How Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence

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1 1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh Teddy Sedenfeld Carnege Mellon Unversty Joseph B.Kadane Abstract We ntroduce two ndces for the degree of ncoherence n a set of lower and upper prevsons: maxmzng the rate of loss the ncoherent boomaer experences n a Dutch Boo, or maxmzng the rate of proft the gambler acheves who maes Dutch Boo aganst the ncoherent boomaer. We report how effcent boomang s acheved aganst these two ndces n the case of ncoherent prevsons for events on a fnte partton, and for ncoherent prevsons that nclude also a smple random varable. We relate the epslon-contamnaton model to effcent boomang n the case of the rate of proft. Keywords. Dutch Boo, coherence, ε-contamnaton model 1 Introducton It s a famlar remar that defnett s Dutch Boo argument provdes a smple dchotomy between coherent and ncoherent prevsons. For our presentaton here, consder the followng verson of hs argument, whch we present as a two-person, zero-sum game between a Booe, who s the subject of the argument, and a Gambler, who s the opponent. Let X be a (bounded) random varable defned on some space S of possbltes. The Booe s requred to offer hs/her prevson p(x) on the condton that the Gambler may then choose a real quantty α X,p(X) resultng n a payoff to the Booe of α X,p(X) [X - p(x) ] wth the opposte payoff to the Gambler a zero-sum game. The Booe s prevsons for a set of random varables are ncoherent f there s a (fnte) selecton of non-zero α s by the Gambler that results, by summng, n a (unformly) negatve payoff to the Booe and a (unformly) postve payoff to the Gambler. The Booe s prevsons are coherent, otherwse. Ths leads to defnett s Dutch Boo Theorem The Booe s prevsons are coherent f and only f they are the expectatons of a (fntely addtve) probablty dstrbuton. defnett extends hs analyss to nclude assessments of condtonal prevsons, gven an event F, through calledoff wagers usng the ndcator for F, χ F, of the form α X,p(X),F χ F [X - p(x) ] Moreover, when the random varables X are restrcted to ndcator functons for events, E, the Booe s prevsons are coherent f and only f they are the condtonal probabltes of a sngle (fntely addtve) probablty. In ths case, the magntude α E,p(E),F s the stae for each wager, and the sgn of α E,p(E),F, postve or negatve, determnes whether the Booe bets respectvely, on or aganst E, called-off f F fals to occur. It s a famlar concern, apprecated by many at ths conference, that defnett s crteron of coherence requres that the Booe posts a sngle prevson, or called-off prevson gven F, for each X. For bettng on events, ths amounts to statng hs/her far (called-off) odds : odds that the Gambler may use regardless the sgn of the coeffcent α. In response to ths concern, the game has been relaxed to permt what C.A.B.Smth [4]

2 called lower and upper pgnc odds. That s, n the case of ndcator varables, the Booe may post one prevson p a lower probablty used wth postve α for wagerng on E, and another prevson q an upper probablty used wth negatve α for wagerng aganst E. In effect, the Booe asserts that at odds of p : 1- p or less he/she wll bet on E, whereas at odds of q : 1- q or greater he/she wll bet aganst E. defnett s Dutch Boo theorem generalzes n ths settng to assert, roughly, that the Booe s lower and upper prevsons are coherent f and only f they are, respectvely, the lower and upper expectatons of a convex set of (fntely addtve) probablty dstrbuton. (See [3] for a precse statement of ths result.) Ths generalzaton, however, retans the ntal dchotomy: the Booe s prevsons are coherent or else the Gambler can mae a Dutch Boo an acheve a sure return. 2 Degrees of Incoherence In [2], we ntroduce two ndces of ncoherence: a rate of loss for the Booe and a rate of proft for the Gambler. These ndex the amount of the Gambler s sure-gan aganst ether of two escrow accounts, accounts that reflect the porton of the total stae each player contrbutes. The rate of loss ndexes the Gambler s guaranteed sure gan (.e., the mnmum of the Booe s assured loss) aganst the proporton of the total stae contrbuted by the Booe. The rate of proft ndexes the Gambler s guaranteed sure gan aganst hs/her own contrbuton to the total stae. In what follows, we focus on the second of these two ndces: the rate of proft acheved by the Gambler. Of course, there are more than these two ways of formalzng degrees of ncoherence. Nau [1] gves a flexble framewor that ncorporates our rate of loss as a specal case, for example. 2.1 Incoherence for events n a partton Let {A: = 1,...n} be a partton of the sure-event by n non-empty events, wth n > 1. Let 0 = p = q =1 be the Booe s lower and upper prevsons for the A ( = 1,..., n). Let s + = Σ q and let s- = Σ p, so that the Booe s ncoherent f ether s + < 1 or 1 < s -. Theorem 2 (from [2]): (1) If s + < 1 then the rate of guaranteed proft equals (1- s + )/s + and s acheved when the Gambler sets all the α = -1/ s +. (2) If s - > 1, then the Gambler maxmzes the mnmum rate of proft by choosng the staes accordng to the followng rule: Let * be the frst such that n p < 1+ ( 1) p = n + 1 n wth * = n f ths equalty always fals. Then the Gambler sets α all equal and postve for > n *+1, and sets α = 0 for all < n-*. The Fgure below llustrates ths result for the case wth n = 3 atoms, a ternary partton. The set of coherent defnett-prevsons s represented by the tranglular hyperplane: the smplex wth extreme values {(1,0,0), (0,1,0), (0,0,1)}. The set of ncoherent lower probabltes, where s - > 1, les above t. The selected hyperplane n the fgure s comprsed of lower probabltes wth s - = 1.5. For those lower prevsons n the whte-regon, outsde the projecton of the coherent smplex, the Gambler maxmzes hs/her rate of proft (whch equals 3/7) by gnorng the Booe s prevson on A3, and achevng boo by havng the Booe bet on each of A1 and A 2, at equal staes. That would be the case f the Booe s lower prevsons were (.6,.7,.2). If, however, the Booe s prevsons were nsde the projecton of the coherent smplex, e.g., (.5,.5,.5), then the Gambler s rate of proft s only 1/3, acheved wth equal staes on each of the three atoms. 2.2 Incoherence wth prevsons for a smple random varable Next, consder the addton of a sngle random varable defned by a (fnte) partton, {A: = 1,...n}, as n the subsecton above. Let X be a (smple) random varable defned on these n events. For the next result, we assume that the the Booe gves prevsons p = p(a () ), ordered to be ncreasng n p, whch are sngly coherent, 0 < p < 1. Also, the Booe gves a prevson for X.. For smplcty, we state the followng result for the case s < 1. Defne these seven quanttes, s = Σ p µ = Σ x p and δ = - µ. p n, 1 s+, = = n + 1 s = = v ( ) = p X x p (1 s, ) x1 v = 1 n ( ) = (1 s xn + + +, ) = n + 1 p x p p X

3 Two atom strategy regon (1,0,1) (0,0,1) (1,1,1) (0,1,1) (0,0,0) (1,0,0) (.6,.7,.4) (.6,.7,.3 (.6,.7,.2) (1,1,0) (0,1,0) Theorem 6 (of [2]) The Gambler acheves the maxmum guaranteed rate of proft, as follows: 1) If δ < (1-s)x 1, let * be the smallest value of such that v - () < 0. Then set α X = -1, set α = x x * for < * and set α = 0 for > *. 2) If δ > (1-s)x n, let * be the smallest value of such that v + () < 0. Then set α X = 1, set α = -x + x n-* for > n-* and α = 0 for < n -*. 3) If (1-s)x 1 < δ < (1-s)x n, then set α X = 0 and apply the prevous theorem,.e., gnore the Booe s prevson for X but, nstead, use solely the ncoherence among the p. A Corollary to ths Theorem s nterestng and ntellgble on ts own. Havng already gven the (possbly ncoherent) prevsons p, and now oblged to provde the addtonal prevson, the Booe can as how to avod ncreasng the rate of proft that the Gambler may acheve. Corollary The Gambler s rate of proft after learnng the Booe s prevson does not ncrease f and only f satsfes: µ + (1-s) x 1 < < µ + (1-s) x n. That s, the corollary dentfes the Booe s mnmax strateges for augmentng the prevsons p for the events A, wth a sngle new prevson for X. Ths corollary apples to called-off bettng as a specal case:

4 Consder the ternary partton and random varable X whose values are gven n the second row of the followng table. Thus, a 1 a 2 a α[1-p(x)] -αp(x) are the three correspondng payoffs to the Booe assocated wth the wager α[x - p(x)]. Then, e.g., wth s < 1, havng already announced the prevsons p ( = 1, 2, 3), the Booe s mnmax strateges for restranng the Gambler s rate-of-proft satsfes: p 1 + p 2 < < p 1 + p s. It s nterestng to note that choosng the pseudo-bayes condtonal value = p 2 /( p 2 + p 3 ) always satsfes these nequaltes. In other words, the ncoherent Booe can tae advantage of the fact that the pseudo-bayes solutonn s mnmax. You don t have to be coherent to le Bayes solutons! Of course, f s = 1, so that the Booe s coherent, the sole mnmax soluton s just the Bayes soluton. 3 Epslon-contamnaton and the rate of guaranteed proft The Gambler s decsons n the frst of the two Theorems, n secton 2.1 above, can be explaned wth an ε-contamnaton model, through the Bayesan dual to the mnmax strateges for ths case. For the Gambler to accept wagers when the Booe offers upper probabltes, the Gambler must fnd these wagers acceptable as lower probabltes n a ratonal decson. Smlarly, for the Gambler to accept wagers when the Booe offers lower probabltes, the Gambler must fnd these wagers acceptable as upper probabltes n a ratonal decson. Gven a fxed probablty dstrbuton, p*, an ε- contamnaton model of probabltes s a set of probabltes Mp* = {(1-ε)p* + εq: 0 < ε < 1}, wth q an arbtrary probablty. Equvalently for fnte algebras, an ε-contamnated model s gven by specfyng a coherent set of lower probabltes for the atoms of the algebra. When the Booe s ncoherent wth upper probabltes, s + < 1, these may be the coherent lower probabltes for the Gambler usng an ε-contamnaton model. In fact, the Gambler maxmzes hs/her expected rate of proft accordng to ths (convex) set by wagerng as ndcated n (1) of the Theorem. When the Booe s ncoherent wth lower probabltes, s - > 1, t s not always the case that these can be the coherent upper probabltes for an ε- contamnaton model. Precsely when the Booe s lower probabltes fall wthn the projecton of the coherent smplex, when they fall wthn the trangular regon llustrated n the Fgure, then the Gambler may use these as the coherent upper probabltes from an e- contamnaton model. Otherwse, the Gambler fts the largest ε-contamnaton model that s allowed by the Booe s offers. Expressed n other words, the strateges reported by the Theorem are those whch gve the Gambler a postve expected value for each component wager used to mae the Dutch Boo, and these relate to an ε-contamnaton model, as just explaned. The same analyss apples to the second of the two Theorems, n secton 2.2 above. Ths case nvolves the Gambler s rate of guaranteed proft when the Booe s prevsons nclude a set of bets on a fnte partton and a prevson for one (smple) random varable defned on that partton. The nequaltes of Theorem 6 correspond, n precsely the same way, to the upper and lower expectatons from an ε-contamnaton model, based on the Booe s ncoherent upper prevsons,.e., when s < 1. The conference presentaton ncludes, also, results for the parallel case when s >1. Then Gambler s maxmn strateges for securng an effcent Dutch Boo, reflect the added complcaton of truncaton of the ε- contamnaton model, just as n the correspondng case (s > 1) for the Theorem of secton Concluson Ths presentaton ntroduces the use of a convex set of coherent probabltes, the ε-contamnaton model, as the Bayes dual solutons to a Gambler s maxmn strateges for what we call the guaranteed rate of proft n mang effcent Dutch Boo aganst an ncoherent Booe. The two cases dscussed here nclude (1) ncoherent upper and lower prevsons for events n a fnte partton, and (2) a context where the Booe ncludes a prevson for a smple random varable defned on ths same partton.

5 Ongong wor (to be reported at the conference) specfces the correspondng convex set of probabltes that are dual to the Gambler s maxmn strateges for maxmzng the Booe s guaranteed rate of loss n each of these two cases. These sets nvolve fxng both upper and lower probablty bounds on the atoms of the fnte algebra, rather than merely fxng the lower probabltes, as s done n an ε-contamnaton model. References [1] Nau, R.F. (1989), Decson analyss wth ndetermnate or ncoherent probabltes. Ann. Oper. Res [2] Schervsh, M.J., Sedenfeld, T., and Kadane, J.B. (1998), Two Measures of Incoherence: How Not to Gamble If You Must, T.R. #660 Dept. of Statstcs, Carnege Mellon Unv., Pgh. PA (A Postscrpt fle s avalable at: [3] Sedenfeld, T., Schervsh, M.J., and Kadane, J.B. (1990), Decsons wthout orderng. In Actng and Reflectng (W.Seg, ed.) Kluwer Academc Publshers, Dorddrecht. [4] Smth, C.A.B. (1961), Consstency n Statstcal Inference and Decson, J.R.S.S. B, 23, 1-25.

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