Statistica Sinica 9(1999), 879-891
ON THE PROBABILITY OF CORRECT SELECTION IN THE
LEVIN-ROBBINS SEQUENTIAL ELIMINATION PROCEDURE
Cheng-Shiun Leu∗† and Bruce Levin†
∗ New
York State Psychiatric Institute and † Columbia University
Abstract: We prove the general validity of a formula conjectured by Levin and
Robbins (1981) to give a lower bound for the probability of correct selection in a
sequential elimination procedure designed to identify the best of c binomial populations. The formula is elementary to calculate and simplifies the design of the
selection procedure.
Key words and phrases: Lower bound formula, probability of correct selection,
sequential elimination procedure.
1. Introduction
Suppose we have c ≥ 2 coins, and for any coin i (1 ≤ i ≤ c), let pi be the
probability of coming up heads on a single toss. We wish to select a coin with
the highest such probability. Levin and Robbins (1981) introduced the following
sequential elimination procedure to accomplish that goal. Begin by tossing the
(n)
(n)
coins vector-at-a-time. For n = 1, 2, . . ., let X(n) = (X1 , . . . , Xc ) be the vector
that reports the number of heads observed for each coin after n tosses, and let
[n]
[n]
[n]
[n]
[n]
X[n] = (X1 , . . . , Xc ) be the ordered X(n) vector with X1 ≥ X2 ≥ · · · ≥ Xc .
Let r be a positive integer chosen in advance of all tosses. Define C to be the
set of coins under consideration at any given time, with the convention that
(C)
c = #(C). Define Nr to be the time of first elimination in a c coin game with
coins C,
(n)
Nr(C) = inf{n ≥ 1 : max {Xi
i,j∈C
(C)
(n)
[n]
− Xj } = X1 − Xc[n] = r}.
If Nr = n, we drop from further consideration, after toss n, any and all coins i
(n)
[n]
satisfying Xi = Xc , i.e. all coins that have fallen r heads behind the leader.
If more than one coin remains the procedure continues, starting from the current
(C )
until
tallies of the remaining subset of coins C ⊂ C, and iterates with Nr
c − 1 coins have been eliminated. Thereupon we declare the remaining coin as
(C)
“best”. Define Pr [i] to be the probability that coin i is selected as best by this
procedure. (We also call this an “e-game” [e- for elimination] as if in a race that
880
CHENG-SHIUN LEU AND BRUCE LEVIN
(C)
coin i wins with probability Pr
by Levin and Robbins:
[i].) The following inequalities were conjectured
Conjecture. For any set of coins C with probabilities {p1 , . . . , pc }, let wi =
pi /(1 − pi ), and suppose, without loss of generality, that p1 ≥ p2 ≥ · · · ≥ pc .
Then for any positive integer r,
Pr(C) [1] ≥ w1r /
c
wir
and Pr(C) [c] ≤ wcr /
i=1
c
wir .
(1)
i=1
Inequality (1) allows determination of r to achieve any desired probability of
correct selection P † , given any specified odds w1 , . . . , wc . In particular, if the
odds ratio w1 /w2 ≥ δ, it suffices to choose r ≥ {log[(c − 1)P † /(1 − P † )]}/log δ.
In the context of medical trials, it is obviously desirable to eliminate the inferior
treatments while maintaining a high probability of correctly selecting the best
treatment.
The conjectured inequalities in (1) arose because of certain properties Levin
and Robbins (1981) proved for a related sequential procedure without elimination
of inferior coins. The stopping rule there was
[n]
[n]
Mr(C) = inf{n ≥ 1 : X1 − X2 = r},
[n]
(C)
with the obvious selection of the coin corresponding to X1 at time Mr = n
as best. Let Pr∗ [i] denote the probability of selecting coin i as best with stopping
(C)
rule Mr . Levin and Robbins proved that (1) holds for Pr∗ [i] and that for i < j,
Pr∗ [i]/Pr∗ [j] ≥ (wi /wj )r .
(2)
They conjectured that inequalities (2) and (1) should hold true for the e-game as
well, on the basis of a rigorous proof for r = 1 together with simulation studies
for “least favorable” parameter configurations of the form p1 > p2 = · · · = pc .
It turns out that the conjecture concerning inequality (2) is not generally
true. Zybert and Levin (1987) proved that the conjecture does hold for c = 3
coins for least favorable configurations, but, surprisingly, there exist p1 > p2 > p3
for which it does not hold. The violations of the inequalities in (2) for the e-game
are numerically small, and the expressions (wi /wj )r are quite good approximations to Pr∗ [i]/Pr∗ [j]. In fact, Zybert and Levin showed that, notwithstanding the
failure of (2) in the e-game, (1) with c = 3 coins does hold for any r and any
p1 ≥ p2 ≥ p3 . This curious situation raises the following questions: if (2) is not
the general reason that (1) is true for c = 3 in the e-game, does (1) even hold for
c > 3, or is the case c = 3 somehow special? If (1) does hold for any c, what is
the fundamental reason?
LEVIN-ROBBINS SEQUENTIAL ELIMINATION PROCEDURE
881
In this paper we prove that conjecture (1) does hold generally for any number
of coins c ≥ 2 and any positive integer r. The idea of the proof is to demonstrate a somewhat stronger result than (1): the conditional probability that coin
(n)
(n)
1 (respectively coin c) wins given that the sample path (X1 , . . . , Xc ) passes
through any symmetric subset of configurations (those left invariant under permutations of the labels of the coins) still obeys the lower (respectively upper)
bound inequalities. Inequality (1) follows simply by choosing the subset containing the single configuration (0, . . . , 0). It is of interest to note that the proof
makes no use of the notion of least favorable configurations. This is similar to
other proofs utilizing fundamental symmetries (see Levin (1984)).
2. Definitions and Notations
In the Levin-Robbins elimination procedure, it is important to consider dif(n)
[n]
ferences of the form Xj − Xc (j = 1, . . . , c), and we define the vector of such
differences as the configuration of X(n) . With a slight abuse of notation, we write
[n]
(n)
[n]
(n)
[n]
the configuration of X(n) as X(n) − Xc = (X1 − Xc , . . . , Xc − Xc ). The
[n]
ordered configuration of X(n) is the configuration of X[n] , i.e. X[n] − Xc . For
example, with c = 3, the configuration (120) indicates coin 1 has one fewer head
than coin 2, and one more than coin 3. The ordered configuration is (210).
In a c coin game, we define for s = 0, . . . , r the sets
Bs(C) = {b = (b1 b2 · · · bc ) | s = b1 ≥ b2 ≥ · · · ≥ bc = 0}.
(C)
Br is the set of all possible ordered configurations of X(n) at the time of first
(C)
elimination, and the union of Bs for s < r comprise all ordered configurations
prior to that time. Before the time of first elimination, the coin tallies pass
through a sequence of configurations which are certain permutations of certain
(C)
ordered configurations in r−1
s=0 Bs . This observation suggests that we ana(C)
[n]
lyze Pr [1] in terms of conditional probabilities given that X[n] − Xc passes
(C)
through any one of the ordered configurations b ∈ Bs for some 0 < s < r.
(C)
For example, with r = 3, c = 3, and b = (210) ∈ B2 , we might consider the
conditional probability of selecting coin 1 given that one of the configurations
{(210), (201), (120), (102), (021), (012)} is reached before the time of first elimi(C)
nation. Also, define Sv = { all distinct permutations of v} for any configuration
(C)
(C)
v. The above set is S(210) , while S(110) = {(110), (101), (011)}.
(C)
Now let Ns be the first time any configuration with maximum component
s > 0 is reached by X(n) in a c coin game. In symbols,
[n]
Ns(C) = inf{n ≥ 1 : X1 − Xc[n] = s}.
882
CHENG-SHIUN LEU AND BRUCE LEVIN
(C)
(C)
Also, for b ∈ Bs and d ∈ Sb , let Ns (d) be the first time a given configuration
(C)
d is reached by X(n) (if ever; if d is never reached, set Ns (d) = ∞):
Ns(C) (d) = inf{n ≥ 1 : X(n) − Xc[n] = d}.
(C)
(C)
We define the stopping times N0 and N0 (0) to be identically zero. The
(C)
(C)
event [Ns = Ns (d)] for 0 < s < r thus consists of all sequences of tosses
in which X(n) reaches configuration d before any other configuration d with
(C)
(C)
maxi di = s. For s = 0, [N0 = N0 (0)] is the entire sample space. We note
(C)
(C)
(C)
(C)
(C)
that events of the form [Ns = Ns (d)] and [Ns = Ns (d) < Nr ] are
(C)
equivalent for 0 ≤ s < r, i.e., stopping time Ns must occur prior to the time of
(C)
(C)
(C)
first elimination Nr . For any b in Bs , we define Wd to be the conditional
probability that permutation d of b is the first to be reached, given that some
(C)
permutation of b is reached at time Ns ,
(C)
Wd
P [Ns(C)
=
=
Ns(C) (d)]
P [Ns(C) = Ns(C) (d )]
(C)
d ∈Sb
For any configuration b and integers i and j in {1, . . . , c}, let bij be the configuration that interchanges components i and j of b. Define wij = wi /wj =
[pi /(1 − pi )]/[pj /(1 − pj )], the odds ratio for coin i and j.
3. Proof of the conjecture
The following lemma and its corollaries apply Wald’s change of measure
(C)
argument to events that occur at time Ns .
(C)
Lemma 1. For any number of coins c, any b ∈ Bs
b −bj
P [Ns(C) = Ns(C) (b)] = wiji
(C)
Proof. By definition, P [Ns
c
(n)
Xk (α)
(C)
= Ns
and any i, j ∈ {1, . . . , c}
P [Ns(C) = Ns(C) (bij )].
(b)] =
n≥1
αP
(n) (α),
where P (n) (α) =
(1 − pk )n is the product-binomial probability function of the first
n tosses, and α represents any sequence of binary outcome vectors in the event
(C)
(C)
[Ns = Ns (b) = n]. For any such α,
k=1 wk
P (n) (α)
(n)
[Xi
= wi
·
k=i,j
X
(n)
wk k
(n)
[Xi
= wij
(n)
(α)−Xj
(α)
(n)
(α)−Xj
(α)]
(n)
[Xj
wj
(1 − pk )n
(α)]
(n)
(α)−Xi
P (n) (αij ),
(α)]
(n)
Xi
(1 − pi )n (1 − pj )n wj
(α)
(n)
Xj
wi
(α)
LEVIN-ROBBINS SEQUENTIAL ELIMINATION PROCEDURE
883
where αij is a sequence of outcomes formed by transposing the outcomes of coins
(C)
(C)
i and j in the sequence α ∈ [Ns = Ns (b) = n]. Note that, on this event,
(n)
(n)
Xi (α) − Xj (α) = bi − bj , and there is a one-one correspondence between
(C)
sequences in [Ns
Therefore
(C)
= Ns
(C)
(b) = n] and those in [Ns
(C)
= Ns
(bij ) = n].
b −bj
P [Ns(C) = Ns(C) (b) = n]/wiji
=
P (n) (αij )
n≥1 α∈[N (C) =N (C) (b)=n]
=
s
s
P (n) (α ) = P [Ns(C) = Ns(C) (bij )].
n≥1 α ∈[N (C) =N (C) (b )=n]
s
s
ij
In particular, Lemma 1 holds for configurations that occur at time of first elimination, which we state as
(C)
Corollary 1. For any number of coins c, any configuration a ∈ Br
elimination, and i, j ∈ {1, . . . , c},
a −aj
P [Nr(C) = Nr(C) (a)] = wiji
at first
P [Nr(C) = Nr(C) (aij )].
Because any permutation d of b can be represented as a sequence of transpo(C)
(C)
(C)
sitions, by using Lemma 1, we can express P [Ns = Ns (d)] as P [Ns =
(C)
Ns (b)] times a permutation constant, defined as the product of the odds ratios
of the coins used in the transpositions raised to the power of the corresponding
differences between the number of heads. The permutation constant is uniquely
defined, because if λ and λ are two such permutation constants corresponding
to different sequences of transpositions leading to the same permutation d of b,
(C)
(C)
(C)
(C)
(C)
(C)
then P [Ns = Ns (d)]= λP [Ns = Ns (b)] = λ P [Ns = Ns (b)] implies
(C)
λ = λ . Therefore, a unique constant Kb can be defined such that
(C)
Wbij = P [Ns(C) = Ns(C) (bij )]
(C)
−(bi −bj )
P [Ns(C) = Ns(C) (d )] = wij
(C)
Kb ,
d ∈Sb
(C)
where Kb is the summation of all appropriate permutation constants. For
example, in the illustration with (r = 3, c = 3) mentioned above,
(C)
(C)
(C)
(C)
(C)
(C)
(C)
(C)
P [N2
1−2
= N2 ((120))] = w12
× P [N2
= N2 ((210))],
P [N2
0−2
= N2 ((012))] = w13
× P [N2
= N2 ((210))],
P [N2
0−1
= N2 ((201))] = w23
× P [N2
= N2 ((210))], . . .
P [N2
0−1 1−2
= N2 ((102))] = w23
w13 × P [N2
(C)
(C)
(C)
(C)
(C)
(C)
(C)
(C)
= N2 ((210))],
884
CHENG-SHIUN LEU AND BRUCE LEVIN
and therefore
(C)
W(120)
=
(C)
P [N2
(C)
N2 ((120))]
=
(C)
(C)
(C)
= N2 (d )]
P [N2
d ∈S(210)
(C)
=
(C)
−1
× P [N2
w12
= N2 ((210))]
(C)
−1
−2
−1
−1 −2
−1 −1
(1 + w12
+ w13
+ w23
+ w23
w12 + w23
w13 ) × P [N2
(C)
= N2 ((210))]
(C)
−1
−1
−2
−1
−1 −2
−1 −1
−1
/(1 + w12
+ w13
+ w23
+ w23
w12 + w23
w13 ) = w12
/K(210) .
= w12
(C)
That is, K(210) , in this case, equals the unique sum of permutation constants
−1
−2
−1
−1 −2
−1 −1
1 + w12
+ w13
+ w23
+ w23
w12 + w23
w13 . In the last term above, if instead of
(210) → (201) → (102) we use (210) → (120) → (102), the permutation constant
−1 −2
−1 −1
w23 equals w23
w13 . We can apply the same argument to arbitrary sequences
w12
of transpositions, and get the following from Lemma 1.
(C)
Corollary 2. P [Ns
(C)
= Ns
(C)
(C)
(C)
(d)] = Kb Wd P [Ns
(C)
= Ns
(b)].
(C)
Now let a ∈ Br be an ordered configuration at first elimination, and let
u ∈ Sa . The same change of measure argument as given in Lemma 1 shows that
u −u
(C)
(C)
(C)
(C)
(C)
(C)
P [Nr = Nr (u) and Ns = Ns (d)] = wiji j × P [Nr = Nr (uij ) and
(C)
Ns
(C)
= Ns
(dij )]. Then Lemma 1 implies.
(C)
(C)
(C)
Corollary 3. For any a ∈ Br , u ∈ Sa , b ∈ Bs for 0 ≤ s < r, and
(u −u )−(di −dj )
(C)
(C)
(C)
(C)
(C)
(C)
· P [Nr =
d ∈ Sb , P [Nr = Nr (u)|Ns = Ns (d)] = wij i j
(C)
Nr
(C)
(uij )|Ns
(C)
= Ns
u −u
(dij )].
(C)
d −d
(C)
(C)
(C)
Since wiji j = Wu /Wuij , and wiji j = Wd /Wdij , if now u is the same
permutation of u as d is of d, Corollary 3 could be written generally as follows.
(C)
(C)
(C)
(C)
Corollary 4. (Wd /Wu )P [Nr = Nr
(C)
(C)
(C)
(C)
P [Nr = Nr (u )|Ns = Ns (d )].
(C)
(u)|Ns
(C)
(C)
= Ns
(C)
(d)] = (Wd /Wu )
We can now prove the main result. The case s = 0 yields the conjectured
inequalities (1).
Theorem 1. For any set of c ≥ 2 coins C with corresponding probabilities
(C)
p1 ≥ · · · ≥ pc , any integers s and r such that 0 ≤ s ≤ r, and any b ∈ Bs ,
(C)
Pr(C) [1|Ns(C) = Ns(C) (d) for some d ∈ Sb ]
=
(C)
d∈Sb
(C)
Wd Pr(C) [1|Ns(C) = Ns(C) (d)] ≥ w1r /
c
i=1
wir
LEVIN-ROBBINS SEQUENTIAL ELIMINATION PROCEDURE
885
and
(C)
Pr(C) [c|Ns(C) = Ns(C) (d) for some d ∈ Sb ]
=
(C)
Wd Pr(C) [c|Ns(C) = Ns(C) (d)] ≤ wcr /
(C)
c
wir .
i=1
d∈Sb
Proof. We use mathematical induction on the number of coins c. Let c = 2
and consider any C = {i, j} for some coin j competing with a better coin i
(i < j). For s = 0, the classical gambler’s ruin problem (Feller (1957)) states
(C)
(C)
(C)
(C)
(C)
Pr [i|N0 = N0 (0)] = Pr [i] = wir /(wir + wjr ) and Pr [j] = wjr /(wir + wjr ).
(C)
For 0 < s ≤ r, the only b ∈ Bs is of the form (s0), and it can be seen
that any sample path leading to a correct selection must pass through (s0) or
(C)
(C)
(C)
(0s) for any s = 0, . . . , r − 1. Therefore Pr [i|Ns = Ns (d) for d = (s0)
(C)
(C)
(C)
or (0s)] = Pr [i] = wir /(wir + wjr ), and Pr [j] = 1 − Pr [i] = wjr /(wir +
wjr ), the classical gambler’s ruin probability, and conclude Theorem 1 is true
when c = 2. Assume, then, that for any k ∈ {2, . . . c − 1}, any subset K of C
with corresponding probabilities pi1 ≥ pi2 ≥ · · · ≥ pik , any s = 0, . . . , r, and
(K) (K)
(K)
(K)
(K)
= Ns (d)] ≥ wir1 / kj=1 wirj and
any b ∈ Bs , d∈S (K) Wd Pr [i1 |Ns
(K)
d∈Sb
(K)
(C)
Wd Pr
b
(K)
[ik |Ns
(K)
= Ns
k
(d)] ≤ wirk /
r
j=1 wij .
We prove the theorem
for c coins with probabilities p1 ≥ · · · ≥ pc . First consider the case s < r. In
(C)
order to be precise in the enumeration of permutations of b in Sb , we define
P erm(c) to be the set of all c! permutation functions σ : (12 · · · c) → σ((12 · · · c))
on c items. For any configuration v, define ν(v) to be the number of permutations
σ ∈ P erm(c) such that σ(v) = v. Thus there are c!/ν(b) distinct permutations of
(C)
b ∈ Sb . Therefore, enumerating the permutations σ ∈ P erm(c) as σ1 , . . . , σc! ,
we have
(C)
(C)
Wd Pr(C) [1|Ns(C) = Ns(C) (d)]
d∈Sb
=
c!
n=1
(C)
Wσn (b) · Pr(C) [1|Ns(C) = Ns(C) (σn (b))] · 1/ν(b).
(3)
c!
(C)
(C)
= Nr (σm (a))|
(C)
m=1 P [Nr
a∈Br
(C)
(C)
(C)
(C)
(C)
Ns = Ns (σn (b))]×Pr [1|Nr = Nr (σm (a))]×1/ν(a) where we have used
(C)
(C)
(C)
(C)
(C)
the Markovian property Pr [1|Nr = Nr (σm (a)), Ns = Ns (σn (b))] =
(C)
(C)
(C)
Pr [1| Nr = Nr (σm (a))]. This follows from the fact that the probability of
(C)
Now Pr
(C)
[1|Ns
(C)
= Ns
(σn (b))] =
ultimately selecting coin 1 (or any other coin) in the e-game given that a specific
886
CHENG-SHIUN LEU AND BRUCE LEVIN
configuration occurs at the time of first elimination does not depend on the path
leading up to that configuration prior to the time of first elimination. Thus (3)
becomes
c!
c!
n=1 a∈B (C) m=1
(C)
Wσn (b) P [Nr(C) = Nr(C) (σm (a))|Ns(C) = Ns(C) (σn (b))]
r
=
×Pr(C) [1|Nr(C)
c!
c! (C)
a∈Br
n=1 m=1
= Nr(C) (σm (a))] · 1/ν(a) · 1/ν(b)
(C)
Wσn (b) P [Nr(C) = Nr(C) (σm (a))|Ns(C) = Ns(C) (σn (b))]
×Pr(C) [1|Nr(C) = Nr(C) (σm (a))] · 1/ν(a) · 1/ν(b)
(4)
Since Sc! is a group under permutation multiplication, for any given σn and σm in
Sc! , there exists one and only one σk ∈ Sc! such that σn−1 σm = σk . Using Corol(C)
(C)
(C)
(C)
(C)
(C)
lary 4, we have (Wσm (a) /Wσk (a) ) × P [Nr = Nr (σk (a))|Ns = Ns (b)] =
(C)
(C)
(C)
Kb Wσn (b) × P [Nr
c!
c! (C)
a∈Br
(C)
= Nr
(C)
(σm (a))|Ns
(C)
= Ns
(σn (b))], and (4) becomes
(C)
1
·
(C)
k=1 m=1 Kb
Wσm (a)
(C)
Wσk (a)
P [Nr(C) = Nr(C) (σk (a))|Ns(C) = Ns(C) (b)]
×Pr(C) [1|Nr(C) = Nr(C) (σm (a))] · 1/ν(a) · 1/ν(b)
c!
=
(C)
a∈Br
· {
c!
1
(C)
k=1 Kb
(C) −1
· Wσk (a)
P [Nr(C) = Nr(C) (σk (a))|Ns(C) = Ns(C) (b)] · 1/ν(b)
(C)
m=1
Wσm (a) Pr(C) [1|Nr(C) = Nr(C) (σm (a))] · 1/ν(a)}.
(C)
(C)
(5)
(C)
It should be clear that expression (5) for Pr [1|Ns = Ns (d) for some d ∈
(C)
(C)
(C)
(C)
(C)
Sb ] can be extended to Pr [i|Ns = Ns (d) for some d ∈ Sb ] for any other
coin i simply by replacing 1 with i in the term in braces in (5). In particular,
the leading terms multiplying the expression in braces are the same for any such
i. We now argue that it suffices to show that the lower bound inequality holds
(C)
in the remaining case s = r, i.e., for any a ∈ Br ,
c!
m=1
(C)
Wσm (a) Pr(C) [1|Nr(C) = Nr(C) (σm (a))] · 1/ν(a) ≥ w1r /
Because
c c!
i=1 m=1
(C)
Wσm (a) Pr(C) [i|Nr(C) = Nr(C) (σm (a))] · 1/ν(a)
c
i=1
wir .
(6)
LEVIN-ROBBINS SEQUENTIAL ELIMINATION PROCEDURE
887
= P [ some coin i wins|Nr(C) = Nr(C) (u) for some u ∈ Sa(C) ] = 1
(proof omitted), we have that (6) is equivalent to (7) below:
c!
(C)
(C)
(C)
=
m=1 Wσm (a) Pr [1|Nr
c c!
(C)
(C)
(C)
i=2
m=1 Wσm (a) Pr [i|Nr
(C)
Nr
(C)
(σm (a)) = Nr
]·
1
ν(a)
(C)
(C)
]·
= Nr
(σm (a)) = Nr
1
ν(a)
wr
≥ c 1
r.
j=2 wj
(7)
Moving the denominator of the left hand side of (7) to the right hand side,
multiplying by all the leading terms of (5) before the braces, and summing over
c!
(C)
k=1 implies that
a∈B
r
(C)
Pr(C) [1|Ns(C) = Ns(C) (d) for some d ∈ Sb ]
≥ (w1r /
c
j=2
wjr )
c
(C)
Pr(C) [i|Ns(C) = Ns(C) (d) for some d ∈ Sb ],
i=2
which concludes the proof that (6) suffices.
(C)
(C)
(C)
(C)
To prove (6), for any u ∈ Sa , by definition, Wu = P [Nr = Nr (u)]/
(C)
(C)
= Nr (u )], and therefore the left hand side (lhs) of (6) is
(C) P [Nr
u ∈S
a
Wu(C) × Pr(C) [1|Nr(C) = Nr(C) (u)]
(C)
u∈Sa
=
(C)
u∈Sa
(C)
P [Nr
(C)
u ∈Sa
(C)
= Nr
(C)
P [Nr
=
(u)]
(C)
Nr (u )]
· Pr(C) [1|Nr(C) = Nr(C) (u)].
(8)
We must now consider the circumstances after the first elimination.
Given
that
(C)
a sequence of tosses reaches a configuration in Sa , there are n0c(a) different
subsets of coins that may still remain in the game, where n0 (a) is defined as
(C)
the number of 0s appearing in a. Therefore, Sa can be divided into n0c(a)
(C)
disjoint subsets that contain all the configurations in Sa with the same subset
of coins remaining in the game. Enumerate the subsets as F1 , . . . , F( c ) . Then
n0 (a)
(8) becomes
(n0c(a))
i=1
u∈Fi
P [Nr(C)=Nr(C) (u)]Pr(C) [1|Nr(C)=Nr(C) (u)]/
P [Nr(C)=Nr(C) (u )]
(C)
u ∈Sa
(n0c(a)) (C)
(C)
P [Nr = Nr (u)]
(C)
(C) [
P [Nr = Nr (u )]] ·
=
(C)
(C)
= Nr (u )]
i=1
u∈Fi
u ∈Fi
u ∈Fi P [Nr
×Pr(C) [1|Nr(C) = Nr(C) (u)] /
(C)
u ∈Sa
P [Nr(C) = Nr(C) (u )].
(9)
888
CHENG-SHIUN LEU AND BRUCE LEVIN
(C)
(C)
(C)
(C)
We wish to re-express the term P [Nr = Nr (u)]/ u ∈Fi P [Nr = Nr (u )]
as a weight in the reduced game that continues after the first elimination. To
do this, for every 1 ≤ i ≤ n0c(a) , choose the configuration u∗i that is ordered
apart from the zero components fixed in Fi . For u ∈ Fi , represent probability
(C)
(C)
(C)
(C)
P [Nr = Nr (u)] as P [Nr = Nr (u∗i )] times the appropriate permutation
(C)
(C)
= Nr (u∗i )] from
constant. The weight that results after cancelling P [Nr
(C )
i
, say, where d (u)
numerator and denominator is identical to the weight Wd (u)
is the reduced configuration from which the c − n0 (a) coin game continues with
coins Ci = {j1 , . . . , jc−n0 (a) }, namely (uj1 − a0 , . . . , ujc−n0 (a) − a0 ), ujk = 0, where
a0 = min1≤j≤c,uj =0 uj . For example, with c = 4, r = 2, one of the subsets Fi is
{(2011), (1021), (1012)}. Choose u∗i = (2011) and u = (1021). Then the term
(C)
(C)
= N2 (u)]/
P [N2
(C)
u ∈Fi
P [N2
(C)
=
(C)
P [N2
(C)
(C)
= N2 (u )]
(C)
= N2 ((1021))]
(C)
(C)
(C)
(C)
P [N2 =N2 ((2011))] + P [N2 =N2 ((1021))] + P [N2 =N2 ((1012))]
(C)
(C)
(C)
−1
= N2 ((2011))]w13
/(P [N2
= P [N2
(C )
(C)
−1
−1
= N2 ((2011))](1 + w13
+ w14
))
(C )
−1
−1
−1
i
i
/(1 + w13
+ w14
) = W(010)
= Wd (u)
,
= w13
where Ci = {1, 3, 4} and d (u) = (010) is the reduced configuration in the game
(C )
i
is identical to the conditional probability that
that continues. That is, W(010)
d (u) = (010) would be the first configuration to be reached among all permuta(C )
in a three
tions of (010), given some such permutation is reached at time N1
(C)
(C)
coin game with coins Ci = {1, 3, 4}. In general, then, writing P [Nr = Nr (u)]/
u ∈Fi
(C)
P [Nr
(C)
= Nr
(C )
i
(u )] = Wd (u)
, left hand side of (6) = (8) = (9) becomes
(n0c(a)) [
P [Nr(C) = Nr(C) (u )]]
i=1
/
u ∈Fi
d (u)∈Fi
(C)
(C )
Wd (u) Pr(C ) [1|Ns
(C )
= Ns
(d (u))]
P [Nr(C) = Nr(C) (u )].
(10)
(C)
u ∈Sa
(C )
where s = maxl (d (u))l and Fi = u∈Fi {d (u)} = Sd (a) comprises the set of all
permutations of reduced configurations from Fi . By the inductive hypotheses,
d (u)∈Fi
(C )
(C )
Wd (u) Pr(C ) [1|Ns
(C )
= Ns
(d (u))] ≥
w1r
c−n
0 (a)
l=1
= wjrl
1
−r .
w1j
j∈Ci
(11)
LEVIN-ROBBINS SEQUENTIAL ELIMINATION PROCEDURE
889
Consider next the leading factor in (10), which we rewrite without primes as
(C)
(C)
1/ u∈S (C) P [Nr = Nr (u)]. Define
a
Ea(C) = {u ∈ Sa(C) |u1 = r}
and
nr (a) =
c
I{aj =r} = #{aj = r}.
j=1
(C)
(C)
We can generate each u ∈ Sa as a u ∈ Ea followed by a transposition of
component 1 with j for j = 1, . . . , c. Then we have
c
P [Nr(C) = Nr(C) (u)] =
(C)
(C)
u∈Ea
j=1
c
i=1
(C)
u∈Fi ∩Ea
j=1
u∈Sa
(n0c(a))
= P [Nr(C) = Nr(C) (a)]
P [Nr(C) = Nr(C) (u1j )]
Ka(C) Wu(C)
1
nr (a)
1
u −u
w1jj 1 ,
nr (a)
(12)
by Corollaries 1 and 2. We want to put the remaining terms of (10) in similar
form. Thus (10) is
(n0c(a))
i=1
u∈Fi
c
n0 (a)
P [Nr(C) = Nr(C) (u)]
(C)
u∈Sa
()
i=1
(C)
u∈Fi ∩Ea
j∈Ci
=
P [Nr(C) = Nr(C) (u)] · {∗}
(C)
P [Nr
(C)
(C)
P [Nr
(C)
(u1j )] ·
= Nr
(C)
= Nr
1
nr (a)
· {∗}
,
(u)]
(13)
u∈Sa
(C)
where {∗} is the left hand side of (11). Since for any u ∈ Fi ∩ Ea and j ∈ Ci ,
(C)
(C)
(C)
(C) u −u
(C)
(C)
P [Nr = Nr (u1j )] = Ka Wu w1jj 1 ×P [Nr = Nr (a)] we have that the
left side of (6) is (10) and is
(C)
P [Nr
=
(C)
Nr (a)]
c
(n
)
0 (a)
(C)
(C)
i=1
u∈Fi ∩Ea
(C)
P [Nr
(C)
u∈Sa
(C)
Ka Wu
(C)
= Nr
1
nr (a)
(u)]
j∈Ci
u −u1
w1jj
· {∗}
.
(14)
890
CHENG-SHIUN LEU AND BRUCE LEVIN
Combining (11) and (14), the left size of (6) is at least
(C)
P [Nr
=
(C)
Nr (a)]
c
)
(n
0 (a)
(C)
(C)
i=1
u∈Fi ∩Ea
(C)
(C)
i=1
=
(C)
(C)
(C)
Ka Wu
u∈Fi ∩Ea
c
(n
)
0 (a)
i=1
(C)
u∈Fi ∩Ea
1
nr (a)
·
j∈C i
(C)
(C)
1
nr (a)
j=1
(u)]
u −u1
c
= Nr
w1jj
j∈C i
Ka Wu
uj −u1
−r
1
w1j
)/(
w1j )
nr (a) (
j∈Ci
j∈Ci
(C)
P [Nr
u∈Sa
c
)
(n
0 (a)
(C)
Ka Wu
−r
w1j
u −u1
w1jj
using (12). It suffices, then, to show that
j∈Ci
u −u1
w1jj
j∈Ci
−r
w1j
≥
c
u −u1
w1jj
c
j=1
j=1
c
for any i and Ci , because then lhs(6) ≥ 1/
(C)
Fi ∩ Ea ,
−r
w1j
−r
j=1 w1j
c
= w1r /
r
j=1 wj .
uj −u1 / j∈C j∈Ci w1j
i
But for
−r
w1j
}
u1 = r, so uj − u1 ≥ −r, whence {
≥ 1.
u∈
Note also that j ∈ Ci means uj > 0 while j ∈ Ci implies uj = 0, because Ci are
the coins not eliminated when configuration u is reached at first elimination.
u −u
−r
; thus
Therefore j∈C w1jj 1 = j∈C w1j
i
j∈Ci
i
u −u1
w1jj
j∈Ci
−r
w1j
≥
u −u1
j∈Ci
w1jj
j∈Ci
+
−r
w1j
+
j∈Ci
j∈Ci
c
u −u1
w1jj
−r
w1j
=
u −u1
w1jj
j=1
c
j=1
−r
w1j
.
(C)
This concludes the proof of the lower bound for Pr [1]. For the upper bound
inequality, the development is entirely analogous, and is omitted for brevity.
4. Discussion
It would be natural to attempt to prove the lower bound inequality in (1) by
proving only (6), instead of the stronger Theorem 1. As it sounds, (6) states that
the conditional probability of correct selection given that the sample path reaches
one of the symmetrical subsets of configurations Sa at the time of first elimination satisfies the lower bound formula, and since all paths must lead to some
such subset, the inequality follows. The stronger theorem is required, though,
LEVIN-ROBBINS SEQUENTIAL ELIMINATION PROCEDURE
891
because the inductive step at (11) takes place in a reduced game where the
current configuration is no longer on an elimination boundary. This is the appropriate generalization of the observation made for the two coin game, where
w1r /(w1r + w2r ) equalled the weighted average of gambler’s ruin probabilities starting at any configuration of the form (s0) or (0s) inside the elimination boundary.
References
Feller, W. (1957). An Introduction to Probability Theory and Its Application. Wiley, New York.
Levin, B. (1984). On a sequential selection procedure of Bechhofer, Kiefer, and Sobel. Statist.
Probab. Lett. 2, 91-94.
Levin, B. and Robbins, H. (1981). Selecting the highest probability in binomial or multinomial
trials. Proc. Natl. Acad. Sci. USA 78, 4663-4666.
Zybert, P. and Levin, B. (1987). Selecting the highest of three binomial probabilities. Proc.
Natl. Acad. Sci. USA 84, 8180-8184.
600 West 168th Street, Columbia University, New York, NY 10032, U.S.A.
E-mail: [email protected]
(Received July 1997; accepted September 1998)