Communication strategy in partnership selection

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.

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

$$ (A_{1}+m_1+az_{10}-R_1)q_{20}>(A_{1}+m_1+az_{11}-R_1)q_{21},$$

(15)

Rearranging the terms in Eq. 15, we obtain an equivalent condition:

$$ A_{1}+m_1<R_1-az_{10}+\frac{aq_{21}\Delta z_1}{\Delta q_2},$$

(16)

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:

$$ z_{21}=\frac{\mbox{Prob}(A_1>R_f-1)-\mbox{Prob}(A_1>R_f+1)} {\mbox{Prob}(A_1>R_f-1)+\mbox{Prob}(A_1>R_f+1)} $$

(17)

and

$$ z_{20}=\frac{\mbox{Prob}(A_1\in(R_a-1,R_f-1))-\mbox{Prob}(A_1\in(R_a+1,R_f+1))} {\mbox{Prob}(A_1\in(R_a-1,R_f-1))+\mbox{Prob}(A_1\in(R_a+1,R_f+1))}, $$

(18)

which, in terms of F _A (·) imply

$$ z_{21}=\frac{F_A(R_f+1)-F_A(R_f-1)}{2-F_A(R_f+1)-F_A(R_f-1)} $$

(19)

and

$$ z_{20}=\frac{F_A(R_f-1)-F_A(R_a-1)-F_A(R_f+1)+F_A(R_a+1)} {F_A(R_f-1)-F_A(R_a-1)+F_A(R_f+1)-F_A(R_a+1)}. $$

(20)

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

$$ \frac{1-F_A(R_f-1)}{1-F_A(R_f+1)}>\frac{F_A(R_f-1)-F_A(R_a-1)}{F_A(R_f+1)-F_A(R_a+1)}.$$

(21)

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 :

$$ z_{1\ell}=\frac{F_A(R_2-az_{2\ell}+1)-F_A(R_2-az_{2\ell}-1)} {2-F_A(R_2-az_{2\ell}+1)-F_A(R_2-az_{2\ell}-1)} $$

(22)

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

$$ \frac{f_A(X)}{1-F_A(X)} \mbox{\ \ increases in $X$,} $$

(23)

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:

$$\begin{array}{lll} &&{\kern-6pt}\frac{(f_A(X\!+\!1)\!-\!f_A(X\!-\!1))(1\!-\!F_A(X\!+\!1))+(F_A(X\!+\!1)-F_A(X\!-\!1))f_A(X+1)}{(1-F_A(X+1))^2}\\ &&{\kern6pt}=\frac{f_A(X+1)(1-F_A(X-1))-f_A(X-1)(1-F_A(X+1))}{(1-F_A(X+1))^2}, \end{array}$$

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

$$ \frac{1-F_A(R_f-1)}{F_A(R_f-1)-F_A(R_a-1)}>\frac{1-F_A(R_f+1)}{F_A(R_f+1)-F_A(R_a+1)}. $$

(24)

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.