INFORMATION DISCLOSURE IN MULTISTAGE TOURNAMENTS∗
MARIA GOLTSMAN AND ARIJIT MUKHERJEE
Abstract. We consider a framework where two agents are participating in a tournament
that has two stages, intermediate and final.
The results of the intermediate stage are
observed by the principal who organizes the tournament, but not by the agents. This paper
attempts to answer the question of how much information about the intermediate results
the principal should disclose back to the agents in order to maximize their aggregate effort.
We find that, when choosing the disclosure policy, the principal faces a tradeoff between
intermediate and final effort. The disclosure policy that resolves this tradeoff in the optimal
way includes two public announcements: one is made if both agents do poorly, and the
other one otherwise. Consequently, both the policy of full disclosure and the policy of no
disclosure are suboptimal.
1. Introduction
Promotion tournaments have been extensively analyzed by economists, starting from Lazear
and Rosen (1981), Green and Stokey (1983) and Nalebuff and Stiglitz (1983). This is not
surprising, since such tournaments are often used in practice.
Most of the existing tour-
nament models are static, in the sense that they assume that the winner is determined by
how well the participants complete a single task. However, one might imagine a situation
where the principal can make the tournament participants complete not one, but several
tasks sequentially, and let the winner be determined based on the completion of all tasks.
In such a dynamic tournament, besides optimally determining the prizes, the principal has
an additional instrument to affect the effort levels of the participants, namely, information
disclosure.
To be precise, suppose that a participant does not observe how well his com-
petitors (or possibly even himself) have completed the intermediate tasks, but the principal
does. Then the principal has the choice of how much information about the intermediate
∗
Goltsman: Department of Economics, University of Western Ontario, Social Science Centre, London, ON
N6A 5C2, Canada. Email: mgoltsma@uwo.ca; Mukherjee: Bates White LLC, 1300 I St, NW, Washington
DC 20005; Email: arijit.mukherjee@bateswhite.com; URL: http://www.amukherjee.net. We would like to
thank Braz Camargo, Zsolt Macskasi, Gregory Pavlov and Michael Whinston for helpful suggestions. The
remaining errors are ours.
1
2
MARIA GOLTSMAN AND ARIJIT MUKHERJEE
results to disclose back to the participants. For example, he might disclose no information
at all (and thus make the dynamic tournament equivalent to a static one); he might tell each
participant exactly how well he and his competitors have fulfilled the intermediate tasks; or
he might choose some intermediate level of information disclosure, pooling the information
or even introducing some stochastic noise. The choice of how much information to disclose
will clearly have an effect on the participants’ incentives.
If the information on how suc-
cessful everyone has been so far is revealed in the middle of the tournament, it will affect
the participants’ future effort incentives. Also, if a participant knows that he can affect the
information being disclosed by his early efforts, he will take this into account at the beginning
of the tournament.1
This paper attempts to answer the question of what is the optimal disclosure policy in a
dynamic tournament where the principal is maximizing the aggregate effort of the participants. We look at the simplest possible model with two ex ante symmetric participants and
two stages, intermediate and final. At each stage, each contestant can either succeed or fail.
After the intermediate stage, the principal, but not the contestants, gets to observe the results
and can disclose a public signal that transmits some information about the results back to
the contestants. After observing the signal, the contestants update their beliefs accordingly
and choose the final-stage effort levels.
The winner is determined as the participant who
gets the highest total number of successes.
First, we compare the disclosure policies that are symmetric (it treats the agents the same
way) and deterministic (the announcement depends deterministically on the intermediate
results). Both restrictions are likely to be satisfied by disclosure policies used in practice.
We find that the optimal disclosure policy in this class takes a simple form: the principal only
tells the agents whether it is the case that both of them have failed at the intermediate stage.
The intuition behind this result is that the principal faces a trade-off between final-stage and
intermediate-stage effort: if he wants to induce high competition at the final stage, he may
have to decrease effort incentives at the intermediate stage. The disclosure policy in question
resolves this trade-off in the optimal way. On the one hand, disclosing that both agents have
failed at the intermediate stage and thus are in the same position stimulates competition at
the final stage. On the other hand, both agents would not want to find themselves in the
1These two effects of information disclosure are called the ex post effect and the strategic effect, respectively,
by Aoyagi (2004).
DISCLOSURE IN MULTISTAGE TOURNAMENTS
3
position where they compete intensively at the final stage. So they have incentives to exert
effort at the intermediate stage in order to decrease the probability of the outcome where
both fail.
We proceed by analyzing all feasible disclosure policies, not necessarily symmetric or deterministic.
We solve the principal’s optimization problem numerically and compare the
resulting maximal total effort to the total effort generated by the optimal symmetric deterministic policy. Our results indicate that the latter policy remains optimal among all feasible
disclosure policies.
The paper is structured as follows.
The next section presents the model.
characterizes the optimal effort choice given any disclosure policy.
optimal choice of a disclosure policy.
Section 3
Section 4 analyzes the
Section 5 presents a discussion of related literature
and possible extensions of the model. All proofs are given in Appendix 1 and omitted in the
text.
2. The Model
2.1. Setup. Two agents, A and B, are competing for a prize. The competition proceeds in
two stages, intermediate and final. At each stage, each agent can exert costly effort. The
principal is maximizing his profit, which is the sum of agents’ efforts exerted at both stages.
After the intermediate effort is exerted, the intermediate results are determined, which
are observed by the principal, but not by the agents. The intermediate result for an agent
can be either success (s) or failure (f).
Thus after the intermediate stage the principal
gets to observe an element of the set Y= {(s, s) , (s, f) , (f, s) , (f, f)} , where, for example,
(s, f ) corresponds to agent A succeeding and agent B failing at the intermediate stage. The
information about the intermediate results can affect how much effort the agents exert at
the final stage and thus their equilibrium behavior at the intermediate stage as well. The
ability to release information about intermediate outputs to the agents gives the principal
an instrument to influence their choice of effort.
Finding the optimal way to release this
information is the aim of this paper.
Formally, the game proceeds as follows:
1) The principal publicly announces a disclosure policy (Z, ζ) , where Z is the set of signals,
and ζ : Y → ∆ (Z);
4
MARIA GOLTSMAN AND ARIJIT MUKHERJEE
2) The agents simultaneously choose intermediate effort levels eIi ∈ [0, 1] , i ∈ {A, B} , and
¡ ¢2
¡ ¢
pay cost of effort c eIi = 12 eIi ;
3) Intermediate outputs yi ∈ {s, f } , i ∈ {A, B} , are realized according to
Pr (yi = s) = αeIi ,
where α ∈ [0, 1] is a parameter;
4) The principal observes (yA , yB ) and sends a public signal z, randomizing according to
ζ (yA , yB ) ;
5) Agents simultaneously choose final effort levels eFi (z) ∈ [0, 1] , i ∈ {A, B} , and pay cost
¡
¢
¢2
¡
of effort c eFi (z) = 12 eFi (z) ;
6) Final outputs Yi ∈ {s, f} , i ∈ {A, B} , are realized according to
Pr (Yi = s) = αeFi (z) ;
7) The winner is determined as the agent who has the highest total number of successes.
If the number of successes is equal for the two agents, agent A is considered the winner with
probability 12 . The winning agent i gets the prize, which is worth 1 to both agents.
Let ζ yA ,yB denote the probability distribution function of ζ (yA , yB ). The expected payoff
of the winning agent i is
1 ¡ I ¢2
e
−
Uw = 1 −
2 i
X
Pr (yA , yB )
(yA ,yB )∈Y
Z
Z
¢2
1¡ F
ei (z) dζ yA ,yB ,
2
and the expected payoff of the losing agent j is
Ul = −
where
1 ¡ I ¢2
e
−
2 j
X
Pr (yA , yB )
(yA ,yB )∈Y



α2 eIA eIB




 αeI ¡1 − αeI ¢
A
B
Pr (yA , yB ) =
¡
¢
I


αeB 1 − αeIA




 ¡1 − αeI ¢ ¡1 − αeI ¢
B
A
Z
Z
¢2
1¡ F
ej (z) dζ yA ,yB ,
2
if (yA , yB ) = (s, s) ,
if (yA , yB ) = (s, f) ,
if (yA , yB ) = (f, s) ,
if (yA , yB ) = (f, f) .
It is implicit in the definition of Uw and Ul that both agents are risk neutral.
These ex-
pressions simply represents the reward of an agent net of his expected cost of effort. The
DISCLOSURE IN MULTISTAGE TOURNAMENTS
5
principal is also assumed to be risk neutral and her payoff is the expected level of aggregate
effort by both agents, i.e.,
Π=
eIA
+ eIB
X
+
Pr (yA , yB )
(yA ,yB )∈Y
Z
Z
¡ F
¢
eA (z) + eFB (z) dζ yA ,yB .
This completes the description of our model. We now introduce the equilibrium notion that
we will be using in our analysis.
2.2. The Equilibrium. We use perfect Bayesian Equilibrium as an equilibrium concept. In
this game, such an equilibrium is defined as follows.
Definition 1. A pair of strategies for the agents,
n ©
o
ª
eIi , eFi (z) z∈Z
i∈{A,B}
together with a
disclosure policy (Z, ζ) constitute an equilibrium, if
o
n ©
ª
(a) For each i ∈ {A, B} , the effort levels eIi , eFi (z) z∈Z solve
(IC)
max
eIi ,(eF
i (z))z∈Z

Z

X
(yA ,yB )∈Y

¸
·
¢2
1¡ F
1 ¡ I ¢2 
ei (z)
e
dζ yA ,yB (z) −
Pr (yA , yB ) Pr (i wins | z) −
2
2 i
where the probability that agent A wins conditional on signal z ∈ Z is
(1)
Pr (A wins | z) =
X
(yA ,yB )∈Y
Pr ((yA , yB ) | z) Pr (A wins | (yA , yB ) , z) ,
and
Pr (B wins | z) = 1 − Pr (A wins | z) ;
(b) The disclosure policy (Z, ζ) solves
max Π = eIA + eIB +
(Z,ζ)
subject to
X
(yA ,yB )∈Y
Pr (yA , yB )
Z
¡ F
¢
eA (z) + eFB (z) dζ yA ,yB (z) ,
n ©
o
ª
eIi , eFi (z) z∈Z solving problem (IC) for i ∈ {A, B} .
6
MARIA GOLTSMAN AND ARIJIT MUKHERJEE
The strategy of the principal is to choose a disclosure policy, while the agents’ strategy
has two elements: a choice of intermediate effort level, and a choice of final effort level given
the realized signal. A three-tuple of strategies by the principal and the two agent is said to
constitute a perfect Bayesian equilibrium is this game if two conditions are satisfied – first,
each agent’s effort choices must maximize his expected payoff given the disclosure policy, and
the strategy of the other agent; second, given the strategies of the two agents, the principal’s
choice of disclosure policy must maximize his expected payoff.
In the following two sections we will characterize the strategies of the agents and the
principal that satisfies these two conditions.
We start with the characterization of the
optimal effort by the agents for a given disclosure policy.
3. Optimal Effort
For every z ∈ Z, let Pyi ,yj (z) = Pr ((yA , yB ) |z) denote the posterior probability of intermediate outcome (yi , yj ) conditional on signal z being observed. Thus
¡
¡
¢¢
Pr z | (s, s) , eIA , eIB
¡
¡ I I ¢¢ ¡
¡
¢¢ ,
Pss (z) = P
Pr (yA , yB ) | eIA , eIB
(yA ,yB )∈Y Pr z | (yA , yB ) , eA , eB
or
¡
¡
¢
¢
Pss (z) = α2 eIA eIB ζ ss (z) Á[α2 eIA eIB ζ ss (z) + αeIA 1 − αeIB ζ sf (z) + 1 − αeIA αeIB ζ f s (z)
¢¡
¢
¡
+ 1 − αeIA 1 − αeIB ζ f f (z)]
z , P z and P z are determined in a similar fashion. Without loss of generality, we
and Psf
fs
ff
will consider only signals for which the denominator is nonzero.
At the final stage, after observing a signal z ∈ Z, the winning probability for agent A is
determined by equation 1, so if the constraints eFi (z) ∈ [0, 1] , ∀z ∈ Z, i ∈ {A, B}, do not
bind, then the first-order conditions are:
(2)
¡
¢
eFA (z) = 12 αPss (z) + 12 α2 Psf (z) eFB (z) + 12 αPf s (z) 1 − αeFB (z) + 12 αPf f (z) ,
¡
¢
eFB (z) = 12 αPss (z) + 12 α2 Pf s (z) eFA (z) + 12 αPsf (z) 1 − αeFA (z) + 12 αPf f (z) .
The second-order conditions hold for both agents, because Pr (i wins|z) is linear in eFi (z)
and the cost of effort is convex. It follows that if the constraints eFi (z) ∈ [0, 1] , ∀z ∈ Z, i ∈
DISCLOSURE IN MULTISTAGE TOURNAMENTS
7
{A, B}, do not bind, then the equilibrium final-stage effort is the ones that solve (2). Here,
we will write the solution denote the solutions as:
eFA∗ (z) = eFA (z; α) ,
(F inal)
eFB∗ (z) = eFB (z; α) .
At the start of the tournament, before any effort is exerted, the probability that agent i
wins is
Pr (i wins)
RP
=
(yA ,yB )∈Y Pr (i wins|z, (yA , yB )) Pr ((yA , yB )) dζ yA ,yB (z)
¡
¡
¢
¢¡
¢¤
R 2 I I £1 2 F
= α eA eB 2 α eA (z) eFB (z) + αeFA (z) 1 − αeFB (z) + 12 1 − αeFA (z) 1 − αeFB (z) ×
+
+
+
R
R
R
dζ ss (z)
¡
¢¤
¡
¢
£
αeIA 1 − αeIB 1 − 12 αeFB (z) 1 − αeFA (z) dζ sf (z)
¡
¢¤
¢
£
¡
1 − αeIA αeIB 12 αeFA (z) 1 − αeFB (z) dζ f s (z)
¡
¢
¡
¢¡
¢£
1 − αeIA 1 − αeIB 12 α2 eFA (z) eFB (z) + αeFA (z) 1 − αeFB (z) +
¡
¢¡
¢¤
1
F
F
2 1 − αeA (z) 1 − αeB (z) dζ f f (z) .
It follows that if the constraints eIi ∈ [0, 1] , i ∈ {A, B}, do not bind, then the first-order
conditions with respect to eIA and eIB can be written as:
¡ I∗ F
¢
I
F
eI∗
A = eA eB , eA (·) , eB (·) ; α ,
¡ I∗ F
¢
I
F
eI∗
B = eB eA , eA (·) , eB (·) ; α .
(Int)
The following proposition asserts that we can indeed use equations (F inal) and (Int) to
find an equilibrium.
Proposition 1. Given any disclosure policy (Z, ζ) , there exists a Bayesian-Nash equilibrium
o
n ©
ª
=
in the game between the two agents that is induced by (Z, ζ) where eIi , eFi (z) z∈Z
i∈{A,B}
n
o
©
ª
F∗
eI∗
.
i , ei (z) z∈Z
i∈{A,B}
8
MARIA GOLTSMAN AND ARIJIT MUKHERJEE
Proposition 1 shows that the intermediate and final effort levels as discussed in (Int) and
(F inal) are in fact the best-response mapping of the agents given the disclosure policy of the
principal. Thus the optimal disclosure policy of the principal is the one that corresponds to
o
n
© F∗ ª
,
e
(z)
mapping which maximizes the expected aggregate
a point in the eI∗
i
i
z∈Z
i∈{A,B}
effort of the two agents. The next sections deals with this problem.
4. Optimal Disclosure Policy
This section analyzes the optimal choice of the disclosure policy.
The first subsection
presents the principal’s problem and proves a number of results that make it more tractable.
The second subsection looks at the class of symmetric and deterministic disclosure policies
and identifies the optimal policy in this class. The last subsection extends the analysis to
all disclosure policies.
4.1. Principal’s Problem. The optimal disclosure policy solves:
P










max(Z,ζ)



s.t.






(
eIA + eIB +
P
(yA ,yB )∈Y
n ©
o
ª
eIi , eFi (z) z∈Z
)
R¡ F
¢
Pr (yA , yB ) eA (z) + eFB (z) dζ yA ,yB (z)
i∈{A,B}
n
o
© F∗ ª
= eI∗
,
e
(z)
i
i
z∈Z
i∈{A,B}
,
∀ (yi , yj ) ∈ Y, ζ yi ,yj is a probability measure on Z.
The following two propositions make the problem more manageable.
Lemma 1. If an optimal disclosure policy exists, then there exists an optimal disclosure
policy with a finite Z.
Using the above lemma, we derive the following result.
Proposition 2. If an optimal disclosure policy exists, then there exists an optimal disclosure
policy with |Z| ≤ 6.
DISCLOSURE IN MULTISTAGE TOURNAMENTS
9
Note that since the final effort depends on the posteriors in a continuous way, it is not
a priori obvious that we can restrict attention to disclosure policies with a finite number
of signals.
However, Lemma 1 and Proposition 2 assert that this is indeed true.
The
reason is that the only way the disclosure policy enters both the objective function and the
constraints in the principal’s problem is through the mathematical expectations of certain
random variables (for example, effort), whose distribution is determined by the disclosure
policy. However, if a probability distribution on some set has a certain expected value, it
is possible to find a finite probability distribution on this set with the same expected value:
this is the intuition behind Lemma 1.
Proposition 2 further bounds the power of the set
Z by treating the principal’s problem as a linear programming problem with nonnegativity
constraints and determining the maximum number of choice variables in this problem that
can take nonzero values at a corner solution.
4.2. Symmetric and Deterministic Disclosure Policies. In this section, we are going
to restrict attention to a certain class of disclosure policies (namely, deterministic and symmetric) and look for the optimal policy in this class.
To better illustrate the trade-offs
involved, we are first going to analyze in detail the case of α = 1 (that is, the case when the
probability of success at each stage is equal to the effort level). The results of this analysis
are summarized by Proposition 3.
First, let us define two restrictions on a disclosure policy.
Definition 2. Disclosure policy (Z, ζ) is symmetric if ∀z ∈ Z, ∃z 0 ∈ Z such that



ζ ss (z) = ζ ss (z 0 ) ;




 ζ (z) = ζ (z 0 ) ;
ff
ff


ζ sf (z) = ζ f s (z 0 ) ;




 ζ (z) = ζ (z 0 ) .
fs
sf
Definition 2 means that the disclosure policy is treating the agents symmetrically ex ante:
if conditional on some signal z, Psf (z) = p, then there should exist a signal z 0 (possibly,
but not necessarily, the same as z), such that Pf s (z 0 ) = p.
We consider it unreasonable
to assume that z 0 should always be the same as z, because this requirement would rule out
some interesting disclosure policies (for example, full disclosure). Symmetry is a reasonable
10
MARIA GOLTSMAN AND ARIJIT MUKHERJEE
requirement in practical applications. It implies that there exists a symmetric equilibrium
where eFA = eFB and ∀z ∈ Z, ∃z 0 ∈ Z : eFA (z) = eFB (z 0 ) .
Definition
3. Disclosure
policy
(Z, ζ)
is
deterministic if ∀ (yA , yB )
∈
Y,
∃z ∈ Z : ζ yA ,yB (z) = 1.
To illustrate the trade-offs involved in choosing the optimal disclosure policy, let us consider
the case of α = 1 and look for the optimum in the class of symmetric deterministic disclosure
policies. There are seven such policies:
1) Full disclosure: Z = {zss , zsf , zf s , zf f } , ζ yA ,yB (zyA ,yB ) = 1;
2) No disclosure: Z = {z} , ζ ij (z) = 1, ∀i, j;
3) Disclosing only if both succeed: Z = {z1 , z2 } , ζ ss (z1 ) = 1, ζ sf (z2 ) = ζ f s (z2 ) =
ζ f f (z2 ) = 1;
4) Disclosing only if both fail: Z = {z1 , z2 } , ζ f f (z1 ) = 1, ζ sf (z2 ) = ζ f s (z2 ) = ζ ss (z2 ) =
1;
5) Disclosing whether the score is the same or not: Z = {z1 , z2 } , ζ ss (z1 ) = ζ f f (z1 ) =
1, ζ sf (z2 ) = ζ f s (z2 ) = 1;
6) Disclosing whether the score is the same, and if it is the same, whether both succeed or
fail:
Z = {z1 , z2 , z3 } , ζ ss (z1 ) = 1, ζ sf (z2 ) = ζ f s (z2 ) = 1, ζ f f (z3 ) = 1;
7) Disclosing who is the leader if the score is not the same:
Z = {z1 , z2 , z3 } , ζ ss (z1 ) = ζ f f (z1 ) = 1, ζ sf (z2 ) = 1, ζ f s (z3 ) = 1.
Comparison of these policies leads to the following proposition:
Proposition 3. When α = 1, the optimal disclosure policy in the class of symmetric deterministic disclosure policies is “disclosing only if both fail”: Z = {z1 , z2 } , ζ f f (z1 ) =
1, ζ sf (z2 ) = ζ f s (z2 ) = ζ ss (z2 ) = 1.
DISCLOSURE IN MULTISTAGE TOURNAMENTS
11
To understand the intuition behind the proposition, note first that the principal is facing
a trade-off between intermediate and final effort: the policy that induces the highest intermediate effort (”disclose only if both fail”) at the same time induces the lowest final effort.
In fact, the ranking of the disclosure policies with respect to final and intermediate effort are
almost the reverse of each other (with the policy of full disclosure as an exception).
To explain this trade-off, let us look at the final stage first. Note that the highest possible
final-stage aggregate effort level (equal to 1) follows a signal that induces a posterior belief
that with probability 1, either both agents succeed or both fail. In other words, the agents
exert the most effort in the final stage if they know for sure that they are in equal position
after the intermediate stage.2 In contrast, when they know for sure that one of the agents
leads after the intermediate stage, both the leader and the follower have less incentive to
exert effort at the final stage. In this case, the marginal benefit of final stage effort (that
is, the marginal increase in winning probability) is smaller both for the leader and for the
follower than in the case when both agents get the same intermediate stage output.
This reflects on the agents’ intermediate-stage effort choice. At the intermediate stage, an
agent takes into account the effect of his effort on the probabilities of different intermediate
outcomes. In particular, suppose that increasing the intermediate-stage effort will make more
probable the intermediate outcomes that lead to intensive competition at the final stage. If
this is the case, there is a disincentive for an agent to exert effort at the intermediate stage.
Thus, the principal might face a trade-off between intermediate and final effort: intermediate
effort can be increased with a policy that creates high competition at the final stage, but
such a policy might reduce effort incentives at the intermediate stage.
The policy “disclose only if both fail” achieves the optimal balance between intermediate
and final effort. On the one hand, the policy includes a signal (namely, z1 ) that creates intensive final-stage competition. On the other hand, the probability of this signal decreases in
intermediate-stage effort, so the agents have incentives to exert high effort in the intermediate
stage in order to avoid competing fiercely at the final stage.
This reasoning also suggests why the policy “disclose only if both succeed,” which seems
symmetric to “disclose only if both fail,” does so much worse. The policy “disclose only if
both succeed” sends a signal that leads to intensive final-stage competition (namely, signal
2This is a common feature of patent race games: see, for example, [7].
12
MARIA GOLTSMAN AND ARIJIT MUKHERJEE
Table 1. Total effort for several values of α
α
0.05
Disclosure Policy
Disclose only if both succeed
0.099876
No disclosure
0.099875
Full disclosure
0.099906
Disclose whether the score is even 0.099929
Disclose only if both fail
0.099930
0.2
0.4
0.6
0.8
1
0.3923
0.3923
0.3942
0.395 4
0.395 6
0.7447
0.7449
0.7568
0.7639
0.7657
1.0401
1.0418
1.0720
1.0861
1.0920
1.2844
1.2932
1.3433
1.3591
1.3709
1.4946
1.5279
1.5841
1.5983
1.6211
z1 ) only after an outcome (s, s). The agents, who dislike exerting effort, would like to avoid
such a signal. This creates disincentives to exert effort at the intermediate stage, because
intermediate effort makes outcome (s, s) together with signal z1 more likely.
We can now check whether Proposition 3 generalizes to the case of α ∈ [0, 1]. Table 4 in
Appendix 2 shows the total effort as a function of α for the five disclosure policies considered
above. Table 1 displays the total effort evaluated at different levels of α.
Table 1 demonstrates that the ranking of disclosure policies in terms of total effort is the
same for all values of α. In particular, the result that “disclose only if both fail” is the best
policy is robust to changes in α. The same is shown in Figure 1 (α < 0.4 is not shown in
this graph as it is hard to visually distinguish the graphs as α gets smaller).
DISCLOSURE IN MULTISTAGE TOURNAMENTS
13
Figure 1: Aggregate expected effort under different symmetric deterministic disclosure
policies – the bold plot represents effort associated with “disclosure if both fail” policy
4.3. General Disclosure Policies. In this section, we extend our analysis to include all
feasible disclosure policies (not only symmetric and deterministic) and search for the optimal
policy in this class for different values of α. We find that “disclose only if both fail” remains
optimal even if more general disclosure policies are allowed.
We resort to numerical analysis to solve problem (P) for different values of α. Proposition
2 allows us to restrict attention to disclosure policies with six signals (any disclosure policy
with less than six signals can be represented as one with six signals, some of which have
probability zero). The optimization problem boils down to a problem of choosing 26 variable
14
MARIA GOLTSMAN AND ARIJIT MUKHERJEE
Table 2. Results of numerical optimization in the class of all feasible disclosure policies
α
0.2
0.4
0.6
0.8
1.0
Highest total effort
under optimal symmetric
deterministic policy
0.3956
0.7657
1.0920
1.3709
1.6211
Highest total effort
under any disclosure policy
(simulation results)
0.3961
0.7662
1.0923
1.3706
1.6178
(2 intermediate effort levels, and 24 probability values representing 4 probability distributions
(each corresponding to a particular (yA , yB ) realization in Y) on the set of signals which has
cardinality of 6) that maximizes the expected aggregate effort level subject to the constraints
that the intermediate effort levels satisfies (Int) and the final effort levels satisfies (F inal).
The numerical simulation was performed using Matlab version 6.5. The outcome of numerical
optimization turned out to be highly dependent on the initial condition, so for each value of
α, 100 simulations were performed, with initial values of these 26 variables are drawn as a
26 × 1 vector from a uniform distribution in [0, 1].
Table 2 summarizes the results.
The middle column reproduces the results of Table 1,
showing the highest effort obtained by any symmetric deterministic disclosure policy for each
value of α. The right column shows the highest total effort obtained from solving problem
(P) numerically.
One can observe that for α equal to 0.8 and 1, the optimal symmetric
deterministic policy generates a higher effort level than the numerical solution to problem
(P). When α equals 0.2, 0.4, or 0.6, the optimal symmetric deterministic disclosure policy
generates lower effort, but the difference is negligible.
The fact that the two columns do not exactly match is due to the numerical nature of the
reported solution to (P). Since the optimization routine in Matlab requires that the initial
conditions must be feasible, it is not possible to use (Int) as a nonlinear equality constraint
¢
¡
(it is rarely possible to draw a tuple eIA , eIB at random that satisfies (Int)). Instead, a
measure of the “error” due to not exactly satisfying constraint (Int) is incorporated as a
“penalty term” in the objective function. With this modification, the optimization routine
¡
¢
can start with an arbitrary eIA , eIB value but converges at a point that “almost” satisfies
DISCLOSURE IN MULTISTAGE TOURNAMENTS
15
the equality constraint.3 Such approximation error is reflected in the discrepancy between
the two columns in Table 2.
5. Discussion
1. Related Literature
There are several recent papers that consider the effect of information transmission on
incentives in dynamic settings.
The one that are closest to the present paper are Aoyagi
(2004) and Ederer (2004). Aoyagi (2004) asks the same question as we do, but considers a
model where the outcome is a sum of effort and a noise term (so that the outcome space at
each stage is continuous) and thus arrives at a very different conclusion. In particular, in his
model the optimal disclosure policy is highly sensitive to the sign of the third derivative of
the cost function: if the third derivative is positive, full disclosure is optimal; if it is negative,
no disclosure is optimal; if the cost function is quadratic, like in the present paper, then all
disclosure policies induce the same effort. Another feature of his model is that the equilibrium
first-stage effort does not depend on the disclosure policy. This is not the case in our model,
and, although we consider only the quadratic cost function, we do not expect the optimal
policy to depend on the sign of the third derivative of the cost function in such a dramatic
way in our model. Ederer (2004) looks at only two disclosure policies, full disclosure and no
disclosure, in a variety of settings that differ with respect to whether agents are characterized
by privately observed ability levels and how an agent’s output depends on his ability level.
He replicates the results of Aoyagi for the case of homogeneous ability, but shows that when
ability is privately known and complementary to effort, then the two policies may generate
different effort levels, and the comparison between them depends on the parameters.
Another related paper is Yildirim (2005), who considers a dynamic contest where the
agents can decide after the intermediate stage whether to reveal the information about their
3More precisely, in order to solve
P0 : min f (x) s.t.g (x) = c,
x
(where c is a scalar) we rewrite the objective function as
P1 : min f (x) + λ (g (x) − c)2
x
where λ is a positive real number. The term λ (g (x) − c)2 represents the “penalty” for not satisfying the
equality constraint. This is always positive by construction. Thus the solution to P1 must sufficiently minimize
this penalty. Therefore, a solution to P1 is a reasonable approximation of the solution to P0 for a suitably
chosen λ. We have used λ = 1000 for all equality constraints in our program.
16
MARIA GOLTSMAN AND ARIJIT MUKHERJEE
performance or not.
In contrast to our paper, Yildirim assumes that the agents cannot
commit to an information disclosure rule and considers only the two extreme revelation
options: whether to reveal everything or to reveal nothing. He finds that in an asymmetric
contest, all the information is revealed in equilibrium, but in an ex ante symmetric one,
there is a continuum of equilibria with different revelation decisions that all lead to the same
outcome. The assumption that the agents cannot commit to an information disclosure rule
has also been considered in the patent race literature (for example, Gordon (2004)) where
the agents can also choose when to reveal the information.
This paper is also related to Dubey and Geanakoplos (2004), who study how coarse should
the optimal grading scheme be in tournaments where a participant’s utility is directly determined by his net status (that is, the number of players with results below his minus the
number of players with results above his).
The main insight of the paper is that coarse
grading schemes may generate better incentives than fine ones: for example, more effort can
be induced if the continuum of outcomes is mapped into a few letter grades, and the participant’s status is determined by the grade he gets, not by his exact outcome. Their results
can be viewed as complementary to ours, since the force that drives their result is not the
effect of information transmission, but the direct dependence of utility on status.
The question of optimal information transmission in a dynamic setting has also been
studied in the context of a principal-agent model by Lizzeri, Meyer and Persico (2002). They
find that, if the prize structure is held fixed, full disclosure generates more effort than no
disclosure, but if the principal is also allowed to optimize with respect to the prize structure,
no disclosure becomes optimal.
2. Possible Extensions
One could consider several possible extensions of the model.
For instance, making the
outcome dependent not only on effort, but also on ability, like in Ederer (2004), will likely
change the optimal disclosure policy.
In such a setting, intermediate output provides in-
formation not only about the relative standing, but also about the opponent’s ability (and
one’s own ability as well in a career concerns - type model). This may strengthen the effect
of disclosure on second-period competition.
If the agents learn that they are close after
the intermediate stage, they will compete even more intensively in the final stage, because
this information also implies that their abilities are likely to be similar. In contrast, if they
learn that they are far apart in the intermediate stage, this will imply that their abilities
DISCLOSURE IN MULTISTAGE TOURNAMENTS
17
are likely to be disparate, which will dampen the incentives to compete in the final stage for
both of them. The effect of introducing heterogeneous abilities on the optimal choice of the
disclosure policy can thus go both ways.
Another potentially interesting extension is to allow the principal to choose how he wishes
to aggregate the results of the two stages into the final score. In particular, the tie-breaking
rule that we are using (split the prize in half if the sum of the results is equal for the
two agents) may be suboptimal. We also cannot rule out the possibility that the optimal
aggregation rule should give less (or more) weight to the intermediate result, or that an
asymmetric tie-breaking rule that favors one agent over the other can generate more effort.
Indeed, the former possibility is suggested by Gershkov and Perry (2006), who find that in a
two-stage tournament, giving equal weight to intermediate and final result can generate less
effort than only conducting a final review.
Finally, in most real-life dynamic tournaments,the principal can send private signals to
participants, telling each competitor only how well he did, but not how well the others did.
Introducing private signals into the model would lead to substantial technical complications,
but make the model more realistic.
6. Appendix 1: Proofs
We present the proofs omitted in the text in this section. Before we do so it is useful to
present the following expressions as we will be using then in our proofs. First, using the law
of total probability, equation (1) can be expanded as below:
¡
¢
Pr (A wins | z) = Pr ((yA , yB ) = (s, s) |z) { 12 α2 eFA (z) eFB (z) + αeFA (z) 1 − αeFB (z)
¡
¢¡
¢
+ 12 1 − αeFA (z) 1 − αeFB (z) }+
¡
¢
Pr ((yA , yB ) = (s, f ) |z) {α2 eFA (z) eFB (z) + αeFA (z) 1 − αeFB (z)
¡
¢¡
¢
¢
¡
+ 1 − αeFA (z) 1 − αeFB (z) + 12 1 − αeFA (z) αeFB (z)}+
¡
¢
Pr ((yA , yB ) = (f, s) |z) { 12 αeFA (z) 1 − αeFB (z) }+
¡
¢
Pr ((yA , yB ) = (f, f) |z) { 12 α2 eFA (z) eFB (z) + αeFA (z) 1 − αeFB (z)
¢¡
¢
¡
+ 12 1 − αeFA (z) 1 − αeFB (z) }.
As mentioned in the text, one can derive closed form solutions for the final effort levels.
18
MARIA GOLTSMAN AND ARIJIT MUKHERJEE
These solutions are:
(F inal)
eFA (z) =
1 3
1 αPss (z)+αPf s (z)+αPf f (z)+ 2 α (Psf (z)−Pfs (z))(Pss (z)+Psf (z)+Pf f (z))
,
2
2
1+ 1 α4 (P (z)−P (z))
eFB
1 3
1 αPss (z)+αPsf (z)+αPf f (z)+ 2 α (Pfs (z)−Psf (z))(Pss (z)+Pf s (z)+Pf f (z))
.
2
2
1+ 1 α4 (P (z)−P (z))
(z) =
4
fs
sf
4
fs
sf
Finally, it is worthwhile to derive the exact form of the first order conditions that pin down
the intermediate effort levels of the two agents. These conditions are:
eIA = α2 eIB
R £1
¡
¢
2 F
F
F
F
2 α eA (z) eB (z) + αeA (z) 1 − αeB (z)
¡
¡
¢¡
¢
¢2 i
dζ ss (z) +
+ 12 1 − αeFA (z) 1 − αeFB (z) − 12 eFA (z)
i
h
¡
¢
¢
¢
R
¡
¡
2
dζ sf (z) −
1 − 12 αeFB (z) 1 − αeFA (z) − 12 eFA (z)
α 1 − αeIB
i
h
¡
¡
¢
¢
R 1 F
2
1
F
F
α2 eIB
dζ f s (z) −
2 αeA (z) 1 − αeB (z) − 2 eA (z)
(Int)
¡
¢
¢ R £1 2 F
¡
F
F
F
α 1 − αeIB
2 α eA (z) eB (z) + αeA (z) 1 − αeB (z) i
¢¡
¢ 1¡ F
¢2
¡
+ 12 1 − αeFA (z) 1 − αeIF
dζ f f (z) ,
(z)
−
(z)
e
B
A
2
R £1
¡
¢
2 eF (z) eF (z) + αeF (z) 1 − αeF (z)
α
A
B
A
B
2
¡
¡
¢¡
¢
¢2 i
dζ ss (z) +
+ 12 1 − αeFA (z) 1 − αeFB (z) − 12 eFA (z)
i
h
¢
¡
¡
¢
¢
¡
R
2
dζ sf (z) −
α 1 − αeIA
1 − 12 αeFB (z) 1 − αeFA (z) − 12 eFA (z)
h
i
R
¡
¡
¢
¢
2
1
1
F
F
F
α2 eIA
dζ f s (z) −
2 αeA (z) 1 − αeB (z) − 2 eA (z)
¢
£
¡
¢
¡
R
1 2 F
F
F
F
α 1 − αeIA
2 α eA (z) eB (z) + αeA (z) 1 − αeB (z) i
¡
¢¡
¢ 1¡ F
¢2
dζ f f (z) .
+ 12 1 − αeFA (z) 1 − αeIF
B (z) − 2 eA (z)
eIB = α2 eIA
Proof of Proposition 1. Given a disclosure policy (Z, ζ) , the game between agents has a
´
³ ¡
¢
that satisfy (F inal) , (Int)
Bayes-Nash equilibrium if there exist eIi , eFi (z) z∈Z
i∈{A,B}
and belong to [0, 1] .
First, let us prove that eFi (z) ∈ [0, 1] for any z ∈ Z, if eFi (z) are defined by (F inal). For
any z ∈ Z,
DISCLOSURE IN MULTISTAGE TOURNAMENTS
19
αPss (z)+αPfs (z)+αPf f (z)+ 12 α3 (Psf (z)−Pf s (z))(Pss (z)+Psf (z)+Pf f (z))
eFA (z) = 12
2
1+ 14 α4 (Pf s (z)−Psf (z))
¤
£
≤ 12 αPss (z) + αPf s (z) + αPf f (z) + 12 α3 (Psf (z) − Pf s (z)) (Pss (z) + Psf (z) + Pf f (z))
£
¤
≤ 12 α + 12 α3 ≤ 34 < 1,
and
αPss (z) + αPf s (z) + αPf f (z) + 12 α3 (Psf (z) − Pf s (z)) (Pss (z) + Psf (z) + Pf f (z))
= α (1 − Psf (z)) + 12 α3 (Psf (z) − Pf s (z)) (1 − Pf s (z))
£
¤
α (1 − Psf ) + 12 α3 (Psf − Pf s ) (1 − Pf s )
≥ min
Psf ,Pf s :
Psf ≥0,Pf s ≥0, Psf +Pf s ≤1
¤¯
£
= α (1 − Psf ) + 12 α3 (Psf − Pf s ) (1 − Pf s ) ¯P
sf =0,
Pf s =1
= 0,
so eFA (z) ≥ 0. Similarly, eFB (z) ∈ [0, 1) , ∀z ∈ Z.
¢
¡
¢
¡
Next, let us prove that equations (Int) have a solution eFA , eFB ∈ [0, 1]2 . Let fA eFA , eFB
¡
¢
and fB eFA , eFB denote the right-hand side of the first and the second equation of (Int) ,
¢
¡
¢
¡
respectively, where eFA (z) , eFB (z) z∈Z are considered as functions of eFA , eFB and deter¡
¢
¡
¢
mined by equations (F inal). Both fA eFA , eFB and fB eFA , eFB are continuous. Moreover,
¢
¢
¡
¢¢
¡
¡ ¡
if eFA , eFB ∈ [0, 1]2 , then fA eFA , eFB , fB eFA , eFB ∈ [0, 1]2 . To see this, note that if
¡ F
¤2
¢ £
eA (z) , eFB (z) ∈ 0, 34 , then:
Kss (z) :=
¡
¢
¢¡
¢
¡
(z) eFB (z) + αeFA (z) 1 − αeFB (z) + 12 1 − αeFA (z) 1 − αeFB (z) −
2
(eFA (z))
2
³
¡ A ¢2 ´
1
F
F
= 2 1 + αeA (z) − αeB (z) − eF (z)
¢¤
£
£7 1 3 ¤ 1¡
, 2 − 8 α , 2 1 + 14 α2 ;
∈ min 32
1 2 F
2 α eA
¡
¢ ¡
¢¡
¢
Ksf (z) := α2 eFA (z) eFB (z) + αeFA (z) 1 − αeFB (z) + 1 − αeFA (z) 1 − αeFB (z)
2
¡
¢
(eF (z))
+ 12 1 − αeFA (z) αeFB (z) − A 2
2
(eF (z))
= 1 − 12 αeFB (z) + 12 α2 eFA (z) eFB (z) − A 2
¢ ¤
£ ¡
∈ 18 5 − 3α + 94 α2 , 1 ;
2
Kf s (z) :=
1
F
2 αeA
(z) − 12 α2 eFA (z) eFB (z) −
(eFA (z))
2
∈
£3 ¡
¢ 1 2¤
3 2
3
8 α − 8α − 4 , 8α ;
20
MARIA GOLTSMAN AND ARIJIT MUKHERJEE
¸ µ
¶¸
·
·
1
1 2
7 1 3
, − α ,
1+ α
.
Kf f (z) := Kss (z) ∈ min
32 2 8
2
4
Similarly,
Mss (z) :=
1 2 F
2 α eA
¡
¢
¢¡
¢
¡
(z) eFB (z) + αeFB (z) 1 − αeFA (z) + 12 1 − αeFA (z) 1 − αeFB (z) −
2
(eFB (z))
£
£7 1 3 ¤ 1¡
¢¤
∈ min 32
, 2 − 8 α , 2 1 + 14 α2 ;
2
¡
¢ ¡
¢¡
¢
Mf s (z) := α2 eFA (z) eFB (z) + αeFB (z) 1 − αeFA (z) + 1 − αeFA (z) 1 − αeFB (z) +
2
F
¢
¡
1
F (z) αeF (z) − (eB (z))
1
−
αe
B
A
2
2
¢ ¤
£ ¡
∈ 18 5 − 3α + 94 α2 , 1 ;
¡ F
¢2 · µ
¸
¶
eB (z)
1 F
1 2 F
3
3 2 3
1 2
F
Msf (z) := αeB (z) − α eA (z) eB (z) −
∈
α− α −
, α ;
2
2
2
8
8
4
8
·
·
¸ µ
¶¸
7 1 3
1
1 2
, − α ,
1+ α
.
Mf f (z) := Mss (z) ∈ min
32 2 8
2
4
Consequently,
¡
¢
R
¢R
¡
Ksf (z) dζ sf (z) −
fA eFA , eFB = α2 eFB Kss (z) dζ ss (z) + α 1 − αeIB
R
¢
R
¡
Kf f (z) dζ f f (z)
α2 eFB Kf s (z) dζ f s (z) − α 1 − αeIB
¡
¢
¡
¢
¡
¢
≤ 12 α2 eFB 1 + 14 α2 + α 1 − αeIB − 38 α2 eFB α − 38 α2 − 34
¡
¢
£7 1 3 ¤
−α 1 − αeIB min 32
, 2 − 8α
¤
¡
¢
¡
¢
£
3 3
7 2
7 1
4
= eFB 17
, 2 − 38 α
+ α − α 1 − αeIB min 32
64 α − 8 α − 32 α
¡
¢
≤ α − 18 α2 1 − 34 α
< 1,
DISCLOSURE IN MULTISTAGE TOURNAMENTS
21
and
£7 1 3 ¤ 1 ¡
¡
¢
¢¡
¢
9 2
fA eFA , eFB ≥ α2 eFB min 32
, 2 − 8 α + 8 α 1 − αeIB 23
−
3α
+
α
−
4
4
¡
¢¡
¢
1 4 F
1
I
1 + 14 α2
8 α eB − 2 α 1 − αeB
¡ 9 4 3 3
¢ 5 3 3 2
£7 1 3 ¤
7 2
7
, 2 − 8 α + eFB − 32
α + 8 α − 32
α + 32
α − 8 α + 32
α
= α2 eFB min 32
¡
¢
9 4
3 2
5 3
7
α + 38 α3 − 32
α + 32
α − 38 α2 + 32
α
≥ eFB − 32
£ 9 4 3 3
¤
3 2
5 3
7
α + 32
α − 38 α2 + 32
α +
≥ 1α< 1 − 32 α + 8 α − 32
3
¤
£5 3 3 2
7
1α≥ 1 32
α − 8 α + 32
α
3
≥ 0
¡
¢
¡
¢
Similarly, fB eFA , eFB ∈ [0, 1] if eFA , eFB ∈ [0, 1]2 . By Brouwer’s fixed-point theorem, it
¢
¡
¢¢
¡ ¡
follows that the function fA eFA , eFB , fB eFA , eFB : [0, 1]2 → [0, 1]2 has a fixed point.
Proof of Lemma 1. Let us change the choice variables in the following way:
(a) If for some z ∈ Z, ζ f f (z) > 0, then the Bayes rule defines a one-to-one correspondence
¡
¢
¡
¢
between ζ yA ,yB (z) (y ,y )∈Y and ζ f f (z) , Pss (z) , Pf s (z) , Psf (z) . In fact, ∀ (yA , yB ) ∈
A
B
(z) , where KyzA ,yB depends only on (Pss (z) , Pf s (z) , Psf (z)). So
¢
¡
for such z the principal can maximize with respect to ζ f f (z) , Pss (z) , Pf s (z) , Psf (z) in¢
¡
stead of ζ yA ,yB (z) (y ,y )∈Y ;
Y, ζ yA ,yB (z) =
KyzA ,yB ζ f f
A
B
(b) If ζ f f (z) = 0 (which implies Pf f (z) = 0) and ζ f s (z) > 0, then, similarly, the principal
¢
¡
¢
¡
can maximize with respect to ζ f s (z) , Pss (z) , Psf (z) instead of ζ f s (z) , ζ sf (z) , ζ ss (z) ;
(c) If ζ f f (z) = ζ f s (z) = 0 and ζ sf (z) > 0, then the principal can maximize with respect
¢
¡
¢
¡
to ζ sf (z) , Pss (z) instead of ζ sf (z) , ζ ss (z) .
So the principal can change the choice variables from (Z, ζ) to
¢
¡
¢
¡
(Z, ζ f f (z) , Pss (z) , Pf s (z) , Psf (z) z:ζ (z)>0 , ζ f s (z) , Pss (z) , Psf (z) z:ζ (z)=0,ζ (z)>0 ,
ff
ff
fs
¢
¡
ζ sf (z) , Pss (z) z:ζ (z)=ζ (z)=0,ζ (z)>0 , (ζ ss (z))z:ζ f f (z)=ζ f s (z)=ζ sf (z)=0,ζ ss (z)>0 ) and add the
ff
fs
sf
P
PyA ,yB (z) = 1.
constraints that PyA ,yB (z) ≥ 0 and
yA ,yB
Let
µ
³
´
∗ (z) , P ∗ (z) , P ∗ (z)
Z ∗ , ζ ∗f f (z) , Pss
fs
sf
¡ ∗
¢
∗ (z)
ζ sf (z) , Pss
z:ζ ∗
z:ζ ∗f f (z)>0
∗
∗
ff (z)=ζ f s (z)=0,ζ sf (z)>0
³
´
∗ (z) , P ∗ (z)
, ζ ∗f s (z) , Pss
sf
, (ζ ∗ss (z))z:ζ ∗
z:ζ ∗f f (z)=0,ζ ∗fs (z)>0
∗
∗
∗
ff (z)=ζ f s (z)=ζ sf (z)=0,ζ ss (z)>0
¶
,
22
MARIA GOLTSMAN AND ARIJIT MUKHERJEE
´
³
¡ F∗ ¢
and eI∗
,
e
(z)
i
i
z∈Z ∗
i∈{A,B}
be an equilibrium, and suppose that Z ∗ is infinite. Then
µ
³
´
³
´
∗ (z) , P ∗ (z) , P ∗ (z)
∗ (z) , P ∗ (z)
,
ζ
(z)
,
P
,
Z ∗ , ζ f f (z) , Pss
f
s
ss
fs
sf
sf
z:ζ f f (z)>0
z:ζ f f (z)=0,ζ fs (z)>0
´
¡
¢
∗ (z)
ζ sf (z) , Pss
, (ζ ss (z))z:ζ f f (z)=ζ f s (z)=ζ sf (z)=0,ζ ss (z)>0
z:ζ (z)=ζ (z)=0,ζ (z)>0
ff
´
³
¡ F∗ ¢
,
e
(z)
and eI∗
i
i
z∈Z ∗
fs
i∈{A,B}
sf
will also be an equilibrium if (Int) holds ((F inal) will hold
automatically, since we haven’t changed Py∗z
and eFi ∗ (z)) and
i ,yj
I∗
eI∗
A + eB +
(2)
eI∗
A
+ eI∗
B
(3)
P
(yA ,yB )∈Y
+
P
(yA ,yB )∈Y
R³
´
eFi ∗ (z) + eFj ∗ (z) dζ ∗yA ,yB (z) =
´
R ³ F∗
Pr (yA , yB )
ei (z) + eFj ∗ (z) dζ yA ,yB (z) ;
Pr (yA , yB )
ζ yA ,yB (z) ≥ 0, ∀ (yA , yB ) ∈ Y, z ∈ Z;
Z
(4)
dζ (yA ,yB ) (z) = 1, ∀ (yA , yB ) ∈ Y.
We will prove that there exist distributions ζ yi ,yj with a finite support such that (Int) , (2), (3)
and (4) hold.
Take (Int), (2) and (4) (7 equations in total) and divide both sides by
R ∗
P
dζ (yA ,yB ) (z). The result will be 7 equations of the form the constant left-hand
(yA ,yB )∈Y
side lies in the convex hull of some set, the points of which are indexed by z ∈ Z.” It follows
that the left-hand side of (Int), (2) and (4) can be represented as a convex combination of
points in this set, where the coefficients of the convex combination are of the form
³¡
¢
¡
¢
R1 ∗
P
×
ζ f f (z) z:ζ (z)>0 , ζ f s (z) z:ζ (z)=0,ζ (z)>0 ,
(z)
dζ
(yA ,yB )∈Y
¡
ζ sf (z)
ff
(yA ,yB )
¢
z:ζ f f (z)=ζ fs (z)=0,ζ sf
ff
fs
, (ζ ss )z:ζ f f (z)=ζ f s (z)=ζ sf (z)=0,ζ ss (z)>0
(z)>0
´
for certain z ∈ Z. This convex combination defines a probability distribution concentrated
on a finite number of points in Z.
By Caratheodory’s theorem, one can choose such a
distribution so that it puts positive probability on not more than 8 points in Z (since we
have 7 equations).
DISCLOSURE IN MULTISTAGE TOURNAMENTS
23
Proof of Proposition 2. By Lemma 1, there exists an optimal disclosure policy that puts
positive probability on at most 8 signals in Z. Let the equilibrium generated by this optimal
disclosure policy be
µ
³
´
∗ (z) , P ∗ (z) , P ∗ (z)
Z ∗ , ζ ∗f f (z) , Pss
fs
sf
z:ζ ∗f f (z)>0
¡ ∗
¢
∗ (z)
ζ sf (z) , Pss
z:ζ ∗
∗
∗
ff (z)=ζ f s (z)=0,ζ sf (z)>0
´
³
¡ F∗ ¢
and eI∗
i , ei (z) z∈Z ∗
i∈{A,B}
µ
¡ ∗
¢
ζ f f (z) z:ζ ∗
f f (z)>0
³
´
∗ (z) , P ∗ (z)
, ζ ∗f s (z) , Pss
sf
z:ζ ∗f f (z)=0,ζ ∗fs (z)>0
, (ζ ∗ss (z))z:ζ ∗
∗
∗
∗
ff (z)=ζ f s (z)=ζ sf (z)=0,ζ ss (z)>0
¶
. Then
¡
¢
, ζ ∗f s (z) z:ζ ∗
∗
ff (z)=0,ζ f s (z)>0
¡
¢
, ζ ∗sf (z) z:ζ ∗
∗
∗
f f (z)=ζ f s (z)=0,ζ sf (z)>0
(ζ ∗ss (z))z:ζ ∗
∗
∗
∗
ff (z)=ζ f s (z)=ζ sf (z)=0,ζ ss (z)>0
,
´
solves
maxζ
(
eI∗
A
+ eI∗
B
+
P
(yA ,yB )∈Y
s.t.
R¡
¢
Pr (yA , yB ) eFA∗ (z) + eFB∗ (z) dζ yA ,yB (z)
)
(Int) , (3) , and (4) .
This is a linear programming problem, and the canonical form of this problem is
max cα s.t. Aα = b, α ≥ 0,
α
where
µ
¢
¡ ∗
α=
ζ f f (z) z:ζ ∗
ff (z)>0
¡
¢
, ζ ∗f s (z) z:ζ ∗
∗
f f (z)=0,ζ fs (z)>0
¡
¢
, ζ ∗sf (z) z:ζ ∗
,
´
(z)=0,ζ ∗ss (z)>0 ) ,
∗
∗
ff (z)=ζ f s (z)=0,ζ sf (z)>0
(ζ ∗ss (z))z:ζ ∗
∗
∗
ff (z)=ζ f s (z)=ζ sf
A is a 6 × 8 matrix (there are 6 constraints, two corresponding to (Int) and four to (6)), b is
a 6 × 1 vector, and c is a 8 × 1 vector. If a solution to this problem exists, then there exists
an extreme point of the feasible region that is optimal.
coordinates of α are strictly positive.
At any extreme point, at most 6
,
24
MARIA GOLTSMAN AND ARIJIT MUKHERJEE
Proof of Proposition 3. Policy (5) and policy (7) generate the same efforts as policy (6) and
policy (1) , respectively. This is because a signal z such that ζ ss (z) = ζ f f (z) = 1, ζ f s (z) =
ζ sf (z) = 0 leads to the same final-stage effort as a signal z 0 such that ζ ss (z 0 ) = 1, ζ f f (z 0 ) =
ζ sf (z 0 ) = ζ f s (z 0 ) = 0, or a signal z 00 such that ζ f f (z 00 ) = 1, ζ ss (z 00 ) = ζ sf (z 00 ) = ζ f s (z 00 ) = 0
(namely, the final-stage effort is
1
2
for both participants following any of these signals). Con-
sequently, policy (5) generates the same final-stage efforts as policy (6) , and, consequently,
the same intermediate-stage efforts, and the same is true for policies (7) and (1). In words,
it does not matter whether only to disclose that the score is even, or to disclose the actual
value of the score, given that it is even. So in effect, we have 5 different cases:
1) Full disclosure
Here PyA ,yB (zyA ,yB ) = 1, so substituting into the FOC for the final-stage effort yields
eFA (zss ) = eFB (zss ) = eFA (zf f ) = eFB (zf f ) = 12 ;
eFA (zsf ) = eFB (zf s ) = 15 ;
eFA (zf s ) = eFB (zsf ) = 25 ;
In a symmetric equilibrium, the first-order conditions at the intermediate stage are:
¢
¢
2
3
41 ¡
3¡
1 − eI − eI −
1 − eI
eI = eI +
8
50
25
8
So
eI =
89
≈ 0.38696
230
The expected final stage effort is
¡
¢
Ez eFA (z) + eFB (z) = (0.38696)2 +(1 − 0.38696)2 +2× 35 ×0.38696 (1 − 0.38696) = 0.81022;
The expected total effort is
2) No disclosure
¢
¡
T E = 2eI + Ez eFA (z) + eFB (z) ≈ 1.5841
With no disclosure, the two periods are symmetric, so eIA = eIB = eFA = eFB = e. Substituting into the first-order conditions for the final stage gives:
DISCLOSURE IN MULTISTAGE TOURNAMENTS
e=
=
2 1
2
2
2
1 e +e(1−e)+(1−e) + 2 (e(1−e)−e(1−e))(e +e(1−e)+(1−e) )
2
1
2
1+ 4 (e(1−e)−e(1−e))
1
2
so
(1 − e (1 − e))
√
3− 5
e=
2
The expected total effort is
³
√ ´
T E = 4e = 2 3 − 5 ≈ 1.5279
3) Disclosing only if both succeed
At the final stage, the effort after signal z1 is
eFA (z1 ) = eFB (z1 ) =
1
2
After signal z2 , in a symmetric equilibrium the posteriors are
Pss (z2 ) = 0,
Psf (z2 ) = Pf s (z2 ) =
eI (1−eI )
1−(eI )2
2
1−eI
(
Pf f (z2 ) =
)
1−(eI )2
;
.
So
eFA (z2 ) = eFB (z2 ) = eF (z2 ) =
1
2
=
2
=
(Pss (z2 ) + Pf s (z2 ) + Pf f (z2 ))
I 2
I
I
1 1−(e ) −e (1−e )
1−(eI )2
1
.
2(1+eI )
At the intermediate stage,
eI =
=
3 I
8e
eI
¢³
¡
+ 1 − eI 1 −
µ
(eF (z2 ))
2
2
−
1
8
¶
eF (z2 )
2
´
³ F
¡ F
¢2 ´
2)
− eI e (z
−
−
e
(z
)
2
2
¶
µ
2
¡
¢ 1 (eF (z2 ))
I
1−e
2 −
2
2
+
(eF (z2 ))
2
−
eF (z2 )
2
+ 12 .
Substituting for eF (z2 ) and reorganizing yields
¡ ¢2
¡ ¢3
9 eI + 14 eI + 2eI − 3 = 0
25
26
MARIA GOLTSMAN AND ARIJIT MUKHERJEE
The only solution that lies between 0 and 1 is
eI ≈ 0.3629
2
(eF (z2 ))
The second-order conditions are satisfied at this point, since
2
−
1
8
< 1.
The expected final-stage effort is
¢ ³
¡ ¢2 ´
¡
Ez eFA (z) + eFB (z) = 1 − eI
The resulting total effort is
¡ I ¢2
¡ I ¢2
1
I
+
e
=
1
−
e
+
e
≈ 0.7688
(1 + eI )
¡
¢
T E = 2eI + Ez eFA (z) + eFB (z) ≈ 1.4946
4) Disclosing only if both fail
At the final stage, the effort after signal z1 is
1
2
eFA (z1 ) = eFB (z1 ) =
After signal z2 , in a symmetric equilibrium the posteriors are
Pf f (z2 ) = 0,
Psf (z2 ) = Pf s (z2 ) =
Pss (z2 ) =
eI (1−eI )
1−(1−eI )2
2
eI
( )
1−(1−eI )2
;
.
So
eFA (z2 ) = eFB (z2 ) = eF (z2 ) =
1
2
=
(Pss (z2 ) + Pf s (z2 ) + Pf f (z2 ))
I 2
I
I
1 1−(1−e ) −e (1−e )
2
=
1
.
2(2−eI )
At the intermediate stage,
µ
¶
2
¢³
¡
(eF (z2 ))
1
I
I + 1 − eI
e =
1−
−
e
2
2
=
eI
µ
2
(eF (z2 ))
2
−
1
8
¶
−
eF (z2 )
2
+ 58 .
1−(1−eI )2
eF (z2 )
2
´
− eI
³
eF (z2 )
2
¡
¢2 ´
−
− eF (z2 )
¡
¢
3
I
8 1−e
DISCLOSURE IN MULTISTAGE TOURNAMENTS
27
Substituting for eFA (z2 ) and reorganizing yields
¡ ¢2
¡ ¢3
9 eI − 41 eI + 53eI − 16 = 0
The only solution that lies between 0 and 1 is
eI ≈ 0.4333
2
The second-order conditions are satisfied at this point, since
(eF (z2 ))
2
−
1
8
< 1.
The expected final-stage effort is
¡
¢2 ´
¢ ³
¡
Ez eFA (z) + eFB (z) = 1 − 1 − eI
The resulting total effort is
¢2
¡ ¢2
¡
1
+ 1 − eI = 1 − eI + eI ≈ 0.7544
I
(2 − e )
¡
¢
T E = 2eI + Ez eFA (z) + eFB (z) ≈ 1.6211
5) Disclosing whether the score is even
After signal z1 , in a symmetric equilibrium the posteriors are
¢2
¡
¡ I ¢2
1 − eI
e
Pf f (z1 ) =
2
2 , Pss (z1 ) =
2
2;
I
I
I
(e ) + (1 − e )
(e ) + (1 − eI )
Psf (z1 ) = Pf s (z1 ) = 0.
The final-stage efforts after signal z1 are
eFA
(z1 ) =
eFB
¡ ¢2 ¡
¢2
1 eI + 1 − eI
1
(z1 ) = e (z1 ) =
=
2
2
I
I
2 (e ) + (1 − e )
2
F
After signal z2 , in a symmetric equilibrium the posteriors are
Pf f (z2 ) = Pss (z2 ) = 0,
Psf (z2 ) = Pf s (z2 ) =
1
2
The final-stage efforts after signal z2 are
eFA (z2 ) = eFB (z2 ) = eF (z2 ) =
1
4
28
MARIA GOLTSMAN AND ARIJIT MUKHERJEE
Table 3. Symmetric Deterministic Policies with α = 1
¡
¢
eI
Ez eFA (z) + eFB (z) Total Effort
0.3870
0.8102
1.5841
0.3820
0.7639
1.5279
0.3629
0.7688
1.4946
0.4333
0.7544
1.6211
0.4211
0.7562
1.598
Disclosure policy
Full disclosure
No disclosure
Disclosing only if both succeed
Disclosing only if both fail
Disclosing whether the score is even
At the intermediate stage,
¢
¢
1
3 I 7¡
3¡
1 − eI − eI −
1 − eI
e +
8
8
16
8
3 I 1
= − e +
16
2
eI =
The solution is
eI =
8
≈ 0.421 05
19
The expected final-stage effort is
¡
¢
Ez eFA (z) + eFB (z) =
µ
8
19
¶2
+
µ
11
19
¶2
+2×
8
11 1
×
× ≈ 0.7562
19 19 2
The resulting total effort is
¡
¢
T E = 2eI + Ez eFA (z) + eFB (z) = 0.42105 ∗ 2 + 0.7562 ≈ 1.598
Table 3 summarizes the equilibrium efforts.
Appendix 2: Symmetric Deterministic Policies when α ∈ [0, 1]
References
[1] Aoyagi, M. (2004): “Information Feedback in a Dynamic Tournament,” working paper.
[2] Dubey, P., and J. Geanakoplos (2004): “Grading Exams: 100,99,...,1 or A, B, C? Incentives in Games of
Status,” Cowles Foundation discussion paper No. 1467.
[3] Ederer, F. (2004): “Feedback and Motivation in Dynamic Tournaments,” working paper.
[4] Gershkov, A., and M. Perry (2006): “Tournaments with Midterm Reviews,” working paper.
[5] Gordon, S. (2004): “Publishing to Deter in R&D Competitions,” working paper.
[6] Green, E., and N. Stokey (1983): “A Comparison of Tournaments and Contracts,” Journal of Political
Economy, 91, 349-364.
DISCLOSURE IN MULTISTAGE TOURNAMENTS
29
Table 4. Total Effort as a function of α
Disclosure policy
Full disclosure
No disclosure
Disclose only if both succeed
Disclose only if both fail
Total³ Effort
´
³ 3
α
2y + (αy)2 + (1 − αy)2 α + 2αy (1 − αy) 4+α
4 +
µ
¶
2
6
α 0.5+ 18 α2 − α 4 +0.5 α 4 2
4+α
(4+α )
µ
¶
where y =
6
α
2α2
1+α2 0.25α2 −0.5
−
(4+α4 )2 (4+α4 )2
√
0.5α2 +1− α2 +1−0.75α4
4
α3
´³
´
³
α
, where y solves
2y + (αy)2 α + 1 − (αy)2 1+αy
¢ 2
¢ 3 ¡
¡1 6
2
5
3
−
0.5α
+
2α
y +¢
8¡α + α y + 0.25α
¢ ¡ 1 5 1 3
4
2
y 0.25α − α + 1 + − 8 α + 4 α − 0.5α = 0
2α
4+α4
´
and y³∈ [0, 1]
´³
´
α
+ (1 − αy)2 α, where y
2y + 1 − (1 − αy)2 2−αy
¢
¡
¢ 2
¡ 2
y +
solves
4α + 0.5α6 ¢ y 3 ¡+ −2.5α5 − 2α3 − 16α
¡
¢
4
2
3
5
y 2.5α + 8α + 16 + 0.5α − 0.5α − 8α = 0
and y³∈ [0, 1]
´
Disclose whether score is even 2y + (αy)2 + (1 − αy)2 α + 2αy (1 − αy) α2
0.5α+ 1 α3 (α2 −1)
where y = 1+ 132α3 + 1 α4
8
16
[7] Harris, C., and J. Vickers (1987): “Racing with Uncertainty,” Review of Economic Studies, 54(1), 1-21.
[8] Lazear, E., and S. Rosen (1981): “Rank Order Tournaments as Optimal Labor Contracts,” Journal of
Political Economy, 89, 841-864..
[9] Lizzeri, A., M. Meyer and N. Persico (2002): “The Incentive Effects of Interim Performance Evaluations,”
CARESS working paper #02-09.
[10] Nalebuff, B., and J. Stiglitz (1983): “Prizes and Incentives: Towards a General Theory of Compensation
and Competition,” Bell Journal of Economics, 14, 21-43.
[11] Yildirim, H. (2005): “Contests with Multiple Rounds,” Games and Economic Behavior, 51(1), 213-227.