Chapter 15 Bargaining with Complete Information

Chapter 15Bargaining with Complete Information • Scope and Scale of Bargaining • Ultimatum Game • Multiple rounds • Multilateral bargaining • Matching

Contents • In this chpater we: • begin with some general remarks about bargaining and the importance of unions • turn to bargaining games where the players have incomplete information • discuss the role of signaling in such games.

Scope and Scale of Bargaining This chapter 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. In this chapter we avoid the complications of asymmetrically informed agents. We begin with some general remarks about bargaining and the importance of unions.

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. • Alternative means include: • Capitulation • Predation and expropriation • Warfare and destruction • Bargaining also has these elements in it.

Examples of bargaining situations • Examples of bargaining situations include: • 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. • Pre nuptial agreements are written by partners betrothed to be married. • No fault divorce law facilitates bargaining over the division of assets amongst divorcing partners.

Unions Unions warrant special mention in discussions of bargaining and industrial relations. They are defined as a continuous associations of wage earners for the purpose of maintaining or improving their remuneration and the conditions of their working lives. In the first half of the 20th century union membership grew from almost nothing to 35% of the labor force, only to decline to less that 15% at the turn of the millennium.

How their composition has changed • Hidden within these gross trends are three composition effects worth mentioning : • Employment in the government sector increased from 5% in the early part of the 20th century to 15% in the 1980s, and then stabilized. Union membership in this sector jumped from about 10% to about 40% between 1960 and 1975. • Employment in agriculture declined from 20% to 3% in the same period. This sector was not unionized at the turn of the 20th century. • Unionization in the nonagricultural private sector has reflected the aggregate trend, declining to about 10% of the workforce down from 35%.

Cross sectional characteristics • Within the U.S. membership is highest in the industrial belt connecting New York with Chicago though Pittsburgh and Detroit (20 – 30%), lower in upper New England and the west (10 – 20%), and lowest in the South and Southwest (10% or less). • Males are 50% more likely to be union members than females, mainly reflecting their occupational choices. • Union membership differs greatly across countries: Canada 35% France 12% Sweden 85% United Kingdom 40%

Industrial breakdown and strikes • Strikes are dramatic and newsworthy, but they are also quite rare: Less than 5% of union members go on strike within a typical work year. Less than 1% of potential working hours of union members are lost from strikes, before accounting for compensating overtime. About 90% of all collective agreements are renewed without a strike, but the threat of a strike affects more than 10%.

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.

The ultimatum game We now analyze the (two person) ultimatum game. Then we shall,extend the game to treat repeated offers, show what happens as we change the number of bargaining parties, and finally broaden the discussion to assignment problems where players match with each other.

Ultimatum games We begin with one of the simplest bargaining games for 2 or more players. One player is designated the proposer, the others are called responders. The proposer makes a proposal. If enough responders agree to this proposal, then it is accepted and implemented. Otherwise the proposal is rejected, and a default plan is implemented instead.

2 player ultimatum games • We consider the problem of splitting a dollar between two players, and investigate three versions of it: • 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.

Solution The game theoretic solution is the same in all three cases. Does the experimental evidence support that hypothesis? The solution is for the proposer to extract (almost) all the surplus, and for the responder to accept the proposal. Observe the same outcome would occur if, right at the outset, the responder had capitulated, or if the proposer had expropriated the whole surplus.

Multiple 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.

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.

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.

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.

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)

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.

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.

When only one player makes offers In this case, the proposer makes an offer in the second round, if his first round offer is rejected. The solution reverts to the canonical one period solution. This simply demonstrates that the rules about who can make an offer affects the outcome a lot.

When the order is random Suppose there is an equal chance of being the proposer in each period. 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.

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

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.

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.

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.

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.

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.

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.

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.

Matching A different type of game where players assign themselves to each other and work out the terms of trade Two issues: assignment terms of trade

Multilateral exchange • In all our previous examples, there is at most one transaction. • In such games if more than two players were involved, they competed with each other for the right to be one of the trading partners. • We now suppose there are opportunities on both sides of the trading mechanism to form a partnership with one of a number of different players. • If the prospective partners were identical, then perhaps a market would form. (But that’s 45-975!)

Examples of assignment problems • We now explore an intermediate case. No two prospective bargaining partners are alike, but matching any two partners from either side of the market might be more productive than not matching them at all. • For example: • How are a pool of MBA graduates assigned to companies as employees? • Who gets tenure at at what university? • How are partners matched up across different law firms? • How are partners paired for marriage and parenting?

A multilateral bargaining game • We consider a bargaining game where a fraction of the players, called publishers, offers royalties to another set of players, called authors, to publish their manuscripts. • Each author has only one manuscript, so can can accept at most one offer. • Each publisher can only handle one manuscript, but can make multiple offers. If more than one offer is accepted, the publisher may select any one of the accepted offers

Valuing job matches • Publishers are not a perfect substitutes. Authors are not identical either. • Each publisher and each author is assigned a quality index, denoted respectively by pi and aj for the ith publisher and jth author. • In this lecture we suppose the value of forming a match between the ith publisher and jth author, denoted vij is the product of the two index values of publisher and the author. That is vij = pi aj

The solution One can show that the best publishers match up with the best authors. This is called positive sorting. It arises because the quality of manuscripts and the reputation of publishers are production compliments. The royalty rate to authors increases with their index. Likewise the net profits to publishers is increasing in their index.

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. • Next chapter we explore the implications of relaxing these two assumptions.

Chapter 15 Bargaining with Complete Information