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Econometrica, Vol. 82, No. 1 (January, 2014), 415–423
COMMENT ON “THE LAW OF LARGE DEMAND
FOR INFORMATION”
YARON AZRIELI
Ohio State University, Columbus, OH 43210, U.S.A.
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Econometrica, Vol. 82, No. 1 (January, 2014), 415–423
COMMENT ON “THE LAW OF LARGE DEMAND
FOR INFORMATION”
BY YARON AZRIELI1
Say that one information structure is eventually Blackwell sufficient for another if,
for every large enough n, an n-sample from the first is Blackwell sufficient (Blackwell
(1951, 1954)) for an n-sample from the second. This note shows that eventual Blackwell sufficiency lies strictly between (one-shot) Blackwell sufficiency and the ordering
of information structures formulated by Moscarini and Smith (2002), and thus offers
a new criterion for comparing experiments. A characterization of eventual Blackwell
sufficiency in terms of the one-shot experiments remains an open question.
KEYWORDS: Information structure, statistical experiment, information ordering,
Blackwell sufficiency.
1. INTRODUCTION
MOSCARINI AND SMITH (2002) (MS HENCEFORTH) derived the demand for
information of a Bayesian decision maker who may observe a sequence of conditionally independent and identically distributed signals before choosing an
action. Their analysis naturally leads to a criterion for comparing information
structures (or experiments) based on their performance when large samples
are drawn: An experiment E1 is better than another E2 , denoted E1 ≥MS E2 ,
if for every decision maker, there is N such that for all n ≥ N, an n-sample
from E1 yields higher expected utility to this decision maker than an n-sample
from E2 . Here, a decision maker means a prior distribution over the set of
states of nature, a set of available actions, and a utility function that maps each
state–action pair into payoff. Note that the threshold N in this definition is not
uniform, that is, it may vary with the decision maker.
MS showed that the above ordering is characterized by the minima of the
experiments’ Hellinger transforms, where the better experiment is the one with
a lower minimum. This in turn implies that ≥MS is “generically” complete (see
Theorem 1 and the following corollary in Moscarini and Smith (2002)).
In this note, we offer the following alternative criterion for comparing experiments, which we call eventual Blackwell sufficiency or simply eventual sufficiency: An experiment E1 is eventually sufficient for E2 , denoted E1 ≥EB E2 , if
there is one threshold N such that for all n ≥ N, every decision maker prefers
an n-sample from E1 over an n-sample from E2 . In other words, E1 ≥EB E2 if we
can find N such that for all n ≥ N, an n-sample from E1 is Blackwell sufficient
for an n-sample from E2 .
Let us denote by ≥B the standard one-shot Blackwell sufficiency criterion
(Blackwell (1951, 1954)). Then it follows from Theorem 12 in Blackwell (1951)
1
I thank Giuseppe Moscarini, Jim Peck, Ching-Jen Sun, a co-editor, and three anonymous
referees for helpful comments and suggestions that improved the quality and clarity of this note.
© 2014 The Econometric Society
DOI: 10.3982/ECTA11127
416
YARON AZRIELI
that E1 ≥B E2 implies E1 ≥EB E2 . In addition, it is clear from the definitions that
E1 ≥EB E2 implies E1 ≥MS E2 . Thus, eventual sufficiency lies between one-shot
sufficiency and the MS criterion. Using a simple example, we will show below
that both inclusions are strict: It may be that E1 ≥EB E2 even though E1 B E2
and it may be that E1 ≥MS E2 even though E1 EB E2 .2
Like the MS criterion, eventual sufficiency compares experiments based on
their performance when large samples are drawn, but it is a “more objective”
criterion in the sense that all decision makers agree on the ranking for large
enough samples. Of course, this objectivity results in a much less complete
ordering than that of MS. Our example suggests that eventual sufficiency differs from the two other criteria on a nontrivial set of pairs of experiments.
A characterization of eventual sufficiency in terms of the underlying one-shot
experiments is thus an interesting open question in the theory of the value of
information.
2. THE EXAMPLE
Let Θ = {θ1 θ2 } be the set of states of nature. Consider two parameterized
experiments Fα and Eβ defined for α ∈ [0 12 ] and β ∈ [0 14 ] as follows. The set
of signals for Fα is Y = {y1 y2 } and for Eβ is X = {x1 x2 x3 }. The conditional
probabilities of observing each signal given the true state are given by the matrices
y1
Fα : θ
1
θ2
α
1−α
y2
1−α
α
Eβ : θ
1
θ2
x1
x2
β
1/2 − β
1/2
1/2
x3
1/2 − β
β
The following four claims combined demonstrate that eventual sufficiency
lies strictly between Blackwell’s (one-shot) sufficiency and the MS criterion.
Figure 1 illustrates these claims.
2
MS suggested (p. 2357) in their discussion that ≥EB is equal to ≥B :
“If E1 is statistically sufficient for E2 , then any n-sample from E1 is sufficient for that from
E2 . But if E1 is not sufficient for E2 , nor conversely, the same is true of their n-replicas,
and Blackwell’s Theorem has nothing to say.”
Our example shows that this is not the case, and, thus, that the relation between statistical sufficiency à la Blackwell and the MS criterion is more complex and subtle. A couple of examples in
which an n-sample from E1 is sufficient for an n-sample from E2 , even though the one-shot experiments are incomparable, already appear in the statistics literature (Torgersen (1970, pp. 233–234),
Hansen and Torgersen (1974, pp. 372–373)). Our example is simpler than these previous ones and
clearly demonstrates the relation between the three orderings.
COMMENT ON “THE LAW OF LARGE DEMAND FOR INFORMATION”
417
FIGURE 1.—Each point inside the rectangle represents a pair of experiments (Eβ Fα ). In the
lightly shaded area Fα ≥MS Eβ but Fα EB Eβ , which shows that ≥EB is strictly stronger than ≥MS .
In the darkly shaded triangle Eβ ≥EB Fα but Eβ B Fα , which shows that ≥B is strictly stronger
than ≥EB .
1
4
CLAIM 1: If α ≤ 2β, then Fα ≥B Eβ ; if α ≥ 14 + β, then Eβ ≥B Fα ; if 2β < α <
+ β, then Fα B Eβ and Eβ B Fα .
CLAIM 2: Denote h(β) = (1 −
1
2
3
4
+ 4β2 − 2β − 2 β( 12 − β)). If α < h(β),
then Fα ≥MS Eβ , and if α > h(β), then Eβ ≥MS Fα .
CLAIM 3: If 2β < α < h(β), then Fα EB Eβ .
CLAIM 4: For (α0 β0 ) = ( 14 321 ), the n-replica Eβn0 is sufficient for the n-replica
F for every n ≥ 2 and, thus, Eβ0 ≥EB Fα0 .
n
α0
Claim 1 characterizes the ≥B ordering in the example, while Claim 2 characterizes the ≥MS ordering. Note that indeed ≥MS is an “almost complete”
ordering (on the graph of h(β) the experiments are incomparable by ≥MS )
and that it is an extension of ≥B . Claims 2 and 3 combined prove that ≥EB is
418
YARON AZRIELI
strictly stronger than ≥MS . Indeed, when 2β < α < h(β), we have Fα EB Eβ by
Claim 3 but Fα ≥MS Eβ by Claim 2. Finally, the combination of Claims 1 and 4
proves that ≥B is strictly stronger than ≥EB since (α0 β0 ) satisfy α0 < 14 + β0 ,
so by Claim 1, Eβ0 B Fα0 , but still Eβ0 ≥EB Fα0 by Claim 4.
Note that, by transitivity of ≥EB , at any point (β α) for which β ≤ β0 and
α ≥ α0 , it holds that Eβ ≥EB Fα . This shows that eventual sufficiency differs
from (one-shot) sufficiency in a nonnegligible set of parameter values (see the
darkly shaded triangle in Figure 1). We conjecture, but could not prove, that
Eβ ≥EB Fα whenever α > h(β). A step in this direction is given by the following
result.
CLAIM 5: If β = 0 and α > h(0) = 12 −
√
3
4
∼
= 0067, then Eβ ≥EB Fα .
3. PROOFS
PROOF OF CLAIM 1: Let Fα and Eβ be the matrices that correspond to the
above information structures, that is,
α
1−α
β
1/2 1/2 − β
Eβ =
Fα =
1−α
α
1/2 − β 1/2
β
If α ≤ 2β, then3
⎛
1 − α − 2β
⎜ 2 − 4α
Mαβ = ⎜
⎝ 2β − α
2 − 4α
1
2
1
2
⎞
2β − α
2 − 4α ⎟
⎟
1 − α − 2β ⎠
2 − 4α
is a stochastic matrix and Fα · Mαβ = Eβ . In other words, Eβ is a “garbling” of
Fα , which implies that Fα ≥B Eβ . Similarly, if α ≥ 14 + β, then4
⎛ 3 − 4α − 4β
⎜ 2 − 8β
⎜
⎜
1
Mβα = ⎜
⎜
2
⎜
⎝ 4α − 4β − 1
2 − 8β
4α − 4β − 1 ⎞
2 − 8β ⎟
⎟
⎟
1
⎟
⎟
2
⎟
3 − 4α − 4β ⎠
2 − 8β
is a stochastic matrix and Eβ · Mβα = Fα , which shows that Eβ ≥B Fα .
For α = 1/2, the matrix Mαβ is not defined, but since α ≤ 2β, this implies β = 1/4, so that
both experiments are completely uninformative and, therefore, sufficient for each other.
4
For β = 1/4, the same argument as in the previous footnote applies.
3
COMMENT ON “THE LAW OF LARGE DEMAND FOR INFORMATION”
419
It is left to show that if 2β < α < 14 + β, then neither experiment is sufficient
for the other. We do that by describing two decision problems such that Fα
gives higher expected utility than Eβ in the first, but the order is reversed in
the second. In both decision problems, the prior over Θ is uniform. In the first
problem, the decision maker can choose between two actions A = {a1 a2 }. The
utility function is given by u(θ1 a1 ) = u(θ2 a2 ) = 1 and u(θ1 a2 ) = u(θ2 a1 ) =
−1. A straightforward computation shows that the expected utility induced by
Fα for this problem is 1 − 2α, while the expected utility induced by Eβ is 12 − 2β.
Thus, as long as α < 14 + β, Fα is better than Eβ for this decision problem.
In the second decision problem, the set of available actions is B = {b1 b2 b3 }
α
and the utility function is given by u(θ1 b1 ) = u(θ2 b2 ) = α−2β
, u(θ1 b2 ) =
α−1
u(θ2 b1 ) = α−2β , and u(θ1 b3 ) = u(θ2 b3 ) = 0. When α > 2β, the expected
utility induced by Fα is 0 and that induced by Eβ is 1/2, so Eβ is better than Fα
for this decision problem. This completes the proof of the claim.
Q.E.D.
PROOF OF CLAIM 2: The MS criterion compares experiments based on the
minima of their Hellinger transforms, where the more informative experiment
is the one with a lower minimum. The Hellinger transform of Fα is given by
α)t α1−t (for 0 ≤ t ≤ 1). It is minimized at t = 1/2
Hα (t) = αt (1 − α)1−t + (1 − √
and its value at this point is 2 α(1 − α). The Hellinger transform of Eβ is given
by Hβ (t) = βt (1/2 − β)1−t + (1/2 − β)t
β1−t + 1/2. Again, it is minimized at t =
Fα and
1/2 and its value at this point is 1/2 + 2 β(1/2 − β). Thus, to compare
√
Eβ according to MS, we need to compare 2 α(1 − α) to 1/2 + 2 β(1/2 − β).
Simple algebra yields the condition in the claim.
Q.E.D.
PROOF OF CLAIM 3: The proof is based on the fact that for every n, there is
a signal realization of Eβn that leads to a posterior that lies outside of the convex
hull of the set of posteriors generated by Fαn . Whenever this is the case, there
exists a decision problem for which Eβn yields higher expected utility than Fαn .
Formally, consider the following sequence of decision problems. The prior
over Θ is uniform and there are three actions {b1 b2 b3 } to choose from in all
decision problems of the sequence. The utility function for the nth problem is
given by
αn
αn + (1 − α)n
un (θ1 b1 ) = un (θ2 b2 ) =
n
α
(2β)n
−
αn + (1 − α)n (2β)n + (1 − 2β)n
αn
−1
α + (1 − α)n
u(θ1 b2 ) = u(θ2 b1 ) =
αn
(2β)n
−
αn + (1 − α)n (2β)n + (1 − 2β)n
n
u(θ1 b3 ) = u(θ2 b3 ) = 0
420
YARON AZRIELI
Note that for n = 1, this coincides with the decision problem in the proof of
Claim 1.
The utility functions are constructed such that when α > 2β, choosing the
action b3 in the nth decision problem is optimal for every signal obtained from
the experiment Fαn . This can be easily seen by considering the extreme cases
where only one of the signals is obtained n times and by recalling that if an
action is optimal at two posteriors, then it is optimal in the interval of posteriors
between them. Thus, the expected utility induced by Fαn in the nth decision
problem is 0. On the other hand, there are signal realizations of Eβn after which
the actions b1 or b2 yield strictly positive expected utility. For instance, if the
signal x1 is observed n times, then choosing the action b2 gives an expected
utility of 1, and if x2 is observed n times, then b1 gives an expected utility of 1.
Thus, the ex ante expected utility induced by Eβn is strictly positive in the nth
decision problem.
Q.E.D.
2
2
≥B F1/4
and
PROOF OF CLAIM 4: To prove the claim, we will show that E1/32
n
n
3
3
that E1/32 ≥B F1/4 . This would imply that E1/32 ≥B F1/4 for every n ≥ 2 by the
following argument: Blackwell (1951, Theorem 12) proved that if I1 ≥B J1 and
I2 ≥B J2 , then the “combination” of I1 with I2 is sufficient for the combination of J1 with J2 .5 Now, for every n ≥ 2, there are nonnegative integers l k
such that n = 2k + 3l. Thus, the n-replica of an experiment is a combination of
several 2-replicas and 3-replicas of the underlying one-shot experiment. A ren
n
≥B F1/4
for every
peated use of Blackwell’s result would then imply that E1/32
n ≥ 2.
The matrices of conditional probabilities for the 2-replicas are given by6
⎛ 1
⎜
2
F1/4
= ⎝ 16
9
16
⎛ 1
9
16
1
16
⎜
2
E1/32
= ⎝ 1024
225
1024
6 ⎞
16 ⎟ ⎠
6
16
225
1
1024 32
1
15
1024 32
15
32
1
32
143 ⎞
512 ⎟ ⎠
143
512
5
By “combination” of experiments it is meant the experiment in which a conditionally independent signal is observed from each of them.
6
Notice that if two signals lead to the same posterior belief, then the corresponding columns
may be combined (added up) without affecting any relevant property of the experiment. In what
follows, we always represent an experiment in its most compact form, combining all signals that
lead to the same posterior.
COMMENT ON “THE LAW OF LARGE DEMAND FOR INFORMATION”
One may verify that
⎛ 1495
⎜ 1792
⎜
⎜ 297
⎜
⎜ 1792
⎜
2
M =⎜
⎜ 207
⎜ 256
⎜
⎜
⎜ 0
⎝
297
1792
1495
1792
0
207
256
0
0
421
⎞
0
⎟
⎟
⎟
0 ⎟
⎟
⎟
49 ⎟
⎟
256 ⎟
⎟
49 ⎟
⎟
256 ⎠
1
2
2
2
2
is a stochastic matrix that satisfies E1/32
· M 2 = F1/4
, so that E1/32
≥B F1/4
.
The 3-replicas of these experiments are represented by the matrices
⎛ 1
9 27 27 ⎞
⎜
3
= ⎝ 64
F1/4
27
64
⎛
64
27
64
1
⎜
32,768
3
E1/32
=⎜
⎝ 3375
32,768
and
⎛
0
⎜
1
⎜
⎜
⎜ 289,280
⎜
⎜ 1,286,166
⎜
⎜ 123
3
M =⎜
⎜
⎜ 2422
⎜
0
⎜
⎜
⎜
0
⎜
⎝
0
64 64 ⎟ ⎠
9
1
64 64
3
813
2048 32,768
675 12,195
2048 32,768
173
1024
173
1024
1
0
0
703,285
1,286,166
1654
2422
0
0
244,758
1,286,166
645
2422
1
0
179
1582
0
0
12,195
32,768
813
32,768
675
2048
3
2048
0
⎞
3375
32,768 ⎟
⎟
1 ⎠
32,768
⎞
⎟
0
⎟
⎟
48,843 ⎟
⎟
1,286,166 ⎟
⎟
⎟
⎟
0
⎟
⎟
⎟
0
⎟
⎟
⎟
1
⎟
1403 ⎠
1582
3
3
is a stochastic matrix that satisfies E1/32
· M 3 = F1/4
. This completes the
proof.
Q.E.D.
PROOF OF CLAIM 5: Assuming a uniform prior over the two states, each
experiment is completely characterized by its “standard measure,” that is, by
422
YARON AZRIELI
the distribution over posteriors it induces (see Blackwell (1951, p. 95)). Let μn0
be the standard measure of the n-replica E0n and let ναn be the standard measure
of the n-replica Fαn .
Consider the following five functions defined for t ∈ [0 1]:
g1 (t) = max{t 1 − t}
g3 (t) = 1 − 2t
g2 (t) = 2t − 1
g4 (t) = 1
g5 (t) = −1
Each gi (t) can be interpreted as the expected utility that some hypothetical
decision maker can obtain when his posterior belief that θ1 is the true state
is t. Thus, it follows from Theorem 1 of MS that there is N such that for all
n ≥ N and for7 1 ≤ i ≤ 5,
n
(1)
gi dμ0 ≥ gi dναn Let g be some convex and continuous function on [0 1]. Then there are
nonnegative numbers a1 a5 such that
(2)
g(t) =
5
ai gi (t)
i=1
holds for t ∈ {0 1/2 1}. Furthermore, since the function
on the intervals [0 1/2] and [1/2 1], it follows that
(3)
g(t) ≤
5
5
i=1
ai gi (t) is linear
ai gi (t)
i=1
for every t ∈ [0 1].
It follows that for every n ≥ N,
5
5
5
n
n
n
g dμ0 =
ai gi dμ0 =
ai gi dμ0 ≥
ai gi dναn
i=1
=
5
ai gi dναn ≥
i=1
i=1
g dναn i=1
where the first equality is by the fact that the support of μn0 is {0 1/2 1} and
by (2), the first inequality is by (1) and nonnegativity of the ai ’s, and the last
7
In fact, we only need the inequality to hold for the first decision maker i = 1. The other
four functions are linear and, therefore, these decision makers are indifferent between any two
experiments.
COMMENT ON “THE LAW OF LARGE DEMAND FOR INFORMATION”
423
inequality is by (3). It follows from Theorem 4 in Blackwell (1951) that E0n ≥B
Fαn , so E0 ≥EB Fα .
Q.E.D.
REFERENCES
BLACKWELL, D. (1951): “Comparison of Experiments,” in Proceedings of the Second Berkeley
Symposium on Mathematical Statistics and Probability. Berkeley: University of California Press,
93–102. [415,420,422,423]
(1954): “Equivalent Comparison of Experiments,” Annals of Mathematics and Statistics,
24, 265–272. [415]
HANSEN, O. H., AND E. N. TORGERSEN (1974): “Comparison of Linear Normal Experiments,”
The Annals of Statistics, 2, 367–373. [416]
MOSCARINI, G., AND L. SMITH (2002): “The Law of Large Demand for Information,” Econometrica, 70, 2351–2366. [415]
TORGERSEN, E. N. (1970): “Comparison of Experiments When the Parameter Space Is Finite,”
Probability Theory and Related Fields, 16, 219–249. [416]
Dept. of Economics, Ohio State University, 1945 North High Street, Columbus,
OH 43210, U.S.A.; [email protected].
Manuscript received October, 2012; final revision received June, 2013.