Received: 20 January 2017 Revised: 15 August 2017 Accepted: 3 November 2017
The optimal control of Axelrod's social norm game
Ahmed S. Jaber
Monica G. Cojocaru
Department of Mathematics and
Statistics, University of Guelph, Guelph,
Department of Mathematics, College of
Science, University of Al-Mustansiriya,
Ahmed S. Jaber, Department of
Mathematics, College of Science,
University of Al-Mustansiriya, Baghdad,
Natural Sciences and Engineering
Research Council of Canada (NSERC)
In this paper, we study the well-known Axelrod's social norm game using repli-
cator dynamics and control theory. In general, a social norm game has 4 pure
strategies: norm following and norm defecting, each of which branching further
into punishing or not punishing others for defection, if observed. Our aim is to
check whether by introducing a control parameter that mimics incentives for
players who follow the norm, we observe changes in the dynamics of the strategy
game, which maximize norm-following strategies. We solve the control problem
numerically and find that incentives and the capacity of individuals to observe
others' defection both have an impact on the emergence of Nash equilibria in
which norm following is dominant.
Axelrod's social norms, Nash strategies, optimal control, replicator dynamics
Robert Axelrod (1986) established a model to compute Nash equilibrium strategies in social norm games by using an
agent-based model of repeated players' interactions. This model became known as the Axelrod model.
In the following
decades, researchers continued to be interested in predicting the behavior of social norms via individual behavior over
The emergence of norms in Axelrod's model led researchers to study the influence and limitations of parameters
such as boldness and vengefulness.
Zaitseva (2010) discussed the no-cheating norm in Axelrod's model by estimating the
average of vengefulness of players.
Mahmoud et al (2011) studied the limitations of omniscience in the emergence of
norms in the Axelrod model and showed how it is possible to overcome these limitations.
Arai and Suzuki (2014) studied
the establishment of a social norm from a different perspective: they introduced the idea of encouraging “right social
norm” to be considered by the population in a coordination game. They used a method of encouraging the individuals
to optimize their performance by using the inverse reinforcement learning.
Khan et al (2015) explored social norms
by introducing intersubjectivity into a centipede game, which is a game with discrete time.
A novel work by De et al
(2017) involves cultural inertia and social norm change in an evolutionary game to explore the societies' sensitivity for
coordination, which might lead to norm change.
Lapinski et al (2017) provide an interesting study about the influence of
financial incentives on the public good game. Their study explores the long-term effect of behavioral payment programs
especially when the financial incentives are offered in a short term.
Andrews et al (2015) looked at Axelrod's norm and metanorm games using classic game theoretic frameworks, where
the population of individuals is divided in fractions given by 4 pure strategies.
Using replicator dynamics, they showed
that all Axelrod's conclusions hold, without the need of the agent-based framework. In this paper, we explore the appli-
cations of the control theory to the social norm game and its associated replicator dynamics, expanding on the work of
Andrews et al.
The optimal control problems started in the 1950s when researchers studied variational problems,
starting with the
works of Pontryagin.
Since then, many researchers have contributed to the concept of optimal control with necessary
Optim Control Appl Meth. 2018;39:949–962. wileyonlinelibrary.com/journal/oca Copyright © 2017 John Wiley & Sons, Ltd. 949