How to commit
Lehrer, E. Kalai,
toE.cooperation
Ehud Lehrer
Ehud Kalai
A. Kalai, D. Samet
www.tau.ac.il/~dsamet
Dov Samet
Adam Kalai
Non-cooperative
Non-Cooperative
game G’
Non-cooperative
game G
• Strategic considerations
• Possible outcomes
Cooperative
• Nash equilibrium
• Enforcing commitment
equ.
equ.
non-equ.
How to cooperate non-cooperatively?
Each player’s strategy
is
How can non-equilibrium
outcomes
a best
response to the
be achieved
non-cooperatively?
other players’ strategies.
The Prisoner’s Dilemma
Bonnie and Clyde are
apprehended after
robbing a bank. The
police have little
incriminating evidence.
Clyde
deny
1 yrs
free
deny
Bonnie
Each of the suspects can
choose to confess or to
deny.
confess
1 yrs
20 yrs
20 yrs
10 yrs
confess
free
10 yrs
Clyde reasons…
Clyde
If Bonnie denies …
… I’d better confess.
deny
1 yrs
If Bonnie confesses …
… I’d better confess.
>
free
deny
Bonnie
1 yrs
20 yrs
20 yrs
No matter what
Bonnie does,
I am better off confessing.
confess
>
confess
free
10 yrs
10 yrs
Bonnie thinks too…
Clyde
deny
… and she reasons exactly
the same way.
confess
1 yrs
free
deny
No matter what
Clyde does,
I am better off confessing.
Bonnie
1 yrs
20 yrs
20 yrs
10 yrs
confess
free
10 yrs
The outcome…
Both Bonnie and Clyde confess.
Clyde
deny
confess
1 yrs
free
deny
Bonnie
1 yrs
20 yrs
20 yrs
10 yrs
confess
free
10 yrs
PD: cooperative perspective
Clyde
deny
confess
1 yrs
free
deny
deny
confess
10
deny
10
12
12
1 yrs
20 yrs
20 yrs
10 yrs
0
2
0
confess
Bonnie
2
confess
free
10 yrs
PD: cooperative perspective
dc
dd
Clyde
cc
deny
confess
10
12
deny
10
0
2
0
confess
12
2
cd
Bonnie
Feasible outcomes
Repeated PD
dc
dd
PD
Clyde
cc
dc
dd
cd
cc
PD
PD
PD
cd
PD
Bonnie
cc
dc
dd
cd
Feasible outcomes
An equilibrium strategy that guarantees dd
Keep denying as long as your opponent does.
Else, keep confessing for ever.
Repeated PD
dc
An equilibrium strategy
PD ½ dc + ½ dd
that guarantees
Bonnie’s role:
Keep denying
cc as Clyde
dc sticks
dd tocd
as long
his
role.
PD keepPD
PDfor ever.
PD
Else,
confessing
Clyde’s role:
Keep denying
odd days
cc ondc
dd cd
and confessing on even days
as long as Bonnie sticks to her
role.
Else, keep confessing for ever.
½ dc + ½ dd
dd
Clyde
cc
cd
Bonnie
Feasible outcomes
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, in which each player
gets at least her individually rational payoff,
is attainable as an equilibrium in the repeated game.
 How can commitments be made without repetition?
 What outcomes can be achieved?
Commitments to act
Choosing which
commitment to make is
a voluntary nonSuppose
Bonnie and Clyde can submit
cooperative
action.
irrevocable commitments.
Bonnie
Clyde
I hereby commit
totoconfess
deny our
our
involvement
in the robbery.
I hereby commit
to deny our
involvement
in the robbery.
Conditional
Commitments
commitments
to act
TheHmmmm......
commitment is
These incomplete.
commitments fail
What
to determine
if Clyde players’
commits
Suppose Bonnie and Clyde can submit
toactions.
confess?
irrevocable commitments.
Bonnie
herebycommit
commit
IIIhereby
hereby commit
to confess
our
to
deny
to deny
involvement
ifif Clyde
Clyde denies,
denies.
into
the
robbery.
confess
if Clyde confesses.
Clyde
IIhereby
herebycommit
commit
to deny
our
to deny
ifinvolvement
Bonnie denies,
in the
robbery.
to confess
if Bonnie confesses.
Conditional commitments
May even be incompatible...
grocer I
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.
grocer II
=
commitment
Bonnie confesses,
Clyde denies.
condition on commitments
Clyde
Bonnie
B1
...
...
...
B2
...
...
...
B3
...
...
...
B2
If C1, confess;
If C2, deny;
If C3, confess.
C1
...
...
...
C2
...
...
...
C3
If B1, deny;
If B2, deny;
If B3, confess.
C3
...
...
...
Nigel Howard (1971)
Paradoxes of Rationality:
Theory of Metagames
and Political Behaviour
John Harsanyi (1967-8)
Games with incomplete
information played by
Bayesian players
A hierarchy of responses
(commitments?)
A hierarchy of beliefs
Types
Player
I: actionstype is her
A player’s
Playerbeliefs
II: Responses
I’s
about to
her
action.opponents’ types
Player I: Beliefs about G
Player II: Beliefs about I’s
beliefs about G.
Player I: Responses to II’s
responses to I’s actions.
Player I: Beliefs about II’s
beliefs about I’s beliefs
about G.
and so on…
and so on…
=
commitment
condition on commitments
Clyde
Bonnie
B1
...
...
...
B2
...
...
...
B3
...
...
...
B2
If C1, confess;
If C2, deny;
If C3, confess.
C1
...
...
...
C2
...
...
...
C3
If B1, deny;
If B2, deny;
If B3, confess.
C3
...
...
...
Good-bye!
They get rid
of me...
Program Equilibria (Tennenholtz)
Text of program
Program
Bonnie
function Act (opp_prog) {
if (opp_prog = this_prog)
then return “deny”;
else return “confess”;
}
=
=
Each of
player’s
program
Name
commitment
is the best response to
the opponent’s
program
Both Bonnie
and Clyde
deny
Both Bonnie
Bonnie
Both
Commitment
to act
and Clyde
Clyde
and
conditioned
denyon
deny
opponent’s commitment
Clyde
function Act (opp_prog) {
if (opp_prog = this_prog)
then return “deny”;
else return “confess”;
}
How to mix pure outcome
Agent II
q
1-q
In
Out
p
Agent I
1
In
10
0
desired agreement
0
10
1
1-p Out
OI
Agent 2
0
II
0
IO
OO
Entry game
Agent I
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, .....
How to mix pure outcome
OI
desired agreement
Jointly controlled lotteries
Agent 2
I
Agent II
Comm-s
Agent I
Comm-s
1/2 g I
1/2
1/2
g II
b II
O
OO
O
O
I
1/2 b I
IO
I
O
I
II
I
Agent I
A “Folk Theorem” for the
commitment submission game
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.
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.