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:

p1(s1, s2) :=          s1(100 − s1)

if s1 < s2

s1(100 − s1) 2

if s1 = s2 if s1 > s2

p2(s1, s2) :=          s2(100 − s2)

if s2 < s1

s2(100 − s2) 2

if s1 = s2 if s2 > s1

The Many Faces of Rationalizability – p.2/27

Bertrand competition, ctd

• s1 = 50 maximizes the value of s1(100 − s1) in the

interval (0, 100]. So 50 is the unique best response of player 1 to any s2 > 50.

• No strategy is a best response to s2

50.

So eliminating all never best responses (NBR) we get

G := ({50}, {50}, p1, p2).

Should we continue?

The Many Faces of Rationalizability – p.3/27

Bertrand competition, ctd

Pearce ’84: No. Bernheim ’84: Yes. In the original game s1 = 49 is a better response to s2 = 50 than s1 = 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:

p1(s1, s2) := s1

if (s1, s2) = (100, 100)

• therwise

p2(s1, s2) := s2

if (s1, s2) = (100, 100)

• therwise

By eliminating all strictly dominated strategies (SDS) we get

G := ({100}, {100}, p1, p2).

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 si = 100 is strictly dominated against

s3−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 G1, G2

G1 ⊆ G2 implies T(G1) ⊆ T(G2).

• 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
• f T, denoted by αT.

T αT 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 αT .

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 := (T1, . . ., Tn, p1, . . ., pn)

with each

pi : T1 × . . . × Tn → R.

• G := (S1, . . ., Sn) a restriction of H if Si ⊆ Ti.
• Strategy si of player i in game H is a best response to

s−i in G if ∀s′

i ∈ Si pi(si, s−i)

• pi(s′

i, s−i).

Important: si does not need to be an element of Si.

• We write si ∈ BRG(s−i).

The Many Faces of Rationalizability – p.12/27

SR operator (Bernheim ’84)

SR(G) := (S′

1, . . ., S′ n),

where

S′

i := {si ∈ Ti | ∃s−i ∈ S−i si ∈ BRH(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 := {si ∈ Ti | ∃s−i ∈ S−i si ∈ BRG(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

• perators coincide.

The Many Faces of Rationalizability – p.15/27

Iterated Elimination of SDS

• Fix initial game

H := (T1, . . ., Tn, p1, . . ., pn). G := (S1, . . ., Sn) a restriction of H.

• si, s′

i: two strategies from Ti.

s′

i strictly dominates si on G if

∀s−i ∈ S−i pi(s′

i, s−i) > pi(si, s−i).

• We write s′

i ≻G si.

The Many Faces of Rationalizability – p.16/27

SS and SS operators

SS(G) := (S′

1, . . ., S′ n),

where

S′

i := {si ∈ Ti | ¬∃s′ i ∈ Ti s′ i ≻G si}.

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 := (S1, . . ., Sn) and

S′

i := {si ∈ Ti | ¬∃s′ i ∈ Si s′ i ≻G si}.

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 := (S1, . . ., Sn) and

S′

i := {si ∈ Si | ¬∃s′ i ∈ Si s′ i ≻G si}.

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 Pi on Ω

({Pi(ω) | ω ∈ Ω} is a partition of Ω and ω ∈ Pi(ω).)

• in each state ω ∈ Ω chooses the strategy si(ω) ∈ Ti.

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

Pi(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

GE := (S1, . . ., Sn) of H

where

Sj := {sj(ω′) | ω′ ∈ E}.

• When player i knows that the state is in Pi(ω),

GPi(ω) represents his knowledge about the players’

strategies.

The Many Faces of Rationalizability – p.22/27

Rationality

• φ(si, G) a property.

Player i is φ-rational in the state ω if φ(si(ω), GPi(ω)).

• Each φ determines an operator on the set of restrictions
• f H:

Tφ(G) := (S′

1, . . ., S′ n),

where

S′

i := {si ∈ Ti | φ(si, G)}.

• φ is monotonic if

G ⊆ G′ and φ(si, G) implies φ(si, G′).

• Note φ is monotonic iff Tφ is monotonic.

The Many Faces of Rationalizability – p.23/27

Epistemic Analysis of ‘S’ Operators

Let

CKφ := (S1, . . ., Sn),

where

Sj := {sj(ω) | ∃ω (for some P1, . . ., Pn it is

common knowledge in ω that each player is φ-rational in ω)}. Theorem Suppose φ is monotonic. Then

CKφ = the largest fixpoint of Tφ.

The Many Faces of Rationalizability – p.24/27

Epistemic Analysis of ‘S’ Operators, ctd

By choosing appropriate φ we get epistemic justification for:

• SR (Bernheim),
• SR,
• SS (Milgrom, Roberts),
• SS (Chen, Van Long, Xuo).

The Many Faces of Rationalizability – p.25/27

Epistemic Analysis of ‘W’ Operators

What about

• WR,
• WR (Pearce),
• WS,
• WS (‘standard definition’)?

Note For each corresponding φ P ∀(s1, . . ., sn) ∈ T1 × . . . × Tn φ(si, ({s1}, . . ., {sn})).

The Many Faces of Rationalizability – p.26/27

Epistemic Analysis of ‘W’ Operators, ctd

Assumption: ∀si ∈ H ∃ω ∈ Ω si = si(ω). Theorem Suppose φ satisfies P. Then

CKφ = H.

Conclusion These procedures cannot be epistemically justified in this framework (as ‘self-contained’ rational procedures.)

The Many Faces of Rationalizability – p.27/27