On the Epistemic Foundation
of Backward Induction?
Geir B. Asheim
Department of Economics, University of Oslo,
P.O. Box 1095 Blindern, N-0317 Oslo, Norway
Abstract. I characterize backward induction in an epistemic model of
perfect information games where rationality is associated with 'rational
preferences' instead of 'rational choice'. In this approach, backward induction corresponds to the rationality principle that a player believes
that his opponents reveal their preferences in each subgame. At an interpretative level this result resembles the one established by Aumann
[4]. By instead imposing that a player believes that his opponents reveal
their preferences only in the whole game, I interpret Ben-Porath's [10]
support of the Dekel-Fudenberg procedure.
1 Introduction
In recent years, two widely cited and in uential contributions on backward induction in nite perfect information games have appeared, namely Aumann [4]
and Ben-Porath [10]. These contributions | both of which consider generic
perfect information games (where all payo s are di erent) | reach opposite
conclusions: While Aumann establishes that Common Knowledge of Rationality (CKR) implies that the backward induction outcome is reached, Ben-Porath
shows that the backward induction outcome is not the only outcome that is consistent with Common Certainty of Rationality (CCR). The models of Aumann
and Ben-Porath are di erent. One such di erence is that Aumann makes use of
'knowledge' in the sense of 'absolute certainty', while Ben-Porath's analysis is
based on 'certainty' in the sense of 'belief with probability one'.
The present paper shows how the conclusions of Aumann and Ben-Porath can
be captured by imposing di erent principles of rationality on the players within
the same general framework. Furthermore, the interpretations of the present
analysis correspond closely to the intuitions that Aumann and Ben-Porath convey in their discussions. Hence, the present contribution may increase our understanding of the di erences between the analyses of Aumann and Ben-Porath, and
thereby enhance our understanding of the rationality requirements underlying
backward induction.
The analysis follows Asheim & Dufwenberg [3] [AD] in arguing that rationality in games should be imposed as a requirement on preferences rather than
?
This paper builds in part on joint work with Martin Dufwenberg, who has contributed
with helpful suggestions. I also thank Stephen Morris for useful comments.
as a requirement on choice. AD show the following result (reproduced as Prop.
1 of Sect. 3): Strategies surviving the Dekel-Fudenberg procedure (see Dekel
& Fudenberg [16]), where one round of weak elimination is followed by iterated strong elimination, can be characterized as maximal elements when it is
commonly known that each player takes into account all vectors of opponent
strategies ('Caution') and believes that his opponents reveal their preferences in
the whole game ('Belief of preference revelation'). For generic perfect information games, Ben-Porath shows that the set of outcomes consistent with CCR at
the beginning of the game corresponds to the set of outcomes that survive the
Dekel-Fudenberg procedure. Hence, maximal elements when 'Caution' and 'Belief of preference revelation' are commonly known correspond to the outcomes
consistent with CCR.
An extensive game allows preference revelation, not only in the whole game,
but in proper subgames. In perfect information games (and, more generally, in
multi-stage games) the subgames constitute an exhaustive set of situations in
which preferences can be revealed. Hence, in perfect information games one can
argue that 'Belief of preference revelation' should be replaced by 'Belief of preference revelation in each subgame': Each player believes that his opponents reveal
their preferences in each subgame. The main results of the present paper (Props.
2 and 3 of Sect. 5) show how, for generic perfect information games, common
knowledge of 'Caution' and 'Belief of preference revelation in each subgame' is
possible and uniquely determines the backward induction outcome. Hence, by
substituting 'Belief of preference revelation in each subgame' for 'Belief of preference revelation', the present analysis provides an alternative route to Aumann's
conclusion, namely that CKR implies the backward induction outcome.
Section 2 presents the formal framework in which extensive games will be
analyzed. AD's characterization of the Dekel-Fudenberg procedure is reviewed
in Sect. 3. Section 4 applies this result to generic extensive games of perfect
information and compares, by means of an example, the present analysis to that
of Ben-Porath [10]. Section 5 introduces 'Belief of preference revelation in each
subgame' as an alternative rationality principle and establishes the paper's main
results. Section 6 interprets the analysis in view of Aumann [4] and Ben-Porath
[10] as well as Battigalli's [8] concept of a 'rationality ordering'.
2 States and Knowledge
The purpose of this section is to present a framework for extensive games where
each player is modeled as a decision maker under uncertainty. The decisiontheoretic analysis builds on Anscombe & Aumann [1]. However, for the analysis
of extensive games, continuity must be relaxed in the fashion of Blume et al.
[11] to allow each player to take into account the possibility that any subgame
can be reached. Moreover, by not imposing completeness, the analysis does not
require subjective probabilities. The framework is summarized by the concept
of a knowledge system. The Appendix contains a presentation of the decisiontheoretic terminology, notation and results that will be utilized.
A Finite Extensive Game of Almost Perfect Information , with n players and K , 1 stages can be described as follows. Any nite multi-stage game
(i.e., a perfect information game or a nitely repeated game) ts this description. The sets of histories is determined inductively: The set of histories at the
beginning of the rst stage 1 is H 1 = f;g. Let H k denote the set of histories at
the beginning of stage k. At h 2 H k , let, for each player i 2 N := f1; : : : ; ng,
i's action set be denoted Ai (h), where i is inactive at h if Ai (h) is a singleton.
Write A(h) := A1 (h) : : : An (h). De ne the set of histories at the beginning of
stage k + 1 as follows: HSk+1 := (h; a) jh 2 H k and a 2 A(h) . This concludes
,1 H k denote the set of subgames and let Z := H k
the induction. Let H := Kk=1
denote the set of outcomes. For each player i, let the von Neumann-Morgenstern
utility function i : Z ! < determine i's payo . Assume that there exist z ,
z 0 2 Z such that i (z ) > i (z 0 ). A pure strategy for player i is a function si
that assigns an action in Ai (h) to any h 2 H . Let Si denote player i's nite
set of pure strategies, and write S := S1 : : : Sn and S,i := j6=i Sj . Write
p, r, and s (2 S ) for pure strategy vectors. For any s 2 S , let z (s) denote the
outcome reached when s is used. De ne ui : S ! < by ui (s) = i (z (s)). Refer to
G = (Si ; ui )i2N as the normal form of the extensive game , . Since ui is a von
Neumann-Morgenstern utility function, we may extend ui to the set of objective
randomizations on S . For any h 2 H [ Z , let S (h) = S1 (h) : : : Sn (h) denote
the set of strategy vectors that are consistent with h being reached. If h0 is the
predecessor of h, then S (h0 ) S (h). If si 2 Si and h 2 H , let si jh denote the
strategy in Si (h) satisfying si jh (h0 ) = si (h0 ) at any h0 2 H except at h0 with
S (h0 ) S (h) where si jh (h0 ) is dictated by si jh being consistent with h.
States and Types. When a normal form game G is turned into a decision
problem for each player (see Tan & Werlang [23]), the uncertainty faced by a
player is the strategy choices of his opponents, the beliefs of his opponents about
the strategy choices of their opponents, and so on. A type of a player corresponds
to a belief about the strategy choices of his opponents, a belief about the beliefs
of his opponents about the strategy choices of their opponents, and so on.
Given an assumption of coherency, models of such in nite hierarchies of beliefs (Boge & Eisele [12], Mertens & Zamir [20], Brandenburger & Dekel [15],
Epstein & Wang [18]) yield S T as the universal state space, where S is the
underlying space of uncertainty and where T = T1 : : : Tn 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 T,i, where T,i := j6=i Tj . In a decision-theoretic
sense, the set of beliefs on S T,i corresponds to the set of "regular" binary
relations on the set of acts on S T,i , where an (Ascombe-Aumann) act on
S T,i is a function that to any element of S T,i assigns an objective randomization on S . In the references above, with the notable exception of Epstein
& Wang [18], a binary relation is "regular" if and only if it is represented by
some subjective probability distribution in (S T,i ).
In conformity with the literature on in nite hierarchies of beliefs, let the set
of states of the world (or simply states) be := S T , and let each type ti of
any player i correspond to a binary relation t on the set of acts on S,i T,i .
However, like AD, I do not construct a universal state space by explicitly modeling in nite hierarchies of beliefs. For tractability I instead directly consider an
implicit model | with a nite type set Ti for each player i | from which in nite
hierarchies of beliefs can be constructed. Moreover, since completeness and continuity are not imposed, the "regularity" conditions on t consist of re exivity,
transitivity and objective independence only, meaning that t is not necessarily
represented by some subjective probability distribution in (S,i T,i ).
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Knowledge. The construction is summarized by the following de nition.
De nition 1. A knowledge system for a normal form game G = (Si; ui)i2N
consists of
for each player i, a nite set of types Ti ,
for each type ti of any player i, a binary relation t (ti 's preferences) on the
set of acts on S,i T,i , where t is re exive and transitive and satis es
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objective independence.
For each player i, i's knowledge can be derived from the knowledge system
by following AD in their adaptation of Brandenburger & Dekel's ([15], p. 197)
knowledge de nition. For each player i and each state ! 2 , let ti (!) denote
the projection of ! on Ti . Associate 'knowledge' of an event with the property
that the complement of the event is Savage-null: If E , write
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Ki E := ! 2 (s; t) 2 nE ) ti 6= ti (!) or
o
ti = ti (!) and (s,i ; t,i ) is Savage-null acc. to t (!) :
Say that i knows the event E given ! if ! 2 Ki E . Write KE := K1 E \ : : : \
KnE . Say that the event E is mutually known given ! if ! 2 KE . Write
CKE := KE \ KKE \ KKKE \ : : : . Say that the event E is commonly
known given ! if ! 2 CKE .
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Theory and Preferences over Strategies. Let tT, denote the marginal of
t on T,i . Since tT, expresses ti 's belief concerning the type vector of i's
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opponents, I follow the terminology of Aumann & Brandenburger ([6], p. 1164)
and refer to tT, as ti 's theory. Likewise, let tS, denote the marginal of t on
S,i . A pure strategy si 2 S,i can be viewed as an act xS, on S,i that assigns
(si ; s,i ) to any s,i 2 S,i . A mixed strategy xi 2 (Si ) corresponds to an act
xS, on S,i that assigns (xi ; s,i) to any s,i 2 S,i. Hence, tS, is a binary
relation also on the subset of acts on S,i that correspond to i's mixed strategies.
Thus, tS, can be referred to as ti 's preferences over i's mixed strategies. The
set of mixed strategies is the set of acts that are at ti 's actual disposal.
Since t is re exive and transitive and satis es objective independence, tS,
shares these properties, and, for any h 2 H ,
Cit (h) := fsi 2 Si (h)jsi is maximal w.r.t. tS, (h) in (Si (h))g
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is non-empty and supports any maximal mixed strategy. Refer to Cit (h) as ti 's
choice set in the subgame h, and refer to Cit : H ! 2S (h) n f;g as ti 's choice
function. Note if si is maximal in a subgame h, then si is maximal in any later
subgame that si is consistent with:
Lemma 1. If si 2 Cit (h), then si 2 Cit (h0 ) for any h0 2 H with si 2 Si (h0) Si (h).
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Proof. Suppose that si is not maximal w.r.t. tS, (h0 ) in (Si (h0 )) . Then there
exists xS, such that xS, tS, (h0 ) yS, , where xS, assigns (xi ; s,i ) to
any s,i 2 S,i with xi 2 (Si (h0 )), and where yS, assigns (si ; s,i ) to any
s,i 2 S,i . By Mailath et al. ([19], Defs. 2 and 3 and the if-part of Theorem
1), S (h0 ) is a strategic independence for i. Hence, xS, can be chosen such that
ui (xS, (s,i )) = ui (yS, (s,i )) for all s,i 2 S,i nS,i (h0 ). By conditional representation, xS, tS, (h) yS, , which contradicts that si is maximal w.r.t. tS, (h)
in (Si (h)).
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Write Cit := Cit (;) . For any h 2 H , write C t (h) := C1t (h) : : : Cnt (h) and
C,t,i (h) := j6=i Cjt (h).
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3 Rational Preferences
Usually rationality is associated with 'rational choice'. This means that rationality is a requirement on a pair (si ; ti ), where si can be said to be a rational
choice by ti if si 2 Cit . See e.g. Epstein ([17], Sect. 6) for a presentation of this
approach in a general context.
The present paper follows AD by letting rationality be associated with requirements on ti only. Since ti corresponds to the preferences t , such requirements will be imposed on t . In support of this alternative approach | which
will be referred to by the term 'rational preferences' | one can note the following: The approach allows
... rationality to be associated with types rather that strategy-type pairs.
... conventional concepts like 'rationalizable strategies' and strategies surviving the Dekel-Fudenberg procedure to be characterized under very weak and
natural conditions (see Prop. 4 of Asheim [2] as well as Prop. 1 below).
... caution to be imposed as a strengthening of rationality without having
to change the de nition of knowledge. Under 'rational choice' the notion of
'knowledge' must be weakened to accommodate caution (see the introduction
of AD as well as Borgers ([13], pp. 266{267) and Epstein ([17], p. 3)). In
contrast, this paper models players that take into account all vectors of
opponent strategies, while still using the term 'knowledge' in the strong
sense described in the above subsection on knowledge. Thus, knowledge of
rationality need not be weakened to allow players to take into account the
possibility that any subgame in the extensive game be reached.
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Here I will focus on showing how 'rational preferences' as an approach to rationality in games can be used to shed light on the analyses of Aumann [4]
and Ben-Porath [10], and thereby enhance our understanding of the rationality
requirements underlying backward induction. For this purpose, it is useful to
reproduce AD's characterization of the Dekel-Fudenberg procedure.
Admissible Rationality. The Dekel-Fudenberg procedure is made up of one
round of elimination of weakly dominated strategies followed by iterated elimination of strongly dominated strategies. AD characterize this procedure by imposing on the preferences t the following rationality principles, where the terms
'conditional representation' is de ned in the Appendix, where weak and strong
dominance are used w.r.t. i's vNM utility function ui , where
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E,i := (s,i ; t,i ) s,i 2 C,t,i and t,i is not Savage-null acc. to tT,
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F,i := (s,i ; t,i ) s,i 2 S,i and t,i is not Savage-null acc. to
and where x and y are two arbitrary acts on S,i T,i .
CR t is Conditionally Represented by ui.
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CAU (CAUtion) If xF, weakly dominates yF, , then x t y.
BPR (Belief of Preference Revelation) If xE, strongly dominates yE, , then
x t y.
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These principles can be explained as follows.
When it is commonly known that all players are of types that satisfy CR,
then it is commonly known that the game G = (Si ; ui )i2N is played, meaning
that G is a game of complete information.
CAU means that
(s,i ; t,i ) is not Savage-null acc. to t if t,i is not Savaget
null acc. to T, . The principle guarantees that all vectors of opponent
strategies are taken into account, implying that the marginal of t on S,i
(i.e., t,i 's preferences over i's mixed strategies tS, ) is admissible.
BPR means that the type 'believes' that the preferences of opponent types
are revealed in the whole game, in the sense that the type always prefers
an act which strongly dominates another act on the event that all opponent types choose maximal elements in the whole game, regardless of what
happens outside this event.
An example of preferences t that satisfy CR, CAU, and BPR can be constructed by letting
1. t be represented by ui and some lexicographic probability system (LPS)
`t = (m1 ; : : : ; mk ) 2 L(S,i T,i ) (see Prop. A2 and Blume et al. [11]),
2. the marginal of t on T,i (i.e., ti 's theory tT, ) be represented by ui and
some subjective probability distribution mtT, 2 (T,i ) (see Prop. A1),
3. supp(`t ) = S,i supp(mtT, ) , and m1 (r,i ; t,i ) > 0 only if r,i 2 C,t,i .
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In this construction, t is complete and the marginal of t on T,i is continuous.
While these properties do not contradict CR, CAU, and BPR, they are not
implied by CR, CAU, and BPR. Note that CAU in conjunction with CR and
BPR rules out that the marginal of t on S,i is continuous.
De nition 2. Type t,i is admissibly rational if t satis es CR, CAU, and
BPR.
Refer to A := A1 \: : :\An as the event of admissible rationality, where, for each i,
Ai := f! 2 jti (!) is admissibly rationalg. The Dekel-Fudenberg procedure can
now be characterized as maximal elements in states where admissible rationality
is commonly known.
Proposition 1 (Asheim & Dufwenberg [3]). A pure strategy ri for i survives the Dekel-Fudenberg procedure if and only if there exists a knowledge system
with ri 2 Cit (!) for some ! 2 CKA.
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4 Generic Games of Perfect Information
Within the formalism of Sect. 2, a nite extensive game is
... of perfect information if, at any h 2 H , there exists at most one player
that has a non-singleton set of actions.
... generic if, for each i, i (z ) 6= i (z 0 ) whenever z and z 0 are di erent outcomes.
Generic extensive games of perfect information has 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, z (pjh ) is the
backward induction outcome in the subgame h, and S (z (pjh)) is the set of strategy vectors consistent with the backward induction outcome in the subgame
h.
Both Aumann [4] and Ben-Porath [10] analyze generic extensive games of
perfect information. As already pointed out, while Aumann establishes that
Common Knowledge of Rationality1 (CKR) implies that the backward induction
outcome is reached, Ben-Porath shows that the backward induction outcome is
not the only outcome that is consistent with Common Certainty of Rationality
(CCR). The purpose of the present section is to interpret the analysis of BenPorath by applying Prop. 1 to the class of generic perfect information games.
Ben-Porath [10] establishes through his Theorem 1 that the set of outcomes
consistent with CCR at the beginning of the game corresponds to the set of
outcomes that survive the Dekel-Fudenberg procedure. Hence, by Prop. 1, maximal elements when admissible rationality is commonly known correspond to the
outcomes consistent with CCR.
1
Aumann's [4]) analysis is based on substantive rationality. See Aumann [4], pp. 14{16,
and Aumann [5].
1
D
2
0
F
2
d
f
1
3
1
D
4
2
3
5
F
Fig. 1. A centipede game
An Example. To illustrate how common knowledge of admissible rationality
is consistent with outcomes other than the unique backward induction outcome,
consider the simple centipede game of Fig. 1 where backward induction implies that down is being played at any decision node. Let T1 = ft01 ; t001 g and
T2 = ft02 ; t002 g. Assume that the preferences of each type of player i is represented
by ui and a 2-level LPS over S,i T,i , implying that all types satisfy CR. In
Table 1. The lexicographic probability systems for each type of the two players
t01 :
t02 :
0
00
00
00
00
, t2 , t2 , t2 , t2 t1 :
d , 35 ; 105 , 0; 101 d , 45 ; 107 , 0; 101 f 0; 101 15 ; 101
f 0; 101 25 ; 103
0
00
, 1t11 , t11 , 2 ; 14 , 01 ; 81 ,0; 81 , 2 ; 14 D
FD
FF 0; 8
0; 8
t002 :
0
, t11 ,1; 21 ,0; 14 D
FD
FF 0; 4
t001
(0; 0)
(0; 0)
(0; 0)
Table 1, the rst numbers in the parentheses express primary probability distributions, while the second numbers express secondary probability distributions.
The strategies DD and DF are merged as their relative likelihood does not
matter. Note that all types satisfy CAU. With these 2-level LPSs each type's
preferences over the player's own strategies are given by
t01 : D FD FF
t001 : FD D FF
t02 : d f
t002 : f d
It is easy to check that all types satisfy BPR (e.g. both t02 and t002 believe that
t01 reveals his preferences in the sense that they deem his maximal element D
in nitely more likely than each of his non-maximal elements FD and FF ).
Hence, preferences consistent with common knowledge of CR, CAU, and BPR
need not re ect backward induction since FD and f are maximal elements.
Note that type t02 , conditional on his decision node being reached (i.e. 1
choosing FD or FF ), updates his beliefs about the type of player 1 and assigns
probability one to 1 being of type t001 . Consequently, the conditional belief of
type t02 about 1's strategy choice assigns probability one to FD. Type t002 , on
the other hand, does not admit the possibility that 1 is of another type than t01 .
Since the choice of F at 1's rst decision node means that t01 has not revealed his
preferences in the whole game, there is no restriction concerning the conditional
belief of type t002 about the choice at 1's second decision node. In the terminology
of Ben-Porath, a "surprise" has occurred. Subsequent to such a surprise, a type
need not believe that the opponent type reveals his preferences.
5 Belief of Preference Revelation in Each Subgame
A simultaneous game o ers only one choice situation in which preferences can
be revealed. Hence, for a game in this class, it seems reasonable that any type of
a player believes that the preferences of opponent types can be revealed in the
whole game only, as formalized by the rationality principle BPR. An extensive
game with a non-trivial dynamic structure, however, o ers such choice situations,
not only in the whole game but also in proper subgames. This implies that, for
such an extensive game, preferences can be revealed in any subgame. Moreover,
for extensive games of almost perfect information, the subgames constitute an
exhaustive set of situations in which preferences can be revealed. This motivates
applying the following rationality principle in this class of games, where
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E,i (h) := (s,i ; t,i ) 2 S,i (h) T,i
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s,i 2 C,t,i (h) and t,i is not Savage-null acc. to tT, :
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BPRS (Belief of Preference Revelation in eacht Subgame) For each h 2 H ,
xE, h strongly dominates yE, h , then x S, h T, y.
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This principle can be interpreted as follows.
BPRS means that the type 'believes' that the preferences of opponent types
are revealed in the each subgame, in the sense that the type always prefers
an act which strongly dominates another act on the event that all opponent
types choose maximal elements in the subgame, regardless of what happens
outside this event.
2
De nition 3. Type ti is admissibly subgame rational if t satis es CR, CAU,
and BPRS.
2
BPRS is a non-inductive analogue to forward knowledge of rationality; see Balkeni
borg & Winter [7].
Refer to A := A1 \: : : \An as the event of admissible subgame rationality, where,
for each i, Ai := f! 2 jti (!) is admissibly subgame rationalg. De nition 3 can
be applied to any nite extensive game of almost perfect information. However,
to relate to Aumann's [4] Theorems A and B, the following analysis is concerned
with generic perfect information games.
The Example Revisited. In the knowledge system of Table 1, type t00 does
not satisfy BPRS. By BPRS any type of player 2 must believe that t0 reveals
2
1
his preferences in the subgame de ned by 2's decision node in the sense that the
type deems the maximal element of t01 in the subgame, FD, in nitely more likely
than the non-maximal element, FF . This means that any type of player 2 prefers
d to f , implying that no type of player 1 satisfying BPRS can prefer FD to D.
Thus, common knowledge of CR, CAU, and BPRS entails that any types of
players 1 and 2 have the preferences D FD FF and d f , respectively,
meaning that if any type of a player reveals his preferences in a subgame, then
his choice is made in accordance with backward induction. Demonstrating that
this conclusion holds in general is the main result of the present paper.
Main results. In analogy with Aumann's [4] Theorems A and B, it is estab-
lished that
... any vector of maximal elements in a subgame of a generic perfect information game, in a state where admissible subgame rationality is commonly
known, leads to the backward induction outcome in the subgame (Prop. 2).
Hence, by substituting BPRS for BPR, the present analysis yields support
to Aumann's conclusion, namely that if an appropriate form of rationality
is commonly known, then the backward induction outcome results.
... for any generic perfect information game, common knowledge of admissible
subgame rationality is possible (Prop. 3). Hence, Prop. 2 is not empty.
Proposition 2. Consider a generic extensive game of perfect information , .
If, for some knowledge system, ! 2 CKA , then, for each h 2 H , C t ! (h) S (z (pjh)), where p denote the unique subgame-perfect equilibrium.
( )
Proof. Some properties of the knowledge de nition (see Sect. 2) must be established for the proof. It is easy to check that Ki = , and, for any events E and
F , Ki E \ Ki F = Ki (E \ F ), Ki E = Ki Ki E , and nKi E = Ki ( nKi E ). Write
K 0E := E and, for each g 1, K g E := KK g,1 E . Since Ki (E \ F ) = Ki E \ Ki F
and Ki Ki E = Ki E , it follows 8g 2, K g E = K1K g,1 E \ : : : \ Kn K g,1 E K1 K1K g,2 E \ : : : \ Kn Kn K g,2 E = K1 K g,2 E \ : : : \ Kn K g,2 E = K g,1 E . The
truth axiom (Ki E E ) is not satis ed, since an event can be known even though
the true state is an element of the complement of the event. However, since
A := A1 \ : : : \ An is an event that concerns the type vector, mutual knowledge
of A implies that A is true: KA = K1 A \ : : : \ Kn A K1 A1 \ : : : \ Kn An =
A1 \ : : : \ An = A since, for each i, Ki Ai = Ai . Hence, it follows that (i)
8g 1, K g A K g,1 A , and (ii) 9g0 0 such that K g A = CKA for g g0
since is nite. In view of these properties, it is sucient to show for any
g = 0; : : : ; K , 2 that if there exists a knowledge system with ! 2 K g A , then
C t(!) (h) S (z (pjh)) for any h 2 H K ,1,g . This is established by induction.
(g = 0) Let h 2 H k,1 . First, consider some j with a singleton action set
at h. Then trivially Cjt (h) = Sj (h) = Sj (z (pjh)). Now, consider some i with a
non-singleton action set at h; since , has perfect information, there is at most
one such i. Let ti = ti (!) for some ! 2 K 0 A = A . Then it follows from CR
and CAU that Cit (h) = Si (z (pjh)) since , is generic.
0
(g = 1; : : : ; K , 2) Suppose that it has been established
0 for g = 0t;(:!:): ; g0 , 1
g
that if there exists a knowledge
system with g 2 K A , then C (h ) S (z (pjh0 )) for any h 2 H K ,1,g0 . Let h 2 H K ,1,g . First, consider some j with a
singleton action set at h. Let tj = tj (!) for some ! 2 K g,1 A . Then, by Lemma
1 and the premise, Sj (h) = Sj (h; a) and Cjt (h) Cjt (h; a) Sj (z (pj(h;a))) if a
T
is a feasible action vector at h. This implies that Cjt (h) a Sj (z (pj(h;a))) Sj (z (pjh )) . Now, consider some i with a non-singleton action set at h; since ,
has perfect information, there is at most one such i. Let ti T
= ti (!) for some
! 2 K g A . The preceding argument implies that C,t,i (h) a S,i (z (pj(h;a) ))
whenever t,i is not Savage-null acc. to tT, since ! 2 K g A Ki K g,1 A . Let
si 2 Si (h) be a strategy that di ers from pi jh by assigning a di erent action at h
(i.e., z (si ; p,i jh ) 6= z (pijh ) and si (h0 ) = pi jh (h0 ) whenever Si (h) Si (h0 )). Write
xS, for the act on S,i that pi jh can be viewed as, and write yS, for the act on
S,i that si can be viewed as. Let x and y be the acts on S,i T,i that satisfy
x(s,i ; t,i ) = xS, (s,i ) and y(s,i ; t,i ) = yS, (s,i ) for all (s,i; t,i ). By backward induction, x\ S, (z(pj ))T, strongly dominates y\ S, (z(pj ))T,
T
since , is generic. Since C,t,i (h) a S,i (z (pj(h;a))) whenever t,i is not Savagenull acc. to tT, , xE, (h) strongly dominates yE, (h) , and, by CR and BPRS,
x tS, (h)T, y and xS, tS, (h) yS, . By Lemma 1 and the premise that
Cit (h; a) Si (z (pj(h;a) )) if a is a feasible action vector at h, it follows that
Cit (h) Si (z (pjh)).
ut
Proposition 3. For any generic extensive games of perfect information , , there
exists a knowledge system with CKA 6= ;.
Proof. Construct a knowledge system with only one type of each player, T =
f(t1 ; : : : ; tn )g, implying that (1) it is commonly known given any ! 2 S T
that the type vector is (t1 ; : : : ; tn ). Write, 8i 2 N , 8k 2 f1; : : : ; K , 1g, P,k i :=
fp,i jh jh 2 H k g and, P,Ki := S,i . Let, 8i 2 N , `t = (m1 ; : : : ; mK ) 2 L(S,i T,i ) satisfy the following requirement: 8k 2 f1; : : :; K g, supp(mk ) = P,k i T,i .
By letting t be represented by ui and the LPS `t , then (2) t satis es CR
and CAU. Let `tS, denote the marginal of `t on S,i , and let, 8h 2 H , `tS, (h) =
(m1S, (h); :::; mKS,0 (h) ) denote the conditional of `tS, given S,i (h) (see Blume et
al. [11], Def. 4.2). By the properties of a subgame-perfect equilibrium, 8h 2 H ,
m1S, (h) (p,i jh ) = 1 and pi jh 2 Cit (h). Hence, since likewise p,i jh 2 C,t,i (h), (3)
tT, satis es BPRS. By (1), (2), and (3), it follows that ! 2 CKA .
ut
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6 Discussion
Consider a generic perfect information game. Say that a type's preferences are in
accordance with backward induction if, in any subgame, a strategy is a maximal
element only if it is consistent with the backward induction outcome. Using this
terminology, Prop. 2 can be restated as follows: Under common knowledge of
admissible subgame rationality in a generic perfect information game, players
are of types with preferences that are in accordance with backward induction.
Furthermore, common knowledge of admissible subgame rationality implies that
players cannot admit the possibility that opponents are of types with preferences
not in accordance with backward induction. This re ects in spirit the conclusion
that can be drawn from Aumann's analysis.
However, since admissible subgame rationality is imposed on preferences,
reaching 2's decision node and 1's second decision node in the centipede game of
Fig. 1 does not contradict common knowledge of admissible subgame rationality.
Of course, these decision nodes will not be reached if players actually reveal their
preferences. But that players are of types satisfying BPRS does not imply that
they will actually reveal their preferences; rather, it means that they 'believe'
(in a sense that Morris [22] shows corresponds to Brandenburger's [14] ' rstorder knowledge') that their opponents will reveal their preferences wherever
the extensive form allows them to.
Knowledge vs. Certainty. The term 'knowledge' (cf. the subsection on knowl-
edge in Sect. 2) signi es, in the present paper, that the complement of a known
event is not taken into account. This notion of 'knowledge' is strong; in fact, it is
the strongest form considered by Morris [22]. Each of the notions 'approximate
knowledge' (Monderer & Samet [21], Borgers [13]), and ' rst-order knowledge'
(Brandenburger [14]) represents an e ective weakening of 'knowledge' exactly
because the complement of a known event is allowed to be taken into account. It
is stronger than Ben-Porath's [10] term 'certainty', which allows a player, while
being certain that his opponents choose rationally, to take into account the complement of this event (i.e. the possibility that they will choose strategy vectors
that are not consistent with rational choice).
'Knowledge', in the sense that the complement of a known event is not taken
into account, can be used here since knowledge of rationality concerns preferences and not choice. When there is common knowledge of admissible (subgame)
rationality (cf. Def. 2 (3)), then each player knows (in the sense of not taking
into account the complement) that each opponent is of a type with preferences
that satisfy CR, CAU, and BPR(S), each player knows (in the sense of not
taking into account the complement) that each opponent knows (in the sense of
not taking into account the complement) that any of his opponents is of a type
with preferences that satisfy CR, CAU, and BPR(S), and so on. By CAU
it is not the case that any player knows that opponents will not play any particular strategy vector. On the contrary, CAU imposes that a player takes into
account all vectors of opponent strategies, implying that he deems possible that
any subgame in the extensive game be reached. By BPR(S) the type of any
player merely 'believes' that (in any subgame) opponents will choose a vector
consisting of maximal elements only, in the sense that the type's preferences
respect strong dominance on the set of opponent vectors of maximal elements.
Rationality Orderings. Consider a two-player game. The constructive proof
of Prop. 3 shows how common knowledge of admissible subgame rationality
may lead a type ti of player i to have preferences over i's strategies that are
represented by an LPS `tS = (m1S ; :::; mKS ) with more than two levels of subjective probability distributions (i.e. K > 2). E.g., in the centipede game of Fig. 1,
common knowledge of admissible subgame rationality implies that any type t2 of
player 2 has preferences that can be represented by `tS = (m1S ; m2S ; m3S ) where
supp(m1S ) = fDg, supp(m2S ) = fD; FDg, and supp(m3S ) = S1 . Within the 'rational choice' approach one may interpret supp(m1S ) to consist of strategies for
S
j that are "most rational", supp(mKS )n k0 <K supp(mkj 0 ) to consist of strategies
S
for j that are "completely irrational", and supp(mkS )n k0 <k supp(mkj 0 ) , for
k = 2; : : : ; K , 1, to consist of strategies for j that are at "intermediate degrees
of
Furthermore, for any k = 2; : : : ; K , ti deems any strategy in
S rationality".
k0 ) in nitely more likely than any strategy not having this propsupp(
m
0
j
k <k
S
erty. This illustrates that (supp(m1S ); : : : ; supp(mKS )n k0 <K supp(mkj 0 )) corresponds closely to what Battigalli [8] calls a rationality ordering for j .
The present construction of such a rationality ordering di ers from the one
proposed by Battigalli along two dimensions: 1. Battigalli considers maximal
elements of players in the whole game only (see his Def. 2.1), while here also
maximal elements in subgames are considered (cf. BPRS). 2. Battigalli considers
maximal elements given preferences where opponent strategies that are less than
"most rational" are given positive primary probability, while here only maximal
elements of (admissibly subgame) rational types can be given positive primary
probability. This means that although Battigalli's construction of a rationality
ordering also promotes the backward induction outcome in any generic perfect
information game, his proof (see Battigalli [9]) is not as directly tied to the
procedure of backward induction. Furthermore, Battigalli's construction of a
rationality ordering promotes the forward induction outcome in an extended
version of the \Battle-of-the-Sexes\ (BoS) game where the BoS game is preceded
by 1 being o ered an outside option that is preferred by 1 to 2's most preferred
outcome in the BoS game. This conclusion is not reached in the present analysis
since there is no choice situation in which 1 under all circumstances can reveal
a particular preference between his BoS strategies.
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Appendix: 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.
Let F be a nite set of states, and let S be a nite set of outcomes. Consider a
decision maker which is uncertain about what state in F will be realized. The decision
maker is endowed with a binary relation over all functions that to each element of F
assigns an objective randomization on S . Any such function x : F ! (S ) is called
an act on F . Write x and y for acts on F . A binary relation on the set of acts on
F is denoted by , where x y means that x is preferred or indi erent to
y . As usual, let (preferred to) and (indi erent to) denote the asymmetric and
symmetric parts of . A binary relation on the set of acts on F is said to satisfy
objective independence00 if x0 (respectively ) x00 i x0 + (1 , )y (respectively ) x + (1 , )y , whenever 0 < < 1 and y is arbitrary.
nontriviality if there exist x and y such that x y .
continuity
if there exist 0 < < < 1 such that x0 + (1 , )x00 y 0
x + (1 , )x00 whenever x0 y x00 .
If F = F1 F2 , say that 1 is the marginal of on F1 if, x 1 1 y 1 i x y
whenever x 1 (f1 ) = x (f1 ; f2 ) and y 1 (f1 ) = y (f1 ; f2 ) for all (f1 ; f2 ).
If E F , let x denote the restriction of x to E . De ne the conditional binary
relation by x0 x00 if, for arbitrary y , (x0 ; y, ) (x00 ; y, ), where ,E
denotes F nE . Say that the state f 2 F is Savage-null if x f g y for all acts x
and y on F . A binary relation is said to satisfy
conditional
continuity if, 8f 2 F , there exist 0 < < < 1 such that x0 + (1 ,
00
)x f g y f g x0 + (1 , )x00 whenever x0 f g y f g x00 .
non-null state independence if x f g y i x f g y whenever e and f are
not Savage-null and x and y satisfy x (e) = x (f ) and y (e) = y (f ).
If e, f 2 F and f g is complete, then e is deemed in nitely more likely than f (e >> f )
if e is not Savage-null and x f g y implies (x,f g; x0f g) f g (y,f g; yf0 g) for all
x0 , y0 . According to this de nition, f may, but need not, be Savage-null if e >> f .
Say that y is maximal w.r.t. if there is no x such that x y .
Let u : S ! < be a vNM utility function, where there exist s0 , s00P2 S such that
u(s0 ) > u(s00 ). If x 2 (S ) is an objective randomization, write u(x) = 2 x(s)u(s).
Say that is conditionally represented by u if (a) is nontrivial and (b) x f g y
i u(x (f )) u(y (f )) whenever f is not Savage-null. If is conditionally represented by u, then satis es conditional continuity and non-null state independence, and, for any f 2 F , f g is complete. Say that x strongly dominates y
if, 8f 2 E , u(x (f )) > u(y (f )). Say that x weakly dominates y if, 8f 2 E ,
u(x (f )) u(y (f )), with strict inequality for some e 2 E . Say that is admissible w.r.t. u if x y whenever x weakly dominates y . If is conditionally
represented by u, then is admissible w.r.t. u i no f 2 F is Savage-null.
Two representation results can now be stated. The rst follows from Anscombe &
Aumann [1] and the properties of a conditionally represented binary relation.
Proposition A1. If is complete, transitive, objectively independent, continuous,
and conditionally represented by u, P
then there is a subjectiveP
probability distribution
m 2 (F ) such that x y i
m
(
f
)
u
(
x
(
f
))
2
2 m (f )u(y (f )).
The second result, due to Blume et al. ([11], Theorem 3.1), requires the notion of a
lexicographic probability system (LPS): If K 1 and, 8k 2 f1; : : : ; K g, m 2 (F ),
then ` = (m1 ; :::; m ) is an LPS on F . Let L(F ) denote the set of LPSs on F .
Proposition A2. If is complete, transitive, objectively
independent, and conditionally represented
` = (m1 ; :::; m ) 2 L(F ) such that
P by u, then there is an LPS P
x y i ( 2 m (f )u(x (f ))) =1 ( 2 m (f )u(y (f ))) =1.3
3
For two vectors v and w, v w i whenever w > v , 9 k0 < k such that v 0 > w 0 .
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