tour is replaced by the necessity to reach any of the customers before the opponent does.