A hero's job

The most popular 2-person non-zero sum conflictive games of cooperation/defection, like the Prisoner's dilemma, the Chicken game etc. have mutual cooperation as a Pareto optimal point, i.e. moving from mutual cooperation to defection harms the opponent. But what happens when this is not the case and yet mutual defection is a "Pareto pessimal", i.e. worst for both players?

Jan and Olga are travelling back home from Mars inside their tiny space capsule. The oxigen generator breaks down and reserves are just enough for only one person to make it to the Earth. Both space travellers glimpse at each other and then stare through the window into the infinite void sourrounding them.

In this game, a hero is needed for the other to obtain some payoff, but mutual cooperation is as bad as mutual defection:

A \ B	Coop	Def
Coop	0\0	0\1
Def	1\0	0\0

Even more than in the Prisoner's dilemma, the rational strategy for this game is to defect. Some have shown how cooperation can be justified in the iterated version of the Prisoner's dilemma, but in our rendering of the hero game no iteration is possible.

In my opinion, if the matrix above actually corresponds to the payoffs obtained by A and B, then there's little more to be said above the game and defecting is indeed the expected strategy. But what motivates a hero is the emotional reward he or she obtains from knowing the good that the heroic action is doing to others. A payoff matrix accounting for this moral compensation would look like this:

A \ B	Coop	Def
Coop	aA\aB	aA\1
Def	1\aB	0\0

Where a_A and a_B are the altruism factors of A and B, respectively. Usually each player does not know the altruism factor of the opponent, so the game becomes one of imperfect information. Also, it can be argued that in the case of mutual cooperation the payoff would still be zero, as the altruism factor should be multiplying the actual outcome of one's heroic action, but in the standard exposition of these type of games players take their choices simultaneously without knowing the opponent's move.

It does not take much Game Theory, anyway, to understand that heroic action is based on intangible payoffs like the love for others or for the country, honor, etc. Raising these altruism factors is arguably one of the goals of the highly ritualistic military culture.

If we extend the hero game to n persons in a way that still it only takes a hero to save the rest, we can see that, from a societal point of view, it is desirable that altruism be not evenly distributed but rather concentrated in a few individuals: otherwise we'd have mass immolation.