Lecture 5 Bargaining

1. Lecture 5Bargaining This lecture focuses on the well known problem of how to split the gains from trade or, more generally, mutual interaction when the objectives of the bargaining parties diverge.

2. Resolving conflict • Bargaining is one way of resolving a conflict between two or more parties, chosen when all parties view it more favorably relative to the alternatives (such as courts, theft, warfare). • For example: • Unions bargain with their employers about wages and working conditions. • Professionals negotiate their employment or work contracts when changing jobs. • Builders and their clients bargain over the nature and extent of the work to reach a work contract.

3. Three dimensions of bargaining • We shall focus on three dimensions of bargaining: • How many parties are involved, and what is being traded or shared? • What are the bargaining rules and/or how do the parties communicate their messages to each other? • How much information do the bargaining parties have about their partners? • Answering these questions helps us to predict the outcome of the negotiations.

4. Two player ultimatum games • Consider the following three versions of the problem of splitting a dollar between two players. In each case, the rejected proposals yield no gains to either party: • The proposer offers anything between 0 and 1, and the responder either accepts or rejects the offer. • The proposer makes an offer, and the responder either accepts or rejects the offer, without knowing exactly what the proposer receives. • The proposer selects an offer, and the responder simultaneously selects a reservation value. If the reservation value is less than the offer, then the responder receives the offer, but only in that case.

5. Solution The solution is the same in all three cases. The solution is for the proposer to extract (almost) all the surplus, and for the responder to accept the proposal.

6. Two rounds of bargaining • Suppose that a responder has a richer message space than simply accepting or rejecting the initial proposal. • After an initial proposal is made, we now assume: • The responder may accept the proposal, or with probability p, make a counter offer. • If the initial offer is rejected, the game ends with probability 1 – p. • If a counter offer is made, the original proposer either accepts or rejects it. • The game ends when an offer is accepted, but if both offers are rejected, no transaction takes place.

7. Solution to a 2 round bargaining game In the final period the second player recognizes that the first will accept any final strictly positive offer, no matter how small. Therefore the second player reject any offer with a share less than p in the total gains from trade. The first player anticipates the response of the second player to his initial proposal. Accordingly the first player offers the second player proportion p, which is accepted.

8. A finite round bargaining game This game can be extended to a finite number of rounds, where two players alternate between making proposals to each other. Suppose there are T rounds. If the proposal in round t < T is rejected, the bargaining continues for another round with probability p, where 0 < p < 1. In that case the player who has just rejected the most recent proposal makes a counter offer. If T proposals are rejected, the bargaining ends. If no agreement is reached, both players receive nothing. If an agreement is reached, the payoffs reflect the terms of the agreement.

9. Sub-game perfection If the game reaches round T - K without reaching an agreement, the player proposing at that time will treat the last K rounds as a K round game in which he leads off with the first proposal. Therefore the amount a player would initially offer the other in a K round game, is identical to the amount he would offer if there are K rounds to go in T > K round game and it was his turn.

10. Solution to finite round bargaining game • One can show using the principle of mathematical induction that the value of making the first offer in a T round alternating offer bargaining is: vT = 1 – p + p2 – . . . + pT = (1 + pT )/(1 + p) where T is an odd number. • Observe that as T diverges, vT converges to: vT = 1 /(1 + p)

11. Infinite horizon • We now directly investigate the solution of the infinite horizon alternating offer bargaining game. • Let v denote the value of the game to the proposer in an infinite horizon game. • Then the value of the game to the responder is at least pv, since he will be the proposer next period if he rejects the current offer, and there is another offer round. • The proposer can therefore attain a payoff of: v = 1 – pv => v = 1/(1+p) which is the limit of the finite horizon game payoff.

12. Alternatives to taking turns • Bargaining parties do not always take turns. We now explore two alternatives: • Only one player is empowered to make offers, and the other can simply respond by accepting or rejecting it. • Each period in a finite round game one party is selected at random to make an offer.

13. When the order is random Suppose there is a chance of being the proposer in each period. How does the solution differ depend on the chance of being selected? We first consider a 2 round game, and then an infinite horizon game. As before p denote the probability of continuing negotiations if no agreement is reach at the end of the first round.

14. Solution to 2 round random offer game • If the first round proposal is rejected, then the expected payoff to both parties is p/2. • The first round proposer can therefore attain a payoff of: v = 1 – p/2

15. Solution to infinite horizonrandom offer game • If the first round proposal is rejected, then the expected payoff to both parties is pv/2. • The first round proposer can therefore attain a payoff of: v = 1 – pv/2 => 2v = 2 – pv => v = 2/(2 + p) • Note that this is identical to the infinitely repeated game for half the continuation probability. • These examples together demonstrate that the number of offers is not the only determinant of the bargaining outcome.

16. Multiplayer ultimatum games We now increase the number of players to N > 2. Each player is initially allocated a random endowment, which everyone observes. The proposer proposes a system of taxes and subsidies to everyone. If at least J < N –1 of the responders accept the proposal, then the tax subsidy system is put in place. Otherwise the resources are not reallocated, and the players consume their initial endowments.

17. Solution to multiplayer ultimatum game Rank the endowments from the poorest responder to the richest one. Let wn denote the endowment of the nth poorest responder. The proposer offers the J poorest responders their initial endowment (or very little more) and then expropriate the entire wealth of the N – J remaining responders. In equilibrium the J poorest responders accept the proposal, the remaining responders reject the proposal, and it is implemented.

18. Another multiplayer ultimatum game Now suppose there are 2 proposers and one responder. The proposers make simultaneous offers to the responder. Then the responder accepts at most one proposal. If a proposal is rejected, the proposer receives nothing. If a proposal is accepted, the proposer and the responder receive the allocation specified in the terms of the proposal. If both proposals are rejected, nobody receives anything.

19. The solution to this game If a proposer makes an offer that does not give the entire surplus to the responder, then the other proposer could make a slightly more attractive offer. Therefore the solution to this bargaining game is for both proposers to offer the entire gains from trade to the responder, and for the responder to pick either one.

20. Heterogeneous valuations As before, there are 2 proposers and one responder, the proposers make simultaneous offers to the responder, the responder accepts at most one proposal. Also as before if a proposal is rejected, the proposer receives nothing. If a proposal is accepted, the proposer and the responder receive the allocation specified in the terms of the proposal. If both proposals are rejected, nobody receives anything. But let us now suppose that the proposers have different valuations for the item, say v1 and v2 respectively, where v1 < v2.

21. Solving heterogeneous valuations game It is not a best response of either proposer to offer less than the other proposer if the other proposer is offering less than both valuations. Furthermore offering more than your valuation is weakly dominated by bidding less than your valuation. Consequently the first proposer offers v1 or less. Therefore the solution of this game is for the second proposer to offer (marginally more than) v1 and for the responder to always accept the offer of the second proposer.

22. Bargaining with full information • Two striking features characterize all the solutions of the bargaining games that we have played so far: • An agreement is always reached. • Negotiations end after one round. • This occurs because nothing is learned from continuing negotiations, yet a cost is sustained because the opportunity to reach an agreement is put at risk from delaying it.

23. Reaching agreement may be costly • Yet there are many situations where conflict is not instantaneously resolved, and where negotiations break down: • In industrial relations, negotiations can be drawn out, and sometimes lead to strikes. • Plans for construction projects are discussed, contracts are written up, but left unsigned, so the projects are cancelled. • Weddings are postponed and called off.

24. The blame game Consider the following experiment in a multi-round bargaining game called BLAME. There are two players, called BBC and a GOVT. At the beginning of the game BBC makes a statement, which is a number between zero and one, denoted N. (Interpret N as a proportion of blame BBC is prepared to accept.) The GOVT can agree with the BBC statement N or refute it. If the GOVT agrees with the statement then the BBC forfeits £N billion funding, and the GOVT loses 1 - N proportion of the vote next election.

25. Counter proposal If the GOVT refutes the statement, there is a 20 percent chance that no one at all will be blamed, because a more newsworthy issue drowns out the conflict between BBC and GOVT. If the GOVT refutes the statement, and the issue remains newsworthy (this happens with probability 0.8), the GOVT issues its own statement P, also a number between zero and one. (Interpret P as a proportion of blame the GOVT offers to accept.) Should the BBC agree with the statement issued by the GOVT, the GOVT loses P proportion of the vote in the next election, and the BBC loses £5(1-P) billion in funds.

26. Endgame Otherwise the BBC refutes the statement of the GOVT, an arbitrator called HUTTON draws a random variable from a uniform distribution with support [0,2] denoted H, the BBC is fined £H billion, and the GOVT loses H/5 proportion of the vote next election. What will happen? The solution can be found using backwards induction. (See the footnotes or read the press!)

27. Evolving payoffs and discount factors Suppose two (or more) parties are jointly liable for a debt that neither wishes to pay. The players take turns in announcing how much blame should be attributed to each player, and the game ends if a sufficient number of them agree with a tabled proposal. If a proposal is rejected, the total liability might increase (since the problem remains unsolved), or decline (if there is some chance the consequences are less dire than the players originally thought). If the players do not reach a verdict after a given number of rounds, another mechanism, such as an independent enquiry, ascribes liability to each player.

28. Summarizing bargaining outcomes when there is complete information If the value of the match is constant throughout the bargaining phase, and is known by both parties, then the preceding discussion shows that it will be formed immediately, or not at all. The only exception occurs if the current value of the match changes throughout the bargaining phase as the players gather new information together.

29. Bargaining with incomplete information If the value of the match is constant throughout the bargaining phase, and is known by both parties, then the preceding discussion shows that it will be formed immediately, or not at all. In the segment on this topic, we will relax the assumption that all the bargaining parties are fully informed. We now modify the original ultimatum game, between a proposer and a responder, by changing the information structure. Suppose the value to the responder of reaching an agreement is not known by the proposer.

30. An experiment • In this game: • The proposer demands s from the responder. • Then the responder draws a value v from the probability distribution F(v). For convenience we normalize v so that v0 v  v1. • The responder either accepts or rejects the demand of s. • If the demand is accepted the proposer receives s and the responder receives v – s, but if the demand is rejected neither party receives anything.

31. The proposer’s objective • The responder accepts the offer if v > s and rejects the offer otherwise. • Now suppose the proposer maximizes his expected wealth, which can be expressed as: Pr{v > s}s = [1 – F(s)]s • Notice the term in the square brackets [1 – F(s)] is the quantity sold, which declines in price, while s is the price itself.

32. Solution to the game • Let so denote the optimal choice of s for the proposer. Clearly v0 so <v1. • If v0<so <v1, then so satisfies a first order condition for this problem: 1 – F’(so) so – F(so) = 0 • Otherwise so = v0 and the proposer receives: [1 – F(v0)]v0 = v0 • The revenue generated by solving the first order condition is compared with v0 to obtain the solution to the proposer’s problem.

33. F(s) is a uniform distribution • Suppose: F(v) = (v – v0)/ (v1 – v0) for all v0 v  v1 which implies F’(v) = 1 /(v1 – v0) for all v0 v  v1 • Thus the first order condition reduces to: 1 – so/(v1 – v0) – (so – v0)/ (v1 – v0) = 0 => v1 – v0 – so – so + v0 = 0 => 2 so = v1

34. Solving the uniform distribution case • In the interior case so = v1/2. It clearly applies when v0 = 0, but that is not the only case. • We compare v0 with the expected revenue from the interior solution v1(v1 – 2v0)/(v1 – v0). • If v0 > 0 define v1 = kv0 for some k > 1. • Then we obtain an interior solution if k > 1 and: k(k – 2) > k – 1 => k2 – 3k + 1 > 0 • So an interior solution holds if and only if k exceeds the larger of the two roots to this equation, that is k > (3 + 51/2)/2 .

35. F(s) is [0,1] uniform • More specifically let: F(v) = v for all 0  v  1 • Then the interior solution applies so so = ½, and F(v) = ½. Thus exchange only occurs half the time it there are gains from trade. • The trading surplus is: • Given our assumption about F(v) it follows that ¼ of the trading surplus is realized, which is ½ of the potential surplus .

36. Counteroffers Since there is only one offer, there is no opportunity for learning to take place during the bargaining process. We now extend the bargaining phase by allowing the player with private information to make an initial offer. If rejected, the bargaining continues for one final round. For convenience we assume throughout this discussion that F(v) is uniform [0,1].

37. Solution when there are counteroffers • The textbook analyzes solutions of the following type: • There is a threshold valuation v* such that in the first found every manager with valuation v > v* offers the same wage w*, and every manager with valuation v < v* offers lower wages. • In the first round the union rejects every offer below w*, and accepts all other offers. • If the bargaining continues to the final round the union solves the first order condition for the one round problem using the valuations of the manager as truncated at v*.

38. Outcomes of two round bargaining game Note that if the probability of continuation is too high, management will not offer anything in the first round, because it would reveal too much about its own private value v. In this case the bargaining process stalls because management find it strategically beneficial to withhold information that can be used against them.

39. Summary • In today’s session we: • began with some general remarks about bargaining and the importance of unions • analyzed the (two person) ultimatum game • extended the game to treat repeated offers • showed what happens as we change the number of bargaining parties • broadened the discussion to assignment problems where players match with each other • turned to bargaining games where the players have incomplete information • discussed the role of signaling in such games.