Abstract
This paper explores the communication and choice strategies of economic agents deciding on a partnership, where agents are uncertain about their payoffs, and payoffs of each agent depend on and are partly known to the potential partner. Business examples of such decisions include mergers, acquisitions, distribution channel partners, as well as manufacturing and brand alliances. Dating and marriage partner selection are also natural examples of this game. The paper shows that (a) when communication is informative, the communication strategy as a function of the expected payoff of the partnership involves pretending fit when expected payoff is high, pretending misfit when expected payoff is low, and telling the truth in the intermediate range, and (b) the condition for informativeness of communication is that the distribution of payoffs has thin tails. Furthermore, the paper shows that the possibility of communication, even when this communication is not restricted to be truthful, can decrease the expected payoff for both the sender and the receiver; in particular, it can decrease the expected social welfare.
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Notes
This example is simplified and is given for illustration only; other expectations can also be consistent in this example. The formal model will lead to unique rational expectations derived from Bayes rule in all cases.
The assumption of only one population being able to talk is needed for the analytical tractability of communication strategies. Section 2.4 discusses the robustness of implications under two-way communication.
See Farrel and Rabin (1996) for a further survey and discussion of cheap talk literature.
I make no assumptions on a, and in fact, one could also assume a = 1 to reduce the number of parameters in the model. However, this parameter will help us to see which part of the payoff affects which part in the strategies, and therefore, will provide greater intuition for the results. Separating the A k part of the payoff and the discrete distribution of the correlated part of the payoffs, makes the model tractable.
To be consistent with the literature, I will sometimes refer to Agent 1 as the sender and to Agent 2 as the receiver.
Section 3 discusses the possibility of Agent 2 also being able to send a message.
The intuition for the results will not change if we would not model where the signals and the common value component are coming from and just assumed that, for example, the common value can take one of the three values and agents would be receiving i.i.d. noisy signals on what the value is.
It is also easy to construct a two-point distribution of A for which condition 23 does not hold.
If agents communicate sequentially, it is the same for the agent communicating last, and therefore, his/her communication strategy as a function of expected payoff is qualitatively the same as the communication strategy of Agent 1 in the main model.
For a more formal discussion, see Kuksov (2007).
One could expect that the right hand side of definition of R f , after substitution, would depend on R f , since the functions that are substituted depend on it. This would give a recursive equation that can be solved for R f . In our case, however, after simplification, R f completely cancels from the right hand side.
This is consistent with Baliga and Morris (2002), who derive a similar result when the communication is about the actions and the communication effect is in coordination on the equilibrium actions rather than about the player type.
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Acknowledgements
I would like to thank the QME review team, Jeroen M. Swinkels, J. Miguel Villas-Boas, Andreas Weingartner, and the seminar participants at Duke University, University of Chicago, University of Toronto for helpful comments and suggestions.
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Appendix
Appendix
1.1 Proof of Lemma 1
The first claim immediately follows from the second claim: if communication is informative and z 10 ≤ z 11, according to the second claim, Agent 1 will always report fit, and thus, the message is not informative of the true fit (a contradiction). To prove the second claim, note that if agent 1 were to report misfit while communication was informative and z 10 ≤ z 11, the probability of acceptance by agent 2 would decreases (since z 21 > z 20, and so, agent 2 expects lower payoff), and the expected value (to agent 1) of match given agent 2’s acceptance decreases by z 11 − z 10 ≥ 0, i.e., does not increase. Hence, Agent 1 always prefers to report fit.
1.2 Proof of Proposition 1
First, note that as we have shown before, “communication is informative” condition implies z 21 > z 20 (and q 21 > q 20) by definition, and implies z 10 > z 11 by Lemma 1. Comparing the expected payoffs to agent 1 when reporting fit and misfit while accepting (i.e., conditional on her expected payoff value higher than her reservation value R 1), one obtains that it is optimal for agent 1 to report misfit (rather than fit) if and only if
Rearranging the terms in Eq. 15, we obtain an equivalent condition:
where the last term is positive (remind that Δz 1 ≡ z 10 − z 11). In other words, we obtain that agent 1 should report fit when she expects high enough payoff. When agent 1 expects lower A 1 + m 1, but still such that A 1 + m 1 ≥ R 1 − az 10, it is optimal for Agent 1 to accept, but report misfit. When A 1 + m 1 is even lower, agent 1 should reject, and hence it does not matter what she reports.
1.3 Proof of Proposition 2
Agent 1’s strategy defined by Eqs. 9 and 16 lead to the following equations for z 2ℓ:
and
which, in terms of F A (·) imply
and
Note that Proposition 1 does not yet imply that if R f > R a , and hence, if there is a truth-telling interval, then z 21 > z 20. However, in order for the message to be informative, we need this satisfied. Simplifying condition z 21 > z 20, we have the following result:
Lemma 2
Conversation can be informative only when
Since Agent 2’s strategy is to accept if and only if A 2 + m 2 + az 2ℓ ≥ R 2, where ℓ is the message received, we have the following equation for z 1l :
The following lemma derives a sufficient condition for z 10 > z 11, which, according to Lemma 1, we need for communication to be informative.
Lemma 3
If
then z 10 > z 11 whenever z 21 > z 20.
Proof
Let X 1 = R 2 − az 21 and X 2 = R 2 − az 20, and assume z 21 > z 20, implying X 2 > X 1. Then, the Eq. 22 imply that for z 10 > z 11, it is sufficient that \(\frac{F_A(X+1)-F_A(X-1)}{2-F_A(X+1)-F_A(X-1)}\) increases in X on [X 1,X 2].
For any two functions g(x) and h(x), we have: h(x)/g(x) is decreasing if and only if (h(x) − g(x))/g(x) ≡ h(x)/g(x) − 1 is decreasing. Therefore, g(x)/h(x) is increasing if and only if g(x)/(h(x) − g(x)) is increasing.
Applying this to the fraction in question, we obtain that the above sufficient condition simplifies to \(\frac{F_A(X+1)-F_A(X-1)}{1-F_A(X+1)}\) is increasing on [X 1,X 2]. A function is increasing if it’s derivative is positive. The derivative of the above fraction with respect to X is equal to:
which is positive if and only if \(\frac{f_A(X+1)}{1-F_A(X+1)}>\frac{f_A(X-1)}{1-F_A(X-1)},\) i.e., it is positive when f A (X)/(1 − F A (X)) is increasing. □
We are now ready to prove the proposition. If f A (x)/(1 − F A (x)) is non-increasing then it is easy to see from the proof of Lemma 3 that z 21 > z 20 leads to z 10 < z 11, and therefore, it is optimal for Agent 1 to always pretend fit, and hence, z 21 = z 20, i.e., communication is not informative.
We now show that condition 23 is sufficient for inequality 21 to hold. Indeed, inequality 21 can be rewritten as
The above inequality is satisfied if the function (1 − F A (x))/(F(x) − F(x − b)) is decreasing, where b = R f − R a > 0. Differentiating this function with respect to x, we obtain that it is decreasing if f A (x − b)/(1 − F A (x − b)) < f A (x)/(1 − F A (x)), which is true if f A (x)/(1 − F A (x)) is increasing.
Hence, we have that if f A (x)/(1 − F A (x)) is increasing, z 21 > z 20 implies that z 10 > z 11, and with the optimal message choice by agent 1, we have z 21 > z 20 satisfied, i.e., communication is informative.
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Kuksov, D. Communication strategy in partnership selection. Quant Mark Econ 7, 267–288 (2009). https://doi.org/10.1007/s11129-009-9070-3
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DOI: https://doi.org/10.1007/s11129-009-9070-3