Generalized Game Trees

We consider two generalizations of the standard two-player game model: different evaluation functions for the players, and more than two players. Relaxing the assumption that players share the same evaluation function produces a hierarchy of levels of knowledge as deep as the search tree. Alpha-beta pruning is only possible when the different evaluation functions behave identically. In extending the standard model to more than two players, the minimax algor i thm is generalized to the maxn algor i thm applied to vectors of N-tuples representing the evaluations for each of the players. If we assume an upper bound on the sum of the components for each player, and a lower bound on each individual component, then shallow alphabeta pruning is possible, but not deep pruning. In the best case, the asymptotic branching factor is reduced to (1 + \ /46 — 3) /2 . In the average case, however, pruning does not reduce the asymptotic branching factor. Thus, alphabeta pruning is found to be effective only in the special case of two players w i th a common evaluation function. 1 I n t r o d u c t i o n Min imax search w i th alpha-beta pruning is the predominant algor i thm employed by two-player game programs[1]. Figure 1 shows a game tree, where squares represent Max nodes and circles correspond to Min nodes, along w i th its min imax value, bounds on interior nodes, and those branches pruned by alpha-beta. There are two assumptions made in this model. One is that there are two players, and the other is that they *This research was supported by an NSF Presidential Young Investigator Award, and NSF Grant IRI-8801939. Thanks to Chris Ferguson for helpful discussions concerning this work, and Valerie Aylett for drawing the figures. both use the same evaluation funct ion. There are, however, games that involve more than two players. Furthermore, the knowledge of different players is l ikely to be quite different in practice. First we wi l l consider the consequences for min imax and alpha-beta of assuming that two players use different evaluation functions. Next we wi l l examine mult i-player game trees. Finally, we wi l l combine the two cases and briefly discuss multi-player games wi th different evaluation functions. 2 D i f f e ren t E v a l u a t i o n Func t i ons Given separate evaluation functions, there are two cases to consider, depending on whether or not each player knows his opponent's funct ion. 2.1 Separa te b u t S h a r e d K n o w l e d g e In the simplest case of separate evaluation functions, each player uses a different function and each player knows his opponent's funct ion. This requires that minimax be modified as follows: Each node now has two evaluations, one for Max and one for M in . In figure 2, the first component is Max's value and the second is Min 's . The player to move at a given node uses his evaluation of the children, and backs up the complete ordered pair for which his component is a maximum or min imum, respectively. In general, alpha beta pruning cannot be used in this case. Compare figure 2 w i th the two-level tree in the lower left corner of figure 1. Using either Max or Min's function exclusively would cause the last node to be pruned, yet its value is the min imax value of the root when both functions are used. The problem is that (12,8) is better than (9,9) for both Max and M i n . Pruning is possible only if the two evaluation functions always agree on the relative ordering of the merits of different positions. In other words, if one node looks better to Max than another, then it also must look worse to M I N . Since the actual values of positions don't matter, but merely their relative order, this constraint implies that both evaluation functions always make the