A Maximally Controllable Indifference-Zone Policy

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1 A Maimally Controllable Indifference-Zone Policy Peter I. Frazier Operations Research and Information Engineering Cornell University Ithaca, NY 14853, USA February 21, 2011 Abstract We consider the indifference-zone formulation of the ranking and selection problem with independent normal samples and common known sampling variance. We construct a continuous-time fully sequential sampling policy that meets the indifference-zone guarantee and is maimally controllable, in the sense that its worst-case probability of correct selection is eactly equal to the target probability. To the author s knowledge, this is the first fully sequential indifference zone policy that is maimally controllable for more than two alternatives. This continuous-time policy has a discrete time analog that also meets the indifference zone guarantee, and whose worst-case probability of correct selection is close to the target in numerical eperiments, even for more than 100,000 alternatives. 1 Introduction A frequently encountered problem in simulation is determining which of several simulated systems is the best. The higher-level problem of deciding how many simulation samples to take from each system to best support this selection of the best is called the ranking and selection (R&S problem. Doing well in R&S requires balancing the amount of time spent sampling against the quality of the ultimate selection. We consider the indifference-zone (IZ formulation of the R&S problem, in which we require a sampling policy to correctly select the best alternative with probability eceeding a target whenever its mean is sufficiently separated from those of the other alternatives. This problem has a rich history, dating to the seminal work [Bechhofer, 1954], with early work summarized in the monograph [Bechhofer et al., 1968]. Research in the area has been active since that time, with a series of papers including [Paulson, 1964, Rinott, 1978, Hartmann, 1988, Paulson, 1994], and active work in the related areas of subset selectian [Gupta, 1965] and multiple comparisons [Hochberg and Tamhane, 1987, Hsu, 1996]. This large body of research is summarized in [Bechhofer et al., 1995], and more recent work is reviewed in [Kim and Nelson, 2006, Kim and Nelson, 2007]. A parallel stream of research into Bayesian formulations of these and related problems 1

2 has also been active in the last 10 years, including, e.g., [Gupta and Miescke, 1996, Chick and Inoue, 2001]. The goal in designing IZ sampling policies is to create a sampling policy that takes as few samples as possible while still holding the IZ guarantee. This goal has been elusive, with early IZ policies introduced in [Bechhofer, 1954, Paulson, 1964, Rinott, 1978, Hartmann, 1991] providing actual probabilities of correct selection (PCS much larger than the desired target probabilites, at the cost of sampling much more than needed to achieve the target. This was especially true when with large numbers of alternative systems. Recent innovations introduced in [Kim and Nelson, 2001, Nelson et al., 2001] offered dramatically improved performance, but even these recently developed IZ policies are overly conservative in many situations, in the sense that they sample more than strictly necessary and over-deliver on PCS targets [Branke et al., 2007]. In this paper, we develop two fully sequential sampling policies that hold IZ guarantees, one in continuous-time and the other in discrete-time. We assume independent normal samples and common known sampling variance. The continuous-time policy has worst-case preference-zone PCS eactly equal to the target probability. We call this property being maimally controllable, and it can also be understood as a kind of optimality or minimaity in the IZ formulation. This is the first stopping rule with this property for more than two alternatives of which the author is aware. The worst-case preference-zone PCS of the discrete-time policy is shown in numerical eperiments to be etremely close to the target PCS, even for as many as 100, 000 alternatives. Surprisingly, these policies are derived using a Bayesian approach, where the Bayesian prior used to derive the policy is concentrated on the so-called least-favorable configurations. The stopping rule computes the Bayesian posterior that one would have given this prior, and then stops as soon as the Bayesian posterior probability of correct selection meets or eceeds the threshold. The resulting sampling policy is shown to hold the IZ guarantee, and to be maimally controllable in the continuous-time case. The proof techniques share similarities with results on the relationship between minima and Bayesian analysis from decision theory (see, e.g., [Berger, 1985]. Thus, this work points to an interesting connection between the IZ and Bayesian formulations of R&S that has recently become an active area of research. The sampling policies derived assume a common known sampling variance, which is an important limitation. In practice, variances are unknown and different across different alternatives, and must be estimated from data. Thus, the author views the current work not as a sampling policy that can be used on its own in practice, but as a solution to a theoretically interesting and long-standing problem that may later be etended and modified to provide better solutions to the practically important heterogeneous unknown sampling variance case. We begin in Section 2 by formally stating the IZ formulation of the R&S problem, as well as its generalization to continuous time. We then introduce the maimally controllable IZ policy for continuous and discrete time in Section 3. Preliminaries for the statement of the main results and their proof are given in Section 4 and the main results themselves, that the MCIZ policies satisfy the IZ guarantee, are the continuous-time MCIZ policy is in fact maimally controllable, are given in. An etension of the MCIZ policy to include a first-stage screening step is described in Section 6 and numerical results, including a comparison with the KN policy of [Kim and Nelson, 2001], is given in Section 7. 2

3 2 Problem Formulation Associated with each alternative is a sampling distribution with mean θ and common sampling variance σ 2. We suppose that the sampling variance is known, but that the sampling means are unknown. Define θ to be the column vector of sampling means θ = (θ 1,..., θ k, and define θ = ma θ. We would like to use information gained through sampling to find the alternative with the largest sample mean. At each point in time t = 1, 2,... we observe a sample from each alternative that is independent (both across time and across alternatives and normally distributed with mean θ and variance σ 2. Let Y t be the sum of all observations from alternative up to time t, so that Y t is a random walk with independent N (θ, σ 2 increments, where N (, denotes the normal distribution with the given mean and variance. We define F t to be the sigma-algebra generated by (Y s s t, and F to be the filtration F = (F t t 0. Because the sampling means are known, Y t is a sufficient statistic for (Y s s t. We choose the number of samples to take in an adaptive way. Upon the decision to stop sampling, we choose an alternative with the goal of choosing the one with the largest sampling mean θ. This description informally defines a policy. Our informal goal is to achieve a given probability of finding the best alternative while minimizing the number of samples taken. Formally, a policy π consists of a stopping time τ of the filtration F and a selection decision ˆ F τ taking values in {1,..., k}. Given a policy π and a true sampling mean vector θ R k, we define the probability of correct selection to be } PCS(π, θ = P θ {ˆ arg ma θ, τ <. We will only consider policies π that choose ˆ arg ma Y t, breaking ties uniformly at random. We consider the indifference-zone formulation of the R&S problem. In this formulation, we seek policies that satisfy a worst-case guarantee on performance over a set of true sampling means. This set of truths over which we would like the guarantee to hold is known as the preference zone, and is parameterized by a parameter δ > 0. The preference zone PZ(δ consists of all sampling means whose gap between the best alternative and all other alternatives is at least δ. That is, { } PZ(δ = θ R k : θ [k] θ [k 1] δ, where θ [k] θ [k 1]... θ [1] are the order statistics of θ. With this definition of the preference zone, we then say that a policy π meets the indifference-zone guarantee at α > 0 and δ > 0 if PCS(π, θ 1 α for all θ PZ(δ. 2.1 Continuous-Time Generalization We now generalize the discrete-time problem from the previous section to include both a discrete-time and a continuous-time version. Let T R + be a measurable subset of R +. For the discrete-time version of the problem, we take T = Z +, and for the continuous-time version we take T = R +. Corresponding to each alternative suppose there is an independent Brownian motion Y t starting at 0 with drift θ and volatility σ. Suppose that we are only allowed to stop at times in T. That is, T is restricted to 3

4 take values only in T. Let Y t denote the column vector (Y t 1,..., Y tk. Because Y t is a sufficient statistic for (Y s s t, it makes no difference whether we observe the process continuously, or only at the times in T. 3 The Maimally Controllable IZ policy In this section we will define two different policies, one for discrete time and the other for continuous time. These policies will be developed using a Bayesian motivation. Later we will see that these policies meet the indifference zone guarantee despite their Bayesian origins. Moreover, the continuous-time policy meets the guarantee eactly, with no overshoot. Let Q be the improper probability measure under which an alternative is chosen uniformly at random from among 1,..., k, θ is uniformly distributed over R, and θ θ = δ for all. Then define q t = Q { arg ma θ Y t } to be the posterior probability at time t that θ is the best of the sampling means. The following epression for q t is a consequence of Lemma 2 (below, ( / δ ( δ q t = ep σ 2 Y t ep σ 2 Y t. (1 Configurations of the form θ θ = δ for all are often called least favorable configurations in the R&S literature, and are commonly considered to be the most difficult configurations under which to select correctly. With this understanding, Q is a probability distribution under which the sampling mean θ is chosen uniformly at random from among these least favorable configurations. Given these definitions, the time-t conditional probability (under the measure Q of a correct selection if one were to stop immediately is ma q t. Thus, an intuitively appealing stopping rule is to stop as soon as this conditional probability meets or eceeds the target 1 α. This informally defines the maimally controllable indifference zone (MCIZ policy. More formally, given a time set T, the corresponding MCIZ policy is τ = inf{t T : ma q t 1 α}. (2 In particular, if T = N then we call the resulting policy the discrete-time MCIZ policy. If T = R + then we call the resulting policy the continuous-time MCIZ policy. These two MCIZ policies may also be defined without reference to their Bayesian intuition and interpretation. This may be done by simply defining q t by (1 and defining the MCIZ policy for a set T by (2. Figure 1 shows the continuation and stopping regions for a problem with k = 3 as a function of the observation process Y t. The MCIZ policy samples as long as the value of (Y t1 Y t3, Y t2 Y t3 is in the region labeled continue sampling, and then stops and chooses the labeled alternative as soon as it eits this central region. The boundaries between regions are curved in the observation process space (left-hand plot, unlike the linear boundaries of many other IZ-procedures. However, the boundaries are linear when transformed to the eponential of the observation process (right-hand plot. 4

5 Y t2 Y t choose 3 choose 2 continue sampling choose Y Y t1 t3 ep(y t2 Y t3 6 choose continue sampling 2 1 choose 3 choose ep(y Y t1 t3 Figure 1: Continuation and stopping regions for k = 3, δ/σ 2 = 1, and 1 α = 0.8. The left-hand figure displays these regions for the observation process Y, while the right-hand figure is transformed to the eponential of the observation process. 4 Preliminaries for the Proofs Before presenting the main results, we give a few definitions and lemmas that will be needed later. We first define CS to be the event of correct selection, CS = {ˆ arg ma θ, τ < }, so that PCS(π, θ = P π θ (CS. We also fi δ > 0 and α (0, 1. Several of the quantities we define will depend implicitly on them, but we do not include this dependence in the notation. We then define a family of improper probability measures that include and generalize Q. For each u R k, let u = ma u and define an improper probability measure Q u as follows. First, let R(1,..., R(k be a collection of random variables, each taking values in {1,..., k}. These random variables are such that under Q u and for each, R( is conditionally uniformly distributed over {1,..., k} \ {R( : < } given R(, <. In other words, R is a uniformly distributed permutation of {1,..., k}. Given this definition of R, let θ be uniformly distributed over R, and let θ θ = u u R(, This defines the distribution of θ under Q u. This definition generalizes the previously defined Q. Defining u = [δ, 0,..., 0], we have Q = Q u. We have defined measures Q u under which the set of differences between the true means, {θ θ }, = 1,..., k, is identical (up to permutations to the set of differences {u u }, = 1,..., k. The components of u have been permuted uniformly at random in order to arrive at the set of differences implied by θ. We will be interested in the posterior probability given prior Q u that a given alternative has the largest sampling mean. If u 1 is strictly bigger than all the other components of u, this event can be written R( = 1. Given a set of data, the following 5

6 lemma provides an epression for the posterior belief that R( = 1. This epression holds in general, not just when u 1 is strictly bigger than the other components of u. Lemma 1. For u R k and {1,..., k}, Q u {R( = 1 Y t } = ( 1 / ( 1 ep σ 2 Y t u r( ep σ 2 Y t u r(, r:r(=1 r (3 where the first sum is over all permutations r with r( = 1, and the second sum is over all permutations r. This lemma has as a consequence the following epression for the posterior under Q that alternative has the best sampling mean. This epression was referenced earlier to justify (1 in the definition of the two MCIZ policies. Lemma 2. { } Q arg ma θ Y t ( / δ ( δ = ep σ 2 Y t ep σ 2 Y t. We will later need the following technical result, which states that, with probability 1, the MCIZ policies take finitely many samples. The proof of this result employs a standard geometric decay argument. Lemma 3. Let θ R k and let π be the discrete-time or continuous-time MCIZ policy. Then τ < almost surely under P π θ. 5 Main Results We begin with three lemmas, which together constitute the proof of the main results. The first, Lemma 4, shows that the non-bayesian probability of correct selection PCS(π, u is identical to the probability of correct selection under the Bayesian prior Q u. The proof follows a symmetry or equalizing argument. Lemma 4. Let π be either the discrete-time or continuous-time MCIZ policy. Then Q π u {CS} = PCS(π, u. Lemma 5. Fi any time t and any u PZ(δ with u 1 u for all. Then ma Q u {R( = 1 F t } ma Q {R( = 1 F t }. Lemma 6. For π either the discrete-time or continuous-time MCIZ policy, Q π {CS} 1 α. If π is the continuous-time MCIZ policy then this inequality is an equality. We now use the three lemmas shown in the previous section to prove our main results, which come in the form of two theorems. The first theorem shows that both the discrete-time and continuous-time MCIZ policies satisfy the indifference zone guarantee. Several other policies from the literature meet the indifference-zone guarantee. In the design of these policies, much effort is spent on minimizing and assessing the etra coverage, i.e., the amount by which the worst-case probability of correct selection, 6

7 inf θ PZ(δ PCS(π, θ, eceeds the requirement of 1 α. The second theorem shows that the continuous-time MCIZ policy is eact, in the sense that this etra coverage is 0. Thus, the etra coverage of the discrete-time policy is due only to overshoot, i.e., the gap between the time at which the continuous-time policy would stop and the net integral time. Theorem 1. Let π be either the discrete-time or continuous-time MCIZ policy, and let θ PZ(δ. Then PCS(π, θ 1 α for each {1,..., k}. Proof. Let u PZ(δ. Then PCS(π, u = Q u (CS Q(CS 1 α where the equality is due to Lemma 4, the first inequality is due to Lemma 5, and the last inequality is due to Lemma 6. Theorem 2. Let π be the continuous-time MCIZ policy. Then inf θ PZ(δ PCS(π, θ = 1 α. Proof. First, Theorem 1 shows that inf θ PZ(δ PCS(π, θ 1 α. Second, consider u = [δ, 0,..., 0] and recall that Q u = Q. We have PCS(π, u = Q u (CS = 1 α, where the first equality is due to Lemma 4, and the second equality is due to Lemma 6 and Q = Q u.. Finally, u PZ(δ implies inf θ PZ(δ PCS(π, θ 1 α. We have shown that inf θ PZ(δ PCS(π, θ is both bounded above and below by 1 α, and hence must be equal to 1 α. 6 MCIZ with First-Stage Screening One drawback of the MCIZ policy is that it allocates samples equally among the alternatives. If we have a large number of clearly suboptimal alternatives, i.e., with large negative θ, and only a few alternatives whose means require a large number of samples to distinguish, then we will spend a great deal of sampling effort on those clearly suboptimal alternatives while trying to distinguish those few that are truly under consideration. One straightforward solution to this problem is to combine the MCIZ policy with a first-stage screening process, as is done, for eample, in [Kim and Nelson, 2001]. A brief first-stage screening can eliminate the most clearly suboptimal alternatives, leaving only those alternatives that are difficult to distinguish for the MCIZ policy. We will see the benefits of this first-stage screenin in the net section. To create this screened version of MCIZ, we use the common known sampling variance version of the first-stage screening process used there, which is in turn a generalization of the screening process from [Gupta, 1965]. The following policy results. We call this policy MCIZ-screen. 1. Given parameters α and δ, fi parameters n 0 2 and α 0 (0, α. 2. First stage sampling: Take n 0 samples from each alternative and let Ȳ0 be the average of the samples from alternative. 7

8 PCS KN MCIZ MCIZ-screen E[N]/k KN MCIZ MCIZ-screen k Number of alternatives (k Figure 2: (SC PCS and E[N]/k as a function of the number of alternatives k under the slippage configuration (SC with 1 α = 0.9, σ = 10, δ = 1, estimated using 1000 independent replications for each policy. PCS KN 0.9 MCIZ MCIZ-screen E[N]/k KN MCIZ MCIZ-screen k Number of alternatives (k Figure 3: (MDM Estimated PCS and E[N]/k as a function of the number of alternatives k under the monotone decreasing means (MDM configuration with 1 α = 0.9, σ = 10, δ = 1, estimated using 1000 replications for each policy. 3. Screening: Let z be the 1 (1 α 0 1/(k 1 normal quantile and let W = zσ n 0 /2. Let { } I = : Ȳ0 ma Ȳi 0 (W δ +. i If I = 1 then select the alternative in I and stop. Otherwise continue to the net step. 4. Fully Sequential Selection: Discard the observations Ȳ01,..., Ȳ0k and use the discrete-time MCIZ procedure to select the best from the alternatives in I using parameters δ, minimum probability of correct selection 1 (α α 0, and number of alternatives I (instead of k. The MCIZ-screen policy can be shown to hold the IZ-guarantee using standard proof techniques, as in [Kim and Nelson, 2001]. 7 Numerical Results We demonstrate the performance of the discrete-time MCIZ and MCIZ-screen policies on two standard test problems, and compare it to one other leading IZ policy the KN 8

9 policy of [Kim and Nelson, 2001]. In summary, we find that the MCIZ and MCIZ-screen policies performs closely to the PCS target under the slippage configuration, even for very large numbers of alternatives, e.g., k > 100, 000. Under the monotone decreasing means configuration, these two policies eceeds the PCS target, but only by a small margin. This demonstrates that the MCIZ policy is etremely controllable. In contrast, the KN policy is much more conservative, taking many more samples than is necessary to achieve the required bound on PCS. MCIZ-screen requires a smaller epected number of samples than KN on both configurations, with the largest difference being on the slippage configuration with many alternatives. The MCIZ-screen procedure also requires fewer samples than MCIZ procedure, with comparable achieved PCS. This difference in epected number of samples between MCIZ-screen and KN is small on the slippage configuration, but is quite dramatic on the monotone decreasing means configuration, where the MCIZ-screen procedure is able to drop those alternatives that clearly perform badly, while the MCIZ must sample all of them until it finally stops. Figure 2 shows the performance of the policies under the slippage configuration, in which θ 1 = δ and θ = 0 for 1, as a function of the number of alternatives k. The slippage configuration is generally understood to be the configuration in the preference zone in which correctly selecting the best is most difficult. Observe that the PCS is etremely close to the target probability. Theoretically we know that the PCS is bounded below by the target probability, and we epect that the actual PCS is slightly above the target because of the decreased fleibility in stopping created by the transition from continuous to discrete time. Surprisingly, the epected number of samples E[N] does not grow much faster than linearly as the number of alternatives increase. Figure 3 shows the performance of the policies under the monotone decreasing means configuration (MDM, in which θ = δ, as a function of the number of alternatives k. We first observe that the PCS is generally higher than the target 1 α. This is because the MDM configuration is not the least favorable, and the MCIZ policy is conversative on PCS under the MDM configuration so that it closely meets the target for minimal PCS at the SC. Observe that the probability of correct selection is roughly constant, and the epected number of samples taken is roughly linear, beyond a certain number of alternatives k. This is because when k is large enough, alternatives where is relatively large have very low θ and thus their sample statistics Y t become etremely low etremely quickly. Their corresponding values ep(y t approach 0 rapidly, and they have little effect on the values q t for small values of, which are the ones that really drive the behavior of the policy. Thus the PCS and stopping time τ remains roughly unchanged, and the total epected number of samples E[N] = ke[τ] scales approimately linearly in k. 8 Conclusion While these theoretical and numerical results are promising, the current policies are limited in several important ways. First, the assumption of common known variance is etremely restrictive, and is seldom met in practice. Generally speaking, variances 9

10 are different across different alternatives. Moreover, these variances are unknown and generally can only be estimated from data. Thus, the most important etension of this work would be to the unknown heterogeneous variance case. In addition, there are many indifference-zone policies that do not require equal allocation, and that choose the sampling allocation adaptively. In many cases, this can be much more efficient. MCIZ-screen is one generalization of MCIZ to take advantage of such adaptive allocations, but other generalizations of this work to allow adaptive sampling allocations could be an avenue toward more efficient maimally controllable indifference-zone policies. Acknowledgments The author would like to thank Rolf Waeber and Shane Henderson for several helpful discussions. Proofs Proof of Lemma 1. Consider any fied permutation r of the integers {1,..., k}. The posterior probability that R = r is proportional to the likelihood of the data given this event. This likelihood is Q u {Y t dy R = r} = Q u {θ ds} Q u {Y t dy R = r, θ = s}. The second term in the integrand may be rewritten R { Q u {Y t dy R = r, θ = s} = Q u Yt dy θ = u r( u + s } [ = ep 1 ( y 2σ 2 (u r( u + s ] ( 2 1 dy = c(s, y ep σ 2 y u r( dy, where c(s, y = ep [ 1 2σ 2 y2 2y ( u + s + (u r( u + s 2]. Because r is a permutation, (u r( u + s 2 does not depend on r and c(s, y may be rewritten as c(s, y = ep [ 1 2σ 2 y2 2y ( u + s + (u u + s 2], which also does not depend on r. This allows us to rewrite the likelihood as ( ( 1 1 Q u {Y t dy R=r} = Q u {θ ds} c(s, y ep σ 2 y u r( dy ep σ 2 y u r( dy. R Now let r vary over the set of all permutations. Because the event {R( = 1} is the union of all the events {R = r} with r( = 1, we have Q u {R( = 1 Y t } = ( 1 / ( 1 ep σ 2 Y t u r( ep σ 2 Y t u r(, r r:r(=1 which recovers the claimed epression (3. 10

11 Proof of Lemma 2. Recall that Q = Q u, where u = [δ, 0..., 0]. Since the first component of u is strictly bigger than the others, {R( = 1} = { arg ma θ }. Thus, Q { arg ma θ Y t } = Q u {R( = 1 Y t }, for which Lemma 1 provides an epression. We have Y tu r( = δy t,r 1 (1 since u r( = 0 for all ecept the alternative r 1 (1 which is best under permutation r, and u r 1 (1 = δ. Then, (3 from Lemma 1 becomes as claimed. Q u {R( = 1 Y t } = r:r(=1 = ep ( / δ ep σ 2 Y t r / ( δ σ 2 Y t ep ( δ ep ( δ σ 2 Y t σ 2 Y t,r 1 (1 Proof of Lemma 3. Define a constant a = σ 2 δ 1 log((k 1(1 α/α. Fi a time t 1 and let be any F t -measurable random variable. Consider the event that Y t Y t a for each. On this event, q t /q t = ep( δ (Y σ 2 t Y t ep( aδ/σ 2 for and q t = 1/(1 + q t/q t 1/(1 + (k 1 ep( aδ/σ 2 = 1 α. The value of a was chosen to make this last equality with 1 α hold. Thus, on the event considered, q t 1 α and τ t. We now define arg ma Y t 1,, which is F t 1 -measurable (and hence also F t - measurable. The previous discussion implies P π θ {τ t F t 1, τ > t 1} P π θ {Y t Y t a F t 1, τ > t 1}. This implies that P π θ {τ t F t 1, τ > t 1} P π θ {Y t, Y t 1,, Y t 1, Y t a F t 1, τ > t 1}, P π θ {Y t, Y t 1,, Y t 1, Y t a F t 1, τ > t 1} = P π θ {Y t, Y t 1, } P π θ {Y t 1, Y t a}. where the second line follows from the fact Y t 1, Y t 1, and the third line follows from the independence of increments of Y t under P π θ. The probability P π θ {Y t, Y t 1, } is the probability of a N (θ, σ 2 random variable, Y t, Y t 1,, eceeding 0. This probability is Φ(θ /σ 2, which is bounded below by Φ(min θ /σ 2. Here, Φ is the normal cumulative distribution function. Similarly, the probability P π θ {Y t 1, Y t a} is the probability of a N ( θ a, σ 2 random variable, Y t 1, Y t a, eceeding 0. This probability is Φ( (a+θ /σ 2, and is bounded below by Φ( (a + ma θ /σ 2. Thus, [ k 1 P π θ {τ t F t 1, τ > t 1} Φ(min θ /σ 2 Φ( (a + ma θ /σ ] 2. Let ɛ be the quantity on the right-hand side of this inequality. ɛ is a strictly positive deterministic quantity that is constant over time. By repeated application of this inequality, we have for any finite t that P π θ {τ > t} (1 ɛ t, which vanishes in the limit as t. This implies that P π θ {τ = } = 0. 11

12 Proof of Lemma 4. First, rewrite the probability of CS under Q π u as Q π u {CS} = Q u {R = r, θ ds} Q π u {CS R = r, θ = s}. r R Conditioning on R = r and θ = s is the same as conditioning on θ = v, where v = u r( u +s for each. Thus, Q π u {CS R = r, θ = s} = Q π u {CS θ = v} = P π v {CS} = PCS(π, v. Furthermore, PCS(π, is invariant to translations and permutations of its second argument, and v is obtained from u by permutation and translation, and so PCS(π, v = PCS(π, u. This implies Q π u {CS} = Q u {R = r, θ ds} PCS(π, u = PCS(π, u, r R because PCS(π, u depends on neither s nor r, and Q u {R = r, θ ds} sums over r and integrates over s to 1. Proof of Lemma 5. Suppose without loss of generality that δ = ma u. If this condition is not met, we may simply translate u to make it true without altering the measure Q u. Fi an alternative arg ma Y t. Because u 1 = ma u the maimum over of the posterior probability that R( = 1 under Q u is achieved at =. Thus, ma Q u {R( = 1 F t } = Q u {R( = 1 F t }. Similarly, u 1 = ma u implies ma Q {R( = 1 F t } = Q {R( = 1 F t }. Define q u ( = Q u {R( = 1 F t } /Q u {R( = 1 F t }. This quantity is well-defined because Q u {R( = 1 F t } > 0. Then, Q u {R( = 1 F t } = Q u {R( = 1 F t } Q u {R( = 1 F t } = 1 q u(. Similarly define q ( = Q {R( = 1 F t } /Q {R( = 1 F t } so that Q {R( = 1 F t } = [ q (] 1. Since q [ q(] 1 is decreasing in each q(, to show that Q u {R( = 1 F t } Q {R( = 1 F t } is enough to show that q u ( q ( for each. This follows trivially for =, so we now consider. For, we have 1 q u ( 1 q ( = r:r( =1 ep( 1 σ Y tu 2 r( r:r( =1 ep( 1 σ 2 Y tu r( ep( δ ep( δ Multiplying through by the strictly positive quantity ep( 1 σ 2 Y t u r( ep( δ σ 2 Y t r:r( =1 σ 2 Y t σ 2 Y t 12

13 shows that the sign of this epression is the sign of ( δ ep σ 2 Y t + 1 σ 2 Y t u r( ( δ ep σ 2 Y t + 1 σ 2 Y t u r( r:r( =1 r:r( =1 = ep δ σ 2 Y t + u 1 σ 2 Y t + u j σ 2 Y t + 1 σ 2 Y t u r( j 1 r:r( =1,r( =j, ep δ σ 2 Y t + u 1 σ 2 Y t + u j σ 2 Y t + 1 σ 2 Y t u r(, = ep δ σ 2 Y t + δ σ 2 Y t + 1 σ 2 Y t u r( [ ( uj ( ep σ 2 Y uj ] t ep σ 2 Y t. j 1 r:r( =1,r( =j, (4 The second line uses that the values of r( for, are the same between the two sums r:r( =1,r( =j and r:r( =1,r( =j. The third line uses that u 1 = δ. Consider the sign of the last term, ep(u j Y t /σ 2 ep(u j Y t /σ 2. Together u 1 = δ, j 1 and u PZ(δ imply u j 0. Furthermore, Y t Y t then implies ep(u j Y t /σ 2 ep(u j Y t /σ 2, which implies ep( 1 Y σ 2 t u j ep( 1 Y σ 2 t u j 0. Thus, the sign of (4 is nonnegative, which shows that q u ( q ( for each, which shows the result. Proof of Lemma 6. Conditioning on θ and using the tower property of conditional epectation shows that Q π {τ < } = E Q {Q π {τ < θ}} = E Q {P π θ {τ < }}. Lemma 3 shows that P π θ {τ < } = 1, and hence Qπ {τ < } = 1 as well. Then we have, Q π {CS} = E π Q {Q π {CS F τ }} = E π { Q Q π } { } {CS F τ } 1 {τ< } = E π Q ma q τ 1 {τ< }. (5 Under either MCIZ policy, ma q τ 1 α on the event {τ < } by construction of the policy. This shows that Q π {CS} E π { } Q (1 α1{τ< } = 1 α. Under the continuous-time MCIZ policy, ma q τ = 1 α on the event {τ < } by the almost-sure continuity of each t q t. This shows that Q π {CS} = E π { } Q (1 α1{τ< } = 1 α under the continuous-time MCIZ policy. References [Bechhofer, 1954] Bechhofer, R. (1954. A single-sample multiple decision procedure for ranking means of normal populations with known variances. The Annals of Mathematical Statistics, 25(1: [Bechhofer et al., 1968] Bechhofer, R., Kiefer, J., and Sobel, M. (1968. Sequential Identification and Ranking Procedures. University of Chicago Press, Chicago. [Bechhofer et al., 1995] Bechhofer, R., Santner, T., and Goldsman, D. (1995. Design and Analysis of Eperiments for Statistical Selection, Screening and Multiple Comparisons. J.Wiley & Sons, New York. 13

14 [Berger, 1985] Berger, J. O. (1985. Statistical decision theory and Bayesian analysis. Springer-Verlag, New York, second edition. [Branke et al., 2007] Branke, J., Chick, S., and Schmidt, C. (2007. Selecting a selection procedure. Management Sci., 53(12: [Chick and Inoue, 2001] Chick, S. and Inoue, K. (2001. New two-stage and sequential procedures for selecting the best simulated system. Operations Research, 49(5: [Gupta, 1965] Gupta, S. (1965. On some multiple decision (selection and ranking rules. Technometrics, 7(2: [Gupta and Miescke, 1996] Gupta, S. and Miescke, K. (1996. Bayesian look ahead onestage sampling allocations for selection of the best population. Journal of statistical planning and inference, 54(2: [Hartmann, 1991] Hartmann (1991. An improvement on paulson s procedure for selecting the population with the largest mean from k normal populations with a common unknown variance. Sequential Analysis, 10(1-2:1 16. [Hartmann, 1988] Hartmann, M. (1988. An improvement on paulson s sequential ranking procedure. Sequential Analysis, 7(4: [Hochberg and Tamhane, 1987] Hochberg, Y. and Tamhane, A. (1987. Multiple comparison procedures. Wiley New York. [Hsu, 1996] Hsu, J. (1996. Multiple Comparisons: theory and methods. CRC Press, Boca Raton. [Kim and Nelson, 2006] Kim, S. and Nelson, B. (2006. Handbook in Operations Research and Management Science: Simulation, chapter Selecting the best system. Elsevier, Amsterdam. [Kim and Nelson, 2007] Kim, S. and Nelson, B. (2007. Recent advances in ranking and selection. In Proceedings of the 39th conference on Winter simulation: 40 years! The best is yet to come, pages IEEE Press, Piscataway NJ. [Kim and Nelson, 2001] Kim, S.-H. and Nelson, B. L. (2001. A fully sequential procedure for indifference-zone selection in simulation. ACM Trans. Model. Comput. Simul., 11(3: [Nelson et al., 2001] Nelson, B., Swann, J., Goldsman, D., and Song, W. (2001. Simple procedures for selecting the best simulated system when the number of alternatives is large. Operations Research, 49(6: [Paulson, 1964] Paulson, E. (1964. A sequential procedure for selecting the population with the largest mean from k normal populations. The Annals of Mathematical Statistics, 35(1: [Paulson, 1994] Paulson, E. (1994. Sequential procedures for selecting the best one of k koopman-darmois populations. Sequential Analysis, 13(3. [Rinott, 1978] Rinott, Y. (1978. On two-stage selection procedures and related probability-inequalities. Communications in Statistics-Theory and Methods, 7(8:

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