Inducing Customers to Try New Goods

Abstract

In recent years, progresses in data mining and business analytics have fostered the advent of recommender systems, behavioral advertising, and other ways of using consumer data to personalize offers and products. We investigate the incentives for sellers to invest in systems that allow the tracking of consumers and then to truthfully report whether potential buyers will enjoy yet untried products. We find that there are two types of equilibria: For some parameter values, sellers will target all potential buyers, hence their targeted ads or purchase recommendations provide no benefit to the consumer. But for other values, ads and recommendations will be accurate. In particular, the incentive for the seller to provide accurate ads and recommendations will be inversely related to the difference between the cost of producing the good and its average market evaluation.

Appendix

Proof of Proposition 1

Let $\pi _s$ be the posterior belief of the buyer about its type upon receiving the signal. There are three cases to consider:

• Case 1 $\pi _s v > c$. The seller must set the price $p_1 \le \pi _s v$ in order to induce purchase. If $p_1 > c$, then the seller would profit from signaling to both types since

  $$\begin{aligned} \pi _s (p_1 - c) < p_1 - c. \end{aligned}$$

  But then the signal is uninformative, so $\pi _s = \pi$. Since $\pi v < c$, this contradicts the premise of this case. If $p_1 \le c$, it will be profitable to signal only type $v$. The seller wants to choose the largest such price, so it picks $p_1 = c$. This is consistent with any $\pi _s$ such that $\pi _s v > c$. The seller would rationally want to signal all type $v$ consumers, since this would maximize second-period sales. Hence a rational buyer must believe $\pi _s=1$.

• Case 2 $\pi _s v < c$. Note that this includes the case $\pi _s = \pi$, in which the buyer regards the seller’s signal as uninformative. In order to induce purchase, the seller must set $p_1 \le \pi _s v$. Since $p_1 \le \pi _s v < c$,

  $$\begin{aligned} \pi _s (p_1 -c ) > p_1 -c, \end{aligned}$$

  so it is not in the seller’s interest to sell to both types. In this case the signal is completely informative, so the buyer should revise its posterior probability to $\pi _s=1$. Since $v > c$, this contradicts the premise. Hence this case cannot arise.

• Case 3 $\pi _{s}v=c$. Consider the strategy where the seller sets a price $p_{1}\le \pi _{s}v=c$ and signals both types. The signal would then be uninformative, so $\pi _{s}=\pi$ which contradicts $\pi _{s}v=c$ since $\pi v<c$. On the other hand, if the seller only signals type $v$, then the signal is informative so $\pi _{s}=1$, which implies $v=c$ , which is a contradiction. $\square$

Proof of Proposition 2

The profit maximizing strategy for the seller is to sell to both types at the highest price that they are willing to accept. This price is $\pi v$ for both types. If the seller’s strategy was to offer the good at a higher price $p_{1}>\pi v$ to all, then the expected utility of the buyer would be negative, and no purchase would result. If the seller’s strategy was to offer the good only to the $v$ type at a price $p_{1\_v}=v>\pi v$, then every buyer upon receiving the signal would accept, in which case the best response of the seller would be to offer that price to all types, thereby making the signal uninformative and the expected utility of the buyer negative once again. Hence, no price above $\pi v$ is sustainable in equilibrium. On the other hand, no price $p_{1}<$ $\pi v$ and no combination of prices $p_{1\_0},p_{1\_v}$ (with $p_{1\_0}\ne p_{1\_v}$) for the $v$ and $0$ types are profit maximizing for the seller. The linear combinations $p_{1\_0}=\frac{\pi }{1-\pi }e,p_{1\_v}=v-e$ (where $e=\left[ 0,v\right]$ and $v-e\ne \frac{\pi }{1-\pi }e$) give both types nonnegative expected utilities but guarantee a lower profit for the seller than selling at a common price $p_{1}=\pi v$. In the knife-edge case where $\pi v=c$, as before the seller makes zero profit so it is again an equilibrium to signal only type $v$, and for type $v$ to buy. $\square$

Proof of Proposition 3

We have already shown that the seller will never charge the same price to both types when $\pi v<c$. The seller will neither sell to both types at different prices. The following linear combinations of prices, $p_{1\_0}=\frac{\pi }{1-\pi }e,p_{1\_v}=v-e$ (where $e=\left[ 0,v \right]$ and $v-e\ne \frac{\pi }{1-\pi }e$, and the seller practically gives away the good to one type of customer), give the buyer a nonnegative expected utility, but guarantee the seller only a profit of $2\pi v-\pi c-c$, which is less than selling only to the $v$ type at $p_{1}=c$. Only the strategy of selling to the $v$ type at $p_{1\_v}=c$ satisfies Bayes’s Law. Let $\pi _{s}$ be the posterior belief of the buyer about its type upon receipt of the signal. When $\pi _{s}v>c$, if the seller sets $p_{1}>c$, then it could sell to both types, thereby making the signal uninformative and $\pi _{s}=\pi$, which contradicts $\pi v<c$. When $\pi _{s}v>c$ and the seller sets $p_{1}=c$ and sells to all, the same contradiction arises. Only if the seller sets $p_{1}=c$ and sells to the $v$ type and $\pi _{s}v>c$, then the signal is informative and $\pi _{s}=1$, which satisfies $\pi _{s}v>c$. $\square$

Proof of Proposition 4

Imagine that $p$ is set to $v$. To understand whether the customers can trust that the seller will only recommend a good sold at this price to those who will like it, we must compare the profit from giving good recommendations, to the profit from recommending the good to all. In the first case the present discounted value of the profit, over an infinite horizon, is $\pi \left( p-c\right) +\frac{\delta }{1-\delta } \pi \left( p-c\right)$. In the second case, if the customer’s trigger strategy is never to buy again from a merchant that recommended a bad product to him, and the supply of customers is finite, the seller’s profits is $\left( p-c\right) +\frac{\theta }{ 1-\theta }\left( p-c\right)$, where $\theta =\frac{\pi }{1+r}$ because at each period $1-\pi$ customers will abandon the seller after discovering that they have been recommended a good that they do not like. Hence, if $\pi \left( p-c\right) +\frac{\delta }{1-\delta }\pi \left( p-c\right) >\left( p-c\right) +\frac{\theta }{1-\theta }\left( p-c\right)$, the honest strategy for the seller is a best response to the customer choosing to purchase. We can rewrite the inequality as:

$$\begin{aligned} \frac{3r\pi +3\pi -2\pi ^{2}-1-r}{1+r-\pi }\left( v-c\right) >0 \end{aligned}$$

(5)

by setting $\frac{\delta }{1-\delta }=\frac{1}{r}$ and $\frac{\theta }{ 1-\theta }=\frac{\pi }{1+r-\pi }$ and $p=v$. Given that it is assumed that $v>c$, and $(1+r)>\pi$ (see denominator in Eq. 5), then this condition is satisfied simply as long as ${3r\pi +3\pi -2\pi ^{2}-1-r}>0$. If this is the case, then the honest strategy is sustainable regardless of whether $\pi v\lessgtr c$ (but as long as $v>c$).$\square$