Tight Upper Bounds for the Expected Loss of Lexicographic Heuristics in Binary Multi-attribute Choice

Transcription

1 Tight Uppe Bounds fo the Expected Loss of Lexicogaphic Heuistics in Binay Multi-attibute Choice Juan A. Caasco, Manel Baucells Except fo fomatting details and the coection of some eata, this vesion matches exactly the vesion published with the same title and authos in Mathematical Social Sciences, vol. 55, no. 2, 2008, pp Abstact Tight uppe bounds fo the expected loss of the DEBA Deteministic-Elimination-By-Aspects lexicogaphic selection heuistic ae obtained fo the case of an additive sepaable utility function with unknown non-negative, non-inceasing attibute weights fo numbes of altenatives and attibutes as lage as 10 unde two pobabilistic models: one in which attibutes ae assumed to be independent Benouilli andom vaiables and anothe one with positive inte-attibute coelation. The uppe bounds impove substantially pevious bounds and extend significantly the cases in which a good pefomance of DEBA can be guaanteed unde the assumed cognitive limitations. Keywods: Multiattibute decision making; Binay attibutes; Lexicogaphic heuistics; DEBA; Pefomance bounds JEL Classification: C44, C63. Depatament d Enginyeia Electònica, Univesitat Politècnica de Catalunya, Diagonal 647, plta. 9, Bacelona, Spain, caasco@eel.upc.edu IESE Business School, Univesidad de Navaa, Avda. Peason 21, Bacelona, Spain, mbaucells@iese.edu

2 1 Intoduction The goal of this pape is the fomal study of the pefomance of lexicogaphic heuistics in multiattibute decision making. Lexicogaphic heuistics ae psychologically appealing because in complex multi-attibute choices povide a decision ule that avoids explicit tade-offs. We have in mind a standad multi-attibute decision poblem Keeney and Raiffa 1993, with an additive, sepaable utility function and m altenatives chaacteized by k attibutes x i,, 1 i m, 1 k. The utility of the altenative i, chaacteized by the pofile x i = x i,1, x i,2,..., x i,k, is defined as U i = w 1 x i,1 + w 2 x i,2 + + w k x i,k, whee the w ae non-negative weights subject to the constaint w 1 + w w k = 1. Cognitive limitations ae intoduced by assuming that the decision make can ode the weights by size, but that the exact values of the weights ae unknown. Theefoe, without loss of geneality, we assume w 1 w 2 w k 0. The poblem is to identify which of the m altenatives is best has the lagest value of U i unde this cognitive limitation. To esolve this poblem, we conside a decision make that follows a lexicogaphic heuistic that elies on the odeing, but not on the magnitude, of the weights. The specific lexicogaphic heuistic that we will examine is DEBA, the deteministic vesion of the elimination-by-aspects model poposed by Tvesky DEBA consides the attibutes in deceasing weight odeing. In the fist step, DEBA eliminates all the altenatives without the maximum values in the fist attibute. If a single altenative emains, it is chosen. Othewise, the values of the second attibute ae examined. This pocedue continues until only one altenative emains o all attibutes have been examined. If, afte examining the last attibute, two o moe altenatives emain, then the choice between them is made at andom. As a pocedue, DEBA genealizes to moe than two altenatives the lexicogaphic binay-choice model Take-The-Best TTB poposed by Gigeenze and Goldstein Thee is a small diffeence, howeve: in TTB, the attibutes ae odeed by thei validities, which ae computed using a database of pevious instances of altenatives, while in DEBA the odeing of the attibutes by deceasing weights is assumed known. Abstacting in TTB the fist step, TTB can be tuly consideed a paticula instance of DEBA with just two altenatives. In this pape, we make the additional assumption that the x i, ae binay, i.e., thei value is eithe 0 o 1. This will be always the case fo attibutes featues that ae eithe pesent o absent, o that take two values. Besides mathematical tactability, the binay setup has anothe impotant advantage. In a binay setup, DEBA agees with othe lexicogaphic heuistics that diffe on the cut-off used to encode the attibute values as high x i, = 1 o low x i, = 0. This binay encoding phase, togethe with the ule to ode the attibutes, pecedes the use of the common lexicogaphic selection ule. Fo example, one could assign a value 1 only to those attibute values with the best level on that attibute; o use instead a low cutoff epesenting a minimum acceptable level. Those two choices yield, espectively, the EBA and the LEX heuistics discussed by Payne et al The binay encoding is also a way to incopoate the bounded ationality of decision-makes that 1

3 easily distinguish between zeo absence and non-zeo pesence values, but ae quite insensitive to the actual magnitude of the attibutes Hsee and Rottensteich The DEBA heuistic is easy to use and popula Böde 2000; Newell and Shanks 2003; Newell et al In many situations, fo example, thee is no need to look beyond the fist one o two attibutes to make a decision. Seveal studies have shown DEBA to be effective in elation to altenative simple decision heuistics Gigeenze and Goldstein 1996; Czelinski et al. 1999; Matignon and Hoffage 1999, 2002 as well as having desiable popeties fo both binay and multivaiate choice Hogath and Kaelaia 2006; Katsikopoulos and Fasolo, Even when attibutes ae continuous vaiables, the model can be quite effective unde some cicumstances Gigeenze et al. 1999; Hogath and Kaelaia Most of these studies give exact pefomance measues fo two o thee altenatives, o povide pefomance estimates based on simulation, o on paticula data sets. An exception is Baucells et al. in pess, hencefoth BCH, whee lowe bounds fo the pobability that DEBA will choose a best altenative one with the lagest utility and uppe bounds fo the expected loss of DEBA have been obtained by exploiting the concept of cumulative dominance Kikwood and Sain Two binay pobabilistic models wee consideed in BCH: one in which attibutes ae assumed to be independent Benoulli andom vaiables and anothe one with positive inte-attibute coelation and attibutes of the same aveage quality. The bounds ae not esticted to DEBA: they apply, espectively, to the so-called cumulative dominance compliant heuistics and fully cumulative dominance compliant heuistics, of which DEBA is an example. The bounds wee obtained using exact computational appoaches allowing the analysis of cases with up to 10 altenatives and 10 attibutes. The lowe bounds fo the pobability that the selection heuistic will choose a best altenative allowed the identification of cases in which the selection heuistic is guaanteed to have a good pefomance unde the assumed cognitive limitations. The uppe bounds fo the expected loss widened that identification. The expected loss, o expected diffeence between the utility of a best altenative and the utility of the altenative chosen by DEBA is the focus of this pape. The uppe bounds fo the expected loss of DEBA obtained in BCH wee deived by noting that, in the pesence of cumulative dominance, DEBA is ensued to make the coect choice and incu zeo utility loss, and, if cumulative dominance holds up to attibute, then the loss is non-geate than the sum of weights fom + 1 to k, which can be shown to be tightly bounded by k /k. That appoach assumes a most pessimistic all-0 path fom attibute + 1 on fo the altenative chosen by DEBA, togethe with a most optimistic all-1 path fom attibute + 1 on fo the best altenative, ignoing the statistical popeties of the pobabilistic models. Ou goal hee is to epai this shotcoming and exploit the peculiaities of DEBA to impove the uppe bounds fo the expected loss deived in BCH. We will conside the same two pobabilistic models as in BCH. The appoach we will follow is to impove the uppe bounds fo the expected loss conditioned on the last attibute index fo which some altenative exhibits cumulative dominance. The esulting new uppe bounds ae tight, impove substantially the uppe bounds obtained in BCH, and widen significantly the identification of cases in which DEBA is guaanteed to have a good pefomance unde the assumed cognitive limitations. Such guaantee can 2

4 be exploited to disegad the use of moe sophisticated and moe costly decision ules, including the obtention of moe accuate estimates fo the weights. To calculate the bounds we use computational appoaches simila to the ones used in BCH yielding exact values fo cases with up to 10 altenatives and 10 attibutes. The est of the pape is oganized as follows. Section 2 pesents the pobabilistic models fo the attibutes which will be consideed they ae the same as those consideed in BCH, eviews the cumulative dominance concept, and summaizes the developments in BCH egading the uppe bounds fo the expected loss unde those pobabilistic models. Sections 3 and 4 ae quite technical, and deive the impoved, new uppe bounds fo the pobabilistic models without and with coelation, espectively. Section 5 includes the numeical calculation of the impoved uppe bounds fo the expected loss of DEBA unde ou two pobabilistic models, the compaison with the uppe bounds obtained in BCH, and the analysis of the tightness of the new uppe bounds. We also show in that section that the new uppe bounds widen significantly the identification of cases in which DEBA is guaanteed to have a good pefomance. Section 6 concludes the pape and highlights futue eseach diections. Finally, the Appendix includes some poofs. Thoughout the pape we use the conventions that n i=n u i = 0, n < n and that 0 by an undefined quantity is equal to 0. 2 Peliminaies Two pobabilistic models fo the values of the attibutes x i,, 1 i m, 1 k will be consideed: ZIAC Zeo Inte-Attibute Coelation model: The x i, ae independent Benoulli andom vaiables with paamete p, 0 < p < 1. PIAC Positive Inte-Attibute Coelation model: The x i, ae obtained as x i, = 1 z i yi, l + z i yi, h, whee the z i, yi, l, yh i, ae independent Benoulli andom vaiables with paametes p, p l = p ρp, and p h = p + ρ1 p, espectively, fo some 0 < p < 1 and some 0 ρ < 1. The ZIAC model is a simple model without need fo justification. We note that E[x i, ] = p. Thus, the paamete p can be looked at as measuing the aveage quality of the attibute : highe values of p model attibutes of highe aveage quality. The PIAC model is intuitively appealing: if thee is positive coelation among the attibutes of a given altenative, it is because thee is some common cause shifting the aveage quality of the attibutes of a given altenative. In the PIAC model, this is captued by the altenatives belonging to a good population with expected values fo the attibute values equal to p h = p + ρ1 p with pobability p and to a bad population with expected values fo the attibute values equal to p l = p ρp with pobability 1 p. In the PIAC model, E[x i, ] = p and the attibute values of any given altenative have positive coelation ρ. The ZIAC model with p = p, 1 k, can be seen as a paticula case of the PIAC model 3

5 X i, 3 2 x 1 =1,0,1,0 x 2 =1,,1 x 3 =0,1,1, Figue 1: Altenative pofiles illustating cumulative dominance in the binay attibute case. with ρ = 0. Since k =1 w = 1, in the PIAC model the expected value of the utility of any given altenative i is E[U i ] = p, so the paamete p measues the aveage quality of an altenative. In the ZIAC model, E[U i ] = k =1 w p, and the unknown weigthed sum k =1 w p is a measue of the aveage quality of an altenative. Of couse, fo p = p, 1 k, E[U i ] = p, and the paamete p is in that case a measue of the aveage quality of an altenative. In BCH, uppe bounds fo the expected loss of DEBA and simila elated heuistics wee obtained fo the ZIAC and the PIAC models using the concept of cumulative dominance. We will stat the eview of those uppe bounds by eviewing the concept of cumulative dominance. The cumulative pofile of an altenative i, 1 i m, is defined as X i,1, X i,2,..., X i,k, whee X i, = s=1 x i,s, 1 k. Altenative i exhibits cumulative dominance ove altenative j up to attibute, denoted by c i, j, if and only if X i,s X j,s, 1 s. Altenative i exhibits cumulative dominance ove altenative j if and only if c k i, j, i.e. if altenative i exhibits cumulative dominance ove altenative j up to attibute k. Fig. 1 illustates cumulative dominance in the binay attibute case. Altenative 2 exhibits cumulative dominance ove altenative 3 up to attibute 2, and altenative 1 exhibits cumulative dominance ove altenatives 2 and 3. Futhe, we can use the fact that weights ae non-inceasing to conclude that U 1 = w 1 + w 3 U 2 = w 1 + w 4 and U 1 = w 1 + w 3 U 3 = w 2 + w 3. It is known that cumulative dominance chaacteizes optimality fo non-inceasing weights Kikwood and Sain 1985: Poposition 1. U i U j fo all weights w 1 w 2 w k 0, k =1 w = 1 if and only if c k i, j. Fo 1 k, let C denote the set of altenatives that exhibit cumulative dominance ove any othe altenative up to attibute, i.e., C = i, 1 i m : c i, j, 1 j m}. Obviously, C 1 C 2 C k. All altenatives in C have identical cumulative attibute pofile up to attibute and, theefoe, they have identical attibute pofile up to attibute. Moe impotantly, if C k is non-empty, then Poposition 1 guaantees that the altenatives in C k will have 4

6 the lagest utility. In the example of Fig. 1, C 1 = C 2 = 1, 2} and C 3 = C 4 = 1}. C 1 will always be non-empty. In the binay attibute case, C 2 will be always non-empty too. This follows by noting that C 2 can only be empty if thee exist two altenatives i, j with x i,1 > x j,1 and x i,1 + x i,2 < x j,1 + x j,2, which, being x i, and x j, binay, is impossible. Fo 3, thee is no guaantee that C will be non-empty. Conside fo instance the case of two altenatives with attibute pofiles x 1 = 1, 0, 0 and x 2 = 0, 1, 1. In that case, we have C 3 =. We say that a heuistic is cumulative dominance compliant if, wheneve C k, the heuistic chooses an altenative fom C k. It follows fom Poposition 1 that: Theoem 1. Fo all weights w 1 w 2 w k 0, k =1 w = 1, if C k is non-empty, then any cumulative dominance compliant heuistic will choose a best altenative. Theoem 1 is not esticted to the binay attibute case. Let denote the highest attibute index fo which some altenative exhibits cumulative dominance ove all othe altenatives. Fomally, = max 1 k : C }. By definition, C =, < k. Of couse, C k is non-empty if and only if = k. In the binay attibute case, 2. Fo non-binay attibutes, could be equal to 1. A heuistic is said to be fully cumulative dominance compliant if it always chooses an altenative fom C. Fully cumulative dominance compliance implies cumulative dominance compliance. The following esult has been shown in BCH: Theoem 2. DEBA fully complies with cumulative dominance. Let c the altenative chosen by DEBA. Then, the loss of the heuistic is L = max 1 i m U i U c. The appoach taken in BCH to uppe bound E[L] fo any fully cumulative compliant heuistic was to compute the pobability mass function of, P = P [ = ], 2 k 1, and to use the uppe bound fo E[L = ], 2 k 1, given by the following theoem, which holds in the moe geneal case of abitay attibutes taking values in the inteval [0, 1]: Theoem 3. Fo any fully cumulative compliant heuistic E[L = ] k /k, 1 k. Since L is 0 when = k and, in the binay attibute case, 2, Theoem 3 allow us to wite: k 1 E[L] E[L] ub = P k k =2. 1 The pocedue poposed in BCH to compute the pobabilities P is as follows. Let Q = P [ ], 2 k. Clealy: P = Q Q + 1, 2 k 1, 5

7 with Q2 = 1, educing the computation of P, 2 k 1 to that of Q, 3 k. The pobabilities Q, 3 k wee computed using ROBDDs Reduced Odeed Binay Decision Diagams with complement 0-edges. ROBDDs Byant 1986 ae canonical epesentations of Boolean functions which only depend on the odeing of the binay vaiables. Given a Boolean function F x 1, x 2,..., x n of n independent Benoulli andom vaiables, we can compute P [F x 1, x 2,..., x n = 1] by building the ROBDD of F as a function of x 1, x 2,..., x n and, then, making a depth-fist tavesal of the ROBDD stating at the oot node. When etuning fom the visit to each non-teminal node n with binay vaiable associated with it x, we obtain the pobability that the Boolean function epesented by n is equal to 1 by multiplying the pobability that the Boolean function epesented by the 0-edge node is equal to 1 by the pobability that x has value 0, multiplying the pobability that the Boolean function epesented by the 1-edge node by the pobability that x has value 1, and adding up those patial esults. The ROBDD of a Boolean function can be built fom a desciption of the function using Boolean opeatos fo instance, NOT, AND, OR, which can be looked at as a combinational cicuit Ecegovac and Lang 1985 having inputs x 1, x 2,..., x n and a single output, by tavesing depth-fist the desciption and using well-known pocedues Byant 1986 to obtain the ROBDD of the output of a NOT gate fom the ROBDD of the input, and the ROBDD of the output of a two-input gate fo instance, AND, OR fom the ROBDDs of the two inputs. ROBDDs with complement 0-edges Bace et al ae a vaiant of ROBDDs in which the oot may epesent eithe the given Boolean function F x 1, x 2,..., x n o its complement and non-teminal nodes may have eithe a 0-edge node and a 1-edge node o a complement 0-edge node an a 1-edge node, the complement 0-edge node of a node n epesenting the complement of the Boolean function obtained fom the Boolean function epesented by n by setting the vaiable x associated with n to 0. Adapting the peviously descibed pocedue to compute P [F x 1, x 2,..., x n = 1] to that vaiant of ROBDDs is tivial. To build ROBDDs with complement 0-edges the well-known CU Decision Diagam package Somenzi 2005 was used. The binay vaiables wee soted taking into account the stuctue of the combinational cicuit epesenting F x 1, x 2,..., x n using the topology heuistic Nikolskaia et al The pobability Q can be computed as the pobability that the indicato function of the event } = C } is equal to 1. Fo the ZIAC model, the independent Benoulli andom vaiables to be consideed ae x i,s, 1 i m, 1 s, and the combinational cicuit was built based on m m 1 C } = 1 Xi,s X j,s }. i=1 j=1 s=1 j i Fo s = 1, the logic used to geneate 1 Xi,1 X j,1 }, 1 i m, 1 j m, j i was 1 Xi,1 X j,1 } = 1 xi,1 x j,1 } = x i,1 x j,1. Fo geneating 1 Xi,s X j,s }, 1 i m, 1 j m, j i, 2 s fom x i,s, 1 i m, 1 j m, j i, 2 s we used m 1 specialized binay cay popagate addes and mm 1 1 binay compaatos see BCH, fo details. Fo the PIAC model, the independent Benoulli andom vaiables to be consideed ae z i, 1 i m and yi,s l, yh i,s, 1 i m, 1 s and the combinational cicuit yielding 1 C } was built as fo the ZIAC model, adding the logic to 6

8 obtain x i,s, 1 i m, 1 s x i,s = z i y l i,s z i y h i,s. Intuitively, it is clea that the uppe bounds fo E[L = ] given by Theoem 3 on which the uppe bound E[L] ub fo E[L] given by 1 ae based can be quite ough fo DEBA. Being DEBA fully cumulative dominance compliant, the altenative c chosen by DEBA will belong to C. Then, fo =, c can only loose fom attibute + 1 on and the uppe bounds given by Theoem 3 ae obtained by assuming that c will have an all-0 patten fom attibute + 1 on and some othe altenative will have an all-1 patten fom attibute +1 on, giving a loss fom attibute +1 on equal to k s=+1 w s, which is tightly bounded by k /k. The impoved uppe bounds fo E[L] fo DEBA deived in this pape ae simply obtained by deiving bette uppe bounds fo E[L = ], 2 k 1 using the popeties of the pobabilistic models unde consideation. 3 Impoved uppe bounds fo the ZIAC model The goal of this section is to deive impoved uppe bounds E[L = ] iub fo E[L = ], 2 k 1 fo the ZIAC pobabilistic model taking advantage of the popeties of DEBA and the ZIAC pobabilistic model. Use of those impoved uppe bounds in conjunction with the ROBDDbased appoach used in BCH and eviewed in the pevious section to compute the pobability mass function of, P = P [ = ], 2 k 1, will yield the impoved uppe bound fo E[L] k 1 E[L] iub = P E[L = ] iub. 2 =2 In the deivation of the impoved uppe bounds fo E[L = ], 2 k 1 we will use the following theoem. Theoem 4. Let, 1 k and let E be any event depending only on x j,s, 1 j m, 1 s 1 such that P [i is chosen by DEBA E ] > 0. Let x i, be the m 1k + 1- vecto with components x j,s x i,s, 1 j m, j i, s k. Let F u 1,..., u m 1k +1 be any function which is non-deceasing on each u l. Then, fo the ZIAC model, E[F x i, i is chosen by DEBA E ] E[F x i, ]. Poof. See the Appendix. Essentially, what Theoem 4 says is that, conditioned on any event E do not depending on the altenatives fom attibute on and which does not peclude any given altenative i to be chosen by DEBA, and in tems of a function which is non-deceasing on the diffeences between the values of the attibutes fom attibute on of the othe altenatives and the coesponding attibutes of 7

9 the altenative chosen by DEBA, fo the ZIAC pobabilistic model, DEBA cannot pefom wose in aveage than a andom selection E[F x i, ] is, by symmety, equal fo all i, 1 i m. The impoved uppe bounds fo E[L = ], 2 k 1 ae given by the following theoem, whee I + x = max0, x}. The theoem also assets that the bounds potentially impove those used in BCH. The sketch of the deivation of the bounds is as follows. Fist, it is elatively easy to show using linea pogamming esults that E[L = ] V, t max, +1 t k t whee V, t is any uppe bound fo E[S, t = ] with } S, t = I + max 1 + x i,s x c,s. 1 i m i c To deive the uppe bound, we exploit the symmeties of the ZIAC model and use Theoem 4 to conclude that Theoem 5. Fo DEBA and the ZIAC model, E[S, t = ] E[Z, t]. whee E[L = ] E[L = ] iub = V, t max, 2 k 1, +1 t k t V, t = E[Z, t], with Z, t = I + max 1 i m Futhemoe, E[L = ] iub k /k, 2 k. x i,s x m,s }}. Poof. Assume =, 2 k 1, and let c be the altenative chosen by DEBA. Fo any altenative i, 1 i m, we can wite k U i = w s w s+1 X i,s + w +1 X i,+1 + w s x i,s s=1 and, since c has cumulative dominance ove all othe altenatives up to attibute and weights ae non-negative and non-inceasing, fo all i, 1 i m, i c, k U i U c w +1 X i,+1 X c,+1 + w s x i,s x c,s, yielding L I + max w +1 X i,+1 X c, i m i c k w s x i,s x c,s }. 3 8

10 We continue by finding the maximum of w +1 X i,+1 X c,+1 + k w sx i,s x c,s subject to the constaints defining the domain of possible values fo w +1, w +2,..., w k : w k 0, w s w s+1, + 1 s k 1, + 1w +1 + k w s 1, whee the last one comes fom w 1 w 2 w w +1 and k s=1 w s = 1. This is a linea pogamming poblem Luenbege, Using the vaiables x s = w s w s+1, + 1 s k 1 and a slack vaiable y, the poblem can be put into standad fom with the estictions: k 1 s=+1 sx s + kw k + y = 1, w k 0, x s 0, + 1 s k 1, y 0, which has basic feasible solutions 1 x +1, x +2,..., x k 1, w k, y =, 0,..., 0, 0, 0, x +1, x +2,..., x k 1, w k, y = 0,,..., 0, 0, 0, x +1, x +2,..., x k 1, w k, y = 0, 0,..., k 1, 0, 0, x +1, x +2,..., x k 1, w k, y = 0, 0,..., 0, 1k, 0, x +1, x +2,..., x k 1, w k, y = 0, 0,..., 0, 0, 1. Since the convex domain defined by the estictions is bounded, the maximum is achieved at some basic feasible solution. Those basic feasible solutions coespond to the points w +1, w +2,..., w k 1 w +1, w +2,..., w k = + 1, 0,..., 0, 1 w +1, w +2,..., w k = + 2, 1 + 2,..., 0, w +1, w +2,..., w k =. 1 k, 1 k,..., 1, k w +1, w +2,..., w k = 0, 0,..., 0, 9

11 and, theefoe, the maximum of w +1 X i,+1 X c,+1 + k w sx i,s x c,s is 1 max X i,+1 X c,+1 + x i,s x c,s. +1 t k t Using that maximum in 3: L I + max 1 i m i c 1 = max +1 t k t I + max +1 t k max 1 i m i c 1 t X i,+1 X c,+1 + X i,+1 X c,+1 + x i,s x c,s x i,s x c,s }. But, because c cumulative dominates all othe altenatives up to attibute and attibutes ae binay, necessaily X i,+1 X c,+1 + 1, i c, and with yielding L S, t = I + max i m i c E[L = ] S, t max +1 t k t whee V, t is any uppe bound fo E[S, t = ]. with Exploiting the symmeties of the ZIAC model, x i,s x c,s }, V, t max, +1 t k t E[S, t = ] = E[Z, t m is chosen by DEBA = ], Z, t = I + max 1 i m x i,s x m,s But, the event = only depends on x j,s, 1 j m, 1 s + 1, P [m is chosen by DEBA = ] > 0, and Z, t is non-deceasing on each x i,s x m,s, 1 i m 1, + 2 s k, and, then, using Theoem 4, E[S, t = ] E[Z, t]. }. It emains to show that E[L = ] iub k /k, 2 k 1. Since Z, t t, we have E[Z, t] t. Then, E[L = ] iub max +1 t k t / = k /k. The E[Z, t] s involved in the impoved uppe bounds fo E[L = ] given by Theoem 4 can be obtained by using the ecuence-based computational scheme given by the following theoem. The ecuences can be easily obtained fom the definition of E[Z, t]. 10

12 Theoem 6. Fo the ZIAC model, E[Z, t], 2 k 1, + 1 t k can be computed using and the ecuences: ψ, t, a = t a =a t E[Z, t] = a ψ, t, a a=1 φm 1,, t, a π, t, a a, 2 k 1, + 1 t k, 1 a t, φ1,, t, a = π, t, a 1, 2 k 1, + 1 t k, 1 a t, φb,, t, a = φb 1,, t, a a 1 a =0 π, t, a + π, t, a 1 a 1 a =1 φb 1,, t, a, 2 b m 1, 2 k 1, + 1 t k, 1 a t, π, + 1, 0 = 1, 2 k 1, π, + 2, 0 = 1 p +2, 2 k 2, π, + 2, 1 = p +2, 2 k 2, π, t, t 1 = p +2 π + 1, t, t 2, 2 k 3, + 3 t k, π, t, a = 1 p +2 π + 1, t, a + p +2 π + 1, t, a 1, 2 k 3, + 3 t k, 1 a t 2, π, t, 0 = 1 p +2 π + 1, t, 0, 2 k 3, + 3 t k. Poof. Let, fo 2 k 1 and + 1 t k, be the andom vaiables H i, t = Ib,, t = max 1 i b 1 + x i,s, 1 i m, x i,s }, 1 b m 1, and let by symmety, all H i, t, 1 i m have the same pobability mass function π, t, a = P [H i, t = a], φb,, t, a = P [Ib,, t = a], ψ, t, a = P [Z, t = a]. Then, the esult follows using elementay pobability theoy by noting that I1,, t = 1 + H 1, t, Ib,, t = maxib 1,, t, 1 + H b, t}, 2 b m 1, Z, t = I + Im 1,, t H m, t. 11

13 4 Impoved uppe bounds fo the PIAC model In this section, we deive impoved uppe bounds E[L = ] iub fo E[L = ], 2 k 1 fo the PIAC pobabilistic model taking advantage of the popeties of DEBA and the PIAC pobabilistic model. The bounds will be obtained in tems of the conditional pobabilities P [A l G = g], 2 k, 0 g m 1, P [A h G = g], 2 k, 1 g m, and P [A l,h G = g], 2 k, 1 g m 1, whee G is the andom vaiable numbe of good altenatives, A l is the event some bad altenative but no good altenative cumulative dominates all othe altenatives up to attibute, A h is the event some good altenative but no bad altenative cumulative dominates all othe altenatives up to attibute, and A l,h is the event some good altenative and some bad altenative cumulative dominates all othe altenatives up to attibute. The main eason why the uppe bounds fo E[L = ], 2 k 1 ae obtained in tems of those conditional pobabilities is that the conditional pobabilities can be computed using ROBDDs which tuned out to have modeate sizes fo values of m and k as lage as 10. The uppe bounds fo E[L = ], 2 k 1 will also depend on the pobability mass function of, P = P [ = ], 2 k 1. That pobability mass function can be easily obtained fom the conditional pobabilities and can be used to obtain the impoved uppe bound E[L] iub fo E[L] fom the uppe bounds fo E[L = ], 2 k 1 using 2 k 1 E[L] iub = P E[L = ] iub. =2 We stat by discussing how the conditional pobabilities can be computed using ROBDDs. Then, we explain how the pobability mass function of, P, 2 k 1 can be computed fom the conditional pobabilities. Finally, we obtain the impoved uppe bounds fo E[L = ], 2 k 1 in tems of the conditional pobabilities and the pobability mass function of, P, 2 k 1. To compute the conditional pobabilities using ROBDDs, we intoduce two Benouilli andom vaiables, a 0 and a 1, independent of z i, 1 i m and yi,s l, yh i,s, 1 i m, 1 s k and define the event B = B l B h, whee B l is the event eithe a 0 = 0 o some bad altenative cumulative dominates all othe altenatives up to attibute and B h is the event eithe a 1 = 0 o some good altenative cumulative dominates all othe altenatives up to attibute. By the definition of the events A l, A h, and A l,h, it is clea that P [A l A l,h G = g] = P [1 B = 1 G = g a 0 = 1 a 1 = 0], P [A h A l,h G = g] = P [1 B = 1 G = g a 0 = 0 a 1 = 1], Futhe, since A l, A h, and A l,h P [A l,h G = g] = P [1 B = 1 G = g a 0 = 1 a 1 = 1]. 4 ae disjoint, P [A l G = g] = P [A l A l,h G = g] P [A l,h G = g] = P [1 B = 1 G = g a 0 = 1 a 1 = 0] P [1 B = 1 G = g a 0 = 1 a 1 = 1] 5 12

14 and P [A h G = g] = P [A h A l,h G = g] P [A l,h G = g] = P [1 B = 1 G = g a 0 = 0 a 1 = 1] P [1 B = 1 G = g a 0 = 1 a 1 = 1] 6 Using 4, 5, and 6, the conditional pobabilities can be computed fom P [1 B = 1 G = g a 0 = 1 a 1 = 0], 2 k, 0 g m 1, P [1 B = 1 G = g a 0 = 0 a 1 = 1], 2 k, 1 g m, and P [1 B = 1 G = g a 0 = 1 a 1 = 1], 2 k, 0 g m. Using the symmeties of the PIAC model, the definition of conditional pobability, and the fact that the undelying Benouilli andom vaiables a 0, a 1, z i, 1 i m, and y l i,s, yh i,s, 1 i m, 1 s ae independent, P [1 B = 1 G = g a 0 = 1 a 1 = 0] can be computed as P [1 B = 1], with the success pobability of the Benouilli andom vaiables z i, 1 i g set to 1, the success pobability of the Benouilli andom vaiables z i, g + 1 i m set to 0, the success pobability of the Benouilli andom vaiable a 0 set to 1, and the success pobability of the Benouilli andom vaiable a 1 set to 0. Similaly, P [1 B = 1 G = g a 0 = 0 a 1 = 1] can be computed as P [1 B = 1], with the success pobability of the Benouilli andom vaiables z i, 1 i g set to 1, the success pobability of the Benouilli andom vaiables z i, g + 1 i m set to 0, the success pobability of the Benouilli andom vaiable a 0 set to 0, and the success pobability of the Benouilli andom vaiable a 1 set to 1; and P [1 B = 1 G = g a 0 = 1 a 1 = 1] can be computed as P [1 B = 1], with the success pobability of the Benouilli andom vaiables z i, 1 i g set to 1, the success pobability of the Benouilli andom vaiables z i, g+1 i m set to 0, the success pobability of the Benouilli andom vaiable a 0 set to 1, and the success pobability of the Benouilli andom vaiable a 1 set to 1. All those expectations can be computed fom a ROBDD epesentation of 1 B as a function of a 0, a 1, z i, 1 i m, y l i,s, yh i,s, 1 i m, 1 s. As in BCH, ROBDDs with complement 0-edges can be built with the help of the CU Decision Diagam Package. A combinational cicuit yielding 1 B as a function of a 0, a 1, z i, 1 i m, yi,s l, yh i,s, 1 i m, 1 s can be built based on [ m m ] [ m m 1 B = a 0 + z i a 1 + z i i=1 j=1 j i s=1 1 Xi,s X j,s } i=1 j=1 j i s=1 1 Xi,s X j,s } whee the indicato functions 1 Xi,s X j,s }, 1 i m, 1 j m, j i, 1 s can be expessed in tems of the vaiables z i, 1 i m, yi,s l, yh i,s, 1 i m, 1 s as explained in Section 2. The following theoem gives expessions fo P, 2 k 1 in tems of P [A l G = g], 0 g m 1, 2 k, P [A h G = g], 1 g m, 2 k, and P [A l,h G = g], 1 g m 1, 2 k. Its poof is immediate fom the definitions of those conditional pobabilities. ], 13

15 Theoem 7. Fo the PIAC model and 2 k 1, P = 1 p m P [A l G = 0] P [A l +1 G = 0] m 1 m + p g 1 p m g g g=1 P [A l G = g] + P [A h G = g] + P [A l,h G = g] P [A l +1 G = g] P [A h +1 G = g] P [A l,h +1 G = g] + p m P [A h G = m] P [A h +1 G = m]. Poof. Conditioning on G and using P [ G = g] = m g p g 1 p m g, 0 g m, P = P [ = ] = = m g=0 m P [ G = g]p [ = G = g] g=0 Using the definitions of A l, A h, and A l,h, fo 2 k 1, = } = A l A h A l,h m p g 1 p m g P [ = G = g]. 7 g A l +1 A h +1 A l,h +1 and, since A l +1 Ah +1 Al,h +1 Al A h A l,h, A l, A h, and A l,h ae disjoint, and A l +1, Ah +1, and A l,h +1 ae disjoint, P [ = G = g] = P [A l G = g] + P [A h G = g] + P [A l,h G = g] P [A l +1 G = g] P [A h +1 G = g] P [A l,h +1 G = g], which used in 7, taking into account P [A l G = m] = P [A l +1 G = m] = P [Ah G = 0] = P [A h +1 G = 0] = P [Al,h G = 0] = P [A l,h +1 G = 0] = P [Al,h G = m] = P [A l,h +1 G = m] = 0, 2 k 1 gives the expession fo P, 2 k 1., The path to the deivation of the impoved uppe bounds fo E[L = ], 2 k 1 stats with the following theoem, which can be looked at as a natual extension to the PIAC model of Theoem 4 in the pevious section. Theoem 8. Let, 1 k, let G 1, 2,..., m}, and let E be any event depending only on x j,s, 1 j m, 1 s 1 such that P [i is chosen by DEBA E G = G ] > 0. Let x i, be the m 1k + 1-vecto with components x j,s x i,s, 1 j m, j i, s k. Let F u 1,..., u m 1k +1 be any function which is non-deceasing on each u l. Then, fo the PIAC model, E[F x i, G = G i is chosen by DEBA E ] E[F x i, G = G ]. 14

16 Poof. See the Appendix. Using Theoem 8, the deivation of the impoved uppe bounds fo E[L = ], 2 k 1 is, oughly, as follows. Fist, etaking a esult obtained in the pevious section, E[L = ] V, t max, +1 t k t whee, c being the altenative chosen by DEBA, V, t is any uppe bound fo E[S, t = ] with S, t = I + max i m i c x i,s x c,s }. That uppe bound is obtained in tems of uppe bounds fo E[S, t G = g = A l ], 0 g m 1, E[S, t G = g = A h ], 1 g m, and E[S, t G = g = A l,h ], 1 g m 1, P, and the conditional pobabilities P [A l G = g], P [A l +1 G = g], 0 g m 1, P [Ah G = g], P [A h +1 G = g], 1 g m, and P [A l,h G = g], P [A l,h +1 G = g], 1 g m 1 using elementay pobability theoy and, A l +1, Ah +1, and Al,h +1. The uppe bounds fo E[S, t G = g = A l ], 0 g m 1, E[S, t G = g = A h ], the elationships between the events = }, A l, A h, A l,h 1 g m, and E[S, t G = g = A l,h ], 1 g m 1 ae obtained by exploiting the symmeties of the PIAC model and using Theoem 8. The following poposition gives the uppe bounds fo E[S, t G = g E[S, t G = g = A h ], and E[S, t G = g = A l,h ]. = A l ], Poposition 2. Let 2 k 1 and + 1 t k. Then, fo DEBA and the PIAC model, E[S, t G = g = A l ] E[Z l,g, t], 0 g m 1, with Z l,g, t = I + max 1 m g>1 max 1 i m g 1 1 g>0 max m g i m y l i,s y l m,s y h i,s y l m,s }, }}, E[S, t G = g = A h ] E[Z h,g, t], 1 g m, with Z h,g, t = I + max 1 m g>0 max 1 i m g 1 g>1 max m g+1 i m y l i,s y h m,s }, y h i,s y h m,s }}, 15

17 and E[S, t G = g = A l,h ] max E[Z l,g, t], E[Z h,g, t]}, 1 g m 1. Poof. Let 0 g m 1, let G l,g be the event altenatives 1,..., m g 1 ae bad, altenatives m g,..., m 1 ae good, and altenative m is bad, and let C l,g, be the event some altenative in the subset 1,..., m g 1, m} but no altenative in the subset m g,..., m 1} cumulative dominates all othe altenatives up to attibute. Exploiting the symmeties of the PIAC model, with E[S, t G = g = A l ] = E[Z, t G l,g m is chosen by DEBA = C l,g, ] Z, t = I + max 1 i m x i,s x m,s But, the event = } C l,g, only depends on x j,s, 1 j m, 1 s + 1, P [m is chosen by DEBA = C l,g, G l,g ] > 0, and Z, t is non-deceasing on each x i,s x m,s, 1 i m 1, + 2 s k, and, then, using Theoem 8, } E[Z, t G l,g m is chosen by DEBA = C l,g, ] E[Z, t G l,g ] = E[Z l,g, t],. yielding E[S, t G = g = A l ] E[Z l,g, t]. Similaly, let 1 g m, let G h,g be the event altenatives 1,..., m g ae bad and altenatives m g + 1,..., m ae good, and let C h,g, be the event some altenative in the subset m g + 1,..., m} but no altenative in the subset 1,..., m g} cumulative dominates all othe altenatives up to attibute. Exploiting the symmeties of the PIAC model, E[S, t G = g = A h ] = E[Z, t G h,g m is chosen by DEBA = C h,g, ]. But, the event = } C h,g, only depends on x j,s, 1 j m, 1 s + 1, P [m is chosen by DEBA = C h,g, G h,g ] > 0, and Z, t is non-deceasing on each x i,s x m,s, 1 i m 1, + 2 s k, and, then, using Theoem 8, E[Z, t G h,g m is chosen by DEBA = C h,g, ] E[Z, t G h,g ] = E[Z h,g, t], yielding E[S, t G = g = A h ] E[Z h,g, t]. 16

18 Finally, let 1 g m 1. Clealy ecall that c denotes the altenative chosen by DEBA, E[S, t G = g = A l,h ] = P [c is bad G = g = A l,h ] E[S, t G = g = A l,h + P [c is good G = g = A l,h ] c is bad] E[S, t G = g = A l,h c is good], and, since P [c is bad G = g = A l,h ] 0, P [c is good G = g = A l,h ] 0, and P [c is bad G = g = A l,h ] + P [c is good G = g = A l,h ] = 1, it suffices to pove that E[S, t G = g = A l,h c is bad] E[Z l,g, t] and that E[S, t G = g = A l,h c is good] E[Z h,g, t]. To pove the fome, let G l,g be the event peviously defined and let C l,g, be the event some altenative in the subset 1,..., m g 1, m} and some altenative in the subset m g,..., m 1} cumulative dominates all othe altenatives up to attibute. Exploiting the symmeties of the PIAC model, E[S, t G = g = A l,h c is bad] = E[Z, t G l,g m is chosen by DEBA = C l,g, ]. But, the event = } C l,g, only depends on x j,s, 1 j m, 1 s + 1, P [m is chosen by DEBA = C l,g, G l,g] > 0, and Z, t is non-deceasing on each x i,s x m,s, 1 i m 1, + 2 s k, and, then, using Theoem 8, yielding E[Z, t G l,g m is chosen by DEBA = C l,g, ] E[Z, t G l,g ] = E[Z l,g, t], E[S, t G = g = A l,h c is bad] E[Z l,g, t]. To pove the second, let G h,g be the event peviously defined and let C h,g, be the event some altenative in the subset 1,..., m g} and some altenative in the subset m g + 1,..., m} cumulative dominates all othe altenatives up to attibute. Exploiting the symmeties of the PIAC model, E[S, t G = g = A l,h c is good] = E[Z, t G h,g m is chosen by DEBA = C h,g, ]. But, the event = } C h,g, only depends on x j,s, 1 j m, 1 s + 1, P [m is chosen by DEBA = C h,g, G h,g] > 0, and Z, t is non-deceasing on each x i,s x m,s, 1 i m 1, + 2 s k, and, then, using Theoem 8, E[Z, t G h,g m is chosen by DEBA = C h,g, ] E[Z, t G h,g ] = E[Z h,g, t], yielding E[S, t G = g = A l,h c is good] E[Z h,g, t]. 17

19 The impoved uppe bounds fo E[L = ], 2 k 1 ae given by the following theoem. Theoem 9. Fo DEBA and the PIAC model, E[L = ] E[L = ] iub = V, t max, 2 k 1, +1 t k t whee V, t = 1 pm P [A l G = 0] P [A l +1 G = 0] E[Z l,0, t] P m m 1 p g 1 p m g g + P g=1 [ P [A l G = g]e[z l,g, t] + P [A h G = g]e[z h,g, t] + P [A l,h G = g] P [A l,h +1 G = g] P [Al +1 G = g] P [A h +1 G = g] maxe[z l,g, t], E[Z h,g, t]} } + min P [A l G = g], P [A l +1 G = g] maxe[z l,g, t], E[Z h,g, t]} E[Z l,g, t] } + min P [A h G = g], P [A h +1 G = g] ] maxe[z l,g, t], E[Z h,g, t]} E[Z h,g, t] + pm P [A h G = m] P [A h +1 G = m] E[Z h,m, t], P with the Z l,g, t and Z h,g, t defined in Poposition 2. Futhemoe, E[L = ] iub k /k, 2 k 1. Poof. Let c be the altenative chosen by DEBA. Fom the poof of Theoem 5, E[L = ] whee V, t is any uppe bound fo E[S, t = ]. V, t max, +1 t k t It emains to deive V, t. To that end, we fist conside the efinement of the event = } into the collection of disjoint non-empty events G = g} = } A l, 0 g m 1, G = g} = } A h, 1 g m, and G = g} = } A l,h, 1 g m 1. Using the definition of conditional expectation, we can expess E[S, t = ] in tems of E[S, t G = g = A l ], 0 g m 1, E[S, t G = g = A h ], 18

20 1 g m, and E[S, t G = g = A l,h ], 1 g m 1 as P E[S, t = ] = P [ G = 0 = A l ] E[S, t G = 0 = A l ] m 1 + P [ G = g = A l ] E[S, t G = g = A l ] g=1 + P [ G = g = A h ] E[S, t G = g = A h ] + P [ G = g = A l,h ] E[S, t G = g = A l,h ] + P [ G = m = A h ] E[S, t G = m = A h ]. Using P [ G = g = A l ] = P [ G = g]p [ = A l G = g], P [ G = g = A h ] = P [ G = g]p [ = A h G = g], P [ G = g = A l,h ] = P [ G = g]p [ = A l,h G = g], and P [ G = g] = m g p g 1 p m g, we get E[S, t = ] 1 pm = P [ = A l G = 0] E[S, t G = 0 = A l P ] m m 1 p g 1 p m g g + P g=1 P [ = A l G = g] E[S, t G = g = A l ] + P [ = A h G = g] E[S, t G = g = A h ] + P [ = A l,h G = g] E[S, t G = g = A l,h ] + pm P P [ = A h G = m] E[S, t G = m = A h ], and using Poposition 2, E[S, t = ] 1 pm P [ = A l G = 0] E[Z l,0, t] P m m 1 p g 1 p m g g + P g=1 P [ = A l G = g] E[Z l,g, t] + P [ = A h G = g] E[Z h,g, t] + P [ = A l,h G = g] maxe[z l,g, t], E[Z h,g, t]} + pm P P [ = A h G = m] E[Z h,m, t], 8 19

21 Fo G = 0, the event = } A l is identical to the event A l A l +1 and Al +1 Al, and we get P [ = A l G = 0] = P [A l G = 0] P [A l +1 G = 0]. 9 Similaly, fo G = m, the event = } A h is identical to the event A h A h +1 and Ah +1 A h +1, and we get P [ = A h G = m] = P [A h G = m] P [A h +1 G = m]. 10 No such simple elationships exist fo P [ = A l G = g], P [ = A h G = g], P [ = A l,h G = g], 1 g m 1, and we will content ouselves by deiving an uppe bound fo T g,, t = P [ = A l G = g] E[Z l,g, t] + P [ = A h G = g] E[Z h,g, t] + P [ = A l,h G = g] maxe[z l,g, t], E[Z h,g, t]}, 1 g m Clealy, fom the definitions of A l, A h, and A l,h, = } A l = A l A l +1 A h +1 A l,h +1, = } A h = A h A l +1 A h +1 A l,h +1, 12 = } A l,h = A l,h A l +1 A h +1 A l,h +1. Also, A l +1 Al A l,h, A h +1 Ah A l,h, and A l,h +1 Al,h. Define A l A l +1 = Al +1 Al,h, A h +1 = Ah +1 Ah, and A h +1 = Ah +1 Al,h shows the elationships between the events A l, A h, A l,h and A l,h +1. Using them in 12, = } A l = A l A l +1, = } A h = A h A h = } A l,h = A l,h +1, and using A l +1 Al, A h +1 Ah, A l +1 Ah +1 Al,h and A l,h +1 ae disjoint, +1 = Al +1 Al,. The Venn diagam of Fig. 2, A l +1, Al +1, Al +1, Ah +1, Ah +1, Ah +1, A l +1 A h +1 A l,h Al,h,, and the fact that A l +1, Ah +1, P [ = A l G = g] = P [A l G = g] P [A l +1 G = g], P [ = A h G = g] = P [A h G = g] P [A h +1 G = g], P [ = A l,h G = g] = P [A l,h G = g] P [A+1 l G = g] P [A h +1 G = g] P [A l,h +1 G = g], 20

22 which used in 11 give T g,, t = P [A l G = g] E[Z l,g, t] + P [A h G = g]e[z h,g, t] + P [A l,h G = g] P [A l,h +1 G = g] maxe[z l,g, t], E[Z h,g, t]} P [A l +1 G = g] E[Z l,g, t] P [A h +1 G = g] E[Z h,g, t] P [A l +1 G = g] maxe[z l,g, t], E[Z h,g, t]} P [A h +1 G = g] maxe[z l,g, t], E[Z h,g, t]} 1 g m 1. The uppe bound fo T g,, t, 1 g m 1, is obtained by finding the maximum of T g,, t as a function of P [A l +1 G = g], P [Ah +1 G = g], P [Al +1 G = g], and P [Ah +1 G = g], subject to the constaints that those vaiables ae known to satisfy. The constaints ae see Fig. 2: P [A l +1 G = g] + P [A l +1 G = g] = P [A l +1 G = g], P [A h +1 G = g] + P [A h +1 G = g] = P [A h +1 G = g], 0 P [A l +1 G = g] P [A l G = g], 0 P [A h +1 G = g] P [A h G = g], P [A l +1 G = g] 0, P [A h +1 G = g] 0, P [A l +1 G = g] + P [A h +1 G = g] P [A l,h G = g] P [A l,h +1 G = g]. This is equivalent to find the maximum of T g,, t = P [A l G = g] E[Z l,g, t] + P [A h G = g] E[Z h,g, t] + P [A l,h G = g] P [A l,h +1 G = g] P [Al +1 G = g] P [A h +1 G = g] maxe[z l,g, t], E[Z h,g, t]} + P [A l +1 G = g] maxe[z l,g, t], E[Z h,g, t]} E[Z l,g, t] + P [A h +1 G = g] maxe[z l,g, t], E[Z h,g, t]} E[Z h,g, t], 1 g m 1 as a function of P [A l +1 G = g] and P [Ah +1 G = g] subject to the constaints 0 P [A l +1 G = g] P [A l G = g], 0 P [A h +1 G = g] P [A h G = g], P [A l +1 G = g] P [A l +1 G = g], P [A h +1 G = g] P [A h +1 G = g], 21

23 A l +1 A l,h A h +1 A l +1 A l +1 A h +1 A h +1 A l,h +1 A l A h Figue 2: Venn diagam showing the elationships between the events A l, A h, A l,h, A l +1, Al A l +1, Ah +1, Ah +1, Ah +1, and Al,h +1. P [A l +1 G = g] + P [A h +1 G = g] P [A l +1 G = g] + P [A h +1 G = g] P [A l,h G = g] + P [A l,h +1 G = g]. Since the function is non-deceasing on both P [A l +1 G = g] and P [Ah +1 G = g], a point a, b at which T g,, t achieves its maximum is yielding 11 a = minp [A l G = g], P [A l +1 G = g]} b = minp [A h G = g], P [A h +1 G = g]} P [ = A l G = g] E[Z l,g, t] + P [ = A h G = g] E[Z h,g, t] + P [ = A l,h G = g] maxe[z l,g, t], E[Z h,g, t]} P [A l G = g] E[Z l,g, t] + P [A h G = g] E[Z h,g, t] + P [A l,h G = g] P [A l,h +1 G = g] P [Al +1 G = g] P [A h +1 G = g] maxe[z l,g, t], E[Z h,g, t]} + minp [A l G = g], P [A l +1 G = g]} maxe[z l,g, t], E[Z h,g, t]} E[Z l,g, t] + minp [A h G = g], P [A h +1 G = g]} maxe[z l,g, t], E[Z h,g, t]} E[Z h,g, t], +1, 1 g m The uppe bound V, t fo E[S, t = ] is obtained by combining 8, 9, 10, and 13. It emains to show that E[L = ] iub k /k, 2 k 1. Since Z l,g, t, Z h,g, t t, we have E[Z l,g, t], E[Z h,g, t] t. Futhemoe, caeful analysis of the pocedue followed to deive V, t eveals that V, t is given by the ight-hand side of 8 with P [ = A l G = g], P [ = A h G = g], and P [ = A l,h G = g], 22

24 1 g m 1 eplaced by quantities Rg, l Rg h, and Rg l,h, 1 g m 1 satisfying R l g, R h g, R l,h g V, t t 0 and Rg l + Rg h + Rg l,h = P [ = G = g]. Then, we have [ 1 1 p m P [ = A l G = 0] P + m 1 g=1 m p g 1 p m g P [ = G = g] g ] + p m P [ = A h G = m], and taking into account that, fo G = 0, = } A l and, fo G = m, = } A h, and using P [ G = g] = m g p g 1 p m g, V, t t 1 P m g=0 P [ G = g]p [ = G = g] = P [ = ] P = 1. Then, E[L = ] iub = max +1 t k V, t/t max +1 t k t /t = k /k. It emains to discuss the computation of the E[Z l,g, t] s and E[Z h,g, t] s involved in the impoved uppe bounds fo E[L = ], 2 k 1 given by Theoem 9. They can be computed using a ecuence-based computational scheme simila to the one given by Theoem 6 to compute the E[Z, t] s involved in the impoved uppe bounds fo the ZIAC model. The ecuence-based computational scheme is given by the following theoem. Theoem 10. Fo the PIAC model, E[Z l,g, t], 0 g m 1, 2 k 1, + 1 t k and E[Z h,g, t], 1 g m, 2 k 1, + 1 t k can be computed using E[Z l,g, t] = W l,g t, E[Z h,g, t] = W h,g t, u W l,g u = a ξ l,g u, a, 0 g m 1, 1 u k 2, a=1 W h,g u = and the ecuences: u a ξ h,g u, a, 1 g m, 1 u k 2, a=1 u ξ l,g u, a = ψm g 1, u, a π l u, a a, 0 g m 1, 1 u k 2, 1 a u, a =a u ξ h,g u, a = ψm g, u, a π h u, a a, 1 g m, 1 u k 2, 1 a u, a =a ψ0, u, a = φ h m 1, u, a, 1 u k 2, 1 a u, ψm 1, u, a = φ l m 1, u, a, 1 u k 2, 1 a u, 23

25 ψb, u, a = φ l b, u, a a φ h m b 1, u, a + φ h m b 1, u, a a =1 1 b m 2, 1 u k 2, 1 a u, φ l 1, u, a = π l u, a 1, 1 u k 2, 1 a u, a 1 a =1 φ l b, u, a, φ l b, u, a = φ l b 1, u, a a 1 a =0 π l u, a + π l u, a 1 a 1 a =1 φ l b 1, u, a, 2 b m 1, 1 u k 2, 1 a u, φ h 1, u, a = π h u, a 1, 1 u k 2, 1 a u, φ h b, u, a = φ h b 1, u, a a 1 a =0 π h u, a + π h u, a 1 a 1 a =1 φ h b 1, u, a, 2 b m 1, 1 u k 2, 1 a u, π l 1, 0 = 1, π l 2, 0 = 1 p l, π l 2, 1 = p l, π l u, u 1 = p l π l u 1, u 2, 3 u k 2, π l u, a = 1 p l π l u 1, a + p l π l u 1, a 1, 3 u k 2, 1 a u 2, π l u, 0 = 1 p l π l u 1, 0, 3 u k 2. π h 1, 0 = 1, π h 2, 0 = 1 p h, π h 2, 1 = p h, π h u, u 1 = p h π h u 1, u 2, 3 u k 2, π h u, a = 1 p h π h u 1, a + p h π h u 1, a 1, 3 u k 2, 1 a u 2, π h u, 0 = 1 p h π h u 1, 0, 3 u k 2. Poof. See the Appendix. 24

26 5 Analysis In this section we will illustate the impoved uppe bounds fo the expected loss of DEBA obtained in this pape, will compae them with those obtained in BCH, and will analyze thei tightness. We will also discuss how the impoved uppe bounds allow us to extend the conclusions in BCH egading the pefomance of DEBA. To illustate the impoved uppe bounds fo the expected loss of DEBA and compae them with those obtained in BCH, we will set as the base case scenaio the ZIAC model with p = 0.5. We will then conside the following six vaiations. Fist, to examine the effect of aveage attibute quality, we will change the pobability level p in the ZIAC model to the values p = 0.2 and p = 0.8. Second, to obseve the effect of vaying p, we will conside a low-to-high patten in which p inceases linealy fom p 1 = 0.2 to p k = 0.8, i.e. p = /k 1, and a high-to-low patten in which p deceases linealy with fom p 1 = 0.8 to p k = 0.2, i.e. p = /k 1. The aveage of the p s is 0.5 in both cases. Thid, to examine the effect of positive coelation, we will set p = 0.5 in the PIAC model and will exploe the coelation levels ρ = 0.2 and ρ = 0.5. Fo these seven scenaios, Fig. 3 plots the uppe bounds fo the expected loss of DEBA, E[L] ub, obtained in BCH fo 2 m 10 and 3 k 10. Fig. 4 plots the impoved uppe bounds, E[L] iub, obtained in this pape. We can fist note that the impoved uppe bounds ae substantially bette than those obtained in BCH: fo lage k they ae about half the peviously obtained uppe bounds. Anothe impotant point is that the impoved uppe bounds seem to incease elatively less apidly with k than the uppe bounds obtained in BCH, implying that the compaison with the uppe bounds obtained in BCH would pobably get bette fo lage values of k. The behavio of E[L] iub with espect to the aveage quality of the attibutes, a vaying aveage attibute quality, and the pesence of inte-attibute positive coelation is simila to the behavio with espect to those chaacteistics of E[L] ub : E[L] iub deceases when the aveage quality of the attibutes inceases beyond 0.5, when thee is a high-to-low patten in the aveage attibute qualities, and in the pesence of inceasing positive inte-attibute coelation. The behavio of E[L] iub with espect to k and m is also simila to the behavio of E[L] ub. Fist, fo fixed m and inceasing k, E[L] iub inceases. This is because in 2 the P ae independent of k and E[L = ] iub is non-deceasing with k see Theoems 5 and 9. Second, fo fixed k, E[L] iub 0 as m. This is because, fo fixed k, fo both the ZIAC and the PIAC pobabilistic models, the pobability that thee will be some all-1 altenative goes to 1 as m see BCH, implying P k 1, which by 2 implies E[L] iub 0. As fo E[L] ub, fo fixed k, thee seems to exist a tuning point fo m, m, befoe which E[L] iub inceases with m and beyond which E[L] iub deceases with m. The tuning point m seems to decease as k deceases, as the aveage quality of the attibutes inceases, and in the pesence of inceasing positive inte-attibute coelation. Befoe analyzing the tightness of E[L] iub, we will compae E[L = ] iub, 2 k 1 with the uppe bounds, E[L = ] ub = k /k, used in BCH and analyze the tightness of the fome. This will allow us to discuss till what extent the uppe bounds obtained in this pape can be futhe impoved. The event =, 2 k 1 implies the existence of some altenative i 25

27 ZIAC p =0.5 0,3 0, m ZIAC p =0.2 ZIAC p =0.8 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k=10 m ,3 0,3 0,1 0, m m ZIAC low-to-high p ZIAC high-to-low p 0,3 0,3 0,1 0, m PIAC p=0.5, ρ= m PIAC p=0.5, ρ=0.5 0,3 0,3 0,1 0, m m Figue 3: E[L] ub as a function of m and k fo seven scenaios base case and six vaiations. 26

28 ZIAC p =0.5 0,3 0, m k=3 k=4 k=5 k=6 k=7 k=8 k=9 k=10 0, 4 0, 2 0 m, ZIAC p =0.2 ZIAC p =0.8 0,3 0,3 0,1 0, m m ZIAC low-to-high p ZIAC high-to-low p 0,3 0,3 0,1 0, m m PIAC p=0.5, ρ=0.2 PIAC p=0.5, ρ=0.5 0,3 0,3 0,1 0, m m Figue 4: E[L] iub as a function of m and k fo seven scenaios base case and six vaiations. 27