Using EvolutionaryGames (2024)

Abstract

This document gives a few use cases for the EvolutionaryGamespackage. EvolutionaryGames provides basic concepts of evolutionary gametheory, like e.g.finding evolutionary stable strategies and computingand drawing evolutionarily stable sets as well as phase diagrams forvarious evolutionary dynamics for single-population games with two,three and four different phenotypes.

Keywords: Evolutionary game theory, evolutionary stablestrategies, evolutionarily stable sets, evolutionary game dynamics.

Motivation

The field of evolutionary game theory is concerned with interactionsamong large populations of strategically interacting agents. Theseagents may adapt their behaviour according to current payoffopportunities. Evolutionary game theory originates from biology (J. M. Smith and Price 1973), but nowadays hasbecome an established tool in other disciplines, in particular ineconomics (see e.g. (Sandholm 2010)) andcomputer science (see e.g. (Suri2007)).

The purpose of our package EvolutionaryGames is to enhance the Recosystem by tools to plot and present evolutionary game dynamics in aboth informative and attractive way. One similar tool in the publicdomain is William Sandholm’s package dynamo (Franchetti and Sandholm 2013) written inMathematica. Dynamo (Franchetti and Sandholm2013) frequently served as an inspiration for our implementation.Still, even though dynamo (Franchetti andSandholm 2013) itself is free, Mathematica is proprietarysoftware and it is nowhere near as widely used as R. Also, the packagestructures in R and the widespread availability of packages via CRANmake it much easier for the user to employ parts of our packageEvolutionaryGames in his or her own software.

The following very brief presentation of concepts in evolutionarygame theory and how EvolutionaryGames addresses them mainly follows thetwo books by Weibull (Weibull 1997) andSandholm (Sandholm 2010). Furthermore, ourvignette provides detailed references to the respective originalarticles. Our point is to show how various concepts from evolutionarygame theory can be computed via the package EvolutionaryGames and whereto find the conceptual, technical and mathematical details. In otherwords, we would like to stress that this little document is by no meansintended to serve as an introduction to the fascinating field ofevolutionary game theory itself. Again, for the latter the reader isreferred to the the two books by Weibull (Weibull1997) and Sandholm (Sandholm 2010).For an in-depth mathematical discussion we also recommend the book byHofbauer and Sigmund (Hofbauer and Sigmund1998).

The package EvolutionaryGames focusses on single-population gameswith two, three or four phenotypes. (The first author hopes to addressmulti-population games in a separate package based on EvolutionaryGamesin the future.) For single-population games we regard EvolutionaryGamesas a powerful and attractive alternative to William Sandholm’sMathematica package dynamo (Franchetti andSandholm 2013).

Computing Evolutionary Stable Strategies (ESS)

The concept of evolutionary stable strategies (ESS) was firstproposed by Maynard Smith and Price in (J. M.Smith and Price 1973). An incumbent strategy \(x\) is said to be evolutionary stableif

• it is either a unique best reply to itself or
  
• in case there exists an alternative best reply \(y\) to \(x\), then \(x\) is a better reply to this mutantstrategy \(y\) than \(y\) is to itself.

If \(x\) is a strict Nashequilibrium, then \(x\) must also be anESS.
Using our package EvolutionaryGames one can compute ESS for both \(2\) x \(2\) and \(3\) x \(3\) symmetric matrix games. The functionESS.R receives three input arguments:

• A : A matrix specifying the symmetric game
• strategies: A vector of strings of length \(2\) or \(3\) specifying the names of allstrategies
• floats: A logical value that handles numberrepresentation. If floats is set to TRUE (default), afloating-point number will be used for the output, otherwise the outputwill be specified as fractions.

Let us first take a look at a classical Hawk-Dove game.

library(EvolutionaryGames)ESS(matrix(c(-1, 4, 0, 2), 2, byrow=TRUE), c("Hawk", "Dove"))

## Hawk Dove## [1,] 0.6666667 0.3333333

ESS(matrix(c(-1, 4, 0, 2), 2, byrow=TRUE), c("Hawk", "Dove"), FALSE)

## Hawk Dove## [1,] 2/3 1/3

It is well known that there are games which do not possess an ESS. Aclassical example is the game Rock-Scissors-Paper. In such a case ourfunction ESS.R will return NULL.

library(EvolutionaryGames)(A <- matrix(c(1, 2, 0, 0, 1, 2, 2, 0, 1), 3, byrow=T))

## [,1] [,2] [,3]## [1,] 1 2 0## [2,] 0 1 2## [3,] 2 0 1

ESS(A, c("Rock", "Scissors", "Paper"))

## NULL

Computing and drawing evolutionarily stable sets

The concept of evolutionarily stable sets was first discussed by B.Thomas in (Thomas 1985). A very readableintroduction can also be found in the book by Weibull (Weibull 1997), section 2.4.1.
An evolutionarily stable set is a nonempty closed set of symmetric Nashequilibrium strategies \(X\) such thateach strategy \(x \in X\) earns atleast the same payoff against any nearby alternative best reply \(y\) as \(y\) earns against itself with equal payoffslimited to the case \(y \in X\). Notethat any evolutionary stable strategy (ESS) constitutes anevolutionarily stable set and that the union of evolutionarily stablesets is again an evolutionarily stable set. See the book by Weibull(Weibull 1997), section 2.4.1, for furtherdetails.
Evolutionarily stable sets are not easy to compute and to plot. Thepackage EvolutionaryGames computes evolutionarily stable sets of a gamewith two players and three strategies in the case that the game has anevolutionary stable strategy (ESS). If the two player three strategygame has no ESS, then the code returns a message stating that ouralgorithm cannot calculate evolutionarily stable sets for models that donot have a proper ESS. The authors are very well aware that there aregames having evolutionarily stable sets but no proper ESS. Still, as ourpackage is not devoted to finding all symmetric Nash equilibria of agame and as there currently is no package for this task on CRAN, wedecided only to handle the case of games with two players and threestrategies possessing at least one proper ESS. Within our algorithm, weneed a proper ESS as a starting point for our computations. The authorsfeel that the possibility of computing and drawing evolutionarily stablesets in such cases is precious for teaching (and research) purposes.However, note that computing and drawing evolutionarily stable sets istime consuming and is clearly the most elaborate task currentlyperformed by the package EvolutionaryGames. Finally, here is an exampleof an evolutionarily stable set together with a plot including sampletrajectories (– the example itself is taken from the book (Broom and Rychtár 2013), p.52):

library(EvolutionaryGames)A <- matrix(c(-2, 5, 10/9, 0, 5/2, 10/9, -10/9, 35/9, 10/9), 3, byrow=TRUE)strategies <- c("Hawk", "Dove", "Mixed ESS")ESS(A, strategies)

## Hawk Dove Mixed ESS## [1,] 0 0 1

ESset(A, strategies)

## [,1] [,2] [,3]## [1,] 0.0000000 0.0000000 1## [2,] 0.5555556 0.4444444 0

The various evolutionary dynamics available inEvolutionaryGames

The basic game-theoretic model of biological natural selection is thereplicator dynamic (Taylor and Jonker1978). Still, various alternative dynamics have been proposed andinvestigated for different applications. Our package currently focusseson continuous dynamics only. The following itemization states whichevolutionary dynamics are currently available in our packageEvolutionaryGames:

• Replicator.R : The standard replicator dynamic(Taylor and Jonker 1978)
• BNN.R : The Brown-von Neumann-Nash dynamic (Brown and Neumann 1950)
• BR.R : The Best Reponse dynamic by Gilboa andMatsui (Gilboa and Matsui 1991)
• ILogit.R : The imitative Logit dynamic by Weibull(Weibull 1997). It requires the parametereta.
• Logit.R : The Logit dynamic by Fudenberg and Levine(Fudenberg and Levine 1998). It requiresthe parameter eta.
• MSReplicator.R : The Maynard-Smith dynamic (J. M. Smith 1982)
• Smith.R : The Smith dynamic (M. J. Smith 1984)

Drawing phase diagrams for two, three and four strategies

library(EvolutionaryGames)A <- matrix(c(-1, 4, 0, 2), 2, byrow=TRUE)phaseDiagram2S(A, Replicator, strategies = c("Hawk", "Dove"))

library(EvolutionaryGames)A <- matrix(c(1, 2, 0, 0, 1, 2, 2, 0, 1), 3, byrow=T)state <- matrix(c(0.7, 0.2, 0.1), 1, 3, byrow=TRUE)RSP <- c("Rock", "Scissors", "Paper")phaseDiagram3S(A, Replicator, NULL, state, FALSE, FALSE, strategies = RSP)

library(EvolutionaryGames)A <- matrix(c(1, 2, 0, 0, 1, 2, 2, 0, 1), 3, byrow=T)state <- matrix(c(0.7, 0.2, 0.1), 1, 3, byrow=TRUE)RSP <- c("Rock", "Scissors", "Paper")phaseDiagram3S(A, Replicator, NULL, state, TRUE, FALSE, strategies = RSP)

library(EvolutionaryGames)A <- matrix(c(1, 2, 0, 0, 1, 2, 2, 0, 1), 3, byrow=T)state <- matrix(c(0.7, 0.2, 0.1), 1, 3, byrow=TRUE)phaseDiagram3S(A, Replicator, NULL, state, TRUE, TRUE, strategies = RSP)

library(EvolutionaryGames)A <- matrix(c(5, -9, 6, 8, 20, 1, 2, -18, -14, 0, 2, 20, 13, 0, 4, -13), 4, 4, byrow=TRUE)state <- c(0.6, 0.15, 0.1, 0.15)phaseDiagram4S(A, Smith, NULL, state, noRGL=TRUE)

How to use our package and write your own dynamics

EvolutionaryGames offers you to create your own dynamics. Inparticular, it is easy to write your own continuous dynamics. First ofall, a dynamic is nothing other than a function that is passed as aparameter to the corresponding function for creating phase diagrams. Thefollowing code fragment shows you the minimum necessary structure of anarbitrary dynamic:

MyDynamic <- function (time, state, parameters) { #...  return(list (dX))}

Any dynamic requires the parameters time,state and parameters. While the parametertime is used internally by deSolve to solve the initialvalue problem, the other two parameters state andparameters are used to specify the desired dynamic and areavailable as numeric vectors. In this context, state standsfor the desired initial state under which the model is to be simulatedand parameters contains further parameters, such as thesymmetric matrix specifying the game and, depending on the dynamic,noise levels or similar parameters.

As can be seen in the example, the return value of a specifieddynamic has to be a numerical list. Each component represents thecorresponding rate of change of a phenotype under the respectivedynamic.

We now show how to implement a very well known dynamic, thereplicator dynamics. Our function definition is as follows:

Replicator <- function (time, state, parameters) { #...  return(list (dX))}

The above game and an initial value is passed as a parameter to afunction for the generation of phase diagrams and made retrievablewithin the dynamic. A small limitation, however, is that our matrixneeds to be converted into a vector due to certain constraints in ourinternal usage of the package deSolve. We recommend that you firsttransfer this vector back into a matrix and maintain the original game.This can be done as follows:

Replicator <- function (time, state, parameters) { a <- parameters states <- sqrt(length(a)) A <- matrix(a, states, byrow = TRUE) A <- t(A) # original symmetric game  return(list (dX))}

We now come to the actual part of the implementation. We firstcalculate the rate of change of each phenotype depending on the otherphenotypes. Immediately afterwards, we calculate the average fitness ofeach phenotype and then set up the actual replicator dynamics:

Replicator <- function(time, state, parameters) { a <- parameters states <- sqrt(length(a)) A <- matrix(a, states, byrow = TRUE) A <- t(A) dX <- c() for(i in 1:states) { dX[i] <- sum(state * A[i, ]) } avgFitness <- sum(dX * state) for(i in 1:states) { dX[i] <- state[i] * (dX[i] - avgFitness) } return(list(dX))}

Our dynamics is now applicable within the predefined functions forgenerating phase diagrams.

Packages used in EvolutionaryGames

We made use of various packages of the CRAN ecosystem. In particular,it was of paramount importance not to write any differential equationsolvers ourselves, but to make use of an established solver,instead.

Literature