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260 CHAPTER 9 Medium Access Control Games 9.3.2.2 Unconstrained Repeated Games with Perfect Monitoring We focus on the repeated game analysis of the 3-transmitter 2-action game with per- fect monitoring (see Chapter 3). In order to characterize the set of utilities of the discounted repeated game, we determine the punishment levels. The vector of pun- ishment level (PL) utilities is u PL = (0, 0, 0), and the set of feasible and individually rational utilities of the one-shot game 3,1,2 is: V = co{(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1)} = {(v 1 , v 2 , v 3 ) R 3 : i {1, 2, 3}, v i 0, v 1 + v 2 + v 3 1}. (9.38) (9.39) The dimension of V is three, i.e., the number of users in interaction. Thus, the full dimensionality condition is satisfied (the interior of V is non-empty). The standard Folk theorem with complete information (see Chapter 3) can be applied here. If E denotes the set of Nash equilibrium utilities of the dynamic discounted game with discount factor 0 < 1 then: 0 lim E = V. (9.40) 9.3.2.3 Unconstrained Repeated Games with a Public Signal In this section, the assumption of perfect monitoring is relaxed, and only a pub- lic signal which indicates the last crowd receiver is assumed to be available to the transmitters. Compared to unconstrained repeated games with perfect monitoring, for which the action profile is observed by all the three transmitters at the end of each time-slot, we now assume that, at the end the each time-slot, the public signal: (t) = {r 1 , r 2 } (9.41) is observed by each user. The public signal is the crowd receiver. Our objective is to characterize the set of equilibrium utilities of the discounted repeated game with the public signal (the crowd receiver). The next realization of the public signal is drawn according to the crowd receiver at the last time-slot. If an action profile a(t - 1) is played at time t - 1 then: t = r 1 r 2 if {j K, a j,t-1 = r 1 } > {i K, a i,t-1 = r 2 } otherwise (9.42) action announced signal r 1 r 1 r 1 , r 1 r 2 r 1 , r 2 r 1 r 1 , r 1 r 1 r 2 r 1 r 1 r 2 r 2 , r 2 r 1 r 2 , r 2 r 2 r 2 , r 2 r 2 r 1 r 2