Win-Stay-Lose-Learn Promotes Cooperation in the
Spatial Prisoner’s Dilemma Game
Yongkui Liu
1,2,3
*, Xiaojie Chen
4
*, Lin Zhang
1
, Long Wang
5
, Matjaz
ˇ
Perc
6
*
1 School of Automation Science and Electrical Engineering, Beihang University, Beijing, China, 2 School of Electronic and Control Engineering, Chang’an University, Xi’an,
China, 3 Center for Road Traffic Intelligent Detection and Equipment Engineering, Chang’an University, Xi’an, China, 4 Evolution and Ecology Program, International
Institute for Applied Systems Analysis, Laxenburg, Austria, 5 State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing,
China, 6 Faculty of Natural Sciences and Mathematics, Department of Physics, University of Maribor, Maribor, Slovenia
Abstract
Holding on to one’s strategy is natural and common if the later warrants success and satisfaction. This goes against
widespread simulation practices of evolutionary games, where players frequently consider changing their strategy even
though their payoffs may be marginally different than those of the other players. Inspired by this observation, we introduce
an aspiration-based win-stay-lose-learn strategy updating rule into the spatial prisoner’s dilemma game. The rule is simple
and intuitive, foreseeing strategy changes only by dissatisfied players, who then attempt to adopt the strategy of one of
their nearest neighbors, while the strategies of satisfied players are not subject to change. We find that the proposed win-
stay-lose-learn rule promotes the evolution of cooperation, and it does so very robustly and independently of the initial
conditions. In fact, we show that even a minute initial fraction of cooperators may be sufficient to eventually secure a highly
cooperative final state. In addition to extensive simulation results that support our conclusions, we also present results
obtained by means of the pair approximation of the studied game. Our findings continue the success story of related win-
stay strategy updating rules, and by doing so reveal new ways of resolving the prisoner’s dilemma.
Citation: Liu Y, Chen X, Zhang L, Wang L, Perc M (2012) Win-Stay-Lose-Learn Promotes Cooperation in the Spatial Prisoner’s Dilemma Game. PLoS ONE 7(2):
e30689. doi:10.1371/journal.pone.0030689
Editor: Ju
¨
rgen Kurths, Humboldt University, Germany
Received November 23, 2011; Accepted December 21, 2011; Published February 17, 2012
Copyright: ß 2012 Liu et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: YL acknowledges financ ial support from the Special Fund for the Basic Scientific Research of Central Colleges of Chang’an University (Grant
CHD2010JC134). LW acknowledges financial support from the National Natural Science Foundation of China (NSFC) (Grants 60736022 and 10972002). MP
acknowledges financial support from the Slovenian Research Agency (ARRS) (Grant J1-4055). This work was additionally supporte d by the National Natural
Science Foundation of China (NSFC) (Grants 61074144, 51005012 and 61103096). The funders had no role in study design, data collection and analysis, decision to
publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: ykliu@chd.edu.cn (YL); chenx@iiasa.ac.at (XC); matjaz.perc@uni-mb.si (MP)
Introduction
Evolutionary game theory provides a powerful mathematical
framework for studying the emergence and stability of cooperation in
social, economic and biological systems [1–5]. The prisoner’s
dilemma game, in particular, is frequently considered as a paradigm
for studying the emergence of cooperation among selfish and
unrelated individuals [6]. The outcome of the prisoner’s dilemma
game is governed by pairwise interactions, such that at any instance
of the game two individuals, who can either cooperate or defect, play
the game against each other by selecting their strategy simulta-
neously and without knowing what the other player has chosen. Both
players receive the reward R upon mutual cooperation, but the
punishment P upon mutual defection. If one player defects while the
other cooperates, however, the cooperator receives the sucker’s
payoff S while the defector receives the temptation T~b.Since
TwRwPwS, there is an innate tension between individual
interests (the rational strategy, yielding an optimal outcome for the
player regardless of what the other player chooses, is defection) and
social welfare (for the society as a whole the optimal strategy is
cooperation) that may result in the ‘‘tragedy of the commons’’ [7].
Five prominent rules for the successful evolution of cooperation,
which may help avert an impeding social decline, are kin selection,
direct and indirect reciprocity, network reciprocity as well as group
selection, as comprehensively reviewed in [8].
Since the pioneering work of Nowak and May [9] spatial games
have received ample attention, and they have become inspirational
for generations of researchers trying to reveal new ways by means of
which cooperation can prevail over defection [10–12]. In the
context of spatial games, network topology and hierarchies have
been identified as a crucial determinant for the success of
cooperative behavior [13–28], where in particular the scale-free
topology has proven very beneficial for the evolution of cooperation.
In fact, payoff normalization [29–31] and conformity [32] belong to
the select and very small class of mechanisms that can upset the
success of cooperators on such highly heterogeneous networks.
Other approaches facilitating the evolution of cooperation include
the introduction of noise to payoffs and updating rules [33–38],
asymmetry between interaction and replacement graphs [39,40],
diversity [41–44], differences between time scales of game dynamics
[44–47], as well as adoption of simultaneous different strategies
against different opponents [48]. Somewhat more personally-
inspired features supporting the evolution of cooperation involve
memory effects [49], heterogeneous teaching activity [50–52],
preferential learning [53,54], mobility [55–59], myopically selective
interactions [60], and coevolutionary partner choice [61–63], to
name but a few examples studied in recent years.
Regardless of the details of mechanisms that may promote the
evolution of cooperation in the spatial prisoner’s dilemma game,
most frequently, it was assumed that individual players learn from
PLoS ONE | www.plosone.org 1 February 2012 | Volume 7 | Issue 2 | e30689