Chapter 15 Bargaining with Complete Information

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Chapter 15Bargaining with Complete Information

• Scope and Scale of Bargaining
• Ultimatum Game
• Multiple rounds
• Multilateral bargaining
• Matching

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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 lt T is rejected, the bargaining
  continues for another round with probability p,
  where 0 lt p lt 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.

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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 gt K round game and it was his
  turn.

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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)

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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 gt v 1/(1p)
• which is the limit of the finite horizon game
  payoff.

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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.

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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.

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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.

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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

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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 gt 2v 2 pv gt 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.

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Multiplayer ultimatum games

• We now increase the number of players to N gt 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 lt 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.

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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.

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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.

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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.

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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 lt v2.

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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.

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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

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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 thats
  45-975!)

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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?

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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

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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

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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.

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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.