E. Lehrer, E. Kalai, A. Kalai, D. Sametwww.tau.ac.il/~dsamet

Ehud

Adam

Kalai

Kalai

Ehud Lehrer

Dov Samet

How to commit to

cooperation

Cooperative

Non-Cooperative

Game Theory

• Strategic considerations

• Nash equilibrium

• Possible outcomes

• Enforcing commitment

Each player’s strategy is a best response to the other players’ strategies.

How to cooperate non-cooperatively?How can non-equilibrium outcomes

be achieved non-cooperatively?

Non-cooperative game G

equ. non-equ.

Non-cooperative game G’

equ.

The Prisoner’s Dilemma

Bonnie and Clyde are apprehended after robbing a bank. The police have little incriminating evidence.

Each of the suspects can choose to confess or to deny.

deny confess

deny

confess

10 yrs

10 yrs1 yrs

1 yrs free

20 yrs20 yrs

free

Bonnie

Clyde

If Bonnie denies …… I’d better confess.

If Bonnie confesses …

… I’d better confess.

>

>No matter whatBonnie does,I am better off confessing.

Clyde reasons…

deny confess

deny

confess

10 yrs

10 yrs

free

20 yrs20 yrs

free

Bonnie

Clyde

1 yrs

1 yrs

Bonnie thinks too…

No matter whatClyde does,I am better off confessing.

… and she reasons exactlythe same way.

deny confess

deny

confess

10 yrs

10 yrs1 yrs

1 yrs free

20 yrs20 yrs

free

Bonnie

Clyde

The outcome…

Both Bonnie and Clyde confess.

confess

deny

confessdeny

10 yrs

10 yrs1 yrs

1 yrs free

20 yrs20 yrs

free

Bonnie

Clyde

PD: cooperative perspective

denyconfess

deny

confess

0

0

2

2

10

10

12

12

confess

deny

confessdeny

10 yrs

10 yrs1 yrs

1 yrs free

20 yrs20 yrs

free

Bonnie

Clyde

denyconfess

deny

confess

0

2

2

12

0

10

10

12

PD: cooperative perspective

Bonnie

Clyde

Feasible outcomes

cc

dcdd

cd

Repeated PD

Bonnie

Clyde

cc

dcdd

cd

cc dc dd cd

PD PD PD

cc dc cd Feasible outcomes

PD

PD

An equilibrium strategy that guarantees ddKeep denying as long as your opponent does.

Else, keep confessing for ever .

dd

cd

Repeated PD

Bonnie

Clyde

cc

dcdd

PD

cc dc dd cd

PD PD PD PD

cc dc dd cd

An equilibrium strategy that guarantees ½ dc + ½ dd

Bonnie’s role: Keep denying as long as Clyde sticks to his role. Else, keep confessing for ever.Clyde’s role: Keep denying on odd days and confessing on even days as long as Bonnie sticks to her role. Else, keep confessing for ever.

Feasible outcomes

½ dc + ½ dd

Time in service of cooperation:• Commitments are long term plans, • Enforcement by punishment, • Enables generation of any frequency of pure outcomes.

The Folk Theorem :Any cooperative outcome, is attainable as an equilibrium in the repeated game.

in which each player gets at least her individually rational payoff,

How can commitments be made without repetition? What outcomes can be achieved?

Commitments to actSuppose Bonnie and Clyde can submit

irrevocable commitments.

Bonnie Clyde

I hereby commit to deny our involvement

in the robbery.

I hereby commit to deny our involvement

in the robbery.

Choosing which commitment to make is

a voluntary non-cooperative action.

I hereby commit to confess our involvement

in the robbery.

An unconditional commitment to act

does not help.

I hereby commit to deny our involvement

in the robbery.

Commitments to actSuppose Bonnie and Clyde can submit

irrevocable commitments.

Bonnie Clyde

I hereby commit to confess our involvement

in the robbery.

Conditional commitments

I hereby commit to deny

if Clyde denies.

The commitment is incomplete.

What if Clyde commits to confess?

I hereby commit to deny

if Clyde denies,to confess

if Clyde confesses.

I hereby commit to deny

if Bonnie denies,to confess

if Bonnie confesses.

Hmmmm......These commitments

fail to determine players’ actions.

Conditional commitmentsMay even be incompatible...

grocer I grocer II

I commit to undersell my competitor.

I commit to undersell my competitor.

Conditional commitments may be incomplete, undefined or incompatible... The problem is that the action is conditioned on the opponent’s action.

commitment condition on commitments

=

Bonnie Clyde

..................

..................

..................

B1 B3B2 C1 C3C2

B2

If C1, confess;If C2, deny;If C3, confess.

C3

If B1, deny;If B2, deny;If B3, confess.

Bonnie confesses,Clyde denies.

Nigel Howard (1971)Paradoxes of Rationality: Theory of Metagames and Political Behaviour

A hierarchy of responses(commitments?)

Player I: actions

Player II: Responses to I’s action.

Player I: Responses to II’s responses to I’s actions.

and so on…

John Harsanyi (1967-8)Games with incomplete information played by Bayesian players

A hierarchy of beliefs

Player I: Beliefs about G

Player II: Beliefs about I’s beliefs about G.

Player I: Beliefs about II’s beliefs about I’s beliefs about G.

and so on…

Types

A player’s type is her beliefs about her opponents’ types

commitment condition on commitments

=

Bonnie Clyde

..................

..................

..................

B1 B3B2 C1 C3C2

B2

If C1, confess;If C2, deny;If C3, confess.

C3

If B1, deny;If B2, deny;If B3, confess.

Good-bye!They get rid

of me...

Each player’s program is the best response to

the opponent’s program

Program Equilibria (Tennenholtz)

function Act (opp_prog) { if (opp_prog = this_prog) then return “deny”; else return “confess”; }

Bonnie Clyde

function Act (opp_prog) { if (opp_prog = this_prog) then return “deny”; else return “confess”;}

Text of program=

Name of commitment

Program=Commitment to act

conditioned on opponent’s commitment

Both Bonnie and Clyde

deny

Both Bonnie and Clyde

deny

Both Bonnie and Clyde

deny

IO

OI

OO

How to mix pure outcome

In Out

In

Out10

0

0

0

10

1

1

0

Entry game

Agent I

Agent II

Agent I

Agent 2

desired agreement

Choose mixed action

Agent I’s commitment:

If II’s commitment is A, play I with probability p and O with prob. 1-p.If II’s commitment is B, .....

p

1-qq

1-p II

IO

OI

OO

How to mix pure outcome

Agent I

Agent 2

desired agreement

II

Agent IComm-s

Agent IIComm-s

g I

g II

b I

b II

I

I

I

O

O

I

I

O

O1/2

1/21/2

1/2

Jointly controlled lotteries

There exist (infinite) commitment sets for player I and II,

such that every feasible outcome (above the individual rational level)

is attained as a mixed Nash equilibrium of the commitment submission game.

A “Folk Theorem” for the commitment submission

game

Some morals…

Base your commitment on the whole scheme of your opponent’s commitment, not on her action.

You can make the choices of your actions unequivocal (deterministic), but…

You should allow for ambiguity concerning the choice of your scheme of commitment.