Bargaining

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					                         Bargaining
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 • Bargaining Games with Perfect Information

 • Take-it-or-leave-it

 • The Ultimatum Game

 • Alternating Offers Model of Bargaining; Finite Bargaining Rounds




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Bargaining

 • A bargaining situation is characterized by a transaction or deal between two parties that
   creates surplus (gains from trade). The parties must come to an agreement on how to
   divide the surplus.

 • All bargaining situations have two things in common:

     o First, the total payoff that the negotiating parties are capable to create as a result of
       reaching an agreement should be greater than the sum of the individual payoffs that
       they could achieve separately. This excess value is called surplus. Without this
       surplus, the negotiation will be pointless

     o The second important point about bargaining follows from the first: It is a non zero
       sum game. When a surplus exists, the negotiation is about how to divide it up. If the
       agreement is not reached, no surplus is created.

 • Examples of bargaining games
      Salary and benefit negotiations between a labor union and a firm
      Trade or environmental agreements between two countries
      Seller and buyer bargaining over the price of a house, used car, etc


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Bargaining Games

Bargaining Game: a game in which two or more players bargain over how to divide
the gains (surplus) from trade


 • Bargaining games can usually be described in extensive form (game tree), which
   gives a better understanding of the bargaining problem in different strategic
   settings


 • If players decide to engage in trade, the total payoff that they create can be
   represented by M


 • If the players in a bargaining game do not reach an agreement and walk away
   from the game, they receive their disagreement value.


 • The disagreement value is also commonly referred to as BATNA (best alternative
   to negotiate agreement)
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Bargaining Games


 • Denote the disagreement value for player 1 as “a” and the disagreement value for
   player 2 as “b”. In many cases the disagreement value for both players is zero



 • The bargaining surplus or gains from trade can be defined as S=M – (a + b).



 • In bargaining games players decide how to split S (or equivalently how to split M).
   In the game players move sequentially making offers, accepting/rejecting offers,
   possibly followed by counteroffers



 • Remember: Engaging in negotiation only makes sense if S>0 (surplus is positive)


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Bargaining Games: Take-it-or-leave-it

  • The take-it-or-leave-it game is the simplest sequential move bargaining game between two
    players, with each player making only one move


  • Player 1 moves first and proposes a division of the total payoff M created by trade


  • For example x for player 1 and M-x for player 2


  • Player 2 moves second and must decide whether to accept or reject player’s 1 proposal


  • If player 2 accepts the proposal is implemented


  • If player 2 rejects then both player receive their disagreement value


  • The game has a simple backward induction solution equilibrium in which player 2 accepts
    the proposal if M-x ≥ b, her disagreement value; and player 1 offers exactly x=M-b.
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Bargaining Games: Used Car Example

 • Used car example: the buyer is willing to pay a maximum price of 8500 for a used
   car, the seller will not sell it by less than 8000


 • The total payoff that trade can create in this game is given by M=500 and the
   disagreement values are 0 and 0. Then the total surplus is also given by S=500


 • Suppose the buyer moves first and knows the maximum value the seller attaches
   to the car, hence the buyer knows the seller will reject any price p<8000 and will
   accept any price p>=8000


 • Buyer maximizes his payoff by offering p=8000 or x=500 (why?) and the seller
   will accept



 • What will happen if the seller moves first?
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The ultimatum game

 • Players 1 and 2 are given the chance to agree over the division of 10 dollars. Player 1
   moves first and proposes a division of $10, and player 2 can accept the proposal or reject it,
   in which case both players get zero: a=b=0. Note that in this case the total surplus is also
   equal to 10.




 • It is optimal for player 2 to accept any offer (10 – x) as long as 10 – x ≥ 0. Hence player 1
   will give player 2 the smallest acceptable share. This unique backward induction
   equilibrium has player 1 proposing x=10 and player 2 accepting all proposals x ≤ 10

 • The player making the offer gets the entire surplus (gains from trade)

 • Take or leave it offers are excessively trivial to represent a real bargaining situation. There
   is no back and forth bargaining
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The ultimatum game

 • Backward induction suggest that player 1 should offer player 2 zero or the minimal unit
   (say 1 cent), and player 2 should accept such an offer. But the one sided outcome of
   surplus division is simply due to the one-sided structure of the game. What happens if the
   players are concerned with fairness?

 • In experiments in which subjects are together in a room, the most common offer is a 50-50
   split. Very few offers to player 2 are smaller than 25; and if made, they are often rejected

 • Even tough the split is more uneven in the dictator game (player 1 proposes and player 2
   can only agree) the majority of offers exceed 30% of the surplus

 • We can include this in the game structure and calculate the subgame perfect equilibrium.

 • But there is most to it, if players 2 rejects player’s 1 offer, is it really believable that both
   players will walk away even though there are potential gains from trade (M > a + b)?; or
   do they continue bargaining?

 • Many real world negotiations are more likely to include back and forth bargaining.



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Alternating Offers Model of Bargaining

  • This is a sequential move game where players have perfect information at each move

  • Players take turns making alternating offers, with one offer per round, which resembles
    real back and forth bargaining

  • The round numbers are given by t=1,2,3,… and we denote by xz the amount that player 1
    asks for in bargaining round z (when is his turn to make an offer), and by ys the amount
    that player 2 asks for in each bargaining round s (when is his turn to make an offer).

  • Player 1 moves first in round one by proposing to keep x1 for himself and giving player 2
    M-x1

  • If player 2 accepts, we have a deal. If player 2 rejects, another bargaining round is played
    in which player 2 proposes to keep y2 for herself an M-y2 for player 1

  • If player 1 accepts we have deal, otherwise player 1 gets to make the next proposal

  • Bargaining continues until an agreement is reached, or if no agreement is reached the
    players earn their disagreement values: (a, b)


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Alternating Offers Model of Bargaining




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Alternating Offers Model of Bargaining



  • The alternating offer bargaining could be of finite or infinite length



  • Why might the bargaining game end at some point?


             Both sides agree to a deadline in advance

             Gains from trade M are zero at a certain date, or the surplus diminishes in value
             over time and may fall below the disagreement values

             The players are impatient and prefer money now to money received in later
             periods

             The discount factor provides a means for evaluating future money amounts in
             terms of current money amounts, If the discount factor is high players are patient
             and if it is low players are impatient

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The basketball fan and the scalper

A scalper and a basketball fan are bargaining over ticket for a final game. The fan has
a value of watching the game equal to $100 ($25 per quarter) and wants to maximize
his net payoff; the scalper wants to earn as much money as possible. If they don’t
reach an agreement both get a payoff of zero (a=b=0).

Alternative offers over four quarters

Suppose that the fan and the scalper bargain for the ticket at the beginning of each
quarter

  • The scalper makes an offer at the beginning of quarters 1 and 3 and the fan makes
    an offer before the start of quarters 2 and 4. Find the outcome and the SPNE of the
    game.

  • Will there be an inefficient delay in reaching the agreement (fan missing part of
    the game)? Does the scalper has the kind of extreme bargaining power as with a
    take-it-or-leave it offer at the beginning of the game?


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The basketball fan and the scalper: SPNE


  • Let’s start at stage 4. In this stage the fan proposes a price p4 and the scalper decides
    whether to accept or reject.

        After the fan has made a proposal p4, if the scalper accepts he gets p4, if he rejects he
        gets 0. Then his best response will be to accept any price such that p4 ≥ 0; and reject
        otherwise

        Note that the fan payoff is given by 25-p4, then his best response is to offer p4=0,
        which gives him a payoff of 25.

  • Now let’s look at stage 3. In this stage the scalper proposes a price p3 and the fan decides
    whether to accept or reject.

        The fan gets 50-p3 if he accepts and gets 25 if he rejects (why?), then his best
        response is to accept an offer if 25≥p3

        The scalper gets p3 if the offer is accepted and he gets 0 if it is rejected, then he
        maximizes his payoff by offering p3=25


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The basketball fan and the scalper: SPNE


  • Now let’s look at stage 2. In this stage the fan proposes a price p2 and the scalper decides
    whether to accept or reject.

        The scalper gets p2 if he accepts the offer and he gets 25 if the offer is rejected. Then
        his best response is to accept p2 if p2>=25, and reject otherwise

        The fan gets 75-p2 if the offer is accepted and he gets 25 if the offer is rejected. Then
        his best response is to offer p2=25 (the scalper will accept and fan gets payoff=50)

  • Now let’s look at stage 1. In this stage the scalper proposes a price p1 and the fan decides
    whether to accept or reject.

        The fan gets 100-p1 if he accepts and he gets 50 if he rejects. Then his best response
        is to accept if p1<=50

        The scalper gets p1 if the offer is accepted and he gets 25 if the offer is rejected. Then
        he maximizes his payoff by offering p1=50




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The basketball fan and the scalper: SPNE

  • SPNE

    Scalper (round one: offer p1*=50; round 2: accept if p2>=25, reject otherwise; round 3:
    after rejection in period 2 offer p3*=25; round 4: accept if p4>=0, reject otherwise)
    Fan (round one: accept if p1<=50, reject otherwise; round 2: after rejection in round 1
    offer p2*=25, round 3: accept if p3<=25, reject otherwise; round 4: after rejection in round
    3 offer p4*=0)

  • Outcome: Scalper offers 50 in the first round and fan accepts.

  • Ability to make counter-offers increases fan bargaining power and results in more even
    surplus split

  • With rational players there are no inefficient delays in reaching an agreement (the fan
    enjoys the entire game)

  • What will happen in the fan is the first to move? Discuss

  • Find the SPNE if for the fan watching the whole game or watching just part of it has the
    same value (equal to $100)

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Discounting: Real state bargaining


  • A seller with a minimum acceptable price of $100,000 is bargaining over the sale
    price of a summer cottage with a buyer with a willingness to pay at most 120.000.
    If they not reach an agreement each gets a payoff of zero (a=b=0). Then S=20000

  • The discount factor for both players is 0.9

  • There are just 2 rounds of bargaining because the seller needs to sell the house by
    a certain date

  • Order of Moves: The buyer makes the first offer and then the seller makes a
    counteroffer in the second round. Find the SPNE

  • As usual the bargaining game can be solved using backward induction starting at
    the last stage

  • What will happen if the seller moves first?


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Discounting: Real state bargaining

  • Let’s look at the second round

    The buyer best response is to accept any offer p2 such that 120,000-p2≥0 and reject
    otherwise. Then the seller best response is to offer p2*=120,000.

  • Now let’s look at the first round

    If the seller accepts he gets p1-100000 if he rejects he gets 20000 in period 2, which is
    equivalent to 20,000*δ=18000 in period one. Then his best response is to accept an offer
    p1 in period one if p1-100000≥18000.
    The buyer gets 120,000-p1 from an offer that is accepted and gets 0 if the offer is rejected.
    Then his bets response is to offer p1*=118,000

  • Outcome: the buyer offer 118,000 in period one and the seller accepts. Despite multiple
    rounds, bargaining ends after the first round (no delays!)

  • With discounting it is always optimal to propose the minimum acceptable amount instead
    of waiting for the next round of bargaining, because the passive player will accept less
    than what she would get in the next round due to impatience. The bargaining power of a
    player is lower the more impatient she is to reach and agreement, or the greater the time
    lag between two rounds of bargaining.
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