Spiteful behavior can make everybody better off

Spiteful behavior can make everybody better off
Robert Philipowski
∗
Abstract
We present an example of a symmetric two-player game admitting a unique Nash equilibrium and a unique evolutionarily stable strategy (ESS) and such that the ESS payoff is
strictly higher than the Nash payoff. In this sense we show that spiteful behavior can make
everybody better off.
Keywords: Nash equilibrium, Evolutionarily stable strategy, Spiteful behavior
JEL classification: C72
In economics one usually assumes that agents (households, firms, local governments, etc.)
behave egoistically in the sense that they try to maximize their own payoff (utility, profit,
social welfare, etc.) in the absolute sense, i.e. without caring about the payoffs of the others.
If moreover they choose their strategic variables simultaneously, this assumption leads to the
well-known concept of Nash equilibrium.
However, as argued by Schaffer (1989) and many subsequent authors (e.g. Al´os-Ferrer and
Ania, 2005, and the references therein), in certain contexts it seems reasonable that instead
of maximizing their payoff in the absolute sense, agents care about their relative performance
compared to their peers. In symmetric games the appropriate equilibrium concept for this
kind of behavior is that of a finite population evolutionarily stable strategy (ESS) introduced by
Schaffer (1988):1
Definition 1. Let (X, u) be a symmetric n-player game, where X is the strategy set of each
player and u : X n → Rn the payoff function, which is assumed to be symmetric in the sense that
ui (sπ(1) , . . . , sπ(n) ) = uπ(i) (s1 , . . . , sn ) for all i ∈ {1, . . . n}, all permutations π of {1, . . . , n} and
all strategy combinations (s1 , . . . , sn ) ∈ X n . We then say that a strategy sE ∈ X is evolutionarily
stable if for every s ∈ X
u1 (s, sE , . . . , sE ) ≤ u2 (s, sE , . . . , sE ).
The interpretation of this notion is as follows: Suppose that all players except the first one
choose the strategy sE . Then the first player will only choose sE as well if there is no strategy s
satisfying u1 (s, sE , . . . , sE ) > u2 (s, sE , . . . , sE ), because by choosing such a strategy the first
player would be better off than the others.
One can easily see that a strategy sE is evolutionarily stable if and only if
sE ∈ arg max (u1 − u2 )(s, sE , . . . , sE ),
(1)
s∈X
cf. Schaffer (1988, Eq. (11)).2 Hence, ESS strategists behave spitefully in the sense that they do
not maximize their own payoff, but the difference between own and other players’ payoffs.
∗
Unit´e de Recherche en Math´ematiques, Universit´e du Luxembourg, 6, rue Richard Coudenhove-Kalergi, 1359
Luxembourg, Grand Duchy of Luxembourg. E-mail address: [email protected]
1
The original ESS concept introduced by Maynard Smith and Price (1973) is only suitable for infinite populations.
2
Proof. If sE is evolutionarily stable, we have (u1 − u2 )(s, sE , . . . , sE ) ≤ 0 for all s ∈ X and, by symmetry,
(u1 − u2 )(sE , . . . , sE ) = 0. Hence sE satisfies (1). If however sE is not evolutionarily stable, there exists a
strategy s satisfying (u1 − u2 )(s, sE , . . . , sE ) > 0, so that sE does not satisfy (1).
1
Intuitively, spiteful behavior should lead to lower payoffs than egoistic (Nash) behavior.3
Of course, due to non-existence and non-uniqueness problems, one cannot expect such a result
to hold in complete generality, but one might at least expect that in games admitting a unique
Nash equilibrium and a unique ESS, the ESS payoff is lower than or equal to the Nash payoff.
The purpose of the present note is to show that this is not true: Under certain circumstances
spiteful behavior can make everybody better off. Namely, we present an example of a symmetric
two-player game admitting a unique Nash equilibrium and a unique evolutionarily stable strategy
and such that the ESS payoff is strictly higher than the Nash payoff.
Example 1. Let X = N0 , and define u : N20 → R2 by
u(n, n) = (n, n)
for all n ∈ N0 ,
u(0, n) = (−n − 1, −n)
for all n ≥ 1,
u(n, 0) = (−n, −n − 1)
for all n ≥ 1,
u(m, n) = (m, m + 1)
for m > n ≥ 1,
u(m, n) = (n + 1, n)
for n > m ≥ 1.
The matrix of this game looks as follows:
2
1
0
1
2
3
4
...
0
1
2
3
4
...
(0, 0)
(−1, −2)
(−2, −3)
(−3, −4)
(−4, −5)
...
(−2, −1)
(1, 1)
(2, 3)
(3, 4)
(4, 5)
...
(−3, −2)
(3, 2)
(2, 2)
(3, 4)
(4, 5)
...
(−4, −3)
(4, 3)
(4, 3)
(3, 3)
(4, 5)
...
(−5, −4)
(5, 4)
(5, 4)
(5, 4)
(4, 4)
...
...
...
...
...
...
...
One can easily see that there is a unique Nash equilibrium, namely (0, 0), and a unique ESS,
namely 1, and that the ESS payoff (which equals 1) is strictly higher than the Nash payoff
(which equals 0).
Concluding remark. Our example is presumably of no immediate economic significance. However, it shows that spiteful (ESS) behavior does not necessarily lead to lower (or equal) payoffs
than egoistic (Nash) behavior, and may serve as a motivation to derive general sufficient criteria
for such a result.
References
Al´os-Ferrer, C. and Ania, A. B. (2005), The evolutionary stability of perfectly competitive
behavior. Economic Theory 26, 497–516.
Ania, A. B. (2008), Evolutionary stability and Nash equilibrium in finite populations, with an
application to price competition. Journal of Economic Behavior and Organization 65, 472–
488.
3
In the context of tax competition this was recently proved under various assumptions by Sano (2012), Wagener
(2013) and Philipowski (2015). It seems that this question has not been addressed explicitly in other branches
of economics. However, Schaffer (1989), Vega-Redondo (1997) and Al´
os-Ferrer and Ania (2005) showed under
various assumptions that in Cournot oligopoly evolutionarily stable behavior leads to the Walrasian outcome and
hence to lower profits than Nash behavior. Moreover, Ania (2008) and Hehenkamp et al. (2010) investigated
conditions under which Nash and evolutionarily stable equilibrium coincide.
2
Hehenkamp, B., Possajennikov, A. and Guse, T. (2010), On the equivalence of Nash and evolutionary equilibrium in finite populations. Journal of Economic Behavior and Organization
73, 254–258.
Maynard Smith, J. and Price, G. R. (1973), The logic of animal conflict. Nature 246, 15–18.
Philipowski, R. (2015), Comparison of Nash and evolutionary stable equilibrium in asymmetric
tax competition. Regional Science and Urban Economics 51, 7–13.
Sano, H. (2012), Evolutionary equilibria in capital tax competition with imitative learning.
Evolutionary and Institutional Economics Review 9, S1–S23.
Schaffer, M. E. (1988), Evolutionarily stable strategies for a finite population and a variable
contest size. Journal of Theoretical Biology 132, 469–478.
Schaffer, M. E. (1989), Are profit-maximisers the best survivors? A Darwinian model of economic
natural selection. Journal of Economic Behavior and Organization 12, 29–45.
Vega-Redondo, F. (1997), The evolution of Walrasian behavior. Econometrica 65, 375–384.
Wagener, A. (2013), Tax competition, relative performance, and policy imitation. International
Economic Review 54, 1251–1264.
3