Fig 1.
Payoff matrix of a three-player zero-sum game shows the payoffs of three players, X, Y and Z. X and Y choose between two options, D and C, whilst Z chooses between L and R.
Fig 2.
Payoff matrix of a coordination game.
Two players are indifferent between two pure-strategy NEs. The probability that any NE is achieved is 0.5 no matter what pure or mixed strategy they choose. Unless some reactive strategies are adopted, the probability of convergence to any NE is 0.5 in a repeated version of this game.
Fig 3.
The relationship between proposition 1 and the folk theorem in infinitely repeated three-player games.
O denotes the minimax payoff profile. Any feasible payoff profile within OABC (it includes the surface of ABC, but excludes the curves AB, BC, and CA.) is a NE according to the folk theorem. Propostion 1 proves that any feasible payoff profile on the ABC surface (including the curves AB, BC, and CA) can be NE if [5] holds. Any type-k equilibrium is a Parteto optimum.
Fig 4.
The payoff matrix of a three-player game is given, where three players are X, Y and Z and each has two options, L and R.
Fig 5.
The set of payoff profiles of all NE is a 3D polyhedron in the payoff space of X, Y, Z players and △ADE is its projection onto the X-Y plane.
△ABC is the projection of the set of feasible profiles. The point A represents the minimax profile. The point G represents a type-3 equilibrium. Any point on the segment DE represents a type-2 equilibrium.