Geometry of payoffs in a 3+ player zero-sum game

Question

In a $2$ player zero-sum game in which each player may mix between $2$ pure actions, each player's action space is an interval in $\mathbb{R}^2$ ($1$ dimension of the ambient space is constrained by the axiom that probabilities sum to $1$, and the other dimension is truncated to an interval by the axiom that probabilities reside between $0$ and $1$). Similarly (sort of), even if we generalize to arbitrarily many pure actions, each player has a real number payoff, but the zero-sum property imposes a constraint such that among all players jointly, the payoff space has $2$ ambient dimensions but $1$ intrinsic dimension, further constrained to an interval by the actual payoff being a convex combination of pure action payoffs.

If we do generalize to arbitrarily many pure actions, then for each player, their action space is a simplex in that many ambient dimensions and $1$ fewer than that many intrinsic dimensions. Can we say something analogous about the (jointly shared by all players) payoff space if we generalize the number of players (presumably independently of their numbers of pure actions)? For example, I hypothesize that the joint payoff space for a $3$ player zero-sum game has this shape...

I will assume it's a simplex, but it may be more appropriate to call it a triangle depending on how conclusions extend to $4+$ player zero-sum games. The idea would be that we assign each player either to a vertex or an edge (one is probably a better design choice than the other though they may both work), and then (analogous to how in the $2$ player case both players are from their local perspectives maximizing an expected payoff in $\mathbb{R}$ yet we globally view one as the maximizer and the other as the minimizer) all players are locally maximizing an expected payoff in $\mathbb{R}$ yet globally (i.e., if we choose to assign players to vertices) each player "pulls" the payoff toward their own vertex, such that one is the "upizer," another the "downrightizer," and the other the "downleftizer."

Often $3+$ player zero-sum games are hard to analyze due to the formation of coalitions between players. A common approach in combinatorial game theory is to stipulate the podium rule, although this may assume that payoffs convey only rank information (i.e., in a 2 player zero-sum game, a winner gets high payoff and a loser gets low payoff, rather than the more expressive framework we probably want to preserve here in which the loser pays the winner some utility in $\mathbb{R}$).

Is this geometry correct, and if so, is it correct with or without some simplifying stipulation eliminating coalitions? Is there a clear way to modify the geometry to correspond the opposite case (to permit coalitions if they are not part of the geometry and to eliminate them if they are)?

Answer 1

The space of expected payoffs need not be convex. Look at the following two-player non-zero-sum game: $$\begin{array} {c|c} & L & R \\ \hline T & 1,0 & 0,0 \\ \hline B & 0,0 & 0,1 \\ \end{array}$$ The payoffs $(1,0)$ and $(0,1)$ can be achieved (using pure actions), but the convex combination $(1/2,1/2)$ can not. The only distribution over action profiles inducing it requires both $(T,L)$ and $(B,R)$ be played with probability $1/2$ each, but this cannot be done when players randomize independently. Indeed, both players need to mix so that both pure actions are played with strictly positive probability, but then the pure action profiles $(T,R)$ and $(B,L)$ will occur with strictly positive probability.

One can turn this example into a three-player zero-sum game in which the third player has any number of pure actions and in which the third player's payoffs are $0$ minus the sum of the payoffs of the other players.