Mathematical Social Sciences 44 (2002) 121–144
www.elsevier.com / locate / econbase
On the epistemic foundation for backward induction
Geir B. Asheim*
Department of Economics, University of Oslo, P.O. Box 1095 Blindern, N-0317 Oslo, Norway
Abstract
Backward induction is characterized in an epistemic model of perfect information games where
players have common certain belief of the consistency of preferences rather than the rationality of
choice. In this approach, backward induction corresponds to common certain belief of ‘belief in
each subgame of opponent rationality’. At an interpretative level this result resembles the one
established by Aumann [Games Econ. Behav. 8 (1995) 6–19]. By instead imposing common
certain belief of ‘belief (only in the whole game) of opponent rationality’, Ben-Porath’s [Rev.
Econ. Stud. 64 (1997) 23–46] support for the Dekel–Fudenberg procedure is interpreted.
 2002 Elsevier Science B.V. All rights reserved.
Keywords: Backward induction
JEL classification: C72
1. Introduction
In recent years, two influential contributions on backward induction in finite perfect
information games have appeared, namely Aumann (1995) and Ben-Porath (1997).
These contributions—both of which consider generic perfect information games (where
all payoffs are different)—reach opposite conclusions: while Aumann establishes that
Common Knowledge of Rationality 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. The models of Aumann and
Ben-Porath are different. One such difference is that Aumann makes use of ‘knowledge’
in the sense of ‘true knowledge’, while Ben-Porath’s analysis is based on ‘certainty’ in
the sense of ‘belief with probability 1’. Another is that the term ‘rationality’ is used in
different senses.
*Tel.: 147-2285-5498; fax: 147-2285-5035.
E-mail address: g.b.asheim@econ.uio.no (G.B. Asheim).
0165-4896 / 02 / $ – see front matter  2002 Elsevier Science B.V. All rights reserved.
PII: S0165-4896( 02 )00011-2
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G.B. Asheim / Mathematical Social Sciences 44 (2002) 121–144
The present paper shows how the conclusions of Aumann and Ben-Porath can be
captured by imposing requirements 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 differences between the analyses of
Aumann and Ben-Porath, and thereby enhance our understanding of the epistemic
conditions underlying backward induction. For ease of presentation, the analysis will be
limited to two-player games. This is purely a matter of convenience as everything can
directly be generalized to n-player games (with n . 2).
Among the large literature on backward induction during the last couple of decades,1
Reny’s (1993) impossibility result is of special importance. Reny associates a player’s
‘rationality’ in an extensive game with what is called ‘weak sequential rationality’; i.e.
that a player chooses rationally in all subgames that are not precluded from being
reached by the player’s own strategy. He shows that there exist perfect information
games where the event that both players satisfy weak sequential rationality cannot be
commonly believed in all subgames. For example, in the centipede game that is
illustrated in Fig. 1 of Section 4.2, common belief of weak sequential rationality cannot
be held in the subgame defined by 2’s decision node. The reason is that if 1 believes that
2 is rational in the subgame, and if 1 believes that 2 believes that 1 will be rational in the
subgame defined by 1’s second decision node, then 1 believes that 2 will choose d,
implying that only D is a best response for 1. Then the fact that the subgame defined by
2’s decision node has been reached, contradicts 2’s belief that 1 is rational in the whole
game.
As a response, Ben-Porath (1997) imposes that common belief of weak sequential
rationality is held in the whole game only. However, weak sequential rationality being
commonly believed in the whole game only, does not imply backward induction. In the
centipede game of Fig. 1, the strategies D and FD for player 1 and d and f for player 2
are consistent with such common belief, while backward induction implies that down is
played at any decision node.
In order to obtain an epistemic characterization of backward induction, Aumann
(1995) considers ‘sequential rationality’ in the sense that a player chooses rationally in
all subgames (see also footnote 5). However, the event that players satisfy sequential
rationality is somewhat problematic. If—in the centipede game of Fig. 1—1 believes or
knows that 2 chooses d, then only by choosing the strategy DD will 1 satisfy sequential
rationality. However, what does it mean that 1 chooses DD in the counterfactual event
that player 2’s decision node were reached? It is perhaps more natural—as suggested by
Stalnaker (1998, Sect. 5)—to consider 2’s belief about 1’s subsequent action if 2’s
decision node were reached. Since Aumann (1995) assumes knowledge of rational
choice in an S5 partition structure, such a question of belief revision cannot be asked
within Aumann’s model.
By imposing a full support restriction (cf. Definition 1) the present paper ensures that
1
Among contributions that are not otherwise referred to here are Basu (1990), Bicchieri (1989), Binmore
(1987, 1995), Bonanno (2001), Clausing and Vilks (2000), Feinberg (2001), Gul (1997), Kaneko (1999),
Rabinowicz (1997), and Rosenthal (1981), as well as Asheim (2001) and Schuhmacher (1999).
G.B. Asheim / Mathematical Social Sciences 44 (2002) 121–144
123
each player takes all opponent strategies into account, having the structural implication
that conditional beliefs are well-defined and the behavioral implication that a rational
choice in the whole game is a rational choice in all subgames that are not precluded
from being reached by the player’s own strategy. Hence, by this restriction, we may
consider ‘rationality’ instead of ‘weak sequential rationality’ (as shown by Lemma 2 and
the subsequent text).
The main distinguishing feature of the present analysis is, however, to consider the
event that a player believes in opponent rationality rather than the event that the player
himself is rational. Asheim and Dufwenberg (2002) (AD) show the following result
(reproduced as Proposition 6 of Section 4.1): strategies surviving the Dekel–Fudenberg
(Dekel and Fudenberg, 1990) procedure, where one round of weak elimination is
followed by iterated strong elimination, can be characterized as maximal strategies when
there is common certain belief that each player believes in the whole game that the
opponent chooses rationally (‘belief of opponent rationality’). For generic perfect
information games, Ben-Porath shows that the set of outcomes consistent with common
belief of weak sequential rationality corresponds to the set of outcomes that survives the
Dekel–Fudenberg procedure. Hence, maximal strategies when there is common certain
belief of ‘belief of opponent rationality’ correspond to outcomes that are promoted by
Ben-Porath’s analysis.
An extensive game offers choice situations, not only in the whole game, but also in
proper subgames. In perfect information games (and, more generally, in multi-stage
games) the subgames constitute an exhaustive set of such choice situations. Hence, in
perfect information games one can replace ‘belief of opponent rationality’ by ‘belief in
each subgame of opponent rationality’: each player believes in each subgame that his
opponent chooses rationally in the subgame. The main results of the present paper
(Propositions 7 and 8 of Section 5.2) show how, for generic perfect information games,
common certain belief of ‘belief in each subgame of opponent rationality’ is possible
and uniquely determines the backward induction outcome. Hence, by substituting ‘belief
in each subgame of opponent rationality’ for ‘belief of opponent rationality’, the present
analysis provides an alternative route to Aumann’s conclusion, namely that common
knowledge (or certain belief) of an appropriate form of (belief of) rationality implies
backward induction.
This epistemic foundation for backward induction requires common certain belief of
‘belief in each subgame of opponent rationality’, where the term ‘certain belief’ is being
used in the sense that an event is certainly believed if the complement is subjectively
impossible (cf. Section 3.2). As shown by a counterexample in Section 5.2, the
characterization does not obtain if instead common belief (in a sense of belief with
probability 1) is applied. Furthermore, the event of which there is common certain
belief—namely ‘belief in each subgame of opponent rationality’—cannot be further
restricted by taking the intersection with the event of ‘rationality’. The reason is that the
full support restriction of Definition 1 is inconsistent with certain belief of opponent
‘rationality’, as the latter prevents a player from taking into account irrational opponent
choices and rules out a well-defined theory of belief revision.
The imposition of ‘belief (in each subgame) of opponent rationality’ restricts the
beliefs of players. Thus, the analysis follows AD by suggesting that in deductive game
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theory, requirements can be imposed on the beliefs of players rather than their choice.
Since the beliefs of players are determined by their preferences, this amounts to
imposing requirements on preferences.2
The paper is organized as follows. Sections 2 and 3 present the formal framework in
which extensive games will be analyzed. Section 4 reviews AD’s characterization of the
Dekel–Fudenberg procedure and applies this result to generic extensive games of perfect
information, thereby comparing the present analysis to that of Ben-Porath (1997).
Section 5 introduces ‘belief in each subgame of opponent rationality’ as an alternative
epistemic condition and establishes the paper’s main results. Section 6 interprets the
present approach in terms of adding belief revision to Aumann’s (1995) analysis, and
discusses the relationship to Battigalli’s (1996) concept of ‘rationality orderings’.
2. States, types, and preferences
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 decision-theoretic analysis
builds on Blume et al. (1991a). The framework is summarized by the concept of a belief
system (cf. Definition 1). Appendix A contains a presentation of the decision-theoretic
terminology, notation, and results that will be utilized.
2.1. An extensive game form
Inspired by Osborne and Rubinstein (1994, Chap. 6), a finite extensive game form of
almost perfect information with two players and M 2 1 stages can be described as
follows. Both perfect information games and finitely repeated games yield game forms
that fit this description. The sets of histories is determined inductively: the set of
histories at the beginning of the first stage 1 is H 1 5 h5j. Let H m denote the set of
histories at the beginning of stage m. At h [ H m , let, for each player i [ I[h1,2j, i’s
action set be denoted A i (h), where i is inactive at h if A i (h) is a singleton. Write
A(h)[A 1 (h) 3 A 2 (h). Define the set of histories at the beginning of stage m 1 1 as
follows: H m 11 [ hsh, aduh [ H m and a [ A(h) j. This concludes the induction. Let
21 m
H[ < M
denote the set of subgames and let Z[H M denote the set of outcomes.
m 51 H
A pure strategy for player i is a function s i that assigns an action in A i (h) to any
h [ H. Let Si denote player i’s finite set of pure strategies, and write S[S1 3 S2 . Write
p, r, and s ( [ S) for pure strategy vectors. Let z: S → Z map strategy vectors into
2
Instead of imposing that a driver chooses to drive on the right side of the road if he believes 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 believes 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., 1991b).
G.B. Asheim / Mathematical Social Sciences 44 (2002) 121–144
125
outcomes.3 Then (S1 , S2 , z) is a finite strategic two-player game form. For any
h [ H < Z, let S(h) 5 S1 (h) 3 S2 (h) denote the set of strategy vectors that are consistent
with h being reached. Note that S(5) 5 S. For any h, h9 [ H < Z, h (weakly) precedes h9
if and only if S(h) $ S(h9). If s i [ Si and h [ H, let s i u h denote the strategy in Si (h)
coinciding with s i except at predecessors of h, where s i u h determines the unique action
leading to h.
2.2. 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 (see Tan and
Werlang, 1988). 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 (Boge
and Eisele, 1979; Mertens and
Zamir, 1985; Brandenburger and Dekel, 1993; Epstein and Wang, 1996) yield S 3 T as
the complete state space, where T 5 T 1 3 T 2 is the set of all feasible type vectors.
Furthermore, for each i, there is a homeomorphism between T i and the set of beliefs on
S 3 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.4 Hence, in the setting of a strategic game form the
beliefs can be restricted to the set of opponent strategy–type pairs, Sj 3 T j . Combined
with a vNM utility function, the set of beliefs on Sj 3 T j corresponds to a set of binary
relations on the set of acts on Sj 3 T j , where an act on Sj 3 T j is a function that to any
element of Sj 3 T j 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 V [S 3 T,
• each type t i of any player i correspond to a binary relation K t i on the set of acts on
Sj 3 T j .
However, as the above results on infinite hierarchies of beliefs are not necessarily
applicable in the present setting, we instead consider an implicit model—with a finite
type set T i for each player i—from which infinite hierarchies of beliefs can be
constructed. Moreover, since continuity of preferences is not imposed, the conditions on
3
A pure strategy s i [ Si can be viewed as an act on Sj that assigns z(s i , s j ) [ Z to any s j [ Sj . The set of pure
strategies Si is partitioned into equivalent classes of acts since a pure strategy s i also determines actions in
subgames which s i prevents from being reached. Each such equivalent class corresponds to a plan of action in
the sense of Rubinstein (1991). As there is no need here to differentiate between identical acts, the concept of
a plan of action would have sufficed.
4
This does not mean that i is not aware of i’s strategy choice. In an interactive setting it signifies that i’s belief
about j’s belief about j’s choice plays no role in the analysis. Note that Tan and Werlang (1988, Sections 2
and 3) characterize rationalizable strategies without specifying beliefs about one’s own choice.
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K t i consist of completeness, reflexivity, transitivity, objective independence, nontriviality, conditional continuity and non-null state independence, meaning that K t i is
represented by a vNM utility function y ti i : Z → R that assigns a payoff to any outcome
and a lexicographic probability system (LPS) l t i 5 ( m 1t i , . . . , m Lt i ) [ LD(Sj 3 T j ) (cf.
Appendix A). Being a vNM utility function, y it i can be extended to objective
randomizations on Z. Assume that K t i has full support on Sj (cf. Appendix A), implying
that, for any subgame, conditional preferences are nontrivial (cf. Section 2.3) and the
conditional belief operator is ‘well-defined’ (cf. Section 3.3).
The construction is summarized by the following definition.
Definition 1. A belief system for a game form (S1 , S2 , z) consists of
• for each player i, a finite set of types T i ,
• for each type t i of any player i, a binary relation K t i (t i ’s preferences) on the set of
acts on Sj 3 T j , where K t i has full support on Sj and is represented by a vNM utility
function y it i on D(Z) and an LPS l t i on Sj 3 T j .
2.3. Conditional preferences and rationality
Let K tSij denote the marginal (cf. Appendix A) of K t i on Sj . A pure strategy s i [ Si
can be viewed as an act x Sj on Sj that assigns z(s i , s j ) to any s j [ Sj . A mixed strategy
x i [ DsSid can be viewed as an act x Sj on Sj that assigns z(x i , s j ) to any s j [ Sj . Hence,
K tSij is a binary relation also on the subset of acts on Sj that correspond to i’s mixed
strategies. Thus, K tSij can be referred to as t i ’s preferences over i’s mixed strategies. The
set of mixed strategies D(Si ) is the set of acts that are at t i ’s actual disposal.
Let, for any subgame h, K tSij (h ) denote the conditional binary relation (cf. Appendix A)
of K tSij on Sj (h). As above it follows that K tSij (h) is t i ’s conditional preferences over i’s
mixed strategies in h. Since K t i is reflexive and transitive and satisfies objective
independence, K tSij (h) shares these properties, and
C ti i (h)[hs i [ Si (h)us i is maximal w.r.t. K tSij (h) in D(Si (h))j
is non-empty and supports any maximal mixed strategy. Refer to C ti i (h) as t i ’s choice set
in the subgame h. Write C ti i [C ti i (5), and write, for any h [ H, C t (h)[C t11 (h) 3 C t22 (h).
By the following lemma, if s i is maximal in a subgame h, then s i is maximal in any
later subgame that s i is consistent with.
Lemma 2. If s i [ C it i (h), then s i [ C ti i (h9) for any h9 [ H with s i [ Si (h9) # Si (h).
Proof. Suppose that s i is not maximal w.r.t. K Sj (h9 ) in D(Si (h9)). Then there exists x Sj
such that x Sj s Sj (h9 ) y Sj , where x Sj assigns z(x i , s j ) to any s j [ Sj with x i [ D(Si (h9)), and
where y Sj assigns z(s i , s j ) to any s j [ Sj . By Mailath et al. (1992, Definitions 2 and 3 and
the if-part of Theorem 1), S(h9) is strategically independent for i. Hence, x Sj can be
G.B. Asheim / Mathematical Social Sciences 44 (2002) 121–144
127
chosen such that x Sj (s j ) 5 y Sj (s j ) for all s j [ Sj \Sj (h9). This implies that x Sj s tSij (h) y Sj ,
which contradicts that s i is maximal w.r.t. K St ij (h) in D(Si (h)). h
Definition 3. The pure strategy s i is a rational choice by t i in h if s i [ C it i (h).
The event that i is rational in h is defined as
[rat i (h)][h(s 1 , s 2 , t 1 , t 2 ) [ V us i [ C it i (h)j.
Write [rat i ][[rat i (5)].
The assumption that K t i has full support on Sj (cf. Section 2.2) has the structural
implication that, for all h, the conditional preferences, K tSij (h ) , are nontrivial. Moreover,
by Lemma 2 it has the behavioral implication that any choice s i that is rational in h is
also rational in any later subgame that s i is consistent with. This means that ‘rationality’
implies ‘weak sequential rationality’. In fact, K St ij (h) is admissible on Sj (h) (cf. Appendix
A), implying that any strategy that is weakly dominated in h cannot be rational in h.
Thus, preference for cautious behavior is induced. However, in the context of generic
perfect information games (cf. Section 4.2) such admissibility has no cutting power
beyond ensuring that ‘rationality’ implies ‘weak sequential rationality’ (see e.g. BenPorath, 1997, Lemma 1.2). Hence, in the class of games considered in our main results it
is of no consequence to use ‘rationality’ combined with full support rather than ‘weak
sequential rationality’.
2.4. An extensive game
Consider an extensive game form (cf. Section 2.1), and let, for each i, yi : Z → R be a
vNM utility function that assigns payoff to any outcome. Then the pair of the extensive
game form and the vNM utility functions (y1 , y2 ) is a finite extensive game of almost
perfect information, G. Let G 5 (S1 , S2 , u 1 , u 2 ) be the corresponding finite strategic
game, where for each i, the vNM utility function u i : S → R is defined by u i 5 yi + z (i.e.
u i (s) 5 yi (z(s)) for any s 5 (s 1 , s 2 ) [ S). Assume that, for each i, there exist r, s [ S such
that u i (r) . u i (s). The event that i plays the game G is given by
[u i ][h(s 1 , s 2 , t 1 , t 2 ) [ V uy ti i + z is a positive affine transformation of u i j,
while [u 1 ] > [u 2 ] is the event that both players play G.
3. Epistemic operators
The results on backward induction in Section 5.2 will be based on two epistemic
operators: ‘certain belief’ about the type of the opponent, and ‘conditional belief’ about
the rationality of the opponent in subgames. These are subjective operators derived from
the LPSs that represent the preferences of player types. It is essential that preferences are
defined also over fictitious acts that do not correspond to actual strategies; in particular,
such acts are needed to elicit beliefs about the type of the opponent.
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G.B. Asheim / Mathematical Social Sciences 44 (2002) 121–144
To state these operators, let, for each player i and each state v [ V, t i (v ) denote the
projection of v on T i . It will follow that at v i ‘certainly believes’ and ‘conditionally
believes’ the event that his type is t i (v ). As it is unnecessary to specify at v i’s belief
about his own strategy choice, we consider only events E # V satisfying that E 5 Si 3
proj Sj 3T i 3T j E. For for any subgame h and any such event E, let
E tj i (h)[h(s j , t j ) [ Sj (h) 3 T j u(s j , t i , t j ) [ proj Sj 3T i 3T j Ej
denote the set of opponent strategy–type pairs that are consistent with h, v [ E and
t i (v ) 5 t i . Write E jt i [E jt i (5).
3.1. Possibility and epistemic priority
An LPS determines both possibility of opponent strategy–type pairs and epistemic
priority of opponent strategy–type pairs (see e.g. Stalnaker, 1998, Section 3). To see
this, consider the LPS l t i 5 ( m 1t i , . . . , m Lt i ) [ LD(Sj 3 T j ) that represents t i ’s preferences,
and construct the following collection of nested sets:
hsupp m t1i , < 2, 51 supp m t,i , . . . , < L, 51 supp m t,i j.
Say that t i deems (s j , t j ) possible (e.g. not Savage-null) if (s j , t j ) [ k jt i , where
k tj i [ < L, 51 supp m t,i ( # Sj 3 Tj ),
while t i gives epistemic priority to (s j9 , t j9 ) over (s j99 , t j99 ) (e.g. (s j9 , t j9 ) is infinitely more
likely than (s 99
j , t j99 ), see Blume et al., 1991a, Definition 5.1) if there exists , 9 [ h1, . . . ,
9
9
Lj such that (s j9 , t 9j ) [ < ,, 51
supp m t,i , while (s j99 , t 99
⁄ < ,, 51
supp m t,i .
j )[
That i’s opponent makes a strategy choice consistent with a subgame h [ H is an
event Si 3 Sj (h) 3 T i 3 T j that is objectively knowable for i if subgame h is reached.
Since Definition 1 entails that t i ’s LPS, l t i , has full support on Sj , t i ’s preferences
conditional on Sj (h) 3 T j are non-trivial and represented by a conditional LPS l t i (h) 5
ti
( m 1t i (h), . . . , m L(h)
(h)) [ LD(Sj (h) 3 T j ) (see Blume et al., 1991a, Definition 4.2). It
follows from Blume et al.’s definition that the set of opponent strategy–type pairs given
the highest epistemic priority under the conditional LPS,
b tj i (h)[supp m t1i (h) ( # Sj (h) 3 T j ),
equals the set of pairs given the highest epistemic priority under the unconditional LPS,
l t i , among opponent strategy–type pairs in Sj (h) 3 Tj .
3.2. Certain belief
Since we will be concerned with what types of j that i deems possible, what types of i
that the possible type of j deem possible, and so on, the operator used for the interactive
epistemology will be based on k tj i .
G.B. Asheim / Mathematical Social Sciences 44 (2002) 121–144
129
Definition 4. If E # V satisfies that E 5 Si 3 proj Sj 3T i 3T j E, then say that at v i certainly
believes the event E if v [ Ki E, where
Ki E[hv [ V uk tj i (v ) # E tj i (v ) j.
The operator Ki corresponds to what Morris (1997) calls ‘Savage-belief’. It is stronger
than Ben-Porath’s (1997) ‘certainty’ as it does not allow the complement of a certainly
believed event to be taken into account, but weaker than Aumann’s (1995) ‘knowledge’
as a certainly believed event need not be true.
If E # V satisfies that E 5 S1 3 S2 3 proj T 1 3T 2 E, then say that there is that, at v, there
is mutual certain belief of E # V if v [ KE, where KE[K1 E > K2 E. Say that, at v,
there is common certain belief of E # V if v [ CKE, where CKE[KE > KKE >
KKKE > . . . .
3.3. Conditional belief
Since Definition 1 entails that all opponent strategies are deemed possible, belief
about the rationality of the opponent in subgames cannot be associated with k tj i . Rather,
it will be based on the set of opponent strategy-pair that has the highest epistemic
priority under the conditional LPS.
Definition 5. If E # V satisfies that E 5 Si 3 proj Sj 3T i 3T j E, then say that at v i believes
the event E conditional on the subgame h if v [ Bi (h)E, where
Bi (h)E[hv [ V u b j i v ) (h) # E tj i (v ) (h)j.
t (
Since b jt i (h)[supp m 1t i (h), Bi (h) corresponds to ‘belief with probability 1’.
If h9 is a predecessor of h (i.e. S(h9) $ S(h)) and b tj i (h9) > (Sj (h) 3 T j ) ± 5, then it
follows from the above definitions that b tj i (h) 5 b tj i (h9) > (Sj (h) 3 T j ). Hence, Bi ( ? )
´ et al., 1985).
satisfies minimal belief revision (see e.g. Alchourron
For any h and at any v, it follows from the property that K t i (v ) has full support on Sj
that i’s belief conditional on the subgame h is ‘well-defined’ (in the sense that the
non-empty set b ti i (v ) (h) is uniquely determined). Hence, a ‘well-defined’ conditional
belief in h is implied by full support alone; it does not require that h is actually being
reached. This means that a requirement on i’s belief conditional on h is a requirement on
the type of player i only; it does not impose that i makes a strategy choice consistent
with h.
Since the conditional belief operator is used only for objectively knowable events that
are subjectively possible, we do not consider hypothetical events. Hence, hypothetical
epistemic operators of the kind developed by Samet (1996) are not needed in the present
framework.
Write Bi E[Bi (5)E for (unconditional) belief. If E # V satisfies that E 5 S1 3 S2 3
proj T 1 3T 2 E, then say that there is mutual belief of E # V if v [ BE, where BE[B1 E >
G.B. Asheim / Mathematical Social Sciences 44 (2002) 121–144
130
B2 E. Say that, at v, there is common belief of E # V if v [ CBE, where CBE[BE >
BBE > BBBE > . . . .
3.4. Properties
It can easily be shown that the following properties hold for any h [ H. Firstly, since
b tj i (h) # k tj i > (Sj (h) 3 Tj ), it follows that certain belief implies conditional belief:
Ki E # Bi (h)E. Moreover, the operators Ki and Bi (h) satisfy
Ki E > Ki F 5 Ki (E > F )
Ki V 5 V
Ki E # Ki Ki E
¬Ki E # Ki (¬Ki E)
Bi (h)E > Bi (h)F 5 Bi (h)(E > F )
Bi (h)5 5 5
Ki E # Ki Bi (h)E
¬Bi (h)E # Ki (¬Bi (h)E).
Since Ki E # Bi (h)E, it follows that Ki 5 5 5, Bi (h)V 5 V, Bi (h)E # Bi (h)Bi (h)E and
¬Bi (h)E # Bi (h)(¬Bi (h)E). Hence, both operators Bi (h) and Ki correspond to KD45
systems.
3.5. Are subjective probabilities needed?
An LPS is a representation of preferences that are based on subjective probabilities.
However, one can convincingly argue that subjective probabilities are not part of the
backward induction argument; indeed, it plays no role in Aumann’s (1995) article. It
might therefore be of interest to point out that the present analysis does not actually
depend on subjective probabilities.
By weakening completeness to conditional completeness while keeping the remaining
conditions on K t i —reflexivity, transitivity, objective independence, nontriviality, conditional continuity and non-null state independence—it follows directly from the vNM
theorem that preferences are conditionally represented by a vNM utility function, y ti i ,
without necessarily determining the relative subjective likelihood of any two opponent
strategy–type pairs.
The preferences still determine, for each subgame, a choice set (e.g. a non-empty set
of maximal pure strategies that supports any maximal mixed strategy) since this only
requires reflexivity, transitivity and objective independence. Moreover, the preferences
still determine possibility (e.g. opponent strategy–type pairs not being Savage-null) and
epistemic priority (e.g. one opponent strategy–type pair being infinitely more likely than
another), meaning we can derive k tj i and, for each h [ H, b tj i (h), and define the certain
and conditional belief operators. In Asheim and Søvik (2001) we develop such
preference-based operators under these weaker conditions and show inter alia that
conditional belief still satisfies minimal belief revision. The family of sets h b tj i (h)uh [ Hj
is a system of conditional filter generating sets (cf. Brandenburger, 1998, Section 5).
However, even when completeness is not imposed, K t i in general encode more
information about t i ’s preferences than can be recovered from such a system of
conditional filter generating sets.
G.B. Asheim / Mathematical Social Sciences 44 (2002) 121–144
131
4. Belief of opponent rationality
Usually requirements in deductive game theory are imposed on choice. For example,
rationality is the requirement that a pair (s i , t i ) satisfies s i [ C ti i . The present paper
follows AD by imposing requirements on t i only. Since t i corresponds to the preferences
K t i , such requirements will be imposed on K t i . Here we focus on showing how such an
approach to deductive game-theoretic analysis can be used to shed light on the analyses
of Aumann (1995) and Ben-Porath (1997), and thereby enhance our understanding of
the epistemic conditions underlying backward induction. For this purpose, it is useful to
reproduce AD’s characterization of the Dekel–Fudenberg procedure.
4.1. Admissible consistency
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 two requirements in addition to ensuring—
what is captured by the assumption of full support on Sj in the present paper—that each
player takes all opponent strategies into account, thereby taking into account the
possibility that any subgame in an extensive game be reached: the first of these ensures
that each player plays the game G, while the second requirement ensures that each
player believes that the opponent chooses rationally (belief of opponent rationality).
Hence, consider the following event
Bi [rat j ] 5 hv [ V u(s j , t j ) [ b jt i (v ) (5) implies s j [ C tj j j.
Since K t i (v ) is represented by y it i (v ) and an LPS l t i (v ) 5 ( m t1i (v ) , . . . , m tLi (v ) ) [ LD(Sj 3
T j ), v [ Bi [rat j ] entails that m 1t i (v ) (s j , t j ) . 0 only if s j [ C jt j .
Say that at v i is admissibly consistent (with the game G and the preferences of his
opponent) if v [ A i , where
A i [[u i ] > Bi [rat j ].
Refer to A[A 1 > A 2 as the event of admissible consistency. The Dekel–Fudenberg
procedure can now be characterized as maximal strategies in states where there is
common certain belief of admissible consistency.
Proposition 6. (Asheim and Dufwenberg, 2002) A pure strategy r i for i survives the
Dekel–Fudenberg procedure in a finite strategic game G if and only if there exists a
belief system with r i [ C it i (v ) for some v [ CKA .
Proof. The assumption of ‘caution’ in AD is here weakened to full support on Sj . The
only-if part follows directly from the first part of Proposition 5.2 in AD. The if part
follows from the second part of Proposition 5.2 in AD, subject to a straightforward
modification due to the weakening of ‘caution’ to full support on Sj . In both parts of the
132
G.B. Asheim / Mathematical Social Sciences 44 (2002) 121–144
proof it is without consequence that complete preferences must be substituted for the
(possibly) incomplete preferences used by AD. h
The characterization result of Proposition 6 is obtained also if CBA is used instead of
CKA; this is essentially the corresponding result by Brandenburger (1992). The result of
AD is more complicated as it involves two different epistemic operators. Still, it yields
the insight that the essential feature in a characterization of the Dekel–Fudenberg
procedure is to let irrational opponent choice be deemed possible. It also turns out to be
a useful benchmark for the analysis of backward induction in Section 5 where common
belief cannot be applied.
4.2. Generic games of perfect information
A finite extensive game is
• . . . of perfect information if, at any h [ H, there exists at most one player that has a
non-singleton action set;
• . . . generic if, for each i, yi (z) ± yi (z9) whenever z and z9 are different 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, z( pu h ) is the backward induction
outcome in the subgame h, and S(z( pu h )) is the set of strategy vectors consistent with the
backward induction outcome in the subgame h.
Both Aumann (1995) and Ben-Porath (1997) analyze generic extensive games of
perfect information. As already pointed out, while Aumann establishes that common
(true) knowledge of (sequential) rationality 5 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 belief (in the whole game) of (weak
sequential) rationality. The purpose of the present section is to interpret the analysis of
Ben-Porath by applying Proposition 6 to the class of generic perfect information games.
Ben-Porath (1997) establishes through his Theorem 1 that the set of outcomes
consistent with common belief (in the whole game) of (weak sequential) rationality
corresponds to the set of outcomes that survive the Dekel–Fudenberg procedure. Hence,
by Proposition 6, maximal strategies when there is common certain belief of admissible
consistency correspond to the outcomes promoted by Ben-Porath’s analysis.
To illustrate how common certain belief of admissible consistency 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 T 1 5 ht 19 , t 199 j and T 2 5 ht 29 , t 299 j. Assume that the preferences of
5
Aumann (1995) uses the term substantive rationality, meaning that for all histories h, if a player were to reach
h, then the player would choose rationally at h. See Aumann (1995, pp. 14–16) and Aumann (1998) as well
as Halpern (1998) and Stalnaker (1998, Section 5).
G.B. Asheim / Mathematical Social Sciences 44 (2002) 121–144
133
Fig. 1. A centipede game.
each type t i of any player i are represented by a vNM utility function y it i satisfying
y ti i + z 5 ui and a two-level LPS on Sj 3 Tj . In Table 1, the first 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; see footnote 3. Note that all types satisfy the full
support requirement of Definition 1. With these two-level LPSs each type’s preferences
over the player’s own strategies are given by
t 91 : D s FD s FF
t 99
1 : FD s D s FF
t 92 : d s f
t 99
2 : f sd
It is easy to check that all types satisfy ‘belief of opponent rationality’ (e.g. both t 92 and
t 99
2 assign positive (primary) probability to an opponent strategy–type pair only if it is a
maximal strategy for the opponent type, i.e. D in the case of t 91 and FD in the case of
t 99
1 ). Thus, with V 5 S 3 T 1 3 T 2 , it follows that V 5 A 5 CKA. Hence, preferences
consistent with common certain belief of admissible consistency need not reflect
backward induction since FD and f are maximal strategies.
Note that type t 92 , 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 (primary) probability 1
to player 1 being of type t 99
1 . Consequently, the conditional belief of type t 9
2 about 1’s
strategy choice assigns (primary) probability 1 to FD. Type t 99
,
on
the
other
hand, does
2
not admit the possibility that 1 is of another type than t 91 . Since the choice of F at 1’s
first decision node is not rational for t 91 , there is no restriction concerning the conditional
belief of type t 99
2 about the choice at 1’s second decision node. In the terminology of
Table 1
A belief system for the game of Fig. 1
t 91 :
t 92
s4 / 5, 7 / 10d
s0, 1 / 10d
t 299
s0, 1 / 10d
s1 / 5, 1 / 10d
t 991 :
d
f
t 91
s1 / 2, 1 / 4d
s0, 1 / 8d
s0, 1 / 8d
t 199
s0, 1 / 8d
s1 / 2, 1 / 4d
s0, 1 / 8d
t 992 :
D
FD
FF
t 92 :
d
f
t 92
s3 / 5, 5 / 10d
s0, 1 / 10d
t 992
s0, 1 / 10d
s2 / 5, 3 / 10d
D
FD
FF
t 91
s1, 1 / 2d
s0, 1 / 4d
s0, 1 / 4d
t 991
s0, 0d
s0, 0d
s0, 0d
134
G.B. Asheim / Mathematical Social Sciences 44 (2002) 121–144
Ben-Porath, a ‘surprise’ has occurred. Subsequent to such a surprise, a type need not
believe that the opponent type chooses rationally among his remaining strategies.
5. Belief in each subgame of opponent rationality
A simultaneous game offers only one choice situation. Hence, for a game in this class,
it seems reasonable that belief of opponent rationality is held in the whole game only, as
formalized by the requirement ‘belief of opponent rationality’. An extensive game with a
nontrivial dynamic structure, however, offers such choice situations, not only in the
whole game, but also in proper subgames. Moreover, for extensive games of almost
perfect information, the subgames constitute an exhaustive set of such choice situations.
This motivates imposing belief in each subgame of opponent rationality. Hence,
consider the event that i believes conditional on h that j is rational in h:
Bi (h)[rat j (h)] 5 hv [ V u(s j , t j ) [ b jt i (v ) (h) implies s j [ C jt j (h)j.
If v [ > h[H Bi (h)[rat j (h)], then at v i believes conditional on any subgame h that j is
rational in h. In other words, > h [H Bi (h)[rat j (h)] is the event that player i believes in
each subgame h that the opponent j is rational in h.6
Consider a finite extensive game G of almost perfect information with corresponding
strategic game G. Say that at v i is admissibly subgame consistent (with G and the
preferences of his opponent) if v [ A *i , where
A *i [[u i ] >s > h [H Bi (h)[rat j (h)]d.
Refer to A*[A 1* > A 2* as the event of admissible subgame consistency. This
definition of admissible subgame consistency can be applied to any finite extensive game
of almost perfect information. However, in order to relate to Aumann’s (1995)
Theorems A and B, the following analysis is concerned with generic perfect information
games.
5.1. The example revisited
In the belief system of Table 1, type t 99
2 does not satisfy ‘belief in each subgame of
opponent rationality’. By ‘belief in each subgame of opponent rationality’, any type of
player 2 must believe, conditional on the subgame defined by 2’s decision node, that 1
6
Note that the requirement of such ‘belief in each subgame of opponent rationality’ allows a player to update
his belief about the type of his opponent. Hence, there is no assumption of ‘epistemic independence’ between
different agents in the sense of Stalnaker (1998); cf. the remark after the proof of Proposition 7 as well as
Section 6.1. Still, the requirement can be considered a non-inductive analogue to ‘forward knowledge of
rationality’ as defined by Balkenborg and Winter (1997), and it is related to the requirement in Samet (1996,
Section 5) that each player hypothesizes that if h were reached, then the opponent would behave rationally at
h.
G.B. Asheim / Mathematical Social Sciences 44 (2002) 121–144
135
chooses his maximal strategy, FD, in the subgame. This means that any type of player 2
prefers d to f, implying that no type of player 1 satisfying ‘belief in each subgame of
opponent rationality’ can prefer FD to D. Thus, common certain belief of admissible
subgame consistency entails that any types of players 1 and 2 have the preferences
D s FD s FF
d sf
respectively, meaning that if any type of a player chooses a maximal strategy in a
subgame, then his choice is made in accordance with backward induction. Demonstrating that this conclusion holds in general for generic perfect information games is the
main result of the present paper.
5.2. Main results
In analogy with Aumann’s (1995) Theorems A and B, it is established that
• . . . any vector of maximal strategies in a subgame of a generic perfect information
game, in a state where there is common certain belief of admissible subgame
consistency, leads to the backward induction outcome in the subgame (Proposition
7). Hence, by substituting > h[H Bi (h)[rat j (h)] for Bi [rat j ], the present analysis yields
support to Aumann’s conclusion, namely that if there is common knowledge (or
certain belief) of an appropriate form of (belief of) rationality, then backward
induction results;
• . . . for any generic perfect information game, common certain belief of admissible
subgame consistency is possible (Proposition 8). Hence, the result of Proposition 7 is
not empty.
Proposition 7. Consider a finite generic extensive game of perfect information G with
corresponding strategic game G. If, for some belief system for G, v [ CKA*, then, for
each h [ H, C t(v ) (h) # S(z( pu h )), where p denotes the unique subgame-perfect equilibrium.
Proof. With KE 5 K1 E > K2 E as the mutual certain belief operator defined on hE #
V uE 5 Si 3 Sj 3 proj T i 3T j Ej, write K 0 E[E and, for each g $ 1, K g E[KK g 21 E. Since
Ki (E > F ) 5 Ki E > Ki F, and since Ki 5 5 5, conjunction, and positive and negative
introspection imply that Ki Ki E 5 Ki E, it follows ;g $ 2, K g E 5 K1 K g21 E >
K2 K g21 E # K1 K1 K g22 E > K2 K2 K g22 E 5 K1 K g22 E > K2 K g22 E 5 K g21 E. Even though
the truth axiom (Ki E # E) is not satisfied, A* # V that can be written as A*[A *1 > A *2
where, for each i, A *i 5 Si 3 Sj 3 proj T i A *i 3 T j , implying that mutual certain belief of
A* implies that A* is true: KA* 5 K1 A* > K2 A* # K1 A 1* > K2 A 2* 5 A 1* > A 2* 5 A*
g
g21
since, for each i, Ki A i* 5 A *i . Hence, (i) ;g $ 1, K A* # K
A*, and (ii) 'g9 $ 0
g
such that K A* 5 CKA* for g $ g9 since V is finite.
In view of these properties, it is sufficient to show for any g 5 0, . . . , M 2 2 that if
G.B. Asheim / Mathematical Social Sciences 44 (2002) 121–144
136
there exists a belief system with v [ K g A*, then C t(v ) (h) # S(z( pu h )) for any h [
H M 212g . This is established by induction.
( g 5 0) Let h [ H M21 . Firstly, consider j with a singleton action set at h. Then
trivially C tj j (h) 5 Sj (h) 5 Sj (z( pu 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 t i 5 t i (v ) for some
v [ K 0 A* 5 A*. Then it follows that C it i (h) 5 Si (z( pu h )) since G is generic, K t i has full
support on Sj , and v [ A* # [u i ].
( g 5 1, . . . , M 2 2) Suppose that it has been established for g9 5 0, . . . , g 2 1 that if
there exists a belief system with v [ K g9 A*, then C t(v ) (h9) # S(z( pu h9 )) for any h9 [
H M 212g9 . Let h [ H M 212g . Part 1. Consider j with a singleton action set at h. Let
t j 5 t j (v ) for some v [ K g21 A*. Then Sj (h) 5 Sj (h, a) and, by Lemma 2 and the
premise, C tj j (h) # C tj j (h, a) # Sj (z( pu (h,a) )) if a is a feasible action vector at h. This
implies that
C tj j (h) # > a Sj (z( pu (h,a) )) # Sj (z( pu h )).
Hence, if s j [ C jt j (h), then s j is consistent with the backward induction outcome in any
subgame (h, a) immediately succeeding h. Part 2. Consider i with a non-singleton action
set at h; since G has perfect information, there is at most one such i. Let t i 5 t i (v ) for
some v [ K g A*. The preceding argument implies that C tj j (h) # > a Sj (z( pu (h,a) )) whenever t j [ T tj i since v [ K g A* # Ki K g 21 A*. Let s i [ Si (h) be a strategy that differs from
pi u h by assigning a different action at h (i.e. z(s i , pj u h ) ± z( pu h ) and s i (h9) 5 pi u h (h9)
whenever Si (h) . Si (h9)). As any strategy can be viewed as an act on Sj (cf. Section 2.3),
write x Sj for the act on Sj that pi u h can be viewed as, and write y Sj for the act on Sj that s i
can be viewed as. Let x and y be the acts on Sj 3 T j that satisfy x(s j , t j ) 5 x Sj (s j ) and y(s j ,
t j ) 5 y Sj (s j ) for all (s j , t j ). Then,
x > a Sj (z( pu (h,a) ))3T j strongly dominates y > a Sj (z( pu (h,a) ))3T j
by backward induction since G is generic and v [ K g A* # [u i ]. Since C jt j (h) # >
ti
ti
a Sj (z( pu (h,a) )) whenever t j [ T j , it follows that, ;t j [ T j ,
x C tj j (h)3ht j j strongly dominates y C tj j (h)3ht j j ,
and, thus, v [ K g A* # Bi (h)[rat j (h)] implies that
t
t
x s Sij (h)3T j y and x Sj s Sij (h ) y Sj .
It has thereby been established that s i [ Si (h)\C it i (h) if s i differs from backward
induction only by the action taken at h. However, by Lemma 2 and the premise that
C ti i (h, a) # Si (z( pu (h,a) )) if a is a feasible action vector at h, it follows that any s i [ C ti i (h)
is consistent with the backward induction outcome in the subgame (h, (s i (h), a j ))
immediately succeeding h. Hence, C ti i (h) # Si (z( pu h )). h
It follows from the proof of Proposition 7 that, for a generic perfect information game
with M 2 1 stages, it is sufficient with M 2 2 order mutual certain belief of admissible
G.B. Asheim / Mathematical Social Sciences 44 (2002) 121–144
137
Fig. 2. A four-legged centipede game.
subgame consistency in order to obtain backward induction. Hence, K M 22 A* can be
substituted for CKA*.
Backward induction will not be obtained, however, if CBA* is substituted for CKA*.
This can be shown by considering a counter-example that builds on the four-legged
centipede game of Fig. 2 and the belief system of Table 2. In the table the preferences of
each type t i of any player i are represented by a vNM utility function y it i satisfying
y ti i + z 5 ui and a one- or three-level LPS on Sj 3 T j , where T 1 5 ht 19 , t 199 , t 1999 j and T 2 5 ht 29 ,
t 99
2 j. While all types satisfy the full support requirement of Definition 1, inspection shows
999 does not satisfy ‘belief in each subgame of
that A* 5 S 3 ht 91 , t 99
1 j 3 ht 9
2 , t 99
2 j, since t 1
opponent rationality’. Provided that i is of a type in ht 9i , t 99
i j, it follows that i believes at
(s, t 1 , t 2 ) that the opponent is of a type in ht j9 , t j99 j. This implies that CBA* 5 A*. Since
99 , it follows that
FD is the maximal strategy for t 99
1 and fd is the maximal strategy for t 2
preferences consistent with common belief of admissible subgame consistency need not
reflect backward induction. However, 2 does not certainly believe at (s, t 1 , t 299 ) that the
opponent is not of type t 999
1 . Therefore, KA* 5 A* 5 S 3 ht 9
1 , t 99
1 j 3 ht 9
2 j, while KKA* 5 5.
Hence, preferences that yield maximal strategies in contradiction with backward
induction are not consistent with common certain belief of admissible subgame
consistency.
The example shows that v [ A* is consistent with t i (v ) updating his beliefs about the
preferences of his opponent conditional on a subgame being reached. That is, if 1 is of
type t 91 , then in the whole game 1 assigns (primary) probability 4 / 5 to 2 being of type t 92
with preferences d s fd s ff, while in the subgame defined by 1’s second decision node 1
assigns (primary) probability 1 to 2 being of type t 99
2 with preferences fd s d | ff. This
Table 2
A belief system for the game of Fig. 2
t 19 :
d
fd
ff
t 199 :
t 992
7
7
,]
s]54 , ]
10 12 d
1
1
]
]
s0, 10 , 12 d
1
s0, 0, ]
12 d
1
1
, ]
s0, ]
10 12 d
1
1
1
]
]
]
s 5 , 10 , 12 d
1
s0, 0, ]
12 d
d
t 19
t 991
t 999
1
s]21 , ]31 , ]41 d
s0, ]16 , ]18 d
s0, 0, ]18 d
s0, ]61 , ]81 d
s]12 , ]13 , ]14 d
s0, 0, ]18 d
s0, 0, 0d
D
s0, 0, 0d
FD
s0, 0, 0d
FF
fd
ff
t 29 :
D
FD
FF
t 999
1 :
t 29
t 92
t 992
5
5
,]
s]35 , ]
10 12 d
1
1
]
]
s0, 10 , 12 d
1
s0, 0, ]
12 d
1
1
, ]
s0, ]
10 12 d
2
3
3
]
]
]
s 5 , 10 , 12 d
1
s0, 0, ]
12 d
t 92
t 992
ff
1
s]
10 d
1
]
s 10 d
3
s]
10 d
1
s]
10 d
1
]
s 10 d
3
s]
10 d
t 91
t 991
t 999
1
s1, ]21 , ]31 d
s0, ]14 , ]16 d
s0, 0, ]16 d
s0, 0, 0d
1
s0, 0, ]
12 d
1
s0, 0, ]
12 d
1 1
s0, ]4 , ]6 d
d
fd
t 299 :
s0, 0, 0d
s0, 0, 0d
138
G.B. Asheim / Mathematical Social Sciences 44 (2002) 121–144
shows that Stalnaker’s (1998) assumption of ‘epistemic independence’ is not made; a
player is in principle allowed to learn about the type of his opponent on the basis of
previous play. However, in a belief system with CKA* ± 5, v [ CKA* implies that 1
certainly believes at v that 2 is of a type with preferences d s fd s ff. In other words, if
there is common certain belief of admissible subgame consistency, there is essentially
nothing to learn about the opponent.
Proposition 8. For any finite generic extensive games of perfect information G with
corresponding strategic game G, there exists a belief system for G with CKA* ± 5.
Proof. Construct a belief system with only one type of each player, and write
V [S 3 ht 1 j 3 ht 2 j. Write, ;i [ N, ;m [ h1, . . . , M 2 1j, P mj [h pj u h uh [ H m j and,
ti
ti
ti
PM
j [Sj . Let, ;i [ N, l 5 ( m 1 , . . . , m M ) [ LD(Sj 3 ht j j) satisfy the following requireti
k
ment: ;m [ h1, . . . , Mj, supp m m 5 P j 3 ht j j. By letting K t i be represented by a vNM
utility function y it i satisfying y it i + z 5 u i and the LPS l t i , then (1) the full support
requirement of Definition 1 is satisfied and, ;i [ N, [u i ] 5 V. Let, ;h [ H, l t i (h) 5
ti
( m 1t i (h), . . . , m L(h)
(h)) denote the conditional of l t i on Sj (h) 3 T j . By the properties of a
subgame-perfect equilibrium, ;h [ H, m t1i (h)( pj u h , t j ) 5 1 and pi u h [ C ti i (h). Hence, since
likewise pj u h [ C tj j (h), we have that (2) ;i [ N, > h [H Bi (h)[rat j (h)] 5 V. By (1) and (2),
it follows that CKA* 5 A* 5 V ± 5. h
The constructive proof of Propostion 8 can be used to show, for any game of almost
perfect information and for any subgame-perfect equilibrium (in strategies that are not
weakly dominated), that the subgame-perfect equilibrium outcome corresponds to a
vector of maximal strategies in a state where there is common certain belief of
admissible subgame consistency.
6. Discussion
In this section we first interpret our analysis in view of Aumann (1995) and then
present a discussion of the relationship to Battigalli (1996).
6.1. Adding belief revision to Aumann’ s analysis
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 rational choice
only if it is consistent with the backward induction outcome. Using this terminology,
Proposition 7 can be restated as follows: under common certain belief of admissible
subgame consistency, players are of types with preferences that are in accordance with
backward induction. Furthermore, common certain belief of admissible subgame
consistency implies that each player deems it impossible that the opponent is of a type
with preferences not in accordance with backward induction.
However, since admissible subgame consistency is imposed on preferences, reaching
G.B. Asheim / Mathematical Social Sciences 44 (2002) 121–144
139
2’s decision node and 1’s second decision node in the centipede game of Fig. 1 does not
contradict common certain belief of admissible subgame consistency. Of course, these
decision nodes will not be reached if players choose rationally. But that players satisfy
‘belief in each subgame of opponent rationality’ is not a requirement concerning whether
their own choice is rational; rather, it means that they believe (with probability 1) in any
subgame that their opponent will choose rationally. Combined with the full support
restriction of Definition 1, which entails that each player deems any opponent strategy
possible, this means that belief revision is well-defined.
Hence, on the one hand, we capture the spirit of a conclusion that can be drawn from
Aumann’s (1995) analysis, namely that when being made subject to epistemic modeling
backward induction corresponds to each player having knowledge (or being certain) of
some essential feature of the opponent. In Aumann’s case, each player deems it
impossible—under common (true) knowledge of (sequential) rationality—that the
opponent makes an action inconsistent with backward induction. The analogous result in
the present case is that each player deems it impossible—under common certain belief of
admissible subgame consistency—that the opponent has preferences not in accordance
with backward induction.
On the other hand, we are still able to present an explicit analysis of how players
revise their beliefs about the opponent’s subsequent choice if surprising actions were to
be made. As noted in the Introduction, this fundamental issue of belief revision cannot
formally be raised within Aumann’s framework.
Stalnaker (1998) argues—contrary to statements made by Aumann (1995, Section
5f)—that an assumption of belief revision is implicit in Aumann’s motivation, namely
that information about different agents of the opponent is treated as epistemically
independent. In the reformulation by Halpern (1998),7 this means that in a state ‘closest’
to the current state when a player learns that the opponent has not followed her strategy,
he believes that the opponent will follow her strategy in the remaining subgame.
There is no assumption of ‘epistemic independence’ in the current interpretation of
Aumann’s result. Instead, we have changed statements ‘about opponents’ from being
concerned with strategy choice to being related to preferences. While it is desirable when
modeling backward induction to have an explicit theory of revision of beliefs about
opponent choice, a theory of revision of beliefs about opponent preferences is
inconsistent with maintaining both (a) that preferences are necessarily revealed from
choice, and (b) that there is common certain belief of the game being played (i.e.
consider the case where A i (5) is non-singleton, and a i [ A i (5) ends the game and leads
to an outcome that is preferred by i to any other outcome). Here we have kept the
assumption that there is common certain belief of the game, meaning that the game is of
‘complete information’, while requiring only conditional belief in each subgame of
opponent rationality, meaning that irrational opponent choices—although being probability zero events—are not impossible.
We have shown how common certain belief of admissible consistency implies that
each player deems it impossible that the opponent has preferences not in accordance
7
See Halpern (1998) for an instructive discussion of the differences between Aumann (1995) and Stalnaker
(1998), as well as how these relate to Samet (1996).
140
G.B. Asheim / Mathematical Social Sciences 44 (2002) 121–144
with backward induction and thus interprets any deviation from the backward induction
path as the opponent not having made a rational choice. In this way we present a model
that combines a result that resembles Aumann (1995) by associating backward induction
with certainty about opponent type, with an analysis that unlike Aumann’s yields a
theory of belief revision about opponent choice.
6.2. Rationality orderings
The constructive proof of Proposition 8 shows how common certain belief of
admissible subgame rationality may lead a type t i of player i to have preferences over i’s
strategies that are represented by a vNM utility function y ti i satisfying y ti i + z 5 u i and an
LPS l St ij 5 ( m 1t i , . . . , m Lt i ) [ LD(Sj ) with more than two levels of subjective probability
distributions (i.e. L . 2). For example, in the centipede game of Fig. 1, common certain
belief of admissible subgame rationality implies that any type t 2 of player 2 has
preferences that can be represented by u 2 and l St 21 5 ( m 1t 2 , m t22 , m 3t 2 ) where supp m 1t 2 5
hDj, supp m t22 5 hD, FDj, and supp m t32 5 S1 . One may interpret supp m t1i to be j’s ‘most
rational’ strategies, supp m Lt i \ < , 9,L supp m ,t i 9 to be j’s ‘completely irrational’ strategies,
and supp m ,t i \ < , 9, , supp m ,t i 9 , for , 5 2, . . . , L 2 1, to consist of strategies for j that are
at ‘intermediate degrees of rationality’. This illustrates that ssupp m t1i , . . . , supp m Lt i \ <
ti
, 9,L supp m , 9d corresponds closely to what Battigalli (1996) calls a rationality ordering
for j.
However, the present construction of such a rationality ordering differs from the one
proposed by Battigalli. This difference is along two dimensions.
1. Battigalli considers best responses in reachable subgames only (see his Definition
2.1), while here belief of opponent rationality is held in all subgames (cf. ‘belief in
each subgame of opponent rationality’).
2. Battigalli considers best responses given beliefs where opponent strategies that are
less than ‘most rational’ are given positive probability, while here each player always
believes that the opponent chooses rationally.
This difference has the following consequences.
• Although Battigalli’s construction of rationality orderings also yields the backward
induction outcome in any generic perfect information game, his proof (cf. Battigalli,
1997) is not tied to the backward induction procedure.
• Battigalli’s construction 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 offered 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 will have a particular
preference between his BoS strategies.8
8
AD show how the concept of admissible consistency can be strengthened so that the forward induction
outcome is promoted in the BoS game with an outside option.
G.B. Asheim / Mathematical Social Sciences 44 (2002) 121–144
141
This also indicates how the epistemic foundation for backward induction offered here
differs from the epistemic foundation for backward (and forward) induction provided by
Battigalli and Siniscalchi (2001).
Acknowledgements
This paper builds in part on joint work with Martin Dufwenberg. I thank him as well
as, among others, Adam Brandenburger, Stephen Morris and Ylva Søvik for helpful
comments and two referees and an associate editor for insightful and detailed
suggestions. A preliminary and abbreviated version of this paper is included in de Swart
(Ed.), Logic, Game Theory, and Social Choice—Proceedings of the International
Conference, LGS’99, Tilburg University Press, 1999.
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 and Aumann (1963), 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 x F : F → D(Z) is called an act on F.
Write x F and y F for acts on F. A reflexive and transitive binary relation on the set of acts
on F is denoted by K F , where x F K F y F means that x F is preferred or indifferent to y F .
As usual, let s F ( preferred to) and | F (indifferent to) denote the asymmetric and
symmetric parts of K F . A binary relation K F on the set of acts on F is said to satisfy
• objective independence if x F9 s F (respectively | F ) x F99 iff g x 9F 1 (1 2 g )y F s F
(respectively | F ) g x F99 1 (1 2 g )y F , whenever 0 , g , 1 and y F is arbitrary.
• nontriviality if there exist x F and y F such that x F s F y F .
If E # F, let x E denote the restriction of x F to E. Define the conditional binary relation
9 , y 2E )KF (x 99E , y 2E ), where 2 E denotes F\E.
K E by x F9 K E x 99
F if, for arbitrary y F , (x E
Say that the state f [ F is Savage-null if x F | h f j y F for all acts x F and y F on F. A binary
relation K F is said to satisfy
• conditional completeness if, ; f [ F, K h f j is complete.
• conditional continuity if, ; f [ F, there exist 0 , g , d , 1 such that d x 9F 1 (1 2
d )x 99
F s h f j y F s h f jg x 9
F 1 (1 2 g )x 99
F whenever x 9
F s h f j y F s h f j x 99
F.
• non-null state independence if x F s hej y F iff x F s h f j y F whenever e and f are not
Savage-null and x F and y F satisfy x F (e) 5 x F ( f ) and y F (e) 5 y F ( f ).
G.B. Asheim / Mathematical Social Sciences 44 (2002) 121–144
142
If e, f [ F and K F is conditionally complete, then e is deemed infinitely more likely
than f (e 4 f ) if e is not Savage-null and x F s hej y F implies (x 2h f j , x h9f j ) s he, f j ( y 2h f j ,
y 9h f j ) for all x F9 , y F9 . According to this definition, f may, but need not, be Savage-null if
e 4 f. Say that y F is maximal w.r.t. K E if there is no x F such that x F s E y F .
If y : Z → R is a vNM utility function, abuse notation slightly by writing y (x) 5
o z [Z x(z)y (z) whenever x [ D(Z) is an objective randomization. Say that x E strongly
dominates y E w.r.t. y if, ; f [ E, y (x E ( f )) . y ( y E ( f )). Say that x E weakly dominates y E
w.r.t. y if, ; f [ E, y (x E ( f )) $ y ( y E ( f )), with strict inequality for some e [ E. Say that
K F is admissible on E ( ± 5) if x F s F y F whenever x E weakly dominates y E .
The following representation result due to Blume et al. (1991a, Theorem 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 $ 1 and, ; , [ h1, . . . ,
Lj, m, [ D(F ), then l 5 ( m1 , . . . , mL ) is an LPS on F. Let LD(F ) denote the set of LPSs
on F, and let, for two utility vectors v and w, v $ L w denote that, whenever w, . v, ,
there exists , 9 , , such that v, 9 . w, 9 .
Proposition 9. If K F is complete, reflexive, and transitive, and satisfies objective
independence, nontriviality, conditional continuity, and non-null state independence,
then there exists a vNM utility function y : Z → R and an LPS l 5 ( m1 , . . . , mL ) [ LD(F )
such that x F K F y F iff
SO
O
m, ( f )y (xF ( f ))D , 51 $ LS
L
f [F
m, ( f )y ( yF ( f ))D , 51 .
L
f [F
Write F 5 F1 3 F2 and let K F be a binary relation on the set of acts on F. Say that
K F 1 is the marginal of K F on F1 if, x F 1 K F 1 y F 1 iff x F K F y F whenever x F 1 ( f1 ) 5 x F ( f1 ,
f2 ) and y F 1 ( f1 ) 5 y F ( f1 , f2 ) for all ( f1 , f2 ). Say that K F has full support on F1 if,
; f1 [ F1 , ' f2 [ F2 such that ( f1 , f2 ) is not Savage-null.
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