Games with Imperfect Information

An information set in a game tree is a set of nodes from which the rules of the game require that the same alternative (i.e., move) be selected. Thus the nodes an information set are indistinguishable to the player moving from that set, thereby reflecting imperfect information, that is, information hidden from that player. Information sets arise naturally in (for example) card gaines like poker and bridge. IIere we focus not on the solution concept for imperfect information games (which has been studied at length), but rather on the computational aspects of such games: how hard is it to compute solutions? We present two fundainental results for imperfect information games. The first result shows that even if there is only a single player, we must seek special cases or heuristics. The second result complements the first, providing an efficient algorithm for just such a special case. Additionally, we show how our special case algorithm can be used as a heuristic in the general case.

[1]  Claude E. Shannon,et al.  Programming a computer for playing chess , 1950 .

[2]  H. W. Kuhn,et al.  11. Extensive Games and the Problem of Information , 1953 .

[3]  G. L. Thompson.  14. Signaling Strategies in n-Person Games , 1953 .

[4]  Steven Vajda,et al.  Games and Decisions. By R. Duncan Luce and Howard Raiffa. Pp. xi, 509. 70s. 1957. (J Wiley & Sons) , 1959, The Mathematical Gazette.

[5]  James R. Slagle,et al.  Experiments With Some Programs That Search Game Trees , 1969, JACM.

[6]  Donald E. Knuth,et al.  An Analysis of Alpha-Beta Pruning , 1975, Artif. Intell..

[7]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[8]  Alfred V. Aho,et al.  Data Structures and Algorithms , 1983 .

[9]  Bruce W. Ballard,et al.  The *-Minimax Search Procedure for Trees Containing Chance Nodes , 1983, Artif. Intell..

[10]  Ki Hang Kim.  Game theory in the social sciences , 1986 .

[11]  Keki B. Irani,et al.  An Algorithmic Solution of N-Person Games , 1986, AAAI.

[12]  David A. McAllester.  Conspiracy Numbers for Min-Max Search , 1988, Artif. Intell..

[13]  Murray Campbell,et al.  Singular Extensions: Adding Selectivity to Brute-Force Searching , 1990, Artif. Intell..

[14]  Jonathan Schaeffer,et al.  Conspiracy Numbers , 1990, Artif. Intell..

[15]  Wolfgang Leiniger,et al.  Games and information: An introduction to game theory: Eric Rasmusen, (Basil Blackwell, Oxford, 1989) , 1991 .

[16]  Richard E. Korf.  Multi-Player Alpha-Beta Pruning , 1991, Artif. Intell..

[17]  Cheng Liu,et al.  Heuristic Search in One-Player Games With Hidden Information , 1992 .

[18]  Michael Mesterton-Gibbons,et al.  An introduction to game-theoretic modelling , 2019 .

[19]  Michael van Lent,et al.  A Pruning Algorithm for Imperfect Information Games , 1993 .