The Payoff Region of a Strategic Game and Its Extreme Points

TL;DR

This paper explores the structure of payoff regions in strategic games using extreme points, providing insights into their shape, convexity, and implications for efficiency concepts in game theory.

Contribution

It introduces a novel approach using extreme points to analyze the payoff regions of n-player strategic games, linking geometric properties to efficiency outcomes.

Findings

Payoff regions can be estimated using extreme points and supporting hyperplanes.

Subregions of payoff regions are non-strictly convex near boundary points.

Extreme points help prove results about Pareto and social efficiency.

Abstract

The range of a payoff function for an $n$-player finite strategic game is investigated using a novel approach, the notion of extreme points of a non-convex set. The shape of a noncooperative payoff region can be estimated using extreme points and supporting hyperplanes of the cooperative payoff region. A basic structural characteristic of a noncooperative payoff region is that any of its subregions must be non-strictly convex if the subregion contains a relative neighborhood of a point on its boundary. Besides, applying the properties of extreme points of a noncooperative payoff region is a simple and effective way to prove some results about Pareto efficiency and social efficiency in game theory.