COMMENTS ON "PARADOX REGAINED"

In the game of Roulette, the gambler has 36 strategies, and the wheel has 37 or 38, depending on whether there is only one zero or both a zero and a double zero. TO fix ideas, let us take the 36 x 37 game. In actual practice the well-balanced wheel plays a mixed strategy, i.e., chooses one of the 37 numbers with probability 1/37. To make Two-person game theory applicable, we must suppose that the wheel, like the gambler, chooses a strategy with a view to maximizing "its," i.e., the casino's, expected gain. The Von Neumann-Morgenstern solution of this game prescribes a dominating strategy to the player if he ha one. Clearly the wheel has such a dominating strategy, namely "Choose 0," since "0" wins regardless of the gambler's choice. A game like that would draw no customers. For this reason, the wheel's choice of strategy is restricted to a single one, namely the above mentioned mixed strategy. In a game for unit stakes, the wheel's expected gain is +1/37; the gambler's -1/37. Since, however, these are only expectations, not actual gains in single plays, gamblers are attracted to play the game in the hope of short-term gains or for the thrill of uncertainty. Let us now examine the meta-game. The strategy "Choose the number that comes up" is not defined, because it cannot be realized if "0" comes up (the gambler cannot choose "0). The best he can do is choose a strategy like "Choose the number that comes up if it is not '0'; if '0' choose m," where 7t = (1, 2, . . . or 36) . In this meta-game, the wheel, being the second player, chooses a meta-meta-strategy, that is, one conditioned on the gambler's meta-strategy. Again the wheel always wins! Suppose now that "0" is eliminated, so that Roulette becomes a "fair game." Now, the expected gain of borh players is zero in the ordinary game, since the gambler loses 1 unit with probability 35/36 and gains 35 units with probability 1/36. Now the metastrategy "Choose the number that comes up" wins against all possible meta-meta-strategies of the wheel. However, meta-game theory prescribes the interchange of roles, whereby the wheel chooses a meta-strategy, while the gambler chooses a meta-meta-strategy. Clearlv the wheel has a meta-strategy that always wins, namely, "Let the number come up that the gambler has not chosen." If the game is played successively with roles interchanged, the gambler and the wheel will win alternately. Naturally, if the original odds are preserved, the meta-game will favor the gambler. To make the meta-game a fair game, the wheel must be given the role of the first player (choosing the meta-strategy rather than the meta-meta-strategy) 35 times out of 36. In my opinion, the meta-game theory presents no interest in the context of zerosum games. The theory was stimulated by paradoxes arising in some non-zerosum games. W e have seen how the paradox of the Prisoner's Dilemma is removed when the game is examined in its meta-game form, that is, beings that conceive all games in their metagame form would not be facing a paradox when playing Prisoner's Dilemma even once. W e have seen, however, that not all ambiguities are removed by the rneta-game theory. For instance, the ambiguity in the game of Chicken remains even when that game is put into meta-game form. This leads us to the question of what sort of paradoxes are removed by the meta-game model and what sort are not and whether the model can be further extended to remove remaining paradoxes. In conclusion, I can only reiterate that the concept of meta-strategy involves no contradiction any more than the concept of strategy. Except in trivial cases, a strategy is always stated in terms of choices conditional on (unknown!) choices of the other player or players. A meta-strategy is no more than an extension of the same idea. Once a game is reduced to normal form, it is always a "refereed" game, not in the sense that some third party makes decisions, but simply in the sense that a third party acts as a "decoder" of the strategic choices of the players, translating them into an actual outcome of the game.