This note uses techniques developed for aggregate games to characterize the set of equilib- ria for a beauty contest or prediction game in which the experts’ preferences are quadratic, but with an otherwise unrestricted information structure for private signals and the state variable. We show that, on aggregate, the experts’ collective estimate of the unknown parameter to be estimated is unbiased for every equilibrium.Aggregate games, beauty contests, prediction games

Martimort, David

Stole, Lars

English

Research Papers in Economics

Working paper
THE COLLECTIVE WISDOM OF BEAUTY CONTESTS1
June 2011
David Martimort and Lars Stole
This note uses techniques developed for aggregate games to characterize the set of equilib-
ria for a beauty contest or prediction game in which the experts' preferences are quadratic,
but with an otherwise unrestricted information structure for private signals and the state
variable. We show that, on aggregate, the experts' collective estimate of the unknown
parameter to be estimated is unbiased for every equilibrium.
Keywords: Aggregate games, beauty contests, prediction games.
1. A BEAUTY-CONTEST GAME
Suppose that the true \beauty" of a person is random variable 0 drawn from a
publicly known, prior distribution with convex support, 0  R. Each player in the
game is an expert (a \ judge" in the context of a beauty contest) who observes a
private signal, si 2 i, which is possibly informative about 0 and the other players'
signals. Player i's strategy is a mapping from i's private signals to a public action
or \prediction" of 0, which we denote as the announcement or prediction, xi(si).
Player i cares both about being correct (i.e., xi close to the true 0) and about his
distance to the average assessment, y = 1
n
P
j xj. Importantly, we do not assume that
player i prefers to be close to the average; indeed, in many strategic settings it seems
likely that a player would benet the most by making an accurate prediction that is
distinctive from the \herd." For example, one can think of this game as being played
by debt rating agencies, each of which desires to correctly estimate a rm's underlying
probability of default. Holding the accuracy jjxi   0jj xed, it seems plausible that
agency i prefers that its report is further away from the average assessment. In short,
a player in a prediction game may want to be both accurate and distinctive. Of
course, one can easily imagine situations where making an inaccurate prediction has
greater costs to a player when the average assessment of the other players is closer to
the truth. This may be the case when the experts are conservative enough and fear
a reputation loss from moving away form that average. Here, a player's payos may
be for accuracy and herding.
Formally, player i's strategy space is Xi = f~ xi j ~ xi : i ! g and xi 2 Xi is
an arbitrary strategy which maps signals to predictions. Denote the set of all signal
proles by  
Q
i2N i and let s = (s1;:::;sn) 2  represent an arbitrary signal
prole. The aggregate prediction space is the set of additively-separable mappings,
1Parts of this note appeared in an earlier version of \Aggregate Representations of Aggregate
Games" (2010) which has been revised without the beauty contests application and is no longer
circulating in its previous form. The authors thank John Birge, Emir Kamenica, David Myatt,
Michael Schwarz and seminar participants at the University of Chicago for helpful conversations and
suggestions.
aParis School of Economics-EHESS, david.martimort@parisschoolofeconomics.eu
bUniversity of Chicago Booth School of Business,lars.stole@chicagobooth.edu
12 D. MARTIMORT, L. STOLE
y :  7! , that can be generated from some strategy prole of the players:
Y 
(
y :  ! 

   
9x 2 X s:t: y(s) =
1
n
X
i2N
xi(si) ; 8s 2 
)
:
Following Morris and Shin (2002), Angeletos and Pavan (2007) and Myatt and Wallace
(2008), we suppose that player i's preferences are quadratic but with the additional
generality that the weights they put in their objective function on being correct may
vary across players and the joint distribution of signals is arbitrary. Formally, player
i's payo is
ui(xi;y) =  
Z

 
(1   i)(xi(si)   0)
2 + i (xi(si)   y(s))
2
dG(0;s);
where G(0;s) is the joint cumulative distribution dened over   . Because the
average includes the player's own assessment, we assume that each i is bounded above
by n2=(2n 1) which guarantees that each player's objective function is concave. This
upper bound exceeds unity and holds generally for any i if n is suciently large;
there is no corresponding lower bound.1 Unlike these cited papers, we emphasize that
we do not assume that the information structure is Gaussian, nor do we require that
i 2 (0;1). For example, i < 0 characterizes a game in which player i prefers to be
correct about  but also prefers to be unique relative to the \actions" of rival players.
2. AGGREGATE CONCURRENCE IN AGGREGATE GAMES
The beauty contest game thus described is an example of an aggregate game as
dened in Martimort and Stole (2011a).2 An aggregate game is a normal form game
with the additional restriction that each player i's payo can be represented by a
function of his own strategy, xi 2 Xi, and a proper (not one-to-one) aggregation
of every player's strategy (including i), y = (x). In the case of beauty contests,
the individual strategies and the aggregate are functions mapping from signal spaces
to reports and the aggregate is simply y(s) = (s) = 1
n
P
i xi(si). The immediate
import of Martimort and Stole (2011a) is that every equilibrium aggregate must
satisfy an aggregate concurrence principle: each player must nd the equilibrium
aggregate optimal over the space of aggregate variations available to that player. In
particular, if x = (x1;:::xn) is an equilibrium of the game with equilibrium aggregate
y = 1
n
P
i xi, then a consequence of Theorem 1 in Martimort and Stole (2011a), is
that for any nonnegative weighting vector (1;:::;n),
y 2 arg max
y2
T
i2N(Xi;x i)
X
i2N
iui(
 1
i (y;x i);y);
where

 1
i (y;x i)  fxi 2 Xi j y = (xi;x i)g:
1If the player's preferences were rewritten as only depending upon the average of other players'
reports, then this additional technical requirement on i would be unnecessary; in such as case,
however, the game would not immediately be a member of the aggregate game class as we have dened
it and so we prefer to dene the beauty contest game as an aggregate game from the beginning.
2See Martimort and Stole (2011b) for applications of aggregate techniques to the class of common
agency games with adverse selection and public contracts.BEAUTY CONTESTS 3
3. A NECESSARY CONDITION
Replacing xi in player i's payo with n(y   y) + xi, player i's payo can be stated
as
ui (n(y   y) + xi;y) =
Z


 
 i
2
y(s)
2 + y(s)(n   i)(ny(s)   xi(si))   y(s)n(1   i)

dG(0;s)
+
Z

Ri(x(s))dG(0;s);
where  i  (n i)2+i(1 i) > 0 and Ri is a remainder function that is independent
of y and xi. Using the weight of i = 1
n(n i) for each player i (this weight is positive
given the assumed upper bound on i), and ignoring the residual terms Ri, it must
be that if x is a Nash equilibrium with aggregate y, then
y 2 arg max
y2Y (y)
 
Z

(
1
2
y(s)
2
 
1
n
X
i2N
 i
n   i
!
  y(s)y(s)(n   1)
 y(s)n
 
1
n
X
i2N
1   i
n   i
!)
dG(0;s);
where Y (y) 
T
i2Nfy j 9zi 2 Xi s.t. y = y+zig is the set of feasible variations in the
aggregates given y. Dene the scalar
 
X
i2N
1   i
n   i
;
and note that  > 1   n. Using the expression for  i, it can be established that
1
n
X
i2N
 i
n   i
  (n   1) =
X
i2N
1   i
n   i
= :
This algebraic fact allows us to simplify the weighted objective function above to
(y;y)   
Z


1
2
y(s)
2 ( + n 1)   y(s)(y(s)(n 1) + 0)

dG(0;s):
The integrand in the denition of  is continuously dierentiable and strictly concave
in y(s) pointwise, and so the necessary condition requires that y(s) maximizes this
objective pointwise over the set of feasible aggregates for each s 2 . Because 
is convex, the local rst-order condition is a necessary condition for any equilibrium
aggregate. Remarkably, aggregate concurrence implies that for any generic preference
vector (i.e.,  6= 0), for any arbitrary information structure (G;), and for every
equilibrium, the aggregate estimate y(s) is an unbiased estimate of 0.
Proposition 1 For any beauty-contest game with  6= 0, every equilibrium aggre-
gate, y, is an unbiased estimate of 0:
(3.1)
Z

(y(s)   0)dG(0;s) = 0:4 D. MARTIMORT, L. STOLE
For any beauty-contest game with  = 0, there is an equilibrium in which the aggregate
is unbiased if E[E[0jsj]jsi] = E[0jsi] for all i 2 N.
The genericity requirement,  6= 0, rules out pathological cases such as when i = 1
for all i 2 N. In this particular case, there is an uncountable number of biased
equilibrium outcomes and any aggregate report can arise in equilibrium. Less obvi-
ous measure-zero pathologies are also ruled out. Consider a two player game with
preferences given by 1 = 7
6, 2 = 3
4 and assume that the players' signals are inde-
pendent, conditional on 0. For every ^  2 , there exists a corresponding equilibrium
in which E[y(s)] = ^ .3 For ^  = E[0], the aggregate is unbiased; otherwise, there are
a uncountable number of biased equilibria. By using an aggregate-game framework
to characterize the equilibria, the key genericity condition,  6= 0, is immediate. If
this genericity condition fails, an unbiased equilibrium aggregate still exists if the
players' signals are independently distributed conditional on 0, but unbiasedness is
no longer assured for all equilibria. Technically, there is a failure of aggregate lower-
hemicontinuity over the parameter space at  = 0.
While this simple \unbiasedness" result does not tell us about the social value of
information, it applies to a much larger class of beauty-contest games than has been
studied in such papers as Morris and Shin (2002), Angeletos and Pavan (2007) and
Myatt and Wallace (2008). Notice that the number of players is nite, so every
player's action has a measurable impact on the aggregate. The players may also
have very dierent preferences about the desirability of being close to the aggregate
y(s). The information structure in the present analysis is not necessarily Gaussian,
nor is it necessarily symmetric across players. Finally, there is no requirement that
the players' signals are independent conditional on 0. This generality makes our
unbiased-aggregate result more surprising.
Suppose, for example, that signals are not independent, conditional on 0. In par-
ticular, consider a two-player game and suppose that player 1's signal is s1 = 0 +"1,
but that player 2's signal is s2 = "1. Hence, player 2 knows nothing about 0 that
is not contained in the public prior, but player 2 knows the exact bias in player 1's
signal. In equilibrium, x1(s1) cannot be everywhere constant, and so if 2 6= 0, player
2's action will depend upon s2 = "1 with the result that x2 will be a biased estimate
of 0 with probability one. The unbiased-aggregate result, however, implies that on
average such individual biases cancel each other out.4
Finally, we emphasize again that aggregate unbiasedness also holds even if i < 0
3The equilibrium strategies are given by x1(s1) = 7
3E[0]  4
3E[0 js1]+ 7
5
^  and x2(s2) = E[0]+ 3
5
^ .
4If the prior distribution of 0 is normal with mean  and variance 2, and if player 1's signal noise
is normally distributed with mean zero and variance 2, then the linear equilibrium of this game is
x1(s1) = a1s1 + b1
and
x2(s2) =
2(1   )
(2   )
 +

(2   )
(a1( + s2) + b1);
where
a1 =
2(1   )(2   )2
4(1   )2 + (2   )22; b1 = 

4(1   )2 + (2   )2
4(1   )2 + (2   )22

:
We have E[1
2(x1(s1)+x2(s2)) 0] = 0 as implied by Proposition 1.BEAUTY CONTESTS 5
for some players. With such preferences, players prefer to make accurate predictions
(i.e., 1    > 0), but also prefer to distinguish themselves from the mob.5 Remark-
ably, equilibrium unbiasedness is also necessary for games in which players prefer to
be dissimilar.
APPENDIX
Proof of Proposition 1: If y is an equilibrium aggregate, it must maximize (y;y)
over the set of feasible aggregates y 2 Y (y). Because (;y) is strictly concave and 
is a convex set, any solution to the SGM program
y 2 arg max
y2Y(y)
(y;y);
must satisfy the rst-order conditions for each permissible variation. It is thus nec-
essary that that for each i 2 N, the rst-order condition for the variation y(s) =
y(s) + zi(si) must be satised at zi = 0. Thus, for each i,
0 2 arg max
zi2Xi
(y + zi;y):
In the context of beauty-contest games, the expression (y + zi;y) specializes to
 
Z


1
2
(y(s) + zi(si))
2 ( + (n   1))
  (y(s) + zi(si))(y(s)(n   1) + 0)

dG(0;s):
Eliminating all terms that do not involve zi, we have the simpler requirement that
0 2 arg min
zi2Xi
Z


zi(si)(y(s)   0) +
1
2
zi(si)
2( + n   1)

dG(0;s):
Suppose that  6= 0. Then the rst-order necessary condition evaluated at zi(si) = 0
yields Z

(y(s)   0)dG(0;s) = 0;
as required.
Suppose instead that  = 0. We will prove that xi(si) = E[0jsi] for i 2 N is an
unbiased equilibrium under the assumption that E[E[0jsj]jsi] = E[0jsi] for all i 2 N.
It is immediate that such a prole of strategies will generate an unbiased aggregate.
To see that it is an equilibrium, we need only consider an agent's individual rst-
order condition with respect to xi; there is no additional value to using an aggregate
approach in this case. Individual i's rst-order condition requires that
iE[xi(si)   0jsi] + (1   i)

n   1
n

E[xi(si)   y(s)jsi] = 0:
5For example, as in Prendergast and Stole (1996), the player prefers to signal his information is
of higher quality by taking more extreme positions.6 D. MARTIMORT, L. STOLE
If xj(sj) = E[0jsj], then E[xj(sj)jsi] = E[0jsi], from which we have
E[y(s)jsi] =
1
n
xi(si) +
1
n
X
j6=i
E[xj(sj)jsi] =
1
n
xi(si) +
n   1
n
E[0jsi]:
Substituting into player i's rst-order condition yields
iE[xi(si)   0jsi] + (1   i)

n   1
n

E[xi(si)   y(s)jsi]
= i (xi(si)   E[0jsi]) + (1   i)

n   1
n
2
(xi(si)   E[0jsi])
which is zero as required for xi(si) = E[0jsi]. Given each player's minimization
program is strictly convex, this is sucient for optimality and proves that the hy-
pothesized strategy prole is an equilibrium. Q.E.D.
REFERENCES
Angeletos, G-M, and A. Pavan, (2007): \Ecient use of information and social
value of information," Econometrica, 75, 1103-42.
Martimort, D. and L. Stole, (2011a): \Aggregate representations of aggregate
games with applications to common agency," Applied Theory Initiative Working
Paper #2011-05, June 2011.
Martimort, D. and L. Stole, (2011b): \Public contracting in delegated common-
agency games ," Applied Theory Initiative Working Paper #2011-06, June 2011.
Morris, S., and H. Shin, (2002): \The social value of public information," Amer-
ican Economic Review, 92, 1521-1534.
Myatt, D. and C. Wallace, (2008): \On the sources of value of information: Pub-
lic announcements and macroeconomic performance," Nueld College, Oxford
University, Working Paper, 27 October 2008.
Prendergast, C. and L. Stole, (1996): \Impetuous youngsters and jaded old timers:
acquiring a reputation for learning," Journal of Political Economy, 104, 1105-
1134.

(2011a): \Aggregate representations of aggregate games with applications to common agency,&quot; Applied Theory Initiative Working Paper #2011-05,

(2011b): \Public contracting in delegated commonagency games ,&quot; Applied Theory Initiative Working Paper #2011-06,

Ecient use of information and social value of information,&quot;

Impetuous youngsters and jaded old timers: acquiring a reputation for learning,&quot;

On the sources of value of information: Public announcements and macroeconomic performance,&quot; Nueld College,

The social value of public information,&quot;

The Collective Wisdom of Beauty Contests

The Collective Wisdom of Beauty Contests

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