Forecasting with Imprecise Probabilities

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1 Foreasting with Impreise Probabilities Teddy Seidenfeld Mark J. Shervish Joseph B. Kadane Carnegie Mellon University Abstrat We review de Finetti s two oherene riteria for determinate probabilities: oherene 1 defined in terms of previsions for a set of events that are undominated by the status quo previsions immune to a sure-loss and oherene 2 defined in terms of foreasts for events undominated in Brier sore by a rival foreast. We propose a riterion of IP-oherene 2 based on a generalization of Brier sore for IP-foreasts that uses 1-sided, lower and upper, probability foreasts. However, whereas Brier sore is a stritly proper soring rule for eliiting determinate probabilities, we show that there is no real-valued stritly proper IP-sore. Nonetheless, with respet to either of two deision rules Γ-Maximin or (Levi s) E-admissibility-+-Γ-Maximin we give a lexiographi stritly proper IP-soring rule that is based on Brier sore. Keywords. Brier sore, oherene, dominane, E-admissibility, Γ-Maximin, proper soring rules. 1. Introdution One important approah to the foundations for subjetive probability is the strategy to redue rational degrees of belief to normative deision theory. Savage s [17] is a lassi among suh theories. De Finetti s Book argument, dating from about 1930 and summarized in [3], is another. De Finetti onsiders personal previsions, whih are an agent s fair pries for buying and selling random variables. These random variables are defined with respet to some ommon spae of possibilities. De Finetti introdues a riterion of oherent previsions: that the agent s fair pries annot be used to form a set of trades that result in a uniform sure loss with respet to that spae of possibilities. Thus in de Finetti s theory, oherene is a normative deision theoreti onstraint on an agent s previsions: avoid sure loss. He established the entral result that a set of previsions is oherent in this sense just in ase there is some (finitely additive) probability against whih the prevision for a random variable is its expeted value. When the random variables are indiator funtions for events, oherent previsions are the agent s personal probabilities for those events, and the agent s fair pries are her/his oherent betting odds. De Finetti thus redued the problem of rational degrees of belief to the problem of oherent previsions. Starting in about 1960, de Finetti emphasized two oherene riteria oherene 1 for previsions (as desribed above), and oherene 2 for foreasts assessed by Brier sore. He established [3, 5] that these two riteria are equivalent for purposes of distinguishing between sets of previsions or sets of foreasts that are undominated versus those that are dominated. Coherene is the ommon requirement that a deision maker avoids dominated alternatives. A set of previsions are oherent 1 i.e., they are undominated by the alternative of the status-quo there is no Book if and only if those same quantities, when used as foreasts evaluated by Brier sore, are oherent 2, i.e., they are undominated by any rival set of foreasts. In his later presentations de Finetti favored oherene 2 over oherene 1 beause, in addition to providing an equivalent riterion for oherene, also proper sores provide a method for inentive ompatible eliitation, unlike the situation with oherene 1 and the prevision game, as we all it. In Setion 2, we make preise and explain these laims. De Finetti s theory of oherent previsions, oherene 1, serves as the basis for numerous IP generalizations see [13, 26, 27, 28] for examples. However, we know of no parallel development of IP theory based on proper soring rules. It is our purpose in this essay to report some basi findings about soring-rule based IP theory. In Setion 3 we explain one approah to an IP version of oherene 2 and illustrate how that approah works. In Setion 4 we present an impossibility result for a real-valued proper IP soring rule. By ontrast, we illustrate a stritly proper, lexiographi (non-standard) IP version of Brier sore. In Setion 5 we onlude with remarks about the approah begun here. 1

2 2. De Finetti s two riteria for oherene 2.1 Coherene 1 and Coherene 2. We begin our review of de Finetti s theory with a reformulation of his first oherene riterion, oherene 1, whih onstrains a rational agent s fair pries for buying and selling random variables. Coherene 1 requires that the rational agent s fair pries annot result in a set of trades that result in a (uniform) sure loss. Coherent 1 previsions annot be dominated by the status-quo, where there are no trades. We reformulate oherene 1 in the ontext of a 2-person, 0-sum game, the prevision game, in order to prepare the reader for onerns about strategi aspets of applying oherene 1. These strategi aspets an distort the rational agent s play, even though that play is oherent 1. That is, the rational agent may have inentives to play oherently 1 in the game while misidentifying his/her degrees of belief. The existene of suh strategi aspets in the prevision game help to motivate de Finetti s seond oherene riterion, oherene 2. Coherene 2 onstrains the rational agent s foreasts for the same set of random variables by requiring that, as assessed by Brier sore, foreasts are undominated relative to eah rival set of foreasts. As we explain, below, oherene 2 is an inentive ompatible riterion for foreasting variables that provides an alternative foundation for subjetive probability. It mitigates the strategi aspets of rational play that threaten to distort the agent s announed pries in the prevision game. The prevision game, is formulated for a lass of bounded variables, X = {X i : i I} eah of whih is measurable with respet to a spae {Ω, B}, where I serves an index set. One player, the bookie, posts a fair, or 2-sided prevision P(X i ) for eah X i X. The bookie s opponent, the gambler, may hoose finitely many non-zero real numbers {α i } where, when the state ω Ω obtains, the bookie s payoff is Σ i α i ( X i (ω) P(X i ) ), and the gambler s payoff is the negative of this quantity, -Σ i α i ( X i (ω) P(X i ) ). That is, the bookie is obliged either to buy (if α > 0), or to sell (if α < 0) α -many units of X at the prie, P(X). Hene, the previsions are desribed as being 2-sided or fair buy/sell pries. The bookie s previsions are inoherent 1 if the gambler has a strategy that insures a uniformly negative payoff for the bookie, i.e., if there exist a finite set {α i } and ε > 0 suh that, for eah ω Ω, Σ i α i ( X i (ω) P(X i ) ) < -ε. Otherwise, the bookie s previsions are oherent 1. De Finetti s Fundamental Theorem of Previsions: The bookie s previsions {P(X): X X} are oherent 1 if and only if there is a finitely additive probability P whose expeted value for X, E P [X], is the bookie s prevision. That is: Coherene 1 obtains if and only if E P [X] = P(X). This result extends to inlude oherene 1 for onditional expetations given non-null events, using the devie of alledoff previsions. Let F be an event with F(ω) its indiator funtion. The bookie s alled-off prevision, P F [X], for X given event F has payoff in state ω to the bookie: F(ω)α( X(ω) - P F (X) ), whih equals 0 the transation is alled-off in ase event F fails. Assuming that the onditioning event is not null, i.e., P(F) 0, then Coherene 1 for alled-off previsions requires that E P [X F] = P F [X]. When the onditioning event F is null, oherene 1 plaes no substantive onstraints on the alled-off prevision P F [X]. That is E P [F(ω)α( X(ω) - P F (X) )] = 0 regardless the real-value of P F [X]. This defet in de Finetti s formulation has been disussed many times in the literature, and with a variety of different proposals to remedy the situation. For different orretions to this defet in oherene 1 see [7, 13, 16, and 28]. In our opinion, the debate over onditional probability given a null event is not yet resolved. The orretives to de Finetti s theory engender other ontroversies. For example, eah of these three proposals underwrites Dubins [6] theory of full onditional probabilities. But Dubins theory of onditional probability produes an asymmetri relevane relation. (See [2].) Beause suh ontroversies about onditioning on null events do not arise for the basi questions about IP-oherene addressed in this essay, we use de Finetti s original version of oherene 1 and sidestep the important hallenge of developing a satisfatory theory of onditional probability given a null event. The problems we do address here are prompted by de Finetti s [3, 4] observation that strategi aspets of betting may affet eliitation of a bookie s fair previsions using the prevision game. For example, when the bookie (believes he/she) knows the gambler s betting odds, then announing a prevision is subjet to strategi play in the game and may fail to reveal the bookie s fair prevision. Example 1: Suppose the bookie s fair (2-sided) prevision for an event G is.50. But suppose the bookie is onfident the 2

3 gambler s fair prevision for G is.75. So the bookie announes P(G) =.70, antiipating that the gambler will find it profitable to buy units of G at the inflated prie. Eliitation using the prevision game fails to identify the bookie s fair prie for G. Note: There are other issues onerning eliitation in the prevision game. Among these is the hallenge of statedependent utilities [20], whih we mention in Setion 5. To mitigate strategi aspets of the prevision game, de Finetti turned to a different oherene riterion: probabilisti foreasting of random variables subjet to Brier sore. In this essay, where our entral goal is to disuss extensions of oherene 2 to Impreise Probabilities for events, we fous on foreasting events, represented by their indiator funtions. E(ω) = 1 if ω E and E(ω) = 0 if ω E. The bookie s previsions serve as probabilisti foreasts subjet to Brier sore: squared-error loss. The penalty for the foreast P(E) when ω Ω is given by two funtions {g 1, g 0 } depending upon the state: g 1 (P(E), ω) = (1 P(E)) 2 if event ω E obtains; g 0 (P(E), ω) = (0 P(E)) 2 if event ω E obtains, whih is summarized by the squared-error penalty sore (E(ω) P(E)) 2 For the onditional (alled-off) foreast P F (E), on ondition that event F obtains, the sore is F(ω)(E(ω) P(E)) 2. And just as in the prevision game, the sore for a finite set of foreasts is the sum of the separate sores. The oherene 2 riterion applies to foreasting real-valued random variables, not just indiator funtions. Definition: A foreast set {P(X): X X} is oherent 2 if, for eah finite subset of X, there is no rival foreast set {Pʹ (X): X X} whose sores uniformly dominates in Ω. The two senses of oherene are equivalent, as de Finetti established [3, Setions ]. Proposition 1: A set of previsions are oherent 1 in the prevision-game if and only if those same set previsions are a oherent 2 set of foreasts under Brier sore. Proof: Here is a geometri version of de Finetti s projetion-argument that establishes oherene 1 and oherene 2 are equivalent oherene riteria. We sketh his argument applied to previsions/foreasts for a omplementary pair events. We use the same geometri presentation in Setion 3 in order to extend oherene 2 to an IP setting. Let X = {X 1, X 2 } be a pair of variables where X 1 is the indiator for an event A and X 2 is the indiator for the omplementary event A. In Figure 1, below, a pair of foreasts, {Q(A), Q(A )} with 0 Q(A), Q(A ) 1, is depited by the point (Q(A), Q(A )) in the unit square. The Brier sore for a pair of suh foreasts depends upon the two possible values of the indiators {X 1, X 2 }, whih are represented by the two points: (1,0), if the event A obtains, and (0,1) if the event A obtains. The Brier sore for the pair of foreasts {Q(A), Q(A )} equals the square of the Eulidean distane between the point (Q(A), Q(A )) and the respetive point, either (1,0) or (0,1), depending upon whih of the two possible values of the indiators {X 1, X 2 } obtains. A foreast pair {Q(A), Q(A )} is inoherent 2 if there is some rival foreast pair {Q (A), Q (A )} whose Brier sore is smaller regardless the realized values of the variables {X 1, X 2 }. Thus, the rival foreast pair dominates if and only if the distane between the two points (Q (A), Q (A )) and (1,0) is less than the distane between the two points (Q(A), Q(A )) and (1,0), and likewise for the respetive distanes to the point (0,1). The oherent 1 foreasts lie along the reverse diagonal, the simplex on two states, where Q(A) + Q(A ) = 1. No suh point is dominated in Brier sore by any other oherent 1 foreast, sine moving along this line segment inreases the distane, and hene inreases the squared error relative to one endpoint or the other. Example 2: Consider, the inoherent 1 previsions: P(A) =.6 and P(A ) =.7. A Book is ahieved against these previsions with the gambler s strategy α 1 = α 2 = 1. Then the net payoff to the bookie is -0.3 regardless whih state ω obtains. In order to see that these are also inoherent 2 foreasts, review Figure 1. If the foreast previsions {Q(A), Q(A )} are not oherent 1, they lie outside the probability simplex. Projet these inoherent 1 foreasts into the simplex. As in Example 2, the point (.60,.70) projets onto the oherent 1 previsions at the point (.45,.55). By elementary properties of Eulidean projetion, the resulting pair of oherent 1 foreasts, represented by the point (.45, 55), are loser to eah endpoint of the simplex than is the pair of inoherent foreasts, represented by 3

4 the point (.60,.70). Thus, the projeted foreasts have a dominating Brier sore with respet to the binary partition of possible values for the variables {X 1, X 2 }. This establishes that the initial foreasts are inoherent 2. Sine no oherent 1 foreast set an be so dominated, we have oherene 1 of the previsions if and only we have oherene 2 of the orresponding foreasts, as required by Proposition 1. (0, 1) (.60,.70) Q(A ) (.45,.55) de Finetti projetion (0, 0) Q(A) (1,0) Figure 1 Notes: If either foreast is outside the unit interval, then it is outside the range for the variable being foreasted. Then it is trivial to dominate that single foreast with a rival foreast hosen to be loser to the nearest endpoint of the range of the variable in question. Also, just as oherene 1 fails to regulate alled-off previsions given a null event, oherene 2 does not regulate alled-off foreasts given a null event. See [7] for a parallel revision to oherene 2 in order to aommodate onditional foreasts given a null event. 2.2 Inentive Compatible Soring Brier sore is just one of an infinite lass of (stritly) proper soring rules. Definition: A soring rule is (stritly) proper just in ase a foreaster (uniquely) minimizes expeted sore by announing her/his previsions. Thus, foreasting with a (stritly) proper soring rule avoids the problem of strategi behavior present in the prevision game: there is no opponent. Even allowing different proper soring rules for different foreasts, by taking the ombined sore for a finite set of foreasts as the sum of the individual sores, the result is again (stritly) proper. Savage [18] and Shervish [19] haraterize the (g 0, g 1 ) pairs for proper soring rules. In [21] we establish that all (proper) soring rules produe the same distintion between oherent 1 and inoherent 1 foreasts as with Brier sore, both for unonditional foreasts and for onditional foreasts given a non-null event. Proposition 2 [21]: (2.1) When the soring rule is proper, finite, and ontinuous, eah inoherent 1 foreast set is dominated by some oherent 1 foreast set. (2.2) When the soring rule is proper, finite, but not ontinuous, eah inoherent 1 foreast set is dominated, but not neessarily by a oherent 1 foreast set. Notes: Result 2.1 an be established by a generalization of de Finetti s geometri argument, where the projetion depends upon the soring rule. See [15]. Gilio and Sanfilippo [8] use a strengthened oherene riterion to extend this analysis to ontinuous soring rules when there is onditioning on null events. The demonstration of result (2.2) in [21] uses game-theoreti reasoning. 3. Coherene 2 with a Brier IP soring rule. One introdution to Impreise Probabilities is provided by C.A.B.Smith s [26] modifiation of de Finetti s prevision 4

5 game, whih provides a riterion of IP-oherene 1 for (losed, onvex) IP sets. Rather than requiring a 2-sided, fair prie, the bookie may fix a pair of 1-sided previsions for eah X X: the bookie may fix separate buy and sell pries. The bookie announes one rate P(X) as a buying prie for use when α > 0, and a possibly different selling prie P (X) for use when α < 0. The result is a generalized Book argument. See [279, hapter 2] for some history and basi results. Proposition 3: (3.1) A bookie s 1-sided previsions avoid sure loss if and only if there is a maximal, non-empty (losed, onvex) set of finitely additive probabilities P where P(X) < infimum P P E P [X] And P (X) > supremum P P E P [X]. When these inequalities are equalities, the 1-sided previsions are said to be IP-oherent 1. (3.2) By requiring lower and upper previsions for suffiiently many variables (from the linear span of X), the 1-sided previsions avoid sure loss if and only if they are also IP-oherent 1. See Theorem 1.ii of [23]. We offer a parallel version for defining IP-oherene 2 based on Brier sore for 1-sided foreasts, as follows: Use a lower foreast to assess a penalty sore when the event foreasted fails; Use an upper foreast to assess a penalty sore when the event foreasted obtains. Let {E i : i = 1,, m} be m events defined over a finite partition Ω = {ω j : j = 1,, n}. The foreaster gives lower and upper probability foreasts {p i, q i } for eah event E i. Soring foreasts with a Brier-styled IP soring rule: Fix a state ω Ω. If ω E i the sore for the foreast of E i is (1-q i ) 2 = g 1 (q i, ω) 2 If ω E i the sore for the foreast of E i is p i = g 0 (p i, ω) That is, use the most favorable foreast value from the pair {p i, q i } for determining the sore. Just as with the other oherene riteria disussed here, the sore for a set of foreasts is the sum of the individual foreast sores. Dominane: A foreast set G (stritly) dominates another F if, for eah ω Ω, the sore for G is (stritly) less than the sore for F. But, sine the vauous {0 = p i, q i = 1} foreast dominates eah rival {0 < p iʹ, q iʹ < 1}, we require an additional restrition on the lass of ompeting foreasts in order to avoid triviality of the resulting theory of IP-oherene 2. Note: This is analogous to a problem that is usually ignored within traditional IP theory. With 1-sided previsions, it remains IP-oherent 1 to be strategi: announe a lower buying (and/or a higher selling) prie than one is prepared to aept. That is, knowing who is the Gambler in the 1-sided Prevision Game, the Bookie may play strategially and mimi having a less determinate IP-oherent 1 set of previsions in order to seure stritly favorable gambles. We propose that IP-oherene 2 takes into aount both a rival model lass M, whih identifies the ompeting lass of rival foreasts, and an index of relative impreision in a foreast set. By allowing the rival foreasts to be restrited to a partiular lass M, we offer a more general approah than when the rival lass is fixed as the maximal lass of all possible lower and upper foreasts. This flexibility permits, also, to link our approah to different theories of Robust Statistis, as illustrated in Example 3, below. Stated informally, a set of 1-sided foreasts F are inoherent 2 when there exists a dominating set of foreasts G that are (i) at least as preise/determinate as F and (ii) where G belongs to the model lass M. We illustrate this idea by filling in the details of the two onepts: the rival model lass M and relative informativeness between foreast sets using the ε-ontamination lass. Example 3: Set M equal to the ε-ontamination lass, defined as follows. Let P be a partiular probability distribution over Ω = {ω 1,, ω n }. Fix 0 ε 1. Let Q be the simplex of all probability distributions on Ω. The ε-ontamination model, P ε, with fous on the distribution P, is the set of probability distributions on Ω defined by P ε = {(1-ε)P + εq: 5

6 Q Q}. This model is popular in studies of Bayesian Robustness. (See Huber [11, 12] and Berger [1].) Also, it is the model obtained from Harsanyi and Selten s [9] trembling hand strategies where P is the target strategy whih an be ahieved with probability 1-ε; otherwise, with probability ε strategy Q obtains. As a third reason for illustrating our ideas with the ε-ontamination model is that the lower probability funtion from this model is also a Dempster-Shafer Belief Funtion, and updating the ε-ontamination model either by Bayes rule or by Dempster s rule yields the same results. For our purposes here, it is useful to know that this lass is haraterized by speifying (IP-oherent 1 ) lower probabilities for atomi events, and then using the largest losed onvex set of distributions satisfying those bounds. (See Seidenfeld [22].) IP-foreasts over a finite partition for Brier-styled, ε-ontamination oherene 2 : Let F = { {p i, q i }: i = 1,, n} be foreasts for eah state ω i Ω ={ω 1,..., ω n }. Define F s sore set S by an ordered n-tuple of n-dimensional points: S = {(q 1, p 2,, p n ), (p 1, q 2,, p n ),, (p 1, p 2,, q n )}. Thus, S ontains at most n-many distint points. Eah point in S has n-many oordinates. Observe that the IP-Brier-style sore for F evaluated at state ω j is the square of the Eulidean distane between the j th point of S and the j th orner of the probability simplex on Ω. Clearly, the IP-sore for a foreast set an be improved merely by moving a lower foreast loser to 0, or by moving an upper foreast loser to 1. So, onsider dominating foreast sets only when the dominating foreast has a sore set that is less indeterminate than the sore set for the dominated foreast. Here is a andidate for relative indeterminay whih, when ombined with our Brier-style IPsore, allows a haraterization of ε-ontamination IP-oherene 2. Definition: Foreast set F 2 is at least as indeterminate as foreast set F 1 (or F 1 is at least as determinate as F 2 ) if the onvex hull of sore set S 1, H(S 1 ), is isomorphi under rigid movements (where both shape and size are held fixed) to a subset of the onvex hull of sore set S 2, H(S 2 ). Note that this relation of relative impreision, or relative indeterminay, is merely a partial order. We opt for suh a onept so that relative indeterminay may be extended to a variety of different real-valued indies of impreision, e.g., by using generalized volume of the sore set to quantify indeterminay. We use these notions to define IP-oherene 2 generally, and then ontinue with our illustration of IP-oherene 2 with respet to the ε-ontamination model. Definition: Given an IP-soring rule, a set F of IP-foreasts is IP-inoherent 2 with respet to the model M provided that there is a dominating set of rival foreasts G from the model M where the set G is at least as determinate than the set F. Say that F is IP-oherent 2 with respet to M if it is not IP-inoherent 2 with respet to M. For onveniene we will write these as M-oherent 2 and M-inoherent 2 Observe that IP-inoherene 2 redues to de Finetti s inoherene 2 when all foreasts in F are determinate, i.e., when p i = q i for eah foreasted event E i (i I), and when M is the lass of all determinate, oherent 1 foreasts. To see this, assume that Ω = k. Then the sore set S is the ordered set with k-many repetitions of the same I -dimensional point. Sine the lower and upper F foreasts for an event are idential, the k-many points in S do not vary with ω. So a dominating rival foreast set G = {p i, q i } must also assign the same lower and upper values to eah event E i (that is, for eah i I, p i = q i }, in order for G to be at least as determinate as F. By Proposition 2.1, then if G dominates F the rival foreast set {p i } establish that F is inoherent 2 and inoherent 1. Next, we provide two basi results for IP-oherene 2 with respet to the ε-ontamination model. Proposition 4: Let 0 p i q i 1, with n-many foreasts F solely for atoms of the algebra, the elements of the partition Ω = {ω 1,, ω n }. (4.1) The sore set S for F lies entirely within the probability simplex on Ω if and only if the lower and upper foreasts F math an ε-ontamination model. And then F annot be dominated by rival foreasts from a more determinate ε-ontamination model. (4.2) If all the elements of a sore set S, assoiated with foreast set F, lie outside the probability simplex on Ω, there is a dominating ε-ontamination foreast model F* with greater determinay than F. F is IP-inoherent 2 against rivals from the ε-ontamination model. 6

7 Proof: Result (4.1) is established by elementary alulations. If and only if eah point of the sore set S belongs to the probability simplex then, when state ω j obtains, orresponding to the j th point of S, 1 = q j + i j p i. This equality obtains for eah j = 1,, n. Then there exists an ε 0 suh that for eah i = 1,, n, q i = p i + ε, whih defines an ε- ontamination model. In the opposite diretion, if foreasts for the atoms are based on an ε-ontamination model, for i = 1,, n, q i = p i + ε, and then 1 = q j + i j p i so that all of the sore set S lies in the probability simplex. Last, if S belongs to the probability simplex and a rival ε-ontamination model F (with orresponding sore set S ) dominates, then H(S) is a proper subset of H(S ) beause for eah j = 1,, n, the j th point of S is loser to the j th extreme point of the probability simplex than is the j th point of S. So, F is less determinate than F. Thus F is IPoherent 2 with respet to the ε-ontamination model. Result (4.2) follows by the Brouwer Fixed-Point Theorem. Begin with a foreast set F = F 0, whose sore set S 0 has eah of its n-many ordered points outside the simplex of oherent 1 foreasts. Reursively reate rival foreast sets as follow. Apply the (de Finetti) projetion to eah of these n-many ordered points of S 0 taking them into the probability simplex of oherent 1 foreasts. This reates a set of (at most) n-points T 1 = {t 1,, t n } where eah t T 1 is a probability distribution P( ) over Ω. Form the new foreast set F 1 = {{p 1i, q 1i }: i = 1,, n} where p 1i = min t T 1{P(ω i )} and q 1i = max t T 1{P(ω i )}. This determines a new sore set S 1. Sine none of the points in S 0 belongs to the probability simplex, by the same reasoning used in de Finetti s analysis for Proposition 1, F 1 dominates F 0. Just in ase S 1 lies in the simplex, when result (4.1) applies, the reursive proedure halts. Otherwise foreast set F 2 is reated from a projetion of sore set S 1 into the probability simplex, et. Sine Eulidean projetions are ontinuous funtions and the probability simplex is ompat, the reursive proess with foreast sets F 0, F 1, F 2,. has a fixed point F* in the lass of ε-ontamination models. By a simple adaptation of de Finetti s argument for Proposition 1, the foreast set F i+1 (weakly) dominates the foreast set F i unless F i is a fixed point of the proess. Note: It may be that F i+1 merely weakly dominates F i for i 1, sine some but not all the points in S 1 may lie in the probability simplex. However, sine all the points of S 0 lie outside the probability simplex, F 1 dominates F 0. Last, the projetion of a losed, onvex set, e.g., the projetion of H(S) into the probability simplex, is isomorphi to a subset of H(S). Thus, assuming that the eah of the points of S 0 is outside the probability simplex on Ω, the fixed point F* of the proess F 0, F 1, F 2,, whih belongs to the ε-ontamination model lass, stritly dominates F 0, and is at least as determinate as F 0. Hene, F 0 is IP-inoherent 2 with respet to the ε-ontamination lass. Example 4: Here is an illustration of Proposition 4, IP-oherene 2 with respet to the ε-ontamination model, using 5 different foreast sets. Let Ω = {ω 1, ω 2, ω 3 }. Foreasts are for the three atoms only. The five foreast sets F j (j = 1,, 5) are presented in the form {{p i, q i } for ω i : i = 1, 2, 3}. The respetive sore sets have three points with oordinates {(q 1, p 2, p 3 ), (p 1, q 2, p 3 ), (p 1, p 2, q 3 )}, as desribed above. Figure 2 diagrams the onvex hull of eah sore set and shows the shaded 2-dimensional, triangular simplex of probability funtions on Ω. 7

8 Figure 2 (for Example 4) The onvex hull of the five sore sets are olor oded. The simplex of probability distributions is shaded. Eah sore set projets onto S 2, the sore set for foreast set F 2, orresponding to an ε-ontamination model. F 1 = { {.55,.80}, {.55,.80}, {.55,.80}} S 1 = {(.80,.55,.55), (.55,.80,.55), (.55,.55,.80)} F 2 = { {.25,.50}, {.25,.50}, {.25,.50}} S 2 = {(.50,.25,.25), (.25,.50,.25), (.25,.25,.50)} F 3 = { {.20,.45}, {.20,.45}, {.20,.45}} S 3 = {(.45,.20,.20), (.20,.45,.20), (.20,.20,.45)} F 4 = { {.10,.35}, {.10,.35}, {.10,.35}} S 4 = {(.35,.10,.10), (.10,.35,.10), (.10,.10,.35)} F 5 = { {.05,.30}, {.05,.30}, {.05,.30}} S 5 = {(.30,.05,.05), (.05,.30,.05), (.05,.05,.30)} The two foreast sets F 1 and F 5 are IP-inoherent 1 in aord with Proposition 3. Their 1-sided previsions lead to sure losses as, respetively, their lower (upper) foreasts are too great (too small). There is no determinate probability distribution agreeing with either set s lower and upper foreasts. Foreast set F 2 orresponds to an ε-ontamination model with fous on the uniform probability P = (1/3, 1/3, 1/3) and ε = 1/4. The onvex hull of the sore set S 2 lies in the probability simplex, as per Proposition (4.1). It is IP-oherent 1 and IP-oherent 2 with respet to the ε-ontamination model lass. 8

9 Foreast set F 3 is IP-oherent 1 as it has lower and upper foreasts agreeing with a losed onvex set of probabilities. Those values agree with a rival ALUP model, but not with an ε-ontamination model. That is, F 3 is IP-oherent 2 with respet to an IP-model lass defined by speifying atomi lower and upper probabilities [ALUP], but not so with respet to the ε-ontamination lass, whih is an IP-model lass determined solely by atomi lower probabilities. (See Appendix 1 for additional details about the ALUP model.) Foreast set F 4 has lower and upper foreasts that do not math those from a losed onvex set of probabilities. Its intervals are too wide. However, the uniform probability agrees with these foreasts, i.e., the probability values (1/3, 1/3, 1/3) fall inside the foreast intervals from F 4. Thus, in aord with Proposition 3, the foreasts from F 4 do not suffer a sure-loss in the 1-sided prevision game; however, F 4 is IP-inoherent 1 and IP-inoherent 2 with respet to the ε- ontamination model lass. As indiated in Figure 2, eah of the other four onvex hulls projets to H(S 2 ). That is, the proess desribed in the proof of Proposition (4.2) has F 2 as its fixed point for eah of the five foreast sets, and the proess terminates after at most one projetion. (See Appendix 2 for an illustration of Proposition (4.2) where the fixed point is merely a limit of the proess.) 4. Inentive ompatible IP-eliitation Reall that de Finetti favored oherene 2 over oherene 1 beause, in addition to serving as an equivalent riterion of oherene, Brier sore provides a stritly proper sore. A deision maker who maximizes expeted utility against Brier sore announes her/his previsions for random variables as their foreasts. Brier sore is an inentive ompatible eliitation for determinate probabilities. It eliminates some of the strategi aspets evident in the prevision game. That is, for a deision maker whose degrees of belief about events are represented by a single probability funtion P( ) and who maximizes expeted utility, she/he has a unique strategy for announing foreasts (and alled-off foreasts) that minimize expeted Brier sore. Announe the probability P(E) as the foreast of event E. If H is not-null, then announe the onditional probability P(E H) for the alled-off foreast of event E, on ondition that H obtains. Reall that when H is null, oherene 2 plaes no restritions on the alled-off foreasts given H. There is no differene to the expeted sore ontributed by any onditional foreast of E, alled-off if H fails, regardless whether that foreast is or is not oherent 2. See [5] for an improved version of oherene 2. However, our presentation in this setion is not affeted by the open problem of how to resolve the problem of reduing onditional probability given a null event to a deision-theoreti riterion of oherene. What an be done to extend Brier sore to an inentive ompatible IP-soring rule? The question is ill-formed without a deision rule that extends maximizing expeted utility to IP deisions. That is, whether a partiular IP-soring rule is proper or not, depends upon what deision rules we allow the IP deision maker to use. The IP ommunity has not agreed on the answer to this question. Here, we onsider only deision rules that redue to the rule of maximizing expeted utility when IP sets of probabilities ollapse onto the speial ase of a singleton set, where upper and lower probabilities are idential and a single probability distribution represents unertainty. Also, we require that deision rules respet the following weak form admissibility. Let S(F, ω) be a real-valued IP-soring rule for foreast set F in state ω. Reall that sores are given in the form of a loss so that smaller is better. Admissibility Priniple: If for eah ω Ω S(F, ω) S(F, ω), then F is admissible in a pairwise hoie between rival foreasts F and F. Moreover, if for eah ω this inequality is strit then F is inadmissible whenever F is an option. In this setion we report two results onerning existene of proper soring rules for eliiting upper and lower probabilities for events, when the foreaster s opinion is represented by a losed, onvex sets of probabilities on a finite state spae and deisions onform to the Admissibility Priniple. The first of the two, Proposition 5, establishes that there is no real-valued IP-ounterpart to a ontinuous soring rule, suh as Brier sore. Proposition 5: There is no real-valued stritly proper IP ontinuous soring rule. 9

10 By ontrast, Proposition 6, if one onsiders soring rules with non-standard values, then stritly proper IP-soring rules exist for eah of two IP-deision rules. The IP-deision rules we investigate in onnetion with Proposition 6 are summarized as follows, with additional details given in Setion 4.2: Γ-Maximin: The admissible options in a deision problem D are those that maximize their lower expeted value. E-admissibility: An option X D is E-admissible if for some P P and eah Y D, E P [X] E P [Y]. E-admissibility-followed-by-Γ-Maximin: Apply Γ-Maximin to the set of E-admissible options in D. Proposition 6: Under either the Γ-Maximin deision rule, or using one of Levi s [13] lexiographi deision rules E-admissibility followed by Γ-Maximin seurity there is a stritly proper lexiographi IP-Brier soring rule. Next, we establish and explain these findings. 4.1 Proof of Proposition 5 The impossibility reported in this result is made evident by onsidering the demands on a real-valued stritly proper IP-soring rule S(F, ω), for foreasting one event, E. Let the interval [p, q], 0 p q 1, represent the foreaster s unertainty for E. In general, the IP-soring rule may be written S([x, x], ω) = g 1 ([p, q], ω) if ω E obtains, and S([x, x], ω) = g 0 ([p, q], ω) if ω E obtains. When p = q, in order to be stritly proper and real-valued, the soring rule must satisfy Theorem 4.2 of Shervish [12]. Speifially, with 0 x 1, the loss for the point foreast S([x, x], ω), x satisfies 1 g 1 (x) = g 1 (1)+ (1 q)λ(dq) if ω E obtains; x x g 0 (x) = g 0 (0)+ qλ(dq) if ω E obtains, 0 where g 1 (1) and g 0 (0) are finite, and λ(dq) is a measure on [0, 1) that gives positive measure to every non-degenerate interval. Continuity of the soring rule results from a ontinuous measure λ with no point masses. For example, Brier sore results by letting λ have the onstant density 2 on the unit interval. When p < q, the impossibility of a stritly proper ontinuous IP-soring rule is a onsequene of the fat that, sine λ is positive on non-degenerate sub-intervals of the unit interval [0,1] and ontinuous, and as there is no ontinuous 1-1 map between the unit square and unit interval, there will be rival interval foreasts [p, q] and [p, q ] with g 1 ([p, q]) g 1 ([p, q ]) 0, and g 0 ([p, q]) g 0 ([p, q ]) 0. Then the interval foreast [p, q ] is admissible against the rival interval foreast [p, q]. When the interval [p, q] is the foreaster s IP-unertainty for event E, she/he will not have reason to announe that interval as her/his foreast rather than the rival foreast [p, q ]. Thus, the IP-soring rule is not stritly proper. If eah of the two inequalities is strit, as illustrated in Examples 5 and 6 (below), then the IP-soring rule is not even proper. Example 5. We illustrate Proposition 5 using the ideas about IP-oherene 2 presented in Setion 3. Consider Brier sore adapted to a foreast interval [p, q] using the favorable end of the foreast interval. That is, let b([p,q], ω) = g 1 ([p, q], ω) = (1-q) 2 if ω E, and b([p,q], ω) = g 0 ([p, q], ω) = p 2 if ω E. Introdue a real-valued index of indeterminay for a foreast set F, I(F), where I agrees with the partial order of relative impreision used to define IP-oherene 2. For instane, let I([p, q]) = q-p. For real values x, y, let H(x,y) be a realvalued funtion inreasing in eah of its arguments, e.g., H(x,y) = x + y. Define an IP-Brier sore for foreast set F by B(F,ω) = H(b(F,ω), I(F)). Then by Proposition 5, B is an improper-ip soring rule. To omplete the example, onsider event E and ompare the two interval foreasts [.25,.75] and [.50,.50]. Then B([.25,.75], ω) = 1/16 + 1/2 = 9/16 and B([.50,.50], ω) = 1/4 + 0 = 1/4, all independent of ω. Hene, the interval foreast [.25,.75] is inadmissible under this IP-Brier soring rule B. That is, under this variant of IP-Brier soring rule, a deision maker whose IP-unertainty about the event E is given by the losed interval of probabilities [.25,.75] stritly prefers announing the point-valued foreast [.50,.50] instead of her/his IP-interval, [.25,.75]. 10

11 4.2 Proof of Proposition 6 First we review the two deision rules mentioned in the result. Let P be a losed, onvex set of probabilities P on the spae {Ω, B}. Let χ be the lass of bounded random variables, X, eah measurable with respet to this spae. For eah X, write X for the infimum over P of the expeted value of X, X = inf P P E P [X], whih identifies the lower expeted value for X with respet to P. Identify a deision problem, D, with a losed subset of χ. That is, the options in a deision problem form a losed set of bounded variables. The two IP-deision rules we investigate in Proposition 6 are defined as follows: Γ-Maximin: The admissible options in D are those that maximize their lower expeted value. Note: By making both P and D losed sets, this max-min operation is well defined. E-admissibility: An option X D is E-admissible if for some P P and eah Y D, E P [X] E P [Y]. E-admissibility-followed-by-Γ-Maximin: Apply Γ-Maximin to the set of E-admissible options in D. In general, these deision rules have very different axiomati haraterizations. Γ-Maximin is represented by a realvalued ordering of χ using X-values to index eah option. But that ordering violates the independene axiom for preferenes. E-admissibility is not represented by an ordering. In fat, it does not even redue to pairwise omparisons. (See [24] for related disussion.) Nonetheless, next we onstrut a lexiographi IP-Brier sore that is stritly proper under either of the two deision rules mentioned in Proposition 6. Proposition 5 preludes a proper IP-soring rule that eliits both endpoint of the interval foreast [p,q] for event E. However, we may eliit either endpoint alone. Define the lower-brier soring rule, b([x,y], ω) = b(x,ω) as: g 1 (x) = (1-x) 2 if ω E g 0 (x) = 1 + x 2 if ω E. and the upper-brier soring rule, b ([x,y], ω) = b (y,ω) as: g 1(y) = (1-y) if ω E g 0 (y) = y 2 if ω E. Eah of these is a stritly proper soring rule for eliiting determinate foreasts. This follows immediately from Shervish s representation (above) where g 1 (1) = g 0 (0) = 0, g 1 (0) = g 1(1) = 1, and λ = 2 is the uniform (Brier) sore density for both rules. Lemma 1: Under the Γ-Maximin deision rule, respetively, the lower- (upper-) Brier sore is stritly proper for the lower (upper) endpoint of the IP-foreast [p,q] of event E. Proof of Lemma 1: We give the argument for the lower-brier sore. The reasoning for the upper-brier sore is similar. Let p = min P P P[E] and q = max P P P[E], so that P P p P(E) q, and these bounds are tight. The lower-brier sore of the foreast [r, s] for E depends solely on r. The P-Expeted sore for foreast [r, s] is: E P [b[r, s]] = P(E)(1-r) 2 + (1-P(E))(1+r 2 ) = (1-r) 2 + 2r(1-P(E)). By simple dominane, 0 r 1. For a given foreast r, this expeted penalty sore is greatest among P P at P(E) = p, when the expeted sore is (1-r) 2 + 2r(1-p). But sine lower-brier sore is stritly proper, this worst value is best, i.e., the worst of these expeted sores is smallest uniquely for a foreast with r = p. Lemma 1 Lemma 2: Under the E-admissibility-followed-by-Γ-Maximin deision rule, respetively, the lower- (upper-) Brier sore is stritly proper for the lower (upper) endpoint of the IP-foreast [p,q] of event E. Proof of Lemma 2: Again, we give the argument only for the lower-brier sore. Sine lower-brier sore is a stritly proper soring rule for determinate foreasts, the E-admissible foreasts are exatly those of the form [r, s] where p r q, and r s. That is, onsider P P with P(E) = r. Only foreasts of the form [r, s] maximizes the P-expeted lower- Brier sore. Hene, relative to lower-brier sore, the set of E-admissible foreasts with respet to the IP-set P are interval foreasts of the form {[r, s]: p r q, and r s}. Then, by Lemma 1, the Γ-Maximin solution from this set is uniquely solved at r = p. Lemma 2 11

12 By Proposition 5, unfortunately, the real-valued omposite sore obtained by adding together these two sores, b ([r,s]) = b([r,s]) + b ([r,s], is not an IP-proper soring rule, whih we verify with Example 6. Example 6: We illustrate the impropriety of the real-valued IP-sore, b ([r,s]), in aord with Proposition 5. Consider an extreme ase where the foreaster is maximally unertain of event E, so that the vauous probability interval [0, 1] represents her/his unertainty. The preise foreast [.5,.5] has onstant b -sore, i.e., b ([.5,.5], ω) = 1 + ¼ + ¼ = 1.5, whih is independent of ω. The straightforward foreast [0,1] has the onstant sore b ([0, 1], ω) = 1+1 = 2, whih also is independent of ω. So foreast [.5,.5] stritly dominates foreast [0,1] under the b -soring rule. Instead of a real-valued IP-soring rule, we use a 2-tier lexiographial (non-standard) omposite soring rule to ombine these two sores in a manner that reates a stritly proper (but non-standard) IP-Brier sore. Definition: The two-tier, lexiographi IP-Brier sore for the interval foreast [p, q] of event E, whih we write as b LU ([r, s]), is the 2-tier lexiographi loss funtion b LU ([r, s], ω) = < b([r, s], ω), b ([r, s], ω) >. That is, lexiographially, first apply the loss funtion b([r, s]), and among those foreasts have equal b-value, then apply the b ([r, s]) loss funtion. By the preeding two lemmas, under the two deision rules named in Proposition 6, only the interval [p, q] is b LU -optimal for foreasting event E when the foreaster s unertainty for that event is the IPinterval [p, q]. Aside: It is evident that the order of the omponents is irrelevant in this 2-tiered, lexiographi IP-Brier sore. To eliit an IP-foreast set F = { {p i, q i }: i = 1,, n} for the events {E 1, E 2,, E n } use, e.g., the 2n tiered lexiographi IP-Brier sore < b 1 ([r 1, s 1 ]), b 1 ([r 1, s 1 ]),, b n ([r n, s n ]), b n ([r n, s n ]) >. Then the following is immediate from Proposition 6. Corollary. The 2n-tiered, lexiographi IP-Brier sore is stritly proper under either the Γ-Maximin or E-admissibilityfollowed-by-Γ-Maximin deision rules. As above, the order of the 2n-terms in the lexiography is irrelevant. In the light of Proposition 5, the theory of IP-oherene 2 developed in Setion 3 does not produe a stritly-proper IPsoring rule. In setion 3 all IP-foreasts have real-valued sores. The stritly proper, lexiographi IP-soring rules identified in Proposition 6 do not satisfy that strutural assumption of the theory in Setion 3. Thus, the analyses of Setions 3 and 4 do not yet provide a unified aount of IP-oherene 2 and IP-proper soring rules. In the next Setion, we disuss our ideas about this hallenge, whih was pointed out to us by one of the Readers. 5. Summary When oherene 1 of 2-sided previsions is not enough, and eliitation also matters, then Brier sore offers an inentive ompatible soring rule with an equivalent oherene riterion: oherene 2 avoid dominated foreasts. This is de Finetti s analysis, Proposition 1. We extend Brier soring to IP-oherene 2 of interval-valued foreasts, analogous to the familiar use of 1-sided (lower and upper) previsions for defining IP-oherene 1. Subjet to an IP-soring rule for foreasting events, the oherent foreaster gives lower and upper probabilisti foreasts for a partiular set of events that haraterize elements of an IPmodel lass M e.g., the ε-ontamination lass is haraterized by IP-foreasts for the atoms of the measure spae Proposition 4. Coherene 2 of the set of IP-foreasts requires that these lower and upper foreasts are not dominated by any more determinate IP model within the model lass M, subjet to the same IP soring rule. However, a distinguishing feature between oherene 1 and oherene 2, namely that Brier sore is inentive ompatible for eliitation of 2-sided (real-valued) foreasts for events, does not extend to 1-sided foreasts. That is, aording to Proposition 5, there is no stritly proper, ontinuous real-valued IP-soring rule for events. However, by relaxing the 12

13 onditions on soring rules to permit a lexiographi utility, subjet to either of two IP-deision rules that we investigate, there do exist stritly proper IP-soring rules for eliiting losed, interval-valued probability foreasts. There are numerous open questions relating to the preliminary work reported in this paper. We list four topis on whih we are urrently at work. 1) The entral results reported here about IP-oherene 2 (Setion 3) and IP-inentive ompatible soring (Proposition 6 of Setion 4) are based on Brier soring, whih is the basis of de Finetti s approah to oherene 2. As we showed in [21], eah stritly proper soring rule an serve in plae of Brier sore to provide a foundation for subjetive probability based on foreasting rather than on the prevision game. We onjeture that the positive results reported here about IPoherene 2 an be dupliated using other stritly proper soring rules in plae of Brier sore. 2) As noted in Setion 2, neither oherene 1 nor oherene 2 onstrains, respetively, a alled-off prevision for an event or a alled-off foreast for an event, given a null-event. However, lexiographi expeted utility [14] is one approah among several others available [7, 16, 28] for improving the treatment of 2-sided onditional probability with alled-off previsions given a null-event. (See [2] for a review of some of the open issues.) Proposition 6 identifies a lass of lexiographi soring rules that satisfy IP-propriety with respet to interval valued foreasts for events. But those lexiographi sores do not onform to the strutural assumption of the theory of IP-oherene 2 developed in Setion 3, whih uses only real-valued sores. Can we use lexiographi soring rules to provide a unified theory that provides both an IP-oherene 2 (inluding alled-off foreasts given a non-empty event) and an IP-inentive ompatible soring rule? 3) A different hallenge to eliitation, even when probability is determinate, is the problem posed by state-dependent utilities. This arises in the hoie of the numeraire that is to be used, either with outomes of previsions for oherene 1, or in soring foreasts for oherene 2. (See [20] for disussion of the problem in the setting of oherene 1.) Does foreasting afford any advantage over betting in this ontext and is there a differene also with IP-eliitation? 4) De Finetti s theory of oherene is designed to aommodate all finitely additive probabilities. That is, ountable additivity is not a requirement of oherene 1 or oherene 2. This is ahieved by insisting that inoherene, i.e., a failure of simple dominane, is ahieved using only finitely many previsions or only finitely many foreasts at one time. In other words, a oherent set of previsions or foreasts may be dominated when more than finitely many are ombined at one, even though they annot be dominated when only finitely many are ombined. It is interesting, we find, that even with determinate probabilities, oherene 1 and oherene 2 are not equivalent in this regard. There are settings where ountably many oherent 2 foreasts may be ombined and remain (simply) undominated by all rival foreasts, though these same previsions may result in a sure-loss when ountably many are ombined into a single option [25]. In order to aommodate all finitely additive probabilities, when does IP-oherene 2 depend upon the restrition that violations of dominane matter only when finitely many foreasts are sored at the same time? Aknowledgements Earlier versions of these results were presented at the University of Warwik s Subjetive Bayes Workshop, the Purdue Wimer Memorial Letures workshop, and CMU s Games and Deisions disussion group, and we thank the partiipants at these meetings for their helpful omments. In partiular we appreiate suggestions from Timos Athanasiou, Lua Rigotti, and Kevin Zollman. Also, we appreiate the onstrutive advie of two, areful, anonymous Readers. Appendix 1 The Atomi Lower-Upper Probability [ALUP] lass. This IP-lass onsists of losed, onvex sets of probabilities defined by lower and upper probabilities for atomi events. That is an ALUP model is the largest (losed) onvex set of distributions that satisfy suh bounds, where the bounds are ahieved by the lower and upper probability values given for the atoms of the spae. See [10] for disussion about this IP-lass of models. IP-oherene 2, where rival foreasts are taken from the ALUP lass, arises when the foreaster is alled upon to give lower-and-upper foreasts for eah atom, ω, and for the omplement to eah atom, ω, in the spae. That is, in order to dupliate Proposition 4 for the ALUP lass the foreaster is alled upon to give 2n-many foreasts when Ω = {ω 1,, ω n }. Example 7 illustrates this. Example 7 (a ontinuation of Example 4): An illustration of ALUP-oherene 2. We provide 3 foreast sets for the 13