I just posted a paper on so-called compromise values for cooperative games here on the game theory research page. The paper is written with my co-author Rene van den Brink, who is a professor of game theory at the VU University Amsterdam.
This new paper, “Construction of Compromise Values for Cooperative Games,” delves into a method for determining fair ways to divide the benefits of cooperation among a group of players. In cooperative games, players work together to achieve a certain outcome, and the question is how to allocate the resulting value. A value in game theory is a rule that assigns a specific payoff to each player.
We focus on compromise values, which are based on the idea of finding a balance between what a player could minimally expect (a lower bound) and what they could maximally hope for (an upper bound). The paper explains that these bounds are represented by mathematical functions. A key example of an early compromise value is the 𝜏-value, which balances a player’s minimal rights with their marginal contribution.
The paper introduces the concept of bound pairs, consisting of upper and lower bound functions that satisfy specific mathematical properties. For a given bound pair, a compromise value can only be meaningfully defined for bound-balanced games, which are games where it’s possible to find a fair allocation that falls between the established lower and upper bounds.
Rene and I explore two primary methods for constructing compromise values. The first involves starting with a well-behaved lower bound function and deriving a corresponding upper bound. Examples of values constructed this way include the Egalitarian Division rule and the CIS-value (Centre of the Imputation Set). The second method begins with a translation covariant upper bound function to construct a related lower bound. The 𝜏-value and, under certain conditions, the CIS-value can be derived using this approach as well.
Furthermore, the paper provides axiomatisations for compromise values, meaning they identify a set of fundamental properties that uniquely define these values. These properties include the minimal rights property and a form of restricted proportionality. By establishing this framework, the authors demonstrate a common underlying structure for many existing and newly proposed values in cooperative game theory, such as the PANSC-, Gately-, and KM-values. The paper highlights that this approach offers a unifying perspective on how to fairly allocate the gains from cooperation.
This paper accompanies my paper on the Gately value for cooperative games that was recently published in Economic Theory.