Communication Networks. A Second Course. Jean Walrand Department of EECS University of California at Berkeley. Repeated, Bargaining, Dynamic. Motivation Repeated Games Bargaining Dynamic Games. Motivation. So far: One-shot (static) games Many games are repeated or dynamic:
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A Second Course
Department of EECS
University of California at Berkeley
Proof:Pick r is in C. Then r rewards of playing the pair (ak,bk) of actions a fraction pk Nk/N of the times, k = 1, …, 4. Strategy of player 1 [2, resp.]: We play (a1, b1) for N1 steps, then … (a4, b4) for N4 steps, and we repeat forever. Ifyou deviate at any time, I play B [R, resp.] forever thereafter.
Proof (continued):Reward of P2 at steps n, n +1 , … if he deviates at time n: < (1 – b)bn5 + bn+1 2 + e =: v if b is large enough.Reward of P2 at steps n, n +1 , … if he does not deviate: > (1 – b)bn1 + bn+1r2 – e =: w if b is large enough.Note that w > v for b large enough since r2 > 2.The strategy is an SPE: Say P2 deviates at time n. ThenP1 plays B forever and, knowing this, P2 must play R forever and P1 must accordingly play B forever since (B, R) is NE.
The SPE that enforces the rewards r is a “threat strategy.”
The key step in the argument is to show that the threatstrategy is an SPE. In other words, the threat is “credible.” This is the case because the threat is torevert to playing a NE forever.
Claim: 2 NEs = (L, L) and (R, R)
If P1 does not choose L:
Fact: Only one SPE: (R, R)
P1: R if P2 = L L otherwise
Equivalent matrix game