Index and stability in bimatrix games
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More precisely, if we consider a strategy for one player in its corresponding simplex, and that player is adjusting its strategy, will immediately cause the trajectory in the second simplex to change, and vice versa. Consequently, it is not straightforward anymore to analyse the dynamics and equilibrium landscape for both players, as any trajectory in one simplex causes the other simplex to change. A movie illustrates what is meant: we show how the dynamics of player 2 changes in function of player 1. We overlay the simplex of the second player with the simplex of the first player; the yellow dots indicate what the strategy of the first player is.
In order to facilitate the process of analysing this game we can apply the counterpart RD theorems here to remedy the problem, and consequently analyse the game in the far simpler symmetric counterpart games that will shed light onto the equilibrium landscape of the Leduc Poker empirical game. As can be observed in Fig. Looking at Fig. Note there is also a rest point at the face formed by strategies D and E , which is not Nash.
Given that there are no mixed equilibria with full support in both games we cannot apply Theorem 1. Using Theorem 2 we now know that we only maintain the two mixed equilibria, i. The other equilibria in the second counterpart game can be discarded as candidates for Nash equilibria in the Leduc poker empirical game since they also do not appear for player 1 when we permute the strategies for player 1 not shown here.
As a final example to illustrate the introduced theory, we examine an asymmetric game, that has one completely mixed equilibrium and several equilibria in its counterpart games. The asymmetric game has a unique completely mixed Nash equilibrium with different mixtures for the two players, i. The two symmetric counterpart game each have seven equilibria. Note that there are also two rest points, which are not Nash, at the faces formed by A and B and B and C.
From these seven equilibria only a , b and c are of interest since these are symmetric equilibria in which both players play with the same strategy or support. Also counterpart game 2 has seven equilibria, i. We observe that only the completely mixed equilibrium of the asymmetric game, i. To apply the theorems we only need to have a look at equilibria a , b and c in counterpart game 1, and d , e and f in counterpart game 2.
As can be observed, a is an unstable mixed equilibrium, b is a stable pure equilibrium, and c is a partly mixed equilibrium at the 2-face formed by strategies A and C. We can make the same observation for the second counterpart game, and see that d , e and f are equilibria in Fig. Equilibrium d , indicated by a yellow oval, is completely mixed, equilibrium e is a pure equilibrium in corner F green oval , and f is a partly mixed equilibrium on the 2-face formed by strategies D and E green ovals as well. If we now apply Theorem 1 we know that we can combine the mixed equilibria of full support of both counterpart games into the mixed equilibrium of the original asymmetric game, in which the mixed equilibrium of counterpart game 1, i.
Both equilibria are unstable in the counterpart games and also form an unstable mixed equilibrium in the asymmetric game. Replicator Dynamics have proved to be an excellent tool to analyse the Nash landscape of multiagent interactions and distributed learning in both abstract games and complex systems 1 , 2 , 4 , 6. The predominant approach has been the use of symmetric replicator equations, allowing for a relatively straightforward analysis in symmetric games.
Many interesting real-world settings though involve roles or player-types for the different agents that take part in an interaction, and as such are asymmetric in nature.
So far, most research has avoided to carry out RD analysis in this type of interactions, by either constructing a new symmetric game, in which the various actions of the different roles are joined together in one population 23 , 24 , or by considering the various roles and strategies as heuristics, grouped in one population as well 2 , 3 , 8. In the latter approach the payoffs due to different player-types are averaged over many samples of the player type resulting in a single average payoff to each player for each entry in the payoff table.
The work presented in this paper takes a different stance by decomposing an asymmetric game into its symmetric counterparts.
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This method proves to be mathematically simple and elegant, and allows for a straightforward analysis of asymmetric games, without the need for turning the strategy spaces into one simplex or population, but instead allows to keep separate simplices for the involved populations of strategies. Furthermore, the counterpart games allow to get insight in the type and form of interaction of the asymmetric game under study, identifying its equilibrium structure and as such enabling analysis of abstract and empirical games discovered through multiagent learning processes e.
Leduc poker empirical game , as was shown in the experimental section. A deeper counter-intuitive understanding of the theoretical results of this paper is that when identifying Nash equilibria in the counterpart games with matching support including permutations of strategies for one of the players , it turns out that also the combination of those equilibria form a Nash equilibrium in the corresponding asymmetric game. However, if you position the second player at a Nash equilibrium, it turns out that player one becomes indifferent between his different strategies, and remains stationary under the RD.
This is what we end up using when establishing the correspondence of the Nash Equilibria in asymmetric and counterpart games, and why the single-simplex plots for the counterpart games are actually meaningful for the asymmetric game - but this is also why they only describe the Nash Equilibria faithfully, but fail to be a valid decomposition of the full asymmetric game away from equilibrium.
These findings shed new light on asymmetric interactions between multiple agents and provide new insights that facilitate a thorough and convenient analysis of asymmetric games. As pointed out by Veller and Hayward 36 , many real-world situations, in which one aims to study evolutionary or learning dynamics of several interacting agents, are better modelled by asymmetric games. As such these theoretical findings can facilitate deeper analysis of equilibrium structures in evolutionary asymmetric games relevant to various topics including economic theory, evolutionary biology, empirical game theory, the evolution of cooperation, evolutionary language games and artificial intelligence 11 , 12 , 37 , 38 , 39 , Finally, the results of this paper also nicely underpin what is said in H.
He also indicates that this aspect of an evolutionary dynamic is often misunderstood. The use of our counterpart dynamics supports and illustrates this statement very clearly, showing that in the counterpart games species play games within a population and as such show an intra-species survival of the fittest, which is then combined into an equilibrium of the asymmetric game. Bloembergen, D.
Evolutionary dynamics of multi-agent learning: A survey. Walsh, W.
Index And Stability In Bimatrix Games A Geometric Combinatorial Approach
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