Conservation Law of Utility and Equilibria in Non-Zero Sum Games

This short note demonstrates how one can define a transformation of a non-zero sum game into a zero sum, so that the optimal mixed strategy achieving equilibrium always exists. The transformation is equivalent to introduction of a passive player into a game (a player with a singleton set of pure strategies), whose payoff depends on the actions of the active players, and it is justified by the law of conservation of utility in a game. In a transformed game, each participant plays against all other players, including the passive player. The advantage of this approach is that the transformed game is zero-sum and has an equilibrium solution. The optimal strategy and the value of the new game, however, can be different from strategies that are rational in the original game. We demonstrate the principle using the Prisoner’s Dilemma example. Let X and Y be a dual pair of ordered linear spaces with respect to bilinear form 〈·, ·〉 : X × Y → R. If x ∈ X is a utility function x : Ω → R, and p ∈ Y , is a probability measure p : Ω → [0, 1], then the expected utility is: Ep{x} = 〈x, p〉 Consider a game such that the space of outcomes of the game is Ω := Ω1 × · · · × Ωm, where Ωi is the set of pure strategies of ith player, and xi : Ω → R are the utility functions of the players. In this case, the game is zero-sum if and only if x1 + · · ·+ xm = 0 If p1, . . . , pm ∈ Y are mixed strategies, then sup p1 inf p2,...,pm Ep1×···×pm{x1} ≤ inf p2,...,pm sup p1 Ep1×···×p2{x1} The famous Min-Max theorem [1] states that in zero-sum games, there always exists a mixed strategy p̄1×· · ·× p̄m, called a solution, such that the above holds with equality, and the common value is called the value of the game. One of the problems in game theory is the existence of a solution to a non-zero sum game. Let us consider a game, such that x1 + · · ·+ xm 6= 0 If the utility functions add up to a constant function, so that ∑ xi ∈ R1 := {β1 ∈ X : β ∈ R}, then one can add a constant function x0 := − 1 m ∑ xi to each xi so that the new utilities x̃i = xi + x0 are zero-sum. We argue that the same can be done in the case when x0 is not a constant function. Thus, we define a bijection T : X → X̃ by T (x) := x+ x0 = x− 1 m m ∑ i=1 xi It is easy to see that the new utility functions are zero sum: x̃1 + · · ·+ x̃m = m ∑