The Bargain

98
The Bargain 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

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Page 1: The Bargain

The Bargain

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

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The BargainBuyer and seller try to agree on a price. Buyer is better off at a price less than b, Seller at a price above s. If b > s, we say there is a positive zone of agreement, or Surplus:

S(urplus) = b(uyer’s reservation price) – s(eller’s reservation price) b

0 50 100 150 200 250

si) b - s > 0 Surplus How to divide?

Page 3: The Bargain

The BargainBuyer and seller try to agree on a price. Buyer is better off at a price less than b, Seller at a price above s. If b > s, we say there is a positive zone of agreement, or Surplus:

S(urplus) = b(uyer’s reservation price) – s(eller’s reservation price) b

0 50 100 150 200 250

si) b - s > 0 Surplus How to divide?

Page 4: The Bargain

The BargainBuyer and seller try to agree on a price. Buyer is better off at a price less than b, Seller at a price above s. If b > s, we say there is a positive zone of agreement, or Surplus:

S(urplus) = b(uyer’s reservation price) – s(eller’s reservation price) b

0 50 100 150 200 250

si) b > s

If b and s are known to both players: How should the surplus be divided?

Page 5: The Bargain

The BargainBuyer and seller try to agree on a price. Buyer is better off at a price less than b, Seller at a price above s. If b > s, we say there is a positive zone of agreement, or Surplus:

S(urplus) = b(uyer’s reservation price) – s(eller’s reservation price) b

0 50 100 150 200 250

si) b > s

If b and s are known to both players: How should the surplus be divided?

Surplus = 50

Page 6: The Bargain

The BargainBuyer and seller try to agree on a price. Buyer is better off at a price less than b, Seller at a price above s. If b = s, we say the price is fully determined, and there is no room for negotiation.

S(urplus) = b(uyer’s reservation price) – s(eller’s reservation price) b

0 50 100 150 200 250

sii) b = s

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The BargainBuyer and seller try to agree on a price. Buyer is better off at a price less than b, Seller at a price above s. If b < s, there is nothing to gained from the exchange.

S(urplus) = b(uyer’s reservation price) – s(eller’s reservation price) b

0 50 100 150 200 250

s(iii) b < s No “zone of agreement”

Page 8: The Bargain

The BargainBuyer and seller try to agree on a price. Buyer is better off at a price less than b, Seller at a price above s. If b < s, there is nothing to gained from the exchange.

S(urplus) = b(uyer’s reservation price) – s(eller’s reservation price) b

0 50 100 150 200 250

s(iii) b < s No “zone of agreement”

What happens if information is incomplete?

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We Play a Game

PROPOSER RESPONDER

Player # ____ Player # ____

Offer $ _____ Accept Reject

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We Play a Game

PROPOSER RESPONDER

Player # ____ Player # ____

Offer $ _____ Accept Reject

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We Play a Game

PROPOSER RESPONDER

Player # ____ Player # ____

Offer $ _____ Accept Reject

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The Ultimatum Game

OFFERS

543210

REJECTEDACCEPTED

N = 20Mean = $1.30

9 Offers > 0 Rejected1 Offer < 1.00 (20%) Accepted

(3/6/00)

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The Ultimatum Game

OFFERS

543210

REJECTEDACCEPTED

N = 33Mean = $1.75

10 Offers > 0 Rejected1 Offer < $1 (20%) Accepted

(2/28/01)

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The Ultimatum Game

OFFERS

543210

REJECTEDACCEPTED

N = 37Mean = $1.69

10 Offers > 0 Rejected*3 Offers < $1 (20%) Accepted

(2/27/02)* 1 subject offered 0

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The Ultimatum Game

OFFERS

543210

REJECTEDACCEPTED

N = 12Mean = $2.77

2 Offers > 0 Rejected0 Offers < 1.00 (20%) Accepted

(7/10/03)

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The Ultimatum Game

OFFERS

543210

REJECTEDACCEPTED

N = 17Mean = $2.30

3 Offers > 0 Rejected0 Offers < 1.00 (20%) Accepted

(3/10/04)

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The Ultimatum Game

OFFERS

543210

REJECTEDACCEPTED

N = 119Mean = $2.28

34 Offers > 0 Rejected5/25 Offers < 1.00 (20%)

Accepted

Pooled data

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The Ultimatum Game

OFFERS

543210

REJECTEDACCEPTED

N = 119Mean = $2.28

34 Offers > 0 Rejected5/25 Offers < 1.00 (20%)

Accepted

Pooled data

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The Ultimatum Game

0 2.72 5 P1

P2

5

2.28

0

2.50

1.00

What is the lowest acceptable offer?

8/8

4/4

21/23

2/22/2

3/3

18/26

13/15N = 119

Mean = $2.2834 Offers > 0 Rejected

5/25 Offers < 1.00 (20%) Accepted

Pooled data

5/6

3/17

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The Ultimatum Game

0 2.72 5 P1

P2

5

2.28

0

2.50

1.00

What is the lowest acceptable offer?

8/8

4/4

21/23

2/22/2

3/3

18/26

13/15N = 119

Mean = $2.2834 Offers > 0 Rejected

5/25 Offers < 1.00 (20%) Accepted

Pooled data

5/6

3/17

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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 <20% are rare and usually rejected

Guth Schmittberger, and Schwarze (1982)Kahnemann, Knetsch, and Thaler (1986)Also, Camerer and Thaler (1995)

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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 <20% are rare and usually rejected

Guth Schmittberger, and Schwarze (1982)Kahnemann, Knetsch, and Thaler (1986)Also, Camerer and Thaler (1995)

How would you advise Proposer?

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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 <20% are rare and usually rejected

Guth Schmittberger, and Schwarze (1982)Kahnemann, Knetsch, and Thaler (1986)Also, Camerer and Thaler (1995)

How would you advise Proposer?

What do you think would happen if the game were repeated?

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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

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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

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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

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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

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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

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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

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We Play Some Games

PROPOSER RESPONDER

Player # ____ Player # ____

Offer 2 or 5 Accept Reject(Keep 8 5)

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We Play Some Games

An offer to give 2 and keep 8 is accepted:

PROPOSER RESPONDER

Player # ____ Player # ____

Offer 2 or 5 Accept Reject(Keep 8 5)

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Fair Play

8 0 5 0 8 0 2 02 0 5 0 2 0 8 0

GAME A GAME B

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Fair Play

8 0 8 0 8 0 10 02 0 2 0 2 0 0 0

GAME C GAME D

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Fair Play

8 0 5 0 8 0 2 02 0 5 0 2 0 8 0

GAME A GAME B

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Fair Play

8 0 8 0 8 0 10 02 0 2 0 2 0 0 0

GAME C GAME D

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Fair Play

A B C D

50%

40

30

20

10

0

3/7

1/4

2/4

0/9

Rejection

Rates,

(8,2) Offer

(5,5) (2,8) (8,2) (10,0) Alternative Offer

4/18/01, in Class.

24 (8,2) Offers 2 (5,5) Offers N = 26

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Fair Play

A B C D

50%

40

30

20

10

0

5/72/3

1/2

2/12

Rejection

Rates,

(8,2) Offer

(5,5) (2,8) (8,2) (10,0) Alternative Offer

4/15/02, in Class.

24 (8,2) Offers 6 (5,5) Offers N = 30

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Fair Play

A B C D

50%

40

30

20

10

0

Source: Falk, Fehr & Fischbacher, 1999

Rejection

Rates,

(8,2) Offer

(5,5) (2,8) (8,2) (10,0) Alternative Offer

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Fair Play

What determines a fair offer?

• Relative shares• Intentions• Endowments• Reference groups• Norms, “manners,” or history

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Fair PlayThese results show that identical offers in an ultimatum game generate systematically different rejection rates, depending on the other offer available to Proposer (but not made). This may reflect considerations of fairness:

i) not only own payoffs, but also relative payoffs matter;

ii) intentions matter.

(FFF, 1999, p. 1)

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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.

Page 42: The Bargain

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.

Page 43: The Bargain

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.

Page 44: The Bargain

Bargaining Games

P2

1

0 1 P1

Disagreement

point

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.

Find the NE of this game.

Divide a Dollar

P1= x; P2 = 1-x.

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Bargaining Games

P2

1

0 1 P1

Disagreement

point

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.

Divide a Dollar

P1= x; P2 = 1-x.

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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.

Subgame Perfection

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(0,0) (3,1)

1

2

Subgame Perfection

(2,2)

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.

Enter Don’t Enter

Fight Don’t Fight

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(0,0) (3,1)

1

2

Subgame Perfection

(2,2)

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.

Enter Don’t Enter

Fight Don’t Fight

Subgame

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(0,0) (3,1)

1

2

Subgame Perfection

(2,2)

Chain Store Game

Enter Don’t

Fight Don’t

0, 0 3, 1

2, 2 2, 2

Fight Don’t

Enter

Don’t

NE = {(E,D), (D,F)}.

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(0,0) (3,1)

1

2

Subgame Perfection

(2,2)

Chain Store Game

Enter Don’t

Fight Don’t

0, 0 3, 1

2, 2 2, 2

Fight Don’t

Enter

Don’t

NE = {(E,D), (D,F)}, but Fight for Player 2 is an incredible threat.

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(0,0) (3,1)

1

2

Subgame Perfection

(2,2)

Chain Store Game

Enter Don’t

Fight Don’t

0, 0 3, 1

2, 2 2, 2

Fight Don’t

Enter

Don’t

NE = {(E,D), (D,F)}. SPNE = {(E,D)}.Subgame Perfect Nash Equilibrium

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A(ccept)

2

H(igh)

1

L(ow)

R(eject)

5,5

0,0

8,2

0,0

Proposer (Player 1) can make

High Offer (50-50%) or Low Offer (80-20%).

Subgame PerfectionMini-Ultimatum Game

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A(ccept)

2

H(igh)

1

L(ow)

R(eject)

H 5,5 0,0 5,5 0,0

L 8,2 0,0 0,0 8,2

5,5

0,0

8,2

0,0

AA RR AR RA

Subgame PerfectionMini-Ultimatum Game

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A(ccept)

2

H(igh)

1

L(ow)

R(eject)

H 5,5 0,0 5,5 0,0

L 8,2 0,0 0,0 8,2

5,5

0,0

8,2

0,0

AA RR AR RA

Subgame PerfectionMini-Ultimatum Game

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A(ccept)

2

H(igh)

1

L(ow)

R(eject)

H 5,5 0,0 5,5 0,0

L 8,2 0,0 0,0 8,2

5,5

0,0

8,2

0,0

AA RR AR RA

Subgame PerfectionMini-Ultimatum Game

Page 56: The Bargain

A(ccept)

2

H(igh)

1

L(ow)

R(eject)

H 5,5 0,0 5,5 0,0

L 8,2 0,0 0,0 8,2

5,5

0,0

8,2

0,0

AA RR AR RA

Subgame PerfectionMini-Ultimatum Game

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A(ccept)

2

H(igh)

1

L(ow)

R(eject)

H 5,5 0,0 5,5 0,0

L 8,2 0,0 0,0 8,2

5,5

0,0

8,2

0,0

AA RR AR RA

Subgame PerfectionMini-Ultimatum Game

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A(ccept)

2

H(igh)

1

L(ow)

R(eject)

H 5,5 0,0 5,5 0,0

L 8,2 0,0 0,0 8,2

AA RR AR RA

5,5

0,0

8,2

0,0

Subgame Perfect Nash Equilibrium

SPNE = {(L,AA)}(H,AR) and (L,RA) involve incredible threats.

Subgame PerfectionMini-Ultimatum Game

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2

H

1

L

2

H 5,5 0,0 5,5 0,0

L 8,2 1,9 1,9 8,2

5,5

0,0

8,2

1,9

AA RR AR RA

Subgame Perfection

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2

H

1

L

H 5,5 0,0 5,5 0,0

L 8,2 1,9 1,9 8,2

5,5

0,0

1,9 SPNE = {(H,AR)}

AA RR AR RA

Subgame Perfection

Page 61: The Bargain

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

Page 62: The Bargain

Alternating Offer Bargaining GameTwo 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.

1

(a,S-a) 2

(b,S-b) 1

(c,S-c) (0,0)

Page 63: The Bargain

Alternating Offer Bargaining Game

1

(a,S-a) 2

(b,S-b) 1

(c,S-c) (0,0)

S = $5.00N = 3

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Alternating Offer Bargaining Game

1

(a,S-a) 2

(b,S-b) 1

(4.99, 0.01) (0,0)

S = $5.00N = 3

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Alternating Offer Bargaining Game 1

(4.99,0.01) 2

(b,S-b) 1

(4.99,0.01) (0,0)

S = $5.00N = 3

SPNE = (4.99,0.01) The game reduces to an Ultimatum Game

Page 66: The Bargain

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 = Discount parameter

Shrinking Pie Game

Page 67: The Bargain

Shrinking Pie Game

S = $5.00N = 3 = 0.5

1

(a,S-a) 2

(b,S-b) 1

(c,S-c) (0,0)

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Shrinking Pie Game

S = $5.00N = 3 = 0.5

1

(3.74,1.26) 2

(1.25, 1.25) 1

(1.24,0.01) (0,0)

1

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Shrinking Pie Game

S = $5.00N = 3 = 0.5

1

(3.74,1.26) 2

(1.25, 1.25) 1

(1.24,0.01) (0,0)

1

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Shrinking Pie Game

S = $5.00N = 3 = 0.5

1

(3.74,1.26) 2

(1.25, 1.25) 1

(1.24,0.01) (0,0)

1

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Shrinking Pie Game

S = $5.00N = 3 = 0.5

1

(3.74,1.26) 2

(1.25, 1.25) 1

(1.24,0.01) (0,0)

1

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Shrinking Pie Game

S = $5.00N = 4 = 0.5

1

(3.13,1.87) 2

(0.64,1.86) 1

(0.63,0.62) 2

(0.01, 0.61) (0,0)

1

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Shrinking Pie Game

0 3.33 5 P1

P2

5

1.67

0

N = 1 (4.99, 0.01) 2 (2.50, 2.50) 3 (3.74, 1.26) 4 (3.12, 1.88)5 (3.43, 1.57)… …This series converges to (S/(1+), S – S/(1+)) =

(3.33, 1.67)

This pair {S/(1+ ),S-S/(1+ )} are the payoffs of the unique SPNE.

for = ½

1

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Shrinking Pie Game

0 3.33 5 P1

P2

5

1.67

0

N = 1 (4.99, 0.01) 2 (2.50, 2.50) 3 (3.74, 1.26) 4 (3.12, 1.88)5 (3.43, 1.57)… …This series converges to (S/(1+), S – S/(1+)) =

(3.33, 1.67)

This pair {S/(1+ ),S-S/(1+ )} are the payoffs of the unique SPNE.

for = ½

1

2

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Shrinking Pie Game

0 3.33 5 P1

P2

5

1.67

0

N = 1 (4.99, 0.01) 2 (2.50, 2.50) 3 (3.74, 1.26) 4 (3.12, 1.88)5 (3.43, 1.57)… …This series converges to (S/(1+), S – S/(1+)) =

(3.33, 1.67)

This pair {S/(1+ ),S-S/(1+ )} are the payoffs of the unique SPNE.

for = ½

1

2

3

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Shrinking Pie Game

0 3.33 5 P1

P2

5

1.67

0

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/(1+), S – S/(1+)) =

(3.33, 1.67)

This pair {S/(1+ ),S-S/(1+ )} are the payoffs of the unique SPNE.

for = ½

1

2

3

4

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Shrinking Pie Game

0 3.33 5 P1

P2

5

1.67

0

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/(1+), S – S/(1+)) =

(3.33, 1.67)

This pair {S/(1+ ),S-S/(1+ )} are the payoffs of the unique SPNE.

for = ½

1

2

3

45

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Shrinking Pie Game

0 3.33 5 P1

P2

5

1.67

0

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/(1+), S – S/(1+)) =

(3.33, 1.67)

This pair {S/(1+ ),S-S/(1+ )} are the payoffs of the unique SPNE.

for = ½

1

2

3

45

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Shrinking Pie GameOptimal Offer (O*) expressed as a share of the total sum to

be divided = [S-S/(1+)]/S

O* = /(1+

SPNE = {1- [/(1+ )], /(1+ )}

Thus both =1 and =0 are special cases of Rubinstein’s model:

When =1 (no bargaining costs), O* = 1/2When =0, game collapses to the ultimatum version and O* = 0 (+)

Page 80: The Bargain

Shrinking Pie GameRubinstein’s solution: If a bargaining game is played in a seriesof alternating offers, and if a speedy resolution is preferred toone that takes longer, then there is only one offer that a rationalplayer should make, and the only rational thing for the opponentto do is accept it immediately! (See Gibbons: 68-71)

Recall that NE is not a very precise solution, because mostgames have multiple NE. Incorporating time imposes aconstraint (bargaining cost) -> selects SPNE from the set of NE.

Even if the final period is unknown (and hence backwardinduction is not possible), it is possible to arrive at a uniqueoutcome that should be (chosen by/agreeable to) rationalplayers.

Page 81: The Bargain

Bargaining GamesBargaining 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”).

Page 82: The Bargain

Acquiring a CompanyBUYER represents Company A (the Acquirer), which is currently considering make a tender offer to acquire Company T (the Target) from SELLER. BUYER and SELLER are going to be meeting to negotiate a price.

Company T is privately held, so its true value is known only to SELLER. Whatever the value, Company T is worth 50% more in the hands of the acquiring company, due to improved management and corporate synergies. BUYER only knows that its value is somewhere between 0 and 100 ($/share), with all values equally likely.

Source: M. Bazerman

Page 83: The Bargain

Acquiring a CompanyBUYER represents Company A (the Acquirer), which is currently considering make a tender offer to acquire Company T (the Target) from SELLER. BUYER and SELLER are going to be meeting to negotiate a price.

Company T is privately held, so its true value is known only to SELLER. Whatever the value, Company T is worth 50% more in the hands of the acquiring company, due to improved management and corporate synergies. BUYER only knows that its value is somewhere between 0 and 100 ($/share), with all values equally likely.

Source: M. Bazerman

Page 84: The Bargain

Acquiring a Company

What offer should Buyer make?

Page 85: The Bargain

Acquiring a Company

5

Source: Bazerman, 1992

9

1 0 4 4

7

27

18

45

123 BU MBA Students

$0 10-15 20-25 30-35 40-45 50-55 60-65 70-75 80-85 90-95Offers

Page 86: The Bargain

Acquiring a Company

$0 10-15 20-25 30-35 40-45 50-55 60-65 70-75 80-85 90-95

5

Source: Bazerman, 1992

9

1 0 4 4

7

27

18

45

123 BU MBA Students

Similar results from

MIT Master’s Candidates

CPA; CEOs.

Offers

Page 87: The Bargain

Acquiring a CompanyThere are many ways to depict the decision tree for this game. Buyer moves first and can make any Offer from to.

Let’s say Buyer offers $60.

O(ffer) = 0 60 100

Buyer

SellerAccept Reject

O – s = -60 30

EP(O) = - 15

The expected payoff of a $60 offer is a net loss of - $15.

Page 88: The Bargain

Acquiring a CompanyThere are many ways to depict the decision tree for this game. Buyer moves first and can make any Offer from to.

Seller accepts if O > s.

O(ffer) = 0 60 100

Buyer

SellerAccept Reject

Chance

s < 60 s > 60

s = 0 60O – s = -60 30

EP(O) = - 15

The expected payoff of a $60 offer is a net loss of - $15.

Page 89: The Bargain

Acquiring a CompanyThere are many ways to depict the decision tree for this game. Buyer moves first and can make any Offer from to.

Let’s say Buyer offers $60.

O(ffer) = 0 60 100

Buyer

SellerAccept Reject

Chance

s < 60 s > 60

s = 0 60s – O = -60 30

EP(O) = - 15

The expected payoff of a $60 offer is a net loss of - $15.

Page 90: The Bargain

Acquiring a Company OFFER VALUE ACCEPT OR VALUE GAIN OR

TO SELLER REJECT TO BUYER LOSS (O) (s) (3/2 s = b) (b - O) $60 $0 A $0 $-60

10 A 15 -45 20 A 30 -30 30 A 45 -15 40 A 60 0 50 A 75 15 60 R - - 70 R - -

Page 91: The Bargain

Acquiring a CompanyThe key to the problem is the asymmetric information structure of the game. SELLER knows the true value of the company (s). BUYER knows only the upper and lower limits (0 < s < 100). Therefore, buyer must form an expectation on s (s').

BUYER also knows that the company is worth 50% more under the new management, i.e., b' = 3/2 s'. BUYER makes an offer (O). The expected payoff of the offer, EP(O), is the difference between the offer and the expected value of the company in the hands of BUYER:

EP(O) = b‘ – O = 3/2s‘ – O.

Page 92: The Bargain

Acquiring a CompanyBUYER wants to maximize her payoff by offering the smallest amount (O) she expects will be accepted:

EP(O) = b‘ – O = 3/2s‘ – O.

O = s' + . Seller accepts if O > s.

Now consider this: Buyer has formed her expectation based on very little information. If Buyer offers O and Seller accepts, this considerably increases Buyer’s information, so she can now update her expectation on s.

How should Buyer update her expectation, conditioned on the new information that s < O?  

Page 93: The Bargain

Acquiring a CompanyBUYER wants to maximize her payoff by offering the smallest amount (O) she expects will be accepted:

EP(O) = b‘ – O = 3/2s‘ – O.

O = s' + . Seller accepts if O > s.

Now consider this: Buyer has formed her expectation based on very little information. If Buyer offers O and Seller accepts, this considerably increases Buyer’s information, so she can now update her expectation on s.

How should Buyer update her expectation, conditioned on the new information that s < O?  

Page 94: The Bargain

Acquiring a CompanyBUYER wants to maximize her payoff by offering the smallest amount (O) she expects will be accepted:

EP(O) = b‘ – O = 3/2s‘ – O.

O = s' + . Seller accepts if O > s.

Let’s say BUYER offers $50. If SELLER accepts, BUYER knows that s cannot be greater than (or equal to) 50, that is: 0 < s < 50. Since all values are equally likely, s''/(s < O) = 25. The expected value of the company to BUYER (b'' = 3/2s'' = 37.50), which is less than the 50 she just offered to pay. (EP(O) = - 12.5.) When SELLER accepts, BUYER gets a sinking feeling in the pit of her stomach.

Page 95: The Bargain

Acquiring a CompanyBUYER wants to maximize her payoff by offering the smallest amount (O) she expects will be accepted:

EP(O) = b‘ – O = 3/2s‘ – O.

O = s' + . Seller accepts if O > s.

Let’s say BUYER offers $50. If SELLER accepts, BUYER knows that s cannot be greater than (or equal to) 50, that is: 0 < s < 50. Since all values are equally likely, s''/(s < O) = 25. The expected value of the company to BUYER (b'' = 3/2s'' = 37.50), which is less than the 50 she just offered to pay. (EP(O) = - 12.5.) When SELLER accepts, BUYER gets a sinking feeling in the pit of her stomach.

THE WINNER’S CURSE!

Page 96: The Bargain

Acquiring a CompanyBUYER wants to maximize her payoff by offering the smallest amount (O) she expects will be accepted:

EP(O) = b‘ – O = 3/2s‘ – O.

O = s' + . Seller accepts if O > s.

Generally:

EP(O) = O - ¼s' (-).

EP is negative for all values of O.

THE WINNER’S CURSE!

Page 97: The Bargain

Acquiring a Company• The high level of uncertainty swamps the potential gains

available, such that value is often left on the table, i.e., on average the outcome is inefficient.

• Under these particular conditions, BUYER should not make an offer.

• SELLER has an incentive to reveal some information to BUYER, because if BUYER can reduce the uncertainty, she may make an offer that leaves both players better off.

Page 98: The Bargain

Bargaining Games• 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).

• Rubinstein’s 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!