Consider a graph of a game board. Rounds in the game result in a token moved fro

Question

Consider a graph of a game board. Rounds in the game result in a token moved from a game board location to a game board location, possibly returning to the same one. Let the game board location at the end of a sequence of rounds determine the equivalence class into which a sequence falls. Note that game board locations are complete and non-overlapping. Using these equivalence classes, assume sequences having an infinite number of rounds can be classified into a finite number of equivalence classes.

(a) If a game board is finite, and we have for data only the final location of the token, can we distinguish, by means of the above equivalence classes, between an infinite number of sequences of rounds?

(b) Consider that the opposite of equivalent is distinct. If a collection of sequences of rounds in the game are all pairwise distinct, and there are an infinite number of such pairs, and the position on the board gives the equivalence class, can the game board be finite?

Explanation / Answer

A play is thus a path through the tree from the root to a terminal node. At any given non-terminal node belonging to Chance, an outgoing branch is chosen according to the probability distribution. At any rational player's node, the player must choose one of the equivalence classes for the edges, which determines precisely one outgoing edge except (in general) the player doesn't know which one is being followed. (An outside observer knowing every other player's choices up to that point, and the realization of Nature's moves, can determine the edge precisely.) A pure strategy for a player thus consists of a selection—choosing precisely one class of outgoing edges for every information set (of his). In a game of perfect information, the information sets are singletons. It's less evident how payoffs should be interpreted in games with Chance nodes. It is assumed that each player has a von Neumann–Morgenstern utility function defined for every game outcome; this assumption entails that every rational player will evaluate an a priori random outcome by its expected utility.

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