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Strategy Selection in Structured Populations
Evolutionary game theory studies frequency dependent selection. The fitness of a strategy is not constant, but depends on the relative frequencies of strategies in the population. This type of evolutionary dynamics occurs in many settings of ecology, infectious disease dynamics, animal behavior and social interactions of humans. Traditionally evolutionary game dynamics are studied in well-mixed populations, where the interaction between any two individuals is equally likely. There have also been several approaches to study evolutionary games in structured populations. In this paper we present a simple result that holds for a large variety of population structures. We consider the game between two strategies, A and B, described by the payoff matrix View the MathML source. We study a mutation and selection process. For weak selection strategy A is favored over B if and only if σa+b>c+σd. This means the effect of population structure on strategy selection can be described by a single parameter, σ. We present the values of σ for various examples including the well-mixed population, games on graphs, games in phenotype space and games on sets. We give a proof for the existence of such a σ, which holds for all population structures and update rules that have certain (natural) properties. We assume weak selection, but allow any mutation rate. We discuss the relationship between σ and the critical benefit to cost ratio for the evolution of cooperation. The single parameter, σ, allows us to quantify the ability of a population structure to promote the evolution of cooperation or to choose efficient equilibria in coordination games.MathematicsOrganismic and Evolutionary Biolog
Evolutionary game theory: Temporal and spatial effects beyond replicator dynamics
Evolutionary game dynamics is one of the most fruitful frameworks for
studying evolution in different disciplines, from Biology to Economics. Within
this context, the approach of choice for many researchers is the so-called
replicator equation, that describes mathematically the idea that those
individuals performing better have more offspring and thus their frequency in
the population grows. While very many interesting results have been obtained
with this equation in the three decades elapsed since it was first proposed, it
is important to realize the limits of its applicability. One particularly
relevant issue in this respect is that of non-mean-field effects, that may
arise from temporal fluctuations or from spatial correlations, both neglected
in the replicator equation. This review discusses these temporal and spatial
effects focusing on the non-trivial modifications they induce when compared to
the outcome of replicator dynamics. Alongside this question, the hypothesis of
linearity and its relation to the choice of the rule for strategy update is
also analyzed. The discussion is presented in terms of the emergence of
cooperation, as one of the current key problems in Biology and in other
disciplines.Comment: Review, 48 pages, 26 figure
Evolutionary Multiplayer Games
Evolutionary game theory has become one of the most diverse and far reaching
theories in biology. Applications of this theory range from cell dynamics to
social evolution. However, many applications make it clear that inherent
non-linearities of natural systems need to be taken into account. One way of
introducing such non-linearities into evolutionary games is by the inclusion of
multiple players. An example is of social dilemmas, where group benefits could
e.g.\ increase less than linear with the number of cooperators. Such
multiplayer games can be introduced in all the fields where evolutionary game
theory is already well established. However, the inclusion of non-linearities
can help to advance the analysis of systems which are known to be complex, e.g.
in the case of non-Mendelian inheritance. We review the diachronic theory and
applications of multiplayer evolutionary games and present the current state of
the field. Our aim is a summary of the theoretical results from well-mixed
populations in infinite as well as finite populations. We also discuss examples
from three fields where the theory has been successfully applied, ecology,
social sciences and population genetics. In closing, we probe certain future
directions which can be explored using the complexity of multiplayer games
while preserving the promise of simplicity of evolutionary games.Comment: 14 pages, 2 figures, review pape
Quantum Replicator Dynamics
We propose quantization relationships which would let us describe and
solution problems originated by conflicting or cooperative behaviors among the
members of a system from the point of view of quantum mechanical interactions.
The quantum analogue of the replicator dynamics is the equation of evolution of
mixed states from quantum statistical mechanics. A system and all its members
will cooperate and rearrange its states to improve their present condition.
They strive to reach the best possible state for each of them which is also the
best possible state for the whole system. This led us to propose a quantum
equilibrium in which a system is stable only if it maximizes the welfare of the
collective above the welfare of the individual. If it is maximized the welfare
of the individual above the welfare of the collective the system gets unstable
and eventually it collapses.Comment: 10 page
The collapse of cooperation in evolving games
Game theory provides a quantitative framework for analyzing the behavior of
rational agents. The Iterated Prisoner's Dilemma in particular has become a
standard model for studying cooperation and cheating, with cooperation often
emerging as a robust outcome in evolving populations. Here we extend
evolutionary game theory by allowing players' strategies as well as their
payoffs to evolve in response to selection on heritable mutations. In nature,
many organisms engage in mutually beneficial interactions, and individuals may
seek to change the ratio of risk to reward for cooperation by altering the
resources they commit to cooperative interactions. To study this, we construct
a general framework for the co-evolution of strategies and payoffs in arbitrary
iterated games. We show that, as payoffs evolve, a trade-off between the
benefits and costs of cooperation precipitates a dramatic loss of cooperation
under the Iterated Prisoner's Dilemma; and eventually to evolution away from
the Prisoner's Dilemma altogether. The collapse of cooperation is so extreme
that the average payoff in a population may decline, even as the potential
payoff for mutual cooperation increases. Our work offers a new perspective on
the Prisoner's Dilemma and its predictions for cooperation in natural
populations; and it provides a general framework to understand the co-evolution
of strategies and payoffs in iterated interactions.Comment: 33 pages, 13 figure
Coveting thy neighbors fitness as a means to resolve social dilemmas
In spatial evolutionary games the fitness of each individual is traditionally
determined by the payoffs it obtains upon playing the game with its neighbors.
Since defection yields the highest individual benefits, the outlook for
cooperators is gloomy. While network reciprocity promotes collaborative
efforts, chances of averting the impending social decline are slim if the
temptation to defect is strong. It is therefore of interest to identify viable
mechanisms that provide additional support for the evolution of cooperation.
Inspired by the fact that the environment may be just as important as
inheritance for individual development, we introduce a simple switch that
allows a player to either keep its original payoff or use the average payoff of
all its neighbors. Depending on which payoff is higher, the influence of either
option can be tuned by means of a single parameter. We show that, in general,
taking into account the environment promotes cooperation. Yet coveting the
fitness of one's neighbors too strongly is not optimal. In fact, cooperation
thrives best only if the influence of payoffs obtained in the traditional way
is equal to that of the average payoff of the neighborhood. We present results
for the prisoner's dilemma and the snowdrift game, for different levels of
uncertainty governing the strategy adoption process, and for different
neighborhood sizes. Our approach outlines a viable route to increased levels of
cooperative behavior in structured populations, but one that requires a
thoughtful implementation.Comment: 10 two-column pages, 5 figures; accepted for publication in Journal
of Theoretical Biolog
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