Every function f defined over the real numbers is the sum of a symmetric function g and an anti-symmetric function h. Simply define g(x) = (f(x) + f(-x))/2 and h(x) = (f(x) – f(-x))/2. It is easy to verify that f(x) = g(x) + h(x) for every x, that g(x) = g(-x), and that h(x) = -h(-x).
Similarly, every two player game G is the sum of a game with common interests G_c and a zero-sum game G_z. If u^1 and u^2 are the payoff functions of the two players in the game G, u^1_c and u^2_c are the payoff functions in the game with common interests G_c, and u^1_z and u^2_z are the payoff functions in the zero-sum game G_z, simply define:
u^1_c(a^1,a^2) = (u^1(a^1,a^2) + u^2(a^1,a^2))/2
u^2_c(a^1,a^2) = (u^1(a^1,a^2) + u^2(a^1,a^2))/2
u^1_z(a^1,a^2) = (u^1(a^1,a^2) – u^2(a^1,a^2))/2
u^2_z(a^1,a^2) = (-u^1(a^1,a^2) + u^2(a^1,a^2))/2
As for the functions f, g, and h of the first paragraph, one can easily verify that u^1 = u^1_c + u^1_z, that u^2 = u^2_c + u^2_z, that u^1_c = u^2_c, and that u^1_z = -u^2_z.
Adam Kalai and Ehud Kalai present this decomposition of games and study its implications here. In this paper they propose a new one-point solution concept for two-player games with transferable utility, which they call the COCO-value (for cooperative-competitive value). The coco-value is defined as follows: the players play the pair of actions that maximizes the sum of their utilities: that is, the pair of actions that is optimal for both in the game G_c with common interests. But the game G that the players actually play is not a game of common interests, so that some transfer of utility has to be made. The game G_z measures the difference in utility between the players; if its value is positive, it means in some sense that player 1 receives in G more than player 2. The value of G_z is then the amount that player 2 pays to player 1. Thus, the coco-value of the game G is a pair of payments, one for each player: each gets the maximal payoff in G_c, and in addition player 2 pays to player 1 the value of G_z.
The two Kalai’s go on and provide axiomatization for their solution concept, as well as its implementation as an equilibrium in a properly defined game.
I like the paper and the intuition that underlies the concept. As a person who likes cooperation, I think that it presents an interesting solution concept for people like me, who prefer cooperation to competition, yet take into account the fact that the interests of the players are not the same.
Happily, few follow-up questions bother me: what about games with more than two players? Do we need to take into account two-player games when defining the coco-value of a three player game? And what about stochastic games? What is the natural extension of the solution concept when states change? Will it involve strategy-proofness?