Presentation on theme: "Arbitration. Introduction In this section we will consider the impact of outside arbitration on coordination games Specifically, we will consider two."— Presentation transcript:
Introduction In this section we will consider the impact of outside arbitration on coordination games Specifically, we will consider two arbitration regimes Standard arbitration Final offer arbitration
Arbitration Suppose that we change the split the surplus game to allow for outside arbitration One common arbitration protocol is to have the arbitrator choose a settlement between the final offers of each side We suppose that the arbitrator gives weight of a to player 1’s final offer
Arbitration Likewise, the arbitrator gives weight (1-a) to player 2’s final offer The weight a is between [0,1] What should each player’s final offer be?
Specifying the Game Under this bargaining protocol, each player chooses a final offer xi Player 1 receives a payoff ax1 + (1 - a)(1 - x2) Player 2 receives a payoff 1 - ax1 - (1 - a)(1 - x2) So what should the players demand?
Best Responses For player 1 the problem is to choose x1 in [0,1] to maximize ax1 + (1 - a)(1 - x2) Clearly, the extreme position x1 = 1 is the best choice Player 2 faces a similar problem and chooses x2 = 1
Equilibrium Given these best responses (in fact these are dominant strategies!) The equilibrium allocation is (a, 1 - a) The imposition of an outside arbitrator actually causes players to entrench in more extreme positions than in the absence of an arbitrator
A Two-stage Game So far, we’ve only considered final offers when arbitration is to be imposed, suppose now that arbitration is imposed only at an impasse in face-to-face bargaining Specifically, suppose that players choose demands in the nash demand game in the first stage. In the event that x1 + x2 >1, then arbitration in imposed
Second Stage If players land in the arbitration stage of the game, suppose that only (1 - d) of the pie remains. That is, arbitration costs d The arbitrator then imposes the solution x = ax1 + (1-a)(1 -x2) and allocates the remaining surplus
Money on the Table Notice that arbitration as introduced the possibility of an inefficient outcome Since arbitration reduces the pie by d, it is in both players’ interests to settle in the first stage So what happens?
Efficient First-stage Agreements Any agreement in the first stage yielding x1 > (1-d)a and x2 > (1-d)(1-a) is preferred by both parties Notice that for all d, such a region exists (1-d)½ ½ 1 - (1-d)½
First Stage Agreements Suppose that the two players agree to split the remaining surplus from avoiding the imposition of arbitration Then, x1 = (1-d)a + gd, and x2 = (1-d)(1-a) +(1-g)d Where g is in [0,1] Notice that this is helps both players compared to the arbitration outcome
Equilibrium But is this an equilibrium? Suppose player 1 decides to “cheat” and go to arbitration, then x1 = 1 This yields the arbitration solution of: a + (1-a)(1 – x2) Discounting, player 1 stands to earn (1-d)(a + (1-a)(1 – x2)
Equilibrium As compared to (1-d)a +gd By conforming to the proposed equilibrium Thus, for player 1 not to deviate requires (1 – d)a + gd > (1 – d)a +(1 – d)(1 – a)(1 – x2) Or g > (1 – a)(1 – x2)(1 – d)/d
Equilibrium For player 2, deviating yields the arbitration solution: (1 – d)(1 - a((1-d)a + gd) Discounting for the costs of arbitration (1-d)(1 - ax1) – (1- d)(1 – a) As compared to (1-d)(1-a) + (1-g)d
Equilibrium Thus, for player 2 not to deviate requires (1-d)(1-a) + (1-g)d > (1-d)(1 - ax1) – (1- d)(1 – a) (1 – g) > (1 – ax1)(1 – d)/d As the costs of arbitration get small these conditions become: g > 1 g < 0 Which obviously cannot both hold
Main Result When arbitration costs are not too large, recourse to arbitration results in inefficient bargaining outcomes! Intuitively, by taking a hard-line, players’ gain in the arbitration stage, when their opponent s conciliatory When costs are small, the efficiency losses make it individually worthwhile
Comments Far from facilitating bargaining, the arbitration option actually increases the chance of an impasse Arbitration can actually reduce the chances of reaching an efficient outcome
Final Offer Arbitration Perhaps we were simply going about the arbitration in the wrong way Consider a different arbitration protocol: Players 1 & 2 make final offers in the usual way The arbitrator must choose the “fairer” of the two offers to impose as the outcome Suppose that the arbitrator’s idea of fair is the Nash bargaining solution.
Final Offer Arbitration Given the Nash bargaining solution, suppose that the arbitrator chooses between x1, x2 such that If x1(1-x1) > x2(1-x2) then x = x1 Otherwise x = x2 That is, the fairer offer is that which is closer to the Nash bargaining solution
Second Stage Game If the players go to arbitration, then the player choosing xi closest to 1/2 “wins” Suppose player 2 chooses x2 > 1/2 then player 1’s best response is to choose x1 = x2 - e where e is a small number Thus undercutting process continues until x1=x2=1/2
So What Happens? Notice that if x1=x2=1/2, then agreement is reached in the first stage and no arbitration is needed. Thus, it cannot be the case that things end up in arbitration
Best Responses Suppose in the first stage, the equilibrium calls for x2>=1/2 Player 1 can choose: x1 = 1-x2 or She can go to arbitration and choose x1 = x2-e
First Stage Settlements For there to be a settlement in the first stage requires: 1-x2 > (1-d)(x2 -e) (but e is close to 0) 1-x2 > (1-d)x2 X2 < 1/(2-d) As d --> 1, this becomes x2 <= 1/2 Hence, x1=x2=1/2
Comments With final offer arbitration players choose more conciliatory stances This conciliation leads to bargains being struck in the first stage game; hence the bargaining outcomes are efficient
Comments The fairness objectives of the arbitrator influence the offers made in the first stage game As costs of arbitration become small, the nash bargaining outcome is implemented as the unique equilibrium in the game
Conclusions Arbitration need not lead to improved bargaining outcomes Standard arbitration can lead to entrenchment in extreme bargaining positions
Conclusions Final offer arbitration creates incentives for conciliatory posturing Final offer arbitration can help to coordinate expectations outside of arbitration making negotiated outcomes easier to obtain