Discrimination And Exploitation As Described By Von Neumann And Morgenstern

This post documents that von Neumann and Morgenstern, in their theory of games, have a formal expression of exploitation, in the context of a theory of discrimination.

They express the three-person, zero-sum game as specified by the characteristic function (section 29.1.2). The three players form coalitions. A coalition of zero or three players has a payout of zero to divide. The one member of a coalition containing one player obtains a payout of -1. A coalition of two players has a payout of 1 to divide among themselves.

For von Neumann and Morgenstern, a solution is a set of imputations, where an imputation assigns a payout to each player. No imputation in the solution dominates another imputation in the set. For every imputation in the set, those imputations outside the set that dominate it are, in turn, dominated by an imputation in the set. An imputation dominates another, roughly, if a coalition of players can form so that they can ensure that their payouts in the dominating imputation exceed that in the dominated imputation.

Nowadays, the von Neumann and Morgenstern solution is said to apply to cooperative games. The Nash equilibrium applies to non-cooperative games.

Von Neumann and Morgenstern find a solution to the three-person, zero-sum game of three non-dominated imputations:

{ (-1, 1/2, 1/2), (1/2, -1, 1/2), (1/2, 1/2, -1) }

But they find a continuum of other solutions besides. Let c be a specified parameter, with:

32:8. -1 ≤ c < 1/2

Then the following set of an uncountably infinite imputations is a solution:

33:1. { (c, a, –c - a ) | -1 ≤ a ≤ 1 - c }

So is the symmetrical set in which the payoff c is assigned to player 2 and the set in which the payoff is assigned to player 3. Von Neumann and Morgenstern have this to say about the solution in Display 33:1:

"The interpretation of this solution consists manifestly of this: One of the players (in this case 1) is being discriminated against by the two others (in this case 2, 3). They assign to him the amount which he gets, c. This amount is the same for all imputations of the solution, i.e. of the accepted standard of behavior. The place in society of player 1 is prescribed by the two other players; he is excluded from all negotiations that may lead to coalitions. Such negotiations do go on, however, between the two other players: the distribution of their share, -c, depends entirely upon their bargaining abilities. The solution, i.e. the accepted standard of behavior, imposes absolutely no restrictions upon the way in which this share is divided between them, - expressed by a, -c – a. This is not surprising. Since the excluded player is absolutely 'tabu', the threat of the partner's desertion is removed from each participant of the coalition. There is no way of determining any definite division of the spoils.

Incidentally: It is quite instructive to see how our concept of a solution as a set of imputations is able to take care of this situation also.

33.1.2. There is more that should be said about this 'discrimination' against a player.

First, it is not done in an entirely arbitrary manner. The c, in which discrimination finds its quantitative expression, is restricted by (32:8) … Now part of (32:8), c ≥ -1, is clear enough in its meaning, but the significance of the other part c < 1/2 is considerably more recondite (cf. however, below). It all comes back to this: Even an arbitrary system of discriminations can be compatible with a stable standard of behavior - i.e. order of society - but it may have to fulfill certain quantitative conditions, in order that it may not impair that stability.

Second, the discrimination need not be clearly disadvantageous to the player who is affected. It cannot be clearly advantageous, - i.e. his fixed value c cannot be equal to or better than the best the others may expect. This would mean ... that c ≥ 1 – c, i.e. c ≥ 1/2 - which is exactly what 32:8 forbids. But it would be clearly disadvantageous only for c = -1; and this is a possible value for c (by (32:8)), but not the only one. c = -1 means that the player is not only excluded, but also exploited to 100 per cent. The remaining c (of (32:8)) with -1 < c < 1/2 correspond to gradually less and less disadvantageous forms of segregation.

33.1.3. It seems remarkable that our concept of a solution is able to express all these nuances of non-discriminatory, and discriminatory, standards of behavior - the latter both in their 100 per cent injurious form, c = -1, and in a continuous family of less and less injurious ones -1 < c < 1/2. It is particularly significant that we did not look for any such thing - the heuristic discussions ... were certainly not in this spirit - but we were nevertheless forced to these conclusions by the rigorous theory itself. And these situations arose even in the extremely simple framework of the zero-sum three-person game!"

Clearly, the mathematics is a metaphor. I do not know of how others built on this. Von Neumann and Morgenstern treat a non-zero sum game as a zero-sum game with a dummy player. The dummy is discriminated against. John Roemer developed a game-theoretic concept of classes exploiting one another, but I do not recall the details, if I ever did understand them.

Reference

• John von Neumann and Oskar Morgenstern. 1955. Theory of Games and Economic Behavior, 3rd ed. Princeton University Press