Epistemic Game Theory Lecture 5 ESSLLI’12, Opole Eric Pacuit Olivier Roy TiLPS, Tilburg University MCMP, LMU Munich ai.stanford.edu/~epacuit http://olivier.amonbofis.net August 10, 2012 Eric Pacuit and Olivier Roy 1
Plan for the week 1. Monday Basic Concepts. 2. Tuesday Epistemics. 3. Wednesday Fundamentals of Epistemic Game Theory. 4. Thursday Tree, Puzzles and Paradoxes. 5. Friday More Puzzles, Extensions and New Directions. • Admissibility continued. • The Brandenburger-Kiesler Paradox. • Nash Equilibrium? • Concluding remarks. Eric Pacuit and Olivier Roy 2
LPS: ( µ 0 , µ 1 , . . . , µ n − 1 ) (each µ i is a probability measure with disjoint supports) Eric Pacuit and Olivier Roy 3
LPS: ( µ 0 , µ 1 , . . . , µ n − 1 ) (each µ i is a probability measure with disjoint supports) ( s i , t i ) is rational provided (i) s i lexicographically maximizes i ’s expected payoff under the LPS associated with t i , and (ii) the LPS associated with t i has full support. Eric Pacuit and Olivier Roy 3
LPS: ( µ 0 , µ 1 , . . . , µ n − 1 ) (each µ i is a probability measure with disjoint supports) ( s i , t i ) is rational provided (i) s i lexicographically maximizes i ’s expected payoff under the LPS associated with t i , and (ii) the LPS associated with t i has full support. A player assumes E provided she considers E infinitely more likely than not - E . Eric Pacuit and Olivier Roy 3
LPS: ( µ 0 , µ 1 , . . . , µ n − 1 ) (each µ i is a probability measure with disjoint supports) ( s i , t i ) is rational provided (i) s i lexicographically maximizes i ’s expected payoff under the LPS associated with t i , and (ii) the LPS associated with t i has full support. A player assumes E provided she considers E infinitely more likely than not - E . The key notion is rationality and common assumption of rationality (RCAR). Eric Pacuit and Olivier Roy 3
But, there’s more... Eric Pacuit and Olivier Roy 4
“Under admissibility, Ann considers everything possible. But this is only a decision-theoretic statement. Ann is in a game, so we imagine she asks herself: “What about Bob? What does he consider possible?” If Ann truly considers everything possible, then it seems she should, in particular, allow for the possibility that Bob does not! Alternatively put, it seems that a full analysis of the admissibility requirement should include the idea that other players do not conform to the requirement.” (pg. 313) A. Brandenburger, A. Friedenberg, H. J. Keisler. Admissibility in Games . Econo- metrica (2008). Eric Pacuit and Olivier Roy 4
Irrationality 2 L C R T 4,0 4,1 0,1 1 M 0,0 0,1 4,1 D 3,0 2,1 2,1 The IA set Eric Pacuit and Olivier Roy 5
Irrationality 2 L C R T 4,0 4,1 0,1 1 M 0,0 0,1 4,1 D 3,0 2,1 2,1 ◮ The IA set Eric Pacuit and Olivier Roy 5
Irrationality 2 L C R T 4,0 4,1 0,1 1 M 0,0 0,1 4,1 D 3,0 2,1 2,1 ◮ All ( L , b i ) are irrational, ( C , b i ), ( R , b i ) are rational if b i has full support, irrational otherwise ◮ D is optimal then either µ ( C ) = µ ( R ) = 1 2 or µ assigns positive probability to both L and R . Eric Pacuit and Olivier Roy 5
Irrationality 2 L C R T 4,0 4,1 0,1 1 M 0,0 0,1 4,1 D 3,0 2,1 2,1 ◮ Fix a rational ( D , a ) where a assumes that Bob is rational. ( a �→ ( µ 0 , . . . , µ n − 1 )) ◮ Let µ i be the first measure assigning nonzero probability to { L } × T B ( i � = 0 since a assumes Bob is rational). Eric Pacuit and Olivier Roy 5
Irrationality 2 L C R T 4,0 4,1 0,1 1 M 0,0 0,1 4,1 D 3,0 2,1 2,1 ◮ Let µ i be the first measure assigning nonzero probability to { L } × T B ( i � = 0). ◮ for each µ k with k < i : (i) µ k assigns probability 1 2 to { C } × T B and 1 2 to { R } × T B ; and (ii) U , M , D are each optimal under µ k . Eric Pacuit and Olivier Roy 5
Irrationality 2 L C R T 4,0 4,1 0,1 1 M 0,0 0,1 4,1 D 3,0 2,1 2,1 ◮ for each µ k with k < i : (i) µ k assigns probability 1 2 to { C } × T B and 1 2 to { R } × T B ; and (ii) T , M , D are each optimal under µ k . ◮ D must be optimal under µ i and so µ i assigns positive probability to both { L } × T B and { R } × T B . Eric Pacuit and Olivier Roy 5
Irrationality 2 L C R T 4,0 4,1 0,1 1 M 0,0 0,1 4,1 D 3,0 2,1 2,1 ◮ D must be optimal under µ i and so µ i assigns positive probability to both { L } × T B and { R } × T B . ◮ Rational strategy-type pairs are each infinitely more likely that irrational strategy-type pairs. Since, each point in { L } × T B is irrational, µ i must assign positive probability to irrational pairs in { R } × T B . Eric Pacuit and Olivier Roy 5
Irrationality 2 L C R T 4,0 4,1 0,1 1 M 0,0 0,1 4,1 D 3,0 2,1 2,1 ◮ µ i must assign positive probability to irrational pairs in { R } × T B . ◮ This can only happen if there are types of Bob that do not consider everything possible. Eric Pacuit and Olivier Roy 5
Brandenburger-Kiesler Paradox The Brandenburger-Keisler Paradox Eric Pacuit and Olivier Roy 6
Brandenburger-Kiesler Paradox 2 b a l c r l 1 t 1 t 4,4 1,1 0,0 1 c 0 m 0 m 1,1 5,5 0,0 r 0 d 0 d 0,1 0,1 6,0 The projection of RCBR is { ( t , l ) } This is not the entire ISDS set “Game independent” conditions Eric Pacuit and Olivier Roy 7
Brandenburger-Kiesler Paradox 2 b a l c r l 1 t 1 t 4,4 1,1 0,0 1 c 0 m 0 m 1,1 5,5 0,0 r 0 d 0 d 0,1 0,1 6,0 ◮ The projection of RCBR is { ( t , l ) } This is not the entire ISDS set “Game independent” conditions Eric Pacuit and Olivier Roy 7
Brandenburger-Kiesler Paradox 2 b a l c r l 1 t 1 t 4,4 1,1 0,0 1 c 0 m 0 m 1,1 5,5 0,0 r 0 d 0 d 0,1 0,1 6,0 ◮ The projection of RCBR is { ( t , l ) } ◮ This is not the entire ISDS set “Game independent” conditions Eric Pacuit and Olivier Roy 7
Brandenburger-Kiesler Paradox 2 b a l c r l 1 t 1 t 4,4 1,1 0,0 1 c 0 m 0 m 1,1 5,5 0,0 r 0 d 0 d 0,1 0,1 6,0 ◮ The projection of RCBR is { ( t , l ) } ◮ This is not the entire ISDS set ◮ “Game independent” conditions and rich type structures Eric Pacuit and Olivier Roy 7
Brandenburger-Kiesler Paradox A Question ◮ For any given set S of external states we can use a Bayesian model or a type space on S to provide consistent representations of the players’ beliefs. Eric Pacuit and Olivier Roy 8
Brandenburger-Kiesler Paradox A Question ◮ For any given set S of external states we can use a Bayesian model or a type space on S to provide consistent representations of the players’ beliefs. ◮ Every state in a belief model or type space induces an infinite hierarchy of beliefs, but not all consistent and coherent infinite hierarchies are in any finite model . It is not obvious that even in an infinite model that all such hierarchies of beliefs can be represented. Eric Pacuit and Olivier Roy 8
Brandenburger-Kiesler Paradox A Question ◮ For any given set S of external states we can use a Bayesian model or a type space on S to provide consistent representations of the players’ beliefs. ◮ Every state in a belief model or type space induces an infinite hierarchy of beliefs, but not all consistent and coherent infinite hierarchies are in any finite model . It is not obvious that even in an infinite model that all such hierarchies of beliefs can be represented. ◮ Which type space is the “correct” one to work with? Eric Pacuit and Olivier Roy 8
Brandenburger-Kiesler Paradox Some Literature A. Brandenburger and E. Dekel. Hierarchies of Beliefs and Common Knowledge . Journal of Economic Theory (1993). Eric Pacuit and Olivier Roy 9
Brandenburger-Kiesler Paradox Some Literature A. Brandenburger and E. Dekel. Hierarchies of Beliefs and Common Knowledge . Journal of Economic Theory (1993). A. Heifetz and D. Samet. Knoweldge Spaces with Arbitrarily High Rank . Games and Economic Behavior (1998). Eric Pacuit and Olivier Roy 9
Brandenburger-Kiesler Paradox Some Literature A. Brandenburger and E. Dekel. Hierarchies of Beliefs and Common Knowledge . Journal of Economic Theory (1993). A. Heifetz and D. Samet. Knoweldge Spaces with Arbitrarily High Rank . Games and Economic Behavior (1998). L. Moss and I. Viglizzo. Harsanyi type spaces and final coalgebras constructed from satisfied theories . EN in Theoretical Computer Science (2004). Eric Pacuit and Olivier Roy 9
Brandenburger-Kiesler Paradox Some Literature A. Brandenburger and E. Dekel. Hierarchies of Beliefs and Common Knowledge . Journal of Economic Theory (1993). A. Heifetz and D. Samet. Knoweldge Spaces with Arbitrarily High Rank . Games and Economic Behavior (1998). L. Moss and I. Viglizzo. Harsanyi type spaces and final coalgebras constructed from satisfied theories . EN in Theoretical Computer Science (2004). A. Friendenberg. When do type structures contain all hierarchies of beliefs? . working paper (2007). Eric Pacuit and Olivier Roy 9
Brandenburger-Kiesler Paradox The General Question Does there exist a space of “ all possible ” beliefs? Eric Pacuit and Olivier Roy 10
Brandenburger-Kiesler Paradox “Conjecture” about Ann “Conjecture” about Bob Ann’s States Bob’s States Is there a space where every possible conjecture is considered by some type? Eric Pacuit and Olivier Roy 11
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