The Many Faces of Rationalizability Krzysztof Apt CWI & - PowerPoint PPT Presentation

The Many Faces of Rationalizability Krzysztof Apt CWI & University of Amsterdam The Many Faces of Rationalizability – p.1/27

Motivation: Example 1 Consider Bertrand competition between two firms. Marginal costs are 0. Prices range over (0 , 100] . Payoff functions:  if s 1 < s 2 s 1 (100 − s 1 )     s 1 (100 − s 1 ) p 1 ( s 1 , s 2 ) := if s 1 = s 2 2    0 if s 1 > s 2   s 2 (100 − s 2 ) if s 2 < s 1     s 2 (100 − s 2 ) p 2 ( s 1 , s 2 ) := if s 1 = s 2 2    0 if s 2 > s 1  The Many Faces of Rationalizability – p.2/27

� � � � ✁ Bertrand competition, ctd s 1 = 50 maximizes the value of s 1 (100 − s 1 ) in the interval (0 , 100] . So 50 is the unique best response of player 1 to any s 2 > 50 . No strategy is a best response to s 2 50 . So eliminating all never best responses (NBR) we get G := ( { 50 } , { 50 } , p 1 , p 2 ) . Should we continue? The Many Faces of Rationalizability – p.3/27

Bertrand competition, ctd Pearce ’84: No. Bernheim ’84: Yes. In the original game s 1 = 49 is a better response to s 2 = 50 than s 1 = 50 . Symmetrically for player 2. So we get the empty game. The Many Faces of Rationalizability – p.4/27

Motivation: Example 2 Production with a discontinuity Two players, each with the set (0 , 100] of strategies. Payoff functions: � s 1 if ( s 1 , s 2 ) � = (100 , 100) p 1 ( s 1 , s 2 ) := otherwise 0 � s 2 if ( s 1 , s 2 ) � = (100 , 100) p 2 ( s 1 , s 2 ) := otherwise 0 By eliminating all strictly dominated strategies (SDS) we get G := ( { 100 } , { 100 } , p 1 , p 2 ) . Should we continue? The Many Faces of Rationalizability – p.5/27

Production with a discontinuity, ctd Usual approach: No. Milgrom, Robert ’90: Yes. In the original game s i = 100 is strictly dominated against s 3 − i = 100 by any other strategy s ′ i . So we get the empty game. The Many Faces of Rationalizability – p.6/27

� � � � Summary For iterated elimination of NBRs and of SDSs various definitions exist. They coincide for finite games but differ on infinite games. The Many Faces of Rationalizability – p.7/27

� � � � � � � � � � Our Approach We analyze such definitions using operators on complete lattices. In general transfinite iterations are needed. Elimination procedures based on monotonic operators admit an epistemic justification (using partition spaces). Pearce’s definition of iterated elimination of NBR is not monotonic. Usual definition of iterated elimination of SDSs is not monotonic either. The Many Faces of Rationalizability – p.8/27

� � � � � � Operators: a recap I Fix a complete lattice ( D, ⊆ ) with the largest element ⊤ . Let T be an operator on ( D, ⊆ ) . T is monotonic if for all G 1 , G 2 G 1 ⊆ G 2 implies T ( G 1 ) ⊆ T ( G 2 ) . T is contracting if for all G T ( G ) ⊆ G. G is a fixpoint of T if T ( G ) = G . The Many Faces of Rationalizability – p.9/27

� � � � � � � � � � Operators: a recap II We define a sequence of elements T α of D , where α is an ordinal: T 0 := ⊤ , T α +1 := T ( T α ) , for all limit ordinals β , T β := � α<β T α . The least α such that T α +1 = T α is the closure ordinal of T , denoted by α T . T α T is then the outcome of (iterating) T . Fixpoint Theorem (Knaster, Tarski) Every monotonic operator on ( D, ⊆ ) has a largest fixpoint. This fixpoint is the outcome of T , i.e., it is of the form T α T . The Many Faces of Rationalizability – p.10/27

Contracting Operators Contracting version of an operator T : T ( G ) := T ( G ) ∩ G. Note If T is monotonic, then T as well and their outcomes coincide. The Many Faces of Rationalizability – p.11/27

� � � � � � � � � Back to Strategic Games Fix initial game H := ( T 1 , . . ., T n , p 1 , . . ., p n ) with each p i : T 1 × . . . × T n → R . G := ( S 1 , . . ., S n ) a restriction of H if S i ⊆ T i . Strategy s i of player i in game H is a best response to s − i in G if ∀ s ′ p i ( s ′ i ∈ S i p i ( s i , s − i ) i , s − i ) . Important: s i does not need to be an element of S i . We write s i ∈ BR G ( s − i ) . The Many Faces of Rationalizability – p.12/27

� � � � SR operator (Bernheim ’84) SR ( G ) := ( S ′ 1 , . . ., S ′ n ) , where S ′ i := { s i ∈ T i | ∃ s − i ∈ S − i s i ∈ BR H ( s − i ) } . So SR ( G ) is obtained by removing from H all strategies that are NBR in H to a joint strategy of opponents from G . Note SR is monotonic and hence SR as well. Theorem The largest fixpoint of SR exists and is its outcome. In some games the closure ordinal of SR can be > ω (Lipman ’94). The Many Faces of Rationalizability – p.13/27

� � � � WR operator (Pearce ’84) WR ( G ) := ( S ′ 1 , . . ., S ′ n ) , where S ′ i := { s i ∈ T i | ∃ s − i ∈ S − i s i ∈ BR G ( s − i ) } . So WR ( G ) is obtained by removing from H all strategies that are NBR in G to a joint strategy of opponents from G . Note The outcome of WR does not need to exist (!). WR is contracting but not monotonic and its largest fixpoint does not need to exist. The Many Faces of Rationalizability – p.14/27

� � � � � � � � When outcomes of SR and WR coincide? B For all s − i ∈ T − i a best response to s − i in H exists. Note B is satisfied for finite games, compact games. In the presence of B the closure ordinals of SR and WR can still be > ω (but not for compact games: Bernheim ’84). Theorem Assume property B . Then the iterations of all 4 operators coincide. The Many Faces of Rationalizability – p.15/27

� � � � � � Iterated Elimination of SDS Fix initial game H := ( T 1 , . . ., T n , p 1 , . . ., p n ) . G := ( S 1 , . . ., S n ) a restriction of H . s i , s ′ i : two strategies from T i . s ′ i strictly dominates s i on G if ∀ s − i ∈ S − i p i ( s ′ i , s − i ) > p i ( s i , s − i ) . We write s ′ i ≻ G s i . The Many Faces of Rationalizability – p.16/27

� � � � SS and SS operators SS ( G ) := ( S ′ 1 , . . ., S ′ n ) , where S ′ i := { s i ∈ T i | ¬∃ s ′ i ∈ T i s ′ i ≻ G s i } . So SS ( G ) is obtained by removing from H all strategies that are strictly dominated on G by some strategy in H (and not in G ). SS operator: Milgrom, Roberts ’90. SS operator: Chen, Van Long, Xuo ’05. Note SS is monotonic and hence SS as well. The Many Faces of Rationalizability – p.17/27

� � � � � � SS operator Theorem (Chen, Van Long, Xuo ’05) The largest fixpoint of SS exists and is its outcome. SS does not remove any Nash equilibria and does not introduce ‘spurious’ Nash equilibria. In some games the closure ordinal of SS can be > ω . The Many Faces of Rationalizability – p.18/27

WS and WS operators WS ( G ) := ( S ′ 1 , . . ., S ′ n ) , where G := ( S 1 , . . ., S n ) and S ′ i := { s i ∈ T i | ¬∃ s ′ i ∈ S i s ′ i ≻ G s i } . So a strategy is removed from H if it is strictly dominated on G by some strategy in G itself. Note The outcome of WS does not need to exist (!). The Many Faces of Rationalizability – p.19/27

� � � � WS operator WS ( G ) := ( S ′ 1 , . . ., S ′ n ) , where G := ( S 1 , . . ., S n ) and S ′ i := { s i ∈ S i | ¬∃ s ′ i ∈ S i s ′ i ≻ G s i } . So a strategy is removed from G if it is strictly dominated on G by some strategy in G itself (we ‘forget’ H ). This is the operator usually considered in the literature (for finite games and dominance by a mixed strategy). Studied for infinite games in Dufwenberg and Stegeman ’02. Theorem For finite games the iterations of all 4 operators coincide. The Many Faces of Rationalizability – p.20/27

� � � � Epistemic Analysis We use partition spaces (Aumann ’87, Brandenburger and Dekel ’87). Assume a space Ω of states. Each player i has a partitional information function P i on Ω ( { P i ( ω ) | ω ∈ Ω } is a partition of Ω and ω ∈ P i ( ω ) .) in each state ω ∈ Ω chooses the strategy s i ( ω ) ∈ T i . The Many Faces of Rationalizability – p.21/27

� � � � � � � � � � Events and Games Event is a subset of Ω . Event F is self-evident if for all ω ∈ F P i ( ω ) ⊆ F for all i ∈ [1 ..n ] . Event E is a common knowledge in ω ∈ Ω if for some self-evident event F we have ω ∈ F ⊆ E . Each event E determines the restriction G E := ( S 1 , . . ., S n ) of H where S j := { s j ( ω ′ ) | ω ′ ∈ E } . When player i knows that the state is in P i ( ω ) , G P i ( ω ) represents his knowledge about the players’ strategies. The Many Faces of Rationalizability – p.22/27

� � � � � � � � Rationality φ ( s i , G ) a property. Player i is φ -rational in the state ω if φ ( s i ( ω ) , G P i ( ω ) ) . Each φ determines an operator on the set of restrictions of H : T φ ( G ) := ( S ′ 1 , . . ., S ′ n ) , where S ′ i := { s i ∈ T i | φ ( s i , G ) } . φ is monotonic if G ⊆ G ′ and φ ( s i , G ) implies φ ( s i , G ′ ) . Note φ is monotonic iff T φ is monotonic. The Many Faces of Rationalizability – p.23/27