Playability Properties in Games of Deterrence and Evolution in the Replicator Dynamics

Since the seminal work of John Maynard Smith (1982), a vast literature has developed on evolution analysis through game theoretic tools. Among the most popular evolutionary systems is the Replicator Dynamics, based in its classical version on the combination between a standard non cooperative matrix game and a dynamic system which evolution depends on the payoffs of the interacting species.
Despite its weaknesses, in particular the fact that it does not take into account emergence and development of species that did not initially exist, the Replicator Dynamics has the advantage of proposing a relatively simple model that analyzes and tests some core features of Darwinian evolution.
Nevertheless, the simplicity of the model reaches its limits when one needs to predict accurately the conditions for reaching evolutionary stability. The reason for it is quite obvious: it stems from the possible difficulties to find an analytical solution to the system of equations modelling the Replicator Dynamics.
An alternative approach has been developed, based on matrix games of a different kind, called Games of Deterrence. Matrix Games of Deterrence are qualitative binary games in which selection of strategic pairs results for each player in only two possible outcomes: acceptable (noted 1) and unacceptable (noted 0). It has been shown (Rudnianski, 1991) that each matrix Game of Deterrence can be associated in a one to one relation with a system of equations called the playability system, the solutions of which determine the playability properties of the players' strategies.
Likewise, it has been shown (Ellison and Rudnianski, 2009) that one could derive evolutionary stability properties of the Replicator Dynamics from the solutions of the playability system associated with a symmetric matrix Game of Deterrence on which the Replicator Dynamics is based.
Thus, it has been established that (Ellison and Rudnianski, 2009):

• To each symmetric solution of the playability system corresponds an evolutionarily stable equilibrium set (ESES)
• If a strategy is not playable in every solution of the playability system, the proportion of the corresponding species in the Replicator Dynamics vanishes with time in every solution of the dynamic system