1,157 research outputs found
Linear algebraic structure of zero-determinant strategies in repeated games
Zero-determinant (ZD) strategies, a recently found novel class of strategies
in repeated games, has attracted much attention in evolutionary game theory. A
ZD strategy unilaterally enforces a linear relation between average payoffs of
players. Although existence and evolutional stability of ZD strategies have
been studied in simple games, their mathematical properties have not been
well-known yet. For example, what happens when more than one players employ ZD
strategies have not been clarified. In this paper, we provide a general
framework for investigating situations where more than one players employ ZD
strategies in terms of linear algebra. First, we theoretically prove that a set
of linear relations of average payoffs enforced by ZD strategies always has
solutions, which implies that incompatible linear relations are impossible.
Second, we prove that linear payoff relations are independent of each other
under some conditions. These results hold for general games with public
monitoring including perfect-monitoring games. Furthermore, we provide a simple
example of a two-player game in which one player can simultaneously enforce two
linear relations, that is, simultaneously control her and her opponent's
average payoffs. All of these results elucidate general mathematical properties
of ZD strategies.Comment: 19 pages, 2 figure
Evolutionary consequences of behavioral diversity
Iterated games provide a framework to describe social interactions among
groups of individuals. Recent work stimulated by the discovery of
"zero-determinant" strategies has rapidly expanded our ability to analyze such
interactions. This body of work has primarily focused on games in which players
face a simple binary choice, to "cooperate" or "defect". Real individuals,
however, often exhibit behavioral diversity, varying their input to a social
interaction both qualitatively and quantitatively. Here we explore how access
to a greater diversity of behavioral choices impacts the evolution of social
dynamics in finite populations. We show that, in public goods games, some
two-choice strategies can nonetheless resist invasion by all possible
multi-choice invaders, even while engaging in relatively little punishment. We
also show that access to greater behavioral choice results in more "rugged "
fitness landscapes, with populations able to stabilize cooperation at multiple
levels of investment, such that choice facilitates cooperation when returns on
investments are low, but hinders cooperation when returns on investments are
high. Finally, we analyze iterated rock-paper-scissors games, whose
non-transitive payoff structure means unilateral control is difficult and
zero-determinant strategies do not exist in general. Despite this, we find that
a large portion of multi-choice strategies can invade and resist invasion by
strategies that lack behavioral diversity -- so that even well-mixed
populations will tend to evolve behavioral diversity.Comment: 26 pages, 4 figure
Small games and long memories promote cooperation
Complex social behaviors lie at the heart of many of the challenges facing
evolutionary biology, sociology, economics, and beyond. For evolutionary
biologists in particular the question is often how such behaviors can arise
\textit{de novo} in a simple evolving system. How can group behaviors such as
collective action, or decision making that accounts for memories of past
experience, emerge and persist? Evolutionary game theory provides a framework
for formalizing these questions and admitting them to rigorous study. Here we
develop such a framework to study the evolution of sustained collective action
in multi-player public-goods games, in which players have arbitrarily long
memories of prior rounds of play and can react to their experience in an
arbitrary way. To study this problem we construct a coordinate system for
memory- strategies in iterated -player games that permits us to
characterize all the cooperative strategies that resist invasion by any mutant
strategy, and thus stabilize cooperative behavior. We show that while larger
games inevitably make cooperation harder to evolve, there nevertheless always
exists a positive volume of strategies that stabilize cooperation provided the
population size is large enough. We also show that, when games are small,
longer-memory strategies make cooperation easier to evolve, by increasing the
number of ways to stabilize cooperation. Finally we explore the co-evolution of
behavior and memory capacity, and we find that longer-memory strategies tend to
evolve in small games, which in turn drives the evolution of cooperation even
when the benefits for cooperation are low
Evolutionary stable strategies in networked games: the influence of topology
Evolutionary game theory is used to model the evolution of competing
strategies in a population of players. Evolutionary stability of a strategy is
a dynamic equilibrium, in which any competing mutated strategy would be wiped
out from a population. If a strategy is weak evolutionarily stable, the
competing strategy may manage to survive within the network. Understanding the
network-related factors that affect the evolutionary stability of a strategy
would be critical in making accurate predictions about the behaviour of a
strategy in a real-world strategic decision making environment. In this work,
we evaluate the effect of network topology on the evolutionary stability of a
strategy. We focus on two well-known strategies known as the Zero-determinant
strategy and the Pavlov strategy. Zero-determinant strategies have been shown
to be evolutionarily unstable in a well-mixed population of players. We
identify that the Zero-determinant strategy may survive, and may even dominate
in a population of players connected through a non-homogeneous network. We
introduce the concept of `topological stability' to denote this phenomenon. We
argue that not only the network topology, but also the evolutionary process
applied and the initial distribution of strategies are critical in determining
the evolutionary stability of strategies. Further, we observe that topological
stability could affect other well-known strategies as well, such as the general
cooperator strategy and the cooperator strategy. Our observations suggest that
the variation of evolutionary stability due to topological stability of
strategies may be more prevalent in the social context of strategic evolution,
in comparison to the biological context
Unexploitable games and unbeatable strategies
Imitation is a simple behavior which uses successful actions of others in
order to handle one's tasks. Because success of imitation generally depends on
whether profit of an imitating agent coincides with those of other agents or
not, game theory is suitable for specifying situations where imitation can be
successful. One of the concepts describing successfulness of imitation in
repeated two-player symmetric games is unbeatability. For infinitely repeated
two-player symmetric games, a necessary and sufficient condition for some
imitation strategy to be unbeatable was specified. However, situations where
imitation can be unbeatable in multi-player games are still not clear. In order
to analyze successfulness of imitation in multi-player situations, here we
introduce a class of totally symmetric games called unexploitable games, which
is a natural extension of two-player symmetric games without exploitation
cycles. We then prove that, for infinitely repeated unexploitable games, there
exist unbeatable imitation strategies. Furthermore, we also prove that, for
infinitely repeated non-trivial unexploitable games, there exist unbeatable
zero-determinant strategies, which unilaterally enforce some relationships on
payoffs of players. These claims are demonstrated in the public goods game,
which is the simplest unexploitable game. These results show that there are
situations where imitation can be unbeatable even in multi-player games.Comment: 6 page
- …