This note relates the mechanisms that are based on mediated contracts of Rahman and Obara (2010) to the mechanisms of Myerson (1982). It shows that the mechanisms in Myerson (1982) are more general in that they encompass the mechanisms based on mediated contracts. It establishes an equivalence between the two classes if mediated contracts are allowed to be stochastic.mediatedcontract; mechanismdesign; revelationprinciple; moralhazard

Strausz, Roland

English

Research Papers in Economics

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Discussion Paper No. 322 
   
Mediated Contracts and 
Mechanism Design 
 
Roland Strausz* 
 
 
 Mediated Contracts and Mechanism Design
Roland Strausz∗
May 25, 2010
Abstract
This note relates the mechanisms that are based on mediated contracts of Rahman
and Obara (2010) to the mechanisms of Myerson (1982). It shows that the mechanisms
in Myerson (1982) are more general in that they encompass the mechanisms based on
mediated contracts. It establishes an equivalence between the two classes if mediated
contracts are allowed to be stochastic.
Keywords: mediated contract, mechanism design, revelation principle, moral hazard
1 Introduction
In an inspiring paper, Rahman and Obara (2010) introduce the concept of mediated contracts.1
The authors clearly show that, in team problems with balanced budgets, mediated contracts
outperform standard, non–mediated ones and even enable a virtual implementation of the ﬁrst
best. What remains less clear is how their concept of mediation relates to other concepts of
mediation in earlier work on mechanism design (e.g., Myerson 1986, 1991, Forges 1986). This
note tries to clarify this link by contrasting mediated contracts to the mechanisms of Myerson
(1982), who extends the revelation principle to settings with moral hazard.2 It argues that the
∗Humboldt-Universit¨ at zu Berlin, Institute for Microeconomic Theory, Spandauer Str. 1, D-10178 Berlin
(Germany), strauszr@wiwi.hu-berlin.de. I thank Andrea Attar, Helmut Bester, Francoise Forges, Daniel
Kr¨ ahmer, Johannes M¨ unster, and David Rahman for very helpful discussions.
1See also Rahman (2009) for the use of mediated contracts in a monitoring context.
2Both Myerson (1982) and Rahman and Obara (2010) allow, next to moral hazard, also for ex ante private
information, but because the conceptual diﬀerence relates to the moral hazard problem, this notes focuses on
the moral hazard and abstracts from ex ante private information.
1mechanisms in Myerson encompass the mechanisms based on mediated contracts and are more
general in that they allow for a larger degree of randomization. An equivalence between the
two classes of mechanisms obtains if one extends mediated contracts to stochastic mediated
contracts that also condition on purely random signals. The equivalence implies that there is
no loss of generality in restricting attention to incentive compatible mediated contracts in the
sense of Rahman and Obara (2010) if such contracts are allowed to be stochastic.
2 Setup
I illustrate my arguments in a version of Holmstr¨ om (1982)’s seminal moral hazard in teams
problem. There are two agents who each have to pick an eﬀort level e ∈ R+.3 The cost of eﬀort
is c(e) = e2 2. The output’s value is linear in eﬀort: y(e1 e2) = e1 + e2. Agents are identical
and have the quasi–linear utility function
u(t e) = t − c(e) 
where t represents a monetary transfer. In this quasi–linear framework, the Pareto eﬃcient pair
of eﬀort levels (e∗
1 e∗
2) is unique. It maximizes y(e1 e2)−c(e1)−c(e2) and exhibits e∗
1 = e∗
2 = 1.
When eﬀort levels are unobservable, the agents cannot implement the eﬃcient eﬀort levels
directly. When output is observable, agents can however write binding contracts how to share
the proceeds from the ﬁnal output. These contracts induce a simultaneous move game in which
the two agents each pick an eﬀort level non–cooperatively. Holmstr¨ om (1982) shows however
that there does not exist a contract pair (t1(y) t2(y)) that satisﬁes the budget balance condition
t1(y) + t2(y) = y
and induces a simultaneous move game with a Nash–equilibrium in which the agents pick the
eﬃcient eﬀort levels e∗
1 = e∗
2 = 1.
3To circumvent measure theoretical complications, Rahman and Obara (2010) derive their formal results for
ﬁnite action spaces, but this example shows their construction works just as well with inﬁnite action spaces.
23 Mediated Contracts
Rahman and Obara (2010) extend the concept of a contract to mediated contracts by introduc-
ing a mediator. In line with earlier literature, the mediator merely facilitates communication
between the two agents; he gives only non–binding recommendations about eﬀort levels. The
new idea of Rahman and Obara (2010) is to condition the contract on the mediator’s rec-
ommendations.4 Hence, the contract no longer conditions transfers only on the output level,
but also on the mediator’s recommendations. In the Holmstr¨ om example above, this means
that, instead of ti(y), contracts are expressed by ti(y er
1 er
2) where er
i represents the mediator’s
recommendation to agent i.
The timing with mediated contracts is as follows: First, the agents agree on some mediated
contract pair (t1(y er
1 er
2) t2(y er
1 er
2)) and a probability distribution over eﬀort pairs (e1 e2).
The mediator then draws an eﬀort pair (er
1 er
2) according to this pre–speciﬁed distribution
and reports to each agent i the drawn eﬀort level er
i as a conﬁdential recommendation. Only
knowing their own recommendation, the two agents choose their eﬀort levels simultaneously.
Finally, the chosen eﬀort levels result in a ﬁnal output y and leads to transfers according to the
pre–determined contract pair (t1(y er
1 er
2) t2(y er
1 er
2)). Rahman and Obara (2010) consider
mediated contracts that balance the budget and are Bayes’ incentive compatible in the sense
that each agent has an incentive to follow the recommendation if the other agent follows the
recommendation.
In order to illustrate the power of mediated contracts in the Holmstr¨ om framework, consider
the following speciﬁc one. With probability 1 − ε the mediator recommends each agent the
eﬃcient eﬀort level: (er
1 er
2) = (1 1). With probability ε 2 the mediator recommends agent
1 the eﬃcient eﬀort level and agent 2 to shirk: (er
1 er
2) = (1 0). Finally, with probability
ε 2 the mediator recommends agent 1 to shirk and agent 2 to work eﬃciently: (er
1 er
2) =
(0 1). Crucially, recommendations are given conﬁdentially so that when agent i receives the
recommendation er
i = 1, he is unsure about the recommendation er
j of the other agent j  =
i. He, however, knows the probability distribution according to which the mediator gives
4The recommendations, therefore, need to be veriﬁable, but this seems unproblematic: The mediator writes
each recommendation on a piece of paper and sends each piece of paper to the respective agent. After the
output occurs, agents are obliged to reveal the pieces of paper publicly.
3recommendations. Using Bayes’ rule, agent i, therefore, expects that er
j = 0 with probability
 0 = Pr{e
j = 0|e
i = 1} =
ε 2
1 − ε + ε 2
=
ε
2 − ε
and er
j = 1 with probability
 1 = Pr{e
j = 1|e
i = 1} = 1 −  0 =
2 − 2ε
2 − ε
 
In addition, let the contract depend on the recommendation (er
1 er
2) as follows. For any
pair of recommendations the output is shared equally, except for the recommendation (1 0)
and (0 1). For these two special cases, the agent who is given the recommendation to work
eﬃciently (er = 1) receives 1 ε times the output as a transfer, whereas the agent who is told
to shirk (er = 0) receives the (negative) transfer (ε − 1)y ε. Formally,
t1(y e
r
1 e
r
2) =

   
   
y ε  if (er
1 er
2) = (1 0) 
(ε − 1)y ε  if (er
1 er
2) = (0 1) 
y 2  otherwise;
t2(y e
r
1 e
r
2) =

   
   
(ε − 1)y ε  if (er
1 er
2) = (1 0) 
y ε  if (er
1 er
2) = (0 1) 
y 2  otherwise.
By construction, the contract’s budget is balanced.
The transfer schedule together with the mediator’s probability distribution over recommen-
dations is Bayes’ incentive compatible: First, given a recommendation to shirk er
i = 0, agent
i is certain that agent j  = i has received the recommendation er
j = 1 so that agent i expects
a negative transfer (ε − 1)y ε from the output y. Given these expectations, it is a dominant
strategy to follow the mediator’s recommendation er
i = 0. Alternatively, given the recommen-
dation er
i = 1, agent i believes that agent j  = i received and follows the recommendation er
j = 0
with probability  0 and the recommendation er
j = 1 with probability  1. In the former case,
his eﬀort level ei yields him a transfer ei ε. In the latter case, the eﬀort level ei yields him a
transfer (1 + ei) 2. Hence, his expected utility from an eﬀort level ei is
 0ei ε + (1 −  0)(1 + ei) 2 − e
2
i 2 =
1 − ε
2 − ε
+ ei − e
2
i 2 
Clearly, ei = 1 maximizes this expression so that it is a best response for agent i to follow the
recommendation er
i = 1.
Because, for any ε ∈ (0 1) the corresponding mediated contract implements the eﬃcient
eﬀort choices e1 = e2 = 1 with probability 1−ε, the mediated contract virtually implements the
eﬃcient outcome as ε approaches 0. This result contrasts sharply with Holmstr¨ om’s ineﬃciency
result and demonstrates the power of mediated contracts.
44 Mechanism Design
The literature on communication and mechanism design has noted the beneﬁcial role of medi-
ators in implementation problems before. In particular, Myerson (1982) introduces a mediator
in a simultaneous move game with moral hazard.5 Similar to Rahman and Obara (2010), the
mediator in Myerson makes conﬁdential recommendations before the agents make their choices.
In contrast, Myerson does not capture contractual commitments by a formal contract, but by
the way the mediator is “programmed” to choose among the contractible variables.
In the context of the Holmstr¨ om model, the contractible variable is the pair of conditional
transfer schedules (t1(y) t2(y)). Hence, Myerson’s interpretation is that the mediator chooses
the recommendations and the transfer schedules, but the picked transfer schedules are revealed
only after the agents have chosen their eﬀort levels. The mechanism describes exactly how
the mediator is to pick the recommendations and the pair of conditional transfer schedules
(t1(y) t2(y)). This reveals the main conceptual diﬀerence between the two frameworks: In
Myerson (1982), a mechanism is a probability distribution over the combination of both recom-
mendations and the pairs of conditional transfer schedules (t1(y) t2(y)). In Rahman and Obara
(2010), a mechanism is a speciﬁc pair of mediated transfer schedules (t1(y er
1 er
2) t2(y er
1 er
2))
and a probability distribution over only the recommendations.6
Hence, Myerson’s framework seems, on the one hand, more general, because the mediator
randomizes over both recommendations and pairs of conditional transfer schedules, whereas
Rahman and Obara (2010) allow only randomizations over recommendations. On the other
hand, Myerson’s framework seems less general, because it allows only transfer schedules that
condition on output, whereas Rahman and Obara (2010) allow transfer schedules that condition
also on the mediator’s recommendations.
Note however that the second suggestion is misleading. Myerson explicitly allows the me-
diator to randomize over combinations of recommendations and contracts. This, in particular,
allows correlation between random draws of recommendations and random draws of transfer
schedules. As a consequence, one can, for any mechanism in the framework of Rahman and
5See also Forges (1986) and Myerson (1986), (1991).
6When the action set of the agents is ﬁnite but transfers are non–countable, the deﬁnition of a mechanism
in Rahman and Obara (2010) is, from a measure theoretical viewpoint, simpler than Myerson’s view, because
Myerson’s deﬁnition also allows mixing over transfers.
5Obara, ﬁnd an equivalent one in the Myerson framework. In the speciﬁc example considered
above, the equivalent mechanism in the sense of Myerson is a mediator who mixes over the
three combinations (er
1 er
2 t1 t2) = (1 1 y 2 y 2), (er
1 er
2 t1 t2) = (1 0 y ε (ε − 1)y ε), and
(er
1 er
2 t1 t2) = (0 1 (ε − 1)y ε y ε) with the respective probabilities 1 − ε, ε 2, and ε 2.
Considering the question whether the Myerson framework is more general, note that me-
diated contracts as deﬁned in Rahman and Obara (2010) do not allow for transfer schedules
that, conditional on the recommendation (er
1 er
2), are random. In contrast, Myerson explicitly
allows for this.7 Because Rahman and Obara (2010) consider a quasi–linear framework, where
all agents are risk neutral with respect to transfers, this distinction does not matter. Any
transfer schedule that, conditional on the recommendation, is random can be replaced by a
deterministic transfer schedule that corresponds to its expected value. This does neither aﬀect
the agent’s utilities nor the balanced budget constraint.
Rasmusen (1987), however, shows that, in teams with risk averse agents, Pareto eﬃcient
allocations are implementable, but this requires random transfers oﬀ–the–equilibrium. Because
mechanisms in the sense of Rahman and Obara (2010) cannot capture such random transfers,
whereas the mechanisms in the sense of Myerson (1982) can, the mechanisms of Rahman and
Obara are, in the setup of Rasmusen (1987), suboptimal in comparison to Myerson.8
A straightforward remedy to the suboptimality is to consider stochastic mediated contracts
that, in addition to recommendations, can also condition on a purely random signal.9 This re-
sults in a class of mechanisms that is equivalent to Myerson (1982) in that there is a one–to–one
correspondence between the two. Stochastic mediated contracts are, therefore, an alternative
representation of the mechanisms in Myerson (1982).
7It is for this reason that, as noted in footnote 6, mechanisms in the sense of Rahman and Obara are simpler
constructs.
8I thank Johannes M¨ unster for pointing out that, with risk aversion, it is even not clear if mediated contracts
allow a virtual implementation of Pareto eﬃcient allocations. E.g., the above construction in Holmstr¨ om’s
framework allows an implementation of the eﬃcient eﬀort levels with a probability arbitrarily close to one, but
with an unbounded variance in the agents’ utility.
9For stochastic mediated contracts see also Rahman (2005).
65 Conclusion
Mechanisms based on stochastic mediated contracts are formally equivalent to the mechanisms
in Myerson (1982). They have, however, two interpretational advantages. First, they make more
explicit the correlation between the mediator’s recommendations and contract choice. Second,
they seem more natural, because the contract choice is explicitly left to the economic agents
and the mediator is limited to providing only recommendations. This representation is also
closer to the fundamental idea of agency theory that contracts are used to provide incentives.
Moreover, because Myerson (1982) derives his class of mechanisms from a revelation principle,
the equivalence further implies that there is no loss of generality in restricting attention to
incentive compatible mediated contracts in the sense of Rahman and Obara (2010) if such
contracts are allowed to be stochastic.
References
[1] Forges, F. (1986), ”An Approach to Communication Equilibria”, Econometrica 54, 1375–
1385.
[2] Holmstr¨ om, B. (1982), ”Moral Hazard in Teams”, Bell Journal of Economics, 13, 324340
[3] Myerson, R. (1982), ”Optimal Coordination Mechanism in Generalized Principal-Agent
Problems”, Journal of Mathematical Economics 10, 67–81.
[4] Myerson, R. (1986), ”Multistage games with communication”, Econometrica 54, 323-358
[5] Myerson (1991), Game Theory: Analysis of Conﬂict, Harvard University Press, Cambridge,
Massachusetts, London, England.
[6] Rahman, D. (2005), ”Optimum Contracts with Public and Private Monitoring”, mimeo
University of California, San Diego.
[7] Rahman, D. (2009), ”But Who Will Monitor the Monitor?”, mimeo University of Min-
nesota.
[8] Rahman, D. and Obara, I. (2010), ”Mediated Partnerships”, Econometrica 78, 285 – 308.
7[9] Rasmusen, E. (1987), ”Moral Hazard in Risk-Averse Teams”, The RAND Journal of Eco-
nomics 18, 428–435.
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