Evolutionary stability and tenable strategy blocks

It is well known that the rock-paper-scissors game has no pure saddle point. We show that this holds more generally: A symmetric two-player zero-sum game has a pure saddle point if and only if it is not a generalized rock-paper-scissors game. Moreover, we show that every finite symmetric quasiconcave two-player zero-sum game has a pure saddle point. Further sufficient conditions for existence are provided. We apply our theory to a rich collection of examples by noting that the class of symmetric two-player zero-sum games coincides with the class of relative payoff games associated with symmetric two-player games. This allows us to derive results on the existence of a finite population evolutionary stable strategies.

It is well known that the rock-paper-scissors game has no pure saddle point. We show that this holds more generally: A symmetric two-player zero-sum game has a pure saddle point if and only if it is not a generalized rock-paper-scissors game. Moreover, we show that every nite symmetric quasiconcave two-player zero-sum game has a pure saddle point. Further sucient conditions for existence are provided. We apply our theory to a rich collection of examples by noting that the class of symmetric two-player zero-sum games coincides with the class of relative payo games associated with symmetric two-player games. This allows us to derive results on the existence of a nite population evolutionary stable strategies.

We observe that a symmetric two-player zero-sum game has a pure strategy equilibrium if and only if it is not a generalized rock-paper-scissors matrix. Moreover, we show that every finite symmetric quasiconcave two-player zero-sum game has a pure equilibrium. Further sufficient conditions for existence are provided. Our findings extend to general two-player zero-sum games using the symmetrization of zero-sum games due to von Neumann. We point out that the class of symmetric two-player zero-sum games coincides with the class of relative payoff games associated with symmetric two-player games. This allows us to derive results on the existence of finite population evolutionary stable strategies.

The continuously stable strategy (CSS) concept, originally developed as an intuitive static condition to predict the dynamic stability of a monomorphic population, is shown to be closely related to classical game-theoretic dominance criteria when applied to continuous strategy spaces. Specifically, for symmetric and non symmetric two-player games, a CSS in the interior of the continuous strategy space is equivalent to neighborhood half-superiority which, for a symmetric game, is connected to the half-dominance and/or risk dominance concepts. For non symmetric games where both players have a one-dimensional continuous strategy space, an interior CSS is shown to be given by a local version of dominance solvability (called neighborhood dominance solvable). Finally, the CSS and half-superiority concepts are applied to points in the bargaining set of Nash bargaining problems.

We provide a classification of symmetric three-player games with two strategies and investigate evolutionary and asymptotic stability (in the replicator dynamics) of their Nash equilibria. We discuss similarities and differences between two-player and multi-player games. In particular, we construct examples which exhibit a novel behavior not found in two-player games.

The concept of evolutionarily stable strategies (ESS) has been central to applications of game theory in evolutionary biology, and it has also had an influence on the modern development of game theory. A regular ESS is an important refinement the ESS concept. Although there is a substantial literature on computing evolutionarily stable strategies, the precise computational complexity of determining the existence of an ESS in a symmetric two-player strategic form game has remained open, though it has been speculated that the problem is $$\mathsf{NP}$$ -hard. In this paper we show that determining the existence of an ESS is both $${\mathsf{NP}}$$ -hard and $${\mathsf{coNP}}$$ -hard, and that the problem is contained in $$\Sigma_{2}^{\rm p}$$ , the second level of the polynomial time hierarchy. We also show that determining the existence of a regular ESS is indeed $${\mathsf{NP}}$$ -complete. Our upper bounds also yield algorithms for computing a (regular) ESS, if one exists, with the same complexities.

One of the most unusual applications of game theory in recent years has been to model how animal behavior evolves from generation to generation. Maynard-Smith [1974] modelled conflicts among animals in their mating behavior, for example, in terms of game theory by introducing the concepts of “strategy” and “fitness” (i.e., payoffs in other terminology) that are considered to be “thought and used” by animals. He defined evolutionary stable strategies(ESS) and showed that animals use ESS in order to secure revolution of their species (this explains why snakes wrestle rather than bite). Now we understand that some examples are observed in the animal world that can be thought of as n-player game rather than two-player game. (Thomas [1984; pp. 186–187]). The aim of the present paper is to extend the two-player ESS originally defined by Maynard-Smith to the n-player ESS, and to apply this concept to some types of n-bidder auction. In Section 2 n-player ESS arc defined by extending the concept of two-player ESS to the n-player case. Three types of n-bidder auctions, i.e., second-bid, first-bid, and sad-loser auctions (in Section 3) and a guessing game, as a variant of auction, (in Section 4), are discussed and a unique mixed-strategy ESS (if it exists) is derived for each of these four examples in auction.

In this paper, we consider the concepts of evolutionarily stable strategy (ESS), neighborhood invader strategy (NIS) and global invader strategy (GIS) in single species with frequency-dependent interactions. We find some general relationships among the three concepts in matrix games. The main conclusion is that ESS and NIS are equivalent to each other and are both equivalent to local superiority; a strategy with global superiority must be a GIS; a GIS may not be equivalent to its global superiority in games with more than two players; and in any two-player matrix game a GIS is just equivalent to its global superiority. In two-player games, globally asymptotic stability in the replicator dynamics has also been shown. Equivalent conditions for the three concepts stated by payoff comparisons are given and are applied to examples involved.

Evolutionary game theory is used to predict the behavior of individuals in populations (either of humans or other species) without relying on a detailed description of how these behaviors evolve over time (e.g., the replicator equation or the best response dynamics). For instance, if these behaviors correspond to a population state that satisfies the static payoff‐comparison conditions of an evolutionarily stable strategy (ESS), then there is typically dynamic (i.e., evolutionary) stability at this state. We begin with a thorough summary of the evolutionary game theory perspective, when there is a finite set of (pure) strategies, for a symmetric two‐player game in either normal or extensive form. The article then briefly discusses generalizations of evolutionary game theory, ESS, and evolutionary stability to several other classes of games. These include symmetric population games where payoffs to pure strategies are nonlinear functions of the current population state as well as asymmetric games where players are assigned different roles (e.g., two‐role bimatrix games). Although terminology borrowed from evolutionary game theory applied to behaviors of individuals in biological species is used throughout, the concepts introduced are equally relevant in games modeling human behavior.

This paper investigates a symmetric two-player game with canyon-shaped payoffs, in which a player’s payoff function is smooth and concave above and below the diagonal, but not differentiable on the diagonal. We demonstrate that there exists a first-mover advantage when the two players move sequentially and a player’s preference to the opponent’s choice is monotonic and identical between a higher strategy player and a lower strategy player. We also show that our symmetric two-player game may yield the first-mover advantage outcome in an endogenous timing game with observable delay.

Finding a Nash equilibrium in a two-player normal form game (2-Nash) is one of the most extensively studied problems within mathematical economics as well as theoretical computer science. Such a game can be represented by two payoff matrices $A$ and $B$, one for each player. $2$-Nash is PPAD-complete in general, while in the case of zero-sum games ($B=-A$) the problem reduces to LP and hence is in P. Extending the notion of zero-sum, in 2005, Kannan and Theobald [Econom. Theory, 42 (2010), pp. 157--174] defined the rank of game $(A, B)$ as $rank(A+B)$, e.g., rank-0 are zero-sum games. They gave an FPTAS for constant rank games and asked if there exists a polynomial time algorithm to compute an exact Nash equilibrium (NE). Adsul et al. [Proceedings of the ACM Symposium on the Theory of Computing, 2011, pp. 195--204] answered this question affirmatively for rank-1 games, leaving rank-$2$ and beyond unresolved. In this paper we show that NE computation in games with rank $\ge3$ is PPAD-hard, settling a decade long open problem. Interestingly, this is the first instance that a problem with an FPTAS turns out to be PPAD-hard. Our reduction bypasses graphical games and game gadgets and provides a simpler proof of PPAD-hardness for NE computation in two-player games. In addition, we show the following: (i) an equivalence between two-dimensional Linear-FIXP and PPAD, improving on a result of Etessami and Yannakakis [SIAM J. Comput., 39 (2010), pp. 2531--2597] on equivalence between Linear-FIXP and PPAD; (ii) NE computation in a two-player game with convex set of Nash equilibria is as hard as solving a simple stochastic game [A. Condon, Inform. and Comput., 96 (1992), pp. 203--224]; (iii) computing a symmetric NE of a symmetric two-player game with rank $\ge 6$ is PPAD-hard; (iv) computing a ${1}/{poly(n)}$-approximate fixed-point of a piecewise-linear function is PPAD-hard."

Evolutionary dynamics over continuous action spaces for population games that arise from symmetric two-player games

We analyze the main dynamical properties of the evolutionarily stable strategy (ℰ𝒮𝒮) for asymmetric two-population games of finite size and its corresponding replicator dynamics. We introduce a definition of ℰ𝒮𝒮 for two-population asymmetric games and a method of symmetrizing such an asymmetric game. We show that every strategy profile of the asymmetric game corresponds to a strategy in the symmetric game, and that every Nash equilibrium (𝒩ℰ) of the asymmetric game corresponds to a (symmetric) 𝒩ℰ of the symmetric version game. We study the (standard) replicator dynamics for the asymmetric game and we define the corresponding (non-standard) dynamics of the symmetric game. We claim that the relationship between 𝒩ℰ, ℰ𝒮𝒮 and the stationary states (𝒮𝒮) of the dynamical system for the asymmetric game can be studied by analyzing the dynamics of the symmetric game.KeywordsNash EquilibriumMixed StrategyPure StrategyEvolutionary GameStable StrategyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We present examples of symmetric two-player games admitting a unique Nash equilibrium and a unique evolutionarily stable strategy (ESS) and such that the ESS payoff is strictly higher than the Nash payoff. In this sense, we show that spiteful behavior can make everybody better off.

Static stability of equilibrium in strategic games differs from dynamic stability in not being linked to any particular dynamical system. In other words, it does not make any assumptions about off-equilibrium behavior. Examples of static notions of stability include evolutionarily stable strategy (ESS) and continuously stable strategy (CSS), both of which are meaningful or justifiable only for particular classes of games, namely, symmetric multilinear games or symmetric games with a unidimensional strategy space, respectively. This paper presents a general notion of local static stability, of which the above two are essentially special cases. It is applicable to virtually all n-person strategic games, both symmetric and asymmetric, with non-discrete strategy spaces.