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Robert Aumann’sGame and Economic Theory
Sergiu Hart
December 9, 2005Stockholm School of Economics
SERGIU HART c©2005 – p. 1
Robert Aumann’sGame and Economic Theory
Sergiu HartCenter of Rationality,
Dept. of Economics, Dept. of MathematicsThe Hebrew University of Jerusalem
[email protected]://www.ma.huji.ac.il/hart
SERGIU HART c©2005 – p. 2
Aumann’s Short CV
1930: Born in Germany
SERGIU HART c©2005 – p. 3
Aumann’s Short CV
1930: Born in Germany
1955: Ph.D. at M.I.T.
SERGIU HART c©2005 – p. 3
Aumann’s Short CV
1930: Born in Germany
1955: Ph.D. at M.I.T.
From 1956: Professor at the HebrewUniversity of Jerusalem
SERGIU HART c©2005 – p. 3
Aumann’s Short CV
1930: Born in Germany
1955: Ph.D. at M.I.T.
From 1956: Professor at the HebrewUniversity of Jerusalem
1991: EstablishingThe Center for Rationality
SERGIU HART c©2005 – p. 3
Aumann’s Short CV
1930: Born in Germany
1955: Ph.D. at M.I.T.
From 1956: Professor at the HebrewUniversity of Jerusalem
1991: EstablishingThe Center for Rationality
1998-2003: Founding President of theGame Theory Society
SERGIU HART c©2005 – p. 3
Aumann’s Short CV
1930: Born in Germany
1955: Ph.D. at M.I.T.
From 1956: Professor at the HebrewUniversity of Jerusalem
1991: EstablishingThe Center for Rationality
1998-2003: Founding President of theGame Theory Society
From 2001: Retired (?)
SERGIU HART c©2005 – p. 3
2005
SERGIU HART c©2005 – p. 4
Major Contributions
Repeated games
SERGIU HART c©2005 – p. 5
Major Contributions
Repeated games
Perfect competition
SERGIU HART c©2005 – p. 5
Major Contributions
Repeated games
Perfect competition
Correlated equilibrium
SERGIU HART c©2005 – p. 5
Major Contributions
Repeated games
Perfect competition
Correlated equilibrium
Interactive epistemology
SERGIU HART c©2005 – p. 5
Major Contributions
Repeated games
Perfect competition
Correlated equilibrium
Interactive epistemology
Cooperative games
SERGIU HART c©2005 – p. 5
Major Contributions
Repeated games
Perfect competition
Correlated equilibrium
Interactive epistemology
Cooperative games
Foundations
SERGIU HART c©2005 – p. 5
Major Contributions
Repeated games
Perfect competition
Correlated equilibrium
Interactive epistemology
Cooperative games
Foundations
. . .
SERGIU HART c©2005 – p. 5
Repeated Games
SERGIU HART c©2005 – p. 6
The President’s Dilemma
SERGIU HART c©2005 – p. 7
The President’s Dilemma
1 to us 4 to them
1 to us
4 to them
SERGIU HART c©2005 – p. 7
The President’s Dilemma
TAU PRESIDENT
1 to us 4 to them
HUJ 1 to us
PRESIDENT 4 to them
SERGIU HART c©2005 – p. 7
The President’s Dilemma
TAU PRESIDENT
1 to us 4 to them
HUJ 1 to us
PRESIDENT 4 to them
SERGIU HART c©2005 – p. 7
The President’s Dilemma
TAU PRESIDENT
1 to us 4 to them
HUJ 1 to us 1 1
PRESIDENT 4 to them
SERGIU HART c©2005 – p. 7
The President’s Dilemma
TAU PRESIDENT
1 to us 4 to them
HUJ 1 to us 1 1 5 0
PRESIDENT 4 to them
SERGIU HART c©2005 – p. 7
The President’s Dilemma
TAU PRESIDENT
1 to us 4 to them
HUJ 1 to us 1 1 5 0
PRESIDENT 4 to them 0 5
SERGIU HART c©2005 – p. 7
The President’s Dilemma
TAU PRESIDENT
1 to us 4 to them
HUJ 1 to us 1 1 5 0
PRESIDENT 4 to them 0 5 4 4
SERGIU HART c©2005 – p. 7
The President’s Dilemma
TAU PRESIDENT
1 to us 4 to them
HUJ 1 to us 1 1 5 0
PRESIDENT 4 to them 0 5 4 4
NASH EQUILIBRIUM
SERGIU HART c©2005 – p. 7
The President’s Dilemma
TAU PRESIDENT
1 to us 4 to them
HUJ 1 to us 1 1 5 0
PRESIDENT 4 to them 0 5 4 4
NASH EQUILIBRIUM
PARETO OPTIMUM
SERGIU HART c©2005 – p. 7
The Duopolist’s Dilemma
PRICE = 20 − QUANTITY, COST = 8 ∗ QUANTITY + 15
SERGIU HART c©2005 – p. 8
The Duopolist’s Dilemma
PRICE = 20 − QUANTITY, COST = 8 ∗ QUANTITY + 15
4 3
4
3
SERGIU HART c©2005 – p. 8
The Duopolist’s Dilemma
PRICE = 20 − QUANTITY, COST = 8 ∗ QUANTITY + 15
QUANTITY(TAU)
4 3
QUANTITY4
(HUJ) 3
SERGIU HART c©2005 – p. 8
The Duopolist’s Dilemma
PRICE = 20 − QUANTITY, COST = 8 ∗ QUANTITY + 15
QUANTITY(TAU)
4 3
QUANTITY4
(HUJ) 3
SERGIU HART c©2005 – p. 8
The Duopolist’s Dilemma
PRICE = 20 − QUANTITY, COST = 8 ∗ QUANTITY + 15
QUANTITY(TAU)
4 3
QUANTITY4 1 1
(HUJ) 3
SERGIU HART c©2005 – p. 8
The Duopolist’s Dilemma
PRICE = 20 − QUANTITY, COST = 8 ∗ QUANTITY + 15
QUANTITY(TAU)
4 3
QUANTITY4 1 1 5 0
(HUJ) 3
SERGIU HART c©2005 – p. 8
The Duopolist’s Dilemma
PRICE = 20 − QUANTITY, COST = 8 ∗ QUANTITY + 15
QUANTITY(TAU)
4 3
QUANTITY4 1 1 5 0
(HUJ) 3 0 5
SERGIU HART c©2005 – p. 8
The Duopolist’s Dilemma
PRICE = 20 − QUANTITY, COST = 8 ∗ QUANTITY + 15
QUANTITY(TAU)
4 3
QUANTITY4 1 1 5 0
(HUJ) 3 0 5 3 3
SERGIU HART c©2005 – p. 8
The Duopolist’s Dilemma
PRICE = 20 − QUANTITY, COST = 8 ∗ QUANTITY + 15
QUANTITY(TAU)
4 3
QUANTITY4 1 1 5 0
(HUJ) 3 0 5 3 3
NASH EQUILIBRIUM / COURNOT
SERGIU HART c©2005 – p. 8
The Duopolist’s Dilemma
PRICE = 20 − QUANTITY, COST = 8 ∗ QUANTITY + 15
QUANTITY(TAU)
4 3
QUANTITY4 1 1 5 0
(HUJ) 3 0 5 3 3
NASH EQUILIBRIUM / COURNOT
PARETO OPTIMUM / CARTELSERGIU HART c©2005 – p. 8
The Folk Theorem
The set ofNASH EQUILIBRIUM
outcomesof the repeated game
SERGIU HART c©2005 – p. 9
The Folk Theorem
The set ofNASH EQUILIBRIUM
outcomesof the repeated game
equals
SERGIU HART c©2005 – p. 9
The Folk Theorem
The set ofNASH EQUILIBRIUM
outcomesof the repeated game
equals
the set ofFEASIBLE and INDIVIDUALLY RATIONAL
outcomesof the one-shot game .
SERGIU HART c©2005 – p. 9
The Folk Theorem
PAYOFF TOPLAYER 1
PAYOFF TO PLAYER 2
SERGIU HART c©2005 – p. 10
The Folk Theorem
PAYOFF TOPLAYER 1
PAYOFF TO PLAYER 2
(5, 0)
(0, 5)
(4, 4)
(1, 1)
SERGIU HART c©2005 – p. 10
The Folk Theorem
PAYOFF TOPLAYER 1
PAYOFF TO PLAYER 2
(5, 0)
(0, 5)
(4, 4)
(1, 1)(1, 1)
Feasible
SERGIU HART c©2005 – p. 10
The Folk Theorem
PAYOFF TOPLAYER 1
PAYOFF TO PLAYER 2
(5, 0)
(0, 5)
(4, 4)
(1, 1)(1, 1)
Feasible
r2 = 1
r1 = 1SERGIU HART c©2005 – p. 10
The Folk Theorem
PAYOFF TOPLAYER 1
PAYOFF TO PLAYER 2
(5, 0)
(0, 5)
(4, 4)
(1, 1)(1, 1)
Feasible
r2 = 1
r1 = 1
Feasible & IR
SERGIU HART c©2005 – p. 10
The Folk Theorem
Idea of proof:
SERGIU HART c©2005 – p. 11
The Folk Theorem
Idea of proof:
Coordination on a feasible “master plan” ...
SERGIU HART c©2005 – p. 11
The Folk Theorem
Idea of proof:
Coordination on a feasible “master plan” ...
... supported by the threat of “punishment”in case of deviation
SERGIU HART c©2005 – p. 11
The Folk Theorem
Idea of proof:
Coordination on a feasible “master plan” ...
... supported by the threat of “punishment”in case of deviation
Proof:
Strategies, payoffs, ...
Aumann 1959
SERGIU HART c©2005 – p. 11
Repeated Games
“The theory of repeated games ... is concernedwith the evolution of fundamental patterns ofinteraction ...
Aumann 1981SERGIU HART c©2005 – p. 12
Repeated Games
“The theory of repeated games ... is concernedwith the evolution of fundamental patterns ofinteraction ... Its aim is to account forphenomena such as cooperation, altruism,revenge, threats (self-destructive or otherwise),etc.
Aumann 1981SERGIU HART c©2005 – p. 12
Repeated Games
“The theory of repeated games ... is concernedwith the evolution of fundamental patterns ofinteraction ... Its aim is to account forphenomena such as cooperation, altruism,revenge, threats (self-destructive or otherwise),etc.—phenomena which may at first seemirrational—
Aumann 1981SERGIU HART c©2005 – p. 12
Repeated Games
“The theory of repeated games ... is concernedwith the evolution of fundamental patterns ofinteraction ... Its aim is to account forphenomena such as cooperation, altruism,revenge, threats (self-destructive or otherwise),etc.—phenomena which may at first seemirrational—in terms of the usual ‘selfish’utility-maximizing paradigm of game theoryand neoclassical economics.”
Aumann 1981SERGIU HART c©2005 – p. 12
The Folk Theorem
Noncooperative strategic behaviorin the repeated game
yields
Cooperative behavior
SERGIU HART c©2005 – p. 13
The Strong Folk Theorem
SERGIU HART c©2005 – p. 14
The Strong Folk Theorem
The set ofSTRONG NASH EQUILIBRIUM outcomes
of the repeated game
SERGIU HART c©2005 – p. 14
The Strong Folk Theorem
Stable relative to deviations by coalitions
The set ofSTRONG NASH EQUILIBRIUM outcomes
of the repeated game
SERGIU HART c©2005 – p. 14
The Strong Folk Theorem
Stable relative to deviations by coalitions
The set ofSTRONG NASH EQUILIBRIUM outcomes
of the repeated game
equals
the COREof the one-shot game .
SERGIU HART c©2005 – p. 14
The Strong Folk Theorem
Stable relative to deviations by coalitions
The set ofSTRONG NASH EQUILIBRIUM outcomes
of the repeated game
equals
the COREof the one-shot game .
Aumann 1959SERGIU HART c©2005 – p. 14
The Perfect Folk Theorem
SERGIU HART c©2005 – p. 15
The Perfect Folk Theorem
The set ofPERFECT NASH EQUILIBRIUM outcomes
of the repeated game
SERGIU HART c©2005 – p. 15
The Perfect Folk Theorem
Stable also off equilibrium (“credible threats”)
The set ofPERFECT NASH EQUILIBRIUM outcomes
of the repeated game
SERGIU HART c©2005 – p. 15
The Perfect Folk Theorem
Stable also off equilibrium (“credible threats”)
The set ofPERFECT NASH EQUILIBRIUM outcomes
of the repeated game
equals
the set ofFEASIBLE and INDIVIDUALLY RATIONAL
outcomes of the one-shot game .
SERGIU HART c©2005 – p. 15
The Perfect Folk Theorem
Stable also off equilibrium (“credible threats”)
The set ofPERFECT NASH EQUILIBRIUM outcomes
of the repeated game
equals
the set ofFEASIBLE and INDIVIDUALLY RATIONAL
outcomes of the one-shot game .
Aumann & Shapley 1976 || Rubinstein 1976SERGIU HART c©2005 – p. 15
The Folk Theorem
Noncooperative strategic behaviorin the repeated game
yields
Cooperative behavior
SERGIU HART c©2005 – p. 16
Asymmetric Information
When players have private information:
SERGIU HART c©2005 – p. 17
Asymmetric Information
When players have private information:
How should an “informed” player takeadvantage of his information?
SERGIU HART c©2005 – p. 17
Asymmetric Information
When players have private information:
How should an “informed” player takeadvantage of his information?
How should an “uninformed” playerbehave?
SERGIU HART c©2005 – p. 17
Asymmetric Information
When players have private information:
How should an “informed” player takeadvantage of his information?
How should an “uninformed” playerbehave?
One-shot game vs. repeated game
SERGIU HART c©2005 – p. 17
Asymmetric Information
When players have private information:
How should an “informed” player takeadvantage of his information?
How should an “uninformed” playerbehave?
One-shot game vs. repeated game
When the information is used
SERGIU HART c©2005 – p. 17
Asymmetric Information
When players have private information:
How should an “informed” player takeadvantage of his information?
How should an “uninformed” playerbehave?
One-shot game vs. repeated game
When the information is used=⇒ The information is revealed
SERGIU HART c©2005 – p. 17
Asymmetric Information
When players have private information:
How should an “informed” player takeadvantage of his information?
How should an “uninformed” playerbehave?
One-shot game vs. repeated game
When the information is used=⇒ The information is revealed=⇒ The advantage disappears?
SERGIU HART c©2005 – p. 17
Asymmetric Information
When players have private information:
How should an “informed” player takeadvantage of his information?
How should an “uninformed” playerbehave?
One-shot game vs. repeated game
When the information is used=⇒ The information is revealed=⇒ The advantage disappears?
How can information be revealed credibly(when mutually advantageous)?
SERGIU HART c©2005 – p. 17
Asymmetric Information
The Folk Theorem ⇒
SERGIU HART c©2005 – p. 18
Asymmetric Information
The Folk Theorem ⇒
Individual Rationality
SERGIU HART c©2005 – p. 18
Asymmetric Information
The Folk Theorem ⇒
Individual Rationality
Feasibility
SERGIU HART c©2005 – p. 18
Asymmetric Information
The Folk Theorem ⇒
Individual Rationality ?
Feasibility
SERGIU HART c©2005 – p. 18
Individual Rationality / Game 1
MATRIX 1 (probability = 1
2):
L RT 4 0
B 0 0
MATRIX 2 (probability = 1
2):
L RT 0 0
B 0 4
SERGIU HART c©2005 – p. 19
Individual Rationality / Game 1
MATRIX 1 (probability = 1
2):
L RT 4 0
B 0 0
MATRIX 2 (probability = 1
2):
L RT 0 0
B 0 4
ROW knows which MATRIX was chosen
SERGIU HART c©2005 – p. 19
Individual Rationality / Game 1
MATRIX 1 (probability = 1
2):
L RT 4 0
B 0 0
MATRIX 2 (probability = 1
2):
L RT 0 0
B 0 4
ROW knows which MATRIX was chosenCOL does not know which MATRIX was chosen
SERGIU HART c©2005 – p. 19
Individual Rationality / Game 1
MATRIX 1 (probability = 1
2):
L RT 4 0
B 0 0
MATRIX 2 (probability = 1
2):
L RT 0 0
B 0 4
ROW knows which MATRIX was chosenCOL does not know which MATRIX was chosen
Actions, but not payoffs, are observedSERGIU HART c©2005 – p. 19
Individual Rationality / Game 1
M1
L RT 4 0
B 0 0
M2
L RT 0 0
B 0 4
SERGIU HART c©2005 – p. 20
Individual Rationality / Game 1
M1
L RT 4 0
B 0 0
M2
L RT 0 0
B 0 4
How much can ROW guaran-tee?
SERGIU HART c©2005 – p. 20
Individual Rationality / Game 1
M1
L RT 4 0
B 0 0
M2
L RT 0 0
B 0 4
How much can ROW guaran-tee?
If ROW playsT when M1
B when M2
SERGIU HART c©2005 – p. 20
Individual Rationality / Game 1
M1
L RT 4 0
B 0 0
M2
L RT 0 0
B 0 4
How much can ROW guaran-tee?
If ROW playsT when M1
B when M2Then COL learns whichMATRIX
SERGIU HART c©2005 – p. 20
Individual Rationality / Game 1
M1
L RT 4 0
B 0 0
M2
L RT 0 0
B 0 4
How much can ROW guaran-tee?
If ROW playsT when M1 (→ R )
B when M2 (→ L )Then COL learns whichMATRIX, and ROW gets 0
SERGIU HART c©2005 – p. 20
Individual Rationality / Game 1
M1
L RT 4 0
B 0 0
M2
L RT 0 0
B 0 4
How much can ROW guaran-tee?
If ROW plays “revealing”T when M1 (→ R )
B when M2 (→ L )Then COL learns whichMATRIX, and ROW gets 0
SERGIU HART c©2005 – p. 20
Individual Rationality / Game 1
M1
L RT 4 0
B 0 0
M2
L RT 0 0
B 0 4
How much can ROW guaran-tee?
If ROW plays “revealing”T when M1 (→ R )
B when M2 (→ L )Then COL learns whichMATRIX, and ROW gets 0
If ROW plays (1
2, 1
2) “non-
revealing” then ROWgets 1
SERGIU HART c©2005 – p. 20
Individual Rationality / Game 1
M1
L RT 4 0
B 0 0
M2
L RT 0 0
B 0 4
How much can ROW guaran-tee?
If ROW plays “revealing”T when M1 (→ R )
B when M2 (→ L )Then COL learns whichMATRIX, and ROW gets 0
If ROW plays (1
2, 1
2) “non-
revealing” then ROWgets 1
Ignoring his information is optimal for ROW
SERGIU HART c©2005 – p. 20
Individual Rationality / Game 2
M1
L RT 4 4
B 4 0
M2
L RT 0 4
B 4 4
SERGIU HART c©2005 – p. 21
Individual Rationality / Game 2
M1
L RT 4 4
B 4 0
M2
L RT 0 4
B 4 4
How much can ROW guaran-tee?
SERGIU HART c©2005 – p. 21
Individual Rationality / Game 2
M1
L RT 4 4
B 4 0
M2
L RT 0 4
B 4 4
How much can ROW guaran-tee?
If ROW playsT when M1
B when M2
SERGIU HART c©2005 – p. 21
Individual Rationality / Game 2
M1
L RT 4 4
B 4 0
M2
L RT 0 4
B 4 4
How much can ROW guaran-tee?
If ROW plays “revealing”T when M1
B when M2
SERGIU HART c©2005 – p. 21
Individual Rationality / Game 2
M1
L RT 4 4
B 4 0
M2
L RT 0 4
B 4 4
How much can ROW guaran-tee?
If ROW plays “revealing”T when M1 (→ R )
B when M2 (→ L )Then COL learns whichMATRIX, and ROW gets 4
SERGIU HART c©2005 – p. 21
Individual Rationality / Game 2
M1
L RT 4 4
B 4 0
M2
L RT 0 4
B 4 4
How much can ROW guaran-tee?
If ROW plays “revealing”T when M1 (→ R )
B when M2 (→ L )Then COL learns whichMATRIX, and ROW gets 4
If ROW plays (1
2, 1
2) “non-
revealing” then ROWgets 3
SERGIU HART c©2005 – p. 21
Individual Rationality / Game 2
M1
L RT 4 4
B 4 0
M2
L RT 0 4
B 4 4
How much can ROW guaran-tee?
If ROW plays “revealing”T when M1 (→ R )
B when M2 (→ L )Then COL learns whichMATRIX, and ROW gets 4
If ROW plays (1
2, 1
2) “non-
revealing” then ROWgets 3
Full revelation is optimal for ROW
SERGIU HART c©2005 – p. 21
Individual Rationality / Game 3
M1
L C RT 6 4 2
B 6 0 2
M2
L C RT 2 0 6
B 2 4 6
SERGIU HART c©2005 – p. 22
Individual Rationality / Game 3
M1
L C RT 6 4 2
B 6 0 2
M2
L C RT 2 0 6
B 2 4 6
How much can ROW guaran-tee?
SERGIU HART c©2005 – p. 22
Individual Rationality / Game 3
M1
L C RT 6 4 2
B 6 0 2
M2
L C RT 2 0 6
B 2 4 6
How much can ROW guaran-tee?
“Revealing” yields 2
T when M1 (→ R )
B when M2 (→ L )
SERGIU HART c©2005 – p. 22
Individual Rationality / Game 3
M1
L C RT 6 4 2
B 6 0 2
M2
L C RT 2 0 6
B 2 4 6
How much can ROW guaran-tee?
“Revealing” yields 2
T when M1 (→ R )
B when M2 (→ L )
“Non-revealing” (1
2, 1
2)
(→ C ) yields 2
SERGIU HART c©2005 – p. 22
Individual Rationality / Game 3
M1
L C RT 6 4 2
B 6 0 2
M2
L C RT 2 0 6
B 2 4 6
How much can ROW guaran-tee?
“Revealing” yields 2
T when M1 (→ R )
B when M2 (→ L )
“Non-revealing” (1
2, 1
2)
(→ C ) yields 2
“Partially revealing”yields 3
SERGIU HART c©2005 – p. 22
Individual Rationality / Game 3
M1
L C RT 6 4 2
B 6 0 2
M2
L C RT 2 0 6
B 2 4 6
How much can ROW guaran-tee?
“Revealing” yields 2
T when M1 (→ R )
B when M2 (→ L )
“Non-revealing” (1
2, 1
2)
(→ C ) yields 2
“Partially revealing”yields 3
Partial revelation is optimal for ROW
SERGIU HART c©2005 – p. 22
Partial Revelation
when M1:
{
play T with probability 0.80
play B with probability 0.20
when M2:
{
play T with probability 0.40
play B with probability 0.60
SERGIU HART c©2005 – p. 23
Partial Revelation
when M1:
{
play T with probability 0.80
play B with probability 0.20
when M2:
{
play T with probability 0.40
play B with probability 0.60
Prob (M1) Prob (M2)
a priori 0.50 0.50
SERGIU HART c©2005 – p. 23
Partial Revelation
when M1:
{
play T with probability 0.80
play B with probability 0.20
when M2:
{
play T with probability 0.40
play B with probability 0.60
Prob (M1) Prob (M2)
a priori 0.50 0.50
after T 0.67 0.33
SERGIU HART c©2005 – p. 23
Partial Revelation
when M1:
{
play T with probability 0.80
play B with probability 0.20
when M2:
{
play T with probability 0.40
play B with probability 0.60
Prob (M1) Prob (M2)
a priori 0.50 0.50
after T 0.67 0.33
after B 0.25 0.75
SERGIU HART c©2005 – p. 23
Individual Rationality
Theorem. The minimax value function of therepeated (zero-sum) game equals the
concavification of the minimax value function ofthe one-shot non-revealing game .
Aumann & Maschler 1966
SERGIU HART c©2005 – p. 24
Individual Rationality
Theorem. The minimax value function of therepeated (zero-sum) game equals the
concavification of the minimax value function ofthe one-shot non-revealing game .
⇒ Optimal strategy of the informed player(precise amount of information to reveal)
Aumann & Maschler 1966
SERGIU HART c©2005 – p. 24
Individual Rationality
Theorem. The minimax value function of therepeated (zero-sum) game equals the
concavification of the minimax value function ofthe one-shot non-revealing game .
⇒ Optimal strategy of the informed player(precise amount of information to reveal)
⇒ Optimal strategy of the uninformed player
Aumann & Maschler 1966
SERGIU HART c©2005 – p. 24
The Amount of Revelation
PROBABILITY
VALUE
bbb
b
SERGIU HART c©2005 – p. 25
The Amount of Revelation
PROBABILITY
VALUE
One-shot gamenon-revealing
bbb
b
SERGIU HART c©2005 – p. 25
The Amount of Revelation
PROBABILITY
VALUE
One-shot gamenon-revealing
Repeated game
bbb
b
SERGIU HART c©2005 – p. 25
The Amount of Revelation
PROBABILITY
VALUE
One-shot gamenon-revealing
Repeated game
b
prior
bb
b
SERGIU HART c©2005 – p. 25
The Amount of Revelation
PROBABILITY
VALUE
One-shot gamenon-revealing
Repeated game
b
prior
bb
b
posteriorsSERGIU HART c©2005 – p. 25
Asymmetric Information
The Folk Theorem ⇒
SERGIU HART c©2005 – p. 26
Asymmetric Information
The Folk Theorem ⇒
Individual Rationality
SERGIU HART c©2005 – p. 26
Asymmetric Information
The Folk Theorem ⇒
Individual Rationality V
SERGIU HART c©2005 – p. 26
Asymmetric Information
The Folk Theorem ⇒
Individual Rationality V
Feasibility
SERGIU HART c©2005 – p. 26
Asymmetric Information
The Folk Theorem ⇒
Individual Rationality V
Feasibility ?
SERGIU HART c©2005 – p. 26
Feasibility / Game 4
M1
L RT 3, 3 0, 1
B 3, 3 0, 1
M2
L RT 4, 0 3, 3
B 4, 0 3, 3
SERGIU HART c©2005 – p. 27
Feasibility / Game 4
M1
L RT 3, 3 0, 1
B 3, 3 0, 1
M2
L RT 4, 0 3, 3
B 4, 0 3, 3
To get (3, 3):
SERGIU HART c©2005 – p. 27
Feasibility / Game 4
M1
L RT 3, 3 0, 1
B 3, 3 0, 1
M2
L RT 4, 0 3, 3
B 4, 0 3, 3
To get (3, 3):
ROW must reveal:T when M1 (→ L )
B when M2 (→ R )
SERGIU HART c©2005 – p. 27
Feasibility / Game 4
M1
L RT 3, 3 0, 1
B 3, 3 0, 1
M2
L RT 4, 0 3, 3
B 4, 0 3, 3
To get (3, 3):
ROW must reveal:T when M1 (→ L )
B when M2 (→ R )
⇒ ROW will play
T also when M2
SERGIU HART c©2005 – p. 27
Feasibility / Game 4
M1
L RT 3, 3 0, 1
B 3, 3 0, 1
M2
L RT 4, 0 3, 3
B 4, 0 3, 3
To get (3, 3):
ROW must reveal:T when M1 (→ L )
B when M2 (→ R )
⇒ ROW will play
T also when M2
Revealing is NOT“incentive compatible”
SERGIU HART c©2005 – p. 27
Feasibility / Game 4
M1
L RT 3, 3 0, 1
B 3, 3 0, 1
M2
L RT 4, 0 3, 3
B 4, 0 3, 3
To get (3, 3):
ROW must reveal:T when M1 (→ L )
B when M2 (→ R )
⇒ ROW will play
T also when M2
Revealing is NOT“incentive compatible”
=⇒ (3, 3) is not feasible !
SERGIU HART c©2005 – p. 27
Equilibria
Equilibria of the repeated (non-zero-sum) game:
Aumann, Maschler & Stearns 1968
SERGIU HART c©2005 – p. 28
Equilibria
Equilibria of the repeated (non-zero-sum) game:
Partial revelation
Aumann, Maschler & Stearns 1968
SERGIU HART c©2005 – p. 28
Equilibria
Equilibria of the repeated (non-zero-sum) game:
Partial revelation
Partial revelation , followed byjoint randomization
Aumann, Maschler & Stearns 1968
SERGIU HART c©2005 – p. 28
Equilibria
Equilibria of the repeated (non-zero-sum) game:
Partial revelation
Partial revelation , followed byjoint randomization
Partial revelation , followed byjoint randomization , followed by
additional partial revelation
. . .
Aumann, Maschler & Stearns 1968
SERGIU HART c©2005 – p. 28
Repeated Games – Summary
The Folk Theorem
SERGIU HART c©2005 – p. 29
Repeated Games – Summary
The Folk Theorem
The Strong Folk Theorem(Aumann 1959)
SERGIU HART c©2005 – p. 29
Repeated Games – Summary
The Folk Theorem
The Strong Folk Theorem(Aumann 1959)
The Perfect Folk Theorem(Aumann & Shapley 1976, Rubinstein 1976)
SERGIU HART c©2005 – p. 29
Repeated Games – Summary
The Folk Theorem
The Strong Folk Theorem(Aumann 1959)
The Perfect Folk Theorem(Aumann & Shapley 1976, Rubinstein 1976)
Asymmetric Information: Individual Rationality(Aumann & Maschler 1966)
SERGIU HART c©2005 – p. 29
Repeated Games – Summary
The Folk Theorem
The Strong Folk Theorem(Aumann 1959)
The Perfect Folk Theorem(Aumann & Shapley 1976, Rubinstein 1976)
Asymmetric Information: Individual Rationality(Aumann & Maschler 1966)
Asymmetric Information: Equilibria(Aumann, Maschler & Stearns 1968)
SERGIU HART c©2005 – p. 29
Perfect Competition
SERGIU HART c©2005 – p. 30
The Market
Pieter Bruegel the Elder (1559)SERGIU HART c©2005 – p. 31
The Market Clears
Ursus Wehrli, Tidying Up Art (2002)SERGIU HART c©2005 – p. 32
Perfect Competition
How should perfect competition be modelled?
SERGIU HART c©2005 – p. 33
Perfect Competition
How should perfect competition be modelled?
“... the influence of an individual participant onthe economy cannot be mathematicallynegligible, as long as there are only finitely manyparticipants.
Aumann 1964SERGIU HART c©2005 – p. 33
Perfect Competition
How should perfect competition be modelled?
“... the influence of an individual participant onthe economy cannot be mathematicallynegligible, as long as there are only finitely manyparticipants. Thus a mathematical modelappropriate to the intuitive notion of perfectcompetition must contain infinitely manyparticipants.
Aumann 1964SERGIU HART c©2005 – p. 33
Perfect Competition
How should perfect competition be modelled?
“... the influence of an individual participant onthe economy cannot be mathematicallynegligible, as long as there are only finitely manyparticipants. Thus a mathematical modelappropriate to the intuitive notion of perfectcompetition must contain infinitely manyparticipants. We submit that the most naturalmodel for this purpose contains a continuum ofparticipants, similar to the continuum of points ona line or the continuum of particles in a fluid.”
Aumann 1964SERGIU HART c©2005 – p. 33
The Equivalence Principle
In markets with a continuum of traders :
SERGIU HART c©2005 – p. 34
The Equivalence Principle
In markets with a continuum of traders :
The set of Walrasian equilibria
coincides with
the solutions of thecorresponding “cooperative” game
SERGIU HART c©2005 – p. 34
The Equivalence Principle
In markets with a continuum of traders :
The set of Walrasian equilibria
coincides with
the solutions of thecorresponding “cooperative” game
(core, value, ...)
SERGIU HART c©2005 – p. 34
The Equivalence Principle
In markets with a continuum of traders :
The set of Walrasian equilibria
coincides with
the solutions of thecorresponding “cooperative” game
(core, value, ...)
Aumann 1964, Aumann & Shapley 1974,Aumann 1975, ...
SERGIU HART c©2005 – p. 34
The Equivalence Principle
The invisible hand (Adam Smith)
The limit contract curve (Edgeworth)
...
SERGIU HART c©2005 – p. 35
The Equivalence Principle
The invisible hand (Adam Smith)
The limit contract curve (Edgeworth)
...
“Intuitively, the Equivalence Principle says thatthe institution of market prices arises naturallyfrom the basic forces at work in a market,(almost) no matter what we assume about theway in which these forces work.”
Aumann 1987SERGIU HART c©2005 – p. 35
Correlated Equilibrium
SERGIU HART c©2005 – p. 36
Correlated Equilibrium
A Correlated Equilibrium is a Nash equilibriumwhen the players receive payoff-irrelevant
signals before playing the game
Aumann 1974
SERGIU HART c©2005 – p. 37
Correlated Equilibrium
A Correlated Equilibrium is a Nash equilibriumwhen the players receive payoff-irrelevant
signals before playing the game
Independent signals
Aumann 1974
SERGIU HART c©2005 – p. 37
Correlated Equilibrium
A Correlated Equilibrium is a Nash equilibriumwhen the players receive payoff-irrelevant
signals before playing the game
Independent signals ⇔ Nash equilibria
Aumann 1974
SERGIU HART c©2005 – p. 37
Correlated Equilibrium
A Correlated Equilibrium is a Nash equilibriumwhen the players receive payoff-irrelevant
signals before playing the game
Independent signals ⇔ Nash equilibria
Public signals (“sunspots”)
Aumann 1974
SERGIU HART c©2005 – p. 37
Correlated Equilibrium
A Correlated Equilibrium is a Nash equilibriumwhen the players receive payoff-irrelevant
signals before playing the game
Independent signals ⇔ Nash equilibria
Public signals (“sunspots”)⇔ convex combinations of Nash equilibria
Aumann 1974
SERGIU HART c©2005 – p. 37
Correlated Equilibrium
A Correlated Equilibrium is a Nash equilibriumwhen the players receive payoff-irrelevant
signals before playing the game
Independent signals ⇔ Nash equilibria
Public signals (“sunspots”)⇔ convex combinations of Nash equilibria
Correlated private signals → new equilibria
Aumann 1974
SERGIU HART c©2005 – p. 37
Correlated Equilibrium
Coordination, communication
SERGIU HART c©2005 – p. 38
Correlated Equilibrium
Coordination, communication
Mechanisms, mediator
SERGIU HART c©2005 – p. 38
Correlated Equilibrium
Coordination, communication
Mechanisms, mediator
Signals (public, correlated) are unavoidable
SERGIU HART c©2005 – p. 38
Correlated Equilibria
"Chicken" game
LEAVE STAY
LEAVE 5, 5 3, 6
STAY 6, 3 0, 0
SERGIU HART c©2005 – p. 39
Correlated Equilibria
"Chicken" game
LEAVE STAY
LEAVE 5, 5 3, 6
STAY 6, 3 0, 0
A Nash equilibrium
SERGIU HART c©2005 – p. 39
Correlated Equilibria
"Chicken" game
LEAVE STAY
LEAVE 5, 5 3, 6
STAY 6, 3 0, 0
Another Nash equilibrium i
SERGIU HART c©2005 – p. 39
Correlated Equilibria
"Chicken" game
LEAVE STAY
LEAVE 5, 5 3, 6
STAY 6, 3 0, 0
L 0 1/2
1/2 0
A (publicly) correlated equilibrium
SERGIU HART c©2005 – p. 39
Correlated Equilibria
"Chicken" game
LEAVE STAY
LEAVE 5, 5 3, 6
STAY 6, 3 0, 0
L S
L 1/3 1/3
S 1/3 0
Another correlated equilibrium :
After signal L play LEAVE
After signal S play STAY
SERGIU HART c©2005 – p. 39
Interactive Epistemology
SERGIU HART c©2005 – p. 40
Common Knowledge
A fact E is common knowledge among a set ofagents if:
SERGIU HART c©2005 – p. 41
Common Knowledge
A fact E is common knowledge among a set ofagents if:
Everyone knows E
SERGIU HART c©2005 – p. 41
Common Knowledge
A fact E is common knowledge among a set ofagents if:
Everyone knows E
Everyone knows that everyone knows E
SERGIU HART c©2005 – p. 41
Common Knowledge
A fact E is common knowledge among a set ofagents if:
Everyone knows E
Everyone knows that everyone knows E
Everyone knows that everyone knows thateveryone knows E
SERGIU HART c©2005 – p. 41
Common Knowledge
A fact E is common knowledge among a set ofagents if:
Everyone knows E
Everyone knows that everyone knows E
Everyone knows that everyone knows thateveryone knows E
Everyone knows that ... ...everyone knows E
SERGIU HART c©2005 – p. 41
Common Knowledge
A fact E is common knowledge among a set ofagents if:
Everyone knows E
Everyone knows that everyone knows E
Everyone knows that everyone knows thateveryone knows E
Everyone knows that ... ...everyone knows E
Lewis 1969 || Aumann 1976SERGIU HART c©2005 – p. 41
Interactive Epistemology
Formal model of knowledge, knowledgeabout knowledge, and common knowledge
Aumann 1976, 1999abSERGIU HART c©2005 – p. 42
Interactive Epistemology
Formal model of knowledge, knowledgeabout knowledge, and common knowledge
Partitions (“semantic”)Sentences (“syntactic”)
Aumann 1976, 1999abSERGIU HART c©2005 – p. 42
Interactive Epistemology
Formal model of knowledge, knowledgeabout knowledge, and common knowledge
Partitions (“semantic”)Sentences (“syntactic”)
The Agreement Theorem :If two people have the same prior,
and their posteriors for an event A arecommon knowledge ,
then their posteriors must be equal .
Aumann 1976, 1999abSERGIU HART c©2005 – p. 42
Rationality
Assume a common prior.
Aumann 1987SERGIU HART c©2005 – p. 43
Rationality
Assume a common prior.
If all players are Bayesian rational ,
Aumann 1987SERGIU HART c©2005 – p. 43
Rationality
Assume a common prior.
If all players are Bayesian rational ,and this is common knowledge ,
Aumann 1987SERGIU HART c©2005 – p. 43
Rationality
Assume a common prior.
If all players are Bayesian rational ,and this is common knowledge ,
then
their play constitutesa correlated equilibrium
Aumann 1987SERGIU HART c©2005 – p. 43
Other Contributions
SERGIU HART c©2005 – p. 44
Other Major Contributions
Cooperative games (NTU, core, value,bargaining set, ...)
SERGIU HART c©2005 – p. 45
Other Major Contributions
Cooperative games (NTU, core, value,bargaining set, ...)
Subjective probability and utility
SERGIU HART c©2005 – p. 45
Other Major Contributions
Cooperative games (NTU, core, value,bargaining set, ...)
Subjective probability and utility
Power and taxes
SERGIU HART c©2005 – p. 45
Other Major Contributions
Cooperative games (NTU, core, value,bargaining set, ...)
Subjective probability and utility
Power and taxes
Coalitions
SERGIU HART c©2005 – p. 45
Other Major Contributions
Cooperative games (NTU, core, value,bargaining set, ...)
Subjective probability and utility
Power and taxes
Coalitions
Foundations
SERGIU HART c©2005 – p. 45
Other Major Contributions
Cooperative games (NTU, core, value,bargaining set, ...)
Subjective probability and utility
Power and taxes
Coalitions
Foundations
Mathematics
SERGIU HART c©2005 – p. 45
Other Major Contributions
Cooperative games (NTU, core, value,bargaining set, ...)
Subjective probability and utility
Power and taxes
Coalitions
Foundations
Mathematics
. . .
SERGIU HART c©2005 – p. 45
The Unified Game Theory
SERGIU HART c©2005 – p. 46
The Unified Game Theory
“Unlike other approaches to disciplines likeeconomics or political science, GAME THEORYdoes not use different, ad-hoc constructs todeal with various specific issues, such as perfectcompetition, monopoly, oligopoly, internationaltrade, taxation, voting, deterrence, animalbehavior, and so on.
Aumann interview 2004SERGIU HART c©2005 – p. 46
The Unified Game Theory
“Unlike other approaches to disciplines likeeconomics or political science, GAME THEORYdoes not use different, ad-hoc constructs todeal with various specific issues, such as perfectcompetition, monopoly, oligopoly, internationaltrade, taxation, voting, deterrence, animalbehavior, and so on.Rather, it develops methodologies that apply inprinciple to all interactive situations , then seeswhere these methodologies lead in each specificapplication.”
Aumann interview 2004SERGIU HART c©2005 – p. 46
Aumann’s Doctoral Students
SERGIU HART c©2005 – p. 47
Aumann’s Doctoral Students
1. Bezalel Peleg
2. David Schmeidler
3. Shmuel Zamir
4. Elon Kohlberg
5. Benyamin Shitovitz
6. Zvi Artstein
7. Eugene Wesley
SERGIU HART c©2005 – p. 47
Aumann’s Doctoral Students
1. Bezalel Peleg
2. David Schmeidler
3. Shmuel Zamir
4. Elon Kohlberg
5. Benyamin Shitovitz
6. Zvi Artstein
7. Eugene Wesley
8. Sergiu Hart
9. Abraham Neyman
10. Yair Tauman
11. Dov Samet
12. Ehud Lehrer
13. Yossi Feinberg
. . .
SERGIU HART c©2005 – p. 47
A Scientist at Play
SERGIU HART c©2005 – p. 48
A Scientist at Worky
SERGIU HART c©2005 – p. 48
A Scientist at Play
SERGIU HART c©2005 – p. 48