Int J Game Theory (2001) 30:453–478
2001
9
99
9
Proper rationalizability in lexicographic beliefs*
Geir B. Asheim
Department of Economics, University of Oslo, P.O. Box 1095 Blindern, N-0317 Oslo, Norway
(e-mail: g.b.asheim@econ.uio.no)
Final version: December 2001
Abstract. Proper consistency is defined by the property that each player takes
all opponent strategies into account (is cautious) and deems one opponent
strategy to be infinitely more likely than another if the opponent prefers the
one to the other (respects preferences). When there is common certain belief of
proper consistency, a most preferred strategy is properly rationalizable. Any
strategy used with positive probability in a proper equilibrium is properly rationalizable. Only strategies that lead to the backward induction outcome are
properly rationalizable in the strategic form of a generic perfect information
game. Proper rationalizability can test the robustness of inductive procedures.
JEL Classification Number: C72.
Key words: Rationalizability, backward induction, strategic form
1. Introduction
Most contributions on the relation between common knowledge/belief of rationality and backward induction in perfect information games perform the
analysis in the extensive form of the game. An exception to this rule is Schuhmacher [24] who – based on Myerson’s [22] concept of a proper equilibrium, but
without making equilibrium assumptions – defines the concept of proper rationalizability in the strategic form and shows that proper rationalizable play
leads to backward induction.
Schuhmacher defines the concept of e-proper rationalizability by assuming
* This paper builds in part on joint work with Martin Dufwenberg and Andrés Perea, who together with a referee have contributed with helpful and detailed suggestions. I also thank Peter
Hammond, Frank Schuhmacher, and Ylva Søvik.
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G. B. Asheim
that players make mistakes, but where more costly mistakes are made with a
much smaller probability than less costly ones. A properly rationalizable strategy can then be defined as the limit of a sequence of e-properly rationalizable
strategies as e goes to zero. For a given e, Schuhmacher o¤ers an epistemic
foundation for e-proper rationalizability. However, for the limiting concept,
i.e. proper rationalizability, there has not been an epistemic foundation available. It is one purpose of the present paper to establish how proper rationalizability can be given an epistemic characterization in strategic two-player
games by means of lexicographic probabilities.
Blume, Brandenburger & Dekel [8] characterize proper equilibrium as a
property of preferences. When doing so they represent a player’s preferences by
a utility function and a lexicographic probability system (LPS; Blume, Brandenburger & Dekel [7]), whereby the player may deem one opponent strategy
to be infinitely more likely than another while still taking the latter strategy
into account. In two-player games, their characterization of proper equilibrium
can be described by the following two properties.
1. Each player is certain of the LPS of his opponent,
2. Each player’s LPS satisfies that the player takes all opponent strategies into
account (is cautious) and that the player deems one opponent strategy to be
infinitely more likely than another if the opponent prefers the one to the
other (respects preferences).
In my characterization of proper rationalizability in two-player games I drop
property 1., which is an equilibrium assumption; instead I assume that there is
common certain belief of property 2., which I call proper consistency.
Since, in my framework, a player is not certain of the LPS of his opponent, player i’s LPS must be defined on Sj Tj , where Sj denotes the set of
opponent strategies and Tj denotes the set of opponent types. Provided that
the utility function assigns to each outcome i’s payo¤ in the game, each type
of player i is simply an LPS on Sj Tj . A type ti is said to be cautious if ti takes
into account all strategies of any opponent type that is not deemed Savagenull. A type ti is said to respect preferences if, for any opponent type that is
not deemed Savage-null, ti deems one strategy of the opponent type to be infinitely more likely than another if the opponent type prefers the one to the
other. A type ti is said to be properly consistent with the preferences of his
opponent if ti both is cautious and respects preferences.
Hence, the present analysis follows Asheim & Dufwenberg [3] (AD) by
suggesting that in deductive game theory, requirements can be imposed on the
beliefs of players rather than their choice. Since the subjective beliefs of players characterize their preferences, this amounts to imposing requirements on
preferences. E.g. instead of imposing that a driver chooses to drive on the right
side of the road if he is certain that his opponent chooses to drive on the right
side of the road, AD suggest to impose that a driver prefers to drive on the
right side of the road if he is certain that his opponent prefers to drive on the
right side of the road. This follows a tradition in equilibrium analysis where
Nash (perfect/proper) equilibrium is defined as an equilibrium in conjectures
(cf. Blume et al. [8]).
A type ti certainly believes the event that his opponent is of a type that
is properly consistent if he deems Savage-null any opponent type that is not
properly consistent. There is common certain belief of proper consistency at
Proper rationalizability in lexicographic beliefs
455
t ¼ ðt1 ; t2 Þ if both t1 and t2 certainly believes the event that his opponent is of
a type that is properly consistent, they both certainly believes that his opponent certainly believes the event that his opponent is of a type that is properly
consistent, etc. A pure strategy si is called properly rationalizable if there is a
set of type vectors T ¼ T1 T2 with common certain belief of proper consistency at t ¼ ðt1 ; t2 Þ such that si is a most preferred strategy given ti ’s LPS.
It is first shown (in Prop. 1) how the event of proper consistency combined
with mutual certain belief of the type vector can be used to characterize the
concept of proper equilibrium. This in turn means that that any pure strategy
used with positive probability in a proper equilibrium is properly rationalizable (Prop. 2), leading to general existence for the latter concept. It is then
established (in Prop. 3) that the present paper’s definition of proper rationalizability corresponds to that of Schuhmacher [24]: A pure strategy is properly
rationalizable in the sense of the present paper if and only if it is used with
positive probability in some properly rationalizable mixed strategy in the sense
of Schuhmacher [24]. Furthermore, my definition is applied to show (in Prop.
4) that only strategies that lead to the backward induction outcome are properly rationalizable in the strategic form of a generic perfect information game.
Thus, Schuhmacher’s Thm. 2 (which shows that the backward induction outcome obtains with ‘‘high’’ probability for any given ‘‘small’’ e) is strengthened,
and an epistemic foundation for the backward induction procedure (as an alternative to Aumann’s [4]) is provided. Lastly, it is illustrated through an example how proper rationalizability can be used to test the robustness of inductive procedures. The proofs of Props. 1–4 are provided in Appendix B.
The analysis is limited to two-player games. The extension to general (nplayer) games raises the issue of whether (and if so, how) each player’s belief
about the strategy choices of the other players are stochastically independent.
This is outside the scope of the present paper.
2. An illustration
The symmetric game of Fig. 1 is an example where common certain belief of
proper consistency is su‰cient to determine completely each player’s preferences over his or her own strategies. The game is due to Blume et al. ([8],
Fig. 1).
In this game, caution implies that player 1 prefers M to U since M weakly
dominates U. Likewise, player 2 prefers C to L. Since 1 respects the preferences of 2 and, in addition, certainly believes that 2 is cautious, it follows that
1 deems C infinitely more likely than L. This in turn implies that 1 prefers D
to U. Likewise, since 2 respects the preferences of 1 and, in addition, certainly
L
C
R
U
1, 1
1, 1
1, 0
M
1, 1
2, 2
2, 2
D
0, 1
2, 2
3, 3
Fig. 1. Illustrating common certain belief of proper consistency
456
G. B. Asheim
believes that 1 is cautious, it follows that 2 prefers R to L. As a consequence,
since 1 respects the preferences of 2, certainly believes that 2 respects the preferences of 1, and certainly believes that 2 certainly believes that 1 is cautious,
it follows that 1 deems R infinitely more likely than L. Consequently, 1 prefers
D to M. A symmetric reasoning entails that 2 prefers R to C. Hence, if there is
common certain belief of proper consistency, it follows that the players’ preferences over their own strategies are given by
U MD
L C R:
The facts that D is the unique most preferred strategy for 1 and R is the unique
most preferred strategy for 2 mean that only D and R are properly rationalizable (cf. Def. 2 of Sect. 4.1). By Prop. 2 of Sect. 4.2, it then follows that the
pure strategy vector ðD; RÞ is the unique proper equilibrium, which can easily
be checked by inspection. However, note that in the argument above, each
player obtains certainty about the preferences of his opponent through deductive reasoning; i.e. such certainty is not assumed as in the concept of proper
equilibrium.
The concept of proper rationalizability yields a strict refinement of (ordinary) rationalizability. All strategies for both players are rationalizable, which
is implied by the fact that, in addition to ðD; RÞ, the pure strategy vectors
ðU; LÞ and ðM; CÞ are also Nash equilibria. The concept of proper rationalizability yields even a strict refinement when compared to the Dekel-Fudenberg
[14] procedure, which consists of one round of weak elimination followed by
iterated strong elimination, and which follows from there being common certain belief of caution and belief of opponent rationality (see Brandenburger
[11] and Börgers [10] as well as AD, Prop. 5.2). When the Dekel-Fudenberg
procedure is employed, only U is eliminated for 1, and only L is eliminated for
2, reflecting that also the pure strategy vector ðM; CÞ is a perfect equilibrium.
It is a general result that proper rationalizability refines the Dekel-Fudenberg
procedure (cf. Thm. 4 of Herings & Vannetelbosch [18] as well as Remark 1
below).
3. States, types, preferences, and belief
The purpose of this section is to present a framework for strategic games where
each player is modeled as a decision-maker under uncertainty. The decisiontheoretic analysis builds on Blume, et al. [7]. The framework is summarized by
the concept of a belief system (cf. Def. 1). Appendix A contains a presentation
of the decision-theoretic terminology, notation and results that will be utilized.
3.1. A strategic game form
With I ¼ f1; 2g as the set of players, let Si denote player i’s finite set of pure
strategies, and let z : S ! Z map strategy vectors into outcomes, where S ¼
S1 S2 is the set of strategy vectors and Z is the set of outcomes. Then
ðS1 ; S2 ; zÞ is a finite strategic two-player game form.
Proper rationalizability in lexicographic beliefs
457
3.2. States and types
The uncertainty faced by a player i in a strategic game form concerns the
strategy choice of his opponent j, j’s belief about i’s strategy choice, and so
on (cf. Tan & Werlang [28]). A type of a player i corresponds to a vNM utility
function and a belief about j’s strategy choice, a belief about j’s belief about
i’s strategy choice, and so on. Models of such infinite hierarchies of beliefs
(Böge & Eisele [9], Mertens & Zamir [20], Brandenburger & Dekel [12], Epstein & Wang [16]) yield S T as the complete state space, where T ¼ T1 T2
is the set of all feasible type vectors. Furthermore, for each i, there is a homeomorphism between Ti and the set of beliefs on S Tj .
For each type of any player i, the type’s decision problem is to choose one
of i’s strategies. For the modeling of this problem, the type’s belief about i’s
strategy choice is not relevant and can be ignored. Hence, in the setting of a
strategic game form the beliefs can be restricted to the set of opponent strategytype pairs, Sj Tj . Combined with a vNM utility function, the set of beliefs on
Sj Tj corresponds to a set of binary relations on the set of acts on Sj Tj ,
where an act on Sj Tj is a function that to any element of Sj Tj assigns an
objective randomization on Z.
In conformity with the literature on infinite hierarchies of beliefs, let
– the set of states of the world (or simply states) be W :¼ S T,
– each type ti of any player i correspond to a binary relation ti on the set of
acts on Sj Tj .
However, as the above results on infinite hierarchies of beliefs are not applicable in the present setting, I instead consider an implicit model – with a finite
type set Ti for each player i – from which infinite hierarchies of beliefs can be
constructed (see Sect. 3.4 for a justification). Moreover, since continuity is not
imposed, ti is assumed to satisfy completeness, transitivity, objective independence, nontriviality, conditional continuity and non-null state independence,
meaning that ti is represented by a vNM utility function uiti : Z ! < that
assigns a payo¤ to any outcome and a lexicographic probability system (LPS)
l ti ¼ ðm1ti ; . . . ; mLti ti Þ A LDðSj Tj Þ (cf. Blume et al. [7] and Appendix A). Being
a vNM utility function, uiti can be extended to objective randomizations on Z.
The construction is summarized by the following definition.
Definition 1. A belief system for a game form ðS1 ; S2 ; zÞ consists of
– for each player i, a finite set of types Ti ,
– for each type ti of any player i, a binary relation ti (ti ’s preferences) on the
set of acts on Sj Tj , where ti is represented by a vNM utility function uiti
on the set of objective randomizations on Z and an LPS l ti on Sj Tj .
3.3. Certain belief
For each player i, i’s certain belief can be derived from the belief system. To
state this epistemic operator, let, for each player i and each state o A W, ti ðoÞ
denote the projection of o on Ti . The operator will be used only for events that
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G. B. Asheim
concern the type vector; i.e. for E J W satisfying E ¼ S1 S2 projT1 T2 E.
For any such event E, let Ejti :¼ fðsj ; tj Þ A Sj Tj j ðt1 ; t2 Þ A projT1 T2 Eg denote
the set of opponent strategy-type pairs that are consistent with o A E and
ti ðoÞ ¼ ti .
Associate ‘certain belief ’ with the property that no element of the complement of the event is assigned positive probability by some probability distribution in l ti : If E J W satisfies E ¼ S1 S2 projT1 T2 E, then say that at o i
certainly believes the event E if o A Ki E, where
t ðoÞ
Ki E :¼ fo A W j kj i
t ðoÞ
J Ej i
g;
with kjti :¼ supp l ti ðJSj Tj Þ denoting the set of opponent strategy-type
pairs that ti does not deem Savage-null. This KD45 operator corresponds to
what Morris [21] calls ‘Savage-belief ’.
Say that there is mutual certain belief of E J W at o if o A KE, where
KE :¼ K1 E X K2 E. Say that there is common certain belief of E J W at o if
o A CKE, where CKE :¼ KE X KKE X KKKE X :
3.4. Preferences over strategies
Let Sti j denote the marginal of ti on Sj . A pure strategy si A Si can be viewed
as an act xSj on Sj that assigns zðsi ; sj Þ to any sj A Sj . Hence, Sti j is a binary
relation also on the subset of acts on Sj that correspond to i’s pure strategies.
Thus, Sti j can be referred to as ti ’s preferences over i’s pure strategies. Since ti
is represented by a vNM utility function and an LPS, Sti j shares these properties. Let Citi :¼ fsi A Si j Esi0 A Si ; si Sti j si0 g denote ti ’s set of most preferred
strategies (i.e. ti ’s choice set).
The use of a finite type set for each player i is justified since (1) the proper
consistency of ti (cf. Sect. 4.1) depends on opponent types only through their
preferences over j’s pure strategies and (2) there is only a finite collection of
such complete and transitive preference relations over Sj (cf. the finite algorithm constructed in the proof of Prop. 3).
3.5. A strategic game
Let, for each i, ui : S ! < be a vNM utility function that assigns payo¤ to any
strategy vector. Then G ¼ ðS1 ; S2 ; u1 ; u2 Þ is a finite strategic two-player game.
Assume that, for each i, there exist s, s 0 A S such that ui ðsÞ > ui ðs 0 Þ. The event
that i plays the game G is given by
t ðoÞ
½ui :¼ fo A W j ui i
z is a positive a‰ne transformation of ui g;
while ½u1 X ½u2 is the event that both players play G.
4. Consistency of preferences
Usually requirements in deductive game theory are imposed on choice. E.g.
rationality is a requirement on a pair ðsi ; ti Þ, where si is said to be a rational
Proper rationalizability in lexicographic beliefs
459
choice by ti if si A Citi . See e.g. Epstein ([15], Sect. 6) for a presentation of this
approach in a general context.
The present paper follows AD by imposing requirements on ti only. Since
ti corresponds to the preferences ti , such requirements will be imposed on
ti . Here I will focus on showing how proper rationalizabiliy – as a nonequilibrium analogue to Myerson’s [22] proper equilibrium – can be defined by
applying such an approach. The analysis is limited to two-player games, as an
extension to games with more than two players raises the issue of independence, which will not be addressed here.
In the same way as Blume et al. [8] characterize proper equilibrium as an
equilibrium in conjectures (which does not entail that players tremble, but that
each player takes into account the possibility that the opponent may tremble),
the present definition of proper rationalizability stems from requirements on
preferences. In particular, it di¤ers from Schuhmacher’s [24] main statement
of his definition by not modeling players that tremble; instead each player
takes into account the possibility that the opponent may tremble, he certainly
believes that his opponent takes into account the possibility that he himself
may tremble, and so on.
In a concluding discussion Schuhmacher also considers defining e-proper
rationalizability by posing requirements on the players’ beliefs rather than on
their mixed strategies. His alternative formulation di¤ers from the one o¤ered
here both in terminology (by having ‘types’ refer to ðsi ; ti Þ pairs) and in analysis (by more costly trembles being deemed much less likely – rather than infinitely less likely – than less costly ones).
4.1. Proper consistency
Proper consistency will be based on three requirements: The first of these ensures that each player plays the game G, the second requirement ensures that
each player takes all opponent strategies into account (is cautious), while the
third requirement ensures that each player deems one opponent strategy to be
infinitely more likely than another if the opponent prefers the one to the other
(respects preferences).
To impose these requirements, consider the following events
t ðoÞ
½caui :¼ fo A W j kj i
t ðoÞ
¼ Sj Tj i
g
½respi :¼ fo A W j ðsj ; tj Þ g ðsj0 ; tj Þ acc: to ti ðoÞ
t ðoÞ
whenever tj A Tj i
t
and sj Sj i sj0 g;
where Tj ti :¼ projTj kjti is the set of opponent types that ti does not deem
Savage-null, and where g means ‘infinitely more likely’ (cf. Appendix A).
– If o A ½caui , then ðsj ; tj Þ is not deemed Savage-null acc. to ti ðoÞ whenever tj
is not deemed Savage-null. It implies that the marginal of ti ðoÞ on Sj (i.e.,
t ðoÞ
ti ðoÞ’s preferences over Si , Si j ) is admissible on Sj .
– If o A ½respi , then ti ðoÞ respects the preferences of any opponent type that
is not deemed Savage-null.
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G. B. Asheim
Say that at o, i is properly consistent (with the game G and the preferences of
his opponent) if o A Aipr , where
Aipr :¼ ½ui X ½caui X ½respi :
Refer to A pr :¼ A1pr X A2pr as the event of proper consistency. The concept of
properly rationalizable strategies is defined as most preferred strategies in
states where there is common certain belief of proper consistency.
Definition 2. A pure strategy si for i is properly rationalizable in a finite stratet ðoÞ
gic two-player game G if there exists a belief system with si A Ci i
for some
pr
o A CKA .
4.2. Results
Blume et al.’s ([8], Prop. 5) characterization of Myerson’s [22] proper equilibrium in two-player games implies the following result, where one should note
that there is mutual certain belief of the type vector ðt1 ðoÞ; t2 ðoÞÞ at o if and
t ðoÞ
t ðoÞ
only if, for each i, Tj i ¼ ftj ðoÞg, and recall that, Ej A I , m1j is the primary
probability distribution in tj ðoÞ’s LPS, l tj ðoÞ .
Proposition 1. Consider a finite strategic two-player game G. A vector of mixed
strategies x ¼ ðx1 ; x2 Þ A DðS1 Þ DðS2 Þ is a proper equilibrium if and only if
there exists a belief system and o A A pr such that (1) there is mutual certain
belief of ðt1 ðoÞ; t2 ðoÞÞ at o, and (2) for each i A I , and for any si A Si , xi ðsi Þ ¼
t ðoÞ
m1j ðsi ; ti ðoÞÞ.
Hence, the event of proper consistency combined with mutual certain belief of
the type vector characterizes proper equilibrium. Mutual certain belief of the
type vector means that each player certainly believes the preferences of his opponent. This is an equilibrium assumption, which is not satisfied in general
when there is common certain belief of proper consistency.
The event of proper consistency combined with mutual certain belief of the
type vector implies that there is common certain belief of proper consistency.
Hence, Prop. 1 implies that any strategy used with positive probability in a
proper equilibrium is properly rationalizable.
Proposition 2. If x ¼ ðx1 ; x2 Þ A DðS1 Þ DðS2 Þ is a proper equilibrium in a finite
strategic two-player game G, then, for each i, any si A supp xi is properly rationalizable.
Since a proper equilibrium always exists, we obtain the following corollary.
Corollary 1. In any finite strategic two-player game G, there exists a belief system with CKA pr 0 q, implying that there exists, for each i, a nonempty set of
properly rationalizable strategies.
t ðoÞ
Remark 1: Substitute the event Bi ½ratj :¼ fo A W j ðsj ; tj Þ A supp m1i implies
t
sj A Cj j g for ½respi . Write Ai :¼ ½ui X ½caui X Bi ½ratj and A :¼ A1 X A2 . Then
Proper rationalizability in lexicographic beliefs
461
a strategy si surviving the Dekel-Fudenberg procedure can be characterized by
t ðoÞ
the property that there exists a belief system with si A Ci i for some o A CKA
(cf. AD, Prop. 5.2). Since ½respi J Bi ½ratj , it follows from Def. 2 that proper
rationalizability refines the Dekel-Fudenberg procedure. The notation reflects
that Bi ½ratj can be interpreted as the event that i believes (with probability one)
that j is rational.
This section is concluded by showing that Def. 2 is equivalent to Schuhmacher’s [24] definition of proper rationalizability, thereby establishing an
epistemic foundation for the concept that he defines.
Proposition 3. Consider a finite strategic two-player game G and let si be a pure
strategy for i. There exists a mixed strategy xi that is properly rationalizable
according to the definition of Schuhmacher [24] with xi ðsi Þ > 0 if and only if si
is properly rationalizable according to Def. 2.
5. Backward induction
Consider the centipede game of Fig. 2. The backward induction argument in
this game goes as follows: If 1’s second decision node is reached and 1 chooses
a most preferred strategy in the subgame, then 1 will choose D. If 2 knows/
believes this and chooses a most preferred strategy in the subgame defined by
her decision node, then 2 will choose d. If 1 knows/believes this and chooses a
most preferred strategy at the beginning of the game, then 1 will choose D at
his first decision node.
It has been hard to provide an unquestionable model of interactive epistemology that supports this simple backward induction argument (cf. Stalnaker [27]). E.g. two influential contributions – Aumann [4] and Ben-Porath
[6] – reach opposite conclusions on whether common knowledge/belief of rationality implies backward induction, while the epistemic analysis of Battigalli
& Siniscalchi [5] provides a foundation for the backward induction outcome in
the whole game, but does not support the backward induction argument. The
problem in the centipede game is that if 1 chooses F at his first decision node
and thus do not play in accordance with backward induction, then 2 may not
believe that 1 will play in accordance with backward induction at 1’s last decision node. If so, she may choose f , which in turn opens for the possibility
that the strategy FD may be a most preferred strategy for 1.
In the game of Fig. 2 the backward induction argument corresponds to
iterated elimination of weakly dominated strategies in the strategic form of
the game (see the right part of Fig. 2): First FF is eliminated for 1, then f is
Fig. 2. A centipede game
462
G. B. Asheim
eliminated for 2, and finally, FD is eliminated for 1. However, a formal model
of interactive epistemology has only recently been provided for this procedure
(cf. Brandenburger & Keisler [13]). A problem in the strategic game of Fig. 2
is that the elimination of f in the second round removes the reason why FF
and not FD was eliminated in the first round. However, it is exactly since FF
and not FD was eliminated in the first round that f can be eliminated in the
second round. And without the elimination of f , FD cannot be eliminated.
It is straightforward to see how common certain belief of proper consistency implies that players have preferences in accordance with backward induction in the sense that, in any subgame, the backward induction outcome is
reached if each player chooses a most preferred strategy: Caution implies that
player 1 prefers FD to FF since FD weakly dominates FF . Since 2 respects the
preferences of 1 and, in addition, certainly believes that 1 is cautious, it follows that 2 deems FD infinitely more likely than FF . This in turn implies that
2 prefers d to f . As a consequence, since 1 respects the preferences of 2, certainly believes that 2 respects the preferences of 1, and certainly believes that 2
certainly believes that 1 is cautious, it follows that 1 deems d infinitely more
likely than f . Consequently, 1 prefers D to FD. Hence, the players’ preferences over their own strategies are given by
D FD FF
d f:
Note that if there is common certain belief of proper consistency, then, in any
subgame, the play of most preferred strategies leads to the backward induction outcome. The remaining part of Sect. 5 shows that this result holds for
any generic extensive game of perfect information.
5.1. Preliminaries
A finite extensive game of almost perfect information G with 2 players and
M 1 stages can be described as follows. The sets of histories is determined inductively: The set of histories at the beginning of the first stage 1 is
H 1 ¼ fqg. Let H m denote the set of histories at the beginning of stage
m. At h A H m , let, for each player i, i’s finite action set be denoted Ai ðhÞ,
where i is inactive at h if Ai ðhÞ is a singleton. Write AðhÞ :¼ A1 ðhÞ A2 ðhÞ.
Define the set of histories at the beginning of stage m þ 1 as follows:
H mþ1 :¼ fðh; aÞ j h A H m and a A AðhÞg. This concludes the induction. Let
M1
H :¼ 6m¼1 H m denote the set of subgames and let Z :¼ H M denote the set
of outcomes.
A pure strategy for player i is a function si that assigns an action in Ai ðhÞ
to any h A H. Let Si denote player i’s finite set of pure strategies. Let z : S ! Z
map strategy vectors into outcomes. Then ðS1 ; S2 ; zÞ is a finite strategic twoplayer game form. A pure strategy si A Si can be viewed as an act on Sj that
assigns zðsi ; sj Þ A Z to any sj A Sj . The set Si is partitioned into equivalent
classes of acts since a pure strategy si also determines actions in subgames
which si prevents from being reached. Each equivalent class corresponds to a
plan of action in the sense of Rubinstein [23]. As there is no need here to differentiate between identical acts, the concept of a plan of action would have
Proper rationalizability in lexicographic beliefs
463
su‰ced. If, for each player i, i’s payo¤ in G is determined by the vNM utility
function ui : Z ! <, then G ¼ ðS1 ; S2 ; u1 ; u2 Þ is the strategic game corresponding to G, where, for each i, ui : S ! < is defined by ui ¼ ui z. For any
h A H W Z, there exists a strategic form structure: Let SðhÞ ¼ S1 ðhÞ S2 ðhÞ
denote the set of strategy vectors that are consistent with h being reached. If h 0
is the predecessor of h, then Sðh 0 Þ K SðhÞ. If si A Si and h A H, let si jh denote
the strategy in Si ðhÞ coinciding with si except at predecessors of h, where si jh
determines the unique action leading to h.
A finite extensive game is of perfect information if, at any h A H, there exists at most one player that has a non-singleton action set. It is generic if, for
each i, ui ðzÞ 0 ui ðz 0 Þ whenever z and z 0 are di¤erent outcomes. Generic extensive games of perfect information have a unique subgame-perfect equilibrium. Moreover, in such games the procedure of backward induction yields in
any subgame the unique subgame-perfect equilibrium outcome. If p denotes
the unique subgame-perfect equilibrium, then, for any subgame h, zð pjh Þ is the
backward induction outcome in the subgame h, and Sðzð pjh ÞÞ is the set of
strategy vectors consistent with the backward induction outcome in the subgame h.
For each type ti of player i, Sti j ðhÞ is ti ’s preferences over i’s pure strategies
Si ðhÞ in any subgame h A H. Let Citi ðhÞ :¼ fsi A Si ðhÞ j Esi0 A Si ðhÞ; si Sti j ðhÞ si0 g
denote ti ’s set of most preferred strategies in the subgame h. Refer to Citi : H !
2 Si ðhÞ nfqg as ti ’s choice function. Note that Citi ¼ Citi ðqÞ, and write, for any
h A H, C t ðhÞ :¼ C1t1 ðhÞ C2t2 ðhÞ.
5.2. Result on backward induction
In analogy with Aumann’s [4] Thm. A, it is established that any vector of most
preferred strategies in a subgame of a generic perfect information game, at a
state where there is common certain belief of proper consistency, leads to the
backward induction outcome in the subgame (Prop. 4). The analogy of Aumann’s Thm. B – that for any generic perfect information game, common certain belief of proper consistency is possible; i.e. that the result of Prop. 4 is not
empty – has already been established through Cor. 1 since any extensive game
of (almost) perfect information G has a corresponding strategic game G.
Proposition 4. Consider a finite generic two-player extensive game of perfect information G with corresponding strategic game G. If, for some belief system,
o A CKA pr , then, for each h A H, C tðoÞ ðhÞ J Sðzðpjh ÞÞ, where p denotes the
unique subgame-perfect equilibrium.
6. Induction in a betting game
The games of Figs. 1 and 2 have in common that the properly rationalizable
strategies coincide with those surviving iterated (maximal) elimination of
weakly dominated strategies. In the present section it will be shown that this
conclusion does not hold in general. Rather, the concept of proper rationalizability can be used to test the robustness of iterated (maximal) elimination of
weakly dominated strategies and other inductive procedures.
464
G. B. Asheim
a
b
c
Player 1
9
6
3
Player 2
9
6
3
1=3
1=3
1=3
Fig. 3. A betting game
yy
yn
ny
nn
YY
2, 2
1, 1
1, 1
0, 0
YN
3, 3
3, 3
0, 0
0, 0
NY
1, 1
2, 2
1, 1
0, 0
NN
0, 0
0, 0
0, 0
0, 0
Fig. 4. The strategic form of the betting game
Figure 3 illustrates a simplified version of a betting game introduced by
Sonsino, Erev & Gilat [25] for the purpose of experimental study; Søvik [26]
has subsequently repeated their experiment in alternative designs. The two
players consider to bet and have a common and uniform prior over the set of
states that determine the outcome of the bet. If the state is a, then 1 loses 9
and 2 wins 9 if betting occurs. If the state is b, then 1 wins 6 and 2 loses 6 if
betting occurs. Finally, if the state is c, then 1 loses 3 and 2 wins 3 if betting
occurs. Player 1 is informed of whether the state of the bet is equal to a or in
the set fb; cg. Player 2 is informed of whether the state of the bet is in the set
fa; bg or equal to c.
As a function of their information, each player can announce to accept the
bet or not. For player 1 the strategy YN means to accept the bet if informed of
a and not to accept the bet if informed of fb; cg, etc. For player 2 the strategy
yn means to accept the bet if informed of fa; bg and not to accept the bet if
informed of c, etc. Betting occurs if and only if both players have accepted the
bet. This yields the strategic game of Fig. 4.
6.1. An inductive procedure
If player 2 naively believes that player 1 is equally likely to accept the bet
when informed of a as when informed of fb; cg, then 2 will wish to accept the
bet when informed of fa; bg. However, the following, seemingly intuitive, inductive procedure appears to indicate that 2 should never accept the bet if informed of fa; bg: Player 1 should not accept the bet when informed of a since
he cannot win by doing so. This eliminates his strategies YY and YN. Player
2, realizing this, should never accept the bet when informed of fa; bg, since –
as long as 1 never accepts the bet when informed of a – she cannot win by
doing so. This eliminates her strategies yy and yn. This in turn means that
player 1, realizing this, should never accept the bet when informed of fb; cg,
Proper rationalizability in lexicographic beliefs
465
since – as long as 2 never accepts the bet when informed of fa; bg – he cannot
win by doing so. This eliminates his strategy NY. This inductive argument
corresponds to iterated (maximal) elimination of weakly dominated strategies,
except that the latter procedure eliminates 2’s strategies yn and nn in the first
round. The argument seems to imply that player 2 should never accept the bet
if informed of fa; bg and that player 1 should never accept the bet if informed
of fb; cg. Is this a robust conclusion?
6.2. Proper rationalizability in the betting game
The strategic game of Fig. 4 has a set of Nash equilibria that includes the pure
strategy vectors ðNN; nyÞ and ðNN; nnÞ, and a set of perfect equilibria that
includes the pure strategy vector ðNN; nyÞ. However, there is a unique proper
equilibrium where player 1 plays NN with probability one, and where player 2
mixes between yy with probability 1=5 and ny with probability 4=5. It is instructive to see why the pure strategy vector ðNN; nyÞ is not a proper equilibrium. If 1 assigns probability one to 2 playing ny, then he prefers YN to NY
(since the more serious mistake to avoid is to accept the bet when being
informed of fb; cg). However, if 2 respects 1’s preferences and certainly believes that 1 prefers YN to NY , then she will herself prefer yy to ny, undermining ðNN; nyÞ as a proper equilibrium. The mixture between yy and ny
in the proper equilibrium is constructed so that 1 is indi¤erent between YN
and NY .
From Prop. 2 it follows that both yy and yn are properly rationalizable
strategies for 2. Moreover, if 1 certainly believes that 2 is of a type with only
yy as a most preferred strategy, then NY is a most preferred strategy for 1,
implying that NY in addition to NN is a properly rationalizable strategy for 1.
That these strategies are in fact properly rationalizable is verified by the belief
system of Table 1. In the table the preferences of each type ti of any player i
are represented by a vNM utility function uiti satisfying uiti z ¼ ui and a 4level LPS on Sj ftj0 ; tj00 g, with the first numbers in the parantheses expressing
primary probability distributions, the second numbers expressing secondary
probability distributions, etc. With W ¼ S ft10 ; t100 g ft20 ; t200 g, it follows that
W ¼ A pr ¼ CKA pr . Since each type’s preferences over his/her own strategies
are given by
Table 1. A belief system for the betting game
t10
yy
yn
ny
nn
t20
YY
YN
NY
NN
t20
ð0; 0; 1; 0Þ
ð0; 0; 0; 1Þ
ð1; 0; 0; 0Þ
ð0; 1; 0; 0Þ
t10
ð0; 0; 0; 0Þ
ð0; 0; 0; 0Þ
ð0; 0; 0; 0Þ
ð0; 0; 0; 0Þ
t200
ð0; 0; 0; 0Þ
ð0; 0; 0; 0Þ
ð0; 0; 0; 0Þ
ð0; 0; 0; 0Þ
t100
ð0; 0; 1; 0Þ
ð0; 0; 0; 1Þ
ð1; 0; 0; 0Þ
ð0; 1; 0; 0Þ
t100
yy
yn
ny
nn
t200
YY
YN
NY
NN
t20
ð0; 0; 0; 0Þ
ð0; 0; 0; 0Þ
ð0; 0; 0; 0Þ
ð0; 0; 0; 0Þ
t10
ð0; 0; 0; 1Þ
ð0; 1; 0; 0Þ
ð0; 0; 1; 0Þ
ð1; 0; 0; 0Þ
t200
ð1; 0; 0; 0Þ
ð0; 1; 0; 0Þ
ð0; 0; 1; 0Þ
ð0; 0; 0; 1Þ
t100
ð0; 0; 0; 0Þ
ð0; 0; 0; 0Þ
ð0; 0; 0; 0Þ
ð0; 0; 0; 0Þ
466
G. B. Asheim
t10 : NN YN NY YY
t100 : NY NN YY YN
t20 : ny nn yy yn
t200 : yy yn ny nn;
it follows that NY and NN are properly rationalizable for player 1 and yy and
ny are properly rationalizable for player 2. Note that YY and YN for player 1
and yn and nn for player 2 cannot be properly rationalizable since these
strategies are weakly dominated and, thus, cannot be most preferred strategies
for cautious players.
The lesson to be learned from this analysis, is that is not obvious that deductive reasoning should lead players to refrain from accepting the bet in the
betting game. The experiments by Sonsino et al. [25] and Søvik [26] show that
some subjects do in fact accept the bet in a slightly more complicated version
of this game. By comparison to Prop. 4, the analysis can be used to support
the argument that backward induction in generic perfect information games is
more convincing than the inductive procedure for the betting game discussed
in Sect. 6.1.
7. Conclusion
Iterated (maximal) elimination of weakly dominated strategies, backward induction in perfect information games, and other inductive procedures have been
subject to critical scrutiny during the last few years. The present paper shows
how proper rationalizability – based on the imposition of common certain belief of the proper consistency of preferences with the game and the preferences
of the opponent – can be used to test the robustness of such procedures.
It has been shown how proper rationalizability in general supports backward induction in generic perfect information games with two players. However, in other games – like the betting game of Sect. 6 – the concept of proper
rationalizability points to fundamental reasons why deductive reasoning may
not coincide with iterated (maximal) elimination of weakly dominated strategies.
8. Appendix A. The decision-theoretic framework
The purpose of this appendix is to present the decision-theoretic terminology,
notation and results utilized and referred to in the main text.
Consider a decision-maker under uncertainty. Let F be a finite set of states,
where the decision-maker is uncertain about what state in F will be realized.
Let Z be a finite set of outcomes. In the tradition of Anscombe & Aumann [1],
the decision-maker is endowed with a binary relation over all functions that to
each element of F assigns an objective randomization on Z. Any such function
xF : F ! DðZÞ is called an act on F. Write xF and yF for acts on F. A complete and transitive binary relation on the set of acts on F is denoted by F ,
where xF F yF means that xF is preferred or indi¤erent to yF . As usual, let
F ( preferred to) and @F (indi¤erent to) denote the asymmetric and symmetric
Proper rationalizability in lexicographic beliefs
467
parts of F . A binary relation F on the set of acts on F is said to satisfy
– objective independence if xF0 F (respectively @F ) xF00 i¤ gxF0 þ ð1 gÞyF F
(respectively @F ) gxF00 þ ð1 gÞyF , whenever 0 < g < 1 and yF is arbitrary.
– nontriviality if there exist xF and yF such that xF F yF .
If E J F , let xE denote the restriction of xF to E. Define the conditional
binary relation E by xF0 E xF00 if, for any yF , ðxE0 ; yE Þ F ðxE00 ; yE Þ, where
E denotes F nE. Say that the state f A F is Savage-null if xF @f f g yF for all
acts xF and yF on F. A binary relation F is said to satisfy
– conditional continuity if, Ef A F , there exist 0 < g < d < 1 such that dxF0 þ
ð1 dÞxF00 f f g yF f f g gxF0 þ ð1 gÞxF00 whenever xF0 f f g yF f f g xF00 .
– non-null state independence if xF feg yF i¤ xF f f g yF whenever e and f
are not Savage-null and xF and yF satisfy xF ðeÞ ¼ xF ð f Þ and yF ðeÞ ¼ yF ð f Þ.
If e; f A F , then e is deemed infinitely more likely than f (e g f ) if e is not
Savage-null and xF feg yF implies ðxf f g ; xf0 f g Þ fe; f g ðyf f g ; yf0 f g Þ for all xF0 ,
yF0 . According to this definition, f may, but need not, be Savage-null if e g f .
If u P
: Z ! < is a vNM utility function, abuse notation slightly by writing
uðxÞ ¼ z A Z xðzÞuðzÞ whenever x A DðZÞ is an objective randomization. Say
that xE strongly dominates yE w.r.t. u if, Ef A E, uðxE ð f ÞÞ > uðyE ð f ÞÞ. Say that
xE weakly dominates yE w.r.t. u if, Ef A E, uðxE ð f ÞÞ b uðyE ð f ÞÞ, with strict
inequality for some e A E. Say that F is admissible on E ð0qÞ if xF F yF
whenever xE weakly dominates yE .
The following representation result due to Blume et al. ([7], Thm. 3.1) can
now be stated. It requires the notion of a lexicographic probability system (LPS)
which consists of L levels of subjective probability distributions: If L b 1 and,
El A f1; . . . ; Lg, ml A DðF Þ, then l ¼ ðm1 ; . . . ; mL Þ is an LPS on F. Let LDðF Þ
denote the set of LPSs on F, and let, for two utility vectors v and w, v bL w
denote that, whenever wl > vl , there exists l 0 < l such that vl 0 > wl 0 .
Proposition A1. If F is complete and transitive, and satisfies objective independence, nontriviality, conditional continuity, and non-null state independence, then there exists a vNM utility function u : Z ! < and an LPS l ¼
ðm1 ; . . . ; mL Þ A LDðF Þ such that xF F yF i¤
!L
!L
X
X
ml ð f ÞuðxF ð f ÞÞ
bL
ml ð f ÞuðyF ð f ÞÞ
:
f AF
l¼1
f AF
l¼1
If F ¼ F1 F2 and F is a binary relation on the set of acts on F, then
say that F1 is the marginal of F on F1 if, xF1 F1 yF1 i¤ xF F yF whenever
xF1 ð f1 Þ ¼ xF ð f1 ; f2 Þ and yF1 ð f1 Þ ¼ yF ð f1 ; f2 Þ for all ð f1 ; f2 Þ.
9. Appendix B. Proofs of Propositions 1–4
Some properties of the certain belief operator (cf. Sect. 3.3) must be established for the proofs of Props. 2, 3, and 4. It can be checked that Ki W ¼ W and
Ki q ¼ q, and, for any events E and F satisfying E ¼ S1 S2 projT1 T2 E
468
G. B. Asheim
(and correspondingly for F ), Ki E X Ki F ¼ Ki ðE X F Þ, Ki E J Ki Ki E, and
sKi E J Ki ðsKi EÞ, implying that, for any such event E, Ki E ¼ Ki Ki E. Write
K 0 E :¼ E and, for each g b 1, K g E :¼ KK g1 E. Since Ki ðE X F Þ ¼ Ki E X
Ki F and Ki Ki E ¼ Ki E, it follows Eg b 2, K g E ¼ K1 K g1 E X K2 K g1 E J
K1 K1 K g2 E X K2 K2 K g2 E ¼ K1 K g2 E X K2 K g2 E ¼ K g1 E. Even though
the truth axiom (Ki E J E) is not satisfied, E ¼ A pr (and likewise in the case of
E ¼ ½e-prop trem) can be written as E ¼ E1 X E2 where, for each i, Ei ¼ Si Sj projTi Ei Tj . For any such event E, mutual certain belief of E implies
that E is true: KE ¼ K1 E X K2 E J K1 E1 X K2 E2 ¼ E1 X E2 ¼ E since, for
each i, Ki Ei ¼ Ei . Hence, (i) Eg b 1, K g E J K g1 E, and (ii) bg 0 b 0 such
that K g E ¼ CKE for g b g 0 since W is finite, implying that CKE ¼ KCKE.
For the proofs of Props. 1 and 3 two results from Blume et al. ([8]; henceforth referred to as BBD) are needed. To state these results, introduce the
following notation. Let l ¼ ðm1 ; . . . ; mL Þ be an LPS on a finite set F and let
r ¼ ðr1 ; . . . ; rL1 Þ A ð0; 1Þ L1 . Then, ral denotes the probability distribution
on F given by the nested convex combination
ð1 r1 Þm1 þ r1 ½ð1 r2 Þm2 þ r2 ½ð1 r3 Þm3 þ r3 ½. . . . . .:
Lemma 1 [Prop. 2 in BBD]. Let ðxðnÞÞn A @ be a sequence of probability distributions on a finite set F. Then, there exists a subsequence xðmÞ of ðxðnÞÞn A @ ,
an LPS l ¼ ðm1 ; . . . ; mL Þ, and a sequence rðmÞ of vectors in ð0; 1Þ L1 converging
to zero such that xðmÞ ¼ rðmÞal for all m.
The following lemma is a variant of Prop. 1 in BBD.
Lemma 2. Consider a type ti of player i whose preferences over acts on Sj Tj are
represented by uiti – with uiti z ¼ ui – and t l ti ¼ ðm1ti ; . . . ; mLti ti Þ A LDðSj Tj Þ.
i
Then, for every sequence ðrðnÞÞn A @ in ð0; 1Þ L 1 converging to zero there is an n 0
such that, Esi , si0 A Si , si Sti j si0 if and only if
XX
XX
ðrðmÞal ti Þðsj ; tj Þui ðsi ; sj Þ >
ðrðmÞal ti Þðsj ; tj Þui ðsi0 ; sj Þ
sj
tj
sj
tj
for all n b n 0 .
Proof: Suppose that si Sti j si0 . Then, there is some l 0 A f1; . . . ; Lti g such that
XX
sj
mlti ðsj ; tj Þui ðsi ; sj Þ ¼
tj
XX
sj
mlti ðsj ; tj Þui ðsi0 ; sj Þ
ð1Þ
mlti0 ðsj ; tj Þui ðsi0 ; sj Þ:
ð2Þ
tj
for all l < l 0 and
XX
sj
tj
mlti0 ðsj ; tj Þui ðsi ; sj Þ >
XX
sj
tj
Let ðrðnÞÞn A @ be a sequence in ð0; 1Þ Li 1 converging to zero. By (1) and (2),
XX
sj
tj
ðrðnÞal ti Þðsj ; tj Þui ðsi ; sj Þ >
XX
sj
tj
ðrðnÞal ti Þðsj ; tj Þui ðsi0 ; sj Þ
Proper rationalizability in lexicographic beliefs
469
if n is large enough. Since Si is finite, this is true if n is large enough for any si ,
si0 A Si satisfying si Sti j si0 . The other direction follows from the proof of Prop.
1 in BBD. r
Proof (of Prop. 1): (Only if.): Let ðx1 ; x2 Þ be a proper equilibrium. Hence, for
each i, there is a sequence of completely mixed strategies ðxi ðnÞÞn A @ converging
to xi , where, for each n, ðx1 ðnÞ; x2 ðnÞÞ is an eðnÞ-proper equilibrium and
eðnÞ ! 0 as n ! y. Then, by the necessity part of Prop. 5 in BBD, there exists a pair of preferences, t1 and t2 , that are represented by u1t1 and l t1 ¼
ðm1t1 ; . . .Þ A LDðS2 ft2 gÞ, and u2t2 and l t2 ¼ ðm1t2 ; . . .Þ A LDðS1 ft1 gÞ, respectively – with u1t1 z ¼ u1 and, Es2 A S2 , m1t1 ðs2 ; t2 Þ ¼ x2 ðs2 Þ, and u2t2 z ¼ u2
and, Es1 A S1 , m1t2 ðs1 ; t1 Þ ¼ x1 ðs1 Þ – such that t1 and t2 are cautious and respect
preferences. This argument involves Lemma 1 (which yields, for each i, the
existence of l ti with full support on DðSj ftj gÞ by means of a subsequence
xj ðmÞ of ðxj ðnÞÞn A @ ) and Lemma 2 (which yields that, for m large enough, i
having the conjecture xj ðmÞ leads to the same preferences over i’s strategies
as Sti j ). Construct a belief system having a state space with a single type
vector, W ¼ S ft1 g ft2 g. By construction, W ¼ ½u1 X ½u2 X ½cau1 X ½cau2 X
½resp1 X ½resp2 ¼ A pr . Furthermore, Eo A W, there is mutual certain belief of
the type vector ðt1 ; t2 Þ at o.
(If.) Suppose there exists a belief system and o A A pr such that (1) there is
mutual certain belief of ðt1 ðoÞ; t2 ðoÞÞ at o, and (2) for each i A I , and for any
t ðoÞ
si A Si , xi ðsi Þ ¼ m1j ðsi ; ti ðoÞÞ. Then, by the su‰ciency part of Prop. 5 in BBD,
there exists, for each i, a sequence of completely mixed strategies ðxi ðnÞÞn A @
converging to xi , where, for each n, ðx1 ðnÞ; x2 ðnÞÞ is an eðnÞ-proper equilibrium and eðnÞ ! 0 as n ! y. This argument involves Lemma 2 (which yields,
for each j, the existence of ðxj ðnÞÞn A @ so that, for all n, i having the conjecture
xj ðnÞ leads to the same preferences over i’s strategies as Sti j ). r
Proof (of Prop. 2): Let ðx1 ; x2 Þ be a proper equilibrium. Follow the ‘only if ’
part of the proof of Prop. 1 to construct a belief system with W ¼ S ft1 g ft2 g ¼ A pr . Hence, CKA pr ¼ W since KW ¼ W. This implies that, Ei A I , any
si A Citi is properly rationalizable. The result follows since respect of opponent
t
preferences implies that, Ei A I , supp xi fti g ¼ supp m1j J Citi fti g. r
To prove Prop. 3, we first need to reproduce Schuhmacher’s [24] definition
of the concept of proper rationalizability. As before, let the set of states be
W ¼ S T, but consider a specialized definition of a belief system.
Definition 3. An -belief system for a game G ¼ ðS1 ; S2 ; u1 ; u2 Þ consists of
– for each player i, a finite set of types Ti ,
– for each type ti of any player i, a pair of a mixed strategy and a probability
distribution ðxiti ; m ti Þ A DðSi Þ DðSj Tj Þ, where supp xiti ¼ Si , supp m ti ¼
Sj Tj ti for some Tj ti J Tj , and, Esj A Sj , Etj A Tj ti ,
m ti ðsj ; tj Þ
t
¼ xj j ðsj Þ:
m ti ðSj ftj gÞ
470
G. B. Asheim
It follows from Def. 3 that, for each type ti of player i, ti ’s preferences over i’s
pure strategies are given by
si Sti j si0 if and only if
XX
XX
m ti ðsj ; tj Þui ðsi ; sj Þ b
m ti ðsj ; tj Þui ðsi0 ; sj Þ:
sj
tj
sj
tj
Given an -belief system, consider, for each type ti of player i, the following
event
t ðoÞ
½e-prop tremi :¼ fo A W j exi i
t ðoÞ
ðsi Þ b xi i
t ðoÞ
ðsi0 Þ whenever si Si j
si0 g:
If o A ½e-prop tremi , then ti ðoÞ is said to satisfy the e-proper trembling condition. Refer to ½e-prop trem :¼ ½e-prop trem1 X ½e-prop trem2 as the event of eproper trembling. Schuhmacher’s [24] definition of e-proper rationalizability
can now be formally stated. Note that I refer to a limit of a sequence of eproperly rationalizable strategies as e goes to zero as an -properly rationalizable strategy.
Definition 4 (Schuhmacher [24]). A mixed strategy xi for i is e-properly rationalizable in a finite strategic two-player game G if there exists an -belief system
t ðoÞ
with xi i ¼ xi for some o A CK½e-prop trem. A mixed strategy xi for i is properly rationalizable if there exists a sequence ðxi ðnÞÞn A @ of eðnÞ-properly rationalizable strategies converging to xi , where eðnÞ ! 0 as n ! y.
Proposition 3 can now be restated.
Proposition 3. Consider a finite strategic two-player game G and let si be a pure
strategy for i. There exists an -properly rationalizable strategy xi with xi ðsi Þ >
0 if and only if si is properly rationalizable according to Def. 2.
Schuhmacher [24] considers a set of type vectors T ¼ T1 T2 , where each
type ti of either player i plays a completely mixed strategy xiti and has a probability distribution on Sj Tj , for which the conditional distribution on Sj t
ftj g coincides with xj j whenever the conditional distribution is defined. His
formulation implies that all types of a player agrees not only on the preferences but also on the relative likelihood of the strategies for any given opponent type. In contrast, Def. 2 of Sect. 4.1 requires the types of a player only to
agree on the preferences of any given opponent type. This di¤erence implies
that expanded type sets must be constructed for the ‘if ’ part of the proof of
Prop. 3.
Proof (of the ‘if ’ part of Prop. 3): Assume that s1 is properly rationalizable
t ðo Þ
for 1 acc. to Def. 2. Hence, there exists a belief system with s1 A C11
for
some o A CKA pr . Let, Ei A I , Ti 0 :¼ fti ðoÞ j o A CKA pr g. Note that, Ei A I
t ðoÞ
and Eo A CKA pr , Tj i J Tj 0 since CKA pr ¼ KCKA pr J Ki CKA pr .
We first construct a sequence, indexed by n, of -belief systems. By Def. 3
this involves, for each n and for each player i, a finite set of types – which we
below denote by Ti 00 and which will not vary with n – and, for each n, for each
Proper rationalizability in lexicographic beliefs
471
i, and for each type ti A Ti 00 , a mixed strategy and a probability distribution
ðxiti ðnÞ; m ti ðnÞÞ A DðSi Þ DðSj Tj 00 Þ that will vary with n.
For either player i and each type ti A Ti 0 of the original belief system,
make as many ‘‘clones’’ of ti as there are members of Tj 0 : Ei A I , Ti 00 :¼
fti ðti ; tj Þ j ti A Ti 0 and tj A Tj 0 g, where ti ðti ; tj Þ is the ‘‘clone’’ of ti associated
with tj . The term ‘‘clone’’ in the above statement reflects that, Etj A Tj 0 , ti ðti ; tj Þ
is assumed to ‘‘share’’ the preferences of ti in the sense that
t ðt ; t Þ
1. the set of opponent types that ti ðti ; tj Þ does not deem Savage-null, Tj i i j ,
is equal to ftj ðtj0 ; ti Þ j tj0 A Tj ti g (JTj 00 since Tj ti J Tj 0 ), and
2. the likelihood of ðsj ; tj ðtj0 ; ti ÞÞ according to ti ðti ; tj Þ is equal to the likelihood
of ðsj ; tj0 Þ according to ti .
t ðt ; t Þ
Since Tj i i j ¼ ftj ðtj0 ; ti Þ j tj0 A Tj ti g is independent of tj , but corresponds to
disjoint subsets of Tj 00 for di¤erent ti ’s, we obtain the following conclusion for
any pair of type vectors ðt1 ; t2 Þ, ðt10 ; t20 Þ A T10 T20 :
Tj
Tj
ti ðti ; tj Þ
ti ðti ; tj Þ
¼ Tj
X Tj
ti ðti0 ; tj0 Þ
ti ðti0 ; tj0 Þ
if ti ¼ ti0 ;
¼q
if ti 0 ti0 :
This ends the construction of type sets in the sequence of -belief systems.
Fix a player i and consider any ti A Ti 00 . Since CKA pr J ½ui , ti can be
represented by a vNM utility function uiti satisfying uiti z ¼ ui and an LPS
l ti on Sj Tj ti . Since CKA pr J ½caui , this LPS yields, for each tj A Tj ti ,
a partition fEjti ð1Þ; . . . ; Ejti ðL ti Þg of Sj Tj ti , where ðsj ; tj Þ g ðsj0 ; tj0 Þ according to ti if and only ðsj ; tj Þ A Ejti ðlÞ, ðsj0 ; tj0 Þ A Ejti ðl 0 Þ and l < l 0 . Since
CKA pr J½respi , it follows that sj is a most preferred strategy for tj in
fsj0 A Sj j ðsj0 ; tj Þ A Ejti ðlÞ W W Ejti ðL ti Þg if ðsj ; tj Þ A Ejti ðlÞ.
Consider any i and ti A Ti 00 . Construct the sequence ðm ti ðnÞÞn A @ as follows.
ti
Choose Eti A fti ðti ; tj Þ j tj A Tj 0 g one common sequence ðr ti ðnÞÞn A @ in ð0; 1Þ L 1
converging to 0 and let the sequence of probability distributions ðm ti ðnÞÞn A @ be
given by m ti ðnÞ ¼ r ti ðnÞal ti . For all n, supp m ti ðnÞ ¼ Sj Tj ti . By Lemma 2
ðr ti ðnÞÞn A @ can be chosen such that, for all n,
XX
sj
m ti ðnÞðsj ; tj Þui ðsi ; sj Þ >
tj
XX
sj
m ti ðnÞðsj ; tj Þui ðsi0 ; sj Þ
tj
if and only if si Stij si0 . Hence, for all n, the belief m ti ðnÞ leads to the same
preferences over i’s strategies as Stij . This ends the construction of the sequences ðm ti ðnÞÞn A @ in the sequence of -belief systems.
Consider now the construction of the sequence ðxiti ðnÞÞn A @ for any i and
t
ti A Ti 00 . There are two cases. Case 1: If there is tj A Tj 00 such that ti A Ti j ,
ti
tj
implying that Si fti g J supp m ðnÞ, then let xi ðnÞ be determined by
xiti ðnÞðsi Þ ¼
m tj ðnÞðsi ; ti Þ
:
m tj ðnÞðSi fti gÞ
472
G. B. Asheim
Moreover, for each n, there exists eðnÞ such that, for each player i, the eðnÞproper trembling condition is satisfied for all such types in Ti 00 : Since
xiti ðnÞðsi0 Þ m tj ðnÞðsi0 ; ti Þ
¼
!0
xiti ðnÞðsi Þ m tj ðnÞðsi ; ti Þ
t
as n ! y
t
if ðsi ; ti Þ A Ei j ðlÞ, ðsi0 ; ti Þ A Ei j ðl 0 Þ and l < l 0 , and since si is a most preferred
t
t
t
strategy for ti in fsi0 A Si j ðsi0 ; ti Þ A Ei j ðlÞ W W Ei j ðL tj Þg if ðsi ; ti Þ A Ei j ðlÞ,
ti
it follows that there exists a sequence ðe ðnÞÞn A @ converging to 0 such that, for
all n,
e ti ðnÞxiti ðnÞðsi Þ b xiti ðnÞðsi0 Þ whenever
XX
XX
m ti ðnÞðsj ; tj Þui ðsi ; sj Þ >
m ti ðnÞðsj ; tj Þui ðsi0 ; sj Þ:
sj
tj
sj
tj
Let, for each n,
eðnÞ :¼ maxfe t1 ðnÞ j bt2 A T200 s:t: t1 A T1t2 g
W fe t2 ðnÞ j bt1 A T100 s:t: t2 A T2t1 g:
Since the type sets are finite, eðnÞ ! 0 as n ! y. Case 2: If there is no tj A Tj 00
t
such that ti A Ti j , then let xiti ðnÞ be any mixed strategy having the property
that ti satisfies the eðnÞ-proper trembling condition given the belief m ti ðnÞ. This
ends the construction of the sequences ðxiti ðnÞÞn A @ in the sequence of -belief
systems.
t
We then turn to the construction of a sequence ðx1 1 ðnÞÞn A @ converging to x1 with x1 ðs1 Þ ¼ 1. Add type t1 to T100 having the property
that
1
t
m t1 ðnÞ ¼ m t1 ðt1 ðo Þ; t2 Þ ðnÞ for some t2 A T20 , but where x1 1 ðnÞ ¼ 1 x1 þ
n
1 t1 ðt1 ðo Þ; t2 Þ
x1
ðnÞ and x1 ðs1 Þ ¼ 1. For all n, we have that the belief m t1 ðnÞ leads
n
t ðo Þ
to the same preferences over 1’s strategies as Si j . This
in turn implies that
t ðo Þ
t1 satisfies the eðnÞ-trembling condition since s1 A C11
.
Consider the sequence, indexed by n, of -belief systems,
– with T100 W ft1 g as the type set for 1 and T200 as the type set for 2,
– with, for each type ti of any player i, ðxiti ðnÞ; m ti ðnÞÞ as the sequence of a
mixed strategy and a probability distribution,
as constructed above. Furthermore, it follows that, for all n, the eðnÞ-proper
trembling condition is satisfied for all types in T100 W ft1 g and all types in T200 ,
where eðnÞ ! 0 as n ! y. Hence, for all n,
CK½eðnÞ-prop trem ¼ S1 S2 ðT100 W ft1 gÞ T200 ;
t
t
in particular, x1 1 ðnÞ is eðnÞ-properly rationalizable. Moreover, ðx1 1 ðnÞÞn A @
converges to x1 with x1 ðs1 Þ ¼ 1. By Def. 4 this shows that s1 is played with
positive probability in some -properly rationalizable strategy. r
Proper rationalizability in lexicographic beliefs
473
In the definition of -proper rationalizability, g in K g ½e-prop trem goes to
infinity for each e, and then e converges to 0. The strategy for the proof of the
‘only if ’ part of Prop. 3 is to reverse the order of g and e, by first noting that eproper -rationalizability implies e-proper g-rationalizability for all g, then
showing that e-proper g-rationalizability as e converges to 0 corresponds to
the gth round of a finite algorithm, and finally proving that any strategy surviving all rounds of the algorithm is properly rationalizable acc. to Def. 2. The
algorithm eliminates preference relations on the players’ strategy sets. It is related to, but di¤ers from, Hammond’s [17] ‘rationalizable dominance relations’, which are recursively constructed by gradually extending a single incomplete binary relation on each player’s strategy set.
Proof (of the ‘only if ’ part of Prop. 3): Say that the mixed strategy xi A DðSi Þ
is an e-properly g-rationalizable strategy for i if there is an -belief system with
t ðoÞ
xi i ¼ xi for some o A K g ½e-prop trem. Since, for all g,
CK½e-prop trem J K g ½e-prop trem
it follows from Def. 4 that if xi is an e-properly rationalizable strategy, then,
t ðoÞ
for all g, there exists an -belief system with xi i ¼ xi for some o A K g
½e-prop trem. Consequently, if a mixed strategy xi for i is -properly rationalizable, then, for all g, there exists a sequence ðxi ðnÞÞn A @ of eðnÞ-properly grationalizable strategies converging to xi , where eðnÞ ! 0 as n ! y. This
means that it is su‰cient to show that if xi satisfies that, for all g, there exists a
sequence ðxi ðnÞÞn A @ of eðnÞ-properly g-rationalizable strategies converging to
xi and eðnÞ ! 0 as n ! y, then any pure strategy si for i used with positive
probability in xi is properly rationalizable acc. to Def. 2. This will in turn be
shown in two steps:
1. If a sequence of eðnÞ-properly g-rationalizable strategies converges to xi
and si A supp xi , then si survives the gth round of a finite algorithm.
2. Any pure strategy surviving all rounds of the algorithm is properly rationalizable acc. to Def. 2.
To construct the algorithm, note that any complete and transitive binary
relation on Si can be represented by a vector of sets ðSi ð1Þ; . . . ; Si ðLÞÞ (with
L b 1) that constitute a partition of Si . The interpretation is that si is preferred or indi¤erent to si0 if and only if si A Si ðlÞ, si0 A Si ðl 0 Þ and l a l 0 . Let,
Ei A I , Si :¼ 2 Si nfqg be the collection of non-empty subsets of Si and
P i :¼ fpi ¼ ðSi ð1Þ; . . . ; Si ðL pi ÞÞ A SiL
pi
j fSi ð1Þ; . . . ; Si ðL pi Þg is a partition of Si g
denote the collection of vectors of sets that constitute a partition of Si . Define
g
the algorithm by, Ei A I , setting P 1
i ¼ P i and determining, Eg b 0, P i as
g
follows: pi ¼ ðSi ð1Þ; . . . ; Si ðL pi ÞÞ A P i if and only if pi A P i and there exists
an LPS l pi A LDðSj P j Þ with supp l pi ¼ Sj P jpi for some P jpi J P jg1 ,
satisfying that
ðsj ; pj Þ g ðsj0 ; pj Þ
acc: to pi
474
G. B. Asheim
if pj ¼ ðSj ð1Þ; . . . ; Sj ðL pj ÞÞ A P jpi , sj A Sj ðlÞ, sj0 A Sj ðl 0 Þ and l < l 0 , and
si Spji si0
if and only if si A Si ðlÞ, si0 A Si ðl 0 Þ and l < l 0 , where pi is represented by uipi
satisfying uipi z ¼ ui and l pi .
Write P :¼ P 1 P 2 and, Eg b 0, P g ¼ P 1g P 2g . Since P 0 J P, it follows by induction that, Eg b 0, P g J P g1 . Moreover, since the finiteness of
S ¼ S1 S2 implies that P is finite, it follows that P g converges to P y in a
finite number of rounds. Say that si survives the gth round of the algorithm if
there exists pi ¼ ðSi ð1Þ; . . . ; Si ðL pi ÞÞ A P ig with Si ð1Þ C si .
Step 1. We first show that any pure strategy si A supp xi survives the gth
round of the algorithm if there exists a sequence ðxi ðnÞÞn A @ of eðnÞ-properly grationalizable strategies converging to xi , where eðnÞ ! 0 as n ! y. Say that
the probability distribution m A DðSj Tj Þ is an e-properly g-rationalizable belief for i if there is an -belief system with m ti ðoÞ ¼ m for some o A K g ½e-prop
trem. It is su‰cient to establish the following result:
If pi ¼ ðSi ð1Þ; . . . ; Si ðL pi ÞÞ A P i satisfies that there exists a sequence ðm pi ðnÞÞn A @
of eðnÞ-properly g-rationalizable beliefs for i, where eðnÞ ! 0 as n ! y, and
where, for all n,
XX
XX
m pi ðnÞðsj ; tj Þui ðsi ; sj Þ >
m pi ðnÞðsj ; tj Þui ðsi0 ; sj Þ
ð3Þ
sj
tj
sj
tj
if and only if si A Si ðlÞ, si0 A Si ðl 0 Þ and l < l 0 , then pi A P ig .
This result is established by induction.
If ðm pi ðnÞÞn A @ is a sequence of eðnÞ-properly g-rationalizable beliefs for i,
then, for each n, there exists an -belief system with T1 ðnÞ T2 ðnÞ as the
set of type vectors, such that m pi ðnÞ A DðSj Tj ðnÞÞ. For the inductive proof
we can w.l.o.g. partition Tj ðnÞ into P j , where pj ¼ ðSj ð1Þ; . . . ; Sj ðL pj ÞÞ A P j
corresponds to the subset of j-types in Tj ðnÞ satisfying that
XX
XX
m tj ðnÞðsi ; ti Þuj ðsj ; si Þ >
m tj ðnÞðsi ; ti Þuj ðsj0 ; si Þ
si
ti
si
ti
if and only if sj A Sj ðlÞ, sj0 A Sj ðl 0 Þ and l < l 0 , since i’s certain belief of j’s
eðnÞ-proper trembling only matters through j-types’ preferences over j’s pure
strategies. Hence, we can w.l.o.g. assume that m pi ðnÞ A DðSj P j Þ.
(g ¼ 0) Let ðm pi ðnÞÞn A @ be a sequence of eðnÞ-properly 0-rationalizable
beliefs for i, where eðnÞ ! 0 as n ! y, and where, for all n, (3) is satisfied. By
Lemma 1, the sequence ðm pi ðnÞÞn A @ contains a subsequence m pi ðmÞ such that
one can find an LPS l pi A LDðSj P j Þ and a sequence of vectors r pi ðmÞ A
ð0; 1Þ L1 (for some L) converging to 0 with
m pi ðmÞ ¼ r pi ðmÞal pi
for all m. By Def. 3, supp l pi ¼ Sj P jpi for some P jpi J P j . Let pi be represented by uipi satisfying uipi z ¼ ui and l pi . Since Def. 3 is the only require-
Proper rationalizability in lexicographic beliefs
475
ment on ðm pi ðnÞÞn A @ for g ¼ 0, we may, for each pj A P jpi , associate pj with
ðSj ð1Þ; . . . ; Si ðL pj ÞÞ A P 1
satisfying that ðsj ; pj Þ g ðsj0 ; pj Þ acc. to pi if sj A
j
Sj ðlÞ, sj0 A Sj ðl 0 Þ and l < l 0 . By Lemma 2, pi yields the same preferences on
Si as m pi ðnÞ (for any n). Hence, pi A P i0 .
(g > 0) Suppose the result holds for g 0 ¼ 0; . . . ; g 1. Let ðm pi ðnÞÞn A @ be
a sequence of eðnÞ-properly g-rationalizable beliefs for i, where eðnÞ ! 0 as
n ! y, and where, for all n, (3) is satisfied. As for g ¼ 0, use Lemma 1 to
construct an LPS l pi A LDðSj P j Þ, where supp l pi ¼ Sj P jpi for some
P jpi J P j , and where pi is represented by uipi satisfying uipi z ¼ ui and l pi .
Since o A K g ½e-prop trem J Ki ð½e-prop tremj X K g1 ½e-prop tremÞ, the induction hypothesis implies that P jpi J P jg1 and ðsj ; pj Þ g ðsj0 ; pj Þ acc. to pi if
pj ¼ ðSj ð1Þ; . . . ; Si ðL pj ÞÞ A P jpi , sj A Sj ðlÞ, sj0 A Sj ðl 0 Þ and l < l 0 . By Lemma 2,
pi yields the same preferences on Si as m pi ðnÞ (for any n). Hence, pi A P ig .
This concludes the induction.
Step 2. We then show that any pure strategy si surviving all rounds of
the algorithm is properly rationalizable acc. to Def. 2. It is su‰cient to show
that one can construct a belief system with W ¼ S T1 T2 ¼ A pr such that,
Ei A I , Epi ¼ ðSi ð1Þ; . . . ; Si ðL pi ÞÞ A P y
i , there exists o A W satisfying that
t ðoÞ
si Si j si0 if and only if si A Si ðlÞ, si0 A Si ðl 0 Þ and l < l 0 . Construct a belief
system with, Ei A I , a bijection pi : Ti ! P y
i from the set of types to the colg
y
0
for g b g 0 , it follows
lection of vectors in P y
i . Since bg such that P ¼ P
g
from the definition of the algorithm ðP Þgb0 that, Ei A I , P y
i is characterized
as follows: pi ¼ ðSi ð1Þ; . . . ; Si ðL pi ÞÞ A P y
if
and
only
if
there
exists ti A Ti such
i
that pi ðti Þ ¼ pi , and an LPS l ti A LDðSj Tj Þ with supp l ti ¼ Sj Tj ti for
some Tj ti J Tj , satisfying for each tj A Tj ti that
ðsj ; tj Þ g ðsj0 ; tj Þ
acc: to ti
if pj ðtj Þ ¼ ðSj ð1Þ; . . . ; Sj ðLpj ðtj Þ Þ, sj A Sj ðlÞ, sj0 A Sj ðl 0 Þ and l < l 0 , and
si Sti j si0
if and only if si A Si ðlÞ, si0 A Si ðl 0 Þ and l < l 0 , where ti is represented by uiti
satisfying uiti z ¼ ui and l ti . Consider any pi ¼ ðSi ð1Þ; . . . ; Si ðL pi ÞÞ A P y
i . By
the construction of the type sets, there exists o A W such that pi ðti ðoÞÞ ¼ pi ,
t ðoÞ
and si Si j si0 if and only if si A Si ðlÞ, si0 A Si ðl 0 Þ and l < l 0 ; in particular,
t ðoÞ
Si ð1Þ ¼ Ci i . It remains to be shown that, Ei A I , W ¼ ½ui X ½caui X ½respi ¼
A pr , implying that CKA pr ¼ W and o A CKA pr .
That W ¼ ½ui follows from the property that, for any ti A Ti , ti is represented by uiti , with uiti z ¼ ui . That W ¼ ½caui follows from the property
that, for any ti A Ti , ti is represented by l ti satisfying supp l ti ¼ Sj Tj ti .
That W ¼ ½respi follows from the property that, for any ti A Ti , ðsj ; tj Þ g
ðsj0 ; tj Þ acc. to ti whenever tj A Tj ti if sj A Sj ðlÞ, sj0 A Sj ðl 0 Þ and l < l 0 , while
t
sj Sj i sj0 if and only if sj A Sj ðlÞ, sj0 A Sj ðl 0 Þ and l < l 0 (where pj ðtj Þ ¼
ðSj ð1Þ; . . . ; Sj ðLpj ðtj Þ Þ). r
The proof of Prop. 4 uses the result that if si is most preferred in a subgame
h, then si is most preferred in any later subgame that si is consistent with.
476
G. B. Asheim
Lemma 3. If si A Citi ðhÞ, then si A Citi ðh 0 Þ for any h 0 A H with si A Si ðh 0 Þ J Si ðhÞ.
The proof of this lemma is based on the concept of a ‘strategically independent set’ due to Mailath, Samuelson & Swinkels [19]. The set S 0 J S is
strategically independent for player i in G ¼ ðS1 ; S2 ; u1 ; u2 Þ if S 0 ¼ S10 S20
and Esi , si0 A Si0 , bsi00 A Si0 such that ui ðsi00 ; sj Þ ¼ ui ðsi0 ; sj Þ for all sj A Sj0 and
ui ðsi00 ; sj Þ ¼ ui ðsi ; sj Þ for all sj A Sj nSj0 . It follows from Mailath et al. (Defs. 2
and 3 and the ‘if ’ part of Thm. 1) that SðhÞ is strategically independent for i
for any subgame h in a finite extensive game of almost perfect information.
Mailath et al. are concerned with the ‘pure strategy reduced strategic form’ of
G. However, since including equivalent strategies leads to additional strategically independent sets, the above result holds even in G. Their proof of the ‘if ’
part of Thm. 1 is based on the property that bsi00 A Si ðhÞ such that zðsi00 ; sj Þ ¼
zðsi0 ; sj Þ for all sj A Sj ðhÞ and zðsi00 ; sj Þ ¼ zðsi ; sj Þ for all sj A Sj nSj ðhÞ. The point
is that i’s decision conditional on j choosing a strategy consistent with h and
i’s decision conditional on j choosing a strategy inconsistent with h can be
made independently.
Proof: Suppose that si is not a most preferred strategy in h 0 . Then there
exists si0 A Si ðh 0 Þ such that si0 Stij ðh 0 Þ si . As noted above, Sðh 0 Þ is strategically
independent for i. Hence, bsi00 A Si ðh 0 Þ such that zðsi00 ; sj Þ ¼ zðsi0 ; sj Þ for all sj A
Sj ðh 0 Þ and zðsi00 ; sj Þ ¼ zðsi ; sj Þ for all sj A Sj nSj ðh 0 Þ. This implies that si00 Stij ðhÞ si ,
which contradicts that si is most preferred in h. r
The proof of Prop. 4 illustrates the importance of defining proper rationalizability by imposing common certain belief of proper consistency, where
‘certain belief ’ of an event means that the complement of the event is deemed
Savage-null. Common belief of proper consistency, where ‘belief ’ means ‘belief with probability one’, would not imply backward induction. The reason is
that (4) would not necessarily hold for all opponent types in Tj ti , while –
conditional on Sj ðhÞ Tj – such types may even be given positive primary
probability. A variant of the belief system for a four-legged centipede game
presented in Table 2 of Asheim [2] constitutes an example where common
belief of proper consistency is consistent with vectors of most preferred strategies that do not lead to the backward induction outcome.
Proof (of Prop. 4): In view of properties of the certain belief operator (cf.
the introductory paragraph to Appendix B), it is su‰cient to show for any
g ¼ 0; . . . ; M 2 that if there exists a belief system with o A K g A pr , then
C tðoÞ ðhÞ J Sðzðpjh ÞÞ for any h A H M1g . This is established by induction.
(g ¼ 0) Let h A H M1 . First, consider j with a singleton action set at h.
t
Then trivially Cj j ðhÞ ¼ Sj ðhÞ ¼ Sj ðzðpjh ÞÞ. Now, consider i with a non-singleton
action set at h; since G has perfect information, there is at most one such i. Let
ti ¼ ti ðoÞ for some o A K 0 A pr ¼ A pr . Then it follows that Citi ðhÞ ¼ Si ðzð pjh ÞÞ
since G is generic and o A A pr J ½ui X ½caui .
(g ¼ 1; . . . ; M 2) Suppose that it has been0 established for g 0 ¼ 0; . . . ; g 1
that if there exists a belief
system with o A K g A pr , then C tðoÞ ðh 0 Þ J Sðzð pjh 0 ÞÞ
0
M1g 0
. Let h A H M1g .
for any h A H
First, consider j with a singleton action set faj g at h. Let tj ¼ tj ðoÞ for
some o A K g1 A pr . Then Sj ðhÞ ¼ Sj ðh; aÞ and, by Lemma 3 and the premise,
Proper rationalizability in lexicographic beliefs
t
477
t
it follows that Cj j ðhÞ J Cj j ðh; aÞ J Sj ðzð pjðh; aÞ ÞÞ if a is a feasible action vector
at h. This implies that
t
Cj j ðhÞ J 7 Sj ðzð pjðh; aÞ ÞÞ J Sj ðzðpjh ÞÞ:
a
t
Hence, if sj A Cj j ðhÞ, then sj is consistent with the backward induction outcome
in any subgame ðh; aÞ immediately succeeding h.
Now, consider i with a non-singleton action set at h; since G has perfect
information, there is at most one such i. Let ti ¼ ti ðoÞ for some o A K g A pr .
t
The preceding argument implies that Cj j ðhÞ J 7a Sj ðzð pjðh; aÞ ÞÞ whenever
ti
g pr
g1 pr
tj A Tj since o A K A J Ki K A . Let si0 A Si ðhÞ be a strategy that di¤ers
from pi jh by assigning a di¤erent action only at h (i.e., zðsi0 ; pj jh Þ 0 zðpjh Þ and
si0 ðh 0 Þ ¼ pi jh ðh 0 Þ if Si ðhÞ I Si ðh 0 ÞÞ. As any pure strategy in Si can be viewed as
an act on Sj (cf. Sect. 3.4), write xSj for the act on Sj that pi jh can be viewed as
(i.e. xSj assigns zðpi jh ; sj Þ to any sj A Sj ), and write ySj for the act on Sj that si0
can be viewed as (i.e. ySj assigns zðsi0 ; sj Þ to any sj A Sj ). Let x and y be the
acts on Sj Tj that satisfy xðsj ; tj Þ ¼ xSj ðsj Þ and yðsj ; tj Þ ¼ ySj ðsj Þ for all ðsj ; tj Þ.
Then,
xXa Sj ðzð pjðh; aÞ ÞÞTj
strongly dominates
yXa Sj ðzð pjðh; aÞ ÞÞTj
t
by backward induction since G is generic and o A K g A pr J ½ui . Since Cj j ðhÞ J
7a Sj ðzðpjðh; aÞ ÞÞ whenever tj A Tj ti , it follows that, Etj A Tj ti ,
xC tj ðhÞft g
j
j
strongly dominates
j
t
t
(4)
yC tj ðhÞft g :
j
t
Note that sj Sj i sj0 if sj A Cj j ðhÞ and sj0 A Sj ðhÞnCj j ðhÞ. Since o A K g A pr J
½respi , it follows that ðsj ; tj Þ g ðsj0 ; tj Þ according to ti whenever tj A Tj ti , sj A
t
t
Cj j ðhÞ and sj0 A Sj ðhÞnCj j ðhÞ, which by (4) implies that x Stij ðhÞftj g y. Since this
holds for all tj A Tj ti , it follows that
x Stij ðhÞTj y
and
xSj Stij ðhÞ ySj :
It has thereby been established that si0 A Si ðhÞnCiti ðhÞ if si0 di¤ers from backward induction only by the action taken at h. However, by the premise that
Citi ðh; aÞ J Si ðzð pjðh; aÞ ÞÞ if a is a feasible action vector at h, it follows that any
si A Citi ðhÞ is consistent with the backward induction outcome in the subgame
ðh; ðsi ðhÞ; aj ÞÞ immediately succeeding h when i plays the action si ðhÞ at h (since
si A Si ðh; ðsi ðhÞ; aj ÞÞ and, by Lemma 3, si A Citi ðh; ðsi ðhÞ; aj ÞÞ). Hence, Citi ðhÞ J
Si ðzð pjh ÞÞ. r
References
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2. Asheim GB (2001) On the epistemic foundation for backward induction. University of Oslo
3. Asheim GB, Dufwenberg M (2001) Admissibility and common belief. University of Oslo and
Stockholm University
478
G. B. Asheim
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