A Game Theoretic Investigation of Selection Methods in Two Population Coevolution

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1 A Game Theoretic Investigation of Selection Methods in Two Poulation Coevolution Sevan G. Ficici Division of Engineering and Alied Sciences Harvard University Cambridge, Massachusetts 238 USA ABSTRACT We examine the dynamical and game-theoretic roerties of several selection methods in the context of twooulation coevolution. The methods we examine are fitness-roortional, linear rank, truncation, and (µ, λ)-es selection. We use simle symmetric variable-sum games in an evolutionary game-theoretic framework. Our results indicate that linear rank, truncation, and (µ, λ)-es selection are somewhat better-behaved in a two-oulation setting than in the one-oulation case analyzed by Ficici et al. [4]. These alternative selection methods maintain the Nasheuilibrium attractors found in roortional selection, but also add non-nash attractors as well as regions of hasesace that lead to cyclic dynamics. Thus, these alternative selection methods do not roerly imlement the Nasheuilibrium solution concet. Categories and Subject Descritors I.2.8 [Comuting Methodologies]: Artificial Intelligence Problem Solving, Control Methods, and Search; I.2.6 [Comuting Methodologies]: Artificial Intelligence Learning General Terms Theory Keywords Coevolution, selection method, oulation dynamics, solution concet. INTRODUCTION Recent work by Ficici et al. [4] examines the gametheoretic and dynamical roerties of several selection methods in the context of single-oulation coevolution. They use simle 2x2 symmetric two-layer games that have olymorhic Nash euilibria. In articular, they focus on olyc ACM, 26. This is the author s version of the work. It is osted here by ermission of ACM for your ersonal use. Not for redistribution. The definitive version was ublished in GECCO 6, VOL, ISBN: , (July, 26) htt://doi.acm.org/.45/ morhic Nash euilibria that are attractors of the canonical relicator dynamics [], in which agents are selected to reroduce in roortion to fitness. Though roortional selection has been extensively studied from the ersective of evolutionary game theory [, ], the many alternative selection methods that are used in evolutionary algorithms have not. Ficici et al. [4] show that most alternatives to roortional selection are unable to converge onto olymorhic Nash euilibria; instead, they find a variety of other dynamics, including convergence to non-nash fixed-oints, cyclic dynamics, and chaos. In this aer, we broaden the scoe of their investigation by looking at 2x2 symmetric variable-sum games in the context of two-oulation coevolution. We consider both classes of 2x2 games that have olymorhic Nash euilibria. We examine the behaviors of linear rank, truncation, and (µ, λ)-es selection, and contrast them to the dynamics of roortional selection. Our emirical results show that the alternative selection methods maintain the attractors found in roortional selection, but also add non-nash attractors as well as regions of hase-sace that lead to cyclic dynamics. Thus, these alternative selection methods do not roerly imlement the Nash-euilibrium solution concet. The aer is organized as follows. Section 2 defines the classes of games that we will use in our investigation. Section 3 details how we embed these games into a two-oulation coevolutionary framework. Section 4 describes how oulations reresent Nash euilibria. Section 5 comletes the descrition of the evolutionary framework. Sections 6 through 9 describe our results, and Section rovides concluding thoughts. 2. 2X2 SYMMETRIC GAMES We examine 2x2 symmetric variable-sum games for two layers. A two-layer game is symmetric when both layers have available the same set of strategies and the identity (i.e., Player or Player 2) of the layer laying a articular strategy has no effect on the ayoffs obtained. Examles of such games include the Hawk-Dove game [] and the Prisoner s Dilemma []. A generic ayoff matrix for a two-strategy symmetric variable-sum game for two-layers is given by Euation. By convention, the ayoffs are for the row layer; thus, if one layer uses Strategy X and the other uses Strategy Y, then the X-strategist earns a ayoff of b and the Y-strategist earns c. (Since we are dealing with symmetric games, the column and row layers are interchangeable.) We can divide the sace of ossible 2x2 games into three mutually exclusive and exhaustive classes,

2 as follows. (We will find that the first two classes hold the most interest for us, below.) G = X Y X a b Y c d 2. Game Tye A The 2x2 games in this class have ayoffs a > c and b < d; these games are known as coordination games [7], since the layers are always better off if they lay the same ure strategy (i.e., coordinate). These coordinated configurations (both laying X or both laying Y) are Nash euilibria in ure strategies: neither layer has incentive to deviate unilaterally from the ure strategy it is using. For examle, if the column layer uses strategy X, then the row layer should also use X; if the column layer instead uses Y, then so too should the row layer. Coordination games also have a mixed-strategy Nash euilibrium where both layers randomize over X and Y with a articular distribution (which we will discuss below). 2.2 Game Tye B The games in this class have ayoffs a < c and b > d; an examle of this tye of game is the Hawk-Dove game []. Games in this class have a single Nash euilibrium in ure strategies: one layer uses the X strategy and the other uses Y; given this configuration, neither layer has incentive to deviate unilaterally. There also exists a mixed-strategy Nash euilibrium where both layers use the same mixed strategy (we discuss this below, as well). 2.3 Game Tye C The final class of 2x2 games has ayoffs a < c and b < d. Here, Strategy X is the referred strategy for a layer regardless of what the other layer does. Thus, we say that X dominates Y. A weaker form of domination can be obtained if one of the ineualities is relaced by euality. (Note that the ayoff structure a > c and b > d is structurally identical, excet that now Strategy Y is referred.) 3. TWO POPULATION SYSTEMS Following Ficici et al. [4], we will use infinitely large oulations of agents; we deart from this earlier work by examining two-oulation systems. Each agent in each oulation will lay one of the two ure strategies of the game (X or Y); agents are not allowed to lay mixed strategies (robability distributions over the ure strategies). Because we have only two ure strategies, we can reresent the state of our two-oulation system with two variables. Let reresent the roortion of X-strategists in Poulation, and the roortion of X-strategists in Poulation 2. We assume comlete mixing, that is, every agent in Poulation interacts with every agent in Poulation 2 and vice versa. Agents accumulate ayoffs as they interact with each other. Let w X and w Y be the cumulative ayoffs received by X- and Y-strategists, resectively, in Poulation ; similarly, let w X2 and w Y2 be the cumulative ayoffs for individuals in Poulation 2. Given the oulation states and, we can calculate cumulative ayoffs with the linear euations in Euation 2. () w X = a + ( )b w Y = c + ( )d w X2 = a + ( )b w Y2 = c + ( )d To gain some insight into the dynamics we might exect, we can lot how the cumulative ayoffs vary over the twodimensional hase sace. Figure rovides such a lot for examle games of tyes A (to) and B (bottom). The horizontal san of an arrow indicates the difference w X w Y, while the vertical san indicates the difference w X2 w Y2 ; thus each arrow indicates, for a articular location in hase sace, the strategy that is most advantageous (and by how much) with resect to each oulation. The ure-strategy Nash euilibria are indicated by the solid circles, and the mixed-strategy Nash euilibrium is indicated by the oen circle. This mixed-strategy Nash euilibrium is also known as a olymorhic Nash euilibrium. 4. MIXED STRATEGIES AND POLYMOR PHIC POPULATIONS Though individual agents cannot lay mixed strategies, we can view a oulation as a whole to lay one. We interret the roortion of X-strategists in Poulation as secifying a mixed strategy where the robability of laying X is and Y is ( ); similarly, the roortion of X- strategists in Poulation 2 secifies a mixed strategy where the robability of laying X is and Y is ( ). Given a two-layer game, we may view Poulations and 2 as reresenting mixed strategies for Players and 2, resectively. The suort of a mixed strategy is the set of ure strategies layed with robability greater than zero. (Note that we may view a ure strategy as a degenerate mixture.) A oulation that contains more than one ure strategy is termed olymorhic; otherwise, the oulation is monomorhic. In a symmetric game with a Nash-euilibrium mixedstrategy m, if Player lays m, then the highest ayoff obtainable by Player 2 is received by also laying m. But Player 2 has other choices, as well. In articular, if Player 2 lays any ure strategy in suort of m, the same ayoff is received [8]. Thus, the strategies in suort of a Nasheuilibrium mixed-strategy will be at fitness euilibrium []. For game tyes A and B, above, this means that Poulation is laying the Nash mixture when w X2 = w Y2 (a similar statement alies to Poulation 2); we need merely solve for to discover the roortion of X-strategists needed to achieve fitness euilibrium (a similar statement alies to ), as shown in Euation 3. Since the game is symmetric, we have = when both oulations are laying the Nash mixture. e = d b a c + d b 5. REPLICATION Once cumulative ayoffs are obtained, we create the next generation of the oulation. Following conventional evolutionary game theory [], offsring are generated asexually and are clones of their arents; that is, we do not use variation oerators. As a result, strategies absent from the (2) (3)

3 Figure : Variation of cumulative ayoffs in twodimensional hase sace for Tye-A (to) and Tye- B (bottom) games. The horizontal san of an arrow indicates the difference w X w Y ; the vertical san indicates the difference w X2 w Y2. For examle, in the Tye-B game, the arrow that originates at =, = indicates that when Strategies X and Y each lay a oulation of all Y-Strategists ( = ), then Strategy X strongly out-scores Strategy Y; further, when Strategies X and Y each lay a oulation of all X-Strategists ( = ), then Strategy Y weakly out-scores Strategy X. The arrows do not locate the oint in hase sace to which the two oulations will go; the arrows merely indicate which ure strategy is most advantageous (and by how much) with resect to each oulation s state. initial oulation cannot aear later in time. The way we calculate the number of offsring each agent receives is determined by the selection method we use. We next discuss the dynamical and game-theoretic roerties of various selection methods, beginning with a conventional relication dynamic that is used in evolutionary game theory. 6. PROPORTIONAL SELECTION Here we review the roerties of roortional selection when alied to symmetric games in a two-oulation framework []. Given states and for Poulations and 2, resectively, Euation 4 shows how to calculate the next oulation states f() and g(); individuals generate offsring in roortion to their fitness (i.e., cumulative ayoff). w X f() = w X + w Y ( ) w X2 g() = w X2 + w Y2 ( ) 6. Tye A Games Figure 2 (to) illustrates the hase lot of roortional selection on a Tye-A game, defined above. The ayoffs we use in this examle are a = 3, b = 2, c = 2, d = 4 (we use this game as our Tye-A exemlar throughout the aer); using Euation 3, we find the Nash-euilibrium mixed-strategy to be at = = 2/3 (oen circle). The dynamical system has two oint-attractors, one at = =. and the other at = =. (closed circles); these attractors corresond to the coordinated ure-strategy Nash euilibria. The basin of attraction for the latter attractor is smaller as a result of the asymmetry in the ayoffs. The mixed-strategy Nash euilibrium is an unstable fixed-oint. Thus, if we begin the system at = = 2/3, then it will remain there; if we erturb either one of the oulation states (or both), then the system will diverge from the unstable fixed-oint and move to one of the two attractors. We can collase this two-dimensional system down to one dimension by setting the initial condition such that = ; this gives us the dynamics of a single oulation. Since all three Nash euilibria lie on the diagonal =, we know that the dynamics of the Tye-A game remain essentially unchanged as we move from one oulation to two. 6.2 Tye B Games Figure 2 (bottom) shows the hase lot of roortional selection on a Tye-B game. The ayoffs here are a = 2, b = 4, c = 3, d = 2 (we use this game as our Tye-B exemlar throughout the aer); this is identical to the Tye-A game, above, excet we have switched the rows of the ayoff matrix. Note that the mixed Nash euilibrium remains an unstable fixed-oint at = = 2/3. The two ure-strategy Nash euilibria, however, have moved off of the = diagonal; now, these oint-attractors are found at =, = and =, =. Though we still have two basins of attraction, note that the basins now have eual size; all oints above the = diagonal go to =, =, oints below go to =, =. If we begin the system on the diagonal = (collasing it to the single-oulation case), we converge to the mixed-strategy euilibrium (i.e., both oulations are olymorhic and have 2/3 X-strategists). The mixed Nash euilibrium is a saddle-oint of the two-oulation system, and the = line is the stable (attracting) manifold. (4)

4 Figure 2: Proortional selection in Tye-A games (to) and Tye-B games (bottom). Each line traces a trajectory in hase sace, with most trajectories leading to one of the ure-strategy Nash euilibria (solid circles). Note that each game-tye includes two non-nash fixed-oints where both oulations are monomorhic. For examle, in the Tye-A game at =, =, we know from Figure that X- Strategists out-score Y-Strategists when they lay against Poulation 2 at =. Nevertheless, Poulation does not contain any X-strategists at =, and so its state cannot change; a similar statement alies to the fact that Y-Strategists out-score X- Strategists when they lay against Poulation at =. The two oulations lack a common strategy on which they can coordinate. For Tye-B games, therefore, we find that the dynamics do change as we move from one oulation to two. This result holds relevance to ractitioners; the mere addition of a second oulation changes the solutions the system can deliver. Given a single oulation, the mixed-strategy Nash euilibrium (olymorhic oulation) is the single attractor; this outcome reresents the Nash euilibrium where Players and 2 adot the same mixed strategy. When two oulations are involved, the Nash-euilibrium mixed strategy is unstable, and one of two ure-strategy Nash euilibria will emerge instead; one oulation will converge to all-x while the other converges to all-y. Note that a single oulation cannot reresent such outcomes. For examle, the Nash euilibrium where Player adots Strategy X and Player 2 adots Strategy Y secifies that each of the two layers uses a different ure strategy. A single oulation cannot simultaneously reresent two distinct (ure or mixed) strategies, one for each layer of the game. A oulation may contain two distinct ure strategies, but such a olymorhic oulation reresents a single mixed strategy. 7. LINEAR RANKING Linear ranking is an alternative to roortional selection that is often used in evolutionary comutation [9, 2]. The general rocess begins by sorting the individuals in a oulation according to their cumulative scores; then, the individual(s) with the lowest cumulative score receives a rank value of., the individual(s) with the next highest cumulative score receives a rank value of 2., and so on. Individuals then reroduce in roortion to their rank values. For our simle 2x2 games, each agent laying the lower-scoring strategy in the oulation receives a rank value of., while agents laying the other strategy receive a rank value of Tye A Games With Tye-A games, linear ranking in a single-oulation will always converge onto one of the two ure-strategy Nash euilibria [4]. With the addition of a second oulation, however, we find that eriod-two cycles are also ossible. Figure 3 (to) shows a hase lot for linear ranking in our examle Tye-A game. We notice that the trajectories are less smooth. The filled suares indicate initial conditions that lead to eriod-two cycles. The cycle-inducing region surrounds the stable (attracting) manifold of the saddle oint located at the mixed Nash euilibrium (oen circle). A cyclic trajectory oscillates across the = diagonal. 7.2 Tye B Games With Tye-B games, linear ranking in a single oulation will always roduce cyclic dynamics [4]. With the addition of a second oulation, we find that linear ranking is somewhat better behaved. Figure 3 (bottom) shows a hase lot for linear ranking in our examle Tye-B game. We notice that most initial conditions manage to converge to one of the two ure-strategy Nash euilibria. On the = diagonal, however, we know that the single-oulation results will emerge; eriod-two cycles will be found. We find that the cycle-inducing region (filled suares) is not only on the diagonal, but also surrounds it. In Tye-B games, the mixed Nash euilibrium is again a saddle oint, but now the = diagonal is the stable (attracting) manifold. Now, a cyclic trajectory oscillates across the unstable manifold (which is stable in Tye-A games).

5 Figure 3: Linear ranking selection in Tye-A games (to) and Tye-B games (bottom). We samle the hase sace at regular intervals to obtain our initial conditions; filled suares indicate initial conditions that lead to cyclic dynamics. For examle, the initial condition =.6, =.5 has a cyclic orbit in the Tye-B game. At this initial condition, Strategy X is the higher-scoring strategy in both oulations; thus, the fitness of X and Y is 2 and, resectively. The next oulation state is =.75,.6755 (this is across the unstable manifold from =.6, =.5); now, Strategy Y is the higher-scoring strategy in both oulations. At this oint, linear rank selection reverses the fitness values and so brings us back to the initial condition. 8. TRUNCATION Truncation selection is often used in evolutionary rogramming [5] and the articular form we examine can be found in [6], for examle. In this selection rocess, we begin by sorting the individuals in a oulation according to their cumulative scores. Each individual in the highest-scoring γ fraction of the oulation will create one offsring to relace one individual in the lowest-scoring γ fraction of the oulation; thus, γ exresses selection ressure. For examle, let us suose that w X > w Y, =.6, and γ =.. The highest-scoring tenth of the oulation is entirely X- strategists and the worst-scoring tenth is all Y-strategists; truncation will cause to increase from.6 to Tye A Games For Tye-A games, one of the two ure-strategy Nash euilibria will always be obtained under truncation selection, given a single-oulation system [4]. We find that, with a two-oulation system, this is not always the case. Indeed, the behavior of truncation selection can be uite similar to that of linear ranking. Figure 4 (to) shows the hase diagram for our examle Tye-A game when γ =.5. We see that most initial conditions converge to one of the two ure-strategy Nash euilibria; there also exists a region of initial conditions (filled suares), surrounding the stable manifold, that lead to cyclic dynamics. 8.2 Tye B Games For Tye-B games, truncation selection cannot converge onto a Nash euilibrium, given a single oulation, regardless of γ > [4]. With the addition of a second oulation, however, the two ure-strategy Nash euilibria become accessible from most initial conditions. Like linear rank selection, there exists a region of initial conditions surrounding the stable manifold that lead to cyclic dynamics. Figure 4 (bottom) shows the hase lot for our examle Tye-B game, when γ =.5. If we set γ =.5, we know that virtually all initial conditions on the = diagonal will cause the system to converge to =., =., which is not a Nash euilibrium [4]. We find that there exists, in addition, a region of initial conditions (filled suares) surrounding the stable manifold ( = diagonal) that also converge to this non-nash fixed-oint. The convergence to =., =. is easily exlained. For examle, let both and be between /2 and 2/3; with these oulation states, it is the case that X-strategists have higher cumulative ayoffs than Y- strategists. Since X-strategists form at least half of the oulation, the next generation will necessarily be comosed entirely of X-strategists. Thus, the largest box on the diagonal reresents the critical region in which we transition to =., =. in the very next time-ste. The remaining initial conditions belong to the iterated re-image of this critical region; that is, they are initial conditions that will eventually fall into the critical region. 9. (µ, λ) ES (µ, λ)-es selection is often used in evolution strategies [2]. Given our simle framework, which lacks variation mechanisms, (µ, λ)-es reduces to a variation of the truncation method we examine above. As with truncation, the selectionressure arameter γ indicates the fraction of the highestscoring individuals in a oulation that will create offsring;

6 Figure 5: Truncation selection in Tye-B game with γ =.5. We samle the hase sace at regular intervals to obtain our initial conditions; filled suares indicate initial conditions that lead to a non-nash attractor at =., = Figure 4: Truncation selection in Tye-A games (to) and Tye-B games (bottom), with γ =.5. We samle the hase sace at regular intervals to obtain our initial conditions; filled suares indicate initial conditions that lead to cyclic dynamics. As with linear rank selection, the cycles obtained under truncation selection involve discontinuities. rather than just relace the lowest-scoring γ fraction of the oulation, we create enough offsring to relace the entire oulation. For examle, let us suose that w X > w Y, =., and γ =.2. The highest-scoring two-tenths of the oulation is comosed of all the X-strategists (the to tenth) and an eual roortion of Y-strategists (the nextbest tenth). The subseuent generation will therefore be comosed of eual numbers of X- and Y-strategists (i.e., will move from. to.5). 9. Tye A Games With Tye-A games, one of the two ure-strategy Nash euilibria will be obtained, given a single oulation, regardless of the selection ressure γ > [4]. We find that, with a two-oulation system, (µ, λ)-es selection can also find these Nash euilibria, rovided that selection ressure is not too severe. For high selection ressures (i.e., low γ), (µ, λ)-es creates non-nash attractors that we have not seen above. While most initial conditions do converge to a Nash euilibrium, many do not. Figure 6 (to) shows the hase lot for our examle Tye-A game with γ =.; the figure indicates initial conditions (filled suares) that converge to the fixed-oint =., =.. This fixed-oint reresents a non-coordinated configuration between the game layers, where one layer uses Strategy X while the other uses Y. Similarly, Figure 6 (bottom) shows the initial conditions that converge onto the fixed-oint =., =., again a non-coordinated state. 9.2 Tye B Games With Tye-B games, (µ, λ)-es cannot converge onto a Nash-euilibrium, given a single oulation, regardless of

7 Figure 6: Tye-A games under (µ, λ)-es selection with γ =.. We samle the hase sace at regular intervals to obtain our initial conditions; filled suares indicate initial conditions that converge onto a non-nash attractor. To: Initial conditions that converge onto =., =.. Bottom: Initial conditions that converge onto =., =.. Figure 7: Tye-B games under (µ, λ)-es selection with γ =.. We samle the hase sace at regular intervals to obtain our initial conditions; filled suares indicate initial conditions that converge onto a non-nash attractor. To: Initial conditions that converge onto = =.. Bottom: Initial conditions that converge onto = =..

8 γ > [4]. We find that with two oulations, (µ, λ)-es is caable of converging onto either of the ure-strategy Nash euilibria, again rovided that selection ressure is not too severe. Figure 7 shows the hase diagram for our examle Tye-B game with γ =.. While the ure-strategy Nash euilibria are still accessible from some initial conditions, we find that most of the hase sace converges onto a non- Nash attractor. The to and bottom grahs in Figure 7 indicate the initial conditions (filled suares) that converge onto = =. and = =., resectively. Note that the initial conditions are on the oosite side of the saddleoint from the fixed-oint they converge onto.. REVIEW AND CONCLUSION We use a simle evolutionary game-theoretic framework to examine the dynamics of several selection methods fitnessroortional, linear rank, truncation, and (µ, λ)-es in the context of two-oulation coevolution. We use simle (2x2) symmetric variable-sum games for two layers. We focus on two classes of game. Games in either class have three Nasheuilibrium fixed-oints in the two-dimensional hase-sace; two of the Nash euilibria involve ure strategies while the third is a mixed-strategy Nash euilibrium. The selection methods we consider have been analyzed by Ficici et al. [4] in the context of single-oulation coevolution; their results show that all selection methods will converge onto Nash euilibrium if the game belongs to the game class Tye-A (see Section 2.). In contrast, only fitnessroortional selection converges to Nash euilibrium if the game belongs to class Tye-B (see Section 2.2); the other methods (that we consider in this aer) cannot converge onto Nash euilibrium. Instead a variety of other dynamics are reorted; these include cyclic dynamics, non-nash attractors, and chaos. Our findings in this aer show that linear rank, truncation, and (µ, λ)-es selection are better-behaved in a twooulation setting than in the one-oulation case analyzed in [4]. The mixed-strategy Nash-euilibrium (in the interior of the hase sace) is well known to be unstable with fitness-roortional selection []; the alternative selection methods we examine are no different in this resect. The ure-strategy Nash euilibria are known to be attractors of fitness-roortional selection [], and they remain attractors for the alternative methods we examine. Nevertheless, the alternative selection methods we consider do introduce some additional behaviors that are not found with fitness-roortional selection. Many initial conditions lead to cyclic dynamics, while others converge onto non-nash fixed-oints. Thus, one cannot use these selection methods with confidence if the desired solution concet is Nash euilibrium. On a more general note, our results further demonstrate that the mechanisms at work in a coevolutionary algorithm (here, we look at selection methods) can rofoundly affect the de facto solution concet that is imlemented, and may cause it to diverge from the solution concet that we intend to imlement [3]. While we use only two exemlar games in our investigation (one each from Tye-A and Tye-B), our results generalize across the games in these classes; results for different games will differ in numerical detail, but be ualitatively identical. Future work will exand our results by moving to higher dimensions (where the games have more than two strategies) and by moving to asymmetric (or bimatrix) games (where the two oulations use different strategy sets).. ACKNOWLEDGMENTS The author thanks Anthony Bucci, Edwin de Jong, Jordan Pollack, Paul Wiegand, Shivakumar Viswanathan, and members of the DEMO Lab. 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