Replicator Equation: Applications Revisited

TL;DR

This paper revisits the replicator equation, a fundamental model in evolutionary game theory, highlighting its diverse applications across biology, social sciences, machine learning, and computer vision.

Contribution

It provides an overview of the replicator equation's applications beyond biology, emphasizing its relevance in modern computational fields.

Findings

Replicator equation models stable Nash Equilibria and ESS.

It has broad applications in machine learning and computer vision.

The paper discusses the equation's expanded role in social and computational sciences.

Abstract

The replicator equation is a simple model of evolution that leads to stable form of Nash Equilibrium, Evolutionary Stable Strategy (ESS). It has been studied in connection with Evolutionary Game Theory and was originally developed for symmetric games. Beyond its first emphasis in biological use, evolutionary game theory has been expanded well beyond in social studies for behavioral analysis, in machine learning, computer vision and others. Its several applications in the fields of machine learning and computer vision has drawn my attention which is the reason to write this extended abstract

Full text

11institutetext: DAIS, Università Ca’ Foscari di Venezia

via Torino 155, 30172 Venezia Mestre, Italy

11email: tinsae.dulecha@unive.it

Replicator Equation: Applications Revisited

Tinsae G.Dulecha

The replicator equation is a simple model of evolution that leads to stable form of Nash Equilibrium, Evolutionary Stable Strategy (ESS). No individuals get an incentive unilaterally deviating from the equilibrium. It has been studied in connection with Evolutionary Game Theory, a theory John Maynard Smith and George R. Price developed to predict biological reproductive success of populations. Replicator equation was originally developed for symmetric games, games whose payoff matrix is skew-symmetric and in general tells us, in a game, how individuals or populations change their strategy over time based on the payoff matrix of the game. Consider a large population of players where each player is assigned a strategy and players cannot choose their strategies which means rationality and consciousness don’t enter the picture (players play based on a pre-assigned set of strategies). Evolutionary game theory assumes a scenario where a non-rational pairs of players, which play based on a pre-assigned set of strategies, repeatedly drawn from a large population plays a symmetric two-player game which drives the strategies with lower payoff to extinction . Since players interacts with another randomly chosen player in the population, a player’s expected payoff is determined by the payoff matrix and the proportion of each strategy in the population. The limiting behavior of the replicator dynamics (i.e., the evolutionary outcome) are the Evolutionary Stable Strategies, a NE with additional stability properties.

Let $x_{i}(t)$ is the proportion of the population which plays strategy $i$ $\in N$(set of strategies) at time $t$. The state of the population at any given instant is then given by $\mathbf{x}_{i}(t)$ = [$x_{1},x_{2},...,x_{n}$]′ where ′ denotes transposition and $n$ refers the size of available pure strategies, $|N|$. With $A$ be the $n\times n$ payoff matrix ( biologically measured as Darwinian fitness i.e reproductive success ), the payoff for the $i^{th}$-strategist, assuming the opponent is playing the $j^{th}$ strategy, is $(a_{ij})$, the corresponding $i^{th}$ row and the $j^{th}$ column of $A$. If the population is in state x, the expected payoff earned by an the $i^{th}$-strategist is $(A\mathbf{x})_{i}$ while the mean payoff over the whole population is $\mathbf{x}^{\prime}A\mathbf{x}$. The growth rate of the population is then computed as the product of the current frequency of strategy with the own fitness relative to the average (the difference in the expected and the average payoff gives us the per capita rate). The game, which is assumed to be played over and over, generation after generation, changes the state of the population over time until equilibrium is reached, a point $\mathbf{x}$ is said to be a stationary (or equilibrium) point of the dynamical system if $\dot{x}$ = 0 where the dot implies derivative. Different formalization of this selection process have been proposed in evolutionary game theory where replicator dynamics is one of the well knowns.

Beyond its first emphasis in biological use, evolutionary game theory has been expanded well beyond in social studies for behavioral analysis, in machine learning, computer vision and others. Its application in the fields of computer science has drawn my attention which is the reason to write this extended abstract.

Torsello *et al. *has shown that dominant sets, a well known clustering notion which generalizes the maximal clique problem to edge weighted graphs, can be bijectively related to Evolutionary Stable Strategies (ESS) [13]. Since the formalization of clustering problem as a game, replicator dynamics has been used to address different problems [13]. Players simultaneously select an object to cluster and receive a payoff proportional to the similarity of the selected objects. Since clusters are sets of objects that are highly similar, the competition induced by the game forces the players to coordinate by selecting objects from the same cluster. Indeed, by doing so, both players are able to maximize their expected payoff. The clustering game is a non-cooperative game between two-players, where the objects to cluster form the set of strategies, while the affinity matrix provides the players’ payoffs. The dynamics is also able to solve multiple affinities by extending the clustering game, which has a single payoff, to a multi-payoff game [10]. Since the introduction of the connection, replicator equation helps dominant set formulation be a robust clustering approach with many interesting and powerful properties such as: it makes no assumption on the underlying data representation, it makes no assumption on the structure of the affinity matrix, being it able to work with asymmetric and even negative similarity functions alike; it does not require a priori knowledge on the number of clusters; able to solve one-class clustering problems (figure ground separation); it allows extracting overlapping clusters [14]; it generalizes naturally to hypergraph clustering problems, in which the clustering game is played by more than two players [11]. The dominant sets formulation, using replicator equation, have proven to be an effective tool for graph-based clustering and have found applications in a variety of different domains, such as bioinformatics [5], computer vision [15], image processing [18],[2], group detection [8][17][16], security and video surveillance [7], [1], [12], [6] etc. Very recently, Eyasu *et al. *, in their two consecutive works [10, 21], generalized the dominant set formulation. The generalized version, using replicator dynamics as a solver, can be applied in different fields of studies such as in computer vision, biomedical analysis, human behavior and social network analysis and others.

The notion of clustering using replicator equation has also be used in [4] where the core idea is to combine an effective diffusion process, based on iteratively approaching evolutionary stable strategies. Similarly, Xingwei *et al. *used it for dense neighbor selection for affinity learning with diffusion on tensor product graph. Few of the most efficient and most recent computer vision applications that uses replicator equations include: Retrieval[19] where replicator is used to different size dense neighbors which improves the quality of affinity propagation; tracking [12] which uses replicator as a solver; interactive image segmentation where the segmentation is guided by the user provided information [21]; large scale image geo-localization [22]

Morteza showed, in his adaptive trajectory analysis of replicator dynamics, how much effective replicator dynamics is for data clustering and structure identification [3]. Extraction of dense subgraphs using game dynamics is done in [9], and in [20] it has been shown that replicator have no difficulty in extracting a few “natural” clusters from heavy background noise. Moreover, Eyasu *et al. *proposed effective replicator based algorithm for simultaneous clustering and outlier detections [23].