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R. Jennings Thus, the optimal choices for a and b are: OPTA (i, t) = arg maxrj=1 EUA(i, j, t) (5) OPTB(i, t) maxrj=1 EUB(i, j, t) (6) = arg We compute the optimal choice for t = 1 by reasoning backwards from t = Nr . At t = 1, if an agent a of type i is the offering agent, then it offers the package that corresponds to b being of type OPTA(i,1). Likewise, if an agent b of type i is the offering agent, then it offers the package that corresponds to a being of type OPTB(i,1). But since OPTA(i,1) and OPTB(i,1) are obtained under uncertainty, an agreement may or may not occur at t = 1.

Hence, F a is: F a (i, j, e, t) = Uia (A(i, j, t)) if Ueb (A(i, j, t)) ≥ EUA(i, t + 1) otherwise EUB(e, γ, t + 1) where γ = OPTB(e, t + 1). The strategy A(i, j, t) for t = Nj is: A(i, j, t) = OFFER [δ n−1 , 0] if a’s turn ACCEPT otherwise and for all time periods t < Nj it is: A (i, j, t) = OFFER TRADEOFFA(EUB(j, t)) if a’s turn if Uia ([xt , y t ], t) ≥ EUA(i, t) ACCEPT else REJECT otherwise where [xt , y t ] is the package offered at t. Analogously, F b is: F b (i, j, e, t) = Uib (B(i, j, t)) if Uea (B(i, j, t)) ≥ EUB(i, t + 1) otherwise EUA(e, α, t + 1) where α = OPTA(e, t + 1).

For the former, a’s equilibrium offer at t = 1 is [1, 0] for each issue. 7. 675] for the first issue and [1, 0] for the second one. 325. Since EUA (1, 1, 1) > EUA (1, 2, 1), OPTA (1, 1) = 1 and a plays the former strategy. Now if b is actually of type 1, then it accepts a’s offer. Thus, the earliest possible time of agreement is t = 1. But if b is of type 2, it rejects a’s offer since it can get a higher expected utility at t = 2. However, since a is of type 1, negotiation ends in a conflict. If agent a’s offer at t = 1 gets rejected it knows that agent b is not of type OPTA(i, 1).

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Digital Image Watermarking using Mixing Systems by Voyatzis Pitas

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