Selective Competition

We consider a dynamic (differential) game with three players competing against each other. Each period each player can allocate his resources so as to direct his competition towards particular rivals -- we call such competition selective. The setting can be applied to a wide variety of cases: competition between firms, competition between political parties, warfare. We show that if the players are myopic, the weaker players eventually loose the game to their strongest rival. Vice versa, if the players value their future payoffs high enough, each player concentrates more on fighting his strongest opponent. Consequently, the weaker players grow stronger, the strongest player grows weaker and eventually all the players converge and remain in the game.