DIVERSIFIED TREATMENT UNDER AMBIGUITY. Charles F. Manski Department of Economics and Institute for Policy Research, Northwestern University

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1 DIVERSIFIED TREATMENT UNDER AMBIGUITY Charle F. Manki Department of Economic and Intitute for Policy Reearch, Northwetern Univerity December 31, 2008 (12:46pm) Abtract Thi paper develop a broad theme about treatment under ambiguity through tudy of a particular deciion criterion. The broad theme i that a planner may often want to cope with ambiguity by diverification, aigning obervationally identical peron to different treatment. Study of the minimax-regret (MR) criterion ubtantiate the theme. The paper ignificantly extend my earlier analyi of one-period planning with individualitic treatment and a linear welfare function. I how that MR treatment allocation are fractional in a large cla of planning problem with nonlinear welfare function, interacting treatment, dynamic with learning, and non-cooperative apect. I alo call attention to ome problem of treatment under ambiguity in which the MR allocation i not fractional. Thi reearch wa upported in part by National Science Foundation grant SES Thi paper revie and upercede an earlier working paper Fractional Treatment Rule for Social Diverification of Indiviible Private Rik, NBER Working Paper W11675, September I have benefitted from the opportunity to preent thi work in eminar at Collegio Carlo Alberto, The Hebrew Univerity of Jerualem, Princeton Univerity, Univeritat Pompeu Fabra, and the Univerity of Wahington.

2 1. Introduction When tudying collective deciion problem, economit have long aked how a planner hould act. A tandard exercie pecifie a et of feaible policie and a welfare function. The planner i preumed to know the welfare achieved by each policy. The objective of the exercie i to characterize the optimal policy. In practice planner often have only partial knowledge of the welfare achieved by alternative policie. Hence, they cannot determine optimal policie. Thi limit the relevance of the tandard exercie to actual policy analyi. In a reearch program that began in Manki (1990, 1995), I have tudied how identification problem that are prevalent in empirical reearch generate ambiguity about the nature of optimal policie. There are myriad ource of ambiguity, many deriving from identification problem that are prevalent in empirical reearch; ee Manki (2007) for expoition. Perhap the mot fundamental identification problem arie from the unobervability of counterfactual policy outcome. At mot one can oberve the outcome that occur under realized policie. The outcome of unrealized policie are logically unobervable. Yet determination of an optimal policy require comparion of all feaible policie. My recent work conider how a planner might cope with ambiguity; ee Manki (2000, 2004a, 2005a, 2006, 2007a, 2007b). The familiar Bayeian precription i to aert a ubjective probability ditribution over the feaible tate of nature and chooe an action that maximize ubjective expected welfare. However, a ubjective probability ditribution i itelf a form of knowledge, and a planner may have no credible bai for aerting one. I have tudied problem of thi type, with particular attention to application of the minimax-regret (MR) criterion. Many of the planning problem that I have tudied to date hare a relatively imple tructure. The planner mut chooe one of two treatment, ay a and b, for each member of a population of obervationally identical peron. The planner can treat population member differentially, aigning ome peron to treatment a and the remainder to b. Treatment i individualitic, each peron outcome depending only on

3 2 the treatment that he receive and not on the treatment of other peron. Outcome take a bounded range of value. The welfare function i linear, umming the outcome of the population member. In thi etting, the optimal policy aign all peron to the treatment that yield the higher mean outcome. A planner face ambiguity when he doe not know which treatment i better. Here are two illutrative application among many that might be cited, the firt being an intance of ocial planning and the econd being one of private planning: Chooing Medical Treatment for a Non-Infectiou Dieae: The planner i a public health agency which chooe treatment for a population of peron who are uceptible to a non-infectiou dieae. The relevant welfare outcome i the health benefit of a treatment minu it cot, meaured in comparable unit. A utilitarian welfare function um net benefit acro the population. The dieae being non-infectiou, each peron outcome depend only on hi own treatment. Medical reearch yield only partial knowledge of treatment repone, o the planner doe not know which treatment i better. An Invetor Aet Allocation Deciion: The planner i an invetor who allocate an endowment between two aet. The population member are dollar of endowment and the treatment are the two aet. The relevant outcome i the return on a dollar inveted in an aet. Welfare i the aggregate return earned by the invetor. At the time of the allocation deciion, the invetor ha only partial knowledge of invetment return. Hence, he doe not know what allocation maximize profit. While aet ditribution ha long been framed a a choice among alternative fractional allocation, medical treatment and other ocial planning problem have commonly been viewed a choice between two ingleton allocation either aign treatment a to all obervationally identical peron or aign b to everyone. Thi pecification of the choice et uffice when a planner know the optimal treatment, but it

4 3 may not when he face ambiguity. Then a planner may reaonably elect a fractional treatment allocation, aigning ome peron to treatment a and the remainder to treatment b. Fractional allocation enlarge the et of feaible policy choice by convexifying the ingleton allocation. They cope with ambiguity through diverification. 1 The broad argument for diverification i that it enable a deciion maker to balance two type of potential error. A Type A error occur when treatment a i choen but i actually inferior to b, and a Type B error occur when b i choen but i inferior to a. The ingleton allocation aigning everyone to treatment a entirely avoid type B error but may yield Type A error, and vice vera for ingleton aignment to treatment b. Fractional allocation make both type of error but reduce their potential magnitude. Hence, it i conceivable that ome fractional allocation may be preferred to the ingleton one. It i well known that a Bayeian planner with a linear welfare function generically chooe a ingleton allocation. A neceary condition for Bayeian diverification i that the welfare function be trictly concave on at leat part of it domain. In previou work (Manki 2005a; 2007a; 2007b, Chapter 11) I have hown that a planner with a linear welfare function who applie the minimax-regret criterion doe diverify. The MR criterion chooe an allocation that balance the potential welfare loe from Type A and Type B error. The MR treatment allocation i fractional whenever a planner face ambiguity. Moreover, it ha a imple explicit form. I build on thi finding here, extending my earlier analyi to planning problem with nonlinear welfare function (Section 3), interacting treatment (Section 4), learning (Section 5), and non-cooperative apect (Section 6). I how that the minimax-regret treatment allocation i fractional in a broad cla of planning problem under ambiguity. Thi contrat with Bayeian and maximin planning, which diverify 1 It i important to ditinguih differential treatment of peron who vary in obervable repect from fractional allocation of peron who are obervationally identical. It i well known that enabling treatment choice to vary ytematically with oberved covariate of population member can improve welfare if treatment repone varie with thee covariate; ee, for example, Manki (2005a, Sec. 1.2). In contrat, fractional allocation randomly differentiate among peron who are obervationally identical.

5 4 treatment choice in a much maller cla of etting. I alo call attention to ome problem of treatment under ambiguity in which the MR allocation i not necearily fractional. Section 2 review the problem of one-period planning with individualitic treatment and linear welfare. After introducing treatment choice under ambiguity, I compare Bayeian, maximin, and minimaxregret planning. The Bayeian and maximin criteria generically yield ingleton allocation, wherea the MR allocation i fractional. I extend the analyi to etting where the planner oberve covariate that ditinguih among member of the population. Section 3 conider a broader cla of welfare function than I have tudied previouly. I firt permit welfare function that monotonically tranform the aggregate um of outcome. I how that the MR treatment allocation remain fractional when a planner face ambiguity, the pecific allocation depending on the tranformation. I then addre planning problem with non-additive cot of treatment, focuing on the polar cae of capacity contraint and fixed cot. I how that the MR allocation under ambiguity i fractional with all capacity contraint and mall fixed cot. However, it i ingleton if fixed cot are ufficiently large. I alo conider planning with deontological welfare function, which enable a planner to expre a poible ethical objection to fractional treatment allocation. Fractional allocation violate one interpretation of the ethical principle calling for equal treatment of equal. They are conitent with thi principle in the ex ante ene that all obervationally identical people have the ame probability of receiving a particular treatment, but they violate it in the ex pot ene that obervationally identical peron ultimately receive different treatment. The ex pot ene of equal treatment expree a deontological conideration that i abent from the conequentialit welfare function uually aumed in economic analyi of planning. I formalize thi idea and how that a pecial cae i mathematically equivalent to planning with fixed cot. Section 4 relaxe the aumption that treatment i individualitic and conider etting where treatment interact, each peron outcome depending both on hi own treatment and on the treatment

6 5 allocation within the population. Problem of thi type are more complex than one with individualitic treatment. Ambiguity i more evere becaue determination of an optimal policy now require knowledge not only of how outcome vary with peron own treatment but alo with the treatment allocation. I ue medical treatment of an infectiou dieae to illutrate. Wherea Section 2 through 4 uppoe a one-period planning problem, Section 5 conider dynamic planning under ambiguity. I uppoe that, in each of a equence of period, a planner chooe treatment for the current cohort of a population. Thi introduce the poibility of learning, with obervation of treatment outcome in earlier period informing treatment choice in later period. I ugget ue of the tractable adaptive minimax-regret (AMR) criterion, which treat each cohort a well a poible in the tatic minimax-regret ene, uing the information available at the time. The reult i a fractional treatment allocation whenever the available knowledge doe not uffice to determine which treatment i better. The criterion i adaptive becaue knowledge of treatment repone accumulate over time, o ucceive cohort may receive different fractional allocation. Fractional allocation randomly aign obervationally identical peron to different treatment. Hence, they automatically create randomized experiment, which are particularly informative for learning treatment repone. I ue medical treatment to illutrate. I explain how the AMR criterion differ from the current practice of randomized clinical trial in medicine and I dicu the drug approval of the U. S. Food and Drug Adminitration. Wherea Section 2 through 5 are written from the perpective of a planner with the power to dictate policy, Section 6 conider ituation in which treatment choice require the agreement of two planner who may have different welfare function and belief about the feaible tate of nature. I how that if both planner face ambiguity and ue the AMR criterion to compare allocation, then there exit fractional allocation that both prefer to the ingleton one. I ue a union-management deciion problem to illutrate. Section 7 conclude.

7 2. One-Period Planning with Individualitic Treatment and Linear Welfare 6 I review here the imple planning problem that form the baeline for thi paper. Section 2.1 et out baic concept and notation. Section 2.2 and 2.3 conider Bayeian, maximin, and minimax-regret planning. Section 2.4 ue entencing of convicted offender to illutrate how treatment choice varie with the deciion criterion ued. Section 2.5 conider treatment choice when the planner oberve covariate that ditinguih among member of the population Baic Concept and Notation The Treatment-Choice Problem There are two treatment, labeled a and b. The et of feaible treatment i T {a, b}. Each member j of a population denoted J ha a repone function y j( ): T Y that map treatment t T into outcome y j(t) Y. The ubcript j in y j( ) indicate that treatment repone may vary acro the population. Let u j(t) u j[y(t), t] denote the net contribution to welfare that occur if peron j receive treatment t and realize outcome y j(t). For example, u j(t) may have the benefit-cot form u j(t) = y j1(t) y j2(t), where y j1(t) i the benefit of treatment t and y j2(t) i it cot. Although treatment repone may vary acro the population, peron are obervationally identical to the planner. Let P[y( )] denote the population ditribution of treatment repone. I uppoe that the population i large in the formal ene of being atomle; that i, P(j) = 0 for all j J. Thi idealization implie that if the planner randomly aign a poitive fraction of the population to a treatment, the ub-population of peron who receive thi treatment i infinite. Thi eliminate ampling variation a an iue when comparing alternative treatment allocation and analyzing treatment repone. The planner tak i to allocate the population between the two treatment. A treatment allocation

8 7 i a number [0, 1] that randomly aign a fraction of the population to treatment b and the remaining 2 1 to treatment a. I aume that the planner want to chooe a treatment allocation that maximize mean welfare in the population. Let E[u(a)] and E[u(b)] be the mean welfare that would reult if a randomly drawn peron were to receive treatment a or b repectively. Welfare with allocation i (1) W( ) = (1 ) + = + ( ). W( ) i a conequentialit welfare function that additively aggregate individual contribution to welfare. If the function u( ) expree private preference, then W( ) i the utilitarian welfare function often aumed in reearch on welfare economic. The optimal treatment allocation i obviou if (, ) are known. The planner hould chooe = 1 if > and = 0 if <. All allocation yield the ame welfare if =. The problem of interet i treatment choice when (, ) i partially known. To formalize the problem, let S index the tate of nature that the planner think feaible. Thu, the planner believe that (, ) lie in the et [(, ), S]. I aume that thi et i bounded and denote the extreme feaible value of and a L min S, L min S, U max S, and U max S. Partial knowledge i unproblematic for deciion making if (, S) or if (, S); chooing = 0 i optimal in the former cae and = 1 in the latter. The planner face ambiguity if both treatment are undominated; that i, if > for ome value of and < for other value. I aume that the planner face ambiguity. 2 Treatment i random from the perpective of population member, each peron having a probability of receiving b and 1 of receiving a. However, a treatment allocation a defined here i not random in the deciion theoretic ene of a mixed trategy. Allocation aign fixed fraction and 1 of the population to treatment b and a repectively. A mixed trategy would make thee fraction random, with their realization determined by an auxiliary randomizing device. Formally, a mixed trategy i a probability ditribution on [0, 1].

9 8 Criteria for Choice Under Ambiguity A planner facing ambiguity doe not know the optimal treatment allocation. Yet he mut omehow chooe an allocation. How might he do o? Deciion theorit have propoed variou way of tranforming the original optimization problem, which cannot be olved, into another one that can be olved. Bayeian recommend that a deciion maker facing ambiguity aert a ubjective ditribution on the tate of nature and chooe an allocation that maximize ubjective mean welfare with repect to thi ditribution. The maximin and the minimax-regret criteria do not ue a ubjective ditribution. Intead they chooe allocation that, in different ene, perform uniformly well over all tate of nature. 3 Bayeian deciion theorit have often aerted pre-eminence for maximization of expected utility (welfare here), aerting not only that a deciion maker might ue thi deciion criterion but that he hould do o. Reference i often made to repreentation theorem deriving the expected utility criterion from conitency axiom on hypothetical choice behavior, famouly von Neumann and Morgentern (1944) and Savage (1954). Thee and other contribution to axiomatic deciion theory conider a deciion maker who ha formed a complete binary preference ordering over a pecified cla of action and, thu, who know how he would behave if he were to face any choice et D. The theorem how that if the preference ordering ha certain propertie, then the agent may be repreented a maximizing expected utility. Thu, the theorem of axiomatic deciion theory are interpretative rather than precriptive. Why then are the N-M and Savage theorem often conidered to be precriptive? Deciion theorit 3 Thee three approache to deciion making under ambiguity are particularly well-known, but they are not the only one that have received attention. A deciion maker who feel able to aert a ubjective ditribution on the tate of nature need not maximize ubjective mean welfare. He could intead maximize ome quantile of the welfare ditribution (ee Manki, 1988). A deciion maker who feel able to aert only a partial ditribution on the tate of nature could maximize minimum expected welfare or minimize maximum expected regret. Thee idea have a long hitory in the literature on tatitical deciion theory, which refer to them a the -maximin and -minimax regret criteria (ee Berger, 1985). The -maximin approach ha alo drawn coniderable attention from economit (e.g., Gilboa and Schmeidler, 1989; Hanen and Sargent, 2008).

10 9 often aert that an agent hould form a complete binary preference ordering on the cla of action and that preference hould have the propertie aumed in the theorem. If one accept thee aertion, the theorem imply that the agent hould behave in a manner repreentable a maximization of expected utility. Thu, the theorem are precriptive if one conider their conitency axiom to be compelling. A famou example i the Chernoff (1954) argument againt the minimax regret criterion. Chernoff oberved that thi criterion can violate the conitency axiom called independence of irrelevant alternative (IIA). The IIA axiom hold that if an agent i not willing to chooe a given action from a hypothetical choice et, then he hould not be willing to chooe it from any larger hypothetical choice et; thu, for any c D E, an agent who would not chooe c from D hould not chooe c from E. Chernoff wrote (p. 426): A third objection which the author conider very eriou i the following. In ome example, the min max regret criterion may elect a trategy d 3 among the available trategie d 1, d 2, d 3, and d 4. On the other hand, if for ome reaon d 4 i made unavailable, the min max regret criterion will elect d 2 among d 1, d 2, and d 3. The author feel that for a reaonable criterion the preence of an undeirable trategy d 4 hould not have an influence on the choice among the remaining trategie. Thi paage i the totality of Chernoff argument. He intropected and concluded that any reaonable deciion criterion hould adhere to IIA, without explaining why he felt thi way. He did not argue that minimax-regret deciion have advere welfare conequence. I do not ue conitency axiom to argue for or againt particular deciion criteria. In Manki (2008b), I have oberved that a deciion maker who want to chooe an optimal policy but lack the knowledge to do o i not concerned with the conitency of hi behavior acro hypothetical choice et. Rather, he want to make a reaonable choice from the choice et that he actually face. Hence, I reaon that precription for deciion making hould repect actuality. That i, they hould promote welfare maximization in the choice problem the agent actually face. Expected utility maximization repect actuality, but it ha no pecial tatu from the actualit perpective.

11 Bayeian and Maximin Deciion Making Baye Rule A Bayeian planner place a ubjective probability ditribution on the tate of nature, compute the ubjective mean value of welfare under each treatment allocation, and chooe an allocation that maximize thi ubjective mean. Thu, the planner olve the optimization problem (2) max E ( ) + [E ( ) E( )], [0, 1] where E ( ) = d and E ( ) = d are the ubjective mean of and. The Baye deciion aign everyone to treatment b if E ( ) > E ( ) and everyone to treatment a if E ( ) > E ( ). All treatment allocation are Baye deciion if E ( ) = E ( ). Thu, a Bayeian planner behave a would a planner who know that the population mean in (1) have the value in (2). Although Bayeian planning i conceptually traightforward, it may not be traightforward to form a credible ubjective ditribution on the tate of nature. The allocation choen by a Bayeian planner depend on the ubjective ditribution ued. Here, a alway, the Bayeian paradigm i appealing only when a deciion maker i able to form a ubjective ditribution that really expree hi belief. The Maximin Criterion To determine the maximin allocation, one firt compute the minimum welfare attained by each allocation acro all tate of nature. One then chooe an allocation that maximize thi minimum welfare. Thu, the criterion i (3) max min + ( ). [0, 1] S

12 11 The olution ha a imple form if (, ) i a feaible value of (, ). Then the maximin allocation i = 0 if L > L, = 1 if L < L, and all [0, 1] if L = L. L L 2.3. The Minimax-Regret Criterion By definition, the regret of treatment allocation in tate of nature i the difference between the maximum achievable welfare and the welfare achieved with thi allocation. The maximum welfare achievable in tate of nature i max (, ). Hence, allocation ha regret max (, ) [ + ( ) ]. The minimax-regret rule compute the maximum regret of each allocation over all tate of nature and chooe an allocation to minimize maximum regret. Thu, the criterion i (4) min max max (, ) [ + ( ) ]. [0, 1] S Let S(a) and S(b) be the ubet of S on which treatment a and b are uperior. That i, let S(a) { S: > } and S(b) { S: > }. Let M(a) max S(a) ( ) and M(b) max S(b) ( ) be maximum regret on S(a) and S(b) repectively. Define M(a) = 0 if S(a) i empty and M(b) = 0 if S(b) i empty. Manki (2007b, Complement 11A) prove that the MR criterion alway make a fractional treatment allocation when both treatment are undominated. The reult i M(b) (5) MR =. M(a) + M(b) The proof i hort and intructive, o I reproduce it here.

13 Proof: The maximum regret of rule on all of S i max [R(, a), R(, b)], where 12 (6a) R(, a) max [(1 ) + ] = max ( ) = M(a), S(a) S(a) (6b) R(, b) max [(1 ) + ] = max (1 )( ) = (1 )M(b), S(b) S(b) are maximum regret on S(a) and S(b). Both treatment are undominated, o R(1, a) = M(a) > 0 and R(0, b) = M(b) > 0. A increae from 0 to 1, R(, a) increae linearly from 0 to M(a) and R(, b) decreae linearly from M(b) to 0. Hence, the MR rule i the unique (0, 1) uch that R(, a) = R(, b). Thi yield (5). Thi proof of (5) how that the MR allocation balance the two type of potential error dicued in the Introduction. Recall that a Type A error occur when treatment a i choen but i actually inferior to b, and a Type B error occur when b i choen but i inferior to a. For any allocation [0, 1], the quantitie R(, b) and R(, a) give the potential welfare loe from Type A and B error repectively. A increae from 0 to 1, the former potential lo decreae from M(b) to 0 and the latter increae from 0 to M(a). The MR criterion chooe to minimize the maximum potential lo, which occur when R(, a) = R(, b). 4 When a planner ue allocation MR, maximum regret i M(a)M(b)/[M(a) + M(b)]. It i intereting to compare thi with the maximum regret that would reult if the planner were only able to chooe one of the ingleton allocation. The olution would be = 0 if M(a) M(b) and = 1 if M(a) M(b). Maximum regret would be min[m(a), M(b)]. Thu, permitting fractional allocation can reduce maximum regret to a little a one-half the value achievable with ingleton allocation, thi occurring when M(a) = M(b). Expreion M(a) and M(b) implify when ( L, U) and ( U, L) are feaible value of (, ). Then 4 Wherea the MR criterion balance the potential welfare loe from Type A and B error, a planner may prefer to weigh thee loe differently. Tetenov (2008) develop thi idea, tudying aymmetric MR criteria.

14 13 M(a) = U L and M(b) = U L. Hence, U L (7) MR =. ( ) + ( ) U L U L Reult (7) implifie further if either or i fully known. In particular, uppoe that i known. Then L = U = and (7) become U (8) MR =. U L Full knowledge of may be realitic if a i the tatu quo treatment and b i an innovation. Suppoe, for example, that treatment a ha been the tandard therapy for a dieae and treatment b i a propoed new therapy. Then one may be able to oberve the outcome experienced when earlier cohort of patient were given treatment a, but no comparable data may be available for treatment b. Hence, the available empirical evidence may reveal but not. The fractional character of the MR treatment allocation contrat harply with the generic ingleton nature of the Bayeian allocation. Wherea the MR allocation minimize maximum regret, a Bayeian 5 allocation minimize ubjective expected regret. It i revealing to conider the pecial cae where i 5 Bayeian deciion making i uually decribed a maximization of expected welfare, but it i mathematically equivalent to minimization of expected regret. Conider an abtract etting where a deciion maker face choice et C, the feaible tate of nature are S, and the objective i to maximize a welfare 1 function w(, ): C S R mapping action a C and tate S into welfare. Let the deciion maker aert a ubjective ditribution on S. The uual decription of the Bayeian criterion i max a C E [w(a, )]. The expected regret of action a i E [max c C w(c, ) w(a, )] = E [max c C w(c, )] E [w(a, )]. The firt term on the right-hand ide doe not vary with action a. Hence, minimization of expected regret i equivalent to maximization of expected welfare.

15 14 known. Bayeian ometime ugget that when a quantity i known only to lie within ome bound, a deciion maker hould aert a uniform ditribution on the quantity and maximize expected welfare. Suppoe that a planner place the uniform ditribution U(, ) on and maximize expected welfare. The ubjective L U mean for i ( L + U)/2, o the Bayeian planner et = 0 if ( L + U)/2 < and = 1 if ( L + U)/2 >. In contrat, a MR planner et MR at the fractional value given in (8). When (, ) i a feaible value of (, ), there i a imilarly harp contrat between the fractional L L character of the MR allocation and the generic ingleton nature of the maximin allocation. However, the maximin allocation may be fractional when (, ) i not feaible. Indeed, the maximin and MR criteria are L L equivalent to one another when max (, ) i contant acro tate of nature. 6 I caution the reader that the MR allocation i not alway fractional when a planner allocate the population among more than two treatment. Conidering a etting with three treatment, Manki (2005b) gave an example in which all three treatment are undominated yet there exit a ingleton MR allocation. Stoye (2007a) ha tudied a cla of planning problem with multiple treatment and ha found that the MR allocation are ubtle to characterize. They often are fractional, but he give an example in which there exit a unique ingleton allocation. Throughout thi paper I retrict attention to planning problem with two treatment Chooing Sentence for Convicted Juvenile Offender To illutrate Bayeian, maximin, and minimax-regret planning, conider the problem of chooing entence for a population of convicted offender. I apply finding reported in Manki and Nagin (1998), 6 Thi i an intance of a general reult. Conider again an abtract etting where a deciion maker face choice et C, the feaible tate of nature are S, and the objective i to maximize a welfare function 1 w(, ): C S R mapping action a C and tate S into welfare. Then the maximin deciion criterion i max a Cmin Sw(a, ) and the MR criterion i min a C max S[max c C w(c, ) w(a, )]. Suppoe that max w(c, ) i contant for all S. Then MR reduce to the maximin criterion. c C

16 15 who tudied the entencing and recidivim of male youth in the tate of Utah who were convicted of offene before they reached age 16. In thi illutration, the planner i the tate of Utah and the population are male under age 16 who are convicted of an offence. Treatment a i the tatu quo policy, thi being a decentralized ytem where judge have dicretion to chooe between reidential confinement and a entence that doe not involve confinement. Treatment b i an innovation mandating confinement for all convicted offender. I take the outcome of interet to be a binary meaure of recidivim. Specifically, y(t) = 1 if an offender who receive treatment t i not convicted of another crime in the two-year period following entencing, and y(t) = 0 if the offender i convicted of a ubequent crime. Let u(t) = y(t). Then = P[y(a) = 1] and = P[y(b) = 1]. Analyzing data on outcome under the tatu quo policy, Manki and Nagin (1998) find that = The data do not fully identify. In the abence of knowledge of how judge chooe entence or how juvenile repond to their entence, the data reveal only that [0.03, 0.92]. Thu, the innovation may be much better or wore than the tatu quo. Manki and Nagin (1998) argue that thi wort-cae bound on i germane to policy making becaue criminologit have found it difficult to learn how entencing affect recidivim. Reearcher have long debated the counterfactual outcome that offender would experience if they were to receive other entence. Conider policy chooe when the tate of Utah know that = 0.61 and [0.03, 0.92]. If the tate applie the Bayeian paradigm, it fully adopt the innovation of mandatory confinement if E ( ) > 0.61 and leave the tatu quo of judicial dicretion in place if E ( ) < If the tate applie the maximin criterion, it leave the tatu quo in place becaue L = 0.03 < If the tate applie the MR criterion, it randomly entence to confinement ( U )/( U L) = ( )/( ) = 0.35 of the offender and leave judicial dicretion in place for the remaining fraction 0.65.

17 Planning with Obervable Covariate Section 2.1 through 2.4 conidered treatment of a population whoe member are obervationally identical to the planner. In practice, peron may have obervable covariate and a planner may be able to differentially treat peron with different covariate. A imple approach to deciion making i to eparate peron by their covariate and apply the finding of Section 2.2 and 2.3 to each group. Thi work when the relevant objective function i eparable in the covariate but not otherwie. I explain here. Suppoe that the planner oberve covariate x j X for each peron j, where the covariate pace X i finite and P(x = ) > 0, X. Then the planner can ditinguih among peron with different covariate, chooing a vector, X of treatment allocation that may vary with. Welfare with thi vector of allocation i (9) W(, X) = + ( ), X where E[u(a) x = ] and E[u(b) x = ]. The welfare function i eparable in. Hence, an allocation vector i optimal if, for each X, it et = 1 when > and = 0 when <. Now conider treatment choice under ambiguity. A Bayeian planner olve the problem (10) max E( ) + [E ( ) E( )], [0, 1], X X where E ( ) = d and E ( ) = d are the ubjective mean of and. The objective function i eparable in. Hence, the Baye deciion aign all peron with covariate to treatment b if E ( ) > E( ) and all uch peron to treatment a if E ( ) > E ( ). Maximin planning i more ubtle becaue the objective function may not be eparable in. The

18 17 optimization problem i (11) max min + ( ). [0, 1], X S X Thi objective function generally i noneparable, and determination of the maximin allocation require joint choice of, X. However, the objective function i eparable if ( L, L), X i a feaible value of (, ), X. Then (11) reduce to (12) max L + ( L L). [0, 1], X X In thi etting, the maximin deciion aign all peron with covariate to treatment b if > and all uch peron to treatment a if >. L L Minimax-regret planning i imilarly ubtle. The optimization problem i L L (13) min max max (, ) [ + ( ) ]. [0, 1], X S X Thi objective function i generally noneparable, and determination of the MR allocation require joint choice of, X. However, the objective function i eparable if ( L, U), X and ( U, L), X are feaible value of (, ), X. Then (13) reduce to (14) min max [( U L), ( U L)(1 )]. [0, 1], X X Hence, the MR allocation for peron with covariate i

19 18 U L (15) MR =. ( ) + ( ) U L U L An important open problem i to characterize the maximin and MR allocation when the et S i uch that the objective function i noneparable. Noneparability generically occur when the planner ha information that relate the value of (, ) acro X. For example, X could be an ordered et and the planner may know that i monotone in. 3. Nonlinear Welfare Thi ection tudie planning when the welfare function i nonlinear in variou way. Section 3.1 conider monotonic tranformation of the aggregate um of outcome. Section 3.2 addree planning problem with non-additive cot of treatment, pecifically capacity contraint or fixed cot. Section 3.3 tudie planning with deontological welfare function Monotone Tranformation of the Welfare Function Conider monotone tranformation of welfare function (1) of the form (16) W( ) = f[ + ( ) ], where f( ) i trictly increaing in it argument. The hape of f( ) i immaterial to treatment choice when one treatment i uperior in all tate of nature. Whatever monotone function f( ) may be, = 0 i optimal if (

20 , S) and = 1 if (, S). However, hape may matter when a planner face ambiguity. 19 In Bayeian planning, the hape of f( ) expree the planner rik preference, with linear f( ) implying indifference between mean-preerving pread of a gamble and concave f( ) implying a preference for gamble with maller pread. However, the Bayeian definition of rik preference doe not carry over to other deciion criteria. Hence, I do not aociate the hape of f( ) with rik preference here. Indeed, I how below that how f( ) matter, if at all, depend on the deciion criterion that the planner ue. Baye Rule A Bayeian planner with welfare function (16) olve the optimization problem (17) max f[ + ( ) ]d. [0, 1] The olution i generically ingleton if f( ) i convex, but it may be fractional if f( ) ha concave egment. Manki and Tetenov (2007, Propoition 5) conider the pecial cae where the planner know and i uncertain only about. We how that the Baye allocation i = 0 if f( ) i concave and E ( ) <. It i fractional if f( ) i continuouly differentiable, E ( ) >, and f( )d < f( ). The allocation may be fractional or the ingleton = 1 if f( )d f( ). The Maximin Criterion The maximin problem (18) max min f[ + ( ) ] [0, 1] S ha the ame olution for all trictly increaing f( ). Thu, the hape of f( ) doe not affect the allocation.

21 20 The Minimax-Regret Criterion The hape of f( ) doe affect the olution to the minimax-regret problem (19) min max max [f( ), f( )] f[(1 ) + ]. [0, 1] S Neverthele, the central qualitative finding of Section 2.3 continue to hold with almot complete generality. I how here that the MR allocation i fractional whenever f( ) i continuou. Proof: Recall that S(a) { S: > } and S(b) { S: > }. Let (20a) R(, a) max f( ) f[(1 ) + ], S(a) (20b) R(, b) max f( ) f[(1 ) + ], S(b) be the maximum regret of allocation on S(a) and S(b) repectively. The maximum regret of on all of S i max [R(, a), R(, b)]. A increae from 0 to 1, R(, a) trictly increae from 0 to R(1, a) > 0 and R(, b) trictly decreae from R(0, b) > 0 to 0. Continuity of f( ) and boundedne of [(, ), S] imply that R(, a) and R(, b) are continuou function of. Hence, there exit a unique (0, 1) uch that R(, a) = R(, b). Thi i the MR allocation. Thi proof, a did the proof to reult (5), how that the MR allocation balance the welfare loe from error of Type A and B. Again, the quantitie R(, b) and R(, a) give the potential welfare loe from Type A and B error repectively. A increae from 0 to 1, the former potential lo decreae from R(0, b) to 0 and the latter increae from 0 to R(1, a). The MR criterion chooe to minimize the maximum

22 potential lo, which occur when R(, a) = R(, b). 21 Logarithmic Welfare Section 2.3 howed that the minimax-regret allocation ha the imple form (7) when f( ) i linear and { L, U), ( U, L)} are feaible value of (, ). The MR allocation typically mut be determined numerically when f( ) i nonlinear. However, a imple form emerge when f( ) i the log function and {( L, U), ( U, L)} are feaible value of (, ). Then (21a) R(, a) = max log{[ /[(1 ) + ]} = log{[ U/[(1 ) U + L]}, S(a) (21b) R(, b) max log{ /[(1 ) + ]} = log{ U/[(1 ) L + U]}. S(b) Hence, the MR allocation olve the equation (22) U/[(1 ) U + L] = U/[(1 ) L + U]. The olution i U( U L) (23) MR =. ( ) + ( ) U U L U U L Comparion of (7) and (23) how that the MR allocation under linear and logarithmic welfare coincide when =, but they otherwie generally differ from one another. In particular, the two U U allocation differ when the planner know and ha partial knowledge of. Then (23) reduce to

23 22 ( U ) (24) MR =. ( ) + ( ) U U L By aumption U >. Hence, the fraction of the population allocated to treatment b when welfare i logarithmic i maller than when welfare i linear. For example, in the entencing illutration of Section 2.4, the MR allocation with logarithmic welfare i 0.26 rather than the 0.35 found with linear welfare. Allocation of an Endowment Between a Safe and a Riky Aet To illutrate the above finding, conider an invetor with concave welfare function who mut allocate an endowment between a afe and a riky aet. Let the afe aet be treatment a, with known return. Let the riky aet be b, whoe return i known to lie in the interval [ L, U], where L < < U. Thu, the afe and riky aet are analogou to a tatu quo treatment and an innovation. A dicued above, a Bayeian invetor allocate fully to the afe aet if E ( ) <, diverifie hi portfolio if E ( ) > and f( )d < f( ), and may either diverify or allocate fully to the riky aet if f( )d f( ). A maximin invetor allocate fully to the afe aet. A minimax-regret invetor alway diverifie, the pecific fractional allocation depending on the hape of f( ). See Brock and Manki (2008) for further analyi of thi invetment deciion in the context of competitive lending Non-Additive Cot of Treatment Treatment may be cotly. The foregoing analyi cover etting where the aggregate cot of a treatment allocation i the um of individual treatment cot. Thi wa alluded to in Section 2.1, where I oberved that u j(t) may have the benefit-cot form u j(t) = y j1(t) y j2(t), where y j1(t) i the benefit when peron j receive treatment t and y j2(t) i the cot. There are many way in which cot might be non-additive. Thi

24 ection conider the polar cae of capacity contraint and fixed cot. 23 Capacity Contraint I have thu far aumed that all treatment allocation [0, 1] are feaible. Capacity contraint may place an upper bound on the fraction of the population who receive each treatment. A capacity contraint i a cot that equal zero when the fraction of peron who receive a treatment i below the upper bound and infinity thereafter. Let the maximum fraction of the population who may receive treatment a and b be (a) and (b) repectively. Then the feaible allocation are [1 (a), (b)]. A contrained Bayeian, maximin, or minimax-regret allocation olve the relevant extremum problem over the feaible. I focu on the MR allocation when welfare ha form (16) and f( ) i continuou. Let denote the uncontrained MR allocation and the contrained MR allocation. A hown MR in Section 2.3, maximum regret at any allocation equal R(, b) for and R(, a) for, where R(, b) i trictly decreaing in and R(, a) i trictly increaing. It follow that the contrained MR allocation i the feaible allocation cloet to the uncontrained MR allocation. That i, CMR MR MR (25) = 1 (a) if < 1 (a), CMR MR MR if [1 (a), (b)], MR (b) if MR > (b). Fixed Cot A fixed cot i a cot component that equal zero when no one receive a treatment and take a contant poitive value when any poitive fraction of the population receive the treatment. Fixed cot give ingleton allocation an advantage relative to fractional one. Suppoe that treatment a and b have non-

25 24 negative fixed cot C(a) and C(b) repectively. Then allocation = 0 and = 1 have fixed cot C(a) and C(b), but any (0, 1) bear the larger fixed cot C(a) + C(b). I how here that the MR allocation i fractional if the fixed cot are mall but i ingleton if they are ufficiently large. For implicity, I uppoe that the welfare function i linear and that the fixed cot have known value that do not vary with the tate of nature. Thu, the welfare function i (26) W( ) = + ( ) C(a) 1[ < 1] C(b) 1[ > 0]. Allocation = 0 i optimal if C(a) C(b) and = 1 i optimal if C(a) C(b). The problem of interet i treatment choice under ambiguity. Let S(a) and S(b) be the ubet of S on which treatment a and b are uperior. That i, S(a) = { S: C(a) > C(b)} and S(b) = { S: C(b) > C(a)}. The planner face ambiguity if S(a) and S(b) are non-empty. A earlier, let M(a) max S(a) ( ) and M(b) max S(b) ( ). Recall from (5) that the MR allocation in the abence of fixed cot i = M(b)/[M(a) + M(b)]. Let denote the MR allocation in the preence of fixed cot. The reult i MR FMR (27) FMR = 0 if M(b) + C(a) C(b) min {M(a) C(a) + C(b), MRM(a) + C(a)[1 MR] + C(b) MR}, C(a) C(b) FMR = MR + M(a) + M(b) if MRM(a) + C(a)[1 MR] + C(b) MR min {M(a) C(a) + C(b), M(b) + C(a) C(b)}, FMR = 1 if M(a) C(a) + C(b) min {M(b) + C(a) C(b), MRM(a) + C(a)[1 MR] + C(b) MR}. Proof: For any [0, 1], the maximum regret of on all of S i max [R(, a), R(, b)], where

26 (28a) R(, a) max C(a) { + ( ) C(a) 1[ < 1] C(b) 1[ > 0]} S(a) 25 = M(a) C(a) 1[ = 1] + C(b) 1[ > 0] (28b) R(, b) max C(b) { + ( ) C(a) 1[ < 1] C(b) 1[ > 0]} S(b) = (1 )M(b) + C(a) 1[ < 1] C(b) 1[ = 0]. are maximum regret on S(a) and S(b). Application of (28) at = 0 and = 1 give the maximum regret value Application of (28) at (0, 1) give max [R(0, a), R(0, b)] = M(b) + C(a) C(b), max [R(1, a), R(1, b)] = M(a) C(a) + C(b). max [R(, a), R(, b)] = max [ M(a) + C(b), (1 )M(b) + C(a)]. The minimum of maximum regret over (0, 1) olve the equation M(a) + C(b) = (1 )M(b) + C(a). Hence, the minimax regret allocation on (0, 1) i M(b) + C(a) C(b) C(a) C(b) = MR + M(a) + M(b) M(a) + M(b) and the minimax regret value on (0, 1) i MRM(a) + C(a)[1 MR] + C(b) MR. The final tep i to minimize maximum regret over (0, 1), = 0, and = 1. Thi yield (27). (27) reduce to Reult (27) implifie if ue of each treatment incur equal fixed cot. Let C C(a) = C(b). Then

27 (29) = 0 if M(b) min {M(a), M(a) + C}, FMR 26 FMR = MR if MRM(a) + C min {M(a), M(b)}, = 1 if M(a) min {M(b), M(a) + C}. FMR MR MR Thu, a common fixed cot maller than min {M(a), M(b)} MRM(a) ha no effect on the minimax-regret allocation. However, a larger fixed cot make the allocation ingleton Deontological Welfare Function I have thu far maintained the traditional conequentialit aumption of welfare economic. That i, policy choice matter only for the outcome they yield. Thi ection conider welfare function that embrace deontological conideration. Deontological ethic uppoe that action may have intrinic value, apart from their conequence. I firt give a brief abtract introduction and then focu on the pecific idea of equal treatment of equal. In Section 3.2, the fixed cot C(a) and C(b) made the treatment allocation affect welfare directly, regardle of the reulting outcome. Although I then decribed C(a) and C(b) in ordinary economic language a fixed cot, welfare function (26) can be interpreted a expreing the deontological idea that any ue of treatment a or b i normatively bad per e, with C(a) and C(b) expreing the repective welfare loe. More generally, a planner might ue a welfare function of the form (30) W( ) = f[ + ( ) + g( )], where g( ) i a planner-pecified function of. Welfare function (30) expree conequentialim through it + ( ) component and

28 27 deontological ethic through it g( ) component. A planner uing uch a welfare function trade off conequentialit and deontological conideration, chooing a deontologically inferior allocation if it yield ufficiently uperior outcome, and vice vera. Permitting trade off among the attribute of an action i almot univerally accepted by economit. However, philoophical dicuion generally take the poition that deontological conideration hould upercede conequentialit one. Thi ugget a lexicographic deciion proce in which one firt retrict attention to action that are deontologically acceptable and only then conider the conequence of thee action. Equal Treatment of Equal When conidering fractional treatment allocation, a particularly alient deontological idea i the normative principle calling for equal treatment of equal. Fractional allocation are conitent with thi principle in the ene that obervationally identical peron have equal probabilitie of receiving particular treatment. They are inconitent with the principle in the ene that obervationally identical peron do not actually receive the ame treatment. Thu, equal treatment hold ex ante but not ex pot. A dramatic illutration of the difference between the ex ante and ex pot ene of equal treatment occur in thi hypothetical problem of treatment choice conidered in Manki (2007b, Section 11.7). Chooing Treatment for X-Pox: Suppoe that a new viral dieae called x-pox i weeping the world. Medical reearcher have propoed two mutually excluive treatment, t = a and t = b, which reflect alternative hypothee, ay H aand H b, about the nature of the viru. If H ti correct, all peron who receive treatment t urvive and all other die. It i known that one of the two hypothee i correct, but it i not known which one; thu, there are two tate of nature, = H and = H. Let welfare be the urvival rate of the population. If a fraction of the population receive treatment b and the remaining 1 receive a b treatment a, the fraction who urvive i (1 ) 1[ = H ] + 1[ = H ]. a b

29 28 The ingleton allocation = 0 and = 1 provide equal treatment in both the ex ante and ex pot ene. Thee allocation alo equalize realized outcome the entire population either urvive or die. The minimax-regret allocation i = ½. Everyone i treated equally ex ante, each peron having a 50 percent chance of receiving each treatment, but not ex pot. Nor are outcome equalized half the population live and half die. 7 If one i concerned only with the ex ante ene of equal treatment, then all value of are deontologically equivalent. In term of welfare function (30), the function g( ) i contant. If one i concerned with the ex pot ene of equal treatment, ingleton allocation have an advantage relative to fractional one. In term of (30), g(0) = g(1) > g( ) for (0, 1). The equal fixed-cot cae conidered at the end of Section 3.2 ha the form g(0) = g(1) = C and g( ) = 2C for (0, 1). Thu, placing value C on the deontological conideration of equal ex pot treatment doe not affect the minimax-regret allocation if C < min {M(a), M(b)} MRM(a). However, it make the minimax-regret allocation ingleton if C i larger. It i worth noting that ocietie concerned with equal treatment occaionally implement policie that ue fractional treatment rule. American example include random drug teting and airport creening, call for jury ervice, and the Green Card and Vietnam draft lotterie. Moreover, randomized clinical trial and other randomized experiment implement fractional rule. Indeed, medical ethic permit randomized clinical trial only under condition of equipoie; that i, when partial knowledge of treatment repone prevent a determination that one treatment i uperior to another. Thee are exactly the circumtance in which the MR allocation i fractional. 7 The maximin allocation alo i = ½, a the maximin and MR criteria are equivalent in thi illutration. In contrat, a Bayeian planner would allocate the entire population to the treatment with the higher ubjective probability of ucce.

30 29 4. Interacting Treatment Thi ection relaxe the aumption that treatment i individualitic and conider etting where treatment interact, each peron outcome depending both on hi own treatment and on the treatment allocation within the population. A good illutration i medical treatment of an infectiou dieae. Let there be two treatment: (a) therapy after infection and (b) vaccination before infection. Suppoe that vaccination of a peron alway prevent hi infection and that the infection rate of unvaccinated peron decreae with the fraction of the population who are vaccinated. However, exiting epidemiological cience yield only partial knowledge of the relationhip between the fraction vaccinated and the infection rate of unvaccinated peron. Then determination of an optimal treatment rule may not be poible. Section 4.1 et up general concept and notation. Section 4.2 conider treatment of an infectiou dieae. See Manki (2008a) for further dicuion Concept and Notation It i eay to extend the concept and notation of Section 2 and 3 to planning problem where outcome depend on a peron own treatment and on the population treatment allocation. Thi only require expanion of the domain of the repone function y ( ) and welfare function u ( ). j Suppoe now that peron j ha a repone function y j(, ): T [0, 1] Y that map own treatment t and population allocation into outcome y j(t, ) Y. Let u j(t, ) u j[y(t, ), t, ] denote the net contribution to welfare. Let ( ) E[u(a, )] and ( ) E[u(b, )] be mean welfare when a randomly drawn peron receive treatment a or b and the population allocation i. Then the welfare function i j (31) W( ) = ( ) + [ ( ) ( )].