Non-cooperative Support for the Asymmetric Nash Bargaining Solution

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conditions for a strategy profile to be an SSPE and Theorem 3.13 shows that an SSPEexists.The analysis in this section resembles Kalandrakis (2004), but some important differencesshould be noted: We admit only pure strategies and require unanimous agreement.Furthermore, we conclude rather than assume that agreement is immediate in SSPE andwe do not impose assumptions on the behavior of players who are indifferent betweenacceptance and rejection of some proposal.Consider a profile of stationary strategies. It can be described by an n × n -matrix Θ,where the entry θj i is the payoff proposed to player j by player i, and a collection A of n 2acceptance sets, where the acceptance set A i j is the set of vectors in V which player j willaccept when proposed by player i. The set of vectors in V proposed by player i and acceptedby player j and his successors is A i S(j) = ∩ k∈S(j)A i k . We refer to Ai = A i S(1) = ⋂ j∈N Ai j asthe social acceptance set for proposer i.Suppose that in period t, the proposal of player i is rejected. With probability 1 − δthe game ends and all players receive zero payoff, and with probability δ period t + 1 isreached and play proceeds according to the profile (Θ, A) of stationary strategies. Theexpected payoff to player j after rejection is rj(Θ, i A). Omitting the argument (Θ, A) fromthe notation wherever possible, we refer to rj i as the reservation payoff of player j when iproposes.Proposition 3.1 The reservation payoff r i belongs to int(V ).Proof: Conditional on the next period being reached, the payoffs are determined by aprobability distribution on V (notice that also 0 ∈ V ), so expected payoffs belong to Vsince V is convex. Since with probability 1 − δ the next period is not reached, theseexpected payoffs equal δ −1 r i , so δ −1 r i ∈ V. Since 0 ∈ int(V ), the convex combination(1 − δ)0 + δδ −1 r i = r i belongs to int(V ). ✷One implication of Proposition 3.1 is that a proposer always has the option to make aproposal that strictly exceeds the reservation payoff of every player.Proposition 3.2 In SSPE, for j ∈ N, if v ∈ A i S(j) , then v k ≥ r i kfor all k ∈ S(j).Proof: Suppose that (Θ, A) is a profile of stationary strategies such that v ∈ A i S(j) butv k < rk i for some player k ∈ S(j). Consider a history in Hr k , where player k responds tothe proposal v made by player i. At that history player k could deviate from (Θ, A) byrejecting v. In that case, an expected payoff of rk i would result. Hence, this deviation isprofitable and (Θ, A) cannot be an SSPE.✷Proposition 3.2 implies that for a vector of payoffs v to belong to the social acceptanceset, it should satisfy v j ≥ r i j for all j ∈ N.6
Proposition 3.3 In SSPE, for j ∈ N, if v ∈ V satisfies v k > rk i for all k ∈ S(j), thenv ∈ A i S(j) .Proof: Suppose that (Θ, A) is a profile of stationary strategies such that v k > rk i for allk ∈ S(j) and v /∈ A i S(j) . Suppose first that v /∈ Ai n. Consider a history in Hn, r where playern responds to the proposal v made by player i.Then, player n could deviate from (Θ, A) byaccepting v. This would yield a payoff of v n > rn, i a contradiction. Consequently, v ∈ A i n.We repeat the argument for players n − 1, n − 2, . . . , j to establish the proposition. ✷Proposition 3.3 established a kind of converse of Proposition 3.2. One implication ofthis proposition is that a vector v ∈ V that satisfies v j > rj i for all j ∈ N belongs to thesocial acceptance set A i j.Proposition 3.4 In SSPE, each player’s proposal θ i lies in the social acceptance set A ifor proposer i.Proof: Suppose by way of contradiction that under some SSPE there is a player i ∈ N suchthat θ i /∈ A i . Consider the subgame starting at a history where player i is the proposer.Since θ i is rejected, r i is the vector of expected payoffs by definition. By Proposition 3.1,r i ∈ int(V ). Consequently, there exists v ∈ V such that v j > rj i for all j ∈ N. By theprevious proposition, v ∈ A i . Hence, it would be a profitable deviation for player i topropose v instead of θ i .✷Proposition 3.5 In SSPE, θj i ≥ 0 and rj i ≥ 0 for all (i, j) ∈ N × N.Proof: Suppose by way of contradiction that (Θ, A) is an SSPE and that θj i < 0 for some(i, j) ∈ N × N. Consider a history where player j has to respond to the proposal θ i . ByProposition 3.4, θ i ∈ A i , so player j will receive a strictly negative payoff if play proceedsaccording to (Θ, A). But then, it would be a profitable deviation for player j to reject theproposal. Consequently, it holds that θj i ≥ 0 for all (i, j) ∈ N × N. It then follows thatrj i ≥ 0 for all (i, j) ∈ N × N.✷The next proposition establishes that an equilibrium proposal of any player gives allother players their respective reservation payoffs. Thus a proposer always extracts theentire surplus from the other players.Proposition 3.6 In SSPE, θj i = rj i for all (i, j) ∈ N × N such that i ≠ j.7
Page 6 and 7: of the current proposer, reservatio
Page 10 and 11: Proof: Since θ i ∈ A i by Propos
Page 12 and 13: θ i implies θ i i > r i i, so the
Page 14 and 15: By Proposition 3.12, χ i (Θ) is u
Page 16 and 17: We denote a column of ¯Θ by ¯θ.
Page 18 and 19: Proposition 4.6 It holds that span(
Page 20 and 21: Proposition 4.8 It holds that ¯θ
Page 22 and 23: ReferencesAlesina, A. (1987),“Mac

Non-cooperative Support for the Asymmetric Nash Bargaining Solution

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