UNIT II: The Basic Theory

• Zero-sum Games• Nonzero-sum Games• Nash Equilibrium: Properties and Problems• Bargaining Games• Review• Midterm 3/23

3/2

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

Bargaining

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

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.

The Ultimatum Game

OFFERS543210

REJECTEDACCEPTED

N = 15Mean = $2.05

4 Offers > 0 Rejected0 Offer < 1.00 (20%) Accepted

(2/25/09)

The Ultimatum Game

0 2.72 5 P1

P2

5

2.28

0

2.50

1.00

What is the lowest acceptable offer?

9/9

4/4

25/27

2/22/2

3/3

20/28

13/15N = 131

Mean = $2.2534 Offers > 0 Rejected

6/26 Offers < 1.00 (20%) Accepted

Pooled data(as of 3/07)

6/7

3/17

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?

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

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

(0,0) (3,1)

1

2

Chain Store Game

(2,2)

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.

Enter Don’t Enter

Fight Don’t Fight

Credibility

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

(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

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

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

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

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

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

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

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)

Alternating Offer Bargaining Game

1

(a,S-a) 2

(b,S-b) 1

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

S = $5.00N = 3

Alternating Offer Bargaining Game

1

(a,S-a) 2

(b,S-b) 1

(4.99, 0.01) (0,0)

S = $5.00N = 3

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

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

Shrinking Pie Game

Shrinking Pie Game

S = $5.00N = 3d = 0.5

1

(a,S-a) 2

(b,dS-b) 1

(c, d2S-c) (0,0)

Shrinking Pie Game

S = $5.00N = 3d = 0.5

1

(3.74,1.26) 2

(1.25, 1.25) 1

(1.24,0.01) (0,0)

1

Shrinking Pie Game

S = $5.00N = 4d = 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

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+d), S – S/(1+d)) =

(3.33, 1.67)

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

for d = ½

1

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+d), S – S/(1+d)) =

(3.33, 1.67)

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

for d = ½

1

2

3

45

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

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

O* = d/(1+d)

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

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

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

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.

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)

Fair Play

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

GAME A GAME B

Fair Play

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

GAME C GAME D

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

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

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

Fair Play

What determines a fair offer?

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

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)

What Counts as Utility?

• Own payoffs Ui(Pi)• Other’s payoffs Ui(Pi+ Pj) sympathy

What Counts as Utility?

• Own payoffs Ui(Pi)• Other’s payoffs Ui(Pi - Pj) envy

What Counts as Utility?

• Own payoffs Ui(Pi)• Other’s payoffs Ui(Pi , Pj)• Equity Ui(Pi + Pi/Pj)• Intentions ?

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.

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!

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