Induced Representations in Cooperative Games with Homogeneous Groups of Players

TL;DR

This paper uses algebraic representation theory to analyze symmetric cooperative games, simplifying computation and revealing structural insights with applications to international security and markets.

Contribution

It introduces a novel algebraic framework using induced representations to analyze homogeneous group games, reducing complexity and characterizing the Shapley value.

Findings

Identifies the spectrum of homogeneous group games via induced representations.

Proves that such games are supported only by irreducible representations.

Demonstrates applications to the UN Security Council and markets.

Abstract

Oftentimes, the Shapley value becomes infeasible for games with many players. However, establishing symmetry allows for polynomial-time computation. To examine this reduction, we identify the spectrum of homogeneous group games by using an induced representation from a Young subgroup. We then prove that such games are supported solely by irreducible representations, via the Littlewood-Richardson rule, where the depth of interactions is strictly bounded by the size of the minority group. Therefore, the algebraic structure of the game filters out the complexities of the general kernel $W$. We then show that this filtration constrains any symmetric linear value to a specific subspace. This recovers the Shapley value uniquely for $m=2$ under standard axioms. Finally, we explore applications to the UN Security Council and complementary goods markets to illustrate the practical power of this approach.