Bargaining

1
Bargaining

• Whoever offers to another a bargain of any kind,
  proposes to do this. Give me that which I want,
  and you shall have this which you want and it
  is this manner that we obtain from one another
  the far greater part of those good offices we
  stand in need of. It is not from the benevolence
  of the butcher, the brewer, or the baker that we
  expect our dinner, but from their regard to their
  own interest.
• -- A. Smith, 1776

2
Bargaining

• We Play a Game
• Bargaining Games
• Credibility
• Subgame Perfection
• Alternating Offers and Shrinking Pies

3
We Play a Game
PROPOSER RESPONDER Player ____ Player
____ Offer _____ Accept Reject
4
The Ultimatum Game

• OFFERS
• 5
• 4
• 3
• 2
• 1
• 0

REJECTEDACCEPTED
N 20 Mean 1.30 9 Offers gt 0 Rejected 1 Offer
lt 1.00 (20) Accepted (3/6/00)
5
The Ultimatum Game
OFFERS 5 4 3 2 1 0
REJECTED ACCEPTED
N 32 Mean 1.75 10 Offers gt 0 Rejected 1
Offer lt 1 (20) Accepted (2/28/01)
6
The Ultimatum Game
OFFERS 5 4 3 2 1 0
REJECTED ACCEPTED
N 38 Mean 1.69 10 Offers gt 0 Rejected 3
Offers lt 1 (20) Accepted (2/27/02)
1 subject offered 0
7
The Ultimatum Game

• OFFERS
• 5
• 4
• 3
• 2
• 1
• 0

REJECTEDACCEPTED
N 12 Mean 2.77 2 Offers gt 0 Rejected 0
Offers lt 1.00 (20) Accepted (7/10/03)
8
The Ultimatum Game

• P2
• 5
• 1.69
• 0

What is the lowest acceptable offer?
2.50 1.00
N 38 Mean 1.69 10 Offers gt 0 Rejected 3
Offers lt 1 (20) Accepted (2/27/02)
0 3.31 5 P1
1 subject offered 0
9
The Ultimatum Game

• Theory predicts very low offers will be made and
  accepted.
• Experiments show
• Mean offers are 30-40 of the total
• Mode 50
• Offers lt20 are rare and usually rejected
• Guth Schmittberger, and Schwarze (1982)
• Kahnemann, Knetsch, and Thaler (1986)
• Also, Camerer and Thaler (1995)

10
The Ultimatum Game

• How can we explain the divergence between
  predicted and observed results?
• Stakes are too low
• Fairness
• Relative shares matter
• Endowments matter
• Culture, norms, or manners
• People make mistakes
• Time/Impatience

11
Bargaining Games

• Bargaining involves (at least) 2 players who face
  the the opportunity of a profitable joint
  venture, provided they can agree in advance on a
  division between them.
• Bargaining involves a combination of common as
  well as conflicting interests.
• The central issue in all bargaining games is
  credibility the strategic use of threats,
  bluffs, and promises.

12
Bargaining Games
Divide a Dollar

• P2
• 1
• 0 1 P1

Two players have the opportunity to share 1, if
they can agree on a division beforehand. Each
writes down a number. If they add to 1, each
gets her number if not they each get 0. Every
division s.t. x (1-x) 1 is a NE.
P1 x P2 1-x.
Disagreement point
13
Chain Store Game
Credibility
A firm (Player 1) is considering
whether to enter the market of a monopolist
(Player 2). The monopolist can choose to fight
the entrant, or not.
1
Enter Dont Enter Fight
Dont Fight
2
(2,2)
(0,0) (3,1)
14
Credibility
Battle of the Sexes
F O F O
Player 1
2, 1 0, 0 0, 0 1, 2
Fight Opera F O
F O (2,1) (0,0) (0,0)
(1,2)
Player 2
Is there a credible threat?
15
Battle of the Sexes
Credibility

• FIGHT OPERA

EP1 2/3 1/3
2, 1 0, 0 0, 0 1, 2
p 2/3
FIGHT OPERA
4/3
p 1/3
q
NE (1, 1) (0, 0) (2/3, 1/3) Prudent 1/3,
2/3)
16
Battle of the Sexes
Credibility

• FIGHT OPERA

P2
2, 1 0, 0 0, 0 1, 2
FIGHT OPERA
BATNA
P1
NE (1, 1) (0, 0) (2/3, 1/3)
Best Alternative to a Negotiated Agreement
17
Subgame Perfection
Subgame a part (or subset) of an extensive game,
starting at a singleton node (not the initial
node) and continuing to payoffs. Subgame Perfect
Nash Equilibrium (SPNE) a NE achieved by
strategies that also constitute NE in each
subgame. eliminates NE in which the players
threats are not credible. selects the outcome
that would be arrived at via backwards induction.

18
Subgame Perfection
Chain Store Game
A firm (Player 1) is considering
whether to enter the market of a monopolist
(Player 2). Player 2 can then choose to fight
the entrant, or not.
1
Enter Dont Enter Fight
Dont Fight
2
(2,2)
(0,0) (3,1)
Subgame
19
Subgame Perfection
Chain Store Game
Fight Dont Enter Dont
0, 0 3, 1 2, 2 2, 2
1
Enter Dont Fight
Dont
2
(2,2)
(0,0) (3,1)
Subgame Perfect Nash Equilibrium
NE (E,D), (D,F). SPNE (E,D).
20
Subgame Perfection
Mini-Ultimatum Game
5,5 0,0 8,2 0,0

• A(ccept)
• 2
• H(igh)
• 1
• L(ow)
• R(eject)

Proposer (Player 1) can make High Offer
(50-50) or Low Offer (80-20).
21
Subgame Perfection
Mini-Ultimatum Game
5,5 0,0 8,2 0,0

• A(ccept)
• 2
• H(igh)
• 1
• L(ow)
• R(eject)

Subgame Perfect Nash Equilibrium
AA RR AR RA
H 5,5 0,0 5,5 0,0 L
8,2 0,0 0,0 8,2
SPNE (L,AA) (H,AR) and (L,RA) involve
incredible threats.
22
Subgame Perfection
5,5 0,0 8,2 1,9

• 2
• H
• 1
• L
• 2

AA RR AR RA
H 5,5 0,0 5,5 0,0 L
8,2 1,9 1,9 8,2
23
Subgame Perfection
5,5 0,0

• 2
• H
• 1
• L

AA RR AR RA
H 5,5 0,0 5,5 0,0 L
8,2 1,9 1,9 8,2
1,9
SPNE (H,AR)
24
Alternating Offer Bargaining Game

• Two players are to divide a sum of money (S) is a
  finite number (N) of alternating offers. Player
  1 (Buyer) goes first Player 2 (Seller) can
  either accept or counter offer, and so on. The
  game continues until an offer is accepted or N is
  reached. If no offer is accepted, the players
  each get zero.
• A. Rubinstein, 1982

25
Alternating Offer Bargaining Game
1 (a,S-a)
2 (b,S-b) 1 (c,S-c)
(0,0)
S 5.00 N 3
26
Alternating Offer Bargaining Game
1 (a,S-a)
2 (b,S-b) 1 (4.99, 0.01)
(0,0)
S 5.00 N 3
27
Alternating Offer Bargaining Game

• 1
• (4.99,0.01) 2
• (b,S-b) 1
• (4.99,0.01) (0,0)

S 5.00 N 3 SPNE (4.99,0.01)
The game reduces to an Ultimatum Game
28
Shrinking Pie Game

• Now consider what happens if the sum to be
  divided decreases with each round of the game
  (e.g., transaction costs, risk aversion,
  impatience).
• Let S Sum of money to be divided
• N Number of rounds
• d Discount parameter

29
Shrinking Pie Game
1 (3.74,1.26) 2
(1.25, 1.25) 1
(1.24,0.01) (0,0) 1
S 5.00 N 3 d 0.5
30
Shrinking Pie Game
1 (3.13,1.87) 2
(0.64,1.86) 1
(0.63,0.62) 2 (0.01, 0.61)
(0,0) 1
S 5.00 N 4 d 0.5
31
Shrinking Pie Game
for d ½
N 1 (4.99, 0.01) 2 (2.50, 2.50)
3 (3.74, 1.26) 4 (3.13,
1.87) 5 (3.43, 1.57) This series converges
to (S/(1d), S S/(1d)) (3.33,
1.67) This pair S/(1 d),S-S/(1 d) are the
payoffs of the unique SPNE.
P2 5 1.67 0
2
4
5
3
1
0 3.33 5 P1
32
Shrinking Pie Game

• Optimal Offer (O) expressed as a share of the
  total sum to be divided S-S/(1d)/S
• O d/(1d)
• SPNE 1- d /(1 d), d /(1 d)
• Thus both d1 and d0 are special cases of
  Rubinsteins model
• When d1 (no bargaining costs), O 1/2
• When d0, game collapses to the ultimatum version
  and O 0 (e)

33
Shrinking Pie Game
34
Bargaining Games

• Bargaining games are fundamental to understanding
  the price determination mechanism in small
  markets.
• The central issue in all bargaining games is
  credibility the strategic use of threats,
  bluffs, and promises.
• When information is asymmetric, profitable
  exchanges may be left on the table.
• In such cases, there is an incentive to make
  oneself credible (e.g., appraisals audits
  reputable agents brand names lemons laws
  corporate governance).

35
Bargaining Negotiation

• In real-world negotiations, players often have
  incomplete, asymmetric, or private information,
  e.g., only the seller of a used car knows its
  true quality and hence its true value.
• Making agreements is made all the more difficult
  when trust and good faith are lacking and there
  is no legal recourse for breach of contract
  (Schelling, 1960 20).
• Rubinsteins solution If a bargaining game is
  played in a series of alternating offers, and if
  a speedy resolution is preferred to one that
  takes longer, then there is only one offer that a
  rational player should make, and the only
  rational thing for the opponent to do is accept
  it immediately!