The Taxpayer Theorem

The finding that extreme concentration of wealth within the United States in the hands of a few thousand people amounts to central planning is empirical evidence in favor of the application to the US economy of a theorem in the Theory of Games and Economic Behavior.

In Chapter 11 of The Theory of Games and Economic Behavior (TGEB), Morgenstern and Von Neumann attempt to reduce the general theory of n-person games to the theory of (n+1)-person zero-sum games through the introduction of a fictitious (n+1)-st player. The payoff to the fictitious (n+1)-st player in the new game equals the negative of the sum of of the payoffs to the n players in the original n-player game. This defines the (n+1)-player zero-sum extension of a general n-player game. In the discussion, Morgenstern and Von Neumann observe that the introduction of the fictitious (n+1)-st player requires some analysis to ensure that the zero-sum extension of an n-player game does not create additional possibilities for interaction among the original n players–that the ‘”fictitious” player does not influence the game in any way. They rule out the absurd possibility of receiving payments from a fictitious player, but have to consider the possibility of self-denial among the n real players in the zero-sum extension.

In the reduction of a general n-player game to an (n+1)-player game in which a fictitious (n+1)-st player is introduced with no ability to form coalitions with the n real players, and whose losses pay for the winnings of the real players, Morgenstern and Von Neumann captured the plight of the lower 99.5% of the population (the "taxpayer"), which collectively serves as the fictitious (n+1)-st player in modern capitalism. The taxpayer is unable to influence the outcome of the games of the real players–the upper one half of one percent of the population (the "central planners"). The taxpayer's role is to passively fork over whatever funds are needed to ensure the maximum winnings for the real players, and the least for himself.

This outcome is established in a proposition on page 513 of TGEB, in which the possibility that the n players may 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 (n+1)-st player, who always receives the minimum.

We might call that proposition the Taxpayer Theorem. Its application to contemporary capitalism, in which the taxpayer insures the central planners against losses, seems not to have been noticed.* In the case of the taxpayer, unlike the fictitious player of the (n+1)-st zero-sum extension of a n-player game, there is some slight possibility of influencing the n real players, but in practice this effect is negligible, and in the strict terms of cooperative game theory, irrational. In the case of the lower 99.5%, side-payments to the upper 0.5% are ruled out because taxpayers are effectively unable to form coalitions, since the concentration of wealth in the upper 0.5% is too extreme. To almost any approximation, the taxpayer is the fictitious player in the zero-sum extension of whatever game the important players in the upper one half of one percent happen to be playing.

*It is inconceivable that Von Neumann would not have foreseen this application, but it is conceivable that he would chose to leave its rediscovery to posterity.