TITLE:
Forming Coalitions in Normal-Form Games
AUTHORS:
Emma Dwobeng, Herbert Corley
KEYWORDS:
Game Theory, Normal-Form Game, Coalitions, Semi-Cooperative Game
JOURNAL NAME:
Theoretical Economics Letters,
Vol.12 No.5,
October
26,
2022
ABSTRACT: For a given n-person
normal-form game G, we consider all
possible sets of mutually exclusive and collective exhaustive coalitions of the n players. For each such set of
coalitions, we define a coalitional semi-cooperative game Γ of G as one in which 1) the coalitions are taken as the players of this
new game, 2) each coalition tries to maximize the sum of its
individual players’ payoffs, and 3) the
players within a coalition cooperate to do so. The purpose of this paper is to
determine an optimal set of coalitions for G for some relevant notion of optimality. To do so, for the payoff matrix of each
possible Γ of G, we determine all
Greedy Scalar Equilibria (GSEs), where a GSE is an analog of the Nash
equilibrium but always exists in pure strategies. For each of these GSEs, we
divide the total payoff for each coalition among its members in the same
proportions as its members average over the entire payoff matrix of G. Doing so gives n modified individual player payoffs associated with each GSE of
all the Γs. For each of these GSEs,
we then compute the geometric mean of its n modified payoffs. A set of coalitions associated with a GSE is deemed optimal
for G if the corresponding geometric
mean is a maximum among all the GSEs for all the Γs. An optimal set of coalitions thus incorporates the selfishness
of the coalitions via the GSE, while the geometric mean of the redistribution
of the players’ payoffs models the cooperation of and the fairness for the
individual players.