A . Matsui Mathematical Social Sciences 39 2000 1 –12
5
T 21 t
t t
t
Ua u p, h 5
O
s p, a, p , a ; r ur 2
T t 51
The decision maker chooses a maximand of U. A past case which is more similar to the possible present case is assigned more weight in evaluating an action. Write
s p, a, c95s p, a, p9, a9; r9. Note that the present formulation is a generalization of the original model of
Gilboa-Schmeidler, in which the value of an action, say, a, is given by
O
s p, qur, 3
q, a, r[M 7
where M is the set of past cases, and p is the present problem. Notice that in Eq. 3 the summation is only over the set of past cases in which action
a was taken. In Eq. 2, the similarity measure is defined over the set of pairs of problems and actions as opposed to the set of problems alone as in Gilboa-Schmeidler.
Their version also does not rely on the past result. The original theory is obtained if it is assumed that s p, a, q, b; r50 for all a ±b, and s p, a, q, a;r5s p, b, q, b;
r9 for all p, q [P, all a, b [ A, and all r, r9[R. These differences are necessary for the equivalence result. For example, if two acts have never been taken in the past, they are
assigned the same utility values in original, which are typically not the case in EU.
The present framework resembles the satisficing theory to the extent that the original CBDT resembles it. The interpretation of the present formulation is that the decision
maker evaluates each action by comparing the past scenarios with possible scenarios instead of comparing problems only. In fact, there is no reason to preclude this
possibility a priori. Throwing a stone at someone is often more similar to kicking him than to shaking hands with him.
3. Embedding
This section shows that CBDT models can be embedded into EU models, and EU models can in turn be embedded into CBDT models. Throughout the section, we assume
¯ that P and R or R , the set of results, are all countable can be finite.
Given a CBDT model, ,P, A, R, s, u ., this model is said to be embedded in an EU ¯
¯ model if there exists a model of the form ,
V, , A, R, f, u, m . such that for all ¯
h [H, p [P, and all a, b [ A, Va uhVbuh holds if and only if Uau p, hUbu p, h
¯ holds where p, h and h correspond to each other as described later. Similarly, given an
¯ ¯
EU model, , V, , A, R, f, u, m ., it is said to be embedded in a CBDT model if
¯ ¯
¯
there exists a model of the form ,P, A, R, s, u . such that for any history h [H \
hh j
1
except the null history, any problem p, and for all a, b [ A, Ua u p, hUbu p, h if and
¯ ¯
only if Va uh Vbuh holds. In the second definition, the null history should be
excluded since CBDT can do little without a case.
7
Gilboa and Schmeidler 1995 discuss several types of generalization, too. Also, see Gilboa and Schmeidler 1997a, proposing act-similarity functions which are ‘similar’ to the present formulation.
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. Matsui Mathematical Social Sciences 39 2000 1 –12
3.1. Embedding CBDT into EU Consider a model of CBDT, ,P, A, R, s, u . with H being the set of all histories.
¯ ¯
Construct its embedded model of EU, , V, , A, R, f, u, m ., in the following
manner. First, let ¯
R 5 R 3 P, and
¯ur, p 5 ur ¯
for all r, p[R. A result specifies the problem of the next period as well. Then let
` P 3 A3H
t
¯ V 5 P 3
P
R
t 51 1
1 2
1
Interpretation is as follows. In each state v 5 p , v , v , . . . [V, p is a possible
problem encountered
by the
decision maker
in the
first period,
and v 5
t t
¯ hr p, a, hj
t 51,2, . . . is a possible realization in the tth period, i.e., p is
p,a,h [P 3 A3H
t
¯ a possible problem encountered by the decision maker in question, and r p, a, h is a
possible result including the problem of the next period, which typically depends on the current problem p and choice a, and history h. Let be the
s-algebra generated by the
t 1
1 t
first t 11 elements of p , v , . . . ,v , i.e., is the smallest s-algebra which contains
t 1
1 1
1
any set cylinder F such that v95 p9 , v9 , . . . is in F if and only if p 5p9 and
v v
t t
`
v 5v9 for all t 51, . . . ,t. Let 5 s , the
s-field generated by ’s.
t 51 t
t
With this preparation, the main task of embedding is to construct a probability
T 1
1 1
measure on V, such that for all T51, 2, . . . , for any history h 5 p , a , r , . . . ,
T 21 T 21
T 21 T
p , a
, r [H , and any current problem p , and any two actions a, b [ A,
T T
T
¯ ¯
¯ Va
uh Vbuh holds if and only if Uau p , h Ubu p , h holds where h is the
T T
T T
T T
T T
1 1
1 2
T 21 T 21
¯ history in EU corresponding to p , h , i.e., h 5 p , a , r , p , . . . ,a
, r ,
T
¯ p [H. The following does this task inductively.
¯ In the beginning of the T th period, the decision maker knows h , or equivalently, a
T T
T 1
T 21 t
t t
t
problem p and the history h 5c , . . . ,c
with c 5 p , a , r . An action a is evaluated by
T 21 T
T t
t
Ua u p , h 5
O
s p , a, c ur 4
T t 51
Let m
satisfy
¯ h
T
1
t t
] u f a, ? 5 r 5
O
s p , a, c 5
¯ ¯
h h
T T
m
t
t [ ht ur 5rj
for all a [ A and all r [R• hr j, where m is given by
T t
m 5max
O O
s p , a, c 6
a [ A
t
r [R t[ ht ur 5rj
A . Matsui Mathematical Social Sciences 39 2000 1 –12
7
which is always finite since only finitely many s? ’s are added. We need this normalization to make the probability of the entire set be 1. For r , define
m f a, ? 5 r 5 1 2
O
m f a, ? 5 r 7
¯ ¯
¯ ¯
h h
h h
T T
T T
r ±r
¯ Now, calculate Va
uh 5o m f a, ? 5 rur. Substituting Eq. 5 into the right-
¯ ¯
T r [R
h h
T T
hand side of this expression, we obtain
T 21
1
T t
t
]
O
s p , a, c ur 8
m
t 51
From Eq. 4, Eq. 8 is equal to 1
T
] Ua u p , h
T
m ¯
¯ for all a [ A where we also make use of ur 50. Therefore, Va
uh Vbuh holds if
T T
T T
and only if Ua u p , h Ubu p , h holds. Repeat this exercise for all p[P and all
T T
h [H. The state space is so large that the above construction is consistent across problems and histories. A CBDT model is embedded into an EU model.
T t
Note that in this embedding, an increase in s p , a,c results in an increase in m f a, ? 5 r, i.e. the more similar the current situation is to the past case where
¯ ¯
h h
T T
action a led to result r, the higher the conditional probability of r given a. In other words, high similarity corresponds, at least in the above sense, to high correlation.
3.2. Embedding EU into CBDT ¯
¯ ¯
Consider a model of EU, , V, , A, R, f, u, m . with H being the set of his-
tories as defined in the previous section. Construct its embedded CBDT model, ,P, A, R, s, u., in the following manner. Let P, the set of problems, be defined as
¯ P 5 H
¯ let R5R, and let
¯ ur 5 ur 1
e ¯
for some e so that ur1e ±0 for all r[R. This linear transformation is needed since
some ratios of utilities are needed later. In this formulation, a problem is considered as a history, i.e. totality of what is known at the time the problem arrives.
2
With this preparation, a similarity function s:P 3 A 3R →
R is inductively con- structed. This task is started with the second period since case-based decision theory is
8 2
¯ meaningless without a case. In the second period, p 5h is the problem encountered by
2
the decision maker. Let s satisfy
8
One can cope with the first period problem if one modifies CBDT so that aspiration level is different across actions.
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. Matsui Mathematical Social Sciences 39 2000 1 –12
ur 1
1
¯ ]]
] sh , a, c 5
a
O
m f a 5 r 1
9
¯ 2
h 2
1
2
2 ur
r [R 1
¯ where
a 5ah , c is constant across actions, but may depend on h and c so that
2 1
2
Eq. 9 is always between 0 and 1. Then it is verified that in the second period,
1 1
¯ ¯
¯ ¯
Ua uh , c Ubuh , c if and only if Vauh Vbuh since
2 2
2 2
1 1
1 1
1
¯ ¯
¯ ]
] Ua
uh , c 5 s h , a, c ur 5 a
O
m f a 5 r ur 1 5
aVa uh 1
¯ 2
2 h
2 2
S D
2
2 2
r [R
holds Suppose that s?, ? has been defined up to period T 21 T 53,4,? ? ? . That is,
¯ s p, a, p9, a9; r9 has been defined for all a, a9[ A, all r9[R, and all p 5h and
t 1
t 1
t 9
¯
9 9
p9 5 h with h 5c ,? ? ?,c and h 5 c9 , . . . ,c9 such that t9,t T 21 T 52,3,? ? ?
t 9 t
t 9 t
t
and c 5c9 for t 51,? ? ?,t9. Now define s?, ? for the problems possibly encountered by
the decision maker in the T th period. Let a
ur 1
]] ]]
] s p, a, p9, a9; r9 5
O
m f a 5 r 1
10
¯ h
T
T
T 2 1 2
ur9
r [R 1
T 21
¯ ¯
for all a, a9[ A, r9[R, and all p 5h and p95h
with h 5c ,? ? ?,c and
T t
T t
t t
9
h 5c9 1,? ? ?,c9 such that t,T and c 5c9 for t 51,? ? ?,t. In Eq. 10, a is, as before,
t
constant across actions and may depend on the past history so that the expression is between 0 and 1. In this manner, the similarity function s is defined for a relevant
domain. To other pairs of cases, simply assign any arbitrary numbers between 0 and 1. ¯
¯ ¯
¯ ¯
It is now verified that for any h [H, Ua uh,hUbuh,h if and only if Vauh
1 1
1 T 21
T 21 T 21
¯ Vb
ub,h. Indeed, for all T51,2, . . . , all h 5 p , a , r , . . . , p , a
, r [H ,
T T
and all a [ A,
T 21 t
t t
t
¯ ¯
Ua uh , h 5
O
sh, a, p , a ; r ur
T T
t 51 T 21
a 1
]] ]
5
O O
m f a 5 rur 1
¯
F G
h T
T 2 1 2
t 51 r [R
1 ]
5 a
O
m f a 5 rur 1 T 2 1
¯ h
T
2
r [R
1 ¯
] 5
aVa uh 1
T 2 1
T
2 holds where the second equality is derived from Eq. 10. Like in the previous
embedding from CBDT to EU, there is a loose correspondence between correlation and similarity: if the previously encountered case gives the decision maker positive resp.
negative utility, then the higher the probability of obtaining a result with positive resp.
A . Matsui Mathematical Social Sciences 39 2000 1 –12
9
negative utility under a certain action, the more similar a pair of the present problem and the action is to the past case.
4. Conclusion
