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Epistemic Logic andGame Theory
A bird-eye view
Olivier Roy
Munich Center for Mathematical Philosophyhttp://olivier.amonbofis.net/
Olivier.Roy@lmu.de
June 27, 2011
Plan for Today
1. A primer in Game Theory
2. “Epistemic” Game Theory
2.1 Common Knowledge of Rationality in Strategic Games2.2 Paradoxes?2.3 Rationality in Extensive games.
A Primer in Game Theory
M. Osborne and A. Rubinstein. A Course in Game Theory. MIT Press, 1994.
R.B. Meyerson. Game Theory: Analysis of Conflict. Harvard UP, 1991.
What is game theory about?I Situations of interdependent decisions.
• Games, of course: chess, checker, Cluedo, football...• More serious (as opposed to fun) things: markets, bargaining,
auctions, international relations, warfare...• But not only about competition: coordination, conventions,
mixed interest (individual and collective)...
I Today: Non-cooperative game theory.
J. von Neumann and O. Morgenstern. A Theory of Games and Economic Be-haviour. Princeton UP, 1944.
J. Nash. Equilibrium points in N-Persons Games. Proceedings of the NationalAcademy of Sciences of the United States of America. 36:48-49,1950.
T. Schelling. The Strategy of Conflict. Harward UP, 1960.
What is game theory about?I Situations of interdependent decisions.
• Games, of course: chess, checker, Cluedo, football...
• More serious (as opposed to fun) things: markets, bargaining,auctions, international relations, warfare...
• But not only about competition: coordination, conventions,mixed interest (individual and collective)...
I Today: Non-cooperative game theory.
J. von Neumann and O. Morgenstern. A Theory of Games and Economic Be-haviour. Princeton UP, 1944.
J. Nash. Equilibrium points in N-Persons Games. Proceedings of the NationalAcademy of Sciences of the United States of America. 36:48-49,1950.
T. Schelling. The Strategy of Conflict. Harward UP, 1960.
What is game theory about?I Situations of interdependent decisions.
• Games, of course: chess, checker, Cluedo, football...• More serious (as opposed to fun) things: markets, bargaining,
auctions, international relations, warfare...
• But not only about competition: coordination, conventions,mixed interest (individual and collective)...
I Today: Non-cooperative game theory.
J. von Neumann and O. Morgenstern. A Theory of Games and Economic Be-haviour. Princeton UP, 1944.
J. Nash. Equilibrium points in N-Persons Games. Proceedings of the NationalAcademy of Sciences of the United States of America. 36:48-49,1950.
T. Schelling. The Strategy of Conflict. Harward UP, 1960.
What is game theory about?I Situations of interdependent decisions.
• Games, of course: chess, checker, Cluedo, football...• More serious (as opposed to fun) things: markets, bargaining,
auctions, international relations, warfare...• But not only about competition: coordination, conventions,
mixed interest (individual and collective)...
I Today: Non-cooperative game theory.
J. von Neumann and O. Morgenstern. A Theory of Games and Economic Be-haviour. Princeton UP, 1944.
J. Nash. Equilibrium points in N-Persons Games. Proceedings of the NationalAcademy of Sciences of the United States of America. 36:48-49,1950.
T. Schelling. The Strategy of Conflict. Harward UP, 1960.
What is game theory about?I Situations of interdependent decisions.
• Games, of course: chess, checker, Cluedo, football...• More serious (as opposed to fun) things: markets, bargaining,
auctions, international relations, warfare...• But not only about competition: coordination, conventions,
mixed interest (individual and collective)...
I Today: Non-cooperative game theory.
J. von Neumann and O. Morgenstern. A Theory of Games and Economic Be-haviour. Princeton UP, 1944.
J. Nash. Equilibrium points in N-Persons Games. Proceedings of the NationalAcademy of Sciences of the United States of America. 36:48-49,1950.
T. Schelling. The Strategy of Conflict. Harward UP, 1960.
Ann/Bob Restaurant A Restaurant B
Restaurant A 1, 1 0, 0
Restaurant B 0, 0 1, 1
I Games in “Strategic” Forms:
• Players.• Actions/strategies.• Payoffs/utilities.
Ann/Bob Restaurant A Restaurant B
Restaurant A 1, 1 0, 0
Restaurant B 0, 0 1, 1
I Games in “Strategic” Forms:
• Players.
• Actions/strategies.• Payoffs/utilities.
Ann/Bob Restaurant A Restaurant B
Restaurant A 1, 1 0, 0
Restaurant B 0, 0 1, 1
I Games in “Strategic” Forms:
• Players.• Actions/strategies.
• Payoffs/utilities.
Ann/Bob Restaurant A Restaurant B
Restaurant A 1, 1 0, 0
Restaurant B 0, 0 1, 1
I Games in “Strategic” Forms:
• Players.• Actions/strategies.
• Payoffs/utilities.
Ann/Bob Restaurant A Restaurant B
Restaurant A 1, 1 0, 0
Restaurant B 0, 0 1, 1
I Games in “Strategic” Forms:
• Players.• Actions/strategies.
• Payoffs/utilities.
Ann/Bob Restaurant A Restaurant B
Restaurant A 1, 1 0, 0
Restaurant B 0, 0 1, 1
I Games in “Strategic” Forms:
• Players.• Actions/strategies.
• Payoffs/utilities.
Ann/Bob Restaurant A Restaurant B
Restaurant A 1, 1 0, 0
Restaurant B 0, 0 1, 1
I Games in “Strategic” Forms:
• Players.• Actions/strategies.
• Payoffs/utilities.
Ann/Bob Restaurant A Restaurant B
Restaurant A 1, 1 0, 0
Restaurant B 0, 0 1, 1
I Games in “Strategic” Forms:
• Players.• Actions/strategies.
• Payoffs/utilities.
Ann/Bob Restaurant A Restaurant B
Restaurant A 1, 1 0, 0
Restaurant B 0, 0 1, 1
I Games in “Strategic” Forms:
• Players.• Actions/strategies.• Payoffs/utilities.
Ann/Bob Restaurant A Restaurant B
Restaurant A 1, 1 0, 0
Restaurant B 0, 0 1, 1
I Games in “Strategic” Forms:
• Players.• Actions/strategies.• Payoffs/utilities.
Cooperate Defect
Cooperate 3, 3 0, 4
Defect 4, 0 1, 1
I The infamous Prisoner’s Dilemma.
• Examples: Traffic jams, arm race, climate change...
Restaurant A Restaurant B
Restaurant A 1, 1 0, 0
Restaurant B 0, 0 1, 1
I Coordination game.
• Coordination, of course, but also conventions, e.g. language.
Restaurant
A
Restaurant
B
Restaurant
A 1, 1 0, 0
Restaurant
B 0, 0 1, 1
I Coordination game.
• Coordination, of course, but also conventions, e.g. language.
(0, 0)
(2, 2)
Bob
Bob
Bob
Ann
Ann
Hi
A
B
a
b
Hi
Lo
Lo
(1, 1)
(1, 1)
(0, 0)
I Games in Extensive Forms
• Same ingredients: players, actions, strategies, payoffs.
• Prefect and imperfect information.
(0, 0)
(2, 2)
Bob
Bob
Bob
Ann
Ann
Hi
A
B
a
b
Hi
Lo
Lo
(1, 1)
(1, 1)
(0, 0)
I Games in Extensive Forms
• Same ingredients: players, actions, strategies, payoffs.• Prefect and imperfect information.
(0,0)
(2,2)
Bob
Bob
Bob
Ann
Ann
Hi
A
B
a
b
Hi
Lo
Lo
(1,1)
(1,1)
(0,0)
I Games in Extensive Forms
• Same ingredients: players, actions, strategies, payoffs.• Prefect and imperfect information.
The big question
C D
C 3, 3 0, 4
D 4, 0 1, 1
(0,0)
(2,2)
Bob
Bob
Bob
Ann
Ann
Hi
A
B
a
b
Hi
Lo
Lo
(1,1)
(1,1)
(0,0)
What will/should (rational) players do?I The classical Answer: “Solution Concepts”.
• Elimination of Strictly dominated strategies, Nash equilibrium,backward induction...
Iterated Elimination of Strictly Dominated Strategies(IESDS)
Ann/ Bob L R
T 1, 1 1, 0
B 0, 0 0, 1
Strictly dominated strategy: one that gives you a worst payoff thananother (or a mixture of others) whatever happen.
1. Start with a game;
2. Eliminate all strictly dominated strategies;
3. Look at the reduced game;
4. Eliminate all strictly dominated strategies here;
5. Repeat 3 and 4 until you don’t eliminate anything.
Iterated Elimination of Strictly Dominated Strategies(IESDS)
Ann/ Bob L R
T 1, 1 1, 0
B 0, 0 0, 1
Strictly dominated strategy: one that gives you a worst payoff thananother (or a mixture of others) whatever happen.
1. Start with a game;
2. Eliminate all strictly dominated strategies;
3. Look at the reduced game;
4. Eliminate all strictly dominated strategies here;
5. Repeat 3 and 4 until you don’t eliminate anything.
Iterated Elimination of Strictly Dominated Strategies(IESDS)
Ann/ Bob L R
T 1, 1 1, 0
B 0, 0 0, 1
Strictly dominated strategy: one that gives you a worst payoff thananother (or a mixture of others) whatever happen.
1. Start with a game;
2. Eliminate all strictly dominated strategies;
3. Look at the reduced game;
4. Eliminate all strictly dominated strategies here;
5. Repeat 3 and 4 until you don’t eliminate anything.
Iterated Elimination of Strictly Dominated Strategies(IESDS)
Ann/ Bob L R
T 1, 1 1, 0
B 0, 0 0, 1
Strictly dominated strategy: one that gives you a worst payoff thananother (or a mixture of others) whatever happen.
1. Start with a game;
2. Eliminate all strictly dominated strategies;
3. Look at the reduced game;
4. Eliminate all strictly dominated strategies here;
5. Repeat 3 and 4 until you don’t eliminate anything.
Iterated Elimination of Strictly Dominated Strategies(IESDS)
Ann/ Bob L R
T 1, 1 1, 0
B 0, 0 0, 1
Strictly dominated strategy: one that gives you a worst payoff thananother (or a mixture of others) whatever happen.
1. Start with a game;
2. Eliminate all strictly dominated strategies;
3. Look at the reduced game;
4. Eliminate all strictly dominated strategies here;
5. Repeat 3 and 4 until you don’t eliminate anything.
Nash Equilibrium
A B
A 1, 1 0, 0
B 0, 0 1, 1
A profile in which no one would do better by choosing otherwise,given what the others are doing.
Nash Equilibrium
A B
A 1, 1 0, 0
B 0, 0 1, 1
A profile in which no one would do better by choosing otherwise,given what the others are doing.
Nash Equilibrium
A B
A 1, 1 0, 0
B 0, 0 1, 1
A profile in which no one would do better by choosing otherwise,given what the others are doing.
Nash Equilibrium
A B
A 1, 1 0, 0
B 0, 0 1, 1
A profile in which no one would do better by choosing otherwise,given what the others are doing.
Backward Induction
(0, 0)
(3, 3)
Bob
Bob
Ann
Hi
A
B
Hi
Lo
Lo
(2, 2)
(1, 1)
Backward Induction
(0, 0)
(3, 3)
Bob
Bob
Ann
Ann
Hi
A
B
Hi
Lo
Lo
(2, 2)
(1, 1)
Backward Induction
(0, 0)
(3, 3)
Bob
Bob
Ann
Ann
Hi
A
B
Hi
Lo
Lo
(2, 2)
(1, 1)
Backward Induction
(0, 0)
(3, 3)
Bob
Bob
Ann
Ann
Hi
A
B
Hi
Lo
Lo
(2, 2)
(1, 1)
Epistemic Game Theory
R.Aumann, Interactive Epistemology (I): Knowledge. International Journal ofGame Theory. 28:263-300, 1999.
A. Brandenburger, The power of paradox: some recent developments in interac-tive epistemology. International Journal of Game Theory. 35:465-492, 2007.
Restaurant A Restaurant B
Restaurant A 1, 1 0, 0
Restaurant B 0, 0 1, 1
Let’s take a step back:
I You’re Ann. What should you do?
• Depends on what you think Bob will do...
First-order information.
• ... but this depends on what you think Bob thinks you’ll do...• ... and this depends on what you think Bob thinks you think
he’ll do...• and so on.
Higher-order information!
Restaurant A Restaurant B
Restaurant A 1, 1 0, 0
Restaurant B 0, 0 1, 1
Let’s take a step back:I You’re Ann. What should you do?
• Depends on what you think Bob will do...
First-order information.
• ... but this depends on what you think Bob thinks you’ll do...• ... and this depends on what you think Bob thinks you think
he’ll do...• and so on.
Higher-order information!
Restaurant A Restaurant B
Restaurant A 1, 1 0, 0
Restaurant B 0, 0 1, 1
Let’s take a step back:I You’re Ann. What should you do?
• Depends on what you think Bob will do...
First-order information.• ... but this depends on what you think Bob thinks you’ll do...• ... and this depends on what you think Bob thinks you think
he’ll do...• and so on.
Higher-order information!
Restaurant A Restaurant B
Restaurant A 1, 1 0, 0
Restaurant B 0, 0 1, 1
Let’s take a step back:I You’re Ann. What should you do?
• Depends on what you think Bob will do...
First-order information.• ... but this depends on what you think Bob thinks you’ll do...• ... and this depends on what you think Bob thinks you think
he’ll do...• and so on.
Higher-order information!
Restaurant A Restaurant B
Restaurant A 1, 1 0, 0
Restaurant B 0, 0 1, 1
Let’s take a step back:I You’re Ann. What should you do?
• Depends on what you think Bob will do...
First-order information.
• ... but this depends on what you think Bob thinks you’ll do...• ... and this depends on what you think Bob thinks you think
he’ll do...• and so on.
Higher-order information!
Restaurant A Restaurant B
Restaurant A 1, 1 0, 0
Restaurant B 0, 0 1, 1
Let’s take a step back:I You’re Ann. What should you do?
• Depends on what you think Bob will do...First-order information.
• ... but this depends on what you think Bob thinks you’ll do...• ... and this depends on what you think Bob thinks you think
he’ll do...• and so on.
Higher-order information!
Restaurant A Restaurant B
Restaurant A 1, 1 0, 0
Restaurant B 0, 0 1, 1
The main idea:
I What should you do in a game?
• Do what’s best...
Choice Rule (Decision Theory)
• given your information.
First AND higher-order.
I This is where epistemic logic comes into play. (knowledgeabout knowledge, beliefs about beliefs, common knowledge,common beliefs...)
Restaurant A Restaurant B
Restaurant A 1, 1 0, 0
Restaurant B 0, 0 1, 1
The main idea:I What should you do in a game?
• Do what’s best...Choice Rule (Decision Theory)
• given your information.
First AND higher-order.
I This is where epistemic logic comes into play. (knowledgeabout knowledge, beliefs about beliefs, common knowledge,common beliefs...)
Restaurant A Restaurant B
Restaurant A 1, 1 0, 0
Restaurant B 0, 0 1, 1
The main idea:I What should you do in a game?
• Do what’s best...Choice Rule (Decision Theory)
• given your information.First AND higher-order.
I This is where epistemic logic comes into play. (knowledgeabout knowledge, beliefs about beliefs, common knowledge,common beliefs...)
Restaurant A Restaurant B
Restaurant A 1, 1 0, 0
Restaurant B 0, 0 1, 1
The main idea:I What should you do in a game?
• Do what’s best...Choice Rule (Decision Theory)
• given your information.First AND higher-order.
I This is where epistemic logic comes into play. (knowledgeabout knowledge, beliefs about beliefs, common knowledge,common beliefs...)
Game G
Strategy Space
Game Model
Rat ¬Rat
b
a
b′
a′
Game G
Strategy Space
Game Model
Rat ¬Rat
b
a
b′
a′
Game G
Strategy Space
Game Model
Rat ¬Rat
b
a
b′
a′
Game G
Strategy Space
Game Model
Rat ¬Rat
b
a
b′
a′
Game G
Strategy Space
Game Model
Rat ¬Rat
b
a
b′
a′
Example 1: Common knowledge of Rationality and IESDS.
D. Bernheim. Rationalizable strategic behavior. Econometrica, 52:1007-1028,1984.
D. Pearce. Rationalizable strategic behavior and the problem of perfection.Econometrica, 52:1029-1050, 1984.
A. Brandenburger and E. Dekel. Rationalizability and correlated equilibria.Econometrica, 55:1391-1402, 1987.
Ann/ Bob L R
T 1, 1 1, 0
B 0, 0 0, 1
I Rationality and Common Belief in Rationality imply...
• If Ann is rational then she will not play B.(Rational here = not choosing strictly dominated strategies)
• If Bob believes that Ann is rational then he will not play R.
I Here Rationality and one level of belief about rationality givesus IESDS.
I In general: Rationality and Common Belief in Rationalityimply IESDS.
Ann/ Bob L R
T 1, 1 1, 0
B 0, 0 0, 1
I Rationality and Common Belief in Rationality imply...
• If Ann is rational then she will not play B.(Rational here = not choosing strictly dominated strategies)
• If Bob believes that Ann is rational then he will not play R.
I Here Rationality and one level of belief about rationality givesus IESDS.
I In general: Rationality and Common Belief in Rationalityimply IESDS.
Ann/ Bob L R
T 1, 1 1, 0
B 0, 0 0, 1
I Rationality and Common Belief in Rationality imply...
• If Ann is rational then she will not play B.(Rational here = not choosing strictly dominated strategies)
• If Bob believes that Ann is rational then he will not play R.
I Here Rationality and one level of belief about rationality givesus IESDS.
I In general: Rationality and Common Belief in Rationalityimply IESDS.
Ann/ Bob L R
T 1, 1 1, 0
B 0, 0 0, 1
I Rationality and Common Belief in Rationality imply...
• If Ann is rational then she will not play B.(Rational here = not choosing strictly dominated strategies)
• If Bob believes that Ann is rational then he will not play R.
I Here Rationality and one level of belief about rationality givesus IESDS.
I In general: Rationality and Common Belief in Rationalityimply IESDS.
Ann/ Bob L R
T 1, 1 1, 0
B 0, 0 0, 1
I Rationality and Common Belief in Rationality imply...
• If Ann is rational then she will not play B.(Rational here = not choosing strictly dominated strategies)
• If Bob believes that Ann is rational then he will not play R.
I Here Rationality and one level of belief about rationality givesus IESDS.
I In general: Rationality and Common Belief in Rationalityimply IESDS.
Paradox (?) of admissibility
L. Samuelson. Dominated Strategies and Common Knowledge. Games andEconomic Behavior (1992).
A. Brandenburger and A. Friedenberg and H. J. Keisler. Admissibility in Games.Econometrica, 76:307-352, 2008.
Admissibility a.k.a. weak dominance.
Bob
Ann
T L R
T 1,1 1,0 U
B 1,0 0,1 U
T weakly dominates B
Admissibility a.k.a. weak dominance.
Bob
Ann
T L R
T 1,1 1,0 U
B 1,0 0,1 U
T weakly dominates B
Admissibility a.k.a. weak dominance.
Bob
Ann
T L R
T 1,1 1,0 U
B 1,0 0,1 U
Then L strictly dominates R.
Admissibility a.k.a. weak dominance.
Bob
Ann
T L R
T 1,1 1,0 U
B 1,0 0,1 U
The Iterated Admissibility set
Admissibility a.k.a. weak dominance.
Bob
Ann
T L R
T 1,1 1,0 U
B 1,0 0,1 U
But, now what is the reason for not playing B?
Admissibility a.k.a. weak dominance.
Paradox?
I Admissibility is equivalent to being a best response whennothing is ruled out.
I But common belief in admissibility does just that, ruling outstrategies.
Solution:I Belief revision.
• Be more subtle about that “ruling out” means. Here:considering very (infinitely) implausible.
• But surprise is possible! (things are not ruled out in a “hard”way).
• ... and they should not! Players still have to be prepared foranything... even the infinitely improbable.
Rationality in Extensive Games
R.J. Aumann. Backward Induction and Common Knowledge of Rationality.Games and Economic Behavior, 8:121-133, 1994.
R. Stalnaker, Knowledge, Belief and Counterfactual Reasoning in Games. Eco-nomics and Philosopy. 12(02):133-163, 1996.
J. Halpern, Substantive Rationality and Backward Induction. Games and Eco-nomic Behavior. 37(02):425-435, 2001.
(0, 0)
(3, 3)
Bob
Bob
Ann
Hi
A
B
Hi
Lo
Lo
(2, 2)
(1, 1)
I If Bob is rational then he will choose A.I If Ann is rational and believes that Bob is rational then she’ll
choose Hi .I If Bob is rational and believes that (Ann is rational and
believes that he is rational), then she’ll play Hi .
(0, 0)
(3, 3)
Bob
Bob
Ann
Ann
Hi
A
B
Hi
Lo
Lo
(2, 2)
(1, 1)
I If Bob is rational then he will choose A.
I If Ann is rational and believes that Bob is rational then she’llchoose Hi .
I If Bob is rational and believes that (Ann is rational andbelieves that he is rational), then she’ll play Hi .
(0, 0)
(3, 3)
Bob
Bob
Ann
Ann
Hi
A
B
Hi
Lo
Lo
(2, 2)
(1, 1)
I If Bob is rational then he will choose A.
I If Ann is rational and believes that Bob is rational then she’llchoose Hi .
I If Bob is rational and believes that (Ann is rational andbelieves that he is rational), then she’ll play Hi .
(0, 0)
(3, 3)
Bob
Bob
Ann
Ann
Hi
A
B
Hi
Lo
Lo
(2, 2)
(1, 1)
I If Bob is rational then he will choose A.
I If Ann is rational and believes that Bob is rational then she’llchoose Hi .
I If Bob is rational and believes that (Ann is rational andbelieves that he is rational), then she’ll play Hi .
(0, 0)
(3, 3)
Bob
Bob
Ann
Ann
Hi
A
B
Hi
Lo
Lo
(2, 2)
(1, 1)
I In general: rationality and common belief in rationality impliesbackward induction.
I Well...
(0, 0)
(3, 3)
Bob
Bob
Ann
Ann
Hi
A
B
Hi
Lo
Lo
(2, 2)
(1, 1)
I In general: rationality and common belief in rationality impliesbackward induction.
I Well...
(0, 4)
(3, 3)
Bob
Bob
Ann
Ann
Hi
A
B
Hi
Lo
Lo
(1, 1)
(4, 0)
I The “centipede” game.
I What should Ann believe about Bob if she gets to play?!?
I If she can revise her beliefs about Bob’s rationality, thenbackward induction is no more ensured.
(0, 4)
(3, 3)
Bob
Bob
Ann
Ann
Hi
A
B
Hi
Lo
Lo
(1, 1)
(4, 0)
I The “centipede” game.
I What should Ann believe about Bob if she gets to play?!?
I If she can revise her beliefs about Bob’s rationality, thenbackward induction is no more ensured.
(0, 4)
(3, 3)
Bob
Bob
Ann
Ann
Hi
A
B
Hi
Lo
Lo
(1, 1)
(4, 0)
I The “centipede” game.
I What should Ann believe about Bob if she gets to play?!?
I If she can revise her beliefs about Bob’s rationality, thenbackward induction is no more ensured.
Conclusion
The “epistemic turn” in game theory:
I Highlights the importance of higher-order information instrategic interaction.
I Arguably one of the strength of epistemic logic.
I More recently: dynamic epistemic logic came into the picture.
I Fruitful collaboration between game theorist, computerscientists, philosophers, and many others!
Some further reading on the connection with epistemic logic:OR and Eric Pacuit, Interactive Rationality. Stanford Encyclopedia of Philoso-phy. In preparation. Email me for details..
J. van Benthem, E. Pacuit and OR. Towards a Theory of Play: the LogicalPerspective on Games and Interaction. Games, 2:1, 52-86, 2011.
... and courses next semester!
Conclusion
The “epistemic turn” in game theory:
I Highlights the importance of higher-order information instrategic interaction.
I Arguably one of the strength of epistemic logic.
I More recently: dynamic epistemic logic came into the picture.
I Fruitful collaboration between game theorist, computerscientists, philosophers, and many others!
Some further reading on the connection with epistemic logic:
OR and Eric Pacuit, Interactive Rationality. Stanford Encyclopedia of Philoso-phy. In preparation. Email me for details..
J. van Benthem, E. Pacuit and OR. Towards a Theory of Play: the LogicalPerspective on Games and Interaction. Games, 2:1, 52-86, 2011.
... and courses next semester!
Conclusion
The “epistemic turn” in game theory:
I Highlights the importance of higher-order information instrategic interaction.
I Arguably one of the strength of epistemic logic.
I More recently: dynamic epistemic logic came into the picture.
I Fruitful collaboration between game theorist, computerscientists, philosophers, and many others!
Some further reading on the connection with epistemic logic:OR and Eric Pacuit, Interactive Rationality. Stanford Encyclopedia of Philoso-phy. In preparation. Email me for details..
J. van Benthem, E. Pacuit and OR. Towards a Theory of Play: the LogicalPerspective on Games and Interaction. Games, 2:1, 52-86, 2011.
... and courses next semester!
Epistemic Logic andGame Theory
A bird-eye view
Olivier Roy
Munich Center for Mathematical Philosophyhttp://olivier.amonbofis.net/
Olivier.Roy@lmu.de
June 27, 2011
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