192 9. Coexisting Wireless LANs as Multi Stage Game

By setting the probability p, that the finite MSG continues, to δ =1 −p , and as-

suming that there is a large number of stages left to be played, i.e., n N MSG ,
the overall payoff VMSGi	a player i observes in an MSG is derived from the

infinite game with discounted payoffs, as shown in Equation (9.1). The discount

version and the probability version of	the		payoff calculation are	equivalent,
hence, in the following the payoff VMSGi	a player i observes in the MSG, is calcu-
lated by the following equation (Fuller, 2002).
!			∞
(1 − p)= δ VMSGi		=	∑δtV i(n) .	(9.3)

n =0

9.1.2Solution of the MSG of two Players: Nash Equilibrium of Strategy Pairs

It can be analyzed whether two strategies, each strategy selected by one player, form a Nash equilibrium of Strategies in the MSG. Here, the concept of Nash equilibrium differs from the definition for the SSG, as it was used in the previous chapters: a Nash equilibrium of Strategies in the MSG is given if no player has the incentive to unilaterally deviate from its strategy for any number of stages, taking into consideration the observed payoff as defined in Equation (9.3).

Players have the intention to establish a steady state in the MSG. The players attempt to influence this steady state to their advantage. The concept of a Nash equilibrium implies best response actions of the players. A result of the analysis of SSGs is that in many scenarios the Nash equilibrium is the Pareto optimal outcome of the SSG. In this context, the equilibrium is a Nash equilibrium of actions in the SSG.

To show that a strategy pair is not a Nash equilibrium in the MSG, at least one other strategy has to be identified that one player would prefer, i.e., a strategy that achieves a higher payoff. In contrast, to show that a strategy pair is a Nash equilibrium in the MSG, it has to be shown that there is no other strategy that one player would prefer to deviate to for any number of stages.

9.2Static Strategies

Static strategies implement one single behavior constantly throughout the MSG. Therefore, static strategies are directly related to behaviors.