mathamalic,~ social

sciences F.I_qEVIER Mathematical Social Sciences 32 (1996) 95-108

The product of capacities and belief functions

Ebbe Hendon , Hans JCrgen Jacobsen, Birgitte Sloth,* Torben Trana~s

Institute of Economics, University of Copenhagen, DK-1455 Copenhagen K, Denmark

Received October 1994; revised December 1995

Abstract

Capacities (monotone, non-additive set functions) have been suggested to describe situations of uncertainty. We examine the question of how to define the product of two independent capacities. In particular, for the product of two belief functions (totally monotone capacities), there is a unique minimal product belief function. This is character- ized in several ways.

Keywords: Uncertainty; Capacities; Non-additive measures; Lower probabilities; Belief functions; Product measure

1. Introduction

There is a growing literature on the representation of uncertainty by capacities; that is, monotone set functions that are not necessarily additive, see, for example, Schmeidler (1989), Gilboa and Schmeidler (1989), Jaffray (1989), and Hendon et al. (1994). There are several open problems in extending definitions and results from measure theory to the theory of capacities, the most notable being Bayes' rule. In this paper we discuss the definition of the product of two independent capacities.

A product of independent capacities is of particular interest in non-cooperative game theory. A mixed strategy of player i can be interpreted as a theory expressing the common expectation of other players concerning the action of i, rather than resulting from an actual randomization by player i. We can then use capacities rather than probability measures to express players' theories. If player 1

* Corresponding author: Fax: +45 3532 3000; e-mail: okosloth@pc.iht.dk.

0165-4896/96/$15.00 (~) 1996 Elsevier Science B.V. All rights reserved Pl l S0165-4896(96)00813-X

96 E. Hendot, et al. / Mathematical Social Sciences 32 (1996) 95-108

holds theories v 2 and v 3 on the actions of players 2 and 3, then she will need to form a product of v, and v3 in order to find a best reply. Given this, one can define an extension of Nash equilibrium. Papers outlining the possibilities of

extending game theory with non-additive measures are Dow and Werlang (1994), Groes et al. (1996), Klibanoff (1993), and Lo (1995).

Without additivity of capacities, the requirement v(E~ × E,_)= v~(E~)vz(Ez) on the Cartesian subsets of the product set is not sufficient to determine uniquely a product capacity v. We show, however, that under this requirement there is a minimal as well as a maximal product capacity between which all other product capacities lie.

If a capacity is interpreted as assigning lower probabilities to events, it is relevant to restrict attention to limited classes of capacities, e.g. to exact capacities (Schmeidler, 1972), or, even further, to totally monotone capacities, or belief functions (Dempster , 1967, and Shafer, 1976). Restricting ourselves to these, it becomes relevant to investigate the existence and properties of a unique minimal product capacity.

We show that, starting from exact marginal capacities, there exists a unique exact product capacity that is minimal among all exact product capacities. This capacity is different from that advocated by Walley and Fine (1982), and Gilboa and Schmeidler (1989).

Totally monotone capacities, or belief functions, is a key concept in formalizing the idea of lower probabilities. Also, for totally monotone marginal capacities, there is a unique totally monotone product capacity that is minimal among all totally monotone product capacities. This product is axiomatized by a few natural requirements on the operation of forming products. Furthermore, it has an intuitive interpretation in terms of the so-called M6bius inverse or mass function. Finally, it turns out that in a certain sense the minimal totally monotone product capacity can be obtained by applying Dempster 's rule of combination to the

marginal capacities.

2. Prel iminaries and a first result

Let S be a finite set, and let 6~ be the set of subsets of S. t Then o : 5 c---~ [0, 1] is

a capacity on S if:

v(O) = o , v (S) = 1 ,

f o r E , F E J , i f E C F , t h e n v ( E ) < ~ v ( F ) .

(1)

(2)

L It is easily seen that the results of Sections 2 and 3 generalize to the case where S is an arbitrary set and .Y is a field on S.

E. H e n d o n et al. / Mathemat ica l Social Sciences 32 (1996) 9 5 - 1 0 8 97

Condition (2) is called monotonicity. The function u is a probability measure if (2) is replaced by a requirement of additivity:

for E, F ~ 6P, v(E 13 F) = v(E) + v(F) - v(E fq F ) .

The set of probability measures on S is denoted by A(S). We let or: 5~1---~ [0, 1], and 02: 5e2--* [0, 1] be two (independent) capacities on

the finite sets S t and S 2. We define S = S t x $2, and let 5e be the set of subsets of S. We want to define a product v: ,5"---)[0, 1] on S of ut and o 2. If the marginal capacities are additive and the product is required to be additive, then the following requirement uniquely defines the product measure:

for all E l ~ 5e I and E 2 E 5e 2, v(Ei x E2) = vt(E1)v2(E2). (3)

We call v a product capacity if it satisfies (1), (2) and (3). Proposition 1 provides existence and a characterization of a unique minimal and maximal element of the set of product capacities (where o <~ v', if o(E) <~ u'(E) for all E ~ ,5").

Proposition 1. Let v t and v 2 be capacities on S t and S 2. Then there are product capacities J2 and 6 such that any product capacity u satisfies v ~ u <<- 6. Furthermore, u and 6 are defined by

v(E) := max ot(Ei)oz(Ez) , (4) E l E,5~I .EzE,3a 2 .g I x E2~E

6(E) := min ot(Et)v2(Ez) . (5) EIE~ t.E2E£e2,E~E t x E 2

Proof. We deal only with the minimal capacity here. The maximal capacity is handled similarly.

Let o be a product capacity. Monotonicity of v implies that if E 1 x E 2 C E ~ b ~, then v(E) >~ v(E t x E2) = v1(EI)v2(E2) , so

v(E) >~ max{vl(Et)u2(Ez) l E t x E 2 C_ E, E l ~ ~l , E2 ~- ~2} = !2(E) .

Thus, we need only show that v thus defined is a product capacity, i.e. that it satisfies (1), (2) and (3). This is trivial for properties (1) and (2). By definition, v(E~ x E z ) ~ v l ( E t ) v z ( E 2 ) . If O~ ~6e t, D2~.,9' 2, and D~ x D2C_E ~ x E 2, then D t C_E t and D z C E 2 (unless E I × E 2 =0 , in which case (3) holds), so by monotonicity of v I and u2, o1(31)o2(32) <~ oI(Et)oz(E2), .and thus o(E l x E2) <~ ot(Et)v2(E2). []

An example shows that v and 6 differ substantially: let the capacities u t and 02 be defined as in Table 1. In Table 2 we list o and 6 at non-Cartesian sets (included also are ~c, v b, and v c to be defined in Sections 3 and 4). To single out a unique product capacity, further requirements are needed.

98 E. Hendon et al. / Mathematical Social Sciences 32 (1996) 95-108

Table 1 Two marginal capacities

1.0 0.1 0.6

0.2 A B

1.0 0.3 C D

Table 2 Some product capacities formed from the capacities of Table 1

A D 0.18 0.20 0.20 0.34 1.00 BC 0.12 0.15 0.15 0.26 1.00 A B C 0.20 0.23 0.28 0.28 1,00 A B D 0.60 0.62 0.68 0.68 1.00 A C D 0,30 0.32 0.37 0.37 1.00 BCD 0,60 0,63 0.72 0.72 1.130

3. The minimal product of exact capacities

The interpretation adopted in the following is that v is a lower probability measure; that is, v(E) is the minimal probability with which the event E occurs. Maintaining this interpretation, it seems a reasonable minimal requirement that there exists a probability measure consistent with v; that is, the set

core(v) = (lr E A I 1r >1 v}

is non-empty. For any capacity v, core(v) is convex and compact. A capacity v is exact if and only if,

co re (v )~O and v(E)= min 1r(E), for a l l E ~ A e . IrEcore(v)

For the case of exact marginal capacities, Walley and Fine (1982) propose the capacity given from the product of the cores of v t and v z by assigning to each event E the capacity:

min ~-(E), "tr Ecore(u i ) ~ core(u2)

where 7rt ® rr 2 is the (ordinary) product of the probability measures ~r~ E A(S~) and zr 2 E A(S z). Gilboa and Schmeidler (1989), independently, propose a capacity given by

min 1r(E). rr ~conv(core(v ! ) ~ core(v 2 ))

E. Hendon et al. / Mathematical Social Sciences 32 (1996) 95-108 99

It is easy to demonstrate that core (v0 ® core(v2) is generally not convex. Nonetheless, the two proposals are identical: the minimal value of a linear function like ~ - ~ or. 1E is not changed by allowing convex combinations of the arguments. Denote by v ~ the resulting capacity. It is immediately verified that v c is indeed an exact product capacity.

If we insist that the product of v~ and v2 should assign as low a probability to each event as possible, the proposals above is not the right answer. Proposition 2 states the existence of a minimal, exact product capacity It', when v~ and v 2 are exact capacities, and characterizes Itc by its core. Proposition 2 shows that It" ~< v ~. In general they will be different. For the exact capacities v~ and v 2 of Table 1, the values of It" and It¢ are given in Table 2. They differ at all non-Cartesian sets.

Proposition 2. Let v~ and v 2 be exact capacities on S~ and S 2. Then there is an exact product capacity v" such that for any exact product capacity v ' , v" <<- v ", and for any product capacity v, core(v) C core(v ') . Furthermore,

core (~ . ' )=t l rEAIVE~C_S, VE2CS2:cr (E , x E 2 ) ~ v , ( E , ) o 2 ( E 2 ) ) . (6)

Proof. Denote the right-hand side of (6) by C. By exactness of v z and v2, C is non-empty: for any ¢q E core(vt) and ¢r 2 E core(v2), we have ¢q ® ¢r 2 E C, since (¢q ® ¢rz)(E l × E2) = lrt(El)cr2(E2) 1> vI(EOv2(E2). Since C is non-empty and compact, the function v ' (E) defined by

u ' (E) := min or(E) ( , ) ~ E C

is well defined. Obviously, It" satisfies (1). If E C F, then for all ¢r ~ C, or(E) <~ or(F), and thus min ,e c or(E) ~< min,,~c or(F), which shows (2). We conclude that It" is a capacity.

By (*), U' is exact if c o r e ( v ' ) = C. If ¢r 'E C, then ¢r'(E)I> min,,~c or (E)= v ' (E ) for any E E 5e, so It ' E core(it '). If or" ~ core(v ' ) , then for all E ~ 5:, ~r"(E) >~it'(E), so for any Cartesian set, by definition of C:

¢r"(E 1 x E2) >~it'(E l x E2) =min of(El x E2) >~ v t ( E 1 ) v z ( E 2 ) , (**) a - ~ C

i.e. ¢r" ~ C. Next, we show that v" satisfies (3). Let E l E 5"1, and E 2 ~ 5¢' 2. By (**),

It ' (E t x E2)>>.vi(Et)vz(E2). However , since vt and v z are exact, there is e p

¢q ® ¢r2 Ecore(vO ® core(v2)C_ C, such that ¢q(E1)=vI (E O, and cry(E2)= v2(E2), i.e.

t s t e ItC(E l × E2) =min ~r(E 1 × E2) ~< (rr I ® ~r2)(E t × E2) = rr,(Et)rrz(E2)

~ E C

= 01 (El)v2(E2).

Finally, we observe that any product capacity v has a smaller core than v ' : pick

lifo E. Hendon et al. / Mathematical Social Sciences 32 (1996) 95-108

• r ~ core(v). By (3), for any El ~5el , and E 2 ~3P:, ~r(E I x E2)~v(E~ x E , )= vl(El)v2(E,.), i.e. ~ -EC. So, c o r e ( v ) C C = c o r e ( t ~ ) . []

Note the relation between g and t~c: since by Proposition 1, t~ ~<!d c, we have core(t~) _~ core(re). Since the inverse inclusion holds by Proposition 2, co re (v )= core(re). An examination of Table 2 shows that v may be strictly increased at non-Cartesian sets without shrinking the core. In the terminology of Schmeidler (1972), we may say that v ~ is the exact hull of 2.

4. The minimal product of belief functions

In this section, we adopt an assumption stronger than exactness of the marginal capacities. A capacity o is K-monotone, K >~ 2, if for all families (Ej)jE J, where E~ ~ 5e and # J = K:

v I_J --/E~'~ ~] (-1)~t+~v E~) . (7)

It is well known that the set of two-monotone, or convex, capacities is strictly included in the set of exact capacities] If v is K-monotone for all K, v is termed totally monotone or a belief function. Since (7) is fulfilled with equality for additive measures (the general inclusion-exclusion equation), these are special instances of totally monotone capacities. Let g~(S) denote the set of belief functions on S.

For any set E ~ 5e~{l~}, the unit capacity v e is defined by ve(D ) = 1 if E C D and r E ( D ) = 0 , otherwise. To each capacity v, associate another mapping m: 5e---> R defined by

m(E):= ~ ( - 1 ) ' e \ ° v ( D ) . D~E

The relation between v and m is one-to-one and has the property v = Eecs.e,, ~ m(E)v E, or, equivalently, for all E ~ 5~:

o ( E ) = ~ m(D) . (8) D~E

The function m is called the mass function or the M6bius inverse of o, m = /x- l (v ) or v =/~(m). The correspondence is also known from game theory, see Shapley (1953).

z Shapley (1971) shows that convex capacities are exact. Strict inclusion is seen, for instance, by consider ing the exact capacity re" of Table 2:

12~(ABC) = 0.23 < 0.28 = 0.2 + 0.1 - 0.02 = 12*(AB) + v ' ( A C ) - lg ' (A) .

E. Hendon et al. / Mathematical Social Sciences 32 (i996) 95-108 101

Shafer (1976) shows that v is a totally monotone capacity if and only if m satisfies (9) and (10) below:

m(O)=O, ~ m(E)= l , (9) E ~

m >~ 0 . (10)

So, v is a belief function if and only if m is a probability measure on the set of non-empty subsets of S. The function m can be interpreted as assigning to each event E ~ 5e the amount of additional (lower) probability that cannot be deduced from the beliefs at proper subsets of E. Two simple examples illustrate this: in the case where an element in E will occur with absolute certainty, yet there is complete uncertainty about which one, the unit capacity v e will apply, and the M6bius inverse/x-l(o~) is equal to the indicator function 1E: ~---> {0, 1}. In the case of an additive measure ~r, the M6bius inverse will satisfy g-1(zr)(E) = 0 if # E > 1; that is, no superset of a singleton adds lower probability.

Proposition 3 offers six equivalent characterizations of our proposed product capacity, o I ® ~o 2. In accordance with the lower probability interpretation, (i) shows that v I ® b y z assigns the lowest possible probability to each event under the restriction that it is a totally monotone product capacity. This is parallel to the definitions of o and v r in Propositions 1 and 2.

In statement (ii) it is demonstrated that the product can be calculated as an 'inner' capacity, similar to te of Proposition 1, Eq. (4), but using the strong monotonicity requirement of (7), rather than the weak requirement of (2).

Part (iii) characterizes the M6bius inverse of o I ® b vz in terms of the M6bius inverses of vl and v z. The characterization shows that o I ® b o z is such that no (lower) probability is added at non-Cartesian subsets of S, only Cartesian subsets have the structure required to form decisive conclusions. Already Walley and Fine (1982) suggest (12). They do not give any justifications or alternative characteri- zations of the corresponding belief function.

For the axiomatic characterization in (iv), note that the affinity axiom (A2) is satisfied by the ordinary product ®. The fundamental axiom is (A1), a simple and easily interpretable formula for unit capacities. This property is fulfilled by the previously defined product capacities I~, l : , and t : . However, they all fail the affinity axiom. Otherwise they would be equal to v I ® b 02, but this is false, see Table 2.

Part (v) of Proposition 3 shows that, in a specific sense, our product capacity can be obtained by applying Dempster's rule of combination to the marginal capacities. Dempster 's rule goes as follows: let v I and v z be totally monotone capacities on the same set S, and with M6bius inverses m~ and m 2. If v I and v z are at all compatible, i.e. there are sets E 1 and E 2 such that E~ tq E 2 ~ 0 and mt(Et)mz(Ez)>O, then the combination of v I and v2, ol ~ v z , is defined by having a M6bius inverse given by

102 E. Hendon et al. / Mathematical Social Sciences 32 (1996) 95-108

~-~(v~ ~v2)(E) =

~, mz(Ei )m2(E/ ) Ei.EIE.~ :E~nEi=E

m , ( E i ) m z ( E j ) Ei,E)~-bC :EiNEj~O

With a lower probability interpretation, the capacity v t on S~ gives information on the algebra {E t × S 2 I E 1 ~ 5¢'}, which can then be extended to the set of all subsets to the capacity v't given by v ' t ( E ) = v t ( Y e t : e t × s ~ c E E l ) . We define v 2 similarly. We then obtain the proposed product by applying Dempster 's rule to v[ a n d v ' 2 , u l ® b o 2 = v i ~ v ' 2 .

Finally, the recursive formula in (vi) shows that v I ® b 02 is obtained by adding at each event the minimal mass sufficient to keep v~ ® bv 2 above v, given in (4), under the restriction that the mass has to be positive. In this sense, u I ® b U2 is the totally monotone hull of _0, just as 0 ~ is the exact hull of v.

Proposition 3. Consider a mapping ®b: ~(Sl ) x ~(S2)--*ff~(S 1 x $2). Then the

fo l lowing statements are equivalent: (i) For all v I and v 2, v I ® b v2 is the unique min imal totally mono tone product

capacity, i.e. vt ® b v2 << v for all totally mono tone product capacities v. (ii) For all v 1 and v 2, v~ ® b V2 is given by

(o, ® b v z ) ( E ) = m a x ~ (-1)'"101(/1~1Eil)02( 2 E~), (11) l~ f lE l ~ J , l ~ " "

where ,~e denotes the set o f families (E~ × EJz))ej, with E~ × E~2 C E for all ] E J. (iii) For all v I and u 2, u I @bY 2 is given by

~ - 1 ( v ~ ) ( E 1 ) ~ - I ( o 2 ) ( E 2 ) , tz - ' (v , ®bv~)(E) = t O ,

i f E = E I x E2, E # O , otherwise.

(12)

(iv) ® b satisfies

(A1) For all E C S t , FC_S 2, v E @b vt:=VE× F. (A2) For all v l, w l , v 2 and w 2, and a E[O, 1] ,

(o,v, + (1 - ,~)w,) ® b Y 2 = a(Vl ®bY2) + (1 - ,~)(w, ® b v 2 ) ,

vl ® b ('~v2 + (1 - ,~)w2) = ,~(vl ® ~ v 2 ) + (1 - ,~)(v, ® b w 0 .

(v) For all v 1 and v 2, let v' l : 5e--. [0, 1] and v~ : 5e-..> [0, 1] be given by

' ( ) v , ( E ) =v~ U E~ , EIE.9"I:EIXS2~E

E2 E,~2:S I x E2C.E

E. Hendon et al. I Mathematical ,Social Sciences 32 (1996) 95-108 103

Then v' t and v' 2 are totally monotone and vl ®by2 =v't ~ v'2, where ~ is Dempster's rule o f combination.

(vi) For all v t and v z, let I2 be given by (4). Then v t ~ b V 2 is given by

I~-I(Vt ~ bV2)(E) = max[O, 12(E) -- D~_~c E Iz-'(V, ~bv2)(D)].3 (13)

Proof. Throughout, we denote by v b and ~b the functions given by the right-hand sides of (11) and (12), respectively, and use m i for/~-l(vi).

(ii)¢~(iii). We must show that /~-l(Vb)=_m b. For any J ~ e , define for any D t x D 2 E ff'l x ,9~2 the set J(D t x D2) := {j E J [D l x D 2 C E: x E~2}. Further- more, it can easily be shown that for any finite set B, ~A{B ( - - 1 ) #'A = 0. NOW for any E E ~ and J E ~ e , we establish the following equalities:

= E (--I)#'+I~ E ml(Ol)m2(O2) ]

L _1 = E ml(Dl)m;(D2)r E (-I) .'+') J(Dt×D2)o$ L l~J(Dt x D2),l~

( -1 ) *~] = ~ ml (Dl )m2(D2)[1 - E J(D 1 x D2)#, ~ I~I(D! x D2)

= E mt (D , )m2(D2) . ](DIxD2)#~

From this rewriting we obtain:

)--max X JESE I~J,l,'O

=max ~ m,(Dl )m2(D2) . .leSE j(Dl × D2),,~

Since m~ and m 2 are positive, the expression is maximized by choosing J as large as possible, i.e. equal to {D~ × D 21 D~ E ~ , D 2 E Se 2, D 1 × D 2 C E}. Then,

vb(E) = ~ ml(Di)m2(D2) = ~ r~b(D), DI×D2~E D~E

so by (8), r~ b is the M6bius inverse of v b. (ii) and (iii)¢:~ (i). We have to show that v b is the minimal totally monotone

3 We use C to denote proper subsets.

104 E. H e n d o n et al. / M a t h e m a t i c a l Soc ia l Sc iences 32 ( 1 9 9 6 ) 9 5 - 1 0 8

product capacity. First we show that t: b is indeed a totally monotone product capacity. Here we use (iii). Observe that for any Cartesian set E 1 x E, ~ 5e z × 5e_~,

ub(EI X E2) = E _mb(D)-~- E . I I ( D I ) F F I 2 ( O 2 )

DC_E 1 ×E 2 D 1 × D2C_E I ×E 2

= u, (G)o , . (E2) . (*)

This immediately shows (9) and (3). As for property (10), if E is not Cartesian then _rob(E) = 0, so consider E = Et x E 2. Since v~ and o 2 are totally monotone, m t and m 2 are non-negative, so rob(E) = ml(Et)m2(E,.) >~ O.

We further have to show that any totally monotone product capacity v dominates v b. By total monotonicity of v, for any family of subsets of E,

ICJ

In particular, for any family J C_ J e of Cartesian subsets of E,

I C J " '

and thus

o(E) ~>max ~ ( - 1 ) # ' + ' v l ( . ( ~ Eil)v~ (~f'~l E'2) =~2b(E). J E Y E I ~ J " - "

(iv) f f (iii). The following calculations show that if (A1) and (A2) prevail then /~-t(v I ®bo2) is given by (12):

UI @bu2 ~-" E @b(~s2,~,orn2(F)vF ) (e¢s,.E~,om,(E)oe)

= ~. rn~(E)m2(F)(ve @6 ue) by (A2) E×FC_S .E×F~O

= ~. ml(E)m2(F)vE× F by (A1) E x F C S , E × F ¢ : O

= ~ m_b(u)oo. D ~ S , D ~ O

( i i i )~ ( i v ) . (A1) tz-~(oe)= 1E and ~-~(vF)= IF, so by (12), ~-~(v E @by r) = le× e, i.e. o e ® b Oe = Ve×r" (A2) We only show affinity in the first component. It is easily shown from (8) that ~z-l(aol +. (1 - a)wl) = a/z- l (ol ) + (1 -- a)~-~(w~), SO

((av I + (1 - a)wl) @bv2)(E )

= Y. + (1 e I x E2~ e

= a ~ lz-t(v,)(El)lz-'(v2)(E,_) E I x E 2 ~ E

E. Hendon et al. I Mathematical Social Sciences 32 (1996) 95-108 105

+ ( 1 - a) ~ tt-l(w,)(Et)la-l(Ve)(E2) E t x E2~E

= a(v, ® b Va)(E) + (1 - a)(w, ® b va)(E).

(iii)¢~(v). Define m: =lz-'(o~). It is easily seen that

fm,(El), i f E = E ~ x S 2 , ITI ~ (E) t 0 , otherwise,

and similarly for m~. Note first that E ~ E s u p p m ~ if and only if there is E~ E s u p p m , , such that E~ = E~ x Sa. Similarly for m~. Consequently, for E: E supp m: we have Ei, i = 1, 2, such that E i FI E~ = E l x E 2 # 0 . We then observe that for any E # 0,

E m',(<)m;(e;) ~,-'(v; ~ 4 ) ( E ) - ~ ' ~ ; ~ : ~ : ~ ' = ~

E m ' l ( E i ) m ; ( E ; ) E',e/~:E:nE:,,~

= E m,(e,)me(g) EiE Sf I ,E,e,,~2: Ei × E]" E

fml(E~)ma(Ea), if E = E 1 x E a

t 0 , otherwise

= ~-'(v, ® ~v:)(E).

(iii}¢V(vi). Let m be given by (13). We show by induction that t~ b = m. This is true for E = 0. Consider a set E, and assume that m = _mb for all D C E. For a Cartesian set E = E~ x E 2 we have

!2(E) - ~_, re(D) = v,(E,)o2(E2) - ~. m,(D,)mE(D2) DCE DI XD2CEI xE2

=(o~E m'(D")(o~e2 m2(D2,)

- ~_~ m,(D,)m2(D=) DzxD2CEIxE2

= m,(E,)m:(E2) ~ O,

i.e. re(E)= r~b(E). If E is not Cartesian, then m b ( E ) = 0, sO by the induction hypothesis we have to show that I~(E)~EDcEmb(D) . Let E~ x E 2 satisfy vl(El)va(E2) = maxt , xE.ze vl(E1)v=(E2), and note that E, x E 2 C E since E is non-Cartesian. Then,

I~(E) = v,(E,)v2(E2) = ~'- ml(D,)m2(D2) DI xD2~I xE2

E m,(D,)m2(O2)= ~ _mb(D). I-7 DIxD2CE DCE

106 E. Hendon et al. / Mathematical Social Sciences 32 (1996) 95-108

Consider again the examples of Table 1, and note that the marginal capacities are totally monotone. Table 2 shows that/2 b differs from o r, and thus that the core of o b is strictly smaller than that of ! : .

5. Discussion

A nice property of the product ® b characterized in Proposition 3, following immediately from the characterization (12), is associativity.

Corollary. A s s u m e that o i E ~(Si), i = 1, 2, 3. Then

/')1 ~ b (/')2 ~ b v 3 ) = (O1 (~ bu2 ) @ b o 3 " (14)

It can be argued that this property should be added to (3) as an axiomatic requirement of a product capacity. Note, however, that the joint requirement of (3) and (14) does not uniquely determine a product capacity. Indeed, the product capacity t7 defined by (5) also fulfils both.

Another nice property easily deduced from (12) is that if the marginal capacities are additive, then ® b yields the usual product measure. This property does not hold for any of the product capacities characterized in Proposition 1, but is valid for v e and v c defined in Section 3.

Another possible axiom could be the requirement of the Fubini-Tonelli result for the Choquet integral: let f s f dv denote the Choquet integral o f f : S---~ • with respect to v, where S = St x $2, and v is a product capacity of vt and v 2. Then we could require that for any v 1, v2, and f,

: :c(!. ) :(j. ) f d v = f ( s t , ' ) d v 2 dv l= f ( ' , s 2 ) d v t dv 2. (15)

S S l S 2

Unfortunately, such a property is not generally true. Consider the capacities of Table 1 and calculate the integral of the indicator function for the set {A, D},

c g c c fs t (fs2 l(a,o}(St, ") d o 2 ) d/) 1 = 0.26, and fs2 ( f s , l{a.o}( ' ,S2) dot) do2 = 0.25. It can, however, be shown that if both marginal capacities are unit capacities, then (15) holds with oE× r being the unique product capacity of the marginal capacities v E and or, lending support to axiom (A1) of Proposition 3.

A case where (15) holds is w h e n f is slice comonotonic , i.e. the set of functions (f( ' , s2) : S t --* R ) , : s 2 are pairwise comonotone for all s 2 E $2, and similarly when holding the first coordinate fixed. For a full development see Ghirardato (1995). Hendon et al. (1993) also obtain this.

We might hope to extend ®b from totally monotone capacities to general capacities. Any capacity v can be written as the difference between two totally

E. Hendon et al. I Mathematical Social Sciences 32 (1996) 95-108 107

Table 3

1.0

1.0 1.0 1.0

1.0 A B

1.0 C D

monotone capacities, v=av÷-~v -, a, ~ > 0 . A possible general product capacity of v I and v z could be

-~,a2(v ? ®by2)+/31/32(v~- ® b y 2 ) . (16)

The simple example in Table 3 demonstrates that this is not feasible. Using (16), we would obtain v*(AD) = 2, which v*(ABCD) = 1, so v* is not monotone.

Turning to applications, research in the use of capacities in decision making under uncertainty has taken two different lines.

In the Choquet Expected Utility theory of Schmeidler (1989) a.o., the interpretation of the endogenously derived capacity is not clear, but probably it expresses both 'belief' in a probabilistic sense, and 'attitude' towards the uncertainty in the decision situation considered. If two independent marginal capacities only express lower probability there is an argument for focusing on a minimal product capacity. If the marginal capacities express both belief and attitude it is not clear which product is appropriate. This represents a serious challenge to applications of the Choquet Expected Utility theory when products must be formed.

The other line of research, including Jaffray (1989), and Hendon et al. (1994), concentrates on belief functions with a lower probability interpretation. The product ® b will be useful under this interpretation.

A final remark: Proposition 3 gives an independent characterization of Dempster's rule of combination for the special case of combining marginal capacities. Since Dempster's rule is merely postulated without any axiomatic underpinning, it is interesting to investigate whether these characterizations can be modified to cover general combinations of totally monotone capacities.

Acknowledgements

We wish to thank Christian Berg for suggesting Proposition 1, and Karl Vind, Peter Wakker and a referee for useful comments and references.

108 E. Hendon et al. / Mathematical Social Sciences 32 (1996) 95-108

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