JOURNAL
OF ECONOMIC
THEORY
42, 94-107 (1987)
Robust
Trading
Mechanisms*
KATHLEEN M. HAGERTY
Department
of Finance, Kellogg
Northwestern
University,
Graduate
School of Management,
Evanston,
Illinois 60201
AND
WILLIAM P. ROGERSON
Department
of Economics,
Northwestern
lJniversit.v,
Evanston,
Illinois
60201
Received September 4, 1985; revised April 29. 1986
We consider the problem of designing a trading institution for a single buyer and
seller when their valuation of the good is private information. It is shown that
posted-price mechanisms are essentially the only mechanisms such that each trader
has a dominant strategy. A posted-price mechanism is one where a price is posted
in advance and trade occurs if and only if all traders agree to trade. Journal of
;c’ 1987 Academic Press. Inc.
Economic
Literature
Classification Numbers: 022,026.
INTRODUCTION
In a very thought-provoking
paper, Myerson and Satterthwaite [8] consider the problem of designing a trading institution for a buyer and seller.’
The problem is that the buyer’s and seller’s valuation of the good are
private information. The only public information is a prior distribution of
values for the two players. The buyer’s and seller’s values are distributed
independently of one another conditional on publicly available data. They
show that it is generally impossible to design a trading mechanism which
satisfies four requirements:
(1) Bayesian-Nash Incentive Compatibility
Given their priors over the other trader’s valuation,
form a Bayesian-Nash equilibrium.
the players’ strategies
* We would like to acknowledge extremely helpful comments and discussions with Roger
Myerson. Rogerson’s work was supported by NSF Grant SES-8504034.
’ See D’Aspermont and Gerard-Varet [2], Laffont and Maskin [S], Myerson [6], Vickery
[9], and Wilson [lo, 11, 121 for related literature.
94
0022-0531/87 $3.00
Copyright
All rights
0 1987 by Academic Press, Inc.
of reproduction
in any form reserved.
ROBUST
TRADING
MECHANISMS
95
(2) Interim Individual Rationality2
Given their priors over the other trader’s valuation, each trader always
prefers to participate in the trading institution than not to participate.
(3) Budget Balancing (BB)
The price paid by the buyer equals the price received by the seller.
(4) Ex-post Efficiency
The mechanism transfers the good to the buyer if and only if the buyer’s
valuation of the good is higher than the seller’s.
They go on to characterize the set of mechanisms which satisfy ( l)--(3)
and then to calculate joint welfare maximizing (second-best) institutions
from this set. For the case of uniform priors, they show that rather simple
“split-the-difference”
rules first investigated by Chatterjee and Samuelson
[ 1 ] are second best. Furthermore, these techniques have the potential to be
applied to problems with different prior distributions or larger numbers of
traders to yield a theory of the design of trading institutions. Some of this
work has been begun,by Gresick and Satterthwaite [3].
The key shortcoming of this approach is that it relies heavily on the
assumption that there exists a common prior over traders’ valuations
known to all participants. In particular, an institution which produces a
very efficient outcome for one prior might perform very poorly under some
other prior. This creates two related problems. First, a social planner may
not be able to ascertain exactly what traders’ priors are when choosing an
institution. Second, given the costs of creating new institutions, a trading
institution (such as a stock exchange, for example) is often chosen with the
intention that it will be used by a variety of traders over a long period of
time. A variety of priors might be expected to occur over this time. These
problems suggest that an important
concern when choosing a trading
institution is that it work “fairly well” over a broad range of priors, i.e.,
that it be robust with respect to changes in the information structure of the
market.
A second, related, shortcoming of this approach is that the techniques
used appear to be very dependent on the assumption that traders’
valuations are distributed independently of one another conditional on
publicly available data. This assumption is clearly violated by many
exchange environments of great interest such as stock or commodity
exchanges. In fact, it is in general violated whenever the traders have
private information concerning some characteristic of the good which is
important to both of them.
’ The terminology
“interim”
individual
Myerson
[4] where they distinguish
it from
they term ex ante and ex post.
rationality
two other
was introduced
possible individual
by Holmstrom
and
rationality
concepts
96
HAGERTY
AND
ROGERSON
These two shortcomings can be totally avoided at the cost of restricting
consideration to a smaller class of mechanisms than those which satisfy
(1 )-( 3), namely those under which each trader has a dominant strategy
independent of other traders’ valuations. Formally, this class is created by
substituting conditions (l*) and (2*), below, for (1) and (2).
( 1*) Dominant Strategy Incentive Compatibility
(DIC)
Each trader has an optimal strategy independent of the other trader’s
strategy.
(2*) Ex-post Individual Rationality (EIR)
Ex Post, no trader is ever made worse off by participating
than by not participating,
in the institution
If a mechanism satisfies these properties, each player will choose to participate and will choose a particular strategy independent of his prior over
the other trader’s valuation. Thus no information about traders’ priors is
necessary in order to predict the outcome of the institution as a function of
traders’ valuations. Of course if a social planner wished to choose an
institution which maximized expected performance he would still have to
select a prior over valuations with which to calculate this expectation.
However, the strategies of traders would not depend on them agreeing with
this prior (or agreeing on any prior). Nor would the prior used by the
planner have to exhibit independent valuations across traders.
Given the desirable properties of this class of institutions, it is of some
interest to investigate how large it is and what levels of efficiency can be
attained while maintaining
this level of robustness to the informational
environment. The purpose of this paper is to characterize the set of all
mechanisms in the Myerson Satterthwaite model which satisfy DIC, EIR,
and BB. Our results suggest that the only such mechanisms are postedprice mechanisms, i.e., the social planner posts a price which does not
depend on any private information and the buyer and seller can trade at
that price or not trade at all. Formally, it is shown that posted-price
mechanisms are the only mechanisms satisfying DIC, EIR, and BB within a
class of mechanisms including3
(i )
(ii)
any set of
(iii)
arbitrarily
all differentiable mechanisms,
all mechanisms such that the probability of trade is 0 or 1 for
valuations,
all mechanisms which are step functions where the steps can be
small.
’ This class is defined
more
precisely
in the body
of the paper.
ROBUST
TRADING
MECHANISMS
97
These results, particularly (iii), strongly suggest that the result is also true
over the set of all mechanisms for two person trading problems. This is
obviously an important question to resolve. An equally interesting direction of research concerns extending the result to n-person trading problems.
We conjecture that posted price mechanisms may be the only ones which
satisfy DIC and EIR in that environment as well.
We interpret this result as essentially negative. Requiring that a
mechanism induce dominant strategies means that no use at all can be
made of traders’ private information. If one hopes to remove all opportunities for strategic interaction
between traders (in the sense that
dominant strategies are induced), the best that can be done is to post a
price independent of traders’ information. The lack of mechanisms satisfying this strong robustness criteria suggests that an interesting avenue of
research would be to investigate whether weaker notions of robustness can
be developed which are satisfied by a much broader range of trading
institutions.
Mechanisms satisfying DIC and EIR might be thought of as being
robust with respect to the informational environment of the traders. Alternative notions of robustness are also possible such as robustness with
respect to traders’ utility functions or the number of traders. Another
interesting avenue of research would be to explore definitions of these alternative notions of robustness, what their relationship
is to strategic
robustness and what sorts of trading institutions exhibit these properties.
Section 2 presents the model and notation. Section 3 presents the key
characterization of robust mechanisms. Then Section 4 shows that postedprice mechanisms are the only ones which satisfy this characterization
within the classes (i)-(iii).
2
The notation used will be the same as in Myerson and Satterthwaite [8].
See Myerson and Satterthwaite [S] for a fuller discussion of the economic
interpretation of this model.
Individual
1 owns an object which individual 2 whishes to buy. Let 11,
denote the value of the object to individual i. Each a, is drawn from the
interval [_v, V] and each individual knows only his own valuation at the
time of trade. It is assumed that individuals are risk neutral and have
additively separable utility in money and the object.
An allocation rule is a pair of functions (p, x) defined over b, 51’ such
that p maps into [O, l] and x is real valued. The function p(u,, u2) is the
98
HAGERTY
AND
ROGERSON
probability that a trade occurs and x(v,, u2) is the expected payment from
the buyer to seller given the types ui and Q.~
The goal of this paper is to investigate what allocation rules can be
achieved by trading institutions where each trader has a dominant strategy
(where not participating is one of his strategy options). By the revelation
principle5 it is sufficient to consider trading institutions where traders
report their type or choose not to play and each trader has a dominant
strategy to truthfully report his type.
A trading mechanism is therefore defined as a pair of functions (p, G)
which determine the probability of a trade and the price of exchange if
trade occurs. Let p(z), , u2) denote the probability of a trade for valuations
U, and u2. Let r denote the price of exchange if trade occurs and let
G(r; v , , u2) denote the distribution of r conditional on valuations u,, v*.~ A
mechanism will be defined to exhibit a price of zero if no trade occurs since
this is an immediate implication of BB and EIR.
Let x(v,, u,) denote the expected payment from the buyer to seller
conditional on valuations u, and u?. This is determined by
Let U;(t),, u2) be the expected return to Mr. i given the values v, and v2.
These are given by
U,(v,, ~‘zl’X(V1,
Lb-do,,
u2) Ul
(2)
and
~,(v,,~,)=P(~,,v,)v,-.~(~,,v,).
(3)
The properties DIC and EIR will now be formally defined. (By
definition, all mechanisms satisfy BB.) A mechanism satisfies DIC if for
every U, , u2 and li in [_v, V],
U,(v,-
u*)
2
x(6
u2)
-P(fi,
u*)
Ul
(4)
4 An allocation
rule could be more fully detined as a triple (p, G, H) where p is as above
and G( ; u,, rz) and H( ; vr. u2) are distribution
functions
determining
the payment
from
buyer to seller contingent
on v, and u2 if, respectively,
trade occurs or trade does not occur.
However,
since all traders
are risk neutral,
all traders
are indifferent
between
any two
allocation
rules with the same (p, I). Therefore
an allocation
rule is simply defined in terms of
the expected payment
it induces; allocation
rules which yield the same probability
of trade
and expected payment
for all types will be viewed as the same rule.
5 See Myerson
[6. 71 and Myerson
and Satterthwaite
[S] for a more complete discussion
of this point.
h G is defmed only for values of (u,, 11~) such that ~(a,, u2) > 0.
ROBUST
TRADING
99
MECHANISMS
and
Equation (4) is the requirement that it is a dominant strategy for the seller
to report his type truthfully. Equation (5) is the similar requirement for the
buyer. A mechanism satisfies EIR if
support(3.; ul, U,)E Cu,,4
(6)
for (or, u2) such that p(u,, u2) > 0 where “support G” denotes the support
of G.
The characteristic of interest can now be defined. An allocation rule,
(p, x), is said to be DIC-EIR
implementable if there exists a mechanism
(p, G) such that
(i) (p, G) satisfies DIC and EIR,
(ii) (p, G) induces the expected payment
satisfy (1).
rule X, i.e., p, G, and x
3
Theorem 1 provides the characterization of allocation rules which are
DIC-EIR implementable.
The results of Section 4 are based on this characterization.
THEOREM
1. The allocation rule (p, x) is DIC-EIR
only if the following six conditions are satisfied.
p is nonincreasing
in u, and nondecreasing
02
P(UI 3%)(Uz - 01) = s P(o,, 5) +P(&
01
~2) dfi
x(u,, u2) =p(uI,
u2) u2 -J^“‘p(u,,
01
x(u,, Q=P(u~,
~2) ~1 +~“‘P(z?
1,I
x(u,, u2)=0
ti) dt;
uJdfi
implementable if and
in u2,
(7)
if
(8)
Ol>U2,
(9)
.for
u1 d
for
0, <u2,
(10)
f or
2’1Gu,,
(11)
for
u,>u2.
(12)
02,
100
HAGERTY
AND
ROGERSON
Proof:
First it will be shown that (7k(12) are necessary for DIC-EIR
implementability.
Suppose that (p, x) is DIC-EIR
implemented
by the
mechanism (p, G). By EIR (8) and (12) are true. By using arguments
similar to those in Myerson and Satterthwaite [S] it can be shown that
DIC implies (7) and, furthermore, that
U,(u,, u2)= U,(O, 112)+ !^” p(& JJ2)di
1’1
(13)
and
(14)
By (8) these can be rewritten as
and
(16)
for u, < u2. Substitution of (2) and (3) into (15) and (16) yields (10) and
(11). Subtract (11) from ( 10) to yield (9).
Now it will be shown that (7t( 12) are sufficient for DIC-EIR implementability. Suppose that (p, x) satisfies (7)-(12). Create a mechanism (p, G)
by defining G( ; ri, u2) to have a single mass point on x(u,, u,)/p(u,, u2).
(Recall that G only need be defined for values of (u, , u2) such that
p(u, , v2) > 0.) Clearly, p, x, and G satisfy ( 1).
It will now be shown that this mechanism satisfies DIC and EIR. First
DIC will be checked. The case of the buyer will be considered; the case for
the seller is similar. It must be shown that (5) is true. Equations (lo), (12),
and (8) imply that
w
x(~,,u*)=P(~l,~2)~2sL’ Au,,fi)d;
for every (0, , u2) E b, L’] 2. Substitute
must be shown that
(17)
(16) and then (17) into (5). Then it
ROBUST
TRADING
MECHANISMS
This follows immediately from the fact that p is nondereasing
argument.
EIR follows immediately from (10) and (11).
101
in its second
Q.E.D.
4
Posted price mechanisms are one obvious class which satisfy DIC and
EIR. They can be described as follows. An auctioneer announces in
advance that he will post some price r, drawn according to a distribution
function, F(r). (The support of F may lie partially or totally outside of
[_v, 61.) After observing the value of r, the buyer and seller announce
whether they are willing to trade. Trade occurs if and only if both are
willing to trade and it occurs at price r.
The formal mechanism generated by this rule is determined as follows.
For any (ul, a,), a trade occurs if and only if r is within the interval with
lower endpoint v, and upper endpoint oz. Whether the interval includes its
endpoints or not depends on whether indifferent traders choose to trade or
not. Therefore
P(U, 7 4 = F(G - F(u, 1
(19)
almost everywhere. Then G(r; ul, u2) is simply F(r) restricted to the interval
with lower and upper endpoints, respectively, of u1 and u2. Once again,
whether the endpoints are included depends on how traders behave when
indifferent. It is straightforward to directly verify through integration by
parts that the x(0,, u2) generated by this mechanism (defined according to
(1)) together with p(v,, u2) satisfy (7)-( 12).
The above discussion motivates the following definition.
DEFINITION.
’
An allocation rule which is DIC-EIR
implementable
is
said to be implementable by posted price if (19) is satisfied almost
everywhere.
The next three corollaries show that all DIC-EIR
implementable
allocation rules are implementable by posted price within various classes of
allocation rules.
’ This definition
ignores one fine point. Namely,
(19) must hold except possibly at points of
discontinuity
of F. Furthermore
at these points of discontinuity
F(o,) must be replaced by
some point in the interval
[lim, _ “, F(u), F(u,)] where the limit is taken from the left. Formally
taking this point into account
does not change the following
results or methods
of proof.
However,
it adds to the notational
and expositional
complexity.
Therefore
they will be
ignored.
102
HAGERTYANDROGERSON
COROLLARY 1. Suppose that (p, x) is DIC-EIR
implementable and that
p is twice continuously difSerentiable. Then (p, x) is implementable by posted
price.
Proof
It is sufficient to show that if p is twice differentiable and satisfies
conditions (7) - (9), that (19) is satisfied almost everywhere. Differentiate
(9) with respect to V, and u2 to yield
-g&-P(W*)(%-u,)=O.
1
Therefore p is additively
(20)
2
separable, i.e., p can be written
P(O,54) = s(Q) - t(u,)
(21)
for two nondecreasing functions s and t. Substitute (21) into (9) to yield
s
“* s(B) - t(C) d6 = 0
L’I
for every u, < u2.
It is clear from (22) that s must equal t almost everywhere.
(22)
Q.E.D.
Many mechanisms of interest are not differentiable but rather map into
{ 0, 1 }. That is, trade never occurs probabilistically.
This, for example, is
true of the optimal mechanisms found by Myerson and Satterthwaite [8]
and studied by Chatterjee and Samuelson [ 11. Corollary 2 shows that the
only mechanisms satisfying DIC and EIR within this class are posted price
mechanisms.
COROLLARY 2. Suppose that (p, x) is DIC-EIR
implementable and that
p maps into (0, l}. Then (p, x) is implementable by posted price.
Proof: It is sufficient to show that if p maps into (0, 1) and satislies
conditions (7))(9) that (19) is satisfied almost everywhere. Consider any
(0 r, u2) such that p(uI, u2) = 1. By (7), there exist values vi* and uz* in
[u,, uz] such that
P(4, C)=
i
0
1
for
for
ti<u:
fi>u,*
0
for
for
I?>u:
d<u:
and
P(C u,)=
i
1
(23)
ROBUST
It will now
impossible.
tradiction,
parts such
TRADING
103
MECHANISMS
be shown that VT = vf. The case of VT < uz will be shown to be
The case of the reverse inequality is similar. Suppose, for conthat VT -CO;. Then the integral in (9) can be broken into three
that the integrand is constant over each part. This yields
(24)
which equals
v~-v,+(vz*-v~).
(25)
This is greater than the 1.h.s. of (9) which equals v2 - ul, thus yielding a
contradiction.
An implication of the fact that v1* = UT is that if p(v,, v2) = 1 there exists
a v* E [v,, v2] such that
for
JJ(v,,v)=
:,
1 for
v>v*
v<v*,
0
P(Ul v,) = *
u>u*
v<v*.
for
for
(26)
It will now be shown that (26) implies that p must have the following form.
There must exist some v* E [_o,U] such that
1
P(V,? 02) = o
for
for
vl<v*
v,>u*
and
or
v2 > v*
t.2 < v*.
(27)
On the boundary lines between the two regions p can be 0 or 1. This is
illustrated by Fig. 1.8
To see this consider the point (0, 1). If ~(0, 1) = 0 we are done trivially. If
~(0, 1) = 1, there exists a v* with the properties defined by (26). By (7) it
must be that p(vl, vq) equals zero if v1 > v* or v2 < v*. The other part of
(27) follows from applying (26) to the points { (0, v): v> v*} and
((v, 1): u<V*}.
Equation (27) is the posted price mechanism such that a price v* is
announced with probability 1, i.e., p satisfies (19) almost everywhere for the
function F defined by
for
for
v<v*
v>v*.
(28)
Q.E.D.
* Figure
1 is drawn
for the case _o= 0.
104
HAGERTYANDROGERSON
“2
/
/
V*
,
/
I
i
“1
FIG. 1. p maps into {0, 1).
Corollary 3 considers the class of functions which will be called block
functions. These are two-dimensional step functions defined as follows:
DEFINITION.
Divide the square [_v, V]’ into n2 equal sized smaller
blocks. Let B, be the interior of the block i steps from the left and j steps
from the bottom, i.e.,
(II,,u~):~+~ i-1
(---_v)<u,<~+~(v-_v)
u
n
(29)
Consider any function p defined over {(a,, u2): v, < u2 and (vi, u2) E
. a block function if it is a constant over each of the
[_v,I?]‘}. Then p IS
blocks { Bu}ici, i.e., there exist numbers (b,i}iGi such that p(u,, u2) = b, for
(01, uz)EBij.
COROLLARY 3. Suppose that (p, x) is DIC-EIR
implementable and that
p restricted to {(u,, v2): u, < v2 and (u,, U~)E [_o,I?]‘} is a block function.
Then (p, x) is implementable by posted price.
ROBUST
TRADING
MECHANISMS
105
Proof:
It is sufficient to show that if p is a block function which satisfies
(7)-(9) that (19) is satisfied almost everywhere. This proof will be in three
steps.
Step 1. hi, = 0 for every i.
Choose any (u,, 11~)in Bii for some i Then, letting b,; be the value of the
block, (9) can be rewritten as
b,,(u?_- u,) = 26ii(v2 - u, ).
(30)
This is satisfied if and only if !I,~ = 0.
Let /I,,;+, = z, for ie ( l,..., n - 1 > where zi is any value. Then
values of p in the remaining
blocks above the diagonal,
jb,,: 1 <i<n--2
and i + 2 <j 6 n }, are determined by
Step 2.
the
(31)
This is illustrated by Fig. 2.9
The proof is by induction. Suppose that (3 1) is true for every (i,j) such
that j < i+ k.” It will now be shown that it is true for (i, i + k + 1). Choose
To make notation less cumbersome, define
any ~~4~4.~+~+~.
i
&=_U+-(G-11)
n
(32)
and
6=_v+ itk
n
(5-y).
(33)
The variables E and 6 are, respectively, the upper value of u, and lower
value of v2 which occur in B, i+k+, . Also let b denote b,,,+k + t and let z
denote Cjz” zi. Then, from Fig. 2 it can be seen that (9) can be written as
(34)
which, in turn can be rewritten as
(6-c)b=(6-c)z.
(35)
Since 6 > E, it follows from (35) that b = z.
’ Figure 2 is drawn for the case D = 0.
“The basis of induction
is established
because
(31) is true for k = 1 by assumption.
106
HAGERTY AND ROGERSON
"1
FIG. 2. p is a block function.
Step 3. Referring to Fig. 2, it is clear that if (ul, u2) E B, for some i <j
that p can be written as
(36)
where
0
F(v) =
i
(37)
(i+ l)(V-_v)
2,
n
j= I
This corresponds to the posted price mechanism where price is announced
Q.E.D.
according to the distribution F.
REFERENCES
I. K. CHATTERJEE AND W. SAMUELSON.
Bargaining
under
incomplete
information,
Oper.
Res.
31 (1983), 835-51.
2. C. D’ASPERMONT AND L. GERARD VARET, Incentives and incomplete information, J. Pub.
Econ. 11 (1979). 2545.
ROBUST TRADING MECHANISMS
107
3. T. GRE~~K AND M. SATTERTHWAI~, “The Rate at Which A Simple Market Becomes
Efficient as the Number of Traders Increases: An Asymptotic Result for Optimal Trading
Mechanisms,” Northwestern Centre for Mathematical Studies in the Social Sciences
Discussion Paper 641.
4. B. HOLMSTROM AND R. B. MYERSON, Eflicient and durable decision rules with incomplete
information. Econometrica
51 (1983), 1799-1820.
5. J.-J. LAFFONT AND E. MASKIN, A Differential Approach to Expected Utility Maximizing
Mechanisms in “Aggregation and Revelation of Preferences” (J.-J. Laffont. Ed.)
pp. 289-308, North-Holland, Amsterdam, 1979.
6. R. MYERSON, Incentive compatibility and the bargaining problem, Econometrica
47
(1979). 61-73.
7. R. MYERSON.Optimal auction design, Math. Oper. Res. 6 (1981), 58-73.
8. R. MYERSONAND M. SATTERTHWAITE, Eflicient mechanisms for bilateral trading, J. Econ.
Theory 29, 265-281.
9. W. VICKREY, Counterspeculation, auctions. and competitive sealed tenders, J. Finance 16
(19611, 8-37.
10. R. WILSON. “Double Auctions,” Stanford Univ. IMSSS Technical Report No. 391,
Stanford, 1982.
It. R. WILSON, “Incentive Efficiency of Double Auctions,” Stanford Univ. IMSSS Technical
Report No. 431. Stanford, 1983.
12. R. WILSON. “Eficient Trading,” Stanford Univ. IMMSS Technical Report No. 432.
Stanford, 1983.