(final-offer arbitration)* Farber (19K0) proposes the follow-ing model of final-offer…

(final-offer arbitration)* Farber (19K0) proposes the follow-ing model of final-offer arbitration. There are three players: a management (i = I), a union (i = 2), and an arbitrator (i = 3). The arbitrator must choose a settlement t ϵ ℝ from the two offers, s1 ϵ ℝ and s2 ϵ ℝ. made by the management and the union respectively. The arbitrator has exogenously given preferences v0 = — (t — s0)2. That is, he would like to be as close to his “bliss point,” s0, as possible. The management and the union don't know the arbitrator's bliss point; they know only that it is drawn from the distribution P with continuous, positive density p on [ s0, s0]. The management and the union choose their offers simultaneously. Their objective functions are u1 = —t and u2 = +t, respectively.

Derive and interpret the first-order conditions for a Nash equilibrium. Show that the two offers are equally likely to be chosen by the arbitrator.