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cooperate, defect or escape, and so on). In other words, we consider the participants
as programmed automata.
A generic program therefore says what to play at hand i, given the results of
previous hands, the identity of the two participants and the internal state (memory).
The choice of the output state can be stochastic or deterministic, but in general
one has to determine, for each hand i, the probability p(i) of outputting 0 (that of
outputting 1 is 1p(i)), and this can be done using a look-up table that constitutes
the genetic information of the individual.
To make things simpler, let us assume that the choice i does depend only on the
past hand, so that a program has to say just how to start (hand 0) and the four
probabilities p(a
x
; a
y
): probability of playing 0 if in the last hand player x played
a
x
and player y played a
y
. More complex strategies can be used [68].
The score of the hand for player x is given by the payo matrix Q(a
x
ja
y
). The
score for player y is the transpose of Q
0
if the game is symmetric. The interesting
situation is the prisoner's dilemma: Q(1j0) > Q((0j0) > Q(1j1) > Q(0j1).
h
For a
single-shot play, standard game theory applies. It can be shown that it is evolution-
ary convenient for both player to defect, since mutual cooperation (Q(0j0)) gives
less payo that exploitation (Q(1j0)); which from the point of view of the opposite
player (transpose) gives the minimal payo. The game is interesting if cooperation
gives the higher average payo (2Q(0j0) > Q(0j1) + Q(1j0)).
Axelrod [68] noted that if the the game is iterated, cooperation can arise: rule
with memory, like TIT-FOR-TAT (TFT: cooperate at rst hand, and then copy
your opponent's last hand) may stabilize the mutual cooperation state, and resist
invasion by other simpler rules (like ALL-DEFECT { ALLD, always play 1). Other
more complex rules may win against TFT (and possibly loose against ALLD).
Nowak [70] gives at least ve ways for which mutual cooperation can arise in an
evolutionary environment:
1. Kin selection. Cooperation is preferred because individuals share a large frac-
tion of the genome, so a gene for cooperation that is common in the population
shares the average payo, even in a single-shot game.
2. Direct reciprocity. This is just Axelrod's iterated game. If the expected length
of the game is large enough, it is worth to try to cooperate for gaining the
higher share.
3. Reputation. Information about past games can be useful in deciding if own
opponent is inclined towards cooperation or towards exploitation.
4. Network reciprocity. In this case one considers the spatial structure of the
group, represented by a graph with local connectivity k. A player plays against
all neighbors, and gather the related score. If the advantage in cooperating in
a cluster of k neighbors is greater than the gain I would have by switching to
defect, the strategy is stable.
h
The asymptotic state of the iterated Prisoner's dilemma game is not given by a \simple" opti-
mization procedure, see Ref. [69]
