Oskar Morgenstern: Almost-symmetric solutions of some symmetric n-person games (1961)

Morgenstern’s published research abstract announces two sharp bounds for a particular class of game-theoretic solutions. It explicitly presents the findings as part of an unfinished collaboration:

This abstract summarizes incomplete joint work by the author and the late J. von Neumann.

The scope is narrowly mathematical. The game is symmetric and simple, with minimal winning coalitions containing exactly n − 1 players. Against this symmetry of the game, Morgenstern considers a solution symmetric in every player except one, designated player n. The central conceptual move is thus to isolate a single exception within an otherwise symmetric solution and ask what restrictions govern that player’s possible payoffs. The abstract states that such solutions exist.

Rather than describe the entire solution set V, Morgenstern projects it onto the exceptional player’s payoff axis:

Let A be the projection of V on the xₙ-axis, i.e., the set of all payoffs to the exceptional player for imputations in V; A is compact, so its complement is open.

This reduction makes both the lowest attainable payoff and gaps in the attainable payoffs objects of analysis. Using the stated (- 1,0) normalization, the abstract reports:

Theorem 1. The minimum of A is < 1/(n - 1). Theorem 2. If (b,a) is a bounded component of the complement of A, then a ≤ (b + 1)(n - 2)/(n - 1). Furthermore, the estimates in these theorems are optimum.

The first result forces the exceptional player’s minimum payoff below a specified threshold. The second constrains the upper endpoint of any bounded gap in terms of its lower endpoint and the number of players. The assertion of optimality gives these bounds their significance: they are presented as sharp restrictions, not merely convenient estimates.

The abstract proceeds from the game’s definition through the projection construction to the two theorems, leaving their proofs to RAND paper P-2169. Its contribution is therefore a precise announcement of how near-symmetry constrains attainable payoffs; it supplies neither the proofs nor a broader interpretation of the results.