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Arbitration

Arbitration. 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.

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Arbitration

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  1. Arbitration

  2. 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

  3. 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

  4. 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?

  5. 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?

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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?

  11. 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 ½ 1 - (1-d)½ (1-d)½

  12. 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

  13. 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)

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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.

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  25. 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

  26. 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

  27. Conclusions • Arbitration need not lead to improved bargaining outcomes • Standard arbitration can lead to entrenchment in extreme bargaining positions

  28. 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

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