The Taxpayer Theorem

Occupy Wall Street has drawn attention to widespread economic injustice. Since there is disagreement over the meaning and significance of economic injustice, it may help to reformulate a central claim of Occupy Wall Street in the “value neutral” language of contemporary game theory: approximately 0.1% of the US population has been systematically winning asymmetric zero-sum games against the lower 99%.

The literature on extensive form asymmetric games, which includes chess and checkers, is vast. Extensive form asymmetric zero-sum games have been applied to coevolution in evolutionary game theory (cf. Oliehoek, de Jong and Vlassis. The Parallel Nash Memory for Asymmetric Games). Asymmetric zero-sum games are considered in the foundational “Theory of Games and Economic Behavior” by John von Neumann and Oskar Morgenstern. Their relevance to economic injustice appears to have been overlooked.

A two-player asymmetric zero-sum game in normal form has the following payoff matrix.

The row player is called Winner, and the column player is called Loser. The payoff to Winner is positive for and . Winner and Loser play the pessimistic maximin strategy against each other; in this game, the pessimistic maximin strategy is the same as the optimistic minimax strategy.

The maximin values for both players sum to zero.

In Chapter 11 of the Theory of Games and Economic Behavior (TGEB), Oskar Morgenstern and John von Neumann attempt to reduce the general theory of -person games to the theory of ()-person zero-sum games through the introduction of a fictitious ()-st player. The payoff to the fictitious ()-st player in the new game equals the negative of the sum of of the payoffs to the players in the original -player game. This defines the ()-player zero-sum extension of a general -player game.

Morgenstern and Von Neumann observe that the introduction of the fictitious ()-st player requires some analysis to ensure that the zero-sum extension of an -player game does not introduce any new possibility for interaction among the original real players. They rule out the absurd possibility of receiving side-payments from a fictitious player, but remark that the possibility of self-denial among the real players in the zero-sum extension requires an argument to exclude it. This outcome is established in a proposition in Chapter 11, Section 56.7 on page 513 of TGEB, in which the possibility that the players might choose self-denial to exploit some coalitional advantage is ruled out. The real players will never deny themselves in any imputation to the advantage of the fictitious ()-st player, who always receives the minimum.

We might call that proposition the Taxpayer Theorem. In the reduction of a general -player game to an ()-player game, in which a fictitious ()-st player is introduced with no ability to form coalitions with the real players, and whose losses pay for the winnings of the real players, Morgenstern and Von Neumann captured the role of the taxpayer, understood as the lower 99% of the population with respect to income. The taxpayer serves as the fictitious ()-st player in game of modern capitalism. The taxpayer is unable to influence the outcome of the games of the real players–the upper one tenth of one percent of the population, and it insures the masters of the universe against losses.

In the case of the taxpayer, unlike the fictitious player of the ()-player zero-sum extension of a -player game, there is some slight possibility of influencing the real players, but in practice this effect is negligible, and in the strict terms of game theory, irrational. In the case of the lower 99%, side-payments to the upper 0.1% are ruled out because taxpayers are effectively unable to form coalitions–the concentration of wealth in the upper 0.1% is too extreme, and the middle 40% of income earners lack sufficient flexibility in discretionary spending. To almost any approximation, the taxpayer is the fictitious player in the zero-sum extension of whatever game the important players happen to be playing.

Update: Steve Randy Waldman applies the stag hunt game to opacity in financial markets. Opacity provides the FIRE sector opportunities to win asymmetric zero-sum games against the public, as Waldman suggests.

Advertisement