# Modified core for cooperative games

In the core of cooperative games, the total cost of forming a coalition is shared by all members in a coalition (efficiency). However for my problem I only require that all members are individually rational in the coalition - that is, if the utility of at least one member of the coalition is less than the utility he would gain by being alone, then he will break away from the coalition.

In other words, what is the solution concept that only requires stability and does not carry the notion of efficiency?

• To me it is not quite clear if you are talking about a concept in cooperative GT (e.g. in NTU games) or if you are talking about non-cooperative GT. – The Almighty Bob Nov 19 '14 at 14:01
• @TheAlmightyBob: Cooperative GT. Is there a concept of coalition in non-coop GT as well? – Bravo Nov 19 '14 at 18:12
• Do you mean rational self-interest? – Jason Nichols Nov 19 '14 at 18:33
• In non-coop GT, there is a solution concept called coalition-proof Nash equilibrium, proposed by Bernheim, Peleg, Whinston (1987, JET). But existence of such an equilibrium is not guaranteed in general. – Herr K. Nov 19 '14 at 18:59

If I understand the question correctly, I believe the neatest way to put it is to simply call you solution concept "individual rationality".

Individual rationality can easily be seen as a cooperative game theory solution concept in its own right. I am not aware of any other name for a solution concept consisting only of an individual rationality requirement.

I suppose it would look something like

$\Pi^m$ is the profit of a monopolist. $\Pi^c$ is the profit under collusion. $\Pi^c=\frac{\Pi^m}{n}$ $\Pi^{comp}$

suppose $\Pi^m> \Pi^c> \Pi^{comp}$. The profit a firm gets for deviating through 1 period is $\Pi^d=\Pi^m-\epsilon=\Pi^m$

The condition to sustain collusion is $\Pi^d +\delta \Pi^{comp} +\delta ^2 \Pi^{comp}+...> \Pi^c+ \delta \Pi^c+ ...$

for collusion to be sustained, time needs to be infinite and discount factor $\delta$ needs to be sufficiently high.