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works.
We consider general two-player games of cooperation given by the payo
matrix
C D
C R S
D T P
:
(16.3)
Thus, a cooperator interacting with another cooperator obtains the reward from
mutual cooperation R. Cooperating against a defector leads to the sucker's payo
S, whereas the defector obtains the temptation to defect T in such an interaction.
Finally, defectors receive the punishment P from interactions with other defectors.
A social dilemma arises when individuals are tempted to defect, although mutual
cooperation would be the social optimum (R > P). We distinguish three generic
cases of 2-player social dilemmas:
Dominance: When T > R > P > S, we enter the realm of the Pris-
oner's Dilemma (PD) [Rapoport and Chammah (1965)], where coopera-
tion is dominated by defection. The opposite scenario, when R > T and
S > P, poses no social dilemma and is referred to as a Harmony Game
(HG) [Posch et al. (1999)].
Coordination: R > T and S < P leads to what is called coordination or
Stag Hunt games (SH) [Skyrms (2003)], in which it is always good to follow
the strategy of the majority in the population. Except for R + S = T + P,
one strategy has a larger basin of attraction. This strategy is called a risk
dominant strategy. For R + S > T + P, cooperation is risk dominant.
Coexistence: In the case of R < T and S > P, known as a Hawk-Dove
[Maynard Smith (1982)] or Snowdrift game (SG) [Sugden (1986); Doebeli
and Hauert (2005); Hauert and Doebeli (2004)], a small minority is fa-
vored. This means that the ultimate outcome in a population of players is
a mixture of strategies C and D.
From the payo matrix, we can calculate the payos of the individuals, depending
on the number of interactions they have with cooperators and defectors.
On a
complete network, the payos are
C
= R(N
C
1) + SN
D
(16.4)
and
D
= TN
C
+ P(N
D
1) :
(16.5)
Often, the payos are scaled by 1=(N1), such that the payos do not increase with
the population size. For the strategy update process dened below, this corresponds
simply to a rescaling of the intensity of selection, i.e. changing the noise intensity, if
all individuals have the same number of interactions. If the number of interactions
is not the same for all players, the heterogeneity between players can lead to new
eects [Santos et al. (2006b); Santos and Pacheco (2006)].
