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Theory and Decision
Published in: Theory and Decision 3/2019

03-06-2019

Strategic communication with reporting costs

Published in: Theory and Decision | Issue 3/2019

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Abstract

A decision maker relies on information of parties affected by her decision. These parties try to influence her decision by selective disclosure of facts. As is well known from the literature, competition between the informed parties constrains their ability to manipulate information. We depart from this literature by introducing a cost to communicate. Our parties trade off their reporting cost against the effect on the decision. Some information is never revealed. In contrast to setups without communication costs, our decision maker can benefit by ex ante committing to an ex post suboptimal decision rule. Moreover, committing ex ante not to listen to one of the parties may also be beneficial for the decision maker.
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Appendix
Available only for authorised users
Footnotes
1
See Milgrom and Roberts (1986), Lipman and Seppi (1995), and Gentzkov and Kamenica (2017). At the end of the section, we review the literature related to our paper.
 
2
Under the adversary system “it is for the parties to determine not only the issues which the court is to decide, but also the material on which the decision will be based. The evidence presented to the court will be that which the parties choose to present and none other. The judge may not require that a particular witness be summoned to give evidence or that a particular document be produced; he may not even question the witnesses himself except for the purpose of clarifying some doubt as to the meaning of what a witness has said under examination by counsel,” Jolowicz (2000, p. 28).
 
3
In the inquisitorial procedure of civil law countries, “it is for the judge to examine the witnesses, if any, it is for the judge to decide whether to summon the parties for interrogation and it is the judge who acts to obtain the assistance of an expert when required,” Jolowicz (2000, p. 220). Our weak form of active adjudication is only one of the multiple instruments an inquisitorial judge has at hand.
 
4
Matthis (2008) provides necessary and sufficient conditions, with respect to preferences and available messages, for full unraveling in one-sender games.
 
5
The irrelevance of commitment is representative of a family of equivalence results between optimal mechanisms and equilibria in one-sender persuasion games; see Glazer and Rubinstein (2004, 2006), Sher (2011), and Hart et al. (2017). In the mechanism design approach, the decision maker moves first and commits to decision rules. The equivalence results state that she does as well when she moves second and plays a sequentially rational strategy. Bhattacharya and Mukherjee (2013) and Bhattacharya et al. (2018) show that the result extends to multiple senders, provided they have extreme agendas (either the same extreme agenda or diametrically opposed agendas).
 
6
Hay and Spier (1997) analyze the allocation of the burden of proof between plaintiff and defendant from the point of view of minimizing litigation costs. Because each party’s submission cost is less than the stake, the trial outcome is always without error. However, litigation expenditures will differ depending on the burden of proof assignment. More distantly related strands of literature deal with costly acquisition of information by the senders or with communication through “fabricated evidence”. See, e.g., Dewatripont and Tirole (1999), Gerardi and Yariv (2008), and Kim (2014) on the first issue; Kartik (2009) and Emons and Fluet (2009, 2019) on the second.
 
7
Sher’s (2011) condition for the irrelevance of commitment in one-sender games is that the decision maker’s utility function is a concave transformation of the sender’s utility function. This is satisfied here with respect to both senders with \(u_{J}(y,x)=-v\left( \left| u_{A}(y)+x\right| \right) =\)\(-v\left( \left| u_{B}(y)-x\right| \right) \).
 
8
Log-concavity of f implies log-concavity of F and is therefore a stronger condition.
 
9
A log-concave f implies that the hazard rate \(f(x)/(1-F(x))\) is increasing. This ensures uniqueness in the sender-A game. The proof is similar to the argument in Bagnoli and Bergstrom (2005).
 
10
To clarify our jargon: If under commitment only one agent reports, we call it a one-sender outcome. Without commitment, when only one party sends a message, be it in the game where both or only one party is allowed to report, we speak of a corner equilibrium.
 
11
With a strictly decreasing f, condition (19) in the proof implies \(\overline{U}_{J}^{\prime }(0)>0\), so \(\hat{y}=0\) cannot be optimal.
 
12
It is obviously never in the adjudicator’s interest to prohibit both parties from reporting.
 
13
Note that the uniform density is log-concave, but nevertheless leads to multiple equilibria in the two-sender game.
 
14
In Table 1, the type of equilibrium is denoted c-A for corner-A, c-B for corner-B, and int for interior.
 
15
Due to the skewness of the distribution, the no-disclosure set of the corner- A equilibrium is about twice the size of the no-disclosure set of the corner-B equilibrium. Yet, the no-disclosure states in the corner-A equilibrium are ex ante very unlikely as compared to the corner-B equilibrium.
 
16
In our examples, the one-sender games always have unique equilibria. Recall that uniqueness obtains when the density f(x) is log-concave, which is the case for \(a\ge 1\) and \(b\ge 1\).
 
17
Equation (16) combines Proposition 1 with the condition in Bhattacharya and Mukherjee (2013) for the case of two opposed experts. Setting \(p_{B}=0\) in (16) yields the equilibrium default decision in the one-sender equilibrium where only A is allowed to report.
 
18
In the corner-B equilibrium reporting costs play no role. Therefore, the optimal default decision under commitment equals the equilibrium default decision without commitment.
 
19
See, e.g., Emons and Fluet (2009, 2019) for such a setup.
 
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Metadata
Title
Strategic communication with reporting costs
Publication date
03-06-2019
Published in
Theory and Decision / Issue 3/2019
Print ISSN: 0040-5833
Electronic ISSN: 1573-7187
DOI
https://doi.org/10.1007/s11238-019-09709-4

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