31,473 research outputs found
Selfishness Level of Strategic Games
We introduce a new measure of the discrepancy in strategic games between the
social welfare in a Nash equilibrium and in a social optimum, that we call
selfishness level. It is the smallest fraction of the social welfare that needs
to be offered to each player to achieve that a social optimum is realized in a
pure Nash equilibrium. The selfishness level is unrelated to the price of
stability and the price of anarchy and is invariant under positive linear
transformations of the payoff functions. Also, it naturally applies to other
solution concepts and other forms of games.
We study the selfishness level of several well-known strategic games. This
allows us to quantify the implicit tension within a game between players'
individual interests and the impact of their decisions on the society as a
whole. Our analyses reveal that the selfishness level often provides a deeper
understanding of the characteristics of the underlying game that influence the
players' willingness to cooperate.
In particular, the selfishness level of finite ordinal potential games is
finite, while that of weakly acyclic games can be infinite. We derive explicit
bounds on the selfishness level of fair cost sharing games and linear
congestion games, which depend on specific parameters of the underlying game
but are independent of the number of players. Further, we show that the
selfishness level of the -players Prisoner's Dilemma is ,
where and are the benefit and cost for cooperation, respectively, that
of the -players public goods game is , where is
the public good multiplier, and that of the Traveler's Dilemma game is
, where is the bonus. Finally, the selfishness level of
Cournot competition (an example of an infinite ordinal potential game, Tragedy
of the Commons, and Bertrand competition is infinite.Comment: 34 page
Entanglement and Dynamic Stability of Nash Equilibria in a Symmetric Quantum Game
We study the evolutionary stability of Nash equilibria (NE) in a symmetric
quantum game played by the recently proposed scheme of applying `identity' and
`Pauli spin flip' operators on the initial state with classical probabilities.
We show that in this symmetric game dynamic stability of a NE can be changed
when the game changes its form, for example, from classical to quantum. It
happens even when the NE remains intact in both forms.Comment: Latex,no figure,submitted to Physics Letters
Sufficient conditions for stable equilibria
A refinement of the set of Nash equilibria that satisfies two assumptions is shown to select a subset that is stable in the sense defined by Kohlberg and Mertens. One assumption requires that a selected set is invariant to adjoining redundant strategies and the other is a strong version of backward induction. Backward induction is interpreted as the requirement that each player's strategy is sequentially rational and conditionally admissible at every information set in an extensive-form game with perfect recall, implemented here by requiring that the equilibrium is quasi-perfect. The strong version requires 'truly' quasi-perfection in that each strategy perturbation refines the selection to a quasi-perfect equilibrium in the set. An exact characterization of stable sets is provided for two-player games.Game theory, equilibrium selection, stability
Evolutionary stability in quantum games
In evolutionary game theory an Evolutionarily Stable Strategy (ESS) is a
refinement of the Nash equilibrium concept that is sometimes also recognized as
evolutionary stability. It is a game-theoretic model, well known to
mathematical biologists, that was found quite useful in the understanding of
evolutionary dynamics of a population. This chapter presents an analysis of
evolutionary stability in the emerging field of quantum games.Comment: 38 pages, 2 figures, contributed chapter to the book "Quantum Aspects
of Life" edited by D. Abbott, P. Davies and A. Pat
Coalition Formation in Political Games
We study the formation of a ruling coalition in political environments. Each individual is endowed with a level of political power. The ruling coalition consists of a subset of the individuals in the society and decides the distribution of resources. A ruling coalition needs to contain enough powerful members to win against any alternative coalition that may challenge it, and it needs to be self-enforcing, in the sense that none of its sub-coalitions should be able to secede and become the new ruling coalition. We first present an axiomatic approach that captures these notions and determines a (generically) unique ruling coalition. We then construct a simple dynamic game that encompasses these ideas and prove that the sequentially weakly dominant equilibria (and the Markovian trembling hand perfect equilibria) of this game coincide with the set of ruling coalitions of the axiomatic approach. We also show the equivalence of these notions to the core of a related non-transferable utility cooperative game. In all cases, the nature of the ruling coalition is determined by the power constraint, which requires that the ruling coalition be powerful enough, and by the enforcement constraint, which imposes that no sub-coalition of the ruling coalition that commands a majority is self-enforcing. The key insight that emerges from this characterization is that the coalition is made self-enforcing precisely by the failure of its winning sub-coalitions to be self-enforcing. This is most simply illustrated by the following simple finding: with a simple majority rule, while three-person (or larger) coalitions can be self-enforcing, two-person coalitions are generically not self-enforcing. Therefore, the reasoning in this paper suggests that three-person juntas or councils should be more common than two-person ones. In addition, we provide conditions under which the grand coalition will be the ruling coalition and conditions under which the most powerful individuals will not be included in the ruling coalition. We also use this framework to discuss endogenous party formation.
Coalition Formation in Political Games
We study the formation of a ruling coalition in political environments. Each individual is endowed with a level of political power. The ruling coalition consists of a subset of the individuals in the society and decides the distribution of resources. A ruling coalition needs to contain enough powerful members to win against any alternative coalition that may challenge it, and it needs to be self-enforcing, in the sense that none of its subcoalitions should be able to secede and become the new ruling coalition. We first present an axiomatic approach that captures these notions and determines a (generically) unique ruling coalition. We then construct a simple dynamic game that encompasses these ideas and prove that the sequentially weakly dominant equilibria (and the Markovian trembling hand perfect equilibria) of this game coincide with the set of ruling coalitions of the axiomatic approach. We also show the equivalence of these notions to the core of a related non-transferable utility cooperative game. In all cases, the nature of the ruling coalition is determined by the power constraint, which requires that the ruling coalition be powerful enough, and by the enforcement constraint, which imposes that no subcoalition of the ruling coalition that commands a majority is self-enforcing. The key insight that emerges from this characterization is that the coalition is made self-enforcing precisely by the failure of its winning subcoalitions to be self-enforcing. This is most simply illustrated by the following simple finding: with simple majority rule, while three-person (or larger) coalitions can be self-enforcing, two-person coalitions are generically not self-enforcing. Therefore, the reasoning in this paper suggests that three-person juntas or councils should be more common than two-person ones. In addition, we provide conditions under which the grand coalition will be the ruling coalition and conditions under which the most powerful individuals will not be included in the ruling coalition. We also use this framework to discuss endogenous party formation.Coalition Formation, Collective Choice, Cooperative Game Theory, Political Economy,Self-Enforcing Coalitions, Stability
Refined best-response correspondence and dynamics
We characterize the smallest faces of the polyhedron of strategy profiles that could possibly be made asymptotically stable under some reasonable deterministic dynamics. These faces are Kalai and Samet's (1984) persistent retracts and are spanned by Basu and Weibull's (1991) CURB sets based on a natural (and, in a well-defined sense, minimal) refinement of the best-reply correspondence. We show that such a correspondence satisfying basic properties such as existence, upper hemi-continuity, and convex-valuedness exists and is unique in most games. We introduce a notion of rationalizability based on this correspondence and its relation to other such concepts. We study its fixed-points and their relations to equilibrium refinements. We find, for instance, that a fixed point of the refined best reply correspondence in the agent normal form of any extensive form game constitutes a perfect Bayesian equilibrium, which is weak perfect Bayesian in every subgame. Finally, we study the index of its fixed point components.Evolutionary game theory, best response dynamics, CURB sets, persistent retracts, asymptotic stability, Nash equilibrium refinements, learning
Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games
Biodiversity is essential to the viability of ecological systems. Species
diversity in ecosystems is promoted by cyclic, non-hierarchical interactions
among competing populations. Such non-transitive relations lead to an evolution
with central features represented by the `rock-paper-scissors' game, where rock
crushes scissors, scissors cut paper, and paper wraps rock. In combination with
spatial dispersal of static populations, this type of competition results in
the stable coexistence of all species and the long-term maintenance of
biodiversity. However, population mobility is a central feature of real
ecosystems: animals migrate, bacteria run and tumble. Here, we observe a
critical influence of mobility on species diversity. When mobility exceeds a
certain value, biodiversity is jeopardized and lost. In contrast, below this
critical threshold all subpopulations coexist and an entanglement of travelling
spiral waves forms in the course of temporal evolution. We establish that this
phenomenon is robust, it does not depend on the details of cyclic competition
or spatial environment. These findings have important implications for
maintenance and evolution of ecological systems and are relevant for the
formation and propagation of patterns in excitable media, such as chemical
kinetics or epidemic outbreaks.Comment: Final submitted version; the printed version can be found at
http://dx.doi.org/10.1038/nature06095 Supplementary movies are available at
http://www.theorie.physik.uni-muenchen.de/lsfrey/images_content/movie1.AVI
and
http://www.theorie.physik.uni-muenchen.de/lsfrey/images_content/movie2.AV
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