John von Neumann and Oskar Morgenstern, Symmetric Solutions of Some General n-Person Games (1961)

This RAND research paper publishes a manuscript dated August 1946, preceded by a preface signed L.S.S. Its subject is a narrowly specified cooperative game: every coalition of at least (n-1) players wins, while every smaller coalition loses. Rather than seek one solution for a broad class of games, the investigation examines restrictions on the solutions of this particular game. Its central result concerns solutions invariant under permutations of the first (n-1) players: the possible payoffs to the remaining player cannot have arbitrarily large gaps, and that player’s minimum payoff must lie below (1/(n-1)). The paper thus makes partial symmetry a means of investigating the structure of solution sets beyond the fully symmetric case.

The preface explains the methodological stakes of such concentrated analysis. Exhaustively understanding a single game may contribute to outstanding questions in solution theory, including the general existence problem. It also places the manuscript’s achievement within clear limits: this is a preliminary investigation that stops after establishing payoff restrictions, not an exhaustive classification of the partially symmetric solutions. The fully symmetric solution is already known; the preface describes it as consisting of imputations in which each payoff above the minimum occurs an even number of times.

The sixteen numbered sections move from the game’s definition and domination relation through a redistribution construction to the final bounds. In the normalization used by the manuscript, an imputation assigns payoffs summing to zero, with each coordinate at least (-1), and consequently at most (n-1). The game is zero-sum for three players and non-constant-sum for larger (n). Domination has an especially tractable form: one imputation dominates another when it gives strictly greater payoffs to all players with at most one exception. This reflects the decisive role of coalitions containing all but one player.

The first conceptual move is to distinguish symmetry of the game from symmetry imposed on its solutions. Symmetry here means invariance under permutations, not equal payoffs within every imputation. The authors represent a whole permutation orbit by ordering the coordinates of the interchangeable players. This reduces domination between orbits to comparisons between ordered coordinates, making partial symmetry mathematically usable.

For symmetry in $1, \ldots, n$, i.e., for $m = n$, all solutions $\vee$ are known. Let us now consider the case of symmetry in $1, \ldots, n-1$, i.e., of $m = n-1$, and investigate its solutions $\vee$.

Leaving the last player outside the imposed symmetry provides the investigation with its principal variable. The authors define (A) as the set of that player’s payoffs across a solution. Instead of immediately describing the entire solution geometrically, they study this one-dimensional projection.

Also, V is compact, hence A is compact.

Compactness allows the proof to work with attained minima and gap endpoints rather than merely limiting values. The argument then asks what happens if (a\in A), but an interval immediately below (a) contains no permitted payoff. Starting from an imputation with last-coordinate payoff (a), the authors transfer an amount (\Delta) away from that player and distribute it among the first (n-1) players. Their construction raises a group of the lowest coordinates to a common level while leaving the higher ones unchanged. Selecting the group through a minimum over candidate averages guarantees that the adjusted coordinates remain ordered. Explicit formulas for their partial sums prepare the later inequalities.

The decisive step connects this redistribution to the two stability requirements of a solution. Because the adjusted last payoff lies in the assumed gap, the new imputation is outside the solution and must be dominated by an imputation within it. Yet members of the solution cannot dominate one another. The ordered-coordinate tests therefore force the dominating imputation to have last payoff exactly (a): a payoff above (a) would produce inadmissible internal domination, while the relevant payoffs below (a) are excluded by the gap assumption.

To turn this restriction into a bound, the authors choose the original imputation by minimizing a maximum of adjusted partial sums. This “minimum maximorum” supplies a nonnegative auxiliary quantity, (\alpha). The redistribution identities and domination inequalities then relate (\alpha), (\Delta), and (a). Particularly important is the treatment of equality: the proof follows the conditions under which the inequalities could become equalities and shows that they would require incompatible choices of a maximizing index. Consequently, whenever the interval ([a-\Delta,a)) is absent from (A) and remains within the admissible payoff range, [ \Delta<\frac{n-1-a}{n-2}. ]

The final section applies this strict bound in two ways. If (a) is the minimum of (A), comparison with the lower admissible payoff (-1) yields (a<1/(n-1)). If ((b,a)) is a gap between two elements of (A), taking (\Delta) arbitrarily close to its width gives [ a-b\leq\frac{n-1-a}{n-2}. ] Thus larger upper endpoints permit smaller gaps. The distinction between the strict minimum-payoff bound and the non-strict gap-width bound follows from how the interval argument is applied.

The estimates of I), II) in 15. are optimum.

This closing claim establishes the intended sharpness of the results, although the manuscript ends without presenting the corresponding constructions. The preface separately explains that solutions approaching the minimum-payoff bound can be obtained through Gillies’s inflation techniques. The paper’s contribution is therefore a precise restriction on partially symmetric stable solutions, supported by a redistribution and extremal argument. Its broader relevance lies in showing how symmetry reduction can expose necessary structural properties even when a complete account of all solutions remains unfinished.