A NEW INTEGRAL FOR CAPACITIES EHUD LEHRER

November 17, 2005

Abstract. A new integral for capacities, different from the Choquet integral, is introduced and characterized here. The main feature of the new integral is concavity, which might be interpreted as uncertainty aversion. The integral is then extended to fuzzy capacities, which assign subjective expected values to random variables (e.g., portfolios) and may assign subjective probability only to a partial set of events. An equivalence between the minimum over sets of additive capacities (not necessarily probability distributions) and the integral w.r.t. fuzzy capacities is demonstrated. The extension to fuzzy capacities enables one to calculate the integral also in cases where the information available is limited to a few events.

School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel. e-mail: [email protected].

A NEW INTEGRAL FOR CAPACITIES

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1. Introduction In many economic activities individuals often face risks and uncertainties concerning future events. The probabilities of these events are rarely known, and individuals are left to act on their subjective beliefs. Since the work of Ellsberg (1961), the conventional theory based on (additive) expected utility has become somewhat controversial, both on descriptive and normative grounds. There is a cumulative indication that individuals often do not use regular (additive) subjective probability. Rather, they exhibit what is referred to as an uncertainty aversion.1 Schmeidler (1989) proposed one of the most influential alternative theories to that of additive subjective probabilities. In Schmeidler’s model, individuals make assessments that fail to be additive across disjoint events. The expected value of utility with respect to a non-additive probability distribution is defined according to the Choquet integral. The decision maker chooses the act that maximizes the expected utility. Following Choquet, a possible non-additive probability is referred to as a capacity. Since Schmeidler’s breakthrough, the Choquet integral has been extensively used in decision theory (see, Gilboa (1987), Wakker (1989), and Sarin and Wakker (1992)). Dow and Werlang (1992, 1994) applied the Choquet integral to game theory and finance. Schmeidler (1986) and Groes et al. (1998) provided a few characterizations of the Choquet integral. Another prominent integral is the Sugeno (or fuzzy) integral (see, Sugeno, 1974). It is expressed in maximum-minimum terms and it corresponds to the notion of the median, rather than to that of the average. As opposed to the Choquet integral, the Sugeno integral and the one introduced here do not coincide with the regular integral when the capacity is additive.2 This paper presents a new integral with respect to capacities, which differs from the Choquet integral. The new integral is axiomatically characterized in two ways. The key property of the new integral is concavity. This means that the integral of the sum of two functions is less than or equal to the sum of the integrals. In the context of decision under uncertainty this property might be interpreted as uncertainty aversion. Three more axioms are necessary in order to characterize the integral. The first one requires that if the capacity is additive, the integral coincides with the conventional 1

A myriad of empirical evidence of choices, that are not consistent with conventional subjective probability and expected utility, have been documented in the literature (see, Camerer and Weber (1992), Starmer (2000)). 2 For further discussion of this issue the reader is referred to Murofushi and Sugeno (1991).

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EHUD LEHRER

integral. The second is an axiom of monotonicity with respect to capacities. It states that if the capacity v assigns to every subset a value which is greater than or equal to that assigned by w, then the integral of any non-negative function with respect to v is greater than or equal to the integral taken with respect to w. The last axiom states that when integrating a function, say X, the integral depends only on the values that the capacity takes on the subset where X is strictly positive. In other words, the integral of a function does not depend on the values that the capacity ascribes to any event where the function vanishes. The integral proposed here makes use of the concavification of a cooperative game that appeared in Weber (1994) and later in Azrieli and Lehrer (2005a). In the last section we introduce an integral w.r.t. fuzzy capacities. Fuzzy capacities assign subjective expected values to some random variables (e.g. portfolios). In particular, a fuzzy capacity may assign subjective probabilities only to some events and not to all. The new integral aggregates all available information, and enables one to calculate an average value also when there is partial information and the capacity does not provide the likelihood of every possible event. The integral w.r.t. fuzzy capacities is inspired by Azrieli and Lehrer (2005b) who used the operational technique (concavification and alike) extensively and employed it to investigate cooperative population games. It turns out that there a strong relation exists between the minimum over additive capacities and the new integral. A full equivalence between the representation of an order over random variables as a minimum over additive capacities3 and a representation by the integral w.r.t. fuzzy capacities is shown in Section 11.

2. The new integral A capacity is a function v that assigns a non-negative real number to every subset of a finite set N and satisfies v(∅) = 0. The capacity v is said to be defined over N . A capacity P is additive if for any two disjoint subsets S, T ⊆ N , P (S) + P (T ) = P (S ∪ T ). Let |N | = n and let v be a capacity defined over it. Definition 1. Fix a capacity v. Define, Z cav Xdv = min{f (X)}, 3See

Gilboa and Schmeidler (1989) for the case of probability distributions.

A NEW INTEGRAL FOR CAPACITIES

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where the minimum is taken over all concave and homogeneous functions f : IRn+ → IR such that4 f (1lR ) ≥ v(R) for every R ⊆ N . Remark 1. Since the minimum of a family of concave and homogeneous functions R cav over IRn+ is concave and homogeneous, so is Xdv, as a function of X. Let v and w be two capacities. We say that v ≥ w if v(S) ≥ w(S) for every S ⊆ N . The following lemma provides an explicit formula for the new integral. Lemma 1. (i) For every X ∈ IRN +, Z cav nX o X (1) Xdv = max αR v(R); αR 1lR = X, αR ≥ 0 . Z (ii)

R⊆N

Z

cav

Xdv =

min

P is additive and P ≥v

R⊆N cav

XdP.

The proof of the lemma is based on the fact that, as a function of X,

R cav

Xdv is

concave. This is rather standard and is therefore omitted. Zhang et al. (2002) discussed expressions similar to that on the right-hand side of eq. (1) with a further restriction that all the sets are required to be mutually disjoint. With this restriction the integral becomes analogous to a Riemann integral. P The summation R⊆N αR 1lR is a decomposition of X w.r.t. v, if αR ≥ 0 for every P R ⊆ N and R⊆N αR 1lR = X. It is an optimal decomposition of X w.r.t. v if it is P a decomposition of X w.r.t. v and intcav Xdv = R⊆N αR v(R). When talking about decompositions, the reference to v will be often dropped. 3. Examples Example 1: Let N = {1, 2, 3}, v(N ) = 1, v(12) = v(23) =

2 , 3

v(13) =

1 4

and

v(i) = 0 for every i ∈ N . A function over N is a 3-dimensional vector. Consider X = (1, 2, 1). Note that (1, 1, 0) + (0, 1, 1) is a decomposition of X. Furthermore, it R cav is an optimal decomposition of X: Xdv = 32 + 32 = 43 . Example 2 (resolving Ellsberg paradox): Suppose that an urn contains 30 red balls and 60 other balls that are either green or blue. A ball is randomly drawn from the urn and a decision maker is given a choice between the two gambles. Gamble X: to receive $100 if a red ball is drawn. GambleY: to receive $100 if a green ball is drawn. In addition, the decision maker is also given the choice between these two gambles: 41l

R

is the indicator of R: 1lR = (1l1R , ..., 1lnR ), where 1liR equals 1 if i ∈ R and 0, otherwise.

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EHUD LEHRER

Gamble Z: to receive $100 if a red or blue ball is drawn. GambleW: to receive $100 if a green or blue ball is drawn. It is well documented that most people strongly prefer Gamble X to Gamble Y and Gamble W to Gamble Z. This is a violation of the expected utility theory. There are three states of nature in this scenario: R, G and B, one for each color. Denote by N the set containing these states. Each of the gambles corresponds to a real function (a random variable) defined over N . For instance, Gamble X corresponds to the random variable X, defined as X(R) = 100 and X(G) = X(Y ) = 0. The probability of four events are known: p(∅) = 0, p(N ) = 1, p({R}) =

1 3

and

2 . 3

p({G, B}) = The probability p is partially specified: it is defined only on a sub-collection of events and not on all events. Although the new integral has been introduced so far in relation to capacities defined over all events, the same idea may be used for what will be later called fuzzy capacities (among which capacities that may be defined only over a sub collection of events). This is explained here in order to resolve Ellsberg paradox and will be elaborated on later in Section 11. The integral of a function X is defined in a fashion similar to eq. (1). When p is defined only over familiar events, X is allowed to be written as a positive linear combination of characteristic functions of familiar events only. Using only the four R cav familiar events, X is optimally decomposed as X = 100 · 1l{R} . And thus, Xdp = 1 100 · 3 . When doing the same for Y , one cannot obtain a precise decomposition of Y (the random variable that corresponds to Gamble Y). The maximal non-negative function which is lower than or equal to Y and can be written only in terms of the four familiar events is 0 · 1lN . The integral of Y is therefore equal to 0. Since, 100 ·

1 3

> 0,

X is preferred to Y . A similar method applied to Z and W yields: Z ≥ 100·1l{R} and the right-hand side R cav is the greatest of its kind. Thus, Zdp = 100 · 13 , while W is optimally decomposed R cav as 100 · 1l{G,B} . Therefore, W dp = 100 · 32 . Since 100 · 13 < 100 · 32 , Gamble W is preferred to Gamble Z . The intuition is that the decision maker bases her evaluation of random variables only on well-known figures: on the probability of the familiar events. The best estimate is provided then by the maximal function that can be expressed using these events and the integral in turn is based on this estimate. 4. The new integral and Choquet integral Let v be a capacity defined over N . The Choquet integral of X w.r.t. v, denoted RC P Xdv, is defined by ni=1 (Xσ(i) − Xσ(i−1) )v(Ri ), where σ is a permutation over

A NEW INTEGRAL FOR CAPACITIES

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N that satisfies Xσ(1) ≤ ... ≤ Xσ(n) and Ri = {σ(i), ..., σ(n)} (X(σ(0)) = 0, by convention). Note that, (2)

X=

where αi = Xσ(i) − Xσ(i−1) . Thus,

X

P

αi 1lR(i) ,

αi 1lR(i) is a decomposition of X. This means

that for the calculation of the Choquet integral, a particular decomposition of X is used. In contrast, the new integral allows all possible decompositions, and as in the definition of the Lebesgue integral (see next section), the one that achieves the maximum of the respective summation is chosen. RC R cav This implies, in particular, that always Xdv ≤ Xdv. Lovasz (1983) and RC R cav Azrieli and Lehrer (2005a) imply that Xdv = Xdv for every X if and only if v is convex (i.e., v(S) + v(T ) ≤ v(S ∪ T ) + v(S ∩ T ) for every S, T ⊆ N ). Example 3: Let N = {1, 2, 3, 4}. The capacity v is defined as the minimum of probability distributions as follows. Denote p1 = ( 81 , 81 , 14 , 12 ), p2 = ( 12 , 18 , 18 , 41 ), p3 = ( 18 , 12 , 14 , 18 ), p4 = ( 18 , 14 , 12 , 18 ) and p5 = ( 18 , 12 , 81 , 14 ). For every S ⊆ N define v(S) = min1≤i≤5 pi (S). Thus, v(j) = v(14) =

1 , 4

v(34) = v(24) =

3 , 8

1 8

for every j = 1, 2, 3, 4, v(12) = v(13) = v(23) =

v(S) =

1 2

if |S| = 3 and v(N ) = 1.

Consider X = (0, 1, 2, 3) and Y = (1, 0, 2, 3). X and Y differ in the values of the first two coordinates: While X assigns the value 0 to the first state and 1 to the second, Y assigns the value 1 to the first state and 0 to the second. The Choquet integral of X RC RC coincides with that of Y : Xdv = Y dv = 12 + 83 + 18 = 1. On the other hand, the valuations of X and Y by the new integral differ. Since X = (0, 1, 0, 1) + 2(0, 0, 1, 1), R cav R cav Xdv = 83 + 2 · 38 = 89 . Moreover, Y dv = 1. Recall that X and Y differ only on the first two coordinates. Note that state 2 is more likely than state 1 in the sense that for every S that does not contain these states, v(S ∪ {1}) ≤ (S ∪ {2}), with a strict inequality when S = {4}. The reason R cav R cav why Xdv > Y dv is that {2, 4} is a part of the decomposition of X and v(2, 4) > v(1, 4). R cav Since both integrals are homogeneous,

10 Y 9

dv =

RC

10 Y 9

dv =

10 . 9

A decision

maker whose preferences are determined by the Choquet integral, would prefer

10 Y 9

over X, while a decision maker whose preferences are determined by the new integral, would prefer X over

10 Y 9

.

The following proposition generalizes this example.

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EHUD LEHRER

Proposition 1. Let v be a capacity. Then, v is convex if and only if for every Z cav Z cav Z C Z C non-negative X and Y , Xdv ≥ Y dv whenever Xdv ≥ Y dv. The proof is postponed to the appendix. 5. The new integral as an extension of Lebesgue integral A function f is simple if it can be written as f =

k X

αi 1lRi , where αi ∈ R.

i=1

For a simple function, the integral of f with respect to a measure µ is defined as Z k k X X αi 1lRi dµ = αi µ(Ri ). And for a non-negative function f it is defined as i=1

Z

i=1

½Z

f dµ := sup

¾ h dµ; h is simple and h ≤ f

Lemma 1 (i) implies that the definition of

R cav

.

Xdv is similar to this definition.

6. Characterization

Z

In this section we characterize the new integral. In what follows

Xdv should

be thought of as a function from pairs (X, v) to the reals. The goal is to find a set of plausible properties of such a function that characterizes it uniquely as the new integral. The first property (including its title) is adopted from Groes et al. (1998). Z Accordance for Additive Measures – (AAM): If v is additive, then Xdv is a regular integral. The following property states that the integral is co-variant with a positive linear rescaling. Homogeneity – (HO): For any v, X and β ≥ 0,

Z

Z βXdv = β

Xdv.

The next axiom is the paramount property of the new integral. In order to explain it consider a situation where a bet of one dollar on horse i yields two dollars if horse i wins the race. A vector, $ 12 X(i), i = 1, ..., n, of bettings (i.e., X(i) on horse i) will be referred to as a bet. The bet 12 X corresponds to the function X, which represents the prizes corresponding to all possible horse winnings. Suppose that the gabler’s assessments about the likelihood of all possible winnings is given by a capacity (nonadditive probability) v. The integral of X tries to capture the notion of “expected” return from the bet 12 X when the probabily considered is v.

A NEW INTEGRAL FOR CAPACITIES

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Suppose now that X and Y are two bets and β ∈ (0, 1). Then, β 12 X, (1−β) 12 Y and β 12 X +(1−β) 12 Y are also bets. The decision maker evaluates the bet β 12 X +(1−β) 12 Y . By splitting it into the bets, β 12 X and (1 − β) 21 Y he can ensure an “expected” return R R of βXdv + (1 − β)Y dv. By splitting β 12 X + (1 − β) 12 Y differently the gambler might guarantee even a higher “expected” return. The particular split into β 21 X and (1 − β) 21 Y ensures that the “expected” return from the bet β 12 X + (1 − β) 12 Y , which is R R R βX + (1 − β)Y dv, is at least βXdv + (1 − β)Y dv. This is the following concavity property.

Z

Concavity – (CAV): For any v, X , Y and β ∈ (0, 1), Z Z βXdv + (1 − β)Y dv.

βX + (1 − β)Y dv ≥

The next axiom refers to two capacities, one of which is additive. It states that if P is additive and P ≥ v, then the integral w.r.t. to P is greater than or equal to that w.r.t. v. In other words, the integral is monotonic in the restrictive sense that if P is additive and it is greater than v, then the integral of any non-negative X w.r.t. P is at least as high as the integral of the seme X taken w.r.t. v. Furthermore, the axiom requires that if P is not greater than v (meaning that there is S such that v(S) > P (S)), then there is an indicator function whose integral w.r.t. v is greater than that w.r.t. P . Monotonicity w.r.t. capacity – (M): For every additive P , P ≥ v if and only if Z Z 1lS dP ≥ 1lS dv for every S ⊆ N . Let S be a subset of N .

The sub-capacity vS is a capacity defined over S:

vS (T ) = v(T ) for every T ⊆ S. The next axiom requires that the integral of an indicator function with respect to v is equal to the integral with respect to subcapacity restricted to S. It suggests that the integral of a function depends on the values that v takes on the subset of N where the function is not vanishing. The following axiom equates two integrals: one w.r.t. v over the domain N , and another w.r.t. vS over a restricted domain, S.

Z

Independence of irrelevant events – (IIE): For every S,

Z 1lS dv =

1lS dvS .

Z Theorem 1. (First Characterization) The integral Xdv satisfies (AAM), (CAV), Z Z cav (HO), (M), and (IIE) if and only if Xdv = Xdv for every non-negative X. The following axiom is a relaxation of (M).

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EHUD LEHRER

WeakZ monotonicity w.r.t. capacity – (WM): For every additive P , if P ≥ v, Z then 1lS dP ≥ 1lS dv for every S ⊆ N . Z Weak Indicator property – (WIP): For every S, 1lS dv ≥ v(S). Schmeidler (1986) and Groes et al. (1998)Z employ a strong version of (WIP), called the indicator property, which states that 1lS dv = v(S). Z Theorem 2. (Second Characterization) The integral Z Z (HO), (WM), and (WIP) if and only if Xdv =

Xdv satisfies (AAM), (CAV), cav

Xdv for every non-negative

X.

Remark 2. Properties (AAM), (HO), (M), and (WIP) are also shared by the Choquet integral.

Remark 3. The property (M) could be divided into two parts: Z(i) (WM), Zand (ii) for every additive P , if P 6≥ v, then there is S ⊂ N such that 1lS dP < 1lS dv. In fact (ii) can be relaxed as Z follows: ZFor every additive P , if P 6≥ v, then there is a non-negative X such that XdP < Xdv. For aesthetic reasons I prefer to leave (M) in its current form. 7. The proof of the Theorems R Proof of Theorem 1. The fact that Xdv satisfies (AAM), (CAV), (HO), (M), and (IIE) is easy to check. As for the inverse direction, (M) implies5 that for every R R additive capacity P that satisfies P ≥ v, 1lS dP ≥ 1lS dv. Lemma 1 (ii) implies R R cav that Xdv, as a function of X, is smaller than or equal to Xdv (recall Definition R 1). By (CAV) and (HO), Xdv is concave and homogeneous. It remains to show R that 1lS dv ≥ v(S) for every S ⊆ N . We proceed by induction on the size of S. For S such that |S| = 1, v is additive, R R and by (AAM) 1lS dv = v(S). Assume that 1lS dv ≥ v(S) for every S ⊆ N with |S| < ` and we prove it for S of size `. R cav R cav P Let S ⊆ N . If v(S) < 1lS dv, then 1lS dv = ki=1 αi v(Ri ), where Ri is a R proper subset of S for every i = 1, ..., k. By the induction hypothesis, 1lRi dv ≥ v(Ri ) 5In

fact, at this point the ‘only if’ direction of (M) suffices.

A NEW INTEGRAL FOR CAPACITIES

for every i = 1, ..., k. And by (CAV),

R

1lS dv ≥

Pk i=1

αi v(Ri ) =

9

R cav

1lS dv > v(S), as

desired.

R cav We can therefore assume that v(S) = 1lS dv and it is greater than or equal to any Pk P combination of the type i=1 αi v(Ri ), where 1lS = ki=1 αi 1lRi , Ri is a proper subset P of S and αi ≥ 0. Assume first that there exists a decomposition 1lS = ki=1 αi 1lRi with P Ri being a proper subset of S (i = 1, ..., k) and v(S) = ki=1 αi v(Ri ). By (CAV), (H) R R P P and by the induction hypothesis, 1lS dv ≥ ki=1 αi 1lRi dv ≥ ki=1 αi v(Ri ) = v(S). P We can therefore assume that for every decomposition 1lS = ki=1 αi 1lRi , with Ri P being a proper subset of S (i = 1, ..., k), v(S) > ki=1 αi v(Ri ). By Shapley-Bordareva theorem6 (see, Bondareva, 1962 and Shapley, 1967) there7 is an additive capacity P over S such that P (S) = v(S) and P (T ) > v(T ) for every proper subset T of S. Let ε > 0 and consider the additive capacity P ε defined by P ε (j) = max(0, P (j) − ε) for every j ∈ S. When ε is sufficiently small, P ε (S) < v(S) and P ε (T ) > v(T ) for every proper subset T of S. Equivalently, Pε (S) < vS (S) and P ε (T ) > vS (T ) for every proper subset T of S.

R R We obtained that P ε 6≥ vS . By8 (M), there is T ⊆ S such that 1lT dvS > 1lT dP ε I claim that T = S. Indeed, if T is a proper subset of S, then PTε > vT . This implies R R R by (M) that 1lT dPTε ≥ 1lT dvT = 1lT dvS (the last inequality is by (IIE)), which R R R is a contradiction. We conclude that 1lS dv = 1lS dvS > 1lS dP ε ≥ P (S) − |S|ε. R Since ε is arbitrary, 1lS dv ≥ P (S) = v(S), as desired and the proof is complete. Proof of Theorem 2. The first part of the proof of Theorem 1 uses (AAM), (CAV), (HO), and only (WM). The second part is devoted to show what (WIP) explicitly assumes. One therefore obtains Theorem 2. 8. Minimum over the core The capacity v has a large core (Sharkey, 1982) if and only if for every S ⊆ N and for every additive capacity Q that satisfies v ≤ Q, there is P in the core of v such that P ≤ Q. The capacity v is exact (Schmeidler, 1972) if and only if for every S ⊆ N , there is P in the core of v such that P (S) = v(S). If v is convex, then v has 6The

core of v consists of all additive capacities P such that P ≥ v and P (N ) = v(N ). ShapleyBordareva theorem characterizes the capacities whose core is non-empty. Pk 7 More precisely, let d = v(S) − max{ i=1 αi v(Ri )}, where the maximum is taken over all decompositions of 1lS that use only proper subsets of S. Define the capacity w as follows: w(T ) = v(T ) if T is a proper subset of S and w(S) = v(S) − d. By Shapley-Bordareva theorem the core of w is not empty: there is an additive capacity Q such that Q(S) = w(S) and Q(T ) ≥ w(T ) for every proper d for every i ∈ S. P is the desired additive capacity. subset T of S. Define P (i) = Q(i) + |S| 8At this point the ‘if’ part of (M) is being used.

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EHUD LEHRER

a large core (see Sharkey, 1982) and if v has a large core, then it is exact (see, Azrieli and Lehrer, 2005a). No two of these three notions are equivalent. The connection between the largeness of the core and the integral is provided in the following statement. Proposition 2. (Azrieli and Lehrer, 2005a) v has a large core if and only if Z Z cav XdP Xdv = min P in the core of v

for every non-negative X. Azrieli and Lehrer (2005a) show that Corollary 1. Z cav

Z X + cdv =

Z

cav

Xdv +

Z

cav

cdv =

cav

Xdv + c · v(N )

for every non-negative X and a constant c if and only if v has a large core. The analogous statement of Proposition 2 for the Choquet integral is due to Schmeidler (1986). He showed that v is convex if and only if Z Z C XdP Xdv = min P in the core of v

for every non-negative X. 9. First order stochastic dominance and concavity Let (v, N ) be a capacity, and X, X 0 be two non-negative functions over N . We say that X 0 (first order) stochastically dominates X w.r.t. v, denoted X 0 ºv X, if for every number t, v(X 0 ≥ t) ≥ v(X ≥ t). The Choquet integral is monotonic w.r.t. stochastic dominance. That is, if X 0 ºv RC 0 RC X, then X dv ≥ Xdv. Example 4: Let N = {1, 2, 3}, v(N ) = 1, v(12) = v(13) = 34 , v(23) = 1 and v(i) = 0 R cav for every i ∈ N . Consider X = (1, 1, 1) and X 0 = (0, 56 , 65 ). Xdv = 54 , while R cav 0 R R cav cav X dv = 65 . In this example X 0 ºv X and nevertheless, X 0 dv < Xdv. R cav Example 4 shows that is not monotonic w.r.t. stochastic dominance. The question arises whether there is a reasonable integral which is monotonic w.r.t. stochastic dominance and concave (i.e., satisfies (CAV)) at the same time. The following example shows that there is no homogeneous (non-trivial) integral which possesses these two properties.

A NEW INTEGRAL FOR CAPACITIES

11

Example 5: Let N = {1, 2, 3}, v(S) = 1 if |S| ≥ 2 and otherwise, v(S) = 0. If R |S| = 2, then 1lS ºv 1lN , and if the integral ·dv is monotonic w.r.t. stochastic domiR R R P nance, then 1lS dv ≥ 1lN dv. However, 1lN = S; |S|=2 12 1lS , and if ·dv is concave R R R P and homogeneous, then 1lN dv ≥ S; |S|=2 12 1lS dv ≥ 32 1lN dv. Therefore, in the presence of homogeneity, monotonicity w.r.t. stochastic dominance and concavity are R not compatible, unless 1lN dv ≤ 0. If, instead, v(S) = 1 − ε (ε > 0) for every S ⊆ N that contains two states, then v becomes monotonic w.r.t. stochastic dominance. The set N can be thought of as a state space and the function 32 1lN can be thought at any state. However, 23 1lN can be P decomposed as an average of three portfolios: 23 1lN = S; |S|=2 31 1lS . Thus, if each of the portfolios 1lS , |S| = 2 (i.e., a payoff of 1 is guaranteed if a state in S is realized) is of as a portfolio that ensures a payoff of

2 3

selected with probability 13 , then, on average, a payoff of

2 3

is guaranteed at any state.

The idea behind concavity is that the value of 23 1lN should be at least the average of Z cav X 1 Z cav 2 the values of the portfolios forming it. That is, 1lN dv ≥ 1lS dv. 3 3 S; |S|=2

10. Properties Properties of the new integral that are not mentioned explicitly in the axioms are listed in this section. Proofs will be provided only to the non-obvious properties. In what follows X and X 0 are non-negative functions over N , or equivalently, points in IRn+ . 10.1. Continuity.

R cav

Xdv is continuous in both, X and v. R cav R cav Xdv ≥ Xdv 0 for 10.2. Monotonicity w.r.t. capacities. If v ≥ v 0 , then every non-negative X. Note that this property is not implied by axiom (M) that refers only the case where one of the two capacities is additive. R cav R cav 0 10.3. Monotonicity w.r.t. functions. If X ≥ X 0 , then Xdv ≥ X dv. R cav 10.4. Characteristic functions. By definition, for every S ⊆ N , 1lS dv ≥ v(S). R cav If 1lS dv > v(S), then there are scalars αi > 0 and Ri which are proper subsets R cav R cav P 1lRi dv = v(Ri ), i = of N , i = 1, ..., k , such that 1lS dv = ki=1 αi v(Ri ) and 1, ..., k. 10.5. Totally balanced capacity. Azrieli and Lehrer (2005a) implies that for any R cav R ⊆ N , the core of the sub-capacity vR is not empty if and only if 1lR dv = v(R). R cav Thus, 1lR dv = v(R) for every R ⊆ N if and only if the capacity is totally balanced (i.e., the core of each of its sub-capacities is not empty).

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10.6. The integral and the totally balanced cover. Let S ⊆ N . Define the R cav capacity v S as follows: v S (R) = v(R) if R 6= S and v S (S) = 1lS dv. Then, R cav R cav Xdv = Xdv S . Thus, increasing the value of the capacity from v(S) to R cav 1lS dv would not change the integral. R cav Let v be a capacity. Define the capacity Bv as follows: Bv (S) = 1lS dv for R cav every S ⊆ N . The capacity Bv is the totally balanced cover of v. Then, Xdv = R cav XdBv for every non-negative X. R cav Xdv > 10.7. The integral and the maximum of a function. It might be that max(X). However, X can be expressed as a positive linear combination of (characteristic) functions whose integral is between their minimum and their maximum. Furthermore, Lemma 2. (i) 1.

R cav

Xdv ≤ max(X) for every non-negative X if and only if Bv (N ) ≤

(ii) If v(N ) = 1, then

R cav

Xdv ≤ max(X) for every non-negative X if and only if

the core of v is non-empty. The proof is deferred to the appendix. 10.8. The integral and the minimum of a function. As stated in Section 4, the new integral is always greater than or equal to the Choquet integral. When v(N ) = 1, RC R cav Xdv ≥ min(X), and therefore Xdv ≥ min(X). R cav

Xdv is piecewise linear in X. That is, the set IRn+ R cav can be divided into finitely many closed cones F1 , ..., F` such that Xdv is linear R cav R cav R cav 0 0 0 in each one: for every X, X ∈ Fi , X + X dv = Xdv + X dv. 10.9. Piecewise linearity.

R cav Xdv is locally 10.10. Local additivity. The previous property implies that additive. That is, every X is included in an open cone, say UX , such that for every R cav R cav R cav 0 X 0 ∈ UX , X + X 0 dv = Xdv + X dv. (It is not true that for every R cav 0 R cav 0 R cav 00 0 00 00 X , X ∈ UX , X + X dv = X dv + X dv.) 10.11. Minimum over a set of capacities. Let C be a set of capacities. Denote R cav m(C)(S) = inf v∈C v(S) for every S ⊆ N . It turns out that for every C, Xdm(C) ≤ R cav minv∈C Xdv. However, if C is the set of all additive capacities that are greater R cav R cav than or equal to v, then Xdm(C) = minv∈C Xdv.

A NEW INTEGRAL FOR CAPACITIES

13

11. An integral w.r.t. a fuzzy capacity 11.1. Fuzzy capacity. Let I = [0, 1]n be the unit square. For every a ∈ I let |a| be the sum of its coordinates. Any subset of N can be identified with its indicator, which is an extreme point of I. Thus, a capacity is a function v that assigns to each extreme point of I a non-negative number and v(0, ..., 0) = 0. The notion of capacity is extended here as follows: Definition 2. (1) The pair (v, A) is a fuzzy capacity if (1, ..., 1) ∈ A ⊆ I, v : A → R+ is continuous, and there is a positive K such that v(a) ≤ K|a| for every a ∈ A. (2) (P, A) is an additive fuzzy capacity if there are non-negative constants, p1 , ..., pn , P such that for every a = (a1 , ..., an ) ∈ A, P (a) = ni=1 ai pi . While a capacity v assigns values (subjective probabilities) to events, a fuzzy capacity assigns values (subjective expected value) to random variables. The data-base of an agent might enable her to evaluate the expected values of some random variables (e.g., portfolios, bets) and not of others. Furthermore, it might enable her to assess the probability of some events, but not of all of them. The set of variables about which the agent has firm assessments is represented by A. Note that A might contain only points of the form 1lS , where S ⊆ N . In this case v is a partially specified nonadditive probability: it evaluates only the probability of events, and not necessarily all of them. The integral aggregates all available information, including individual assessments of the likelihood of events and expected values of variables, into a comprehensive picture. Upon observing the comprehensive picture the agent might re-evaluate the likelihood of events or the expected values she assigns to random variables and change her mind. We say that (x1 , ..., xn ) ≥ (y1 , ..., yn ) if xi ≥ yi , i = 1, ..., n. A function f over IRn+ is said to be monotonic if for every X, Y ∈ IRn+ , X ≥ Y implies f (X) ≥ f (Y ). Similar to the definition in Section 2 we define the integral of a non-negative X w.r.t. a fuzzy capacity (v, A). Let L be the set of all concave, monotonic and homogeneous functions f : IRn+ → IR such that f (a) ≥ v(a) for every a ∈ A. The integral w.r.t. (v, A) is defined as9 Z

cav

Xdv = min f (X). f ∈L

9The

set A is dropped from the notation.

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for every non-negative X. The minimum of all concave, monotonic and homogeneous functions is well defined and possesses the same properties. Similarly to Lemma 1 one obtains, Z (3)

cav

Xdv = max

k nX

o αi v(ai ) ,

i=1

where the maximum is taken over all ai ∈ A, αi ≥ 0, i = 1, ..., k that satisfy Pk Denote by coneA the convex cone generated by A. That is, i=1 αi ai ≤ X. P P coneA = { αi ai ; ai ∈ A and αi ≥ 0}. Note that in eq. (3) ki=1 αi ai is allowed to be less than or equal to and not necessarily equal to X as in Lemma 1. Inequality is allowed since coneA might be a strict subset of IRn+ . Note also that if (P, A) is R cav R additive, then XdP = XdP (the regular integral of X) for every X ∈ coneA. Example 6: Let N = {1, 2}. Thus, I = [0, 1] × [0, 1]. Define the fuzzy capacity (v, A) as follows: A = {(1, 1), ( 21 , 14 )}, v(1, 1) = 1 and v( 12 , 41 ) = 13 . Consider X = (1, 3 ). X = 12 (1, 1) + ( 21 , 14 ) and this is an optimal decomposition of X. Thus, R cav4 Xdv = 12 · 1 + 31 = 56 . Now let Y = (2, 3). Y = (2, 3) ≥ 2(1, 1) and this attains R cav the maximum of the right-hand side of eq. (3). Therefore, Y dv = 2. Example 2 revisited: Ellsberg paradox was analyzed in Example 2. In order to phrase this analysis in terms of fuzzy capacities, let N = {R, G, B}. Thus, I = [0, 1]3 , the three dimensional unit cube. Set A = {(1, 1, 1), (1, 0, 0), (0, 1, 1)} and v(1, 1, 1) = and v(0, 1, 1) = 32 . Let X be (100, 0, 0). Since 100(1, 0, 0) is an R cav optimal decomposition of X, Xdv = 100 · 13 . Also define, Y = (0, 100, 0) the R cav right-hand side of eq. (3) is attained by 0(1, 1, 1), and therefore, Y dv = 0. 1, v(1, 0, 0) =

1 3

The core of (v, A) (see also10 Aubin (1979) and Azrieli and Lehrer (2005b)) consists of all the additive fuzzy capacities P such that P (1, ..., 1) = v(1, ..., 1) and for every a ∈ A, P (a) ≥ v(a). The fuzzy capacity (v, A) is exact if for every a ∈ A there is P in the core of v such that P (a) = v(a). 11.2. Minimum over additive capacities and the integral. Let P be a compact set of additive capacities. Denote the fuzzy capacity (vP , I) as follows: vP (a) = R cav minP ∈P adP for every a ∈ I. Remark 4. For any compact set of additive capacities, P, denote by convP the convex hull of P. For any a ∈ A, the value vconvP (a) is attained at an extreme point of convP, which is in P. Therefore, vP = vconvP . 10Both

referred to the special case where A = I.

A NEW INTEGRAL FOR CAPACITIES

15

The following example illustrates the main idea demonstrated in this section. Example 7: Let N = {1, 2, 3} and consider the set P which consists of the probability distributions P1 = ( 12 , 14 , 41 ), P2 = ( 14 , 12 , 14 ) and P3 = ( 14 , 41 , 12 ). Denote by w the capacity vP restricted to A = {1lS ; S ⊆ N }. Thus,11 w(N ) = 1 and w(S) = |S| 14 for R cav |S| ≤ 2. In this case for every non-negative X, minP ∈P EP (X) = Xdw. 2 7 7 Now consider P4 = ( 16 , 16 , 16 ) and P 0 = {P1 , P2 , P3 , P4 }. Denote by u the capacity vP 0 restricted to A. Thus, u(N ) = 1, u(S) = 12 if |S| = 2, u(1) = 18 , and u(2) = R cav u(3) = 41 . In order to show that minP ∈P 0 EP (X) 6= Xdu for some non-negative

X, consider X = ( 35 , 25 , 0). On one hand, minP ∈P 0 EP (X) = 41 , and on the other, R cav 9 Xdu = 15 u(1, 0, 0) + 25 u(1, 1, 0) = 15 18 + 25 12 = 40 < 14 . In other words, in order to R cav get equality between XdvP 0 and minP ∈P EP (X), one cannot restrict oneself to A. We enlarge A: let A0 = A ∪ {( 35 , 25 , 0), ( 35 , 0, 52 )}. Define the fuzzy capacity (w0 , A0 ) as follows: it coincides with u on A, and w0 ( 53 , 52 , 0) = w0 ( 35 , 0, 25 ) = 41 . We obR cav Xdw0 . For instance, let tained that for every non-negative X, minP ∈P 0 EP (X) = R cav 7 1 7 1 1 2 3 X = ( 53 , 15 , 15 ). minP ∈P 0 EP (X) = EP4 (X) = 16 + + = and Xdw0 = 5 16 5 16 5 4 1 0 3 2 w ( 5 , 5 , 0) + 12 w0 ( 35 , 0, 25 ) = 14 . 2 The information embedded in P 0 cannot be compressed into a capacity defined only over the extreme points of I (i.e., to subsets on N ). The values of w0 over the points ( 35 , 25 , 0) and ( 35 , 0, 25 ) are necessary. On the other hand, the values of w0 on A0 are sufficient to provide all the information needed to obtain minP ∈P EP (X) through the integral. The following lemma (without a proof) connects between the minimum over a set of capacities and exactness. Lemma 3. If for any P, P 0 ∈ P, P (1, ..., 1) = P 0 (1, ..., 1), then vP is exact. Gilboa and Schmeidler (1989) characterized those preference orders over acts (which are translated to functions over N ) that can be represented by a minimum over a compact and convex set of probability distributions. It turns out that the representations as a minimum over additive capacities (not necessarily probability distributions) and as an integral w.r.t. a fuzzy capacity are equivalent. Formally, Proposition 3. (1) Let P be a compact set of additive capacities. Then, Z cav XdvP = min EP (X) P ∈P

11In

this example we identify a subset of N with its indicator.

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EHUD LEHRER

for every non-negative X. Furthermore, if P is either finite or a polygon, then there is a fuzzy capacity (v, A) with A being finite such that minP ∈P P (X) = R cav Xdv. (2) For every fuzzy capacity (v, A), if (v, A) is Lipchitz (i.e., there is a constant L > 0 such that for every a, a0 ∈ A, |v(a) − v(a0 )| ≤ Kka − a0 k2 ), then there is a compact and convex set of additive capacities (not necessarily probability distributions), P, such that Z cav Xdv = min P (X). P ∈P

Moreover, if (v, A) is exact, then P (1, ..., 1) = P 0 (1, ..., 1) for every P, P 0 ∈ P. The proof12 is rather standard and is therefore omitted. The following example shows that in Proposition 3 (2) the Lipchitz condition is necessary. √ Example 8: Let I = [0, 1]2 and v(x, y) = xy. The fuzzy capacity (v, I) is concave. However, at the boundary point (0, 1) there is no supporting hyper-plane to the graph of v. Therefore, there is no (non-trivial) additive capacity P such that √ P (0, 1) = v(0, 1) = 0 and at the same time P (x, y) ≥ v(x, y) = xy for every q p (x, y) ∈ I. However, for every (a, b) ∈ I with a, b > 0 let P(a,b) = 21 ( ab , ab ). On one q √ p hand, P(a,b) (a, b) = ab = v(a, b) and on the other, P(a,b) (x, y) = 21 ab x + 12 ab y ≥ √ xy = v(x, y) for every (x, y) ∈ I. In other words, P(a,b) corresponds to a supporting hyper-plane of the graph of v at the point (a, b). Finally notice that Z

cav

(0, 1)dv = v(0, 1) = 0 r a = inf = a,b>0 and b (a,b)→(0,1)

inf

a,b>0 and (a,b)→(0,1)

P(a,b) (0, 1).

In this case the infimum cannot be replaced by a minimum. The reason is that v does not satisfy the Lipchitz condition stated in Proposition 3 (2). References [1] Aubin, J.P. (1979) Mathematical methods of game and economic theory, North-Holland. [2] Azrieli, Y. and E. Lehrer (2005a) “A note on the concavification of a cooperative game,” http://www.math.tau.ac.il/∼lehrer/Papers/Note-concav-cooperative-4.pdf. 12It

is based on the fact that any concave function over a compact and convex set D, that can be extended as a concave function to an open set that contains D, is the minimum of all its supporting linear functions.

A NEW INTEGRAL FOR CAPACITIES

17

[3] Azrieli, Y. and E. Lehrer (2005b) “Investment games or population games,” http://www.math.tau.ac.il/∼lehrer/Papers/population-games-10.pdf. [4] Bondareva, O. (1962) “The theory of the core in an n-person game (in Russian),” Vestnik Leningrad Univ. 13, 141-142. [5] Camerer, C. F. and M. Weber (1992) “Recent developments in modelling preferences: Uncertainty and ambiguity,” Journal of Risk and Uncertainty 5, 325-370. [6] Dow, J. and S. Werlang (1992) “Uncertainty aversion, risk aversion, and the optimal choice of portfolio,” Econometrica 60, 197-204. [7] Dow, J. and S. Werlang (1994) “Nash equilibrium under Knightian uncertainty,” Journal of Economic Theory 64, 305-324. [8] Gilboa, I. (1987) “Expected utility with purely subjective non-additive probabilities,” Journal of Mathematical Economics 16, 65-88. [9] Gilboa, I. and D. Schmeidler (1989) “Maxmin expected utility with non-unique prior,” Journal of Mathematical Economics 18, 141-153. [10] Groes, E. Jacobsen, H. J. Sloth, B. and T. Tranæs (1998) “Axiomatic characterizations of the Choquet integral,” Economic Theory 12, 441-448. [11] Kalai, E. and E. Zemel (1982) “Generalized network problems yielding totally balanced games,” Operations Research 30, 998-1008. [12] Lovasz, L. (1983) “Submodular functions and convexity,” in: A. Bachem et al. (Eds.), Mathematical Programming: The State of the Art. Springer-Verlag, Berlin, 235-257. [13] Murofushi, T. and M. Sugeno (1991) “Fuzzy t-conorm integral with respect to fuzzy measures: generalization of Sugeno integral and Choquet integral,” Fuzzy Sets and Systems 42, 1 57-71. [14] Sarin, R. and P. P. Wakker (1992) “A simple axiomatization of non-additive expected utility”, Econometrica 60, 1255-1272. [15] Schmeidler, D. (1972) “Cores of exact games I,” Journal of Mathematical Analysis and Applications 40, 214-225. [16] Schmeidler, D. (1986) “Integral representation without additivity,” Proceedings of the American Mathematic Society 97, 255-261. [17] Schmeidler, D. (1989) “Subjective probabilities without additivity,” Econometrica, 57, 571-587. [18] Shapley, L. S. (1967) “On balanced sets and cores,” Naval Research Logistics Quarterly, 14, 453-460. [19] Sharkey, W. W. (1982) “Cooperative games with large cores,” International Journal of Game Theory, 11, 175-182. [20] Starmer, C. (2000) “Developments in non-expected utility theory: The hunt for a descriptive theory of choice under risk,” Journal of Economic Literature 38, 332-382. [21] Sugeno, M. (1974) “Theory of fuzzy integrals and its applications,” (PhD Dissertation). Tokyo Institute of Technology, Tokyo, Japan. [22] Wakker, P. P. (1989) “Continuous subjective expected utility with nonadditive probabilities,” Journal of Mathematical Economics 18, 1-27. [23] Weber, R. J. (1994) “Games in coalitional form,” in: R. J. Aumann, S. Hart (Eds.), Handbook of Game Theory, Vol. II. Elsevier. [24] Zhang, G., W. Yonghong and L. Jie (2002) “Lattice-valued Zp-pan-integrals I: For lattice-valued simple functions on lattice,” J. of Fuzzy Mathematics 10, 1, 213-226.

12. Appendix Proposition 1.

Let v be a capacity. Then, v is convex if and only if for every R cav R cav RC RC non-negative X and Y , Xdv ≥ Y dv whenever Xdv ≥ Y dv. R cav RC Proof. If v is convex, then Xdv = Xdv for every non-negative X. Conversely, if v is not convex, then in particular v is not identically 0. Moreover, by Lovasz (1983, R cav RC Proposition 4.1, p. 249) there is a non-negative X such that Xdv 6= Xdv.

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R cav RC R cav RC Since Xdv ≥ Xdv, Xdv > Xdv. By the definition of the new integral, R cav there is S ⊆ N such that 1lS dv = v(S) > 0. There is a constant c > 0 such R cav R cav RC R cav RC Xdv > c1lS dv > Xdv. Since c1lS dv = c1lS dv, we obtain, that R cav R cav RC RC Xdv > c1lS dv and c1lS dv > Xdv, as desired. R cav Lemma 2. (i) Xdv ≤ max(X) for every non-negative X if and only if Bv (N ) ≤ 1. (ii) If v(N ) = 1, then

R cav

Xdv ≤ max(X) for every non-negative X if and only if

the core of v is non-empty. Proof. (i) Suppose first that Bv (N ) ≤ 1 and suppose to the contrary that there is a R cav non-negative X such that Xdv > max(X). Since the integral is homogeneous, it can be assumed without loss of generality that max(X) = 1. In particular, 1lN ≥ X. R cav R cav R cav By monotonicity w.r.t. functions, 1lN dv ≥ Xdv and therefore, 1lN dv > 1. R cav R cav However, 1lN dv = 1lN dBv = Bv (N ). Thus, Bv (N ) > 1, which contradicts the assumption.

R cav Conversely, suppose that Xdv ≤ max(X) for every non-negative X. It implies R cav R cav in particular that 1lN dv ≤ 1. However, 1lN dv = Bv (N ), which implies that Bv (N ) ≤ 1. (ii) When v(N ) = 1, Bv (N ) ≤ 1 means that Bv (N ) = v(N ), which by BondarevaShapley theorem (see Bondareva, 1962 and Shapley, 1967) is equivalent to the nonemptiness of the core.