 
Summary: SUPPORTING INFORMATION FOR J.B. ANDR´E AND N.
BAUMARD "SOCIAL OPPORTUNITIES AND THE EVOLUTION
OF FAIRNESS"
JEANBAPTISTE ANDR´E AND NICOLAS BAUMARD
1. The ultimatum game
1.1. Nash equilibria. As mentioned briefly in the main text, there is an infinite2
number of Nash equilibrium strategies (i.e. strategies that are best reply to them
selves) in the ultimatum game (UG). In fact, any strategy with p = q [0, 1] is4
a Nash equilibrium. If one's partner requests x (respectively if she offers x), there
is nothing better to do than offering x (respectively, requesting x). Evolutionar6
ily speaking, when every individual in the population offers and requests exactly
x [0, 1], then any mutant is either neutral or counterselected when playing in8
front of the resident.
Polymorphic states of the population can also be stable. A polymorphic Nash10
equilibrium is a state of the population such that (i) every strategy present in the
population obtains the same payoff G, and (ii) every absent strategy obtains G G12
when confronted to a representative sample of the resident population. Note that,
in the remaining of the Supporting Information, we sometimes use the expression14
"Nash equilibrium"or even sometimes only"equilibrium", for both polymorphic Nash
equilibria and Nash equilibria stricto sensu.16
