# Shapley Value when players can opt out

I've been reading up on some game theory concepts in order to better understand good ways to allocate benefits across a coalition (e.g. https://www.casact.org/library/astin/vol14no1/61.pdf). I've learned about Shapley Value, Nucleolus and some similar methods of specifying a unique point within the core.

However, I have a slightly different problem that I don't know how to approach. Suppose I have a game where I want to ensure an optimal group result, and I want a stable equilibrium (so I still want some features such as individual rationality and collective rationality), but in this game players can choose not to play at all. For example:

Suppose that Player 1 acting alone has utility -5 and Player 2 acting alone has utility -10. Also suppose that the coalition of Player 1 and Player 2 has utility of 1. Players 1 and 2 can both choose to play or not. What are some methods for defining a unique value within the core? Is there a direct translation of this problem into the more common problem where everyone has positive utility and we are merely allocating the grand coalition's benefit?

Thanks, Patrick

• I'd suggest making a payoff matrix or a markov decision making model. Also, this looks pretty similar to the prisoner's dilemma – Joseph Eck May 2 '18 at 1:23
• I don't get your point. By your example, the Shapley value distributes $(3,-2)$, which is a core selection. Is it rational for a player not to participate at the game? Certainly not! Moreover, if you are looking on positive utilities, you can transform the game into a zero-normalized one, for instance. Finally, be careful with the Lemaire article, it is well known that the manuscript contains a lot of mistakes. For instance, the nucleolus on p. 79 is wrong, the correct one is $(105375/2,97875/4,51375/4)$. – Holger I. Meinhardt May 2 '18 at 8:48