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Dynamic consistency, consequentialism and reduction of compound lotteries
economics letters Economics
ELSEVIER
Letters
46 (1994) 121-129
Dynamic consistency, consequentialism compound lotteries
and reduction
of
Oscar Volij * CentER for Economic
Research,
PO Box 90153, 5000 LE Tilburg,
Netherlands
Received 10 February 1994; accepted 24 February 1994
Abstract We use Karni and Schmeidler’s (Journal of Economic Theory, 1991, 54, 401-408) formalization of the concepts of dynamic consistency, consequentialism and reduction of compound lotteries in atemporal sequential decisions to show that given dynamic consistency and any of the other two concepts, the third is equivalent to the independence axiom of choice under risk in one-stage decision problems. JEL classification:
D81
1. Introduction Much has been written for and against the expected utility theory of choice under risk in general, and about the independence axiom in particular. One of the most formidable arguments against non-expected utility models is that non-expected utility maximizers behave in a dynamically inconsistent manner. Some variations of the argument state that this kind of behavior is subject to manipulation [see Green (1987) and Yaari (1985)]. Other versions assert that the value of information to individuals who do not maximize expected utility may be negative in some special cases [see, for example, Schlee (1990) and Wakker (1988)]. In a way, these two versions question the rationality of non-expected utility maximizers. Both the possibility of being systematically exploited and the fact that the value of information may be negative, contradict our intuitive notion of rational behavior. However, the independence axiom is a property of preferences over one-stage lotteries, that is, lotteries whose prizes are basic consequences, while the validity of the previous claims rests on a sequential decision setup where the individual is supposed to choose among compound lotteries. Hence, it is not so clear that non-expected utility maximizers behave irrationally. On the other hand, * This paper presents some elements of the first chapter Ph.D. seminar in Jerusalem for their comments. 0165-1765/94/$07.00 0 1994 Elsevier SSDZ: 0165-1765(94)00456-C
Science
of my Ph.D.
B.V. All rights reserved
thesis.
I wish to thank all
participants of the
122
0. Volij I Economics
Letters 46 (1994)
121-129
Hammond (1988a,b), McClennen (1990), Karni and Schmeidler (1991) and Machina (1989) made it clear that there is some connection between choice behavior among one-stage lotteries and choice behavior in sequential decision problems. Specifically, Karni and Schmeidler (1991) showed that if a choice behavior in sequential decision problems satisfies consequentialism, dynamic consistency and reduction of compound lotteries, then the induced behavior or in other words the induced preference relation over on one-stage choice problems, one-stage lotteries, satisfies the independence axiom. The importance of this result lies in the fact that it makes it clear that a model of choice behavior under risk that assumes away the independence axiom must specify which of the three properties of sequential behavior mentioned above is not satisfied. Roughly speaking, consequentialism means that choice behavior under risk is not influenced by the past or by risks already borne. Dynamic consistency means that planned behavior equals actual behavior even after reconsidering the plan before carrying out each action. Reduction of compound lotteries means that the ultimate probabilities of the basic outcomes are important for making a choice and not the way those probabilities are obtained. Even though these three properties are restrictions on individual behavior, they define three different approaches to extending the basic concepts in game theory to games in which players’ preferences do not satisfy the independence axiom. For example, Dekel et al. (1991) extended the notion of Nash equilibrium in normal form games to the case in which players do not satisfy the reduction of compound lotteries axiom. Crawford (1990), on the other hand, extended the Nash equilibrium concept and developed a new equilibrium concept for a case which is consistent with players not satisfying dynamic consistency or consequentialism. Fershtman et al. (1991) used a model with dynamically inconsistent players to explain delay in contracts. Since it is not clear at all which is the most appropriate way to model non-expected behavior, it is important to further analyze the relations among the different axioms. It turns out that given any two of the properties of choice behavior in sequential decision problems, the third one is equivalent to independence. Karni and Schmeidler (1991) showed that in the presence of consequentialism and reduction of compound lotteries, dynamic consistency is equivalent to independence. What I do in this paper is to show the other two relations, namely that consequentialism and reduction of compound lotteries are each equivalent to the independence axiom if the other two properties are also present, respectively. The paper is organized as follows. Section 2 gives the basic definitions, section 3 presents the results and section 4 concludes.
2. Basic definitions 1 We follow Karni and Schmeidler in describing a sequential decision as a compound lottery and not as a decision tree and an act that assigns to each node in it a probability distribution to the set of its immediate successors. It may be helpful, however, to think of a compound lottery as the unique one that corresponds to some decision tree and act. Let X be a set of basic consequences. Given a set S, A(,S) denotes the set of all simple probability measures on the ’ This part is based
on Karni
and Schmeidler
(1991).
123
0. Volij I Economics Letters 46 (1994) 121-129
family of subsets of S. Define inductively for k > 0, A”(X) = A(A”-l(X)), where A”(X) = X. lotteries of any number of Define also T(X) = U k_,oAk(X). T(X) is the set of all compound stages on X. We want to identify any compound lottery y E A”(X) with 6, E Ak+‘(X), where 6, is the probability U
measure
concentrated on lottery y. As a result we have that for all k, Ak+‘(X) 2 Tk(X) = {y ET(X) 1y E Ak(X)\dk-‘(X)}, then we have T(X) =
F=,A’(X). If we define
uk>”
rk(X)*
Given two compound lotteries, y and z, by z Ey we mean that if the compound lottery y is played out, there is a positive probability that at some stage the compound lottery z will be obtained. Formally, z E y if z = y or if there are k > m L 0 s.t. y E Tk(X), z E rm(X) and for some lljlk-m there are yiET(X), where i=O, 1,. . . , j y,=y; y,=z, and for i=O, 1 . . > j, y,(~,~,) > 0, where yi(yi_,) is the probability that the compound lottery yi assigns to y::, . To formalize the various properties of choice behavior in sequential decision problems, a preference relation on sublotteries conditioned on the compound lottery they belong to is needed. If z my, we call z a sublottery of y and denote (z 1y) the sublottery z given y; for convenience we identify y with (y 1y). Our primitive will be a transitive preference relation over the set of sublotteries conditioned on the lotteries they belong to, i.e. over the set ~(X)={(zIy)IYE~(X)andzEY). This preference relation need not be complete. We only require that given some past history (after some of the uncertainty is resolved), the agent has complete preferences over continuation lotteries, i.e. he need not have well-defined preferences over continuation lotteries of different past histories. To put it formally, we need some definitions. Given y E Tk(X) and z E y, if z appears in y is obtained from y by replacing z with z’ if the only only once, we say that y’ ET(X) difference between y and y’ is that y, = z is replaced by y, = z’ in the definition of z E y. If z occurs in y more than once, we replace z with z’ in one place only, assuming that there is no confusion regarding which place it is. Now, we require from the preference relation 2 on W(X) that for any y, y’~r(X) and for any z Ey z’ Ey’; if y ’ is obtained from y by replacing z with z’, then either (z 1y)>(z’ I y’) or (z’ I y’))(z I y). The potential incompleteness of the preference relation need not bother us since even if the agent had a complete preference relation on P(X), we have no way to empirically verify it. Hence, the empirically unverifiable part of it would be, for all practical purposes, irrelevant. 2 From now on, whenever we write (z I y) 2 (z’ I y’), 1‘t must be understood that y’ is obtained from y by replacing z with z’. That is, the stated relation is empirically verifiable. We assume that the set of outcomes is sufficiently rich in the sense that for any simple lottery z E A(X) there is a sure outcome CE(z) E X such that z - CE(z).
Definition 2.1. A preference (z’ I Y’)i(Z
relation
12; (z’I y”‘> E wf>,
2 on q(X)
satisfies
’ Another way to deal with the incompleteness of the preference obtained from y by replacing z with z’, then (z 1y) - (z’ 1y’).
consequentialism
relation
is to assume
if, for all (z I y);
that
when
y’ is not
124
0. Volij I Economics
(2 IYMZ' whenever
IY'>
G (2 IFMZ’
it is empirically
Letters 46 (1994) 121-129
Iy”‘> 3
verifiable.
Consequentialism means that if the sublottery z is preferred to z’, given some history, then it must be preferred given any history. What does consequentialism mean in terms of decision trees? It means that when the agent is at some decision node, his choice of steps from then on will not take into account the past history that brought him to that node. In particular, the risks already borne by the agent have no influence on his future choices. No matter which history brought him to that node, and no matter what other paths could have taken place, his decision at that node will always be the same. Definition (2’
2.2. A preference
relation
2 on P(X)
satisfies dynamic
consistency
if, for all (z ( y) ,
I Y’> E wn (YIYMY’IY’)
whenever
= (4YkVlY’)
it is empirically
verifiable.
Roughly speaking, dynamic consistency says that if the individual prefers y to y’, then whenever he reaches lottery z he will not exchange it for 2’. In terms of decision trees, dynamic consistency says that whenever the individual reaches a decision node, he will always stick to his original strategy, even if he is given the opportunity to reconsider his choice. Before we go on we introduce the following notation. For every y ET(X), jE A(X) denotes the simple, single-stage lottery obtained from y by application of the calculus of probabilities. Simple lottery j is called the reduction of y.
Definition 2.3. A preference relation > on P(X) satisfies reduction of compound lotteries if, for all (zly), (z’ly’)~?P(X), (zIy)>(z’ly’) G (zlj)>(t’Iy”‘) when it is empirically verifiable and where y” is obtained from y by replacing z with Z and y”’ is obtained from y’ by replacing z’ with 2’. Reduction of compound lotteries is what Hammond (1988a,b), in the context of behavior norms, calls consequentialism. It means that when looking forward, the individual only cares about the ultimate probabilities of each outcome and not about the way these probabilities are obtained. Take any preference relation > on q(X). We can define an induced preference relation 2’ on T(X) in the following way:
Definition 2.4. Given induced
by 2,
denoted
for all ZJ’ E A(X)
a preference relation 2 on V(X), by >‘, is defined as follows: ,
z)‘z’
e
(ZIZ))(Z’IZ’).
the preference
relation
on A(X)
0. Volij I Economics Letters 46 (1994) 121-129
12.5
The induced preference relation 2’ represents the behavior of an agent whose preferences are given by 2, when he faces a choice between one-stage lotteries. Note that even though 2 is not a complete preference relation on ?P(X), 2’ is in fact complete, since the restriction of 2 to the elements of the form (y ) y) is complete. Let Z and 2 be two one-stage lotteries. If 2 satisfies consequentialism, it is easy to see that Z>’ 2’ e (21 y))(Z’ ] y’) for all ZEY, 2’ E y’, where y’ is obtained from y by replacing 5 with 2’. If, moreover, 2 satisfies reduction of compound lotteries, (z ] y) 2 (z’ ] y ‘) for all z E y and z’ E y’ such that Z is the reduction of z and 2’ is the reduction of z’, when y’ is obtained from y by replacing z with z’. This proves that given any complete preference relation _L’ on A(X), there is a unique (up to empirically unverifiable modifications) preference relation 2 on P(X) that satisfies reduction of compound lotteries and consequentialism and such that 2’ is the preference relation induced by 2. On the other hand, there are many preference relations that do not satisfy either consequentialism or reduction of compound lotteries, which induce a given preference relation 2’ on A(X). Independence is a property that sets restrictions on preferences defined on A(X). In order to define independence we need some more notation. For each y, y’ in A(X) and (YE [O;l] we define cxy + (1 - cx)j’ E A(X) as follows: for all x E X, (cyy + (1 - cr)y’)(x) = cyy(x) + (1 - c-u)y’(x). Definition 2.5. A preference relation 2’ on A(X) satisfies independence if, and all w, u and z En(X), and all cz E (O;l], wz’u if and only if (YW+ (1 - cr)z>‘av + (1- (;Y)z.
3. Relations among the axioms Karni and Schmeidler
(1991) proved the following:
Lemma 3.1. Let y E Z(X) and z Ey, then there exists t E A(X) and a E (0;l) such that y = (~2 + (1 - a)t. Furthermore, if y’ is obtained from y by replacing z with z’, then 7’ = at’ +
(1 - a)t. They use this lemma to show the following: Theorem 3.2. If a preference relation 2 on P(X) satisfies reduction of compound lotteries and consequentialism, then it satisfies dynamic consistency if and only if the induced preference relation 2’ on A(X) satisfies independence.
We can now state and prove the following result in terms of our definitions: Theorem 3.3. Zf a preference relation 2 on !P(X) satisfies reduction of compound lotteries and dynamic consistency, then it satisfies consequentialism if and only if the induced preference relation 2’ on A(X) satisfies independence.
126
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Letters 46 (1994)
121-129
Proof. The proof of the sufficiency part is the same as in Karni and Schmeidler (1991). Necessity: Let (z ( y), (z’ ( y’), (z 19, (z’ Iv”‘) E!P(X), where y’ is obtained from y by replacing z with z’ and y”’ is obtained from y” by replacing z with z’. We must show that (z 1y))(z’ 1y’) a (z ) g)(z’ 1y”‘). By dynamic consistency, (z 1y)>(z’ 1y’) G (y I y>> (y’ I y’). By reduction of compound lotteries, (y 1y)>(y’ I y’) e (YIY)>(Y’ I Y’). By definition, (7) >‘>>(Y’ / 9’) w y>’ 7’. By the lemma, there exist t E A(X) and (YE (0,l) such that Y = a.2 + (1 - a)t and j’ = (YZ’+ (1 - a)t. By independence, Y>’ j’ @ 2,’ 5’. On the other hand, by the same lemma, there exists t’ E A(X) and CX’E (0;l) such that y”=a’Z+(l-cr’)t’ and y”‘=a’Z’+(l-cz’)t’. By independence, 2,‘F’ e $2’7’. By definition, r>’ y”’ G (y”la> (y”’ I F’). By reduction of compound lotteries, (y” ( y^) ~(y”’ I y”‘) G (y”\fi>(y”‘)y”‘). By dynamic consistency, (y”lj)>(y”‘Iy”‘) e (z jg~(z’)y”‘). !J The axioms we have seen so far set restrictions on the overall preference relation 2. The following axiom sets restrictions only on ex ante preferences in the sense that it deals only with preferences over lotteries after the null history. It sets no restriction on preferences over lotteries conditioned on some non-trivial history. Definition 3.4. A preference (2 1Y)&z’ verified:
l Y’) E WX),
(ZlZ)L(Z’IZ’)
relation
2 on P(X) satisfies compound independence if, for all from y by replacing z with z’, the following is
w h ere y’ is obtained @ (Y
IYMY’IY’)~
This axiom says that the ex ante preference between two compound lotteries that differ in only one sublottery is determined exclusively by the ex ante preference between these sublotteries. Theorem 3.5. The three axioms, compound independence, dynamic consistency and consequential&m, are pairwise independent, but any two of them imply the third. Proof. CI and DC 3 C: Let (z 1y),(z’ 1y’),(z I y7,(z’ I y”‘) E 4(X), where y’ and y”’ are obtained from y and y”, respectively, by replacing z with z’. By dynamic consistency, (z I y)> (2’ I Y’) G (Y I Y)>(Y’ I Y’). By compound independence, (Y I Y)Z(Y’ I Y’) e (z Iz)> (z’ l z’). By compound independence, (z I z)) (z’ I z’) G (y” I y7 2 (y”’ I y”‘). By dynamic consistency, (Y”IjQ(y”‘IY”‘) e (+)1(z’Iy”‘). C and DC 3 CI: Let (z I y),(z’ I y’) E !P(X), where y’ is obtained from y by replacing z with 2’. (z 1Z)L (z’ I z’) e (z I y)> (z’ ( y’). By dynamic consistency, By consequentialism,
(4Yk(Z’IY’)
=
(YIY)L(Y’lY’).
CI and C + DC: Let (z ) y),(z’ 1y’) E q(X), z’. By consequentialism, (z I y)> (z’ I y’) a
(4~MZ’l~‘) = (Y IY)k(Y’lY’).
where y’ is obtained from y by replacing z with (z I z)>(z’ I z’). By compound independence,
We now define three preference relations on q(X), each satisfying only one of the three axioms. Let 2’ be a preference relation on A(X) that does not satisfy independence and let 2” be another preference relation on A(X) that does satisfy independence. We define L1, k2 and & as follows:
127
0. Volij I Economics Letters 46 (1994) 121-129
forw(zly), (z'IY'F~(X), (2 IYMZ I Y'> e Y”Y’ ; (2 1y),,(z’
1y') a
2,’
Z’ ;
Applying the definitions of these preference relations twice, it is easy to see that & satisfies dynamic consistency, and that k2 satisfies consequentialism. Applying the definition of 2x and taking into account that >” satisfies independence, we can see that & satisfies compound independence. Since by the first part of the proof any two axioms imply the third, in order to complete the proof it is enough to show that >I does not satisfy consequentialism and that neither k2 nor z3 satisfy dynamic consistency. Since 2’ does not satisfy independence, we can find z, z’ and Tad and cxE (O,l] such that z >’ z’ but (YZ’ + (1 - c+,’
(YZ+ (1 - a)t .
Let y be the two-stage lottery that assigns probability (Y to lottery z and probability 1 - (Y to lottery t. Analogously, let y’ be the two-stage lottery that assigns probability (Y to z’ and probability 1 - cy to t. Obviously, 7 = CYZ+ (1 - &)t and y’ = (YZ’ + (1 - a)t. From y>’ 7’ it follows that (z’ I y’) >, (t I y) and from z > ‘z’ it follows that (z I z)zl (z’ I z’), showing that ?I does not satisfy consequentialism. From z > “z’ it follows that (z I y)k2 (z’ I y’) but from 7,’ y’ it follows that(y’ I y’) k2 (y I y), showing that z2 does not satisfy dynamic consistency. From the definition of &, it is immediate that it does not satisfy dynamic consistency. 0 Theorem 3.6. Let 2 be a preference relation on P(X) that satisfies consequentialism and dynamic consistency. Then, > satisfies reduction of compound lotteries if and only if the induced preference relation 2’ on A(X) satisfies independence. Proof. The ‘only if’ part has been proved by Karni and Schmeidler show the ‘if’ part. The following lemmas will be useful.
(1991) so we need only to
Lemma 3.7. Let z = {zi, p,}y= 1 and z’ = {z :., pi};=, be two compound lotteries (z is the compound lottery that yields lottery zi with probability p,). Suppose that 2 satisfies compound independence and that for all i, z, )z i. Then z 2 z’. Proof.
Since zi 2 z :, by compound (z;,pl;*
independence
*. ;zI=l,pi-l;zi,Pi;zi+l,Pi+l;“’
z-G;> PIi.
’ * ;Z:,Pi;Zi+l,Pi+l;“‘;Zn,
JznY Pn>
p,),
i=l,...,
For i = 1 the left-hand side is z and for i = n the right-hand of 2 we get the desired result. Lemma 3.8. For any compound
lottery z E l(X),
z - t.
n. side is z’. Therefore,
by transitivity
128
0. Volij I Economics
Letters 46 (1994)
121-129
Proof. We will show this by induction. Let z = {zi, P,}:,~ E r2(X) be a two-stage lottery. For each zi E A(X) let CE(zi) denote a sure outcome that satisfies CE(z;) - ‘z;. By the definition of -’ and the previous lemma, z - (CE(z,),
~1; . . . ; CJXQ,
P,>
By independence,
W(q),
~1;
. . . ; CE(z,,), P,> -
rL$l=’ . PiZi
By the definition of -’ and transitivity we get z - 2. Now suppose that the claim is true for any compound lottery z E z = {zi, P,}:,~ be a (k + 1)-stage lottery (each z, is a k-stage lottery). hypothesis, zi - Zi, hence by the previous lemma, z - {Z,, pi}:=, . {Z,, p,} lottery, so by the first part of the proof { Zi, pi}:=, - 2. And by transitivity proves the lemma.
U
Lzlrm(X). Let By the induction YE1 is a two-stage we get z - 2. This
Assume now that (z 1y) 2 (z’ 1y’). By consequentialism, z > z’, which implies by the previous (2 1~3, (2’ 1y”‘) where y” is obtained lemma that 222’. Then, again by consequentialism, from y by replacing z with t and y”’ is obtained from y’ by replacing z’ with Z’. This concludes 0 the proof of the theorem.
4. Concluding
remarks
Our results provide two different interpretations of the independence axiom of the expected utility theory of choice under risk. They state that this property can be seen alternatively as equivalent to consequentialism and to the reduction of compound lotteries axiom. These results, together with Karni and Schmeidler’s (1991) main result, stress the relation of the independence axiom to properties of choice behavior in sequential decision problems. As a conclusion we can say that, even though each of the three properties we have dealt with, namely consequentialism, reduction of compound lotteries and dynamic consistency, each have a different normative appeal, they play a symmetric role in the interpretation of the independence axiom.
References Crawford, V., 1990, Equilibrium without independence, Journal of Economic Theory 50, 127-1.54. Dekel, E., Z. Safra and U. Segal, 1991, Existence of dynamic consistency of Nash equilibrium with non-expected utility preferences, Journal of Economic Theory 55, 229-246. Fershtman, C., Z. Safra and D. Vincent, 1991, Delayed agreements and nonexpected utility, Games and Economic Behavior 3, 423-437. Green, J., 1987, ‘Making book against oneself’, the independence axiom and nonlinear utility theory, Quarterly Journal of Economics 102, 785-796. Hammond, P.J., 1988a, Consequentialist foundations of expected utility, Theory and Decision, 25, 25-78.
0. Volij I Economics Letters 46 (1994) 121-129
129
Hammond, P.J., 1988b, Consequentialism and the independence axiom, in: B.R. Munier, ed., Risk, decision and rationality (Reidel, Dordrecht/Holland) 339-344. Karni, E. and D. Schmeidler, 1991, Atemporal dynamic consistency and expected utility theory, Journal of Economic Theory 54, 401-408. Machina, M., 1989, Dynamic consistency and non-expected utility models of choice under uncertainty, Journal of Economic Literature 27, 1622-1668. McClennen, E., 1990, Rationality and dynamic choice (Cambridge University Press, Cambridge). Schlee, E., 1990, The value of information in anticipated utility theory, Journal of Risk and Uncertainty 3, 83-92. Wakker, P., 1988, Nonexpected utility as aversion to information, Journal of Behavioral Decision Making 1, 169-175. Yaari, M., 1985, On the role of Dutch books in the theory of choice under risk, Nancy L. Schwartz Memorial Lecture, J.L. Kellog Graduate School of Management, Northwestern University.
ELSEVIER
Letters
46 (1994) 121-129
Dynamic consistency, consequentialism compound lotteries
and reduction
of
Oscar Volij * CentER for Economic
Research,
PO Box 90153, 5000 LE Tilburg,
Netherlands
Received 10 February 1994; accepted 24 February 1994
Abstract We use Karni and Schmeidler’s (Journal of Economic Theory, 1991, 54, 401-408) formalization of the concepts of dynamic consistency, consequentialism and reduction of compound lotteries in atemporal sequential decisions to show that given dynamic consistency and any of the other two concepts, the third is equivalent to the independence axiom of choice under risk in one-stage decision problems. JEL classification:
D81
1. Introduction Much has been written for and against the expected utility theory of choice under risk in general, and about the independence axiom in particular. One of the most formidable arguments against non-expected utility models is that non-expected utility maximizers behave in a dynamically inconsistent manner. Some variations of the argument state that this kind of behavior is subject to manipulation [see Green (1987) and Yaari (1985)]. Other versions assert that the value of information to individuals who do not maximize expected utility may be negative in some special cases [see, for example, Schlee (1990) and Wakker (1988)]. In a way, these two versions question the rationality of non-expected utility maximizers. Both the possibility of being systematically exploited and the fact that the value of information may be negative, contradict our intuitive notion of rational behavior. However, the independence axiom is a property of preferences over one-stage lotteries, that is, lotteries whose prizes are basic consequences, while the validity of the previous claims rests on a sequential decision setup where the individual is supposed to choose among compound lotteries. Hence, it is not so clear that non-expected utility maximizers behave irrationally. On the other hand, * This paper presents some elements of the first chapter Ph.D. seminar in Jerusalem for their comments. 0165-1765/94/$07.00 0 1994 Elsevier SSDZ: 0165-1765(94)00456-C
Science
of my Ph.D.
B.V. All rights reserved
thesis.
I wish to thank all
participants of the
122
0. Volij I Economics
Letters 46 (1994)
121-129
Hammond (1988a,b), McClennen (1990), Karni and Schmeidler (1991) and Machina (1989) made it clear that there is some connection between choice behavior among one-stage lotteries and choice behavior in sequential decision problems. Specifically, Karni and Schmeidler (1991) showed that if a choice behavior in sequential decision problems satisfies consequentialism, dynamic consistency and reduction of compound lotteries, then the induced behavior or in other words the induced preference relation over on one-stage choice problems, one-stage lotteries, satisfies the independence axiom. The importance of this result lies in the fact that it makes it clear that a model of choice behavior under risk that assumes away the independence axiom must specify which of the three properties of sequential behavior mentioned above is not satisfied. Roughly speaking, consequentialism means that choice behavior under risk is not influenced by the past or by risks already borne. Dynamic consistency means that planned behavior equals actual behavior even after reconsidering the plan before carrying out each action. Reduction of compound lotteries means that the ultimate probabilities of the basic outcomes are important for making a choice and not the way those probabilities are obtained. Even though these three properties are restrictions on individual behavior, they define three different approaches to extending the basic concepts in game theory to games in which players’ preferences do not satisfy the independence axiom. For example, Dekel et al. (1991) extended the notion of Nash equilibrium in normal form games to the case in which players do not satisfy the reduction of compound lotteries axiom. Crawford (1990), on the other hand, extended the Nash equilibrium concept and developed a new equilibrium concept for a case which is consistent with players not satisfying dynamic consistency or consequentialism. Fershtman et al. (1991) used a model with dynamically inconsistent players to explain delay in contracts. Since it is not clear at all which is the most appropriate way to model non-expected behavior, it is important to further analyze the relations among the different axioms. It turns out that given any two of the properties of choice behavior in sequential decision problems, the third one is equivalent to independence. Karni and Schmeidler (1991) showed that in the presence of consequentialism and reduction of compound lotteries, dynamic consistency is equivalent to independence. What I do in this paper is to show the other two relations, namely that consequentialism and reduction of compound lotteries are each equivalent to the independence axiom if the other two properties are also present, respectively. The paper is organized as follows. Section 2 gives the basic definitions, section 3 presents the results and section 4 concludes.
2. Basic definitions 1 We follow Karni and Schmeidler in describing a sequential decision as a compound lottery and not as a decision tree and an act that assigns to each node in it a probability distribution to the set of its immediate successors. It may be helpful, however, to think of a compound lottery as the unique one that corresponds to some decision tree and act. Let X be a set of basic consequences. Given a set S, A(,S) denotes the set of all simple probability measures on the ’ This part is based
on Karni
and Schmeidler
(1991).
123
0. Volij I Economics Letters 46 (1994) 121-129
family of subsets of S. Define inductively for k > 0, A”(X) = A(A”-l(X)), where A”(X) = X. lotteries of any number of Define also T(X) = U k_,oAk(X). T(X) is the set of all compound stages on X. We want to identify any compound lottery y E A”(X) with 6, E Ak+‘(X), where 6, is the probability U
measure
concentrated on lottery y. As a result we have that for all k, Ak+‘(X) 2 Tk(X) = {y ET(X) 1y E Ak(X)\dk-‘(X)}, then we have T(X) =
F=,A’(X). If we define
uk>”
rk(X)*
Given two compound lotteries, y and z, by z Ey we mean that if the compound lottery y is played out, there is a positive probability that at some stage the compound lottery z will be obtained. Formally, z E y if z = y or if there are k > m L 0 s.t. y E Tk(X), z E rm(X) and for some lljlk-m there are yiET(X), where i=O, 1,. . . , j y,=y; y,=z, and for i=O, 1 . . > j, y,(~,~,) > 0, where yi(yi_,) is the probability that the compound lottery yi assigns to y::, . To formalize the various properties of choice behavior in sequential decision problems, a preference relation on sublotteries conditioned on the compound lottery they belong to is needed. If z my, we call z a sublottery of y and denote (z 1y) the sublottery z given y; for convenience we identify y with (y 1y). Our primitive will be a transitive preference relation over the set of sublotteries conditioned on the lotteries they belong to, i.e. over the set ~(X)={(zIy)IYE~(X)andzEY). This preference relation need not be complete. We only require that given some past history (after some of the uncertainty is resolved), the agent has complete preferences over continuation lotteries, i.e. he need not have well-defined preferences over continuation lotteries of different past histories. To put it formally, we need some definitions. Given y E Tk(X) and z E y, if z appears in y is obtained from y by replacing z with z’ if the only only once, we say that y’ ET(X) difference between y and y’ is that y, = z is replaced by y, = z’ in the definition of z E y. If z occurs in y more than once, we replace z with z’ in one place only, assuming that there is no confusion regarding which place it is. Now, we require from the preference relation 2 on W(X) that for any y, y’~r(X) and for any z Ey z’ Ey’; if y ’ is obtained from y by replacing z with z’, then either (z 1y)>(z’ I y’) or (z’ I y’))(z I y). The potential incompleteness of the preference relation need not bother us since even if the agent had a complete preference relation on P(X), we have no way to empirically verify it. Hence, the empirically unverifiable part of it would be, for all practical purposes, irrelevant. 2 From now on, whenever we write (z I y) 2 (z’ I y’), 1‘t must be understood that y’ is obtained from y by replacing z with z’. That is, the stated relation is empirically verifiable. We assume that the set of outcomes is sufficiently rich in the sense that for any simple lottery z E A(X) there is a sure outcome CE(z) E X such that z - CE(z).
Definition 2.1. A preference (z’ I Y’)i(Z
relation
12; (z’I y”‘> E wf>,
2 on q(X)
satisfies
’ Another way to deal with the incompleteness of the preference obtained from y by replacing z with z’, then (z 1y) - (z’ 1y’).
consequentialism
relation
is to assume
if, for all (z I y);
that
when
y’ is not
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(2 IYMZ' whenever
IY'>
G (2 IFMZ’
it is empirically
Letters 46 (1994) 121-129
Iy”‘> 3
verifiable.
Consequentialism means that if the sublottery z is preferred to z’, given some history, then it must be preferred given any history. What does consequentialism mean in terms of decision trees? It means that when the agent is at some decision node, his choice of steps from then on will not take into account the past history that brought him to that node. In particular, the risks already borne by the agent have no influence on his future choices. No matter which history brought him to that node, and no matter what other paths could have taken place, his decision at that node will always be the same. Definition (2’
2.2. A preference
relation
2 on P(X)
satisfies dynamic
consistency
if, for all (z ( y) ,
I Y’> E wn (YIYMY’IY’)
whenever
= (4YkVlY’)
it is empirically
verifiable.
Roughly speaking, dynamic consistency says that if the individual prefers y to y’, then whenever he reaches lottery z he will not exchange it for 2’. In terms of decision trees, dynamic consistency says that whenever the individual reaches a decision node, he will always stick to his original strategy, even if he is given the opportunity to reconsider his choice. Before we go on we introduce the following notation. For every y ET(X), jE A(X) denotes the simple, single-stage lottery obtained from y by application of the calculus of probabilities. Simple lottery j is called the reduction of y.
Definition 2.3. A preference relation > on P(X) satisfies reduction of compound lotteries if, for all (zly), (z’ly’)~?P(X), (zIy)>(z’ly’) G (zlj)>(t’Iy”‘) when it is empirically verifiable and where y” is obtained from y by replacing z with Z and y”’ is obtained from y’ by replacing z’ with 2’. Reduction of compound lotteries is what Hammond (1988a,b), in the context of behavior norms, calls consequentialism. It means that when looking forward, the individual only cares about the ultimate probabilities of each outcome and not about the way these probabilities are obtained. Take any preference relation > on q(X). We can define an induced preference relation 2’ on T(X) in the following way:
Definition 2.4. Given induced
by 2,
denoted
for all ZJ’ E A(X)
a preference relation 2 on V(X), by >‘, is defined as follows: ,
z)‘z’
e
(ZIZ))(Z’IZ’).
the preference
relation
on A(X)
0. Volij I Economics Letters 46 (1994) 121-129
12.5
The induced preference relation 2’ represents the behavior of an agent whose preferences are given by 2, when he faces a choice between one-stage lotteries. Note that even though 2 is not a complete preference relation on ?P(X), 2’ is in fact complete, since the restriction of 2 to the elements of the form (y ) y) is complete. Let Z and 2 be two one-stage lotteries. If 2 satisfies consequentialism, it is easy to see that Z>’ 2’ e (21 y))(Z’ ] y’) for all ZEY, 2’ E y’, where y’ is obtained from y by replacing 5 with 2’. If, moreover, 2 satisfies reduction of compound lotteries, (z ] y) 2 (z’ ] y ‘) for all z E y and z’ E y’ such that Z is the reduction of z and 2’ is the reduction of z’, when y’ is obtained from y by replacing z with z’. This proves that given any complete preference relation _L’ on A(X), there is a unique (up to empirically unverifiable modifications) preference relation 2 on P(X) that satisfies reduction of compound lotteries and consequentialism and such that 2’ is the preference relation induced by 2. On the other hand, there are many preference relations that do not satisfy either consequentialism or reduction of compound lotteries, which induce a given preference relation 2’ on A(X). Independence is a property that sets restrictions on preferences defined on A(X). In order to define independence we need some more notation. For each y, y’ in A(X) and (YE [O;l] we define cxy + (1 - cx)j’ E A(X) as follows: for all x E X, (cyy + (1 - cr)y’)(x) = cyy(x) + (1 - c-u)y’(x). Definition 2.5. A preference relation 2’ on A(X) satisfies independence if, and all w, u and z En(X), and all cz E (O;l], wz’u if and only if (YW+ (1 - cr)z>‘av + (1- (;Y)z.
3. Relations among the axioms Karni and Schmeidler
(1991) proved the following:
Lemma 3.1. Let y E Z(X) and z Ey, then there exists t E A(X) and a E (0;l) such that y = (~2 + (1 - a)t. Furthermore, if y’ is obtained from y by replacing z with z’, then 7’ = at’ +
(1 - a)t. They use this lemma to show the following: Theorem 3.2. If a preference relation 2 on P(X) satisfies reduction of compound lotteries and consequentialism, then it satisfies dynamic consistency if and only if the induced preference relation 2’ on A(X) satisfies independence.
We can now state and prove the following result in terms of our definitions: Theorem 3.3. Zf a preference relation 2 on !P(X) satisfies reduction of compound lotteries and dynamic consistency, then it satisfies consequentialism if and only if the induced preference relation 2’ on A(X) satisfies independence.
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Proof. The proof of the sufficiency part is the same as in Karni and Schmeidler (1991). Necessity: Let (z ( y), (z’ ( y’), (z 19, (z’ Iv”‘) E!P(X), where y’ is obtained from y by replacing z with z’ and y”’ is obtained from y” by replacing z with z’. We must show that (z 1y))(z’ 1y’) a (z ) g)(z’ 1y”‘). By dynamic consistency, (z 1y)>(z’ 1y’) G (y I y>> (y’ I y’). By reduction of compound lotteries, (y 1y)>(y’ I y’) e (YIY)>(Y’ I Y’). By definition, (7) >‘>>(Y’ / 9’) w y>’ 7’. By the lemma, there exist t E A(X) and (YE (0,l) such that Y = a.2 + (1 - a)t and j’ = (YZ’+ (1 - a)t. By independence, Y>’ j’ @ 2,’ 5’. On the other hand, by the same lemma, there exists t’ E A(X) and CX’E (0;l) such that y”=a’Z+(l-cr’)t’ and y”‘=a’Z’+(l-cz’)t’. By independence, 2,‘F’ e $2’7’. By definition, r>’ y”’ G (y”la> (y”’ I F’). By reduction of compound lotteries, (y” ( y^) ~(y”’ I y”‘) G (y”\fi>(y”‘)y”‘). By dynamic consistency, (y”lj)>(y”‘Iy”‘) e (z jg~(z’)y”‘). !J The axioms we have seen so far set restrictions on the overall preference relation 2. The following axiom sets restrictions only on ex ante preferences in the sense that it deals only with preferences over lotteries after the null history. It sets no restriction on preferences over lotteries conditioned on some non-trivial history. Definition 3.4. A preference (2 1Y)&z’ verified:
l Y’) E WX),
(ZlZ)L(Z’IZ’)
relation
2 on P(X) satisfies compound independence if, for all from y by replacing z with z’, the following is
w h ere y’ is obtained @ (Y
IYMY’IY’)~
This axiom says that the ex ante preference between two compound lotteries that differ in only one sublottery is determined exclusively by the ex ante preference between these sublotteries. Theorem 3.5. The three axioms, compound independence, dynamic consistency and consequential&m, are pairwise independent, but any two of them imply the third. Proof. CI and DC 3 C: Let (z 1y),(z’ 1y’),(z I y7,(z’ I y”‘) E 4(X), where y’ and y”’ are obtained from y and y”, respectively, by replacing z with z’. By dynamic consistency, (z I y)> (2’ I Y’) G (Y I Y)>(Y’ I Y’). By compound independence, (Y I Y)Z(Y’ I Y’) e (z Iz)> (z’ l z’). By compound independence, (z I z)) (z’ I z’) G (y” I y7 2 (y”’ I y”‘). By dynamic consistency, (Y”IjQ(y”‘IY”‘) e (+)1(z’Iy”‘). C and DC 3 CI: Let (z I y),(z’ I y’) E !P(X), where y’ is obtained from y by replacing z with 2’. (z 1Z)L (z’ I z’) e (z I y)> (z’ ( y’). By dynamic consistency, By consequentialism,
(4Yk(Z’IY’)
=
(YIY)L(Y’lY’).
CI and C + DC: Let (z ) y),(z’ 1y’) E q(X), z’. By consequentialism, (z I y)> (z’ I y’) a
(4~MZ’l~‘) = (Y IY)k(Y’lY’).
where y’ is obtained from y by replacing z with (z I z)>(z’ I z’). By compound independence,
We now define three preference relations on q(X), each satisfying only one of the three axioms. Let 2’ be a preference relation on A(X) that does not satisfy independence and let 2” be another preference relation on A(X) that does satisfy independence. We define L1, k2 and & as follows:
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0. Volij I Economics Letters 46 (1994) 121-129
forw(zly), (z'IY'F~(X), (2 IYMZ I Y'> e Y”Y’ ; (2 1y),,(z’
1y') a
2,’
Z’ ;
Applying the definitions of these preference relations twice, it is easy to see that & satisfies dynamic consistency, and that k2 satisfies consequentialism. Applying the definition of 2x and taking into account that >” satisfies independence, we can see that & satisfies compound independence. Since by the first part of the proof any two axioms imply the third, in order to complete the proof it is enough to show that >I does not satisfy consequentialism and that neither k2 nor z3 satisfy dynamic consistency. Since 2’ does not satisfy independence, we can find z, z’ and Tad and cxE (O,l] such that z >’ z’ but (YZ’ + (1 - c+,’
(YZ+ (1 - a)t .
Let y be the two-stage lottery that assigns probability (Y to lottery z and probability 1 - (Y to lottery t. Analogously, let y’ be the two-stage lottery that assigns probability (Y to z’ and probability 1 - cy to t. Obviously, 7 = CYZ+ (1 - &)t and y’ = (YZ’ + (1 - a)t. From y>’ 7’ it follows that (z’ I y’) >, (t I y) and from z > ‘z’ it follows that (z I z)zl (z’ I z’), showing that ?I does not satisfy consequentialism. From z > “z’ it follows that (z I y)k2 (z’ I y’) but from 7,’ y’ it follows that(y’ I y’) k2 (y I y), showing that z2 does not satisfy dynamic consistency. From the definition of &, it is immediate that it does not satisfy dynamic consistency. 0 Theorem 3.6. Let 2 be a preference relation on P(X) that satisfies consequentialism and dynamic consistency. Then, > satisfies reduction of compound lotteries if and only if the induced preference relation 2’ on A(X) satisfies independence. Proof. The ‘only if’ part has been proved by Karni and Schmeidler show the ‘if’ part. The following lemmas will be useful.
(1991) so we need only to
Lemma 3.7. Let z = {zi, p,}y= 1 and z’ = {z :., pi};=, be two compound lotteries (z is the compound lottery that yields lottery zi with probability p,). Suppose that 2 satisfies compound independence and that for all i, z, )z i. Then z 2 z’. Proof.
Since zi 2 z :, by compound (z;,pl;*
independence
*. ;zI=l,pi-l;zi,Pi;zi+l,Pi+l;“’
z-G;> PIi.
’ * ;Z:,Pi;Zi+l,Pi+l;“‘;Zn,
JznY Pn>
p,),
i=l,...,
For i = 1 the left-hand side is z and for i = n the right-hand of 2 we get the desired result. Lemma 3.8. For any compound
lottery z E l(X),
z - t.
n. side is z’. Therefore,
by transitivity
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Proof. We will show this by induction. Let z = {zi, P,}:,~ E r2(X) be a two-stage lottery. For each zi E A(X) let CE(zi) denote a sure outcome that satisfies CE(z;) - ‘z;. By the definition of -’ and the previous lemma, z - (CE(z,),
~1; . . . ; CJXQ,
P,>
By independence,
W(q),
~1;
. . . ; CE(z,,), P,> -
rL$l=’ . PiZi
By the definition of -’ and transitivity we get z - 2. Now suppose that the claim is true for any compound lottery z E z = {zi, P,}:,~ be a (k + 1)-stage lottery (each z, is a k-stage lottery). hypothesis, zi - Zi, hence by the previous lemma, z - {Z,, pi}:=, . {Z,, p,} lottery, so by the first part of the proof { Zi, pi}:=, - 2. And by transitivity proves the lemma.
U
Lzlrm(X). Let By the induction YE1 is a two-stage we get z - 2. This
Assume now that (z 1y) 2 (z’ 1y’). By consequentialism, z > z’, which implies by the previous (2 1~3, (2’ 1y”‘) where y” is obtained lemma that 222’. Then, again by consequentialism, from y by replacing z with t and y”’ is obtained from y’ by replacing z’ with Z’. This concludes 0 the proof of the theorem.
4. Concluding
remarks
Our results provide two different interpretations of the independence axiom of the expected utility theory of choice under risk. They state that this property can be seen alternatively as equivalent to consequentialism and to the reduction of compound lotteries axiom. These results, together with Karni and Schmeidler’s (1991) main result, stress the relation of the independence axiom to properties of choice behavior in sequential decision problems. As a conclusion we can say that, even though each of the three properties we have dealt with, namely consequentialism, reduction of compound lotteries and dynamic consistency, each have a different normative appeal, they play a symmetric role in the interpretation of the independence axiom.
References Crawford, V., 1990, Equilibrium without independence, Journal of Economic Theory 50, 127-1.54. Dekel, E., Z. Safra and U. Segal, 1991, Existence of dynamic consistency of Nash equilibrium with non-expected utility preferences, Journal of Economic Theory 55, 229-246. Fershtman, C., Z. Safra and D. Vincent, 1991, Delayed agreements and nonexpected utility, Games and Economic Behavior 3, 423-437. Green, J., 1987, ‘Making book against oneself’, the independence axiom and nonlinear utility theory, Quarterly Journal of Economics 102, 785-796. Hammond, P.J., 1988a, Consequentialist foundations of expected utility, Theory and Decision, 25, 25-78.
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Hammond, P.J., 1988b, Consequentialism and the independence axiom, in: B.R. Munier, ed., Risk, decision and rationality (Reidel, Dordrecht/Holland) 339-344. Karni, E. and D. Schmeidler, 1991, Atemporal dynamic consistency and expected utility theory, Journal of Economic Theory 54, 401-408. Machina, M., 1989, Dynamic consistency and non-expected utility models of choice under uncertainty, Journal of Economic Literature 27, 1622-1668. McClennen, E., 1990, Rationality and dynamic choice (Cambridge University Press, Cambridge). Schlee, E., 1990, The value of information in anticipated utility theory, Journal of Risk and Uncertainty 3, 83-92. Wakker, P., 1988, Nonexpected utility as aversion to information, Journal of Behavioral Decision Making 1, 169-175. Yaari, M., 1985, On the role of Dutch books in the theory of choice under risk, Nancy L. Schwartz Memorial Lecture, J.L. Kellog Graduate School of Management, Northwestern University.