About Game Theory

Game theory is a branch of mathematics and economics dedicated to formal models of strategic interaction and its outcomes. Noncooperative game theory models strategic interaction. The actors in games are players. Cooperative game theory allows players to make binding agreements and form coalitions. Cooperative game theory tends to focus on outcomes. The proportional value, a key ingredient in proportional marginal variance decomposition, is a rule for determining the outcomes of a cooperative game given what every coalition could achieve on its own. The proportional value was identified independently by Barry Feldman and by K. M. Ortmann.

The proportional value is an analog of Shapley value, a linear value and the most important value in cooperative game theory. The Shapley value is derived from three basic axioms, two of which are linearity and symmetry. The Shapley value is the expected marginal contribution a player makes when all players join a growing coalition of all players in random order. The Shapley value is also the equilibrium outcome of a number of noncooperative bargaining games. In these games, all players have an equal probability of being selected to propose a bargaining outcome.

Game theorists can solve only very simplified models of bargaining situations. Feldman's identification of the proportional value started with the premise that it is unrealistic to assume that strong and weak players alike should always have equal opportunity to take the initiative in bargaining.

A simple stylized alternative to the notion that all players should have equal bargaining power is that the probability of getting the opportunity to make a proposal is proportional to a player's perceived power in the game. The logical measure of perceived power is expected payoff in a noncooperative game. Feldman shows that the proportional value is the equilibrium outcome in a particular bargaining game when the probability of a player having the opportunity to propose at any point in the game is proportional to her expected payoff at that time.

  Copyright © 2006 Prism Analytics