The methodology of SI in game theory

Once the idea of a coalitional better response is defined, it is easy to add this behaviour to a dynamic process of strategy updating. Each period, rather than simply choose an individual to update his strategy, instead choose a group of individuals who update their strategies such that all individuals in the group increase their payoffs by doing so.

However, if coalitional moves are included in a dynamic model of behaviour, then there is less chance of an equilibrium being reached. The model may predict cyclic behaviour. An example is the two player prisoner’s dilemma, in which individual optimization will cause players to play Defect, but collective optimization will cause players to play Cooperate.

Cooperate Defect
Cooperate 3,3 1,4
Defect 4,1 2,2

The paper Coalitional Stochastic Stability shows how, if so desired, cyclic behaviour driven by different levels of agency (collective versus individual) can be eliminated by considering vanishing amounts of coalitional behaviour.

Coalitional Stochastic Stability, J.Newton, Games and Economic Behavior (2012).

The literature on stochastic stability considers which Nash equilibria are most likely to be selected by behavioural dynamics perturbed by random shocks to strategy choice. Coalitional Stochastic Stability considers coalitional behaviour as a perturbation. The idea is that even a small amount of coalitional behaviour in a behavioural rule is more realistic than none at all.

In the Prisoner’s dilemma, as coalitional behaviour becomes rare, the players will coordinate on the (Defect, Defect) Nash equilibrium almost all of the time.

For the Stag Hunt, coalitional behaviour will lead players to coordinate on the high payoff (Stag, Stag) equilibrium. This contrasts with the case of individualistic updating perturbed by random shocks, where in the long run the inefficient (Hare, Hare) equilibrium is played almost all of the time.

Stag Hare
Stag 3,3 0,2
Hare 2,0 2,2

For more results, deeper analysis and further applications, please read the cited paper.

Work in this area has been extended by Ryoji Sawa in his paper:

Coalitional stochastic stability in games, networks and markets, R.Sawa, Games and Economic Behavior (2014).

Here, ideas of coalitional stochastic stability are incorporated into the logit choice rule, which is one of the most commonly used models of random choice. The paper shows how methods for determining the rarity of transitions under the logit choice rule extend in an intuitive way to coalitional models.