2. Philosophy - Stefano Franchi

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C HESS, GAMES, AND FLIES
The matrix representation is usually called the “normal” or “normalized” form of a
game, as opposed to the “extensive” form (e.g. the tree-like representation). The latter was
actually first devised by Tucker some ten years after the publication of Theory of Games.
Notice that I am not accusing von Neumann and Morgenstern of having used a some-
how “wrong” representation. That would not only be silly, but also plainly wrong, from the
point of view of game theory. In fact, many of the sophisticated distinctions and concepts
provided by the theory (dominant strategy, zero-sum, equilibrium, etc.), are immediately
evident when the game is represented in normal form and rather obscure in the extensive
form. The point here is to show the relationship between game-theory and chess (as an in-
stance of a large class of games) and how the former basic concepts make the latter quite
irrelevant. In this context, moreover, the issue is not that the matrix does not permit to ex-
press the complete game of chess; the tree-like one does not either, since in both case we
would have to do with immensely large objects. The point is that the normalized form does
not allow to think about the complexity, since there are no moves in it, only outcomes. In
the case of games whose complexity lies along the temporal axis (e.g. the moves), the
games itself escapes completely the formalism.
We have now come to the last basic concept of the theory. A solution for a game is,
intuitively, a strategy that can maximize, for a player, his minimum guaranteed payoff: the
payoff that a player may won no matter what the opponent(s) strategy may be. The problem
of game theory, then, is to find if, for a given game, a solution exists. On the basis of the
previous definitions, von Neumann and Morgenstern are able to provide a taxonomy of
games along different, independent, dimensions: number of players, perfect or imperfect
information (like most card-games), cooperative vs. non-cooperative, zero-sum (one play-
er’s loss is the other player’s win) and non-zero-sum. For each class, they set to solve the
problem of game theory. The simplest case turns out to be the class of 2-person, zero-sum,
perfect information game, whose best example is, yet again, chess. For this class of games,
the theory provide a theorem, sometimes called the Fundamental theorem or the maximin
theorem, that assures of the existence of a solution in any case. 19