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148 CHAPTER 5 Partially Distributed Learning Algorithms · · ¯ Denote f j,t as the relative mean use of actions by player j up to t - 1. According to fictitious play, players choose, a 1,t (resp. a 2,t ): a 1,t arg max a 1 ¯ a 2 , f 2,t (a 2 )>0 ¯ f 2,t (a 2 )~ 1 (a 1 )1 C (a 1 , a 2 ) u l · That is, in each period, each player chooses his best response to the observed empirical frequency of his opponent's choices. We analyze the fictitious play outcome for the constraint set: C = {a = (a 1 , a 2 ) | 0 a 1 a 1 , 0 a 2 a 2 , 0 a 1 + a 2 v({1, 2}), ¯ ¯ with a 1 + a 2 > v({1, 2}), a j < v({1, 2}). ¯ ¯ ¯ ·························································································· · · · · · · · · For any (a 1,1 , a 2,1 ) C the following holds: (i) a 1,2 = v({1, 2}) - a 2,1 , (ii) a 2,2 = v({1, 2}) - · · a 1,1 · · · · For any t 3, a 1,t must be either a 1,1 or (v({1, 2}) - a 2,1 ), and a 2,t {a 2,1 , v({1, 2}) - a 1,1 } · · ¯ ¯ ¯ ¯ · · f 1,t (a 1,1 ) + f 1,t (v({1, 2}) - a 2,1 ) = 1, f 2,t (a 2,1 ) + f 2,t (v({1, 2}) - a 1,1 ) = 1 · · w 1 w 2 · · (a 1 , a 2 ) arg max a u 1 (a 1 ) ~ u 2 (v({1, 2}) - a 1 ) ~ , a 2 = (a 1 ), defines the Pareto fron- · 1 · · tier and satisfies the implicit equation: · · · · [~ 1 (a 1 )] w 1 [~ 2 (v({1, 2}) - a 1 )] w 2 = [~ 1 (v({1, 2}) - (a 1 ))] w 1 [~ 2 ((a 1 ))] w 2 u u u u · · · · · · Theorem 156 Example 157: Rate Control in AWGN Channel We consider a system consisting of one receiver and its uplink additive white Gaussian noise (AWGN) multiple access channel with two senders. The signal at the receiver is given by Y = + m h j X j where X j is the transmitted signal of user j, and is zero-mean Gaus- j=1 sian noise with variance N 0 . Each user has an individual power constraint E(X j 2 ) p j . The optimal power allocation scheme for Shannon capacity is to transmit at the maximum power available, i.e., p j , for each user. Hence, we consider the case in which maximum power is used. The decisions of the users then consist of choosing their communication rates, and the receiver's role is to decode, if possible. The capacity region is a set of all vectors a R 2 + such that senders j {1, 2} can reliably communicate at rate a j , j {1, 2}. The capacity region C for this channel is the set: jJ p j w j 2 , J {1, 2} (5.21) C = a R + a j log 1 + N 0 jJ p w p w where w j = |h j | 2 . Let a 1 = log 1 + 1 1 , a 2 = log 1 + 2 2 , and v({1, 2}) = ¯ ¯ N 0 N 0 p j w log 1 + j{1,2} N j . Under the constraint C, if sender j wants to communicate at 0 a higher rate, one of the other senders has to lower his rate; otherwise, the capacity constraint is violated. Let: r j = log 1 + p j w j N 0 + j {1,2},j =j p j w j (5.22)