Editing Evolutionary game theory (section)

===Prisoner's dilemma===
{{main|Prisoner's dilemma}}

A difficulty of the theory of evolution, recognised by Darwin himself, was the problem of [[altruism]]. If the basis for selection is at an individual level, altruism makes no sense at all. But universal selection at the group level (for the good of the species, not the individual) fails to pass the test of the mathematics of game theory and is certainly not the general case in nature.<ref>{{cite book |author=Okasha, Samir |title=Evolution and the Levels of Selection |date=2006 |publisher=Oxford University Press |isbn=978-0-19-926797-2 }}</ref> Yet in many social animals, altruistic behaviour exists. The solution to this problem can be found in the application of evolutionary game theory to the [[prisoner's dilemma]] game – a game which tests the payoffs of cooperating or in defecting from cooperation.  It is the most studied game in all of game theory.<ref>{{cite journal |author1=Pacheco, Jorge M. |author2=Santos, Francisco C. |author3=Souza, Max O. |author4=Skyrms, Brian  |title=Evolutionary dynamics of collective action in N-person stag hunt dilemmas |journal=Proceedings of the Royal Society |date=2009 |doi=10.1098/rspb.2008.1126 |pmid=18812288 |volume=276 |issue=1655 |pages=315–321|pmc=2674356 }}</ref>

The analysis of the prisoner's dilemma is as a repetitive game. This affords competitors the possibility of retaliating for defection in previous rounds of the game. Many strategies have been tested; the best competitive strategies are general cooperation, with a reserved retaliatory response if necessary.<ref>{{cite book |author=Axelrod, R. |date=1984 |title=The Evolution of Cooperation|publisher=Basic Books |isbn=0-465-02121-2|edition=1st}} {{cite book |author=Axelrod, R. |date=2009 |title=The Evolution of Cooperation|edition=Revised |isbn=<!--0-465-02121-2-->978-0-14-012495-8}}</ref> The most famous and one of the most successful of these is [[tit-for-tat]] with a simple algorithm.

<syntaxhighlight lang="python">
def tit_for_tat(last_move_by_opponent):
    """Defect if opponent defects, else cooperate."""
    if last_move_by_opponent == defect:
        defect()
    else:
        cooperate()
</syntaxhighlight>

The pay-off for any single round of the game is defined by the pay-off matrix for a single round game (shown in bar chart 1 below). In multi-round games the different choices – co-operate or defect – can be made in any particular round, resulting in a certain round payoff.  It is, however, the possible accumulated pay-offs over the multiple rounds that count in shaping the overall pay-offs for differing multi-round strategies such as tit-for-tat.

[[File:PrisonersPayoff.jpg|thumb|400px|Payoffs in two varieties of prisoner's dilemma game
<br>Prisoner's dilemma: co-operate or defect
<br>Payoff <sub>(temptation in defecting vs. co-operation)</sub> > Payoff <sub>(mutual co-operation)</sub> > Payoff<sub>(joint defection)</sub> > Payoff<sub>(sucker co-operates but opponent defects)</sub>]]

Example 1: The straightforward single round prisoner's dilemma game.  The classic prisoner's dilemma game payoffs gives a player a maximum payoff if they defect and their partner co-operates (this choice is known as ''temptation''). If, however, the player co-operates and their partner defects, they get the worst possible result (the suckers payoff). In these payoff conditions the best choice (a [[Nash equilibrium]]) is to defect.

Example 2: Prisoner's dilemma played repeatedly. The strategy employed is ''tit-for-tat'' which alters behaviours based on the action taken by a partner in the previous round – i.e. reward co-operation and punish defection.  The effect of this strategy in accumulated payoff over many rounds is to produce a higher payoff for both players' co-operation and a lower payoff for defection.  This removes the temptation to defect.  The suckers payoff also becomes less, although "invasion" by a pure defection strategy is not entirely eliminated.