## von Neumann's 4-player {1/3, 1/3, -1/3, -1/3} imputation

In Theory of Games and Economic Behavior, von Neumann discusses solutions to some kinds of zero-sum four-person games. See section 37.4.2, page 317. There, he finds that one set of imputations is incomplete, and must have at least another imputation added to it. He writes that [it] seems very difficult to find a heuristic motivation for the steps which are now necessary before giving the imputation as:

$$\vec a^{IV} = \left\{1/3, 1/3, -1/3, -1/3\right\}$$

The situation is unusual in that the first three players have formed a coalition against the fourth. So, why does the third player have the same loss as the fourth? This is the heuristic that von Neumann didn’t provide, and he concludes by saying only that [if] a common-sense interpretation of this solution… is wanted, … it seems to be some kind of compromise between a part (two members) of a possible victorious coalition and the other two players.

However, there’s an intriguing possibility.

After the first coalition of players 1 and 2 has formed, and it’s known (in the case von Neumann was considering) that player 4 would lose, player 3 has a choice to make: join the coalition, stand on his own, or join with player 4.

Suppose that player 3 has a true allegiance to player 4. If he joins him in the loss, they both suffer $$-1/2$$. But he has an alternative: join the coalition, share in the win, then split the outcome with player 4.

After joining, the winnings would be $$\{1/3, 1/3, 1/3, -1\}$$, but players 3 and 4 then combine their winnings to total $$-2/3$$, and then split that evenly between them, for $$-1/3$$ each, arriving at von Neumann’s imputation.