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KNOWLEDGE OF THE GAME,
RELATIVE RATIONALITY,
AND BACKWARDS INDUCTION
WITHOUT COUNTERFACTUALS
Arnis Vilks
Handelshochschule Leipzig
- Leipzig Graduate School of Management Jahnallee 59
D-04109 Leipzig
Germany
e-mail: [email protected]
Tel.: +49-341-9851670
Fax: +49-341-9851674
ABSTRACT. We analyse epistemic conditions for the backwards induction play in games of perfect
information. Unlike most previous literature on the subject, we explicitly pay attention to players’
knowledge of the game, avoid counterfactuals, and use the syntactic approach of the epistemic logic
KT. Moreover, taking doxastic possibility to be the dual of knowledge, we introduce a concept of
relative rationality in the sense of rational choice from the moves one considers possible. The main
result says that the backwards induction play is implied by sufficiently high order mutual knowledge of
(1) the structure of the game, (2) relative rationality, and (3) conditional doxastic possibility of all
moves which belong to the backwards induction profile. For a certain class of games, we also show
that replacing (3) by conditional possibility of some other profile S, does not imply play according to
S. Moreover, we show that our sufficient condition for the BI play is weaker than the one of Aumann
(1995).
Journal of Economic Literature Classification Numbers: C72, D80.
This is a much revised and extended version of HHL Working Paper no. 9. Previous versions have
been presented at CORE, at the LOFT 2 conference in Torino, the 11th Conference on Game Theory
and Applications in Milan, at FUR VIII in Mons, ESEM 97 in Toulouse, and the Workshop on
Entomological Game Theory in Uppsala. I have benefitted from comments by Geir Asheim, Dieter
Balkenborg, Giacomo Bonanno, Thorsten Clausing, Peter Hammond, Aviad Heifetz, Mamoru
Kaneko, Bart Lipman, Philippe Mongin, Klaus Nehring, Wlodek Rabinowicz, Jochen Runde, Robert
Stalnaker, Harald Wiese, and two referees.
- 2 1. Introduction
In game theory it is typically assumed (see, for instance, Myerson, 1991, pp. 1-5) that
(a) players are rational,
(b) players know the game to be played, and
(c) players are logically omniscient.
In fact, it is typically assumed that (a), (b), and (c) not only hold, but are common knowledge (CK)
among the players. However, the exact content and the implications of these assumptions continue to
be the source of controversy. In particular, this is so for the intuitively simple, and seemingly compelling
argument that for a perfect information (PI) game CK (or in fact sufficiently high order mutual
knowledge) of (a), (b), and (c) implies the backwards induction (BI) play. (See, for instance, Aumann,
1995, 1996, Binmore, 1996, 1997, Dekel and Gul, 1997). The present paper provides a formal
account of this argument which differs in several important ways from the previous literature.
a. Making “knowledge of the game” explicit. Almost without exceptions, the literature on the logic
of BI has focussed on (a), leaving (b) and (c) implicit. For instance, Aumann (1995,p.7) paraphrases
his result as “in PI games, common knowledge of rationality implies backward induction”. In all fairness
to Aumann’s clarity and precision, this is hardly an adequate statement of the usual intuition, unless
something like CK of (b) and (c) is implicitly taken for granted. For, quite obviously, the usual BI
argument does not make sense if players might be ignorant or mistaken about the game they are
playing, or if they fail to draw all relevant conclusions from what they know. And, at least in economic
applications of game theory, CK of (b) and (c) are hardly less problematic assumptions than CK of
(a).
While the logical omniscience assumptions embodied in the usual state-space representation of
knowledge may be said to be fairly well-understood, this is much less the case for players’ knowledge
of the game. The following remark in a well-known textbook is symptomatic: “Throughout our
discussion, we assume that the structure of the game is common knowledge in an informal sense.
Applying formal definitions of common knowledge to the structure of the game leads to technical and
philosophical problems that we prefer not to address.” (Fudenberg and Tirole, 1991, pp. 546-7.)
Actually, quite apart from any specific problems with common knowledge of the game, or with
knowledge of the players’ payoff functions, it is not even clear in which sense the usual framework
ensures that players know the rules of the game. To see the problem, recall the specific version of the
framework in Aumann (1995). There, a knowledge system for the PI game ' is defined to consist
of a state space S, a function s from S to the set of strategy profiles for ', and for each player i, a
partitional possibility correspondence Ki from S to S. Such a knowledge system (S, s, (Ki)) is
interpreted as representing the players’ knowledge before the game starts. Apparently, the players are
informally assumed to know the game ', i.e., its rules as given by the game tree and the player
assignment, and the preferences as given by the payoff functions. However, any knowledge system for
a given game ' is formally a knowledge system for many other games as well. To illustrate, consider
the two simple PI games between Ivy and Joe, depicted in Figures 1 and 2.
- 3 -
Clearly, both strategy profiles for '2 are also strategy profiles for '1, and thus any knowledge system
for '2 is formally also a knowledge system for '1. But should we conclude that in such a knowledge
system the players have, informally, knowledge of both games? While this conclusion may seem absurd
at first, note that Ivy might explain the first step of her BI reasoning as follows: “I know that one and
only one of the plays of '1 will come about. I also know that Joe is rational, and thus he will neither
make the move from b to d, nor the one from c to f. Therefore I am actually about to play game '2.”
In a sense, it seems quite reasonable here to say that Ivy knows of both games that they will be played.
Similarly, if we consider a knowledge system for '1 where, at some state T, it is common knowledge
that only strategies of '2 will be taken, it seems quite natural for Ivy to insist at T that '2 is the relevant
game, and that she need not worry about the “extra” moves included in '1. Again, it does not seem
absurd to say that Ivy knows of two distinct games that they will be played.
- 4 In fact, it is easy to say in which precise sense Ivy can know both “'1 will be played” and “'2 will be
played”. If these statements are construed as “one and only one play through the tree of '1 (or '2,
respectively) will be followed”, they correspond to perfectly well-defined events in any knowledge
system for either '1 or '2, and thus the standard formal definition of knowledge can be applied. To be
sure, in many a context games with clearly “redundant” moves may be disregarded, but in our
reconstruction of the BI argument players will be required to know “the rules of the game” simply in the
sense of the event just defined.
b. Taking account of doxastic possibility. By definition, an event E is doxastically possible for agent
i, if he does not know that E is not the case. As the “dual” of knowledge, doxastic possibility can be
expected to play an important role in epistemic conditions for solution concepts. For instance, Dekel
and Gul (1997, p. 124) have recently suggested that doxastic possibility of opponents’ strategies
represents a weak form of players’ “caution”. In this paper, we use assumptions about doxastic
possibility of one’s own moves. Actually, it seems presupposed as a matter of course in informal BI
reasoning, that if a player’s decison node a is reached, it is consistent with his knowledge that he will
make his BI move at a. If one assumed, to the contrary, that the player could have knowledge which
implies that he will not make his BI move at a reached node a, the BI argument obviously cannot go
through. In fact, as with rationality, the informal BI argument seems to rely on something like common
knowledge, or at least iterated mutual knowledge, of the BI moves’ perceived possibility.
Actually, there is also a formally precise sense in which sufficient epistemic conditions for BI must also
entail CK of the BI moves’ possibility. To see this, let I denote the event that game ' is played
according to BI, and assume that CK of an event E* is sufficient for the BI play, symbolically
CK(E*)dI. If the knowledge operators Ki have the usual properties, Id¬Ki(¬I) holds, and thus we
have CK(E*)=CK(CK(E*))dCK(¬Ki(¬I)). This argument can be extended to cover BI moves off
the BI path by defining conditional possibility of a move m at node a as the event “if node a is
reached, the player whose node it is considers it possible to make move m at a”. It is then easily seen
that a sufficient condition for the BI play which has the form CK(E*), must imply CK of conditional
possibility of all BI moves (be they on the BI path or off it).
By itself, the assumption “CK of conditional possibility of the BI moves” is quite weak. For instance,
it is satisfied whenever it is CK that “some path through ' will be played, and any of them is
doxastically possible”. It is also consistent with some players’ having knowledge which allows them to
exclude some or all non-BI moves. What we will show is that its combination with CK of “some path
through ' will be played”, and of a certain weak form of rationality implies the BI play of '.
Before we turn to the required notion of rationality, a word of caution seems appropriate concerning
the standard practice of reading ¬Ki(¬E) as “i considers E possible”. (E.g., Lenzen, 1980; Binmore,
1992; Fagin, Halpern, Moses, and Vardi, 1995; Samet, 1996; Dekel and Gul, 1997). For the sake of
brevity, we will follow that practice, but it is worth noting that doxastic possibility is not the same thing
as perceived “physical” or “objective” possibility. If a player knows he will not make a particular move
m, this move is not doxastically possible for that player, but he may very well continue to believe that
m is possible in some physical or objective sense. Arguably, something like perceived possibility of all
moves of the game must be taken for granted if one assumes “CK of the game”. However, this
- 5 stipulation will not be adopted here, and doxastic possibility of all moves is clearly not compatible with
players’ having sufficient knowledge to rule out some of them.
c. A concept of relative rationality. Consider the following knowledge system for the game of Figure
3: S={8,:,D}, s(8)=l, s(:)=m, s(D)=r, KIvy(8)=KIvy(:)={8,:}, KIvy(D)={D}. Clearly, D is the only
state of this knowledge system where Ivy is rational with respect to '3. However, at state :, Ivy might
explain her situation as follows: “For all that I know, r is not possible. Thus I have only the two options
l and m, whereof I prefer the latter.” If we now observe her choosing m, it seems natural to say that she
chose rationally relative to what she considered possible. In what follows, we will use a concept of
“relative” rationality, which is satisfied at : and D, but not at 8 of our example:
In order to be relatively rational, a player must not take an action if he considers possible an
alternative one, whereof he knows that it implies a higher payoff.
By itself, this condition of relative rationality is quite weak. For instance, it is satisfied at any state where
the player knows what he is going to choose at each of his/her decision nodes. Nevertheless, when
combined with assumptions about doxastic possibility of moves, relative rationality has interesting
implications. For instance, in any knowledge system for '3, relative rationality combined with doxastic
possibility of r implies that r will be played. It is easily seen that this implication continues to hold when
the possibility correspondences are allowed to be non-partitional. By contrast, doxastic possibility of
m and relative rationality do not imply that m is played. (Let S and s be defined as above, but
KIvy(8)=KIvy(:)=KIvy(D)=S. At D, Ivy is relatively rational, but considers m possible.)
d. Avoiding counterfactuals. Using conditional possibility of moves and relative rationality will allow
us to formulate sufficient conditions for the BI play which do not require players to entertain beliefs
about counterfactual or subjunctive conditionals. While it is often claimed that such conditionals are
essential for a discussion of rationality in extensive games, Dekel and Gul (1997), and Arlo-Costa and
Bicchieri (1998) have recently suggested that their importance is more limited than conventional
wisdom holds. We will support this suggestion by showing that (at least a version of) the BI argument
does not depend at all on players’ having beliefs or knowledge about “counterfactuals”.
- 6 To avoid counterfactuals means, however, to avoid the usual notion of strategy, too. After all, to have
a strategy for an extensive game may require one to “plan” moves even for nodes whereof one knows
that they will not be reached. Almost all the literature on epistemic conditions in game theory takes it
for granted without further ado that players must have a strategy for the exogenously given game. (An
exception is Samet, 1996, who derives the existence of strategies from a specific theory of hypothetical
reasoning.) Consequently the notion of rationality is typically defined in terms of strategies, by
comparing the “chosen” strategy with alternative strategies which are “available” in the given game.
However, the usual blackboard demonstration that CK of (a), (b), and (c) must lead to the BI play
does not directly refer to complete strategies at all, but instead proceeds by eliminating individual
moves.
e. A syntactic approach. To bring out clearly that our BI argument does not require players’ to use
or even “understand” conditionals other than the material implication, we will adopt a syntactic
approach. That is, we fully specify the formal system or ‘logic’ we are using - its alphabet, its formation
rules, its axioms, and its rules of inference. Being the standard approach in formal logic, the syntactic
approach is thus somewhat more pedantic than the event-based one, which is better known among
economists. Of course, our results can be easily restated in terms of the usual state-space framework,
and we indicate below how this may be done. However, given that the problems are inherently ones
of logic and conceptual clarification, we feel that the more direct and explicit approach is also more
adequate. A key idea (adopted from Bacharach, 1987, 1994) is that once “the game” and “rationality”
are represented by suitable propositions (instead of set-theoretic structures), one can immediately apply
a standard system of epistemic logic to express the proposition that the game and rationality are known
to the players. With respect to the required propositional representation of an extensive game we build
on the work of Bonanno’s (1991, 1993).
The paper is organized as follows. Section 2 presents the formal system and its event-based
counterpart. Section 3 explains our BI argument, both formally and informally, for the simple example
of Figure 1. For PI games in general, Section 4 gives the definitions needed, and Section 5 the main
results. Section 6 compares our main result to the one of Aumann (1995), Section 7 contains some
discussion, and Section 8 the proofs.
2. The logic KT’
a. The formal system. Throughout, we work in a formal logic KT’ that is defined as follows. The
alphabet of KT’ consists of: nodes: a, b, c,...; players: i, j, k,...; the preference sign: ™;
propositional connectives: v, ¬; the knowledge operator: K. The well-formed formulas of KT’
(wffs for short) are defined thus: If i is a player, and a, b are nodes then the expressions iab and a™ib
are wffs. If n and R are wffs, and i is a player, then ¬n, nvR, and Kin are wffs. By M we denote
the set of all “primitive” wffs, that is, all wffs of either the form iab or the form a™ib.
The intended interpretation of iab is: "player i will move from a to b", the interpretation of Kin is:
"player i knows that n", and a™ib means: "i prefers a to b". We will freely make use of the
propositional connectives w,6,and :, which may be defined in terms of ¬ and v in the usual manner,
- 7 and of brackets with the convention that Ki and ¬ are “stronger” than v and w, and these in turn are
stronger than 6 and :.
Apart from the alphabet and formation rules, our logic KT’ is just the multi-agent version of the familiar
modal logic KT (or T, cf., e.g., Chellas, 1980, Bacharach, 1994, or Fagin, Halpern, Moses, Vardi,
1995). Its rules of inference are:
(MP) If |n, and |n6R, then |R
(E)
If | n then | Kin
(“modus ponens”).
(“epistemization”).
(The “turnstile” | means “it is provable in KT’ that...”) The axioms of KT’ are given by the three
schemes:
(PL)
(K)
(T)
All tautologies of propositional logic.
Ki(n) v Ki(n6R) 6 Ki(R) (“distribution”).
Ki(n) 6 n
(“veridicality”).
The meaning of these rules and axioms should be clear: By (E) we are assuming that players know all
theorems of KT’, while (K) expresses that they know all deductive consequences of whatever they
know. Finally, (T) says that something can be known only if it is the case.
b. Semantics. The formal system KT’ can be interpreted by means of reflexive Kripke models. A
reflexive Kripke model is here taken to be a structure M=(S, (ö:):0N, f), where
S is some non-empty set,
N is the set of all players of KT’,
ö: is a correspondence from S to S, such that T0ö:(T) for all T0S, and
f is a correspondence from S to M .
A wff n is said to be satisfied (or true) in state T0S of model M, symbolically: M,TÖn, according
to the following definition:
For n0M : M,TÖn iff n0f(T);
M,TÖ¬n iff not: M,TÖn;
M,TÖnvR iff both M,TÖn and M,TÖR;
M,TÖK:n iff M,T’Ön for all T’0ö:(T).
Intuitively, f(T) consists of all “primitive” facts which hold in state T, while ö:(T) consists of all states
which are considered possible by :. It is a standard result of modal logic that a wff is a theorem of KT’
if and only if n is satisfied in all states of all reflexive Kripke models (cf. Fagin, et al., 1995, 47-62.)
Given a model M and a wff n, we can define the event [n], or “set of states of M where n holds”,
as follows:
- 8 [n]:={T0S| n0f(T)} for n0M ;
[nvR]:=[n]1[R];
[¬n]:=S([n];
[K :n]:={T0S|ö:(T)d[n]}.
By means of this definition all wffs of KT’ can be translated into the language of events, and a wff n is
a theorem of KT’ iff [n]=S for all reflexive Kripke models. As results about information structures are
often expressed in the form “AdB” in the literature, we add that [n]d[R] is equivalent to
[¬n]c[R]=S.
3. BI in a simple example
a. The structure of the game. For convenience, we reproduce the game of Figure 1, translating the
names of Ivy and Joe into the language of KT’. In this Section, we call it ' for simplicity. In the
language of KT’, the four plays of ' may be represented by the following four wffs:
iabv¬iacvjbdv¬jbev¬jcfv¬jcg
iabv¬iacv¬jbdvjbev¬jcfv¬jcg
¬iabviacv¬jbdv¬jbevjcfv¬jcg
¬iabviacv¬jbdv¬jbev¬jcfvjcg
(p1)
(p2)
(p3)
(p4)
In the spirit of Bonanno(1991, 1993), we take the propositional game form of ' to be the formula
p1wp2wp3 wp4. It expresses that one and only one of the four plays of ' will be played. In addition,
we need the following formula that describes the agents’ preferences, and will be called the preference
formula of ' :
d™ig v d™ie v d™if v g™ie v g™if v e™if v e™jg v e™jd v e™jf v g™jd v g™jf v d™jf
- 9 The Bonanno formula of ' , denoted by G, is defined as the conjunction of its propositional game
form and its preference formula.
b. Conditional possibility of the BI moves. Abbreviating formula jbdwjbe by “(b)”, formula jcfwjcg
by “(c)” , and iabwiac by “(a)”, we define the conditional possibility formula for the BI moves in
' , denoted by CPBI to be the following:
((b)6¬Kj¬jbe) v ((c)6¬Kj¬jcg) v ((a)6¬Ki¬iac).
This expresses the natural assumption that if a particular decision node of ' is reached, the respective
player’s knowledge is consistent with the BI move at that node.
c. Relative rationality in ' requires that the following wffs be satisfied:
e™jd v ¬Kj¬jbe 6 ¬jbd
g™jf v ¬Kj¬jcg 6 ¬jcf
Ki(iab:jbe) v Ki(iac:jcg) v g™ie v ¬Ki(¬iac) 6 ¬iab
Ki(iab:jbe) v Ki(iac:jcf) v e™if v ¬Ki(¬iab) 6 ¬iac.
(In the next Section, we define the relative rationality formula for ' , denoted by RR, to be the
conjunction of these and a number of further similar conditions.)
d. A backward induction argument. We can now explain what our main result says about the simple
example ': The formula Ki(GvCPBIvRR)vKj(GvCPBIvRR)6 p4 is a theorem of (KT’). In words,
if both players know the structure of the game, that both players are relatively rational, and that the BI
moves are conditionally possible, then the game will be played according to BI. In order to emphasize
that we regard the result as a formal reconstruction of the “ordinary” informal BI argument, we now
sketch a formal derivation of p4 from the assumption that the players know GvCPBIvRR, introducing
each step of the derivation with an informal statement of the respective assumption or inference.
Propositional logic and the truth axiom (T) will be used freely.
Assume that Ivy knows that one and only one of '’s plays will be played, and that preferences
are as in '. Formally:
(1)
Ki(G')
(by assumption)
Assume also that Ivy knows that if node b is reached, Joe considers jbe possible. Formally:
(2)
Ki( (b)6 ¬Kj¬jbe)
Moreover, assume that Ivy knows Joe to be relatively rational. Formally:
(by assumption)
- 10 (3)
Ki( e™jd v ¬Kj¬jbe 6 ¬jbd )
(by assumption)
Because of the previous assumptions, and axiom (K), Ivy must know that she will make move
iab if and only if Joe makes the move jbe. Formally:
(4)
Ki( iab : jbe )
(from (1), (2), (3), and axiom (K))
Treating the lower half of the tree analogously, Ivy must know that she makes move iac if and
only if Joe makes move jcg. Formally:
(5)
Ki( (c)6 ¬Kj¬jcg)
(by assumption)
(6)
Ki( g™jf v ¬Kj¬jcg 6 ¬jcf )
(by assumption)
(7)
Ki( iac : jcg )
(from (1), (5), (6), and axiom (K))
Now assume that Ivy knows her own BI move to be possible if node a is reached. Formally:
(8)
(a)6¬Ki¬iac)
(by assumption and axiom (T))
As Ivy knows that node a will be reached, she considers her BI move possible. Formally:
(9)
¬Ki¬iac
(from (1), and the fact that G6(a) is a theorem of KT’)
Assume that Ivy is relatively rational. Formally:
(10)
Ki(iab:jbe) v Ki(iac:jcg) v g™ie v ¬Ki(¬iac) 6 ¬iab
(by assumption)
It follows that Ivy will not make move iab. Formally:
(11)
¬iab
(from (4), (7), (9), (10) by modus ponens)
This means she will play iac.
(12)
iac
(from (11) and (1))
We already know that this happens if and only if Joe plays jcg.
(13)
iac v jcg
(from (12) and (7))
Thus, we can derive the BI play p4 from the assumption Ki(GvCP B I v RR), and a fortiori from
Ki(GvCPBIvRR)vKj(GvCPBIvRR). The deduction theorem for systems of modal logic (cf. Chellas,
1980) ensures that Ki(GvCPBIvRR)vKj(GvCPBIvRR)6p4 is a theorem of (KT’).
- 11 One may wonder what happens if conditional possibility of the BI moves is replaced by conditional
possibility of the moves from some other strategy profile. Let S denote the profile which assigns jbe to
node b, and jcg to node c (as in BI), but iab to node a, and let CPS be defined in analogy to CPBI.
Obviously, the play that results from S is described by p2. However, it follows from a second result
proved below that Ki(GvCPSvRR)vKj(GvCPSvRR)6p2 is not a theorem of (KT’). Instead,
Ki(GvCPSvRR)vKj(GvCPSvRR)vp4 is consistent. To see this, define the following Kripke model.
S:={$,(}, f($):={iab, jbe}cA, where A={d™ig, d™ie,..., g™jf, d™jf} contains all primitive facts
about preferences according to ', f(():={iac, jcg}cA, and öi($)=öj($)={$}, öi(()=öj(()=S.
It is easy to check that Ki(GvCPSvRR)vKj(GvCPSvRR)vp4 is satisfied in state (. One can think
of this state as a world where the players know that Joe will only make moves of his BI strategy, and
where Ivy makes her BI move iac, but does not know - before the game starts - that she will do this.
Instead, she considers it possible that she will “feel forced” to choose iab.
4. Definitions
We limit ourselves to perfect information games with more than one play, without moves by nature, and
assume that the payoffs are in general position. (I.e., for any player any two distinct terminal nodes have
different payoffs.) For extensive games of this class we use the abbrevation PI game.
Definition 1. Let ' be a PI game. For each non-terminal node a of ' we define a's node formula as
follows. If i is the player who moves at node a, and b1,...,bm are the immediate successors of a, the wff
(iab1wiab2w...wiabm) v ¬(iab1viab2) v ... v ¬(iabm-1viabm)
is called a's node formula, if a is the origin of '; if a has an immediate predecessor c, and j is the player
who moves at c, then a's node formula is
(jca:(iab1wiab2w...wiabm)) v ¬(iab1viab2) v ... v ¬(iabm-1viabm).
The propositional game form of ' is the conjunction of all its node formulas.
Definition 2. Let V denote the set of all nodes of ', and o its origin. For any node b0V({o}, let :(b)
stand for the formula iab, where a is the immediate predecessor of b, and i the player who makes a
move at a. If t is a terminal node of ', let VtdV consist of all nodes on the path from o to t (including
o and t). A play formula of ' is a wff of the form
Remark. If M is the number of plays of ', and n1,...,nM the corresponding play formulas, the formula
- 12 -
n1 w n2 w...w nM,
is logically equivalent to the propositional game form of '. (It is its so-called disjunctive normal form.
See, e.g., Hamilton, 1988, p.17.)
Definition 3. If t1, t2 are distinct terminal nodes of ', and Bk(t1), Bk(t2) are the respective payoffs for
player k, then his comparison formula for {t1, t2} is t1™kt2, if Bk(t1)>Bk(t2), but t2™kt1 if Bk(t2)>Bk(t1).
The preference formula of ' is the conjunction of all its comparison formulas (for all players of ',
and all sets of two distinct terminal nodes of ').
(As we assume payoffs to be in general position, comparison formulas are well-defined. As we assume
that ' has distinct terminal nodes, there is at least one comparison formula.)
Definition 4. The Bonanno formula of ' , denoted by G', is the conjunction of its propositional game
form and its preference formula.
Definition 5. If i, j, k are players of ', if a, b, c, d, e are nodes of ', and t1, t2 are terminal nodes of
' such that (a,b) and (a,c) are moves of player i, while (d,t1) and (e,t2 ) are moves of players j and k,
respectively, then the following formula is a relative rationality condition for player i in ' :
Ki[(iab:jdt1)v(iac:ket2)] v t1™it2 v ¬Ki(¬iab) 6 ¬iac.
The relative rationality formula for i in ' , denoted by RRi,', is the conjunction of all such rationality
conditions. The relative rationality formula for ' , denoted by RR' is the conjunction of all the
RRi,'s.
The notion of m-th order mutual knowledge of a wff n is defined as follows.
Definition 6: If i1,...,in are the players in ', then the mutual knowledge formula for n in ' , denoted
by E'(n), is
Ki1n v Ki2n v...v Kinn,
the m-th order mutual knowledge formula for n in ' , E'm(n), is defined recursively as follows:
E'1(n) = E'(n),
E'm(n) = E'(E'm-1(n)) for m>1.
For convenience, we define E'0(n)=n.
Definition 7. Let ' be a PI game. For each non-terminal node a of ' we define conditional
possibility formulas for the moves at a as follows. If i is the player who moves at node a, and
b1,...,bm are the immediate successors of a, the wff
- 13 (iab1wiab2w...wiabm) 6 ¬Ki(¬iab:)
is the conditional possibility formula for move iab:. If S is a strategy profile for ', CP'(S) will
denote the conjunction of all conditional possibility formulas for moves which are prescribed at some
(possibly unreached) node by S. We will write CP'(BI) with the understanding that “BI” here stands
for the backwards induction strategy profile of the game '.
5. Results
Let ' be an arbitrary PI game, and let m be the maximal number of moves in a play of '. Our first
observation is that mutual knowledge of the Bonanno formula for ' and relative rationality places no
restriction whatsoever on the play of '. Formally:
Observation 1. For any move iab in ' and any number n, the formula
E'n(G' v RR' ) v iab
is consistent in KT’ (that is, its negation is not provable in KT’).
The intuitive reason for this result can be explained as follows: Knowing that one of the plays of ' will
be played is perfectly compatible with knowing that an arbitrary particular one of these plays will be
played. If the players know, for instance, that they will be physically forced to follow a particular nonBI path, they will do so and still be relatively rational.
Our main theorem shows that the situation changes dramatically, when the players have mutual
knowledge of not only the Bonanno formula and relative rationality, but also of conditional possibility
of the BI moves. If the order of mutual knowledge is sufficiently high, the BI play of ' logically follows.
Theorem 1. For any move iab on the BI path of ':
| E'm-1(G' v RR' v CP'(BI)) 6 iab.
It might perhaps be suspected that an analogous theorem holds for any strategy profile. This is not the
case, however. For PI games where each play contains at most one decision node of each player, we
can show that Theorem 1 holds only for the BI profile.
Theorem 2. Let ' be a PI game where each play contains at most one decision node per player.
Let S be a strategy profile for ', n an arbitrary number, and P a play formula of '. The formula
E'n(G' v RR' v CP'(S)) 6 P
is a theorem of KT’ only if P is the BI-play formula of '.
- 14 For the class of games considered, it follows immediately that the antecedent of the formula in Theorem
2 is consistent. In fact, a Kripke model with just one state, and play according to profile S shows that
this is true for arbitrary PI games.
Observation 2. For any strategy profile S (for '), and any number n,
the formula E'n(G' v RR' v CP'(S)) is consistent in KT’.
Informally, we can rephrase the above results as follows. It is consistent to assume that players have
CK of the game, and of relative rationality, and of conditional possibility of the moves from some
arbitrary strategy profile. For the games considered in Theorem 2, an assumption of this form is either
too weak (in KT’) to imply a particular play, or it implies the BI play.
Now, as the rule of epistemization ensures that players know Theorem 1, mutual knowledge of its
antecedent implies that they will foreknow the BI play. In this case, some moves will not be doxastically
possible - and not even conditionally so. It follows that we cannot replace, in Observation 2, CP'(S)
by conditional possibility of all moves. To make this consideration precise, note that we know from
Theorem 1, that E'm-1(G' v RR' v CP'(BI)) implies the BI play, but it need not imply knowledge
thereof. However, as soon as we assume m-th order mutual knowledge of G'v RR' vCP'(BI), it
follows that players know what will happen, and what will not happen in '.
Corollary to Theorem 1. Let ' be an arbitrary PI game, and let j be a player of '. For any
move iab on the BI path of ':
| E'm(G' v RR'v CP'(BI)) 6 Kj(iab),
and for any move iab off the BI path of ':
| E'm(G' v RR' v CP'(BI)) 6 Kj(¬iab).
Now, let CP' denote the conjunction of all conditional possibility formulas for moves of '. As
Kj(¬iab) contradicts conditional possibility of iab, whenever node a is reached, we have
Theorem 3. For any PI game ', the formula E'm(G' v CP' v RR' ) is inconsistent in KT’.
Clearly, mutual knowledge of G' v CP' as such is not inconsistent.
Observation 3. For any number n, the formula E'n(G' v CP' ) is consistent in KT’.
6. Relation to Aumann’s result
As our Theorem 1 provides a sufficient condition for the BI play, it is instructive to compare it to
Aumann’s (1995) result. It turns out that our condition is weaker than Aumann’s. To see this, first recall
that Aumann only considers knowledge systems (S, s, (Ki)) for a given game ', where s is a function
- 15 that assigns to each T0S the profile s(T) of players’ strategies at T. In this subsection, we
accordingly keep the game fixed, and translate the formulas of our syntactical approach into events, i.e.
subsets of S. For events A and B we write ~A for S(A, and A/B for the event (~AcB)1(~BcA)
(which can also be written (A1B)c(~A1~B)). In this subsection only, we write KiA for the event
{T0S| Ki(T)dA}. For the given game, let Vi denote the set of player i’s decision nodes, Z the set of
terminal nodes, and for each node a, let " (a) be the set of a’s immediate successors. We write Gi for
the set of player i’s strategies, and Bia(s) for player i’s payoff in the subgame with origin a when it is
played according to strategy profile s. For a node a, [a] denotes the event “node a is reached”, i.e., the
set of all T0S such that s(T) induces a play through a. For a strategy si0Gi, we follow Aumann in
writing [s i=si] for {T0S|s i(T )=si}, and [Bia(s;si)>Bia(s)] for {T0S|Bia(s -i(T);si)>Bia(s(T))}.
Aumann’s condition for BI is made up of two elements. He assumes that each player knows his own
strategy. This corresponds to the event
Moreover, “substantive” rationality is given by the event
Defining the common knowledge operator CK as usual, and denoting by I the event that the BI path is
played, Aumann’s result can be expressed as follows:
CK(A1R) d I
By contrast, our relative rationality conditions translate into events of the form
where a and a’ are immediate successors of some decision node of player i, and t’ is a terminal node
where i gets a lower payoff than at t. Thus our relative rationality formula corresponds to the event
In order to express conditional possibility of the BI moves, we write 4(v) for that immediate successor
of node v which is chosen at v according to the BI profile. Conditional possibility of BI moves is then
- 16 represented by the event
Our Theorem 1 now boils down to
CK(RR1CPBI) d I
It is perhaps not obvious that Aumann’s condition for BI is stronger than ours.
Theorem 4.
CK(A1R) d CK(RR1CPBI) d I
Actually, for most games ' one can find a knowledge system where the left inclusion is strict. (Less
interestingly, the same is true for the right inclusion.) For instance, consider the example of Figure 1
again, and a knowledge system with just one state T, where Ivy has the strategy c, and Joe has the
strategy (d;g). It is easy to check for this example that {T}=CK(RR1CPBI), but R=i. Aumann’s
condition of “substantive” rationality is violated, because Joe knowingly continues with a strategy that
would yield him less than he could have gotten with a different strategy, if node b, whereof he knows
that it will not be reached, were reached.
7. Discussion
a. Counterfactuals and the standard, informal BI argument. The main “message” of the present
paper runs counter to many authoritative assertions about the unavoidability of counterfactual reasoning
in BI: We have given a sufficient condition for BI play in a fully explicit formal language which contains
only the truth-functional connectives, and normal (Kripkean) knowledge operators. Counterfactual
conditionals do not appear in our epistemic condition for BI - in fact, they cannot even be expressed
in the language of KT’. One of our reasons for trying to avoid counterfactuals was that the meaning and
logic of them is highly controversial. Actually, this is true to such an extent that we have to add a caveat
to our claim that our formal language cannot express counterfactual conditionals. To wit, one might
insist that Ki(n6R) does express a counterfactual of sorts. After all, it might be read as “if n were the
case, then, for all that i knows, R would hold”. Clearly, this conditional may be false even if its
antecedent, n , is false, too, and this seems to be almost the only uncontroversial feature of
counterfactual conditionals. To the extent that this feature is regarded as a sufficient reason for counting
Ki(n6R) as a counterfactual, we admit that our reconstruction of the BI argument does make use of
counterfactuals.
However, we hope to have shown that our version of the BI argument does not require any of the more
elaborate notions of counterfactual conditional that have been suggested in the literature. This is not to
deny that other interesting sufficient conditions for BI make essential use of such counterfactuals. For
instance, this is undeniably the case for Aumann’s condition, which not only implies the BI play, but also
- 17 the clearly stronger conclusion that all players adopt their respective BI strategies. However, we feel
that the “standard” informal account of BI as it is presented in countless introductory courses in
economics or game theory, has little need for counterfactuals, or, in fact, for the notion of strategy. After
all, it proceeds by eliminating individual moves from the game tree, and then considering the result
of the pruning as a new game tree, where again some moves can be eliminated on the basis of
rationality considerations, and so on. (Our claim that the “classroom” BI argument does not depend on
the counterfactual interpretation of strategies is also buttressed by the observation that this argument can
be convincingly taught to students before introducing them to the notion of a strategy for an extensive
game.)
Inspection of the proof of Theorem 1, exemplified in Section 3 above, should also convince the reader
that it is essentially a formal restatement of the familiar, but informal BI reasoning. However, it may well
be that a similar proof can be given with still weaker epistemic conditions than the one of Theorem 1.
For instance, the example of Section 3 shows that the BI argument does not require the full force of
mutual knowledge of the players’ rationality. In that example, it was neither assumed that players have
knowledge of their own rationality, nor that the second player knows about the first player’s rationality.
This indicates that - instead of mutual or even common knowledge - the BI argument only requires
some kind of ‘forward knowledge’ of later players’ knowledge of the game and rationality. This idea
has been studied in different settings by Balkenborg and Winter (1997), Rabinowicz (1998), and Sobel
(1998).
b. Modifying the logic. By working with the logic KT’, we have shown that the BI argument does not
depend on positive or negative introspection. Clearly, our Theorem 1 remains valid when a logic with
additional axioms - such as S5 - is used. However, it has been pointed out by Stalnaker (1996) that the
combination of negative introspection and the truth axiom rules out mistaken beliefs: If an agent i
believes that n, but n is actually false, the truth axiom requires that we must have ¬Ki(n), and by
negative introspection we get Ki(¬Ki(n)). But, given that the agent believes n, it seems he should at
least consider it possible that he knows n.
Clearly, even KT’ has a concept of knowledge that may seem overly strong. In particular, it seems that
the full force of the truth axiom is not required for the BI argument: As long as it is explicitly assumed
that relative rationality and possibility of BI moves hold along the actual path, it does not seem to
matter for the argument whether the other beliefs are correct or not. We therefore conjecture that a
suitable modification of Theorem 1 is valid even in the weakest “Kripkean” epistemic logic K.
Even the epistemic logic K stipulates in its multi-agent version that all theorems are commonly known,
and that players’ ability to draw all inferences from whatever they know, is commonly known, too. To
the extent that these stipulations are unrealistic in applications, it is an interesting open question how
much of the BI argument remains valid in less demanding systems of multi-agent epistemic logic. (Cf.
Lismont and Mongin, 1994, for a discussion of common belief and common knowledge in various
logical systems. In Vilks, 1997, 1998, we explore the idea of distinguishing formally between different
stages in a player’s reasoning process.)
c. Replacing doxastic possibility by known feasibility. Our notion of relative rationality bears a close
resemblance to the textbook definition of economic rationality, according to which rationality means
- 18 choice of a best alternative from the set of feasible ones (cf., for instance, Gravelle and Rees, 1992,
pp. 6-7). In fact, the two notions of rationality would coincide if the notions of feasibility and doxastic
possibility did. In spite of its apparently fundamental role for economic theory, the notion of feasibility
is strangely ill-explored. However, it seems safe to say that most economists think of feasibility as of
something more objective than doxastic possibility. Moreover, if doxastic possibility and feasibility were
identified, our Theorem 3 might appear somewhat paradoxical. After all, many game theorists would
like to insist on the logical possibility of players having mutual knowledge (of arbitrarily high order) of
the feasibility of all moves, rationality, and the game.
In fact, it may seem that a simple and straightforward extension of our logic allows us to distinguish
formally between what is doxastically possible and what is known to be feasible, and that this might
suffice to get rid of the inconsistency result. All that is required seems to be a formal system which has
an standard ontic modality •, representing “objective feasibility”, in addition to the doxastic modalitites
of KT’. Let us consider such an extended system KT* with mixed modalities. In addition to the axioms
of KT’, KT* will have the following two axioms schemes: n6•n, and ~nv~(n6R)6~R, where
~n:=¬•¬n. Moreover, KT* has one additional rule of inference: If |n, then |~n.
In KT*, the assumption “player i knows that iab is feasible” could now be formally expressed as
Ki•iab, and arguably this formula should replace ¬Ki¬iab both in our relative rationality conditions,
and in our conditional possibility formulas. Define RR'* as the wff of KT* that results from RR' by
replacing all its subformulas of the form ¬Ki¬iab by the corresponding Ki•iab. Define CP'(BI)* and
CP'* similarly. It now seems that the proof of Theorem 1 should go through analogously for the starred
versions of rationality and feasibility. In fact, it does, but an analogue of Theorem 3 still holds. We do
not attempt a general treatment here, but limit ourselves to the example of Figure 4.
For the game ' of Figure 4, we get, in KT*, the theorem E2(G'vRR'*vCP'*)6(iacvjcg). From this,
we infer E3(G'vRR'*vCP'*)6Ki(iac:jcg). However, we also get the theorem
E2(G'vRR'*vCP'*)6(¬iabv¬jbd), and from this we infer E3(G'vRR'*vCP'*)6Ki(iab:jbd). But
RR'* includes the following (starred) rationality condition:
Ki[(iab:jbd)v(iac:jcg)] v d™ig v Ki•iab 6 ¬iac
and CP'* includes (iabwiac)6Ki•iab, the antecedent of which is implied by G'. Thus we get, in KT*,
the following theorem:
E3(G'vRR'*vCP'*)6(iacv¬iac).
The argument here is similar to ‘Bonanno’s paradox’ (described in Bonanno, 1991, and discussed by
Dekel and Gul, 1997, and Vilks, 1997). One has the impression that the starred rationality conditions
should not be used for moves which are known to be equivalent only because it is known that they will
not be made. To put it differently: If a player knows - without the possibility of error - that he will not
make a particular move, he should no longer judge the merits of other moves by comparing them to it.
It is precisely such irrelevant comparisons which give rise to Bonanno’s paradox, and its epistemic
variant considered here. By contrast, requiring doxastic possibility instead of known feasibility rules out
such irrelevant comparisons.
- 19 At any rate, we have to conclude that the logical consistency of ‘CK of the game, rationality, and
possibility of all moves’ cannot be restored by simply replacing doxastic possibility by some
independent notion of feasibility. To be sure, more far-reaching modifications of our conditions for BI
and/or of the logic used may lead to different results. For instance, Wlodek Rabinowicz has suggested
(in personal communication) that one might replace our rationality conditions by conditions of the form
Ki[(iab´jbd)v(iac´jcg)] v d™ig v Ki•iab 6 ¬iac,
where ´ stands for some strong - perhaps counterfactual - conditional. As the main aim of the present
paper is to show that a BI argument very close to the standard, informal one can be formulated without
resorting to counterfactual implications, we leave an analysis of this idea to some future occasion.
After all, Theorem 3 is not in any way paradoxical unless doxastic possibility is identified with feasibility.
It highlights that assumptions about what players know and what they do not know may easily conflict
in a logic which stipulates that all logical consequences of whatever is known must already be also
known. In this respect it is quite in line with arguments such as those by Bicchieri (1989) or Reny
(1992, 1993).
Incidentally, the fact that the antecedent of Theorem 1 does not require players to believe that the nonBI moves are possible, shows that the BI argument remains valid when some players know that some
non-BI moves will not be made because they are physically impossible.
d. What is “the game to be played”? In our reconstruction of the BI argument, we have construed
“the rules of the game '” as specifiying only that exactly one path through the tree of ' will be played.
As we pointed out in the introduction this construal means that “the game to be played” is not
necessarily unique, but that instead the rules of various different games may hold at the same time, and
for the same players. For instance, coming back to the example of the introduction, we can now say
that Ivy and Joe may very well have mutual knowledge of the Bonanno propositions of both the game
of Figure 1, and that of Figure 2.
This non-uniqueness of “the game to be played” certainly seems at variance with some of game
theorists’ habits of speech. Thus, for instance, the definite description in the very phrase “the game to
be played” suggests that there can be only one such game. However, there are also habits of speech
which suggest that one can say of different game trees that they represent rules that must be followed.
For instance, when a subgame of a given tree ' is assumed to be reached, it seems natural to assume
that the players know that the rules of the subgame must be followed. This assumption does not seem
to conflict with also assuming that the players still know the structure of the original game '. If one
follows the rules of the subtree, one can hardly violate the rules of '. Similarly, if two chess players
have agreed to play the opening Ruy Lopez, it seems natural to assume that they are about to play Ruy
Lopez, and at the same time that they are about to play chess.
Given that there is no established way of formally representing the players’ knowledge of the structure
of the game, the Bonanno proposition at least seems to be a promising candidate, and as we hope to
have shown, in some cases it is all that is needed.
- 20 8. Proofs
Proof of Observation 1. Fix an arbitrary move iab, and an arbitrary play Y of ' that contains iab. In
order to show that the formula E' n (G' v RR' ) v iab is consistent in KT’, we define a reflexive
Kripke model M=(S, (ö:):0N), f) for which the formula is valid (=satisfied in all states). Intuitively,
it will interpret the players as knowing that Y will be played. To define M formally, let S={s0}, and
ö: (s0)={s0} for all :. The facts correspondence f is defined as follows: For moves :xy let :xy0f(s0)
iff :xy belongs toY, and let y™:y’0f(s0) iff y and y’ are terminal nodes of ' such that B:(y)>B:(y’).
One can now check that the formula E'n(G' v RR') v iab is valid in M. QED.
Observations 2 and 3 are proved analogously.
Theorem 1 is an immediate corollary of the following lemma. For its formulation and proof we keep the
given PI game fixed, accordingly suppress the subscript ', and introduce the notation $(a) for the
terminal node that ends the subgame of ' with origin a if it is played according to backwards induction.
If iab is player i’s backward induction choice at a, it follows that $(a)=$(b).
Lemma. Let a be a non-terminal node that is immediately preceded by player j’s node p, and let m be
the maximal length of the subgame with origin a. Then
| Em-1(G v RR v CP(BI)) 6 (jpa:kq$(a)),
where q is the node immediately preceding $(a), and k is the player who has to move at q.
Proof. By induction on m. For m=1, the node formula of a implies jpa:(iat1 wiat2w...wiath), where t1,
t2,..., th are the terminal nodes following a. Thus we have
| GvRRvCP(BI) 6 (jpa:(iat1 wiat2w...wiath))
By (PL), (E), and (MP) we have, for x=1,...,h:
| GvRRvCP(BI) 6 (jpa 6 Ki(iatx :iatx))
Without loss of generality, we can assume that h>1 and $(a)=t 1 . For x>1, the preference formula
implies t1™i tx. Hence
| GvRRvCP(BI) 6 (jpa 6 (t1™i tx))
Moreover
| GvRRvCP(BI) 6 (jpa 6 ¬Ki(¬iat1))
Combining the last three assertions, we get
- 21 -
| GvRRvCP(BI) 6 (jpa 6 [K i(iat1 : iat1)vKi(iatx :iatx)v (t1™i tx)v ¬Ki(¬iat1)])
Now we use RR to get
| GvRRvCP(BI) 6 (jpa 6 ¬iatx)
As this holds for all x>1, we can combine these h-1 assertions with the first one of the proof to get
| GvRRvCP(BI) 6 (jpa 6 iat1)
and as G implies iat16jpa, we get the required
| GvRRvCP(BI) 6 (jpa : ia$(a)).
Now let m>1, and assume the Lemma to be true for all subgames shorter than m. Now a’s node
formula implies jpa:(iac1 wiac2w...wiach), where c1, c2,..., ch are the immediate successors of a. Thus,
using (T), we have
| Em-1(GvRRvCP(BI)) 6 (jpa:(iac1 wiac2w...wiach))
By the induction hypothesis we have, for x=1,...,h:
| Em-2(GvRRvCP(BI)) 6 (iacx : kxqx$(cx))
By (E) and (K) we get
| Em-1(GvRRvCP(BI)) 6 E(iacx : kxqx$(cx)).
Hence
| Em-1(GvRRvCP(BI)) 6 (jpa6Ki(iacx : kxqx$(cx)))
If iac1 is the BI choice at a, and x>1, the preference formula implies $(c1)™i $(cx). Hence
| Em-1(GvRRvCP(BI)) 6 (jpa 6 ($(c1)™i $(cx)))
Moreover
| Em-1(GvRRvCP(BI)) 6 (jpa 6 ¬Ki(¬iac1))
Combining the assertions, we get
| Em-1(GvRRvCP(BI)) 6 (jpa 6 [K i(iac1 : k1q1 $ (c 1))vKi(iacx:kxqx$(cx))v($(c1)™i $(cx))v
¬Ki(¬iac1)])
- 22 Again we use RR to get
| Em-1(GvRRvCP(BI)) 6 (jpa 6 ¬iacx),
and as in the base step we get
| Em-1(GvRRvCP(BI)) 6 (jpa : iac1)
Using the induction hypothesis again, and remembering that $(a)=$(c1), we get
| Em-1(GvRRvCP(BI)) 6 (jpa :kq$(a)). QED.
Proof of Theorem 1. Let o be the origin of ', and i the player who moves at o. We have
| Em-1(GvRRvCP(BI)) 6 (ioc1wioc2w...wioch),
where, again, c1, c2,..., ch are the immediate successors of o. Assume that ioc1 is the BI choice at o.
From the Lemma it follows by (E) and (K) that, for all x=1,...h,
| Em-1(GvRRvCP(BI)) 6 Ki(iotx:kxqx$(tx).
Hence, for all x>1:
| Em-1(GvRRvCP(BI)) 6
[K i(ioc1: k1q1$(c1)vKi(iocx : kxqx$(cx))v ($(c1)™i $(cx))v
¬Ki(¬ioc1)].
We conclude that, for all x>1,
| Em-1(GvRRvCP(BI)) 6 ¬iocx,
and combine these h-1 assertions to get
| Em-1(GvRRvCP(BI)) 6 ioc1.
Using the Lemma again, it follows that
| Em-1(GvRRvCP(BI)) 6 kq$(c1),
and as Gvkq$(c1) implies any move on the BI path, we have completed the proof. QED.
Proof of Theorem 2. Let BIP denote the BI-play formula. It suffices to show that for any profile S, the
formula En(GvRRvCP(S)) v BIP is satisfied at some state of some reflexive Kripke structure. Fix an
arbitrary profile S. The required Kripke structure can be defined as follows: S is the set of those plays
of ' which contain only moves from S or the BI profile. For any such T0S, define f(T) to consist of
all iab such that player i’s move from a to b belongs to T, and all t™it’ such that Bi(t)>Bi(t’) according
- 23 to i’s payoff function Bi(.). Player i’s possibility correspondence öi is defined as follows (plays are
considered as sets of nodes, and player i’s strategies Si(.) and BIi(.) as mappings into the set of nodes):
öi(T):=S({T’| i owns a node a0T1T’ such that Si(a)0T, BIi(a)0T’, and Si(a)…BIi(a)}
I.e., a player i considers all plays (made up from S-moves and/or BI-moves) possible, except that in
a state T, where he makes a move at some node a which is prescribed by S, but not by the BI profile,
he does not consider possible that this node a is reached, and he makes his BI choice there. Intuitively,
when i plays according to S, he does not consider it possible to choose otherwise.
It is now easy to check that the formula GvRRvCP(S) is satisfied at every state of S. This is clear for
G and for CP(S). To check validity of RR, consider an arbitrary relative rationality condition
Ki[(iab:jdt1)v(iac:ket2)] v t1™it2 v ¬Ki(¬iab) 6 ¬iac.
If node a is not reached by the play T, the condition is satisfied because ¬iac is. If iac is played at T,
then c=Si(a) or c=BIi(a). In the first case, öi(T) contains no T’ with a0T’ and c … S i(a). Hence,
¬Ki(¬iab) is satisfied only for b=c. But then Ki[(iab:jdt1)v(iac:ket2)] can only be satisfied if both jdt1
and ket2 are. By construction of the states of S this implies t1=t2, and this implies that the formula t1™it2
is not satisfied. Thus the relative rationality condition is satisfied. In the second case, where iac is played
at T, and c=BIi(a), the same reasoning applies for b=c. For b…c, ¬Ki(¬iab) is satisfied only if b=Si(a).
As öi(T)=S, the formula Ki[(iab:jdt1)v(iac:ket2)] can only be satisfied if all plays from S which
go through b, end with t1, and all plays which go through c, end with t2. In particular, the BI-play of the
subgame with origin b must end with t1, and the BI-play of the subgame with origin c must end with t2.
As c=BIi(a), this means that Bi(t2)>Bi(t1), and hence t1™it2 is not satisfied at T. Thus RR, and hence
GvRRvCP(S), is satisfied throughout S. At the BI-play T*, En(GvRRvCP(S)) v BIP is satisfied for
arbitrary n. QED.
Proof of the Corollary to Theorem 1. For the first part, apply rule (E), and Definition 5 to the wff in
Theorem 2 to get
| E'[E'm-1(G' v CP'(BI) v RR') 6 iab].
As
E'(iab))
| E'[E'm-1(G' v CP ' (BI) v RR') 6 iab] 6 (E'[E'm-1(G' v CP'(BI) v RR')] 6
can be derived from (K), Definition 5, and propositional logic, (MP) gives
| E'[E'm-1(G' v CP'(BI) v RR')] 6 E'(iab)
which implies the assertion.
For the second part, let BIP denote the conjunction of all moves on the BI path. For an arbitrary move
iab off the BI path, G'vBIP6¬iab is a tautology. Hence, by (E), (K), and (MP):
- 24 -
| Kj(G'vBIP) 6 Kj(¬iab).
As the first part of the corollary implies
| E'm(G' v CP'(BI) v RR') 6 Kj(G'vBIP),
the second assertion of the corollary follows. QED.
Proof of Theorem 3. Let a be the first node of ' at which there is more than one alternative, and let
iab be a move other than that prescribed by BI. We then have | G' v F' 6 ¬Ki(¬iab), and thus
| E'm(G' v F' v RR') 6 ¬Ki(¬iab).
As
| E'm(G' v F' v RR') 6 E'm(G' v CP'(BI) v RR'),
we can use the corollary to Theorem 2 to get:
| E'm(G' v CP' v RR') 6 (¬Ki(¬iab) v Ki(¬iab)). QED.
Proof of Theorem 4. First, we show that AdRR. Assume to the contrary that there is an T0A such
that TóRR. Thus some relative rationality condition must be violated at T. I.e., there must be a player
i, a node v0Vi, nodes a and a’ from " (v), and terminal nodes t, t’ such that Bi(t’)<Bi(t), and
From
and Bi(t’)<Bi(t) it follows that a and a’ must be distinct. But then it follows from T0[a’]1A that
T0Ki(~[a]) which contradicts T0~Ki(~[a]). Thus we have proved AdRR, and hence
CK(A1R)dCK(RR). Next, we observe that IdCPBI. For whenever node v is reached at T0I, so is
4(v), and as [4(v)]d~Ki~[4(v)], T0I implies T0~[v]c~Ki~[4(v)]. Thus Aumann’s result
CK(A1R)dI implies CK(A1R)dCPBI, and fromthis we infer CK(A1R)dCK(CPBI). As we already
know that CK(A 1 R) d CK(RR), it follows that CK(A 1 R) is a subset of
CK(RR)1CK(CPBI)=CK(RR1CPBI). QED.
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