Localized price promotions as a quality signal in a publicly observable network

Abstract

This paper considers a quality signaling game in which consumers share information on quality with their neighbors in a social network. The firm chooses a set of consumers to whom to offer their product and a price across two stages of sales. When this publicly-observable localization is available, the cheapest way for the high quality seller to separate herself from a low quality seller is to initially offer the product to the smallest set of consumers who will saturate the network with word-of-mouth; targeting a set that generates less word-of-mouth will require more money-burning to successfully signal. The set is idiosyncratic and identifying it requires considering the connectivity of sets of nodes in the network collectively, rather than relying on individual measures of connectivity. Observable networks therefore enable credible signaling that is less socially wasteful than would otherwise be the case, and this benefit is greater the smaller is the set of consumers which will cover the network with word-of-mouth.

Appendix: Proofs

1.1 A.1 Lemma 1

Proof

First we will demonstrate that at most one high type payoff value is realized in S1 equilibria that survive the application of the Intuitive Criterion.

We require $\hat {p}$ such that a low-type firm imitating the high-type strategy completely would receive zero profit in expectation. The potential credible imitator would make γ ₁ sales at $\hat {p}$ in round 1 and (n − γ ₁−ω(Γ₁)) sales at q _H in round 2 (since neighbors to consumers in Γ₁ observe the true type, they will not buy from the imitator in round 2). $\hat {p}$ must thus satisfy:

$$ \pi_{L,imitate}=\gamma_{1}\hat{p}+(n-\gamma_{1}-\omega({\Gamma}_{1}))q_{H} $$

(8)

$$ \leq0 $$

(9)

$$ \Rightarrow \hat{p}\leq-\frac{(n-\gamma_{1}-\omega({\Gamma}_{1}))q_{H}}{\gamma_{1}} $$

(10)

The profit to the high type is thus:

$$\begin{array}{@{}rcl@{}} \pi_{H}&=&\gamma_{1} \hat{p}+(n-\gamma_{1})q_{H} \end{array} $$

(11)

All S1 equilibria that do not maximize this profit with respect to visitation γ ₁ and price $\hat {p}$ fail to survive the application of the Intuitive Criterion. By the Intuitive Criterion consumers will place zero probability on a deviation from any such equilibrium to the profit-maximizing S1 equilibrium coming from the low type firm; the deviation is profitable for the high type and will be made.

Next we will go on to characterize this unique level of profit made by the high type firm in S1 equilibrium. Consider the S1 equilibrium in which Γ₁ = Γ^∗. In that case, ω(Γ₁), the number of neighbors to consumers in Γ₁, is equal to n − γ ^∗, since by the definition of Γ^∗ all remaining consumers are neighbors of those in Γ^∗. The maximal price in that case is:

$$ \hat{p}=-\frac{(n-\gamma^{*}-\omega({\Gamma}_{1}))q_{H}}{\gamma^{*}}$$

(12)

$$ \hat{p}=-\frac{(n-\gamma^{*}-(n-\gamma^{*}))q_{H}}{\gamma^{*}} $$

(13)

$$ \hat{p}=0 $$

(14)

The profit to the high type is thus:

$$\begin{array}{@{}rcl@{}} \pi_{H,1}=(n-\gamma^{*})q_{H} \end{array} $$

(15)

Now consider any other S1 equilibrium, with generic Γ₁. This has a maximal price:

$$\begin{array}{@{}rcl@{}} \hat{p}=-\frac{(n-\gamma_{1}-\omega({\Gamma}_{1}))q_{H}}{\gamma_{1}} \end{array} $$

(16)

Which yields profit to the high type:

$$\begin{array}{@{}rcl@{}} \pi_{H,2}=-(n-\gamma_{1}-\omega({\Gamma}_{1}))q_{H}+(n-\gamma_{1})q_{H} \end{array} $$

(17)

Now comparing the two:

$$ \pi_{H,2}>\pi_{H,1} $$

(18)

$$ \text{if }\omega({\Gamma}_{1})>n-\gamma^{*} $$

(19)

But ω(Γ₁) ≤ n − γ ^∗ if γ ₁ ≥ γ ^∗ and ω(Γ₁) < n − γ ^∗ if γ ₁ < γ ^∗, so it is never true that π _{H, 2} > π _{H, 1}. The most profitable S1 equilibrium thus features Γ₁ = Γ^∗ and has profit given by:

$$\begin{array}{@{}rcl@{}} \pi_{H}=(n-\gamma^{*})q_{H} \end{array} $$

(20)

□

1.2 A.2 Lemma 2

Proof

The S1 strategy σ _{F, H} : {0, q _H , Γ^∗, C − Γ^∗} is profitable for the high type but not profitable for the low type firm, so when consumers see this strategy they must place zero probability on it coming from the low type (by the Intuitive Criterion). The high type earns payoff δ(n − γ ^∗)q _H by playing this strategy, so will prefer to deviate to it from any other strategy that yields a lower equilibrium payoff. □

1.3 A.3 Lemma 3

Proof

Assume not, so that there exists an equilibrium with some Γ₁ and Γ₂ = ∅. Consider three cases:

• 1. Take Γ₁ = C. Consider a deviation to Γ₁ = C − i, Γ₂ = i by the high type firm. Consumer i will buy since she will learn that quality is q _H ; this is profitable for the high type firm since $q_{H}>\bar {q}$, but not profitable for the low type firm, which would lose consumer i.

• 2. Take Γ₁⊂C, Γ₁ ≠ ∅. Consumers in Ω(Γ₁) learn that quality is q _H . A deviation to Γ₂ = Ω(Γ₁), p ₂ = q _H is profitable for the high type firm, but (weakly) not profitable for the low type firm since those consumers learn true quality before stage 2 and so will not buy from the low type.

• 3. Take Γ₁ = ∅. Consider a deviation to p ₁ = 0, Γ₁ ≠ ∅, Γ₂ = Ω(Γ₁), p ₂ = q _H . This is profitable for the high type firm, but (weakly) not profitable for the low type firm, which makes no profit in stage 1 and no sales in stage 2.

In each case there is a deviation to Γ₂ ≠ ∅ that is profitable for the high type but not the low type. By the Intuitive Criterion, consumers must place zero probability on that deviation coming from the low type and so will accept the deviation’s offers; this is profitable for the high type. Γ₂ = ∅ could not have been part of an equilibrium. □

1.4 A.4 Lemma 4

Proof

Assume not, so that γ ₁ > γ ^∗ in an equilibrium that survives the Intuitive Criterion. Consider two cases:

• 1. If consumers in Γ₁ choose not to buy in round 1 or if p ₁ < 0, the high type firm cannot realize a higher payoff than in the equilibria of Lemma 1. By that result, the high type can profitably deviate to an S1 equilibrium, and since the low type cannot profit from such a deviation consumers place zero probability on the deviant being of low type.

• 2. If consumers in Γ₁ choose to buy in round 1, the high type firm can deviate to Γ₁ = Γ^∗, Γ₂ = C − Γ^∗, p ₂ = q _H and p ₁ unchanged. This deviation is profitable for the high type firm since $q_{H}>\bar {q}$, but is not profitable for the low type since it reduces sales in round 1 and does not increase sales in round 2, since word-of-mouth reveals type to all in C − Γ^∗.

γ ₁ > γ ^∗ could not have been part of an equilibrium surviving the Intuitive Criterion. □