A Signaling Theory of the Online Consumer Review Policy

4.1 Separating Equilibria

We first consider separating equilibria. Since the private information of the seller is revealed with probability one in a separating equilibrium, we will denote the consumer’s belief about the seller’s type as q ^e (for expected type).

The second-period consumer updates his belief after observing whether a review is available (i.e. whether the first-period consumer bought it given the assumption that r ≥ k so that a consumer reviews whenever he buys) and after observing the signal from the review (whether s = H or L). Let λ ̂ 2 , b and λ ̂ 2 , s be the posterior beliefs respectively. We will consider two cases. If a second-period consumer updates his belief only based on s with ignoring the first observation, we will call him a naive consumer. Then, the resulting posterior belief updated by the naive consumer will be denoted simply by λ ̂ 2 . If he updates his belief twice, he will be called rational consumer.

In this section, we only consider the case that the second-period consumer is so naive that he updates his belief solely based on the review outcome without making any inference from other observations. The analysis for the case of a rational consumer will be provided in Section 5.1.

Let Π(r, q, q ^e ) be the discounted sum of the seller’s first period profit and the second period profit when the seller whose actual type is q is perceived to be a type of q ^e after it chooses the cashback amount r. Under the review regime (r ≥ k), the second-period demand is determined by the review signal s, whereas the first-period demand is determined by the quality that the first-period consumer perceives from the signal r (q ^e ). Let D ( r , q e ) = 1 − p − r + k q e be the first-period demand function. After observing s, the second-period consumer updates his posterior belief to

Note that λ ̂ 2 ( H ) > λ > λ ̂ 2 ( L ) , since 1 − α α < 1 . By using them, we can compute

where E ( q | H ) = λ ̂ 2 ( H ) H + ( 1 − λ ̂ 2 ( H ) ) L , E ( q | L ) = λ ̂ 2 ( L ) H + ( 1 − λ ̂ 2 ( L ) ) L , and M ≡ λH + (1 − λ)L are expected qualities conditional on s = H, s = L, and unconditional expected quality, respectively. It is clear that E(q|H) > M > E(q|L) because λ ̂ 2 ( H ) > λ > λ ̂ 2 ( L ) .

Equation (5) can be explained as follows. The first term is the first-period profit of a high-type seller. In the second period, there are three possibilities. If the first-period consumer purchases the good and writes a review saying “H” (or “L” respectively), the second-period consumer believes that q ^e  = H (or q ^e  = L, respectively). If he does not purchases the good so no review is available, the second-period consumer maintains his prior belief λ, i.e. q ^e  = λH + (1 − λ)L ≡ M. Equation (6) can be similarly interpreted.

Let Π*(q, q ^e ) = max _r Π(r, q, q ^e ) and r*(q, q ^e ) = arg max _r Π(r, q, q ^e ). Also, let ρ ≡ p − c be the margin. Then, it is easy to prove the following lemma from (5) and (6).

If q = H, the second-period average demand when the first-period consumer purchases and reviews is higher than the second-period demand when no review is available. This is intuitive, because the review is more likely to be good. Similarly, if q = L, the second-period average demand when a review is available is lower than the second-period demand when a review is unavailable, because the review is more likely to be bad. So, in the first case, the seller will offer a higher amount of cashback r to induce the first-period consumer to buy, whereas he will offer a lower amount of cashback to discourage the first-period consumer from purchasing the good or providing a review. It is noteworthy that if α is too large, a low type may earn a lower profit by pretending to be a high type, because being perceived as a high type increases the probability of the first-period sales and thus decreases the probability of the second-period sales due to the likelihood of negative reviews. So, there is a tradeoff between the first-period sales and the second-period sales for a low type. If α is low, the first-period effect dominates the second-period effect. In this case, the low type will want to be perceived as a high type.

If we define the first best solution of H type and L type by r ^f (H) and r ^f (L) respectively, Lemma 1 (ii) implies that r ^f (H) > r ^f (L) since r ^f (H) ≡ r*(H, H) and r ^f (L) ≡ r*(L, L).

To avoid the trivial case, we assume that:

This assumption implies that a low-quality seller prefers imitating a high-type seller’s cashback amount if the high-quality seller pays r ^f (H) for a review, rather than choosing a different cashback amount r ^f (L) which is his first-best cashback amount. This condition will be called [DC] condition.^[17] If we define α ₁ by α making (7) binding, i.e. satisfying Π(r ^f (H), L, H; α ₁) = Π(r ^f (L), L, L; α ₁), then [DC] condition implies that for any α ∈ 1 2 , α 1 , [DC] condition is satisfied. As α gets larger, a low type’s profit when he imitates a high type gets lower, because the second-period consumer is less likely to buy the product. This means that Π(r ^f (H), L, H; α) < Π(r ^f (L), L, L; α) for α > α ₁. In this case, a low type has no incentive to pretend to be a high type, so he will chooses r = r ^f (L) and [DC] condition is not satisfied. Hence, no costly signaling by a high type would be necessary.

To characterize the whole set of separating equilibria, we will assume the most pessimistic off-the-equilibrium belief λ ̂ 1 ( r ) = 0 for any off-the-equilibrium refund rate r, which gives the most severe punishment when the seller deviates from an equilibrium refund amount. The following lemma turns out to be useful for characterizing the set of all possible separating equilibria.

This lemma implies that a low type’s equilibrium refund amount will not be distorted from r ^f (L). This is clear because, if r _L  ≠ r ^f (L), then a low-type seller would prefer deviating to r ^f (L) insofar as its type is revealed in equilibrium.

The high type’s equilibrium refund amount r _H must satisfy the following two incentive compatibility conditions:

The first inequality, which is labeled as [ICL1], is the L type’s incentive compatibility condition. It requires that r _H is too costly for a low-type seller to imitate (i.e. it ensures that the low-type seller does not find it profitable to imitate the high type’s refund amount r _H .) On the other hand, [ICH1], which is the H type’s incentive compatibility condition, requires that r _H is not so costly for a high-type seller that it would prefer deviating from r _H to be perceived as a low type.

Figure 1 shows the region of r _H that satisfies both incentive compatibility conditions. In Figure 1, r ̲ L and r ̄ L are values of r that satisfy [ICL1] with equality, and r ̲ H and r ̄ H are values of r that satisfy [ICH1] with equality. Figure 1a illustrates the possibility that a high-quality good is signaled by a high cashback amount. In contrast, Figure 1b illustrates both possibilities: that a high-quality good may be signaled either by a high refund amount or by a low refund amount. If we invoke the stronger equilibrium concept of Cho and Kreps’ (1987) Intuitive Criterion (abbreviated as “C–K”), however, then high quality is signaled only by a high refund amount.

Figure 1:

Separating equilibria.

We say that a weak perfect Bayesian equilibrium (r _H , r _L ) fails to pass the C–K Intuitive Criterion if there exists a cashback amount r( ≠ r _H , r _L ) such that:

Inequality (8) implies that an off-the-equilibrium r is equilibrium-dominated for type L. Inequality (9) implies that whenever the first-period consumer believes that the refund amount r was chosen by H (for whom r is not equilibrium-dominated), then H would have an incentive to deviate to r from r _H . If there exists a refund amount (r) that satisfies these two conditions, then (r _H , r _L ) cannot pass the C–K Intuitive Criterion because H would have an incentive to deviate from the equilibrium. Applying C–K Intuitive Criterion, we obtain the following proposition.

Proposition 1 implies that a high refund amount for a review signals the seller’s high quality. Although the seller’s quality is not signaled to the second-period consumer due to his limited ability to observe, it is possible that he can get some signal from the review of the first-period consumer. The assumption that α > 1 2 makes it less costly (or more profitable) for a high-quality type to pay a consumer back for a review because it induces a higher expected profit in the second period, due to the higher probability that the review outcome is good. Therefore, a low type cannot imitate a high type’s generous payback.

Proposition 1 also says that the high-type seller does not need to use any signal that costs more than r ̄ L , which is the well-known “Riley outcome.” The Riley outcome r ̄ L is the most efficient separating equilibrium because it incurs the least signaling cost. To see why applying the Intuitive Criterion can eliminate all the separating equilibria except the most efficient one ( r ̄ L ) for any equilibrium refund amount of the high type, r H ∈ ( r ̄ L , r ̄ H ] , consider the following off-the-equilibrium message r ′ ∈ ( r ̄ L , r H ) as depicted in Figure 2. In Figure 2, it is easy to see that Π ( r f ( L ) , L , L ) = Π ( r ̄ L , L , H ) > Π ( r ′ , L , L ) , Π ( r ′ , L , H ) , which implies that r′ is equilibrium-dominated for low-quality types. Once the consumer observes r′ and therefore eliminates the possibility that the seller is a low type, it follows that Π(r _H , H, H) < Π(r′, H, H), since r f ( H ) < r ̄ L < r ′ < r H . Therefore, C–K Intuitive Criterion requires that λ ̂ 1 ( r ′ ) = 1 . This result implies that the equilibrium rate for a high type, r H ∈ ( r ̄ L , r ̄ H ] , fails to pass the Intuitive Criterion.

Figure 2:

C–K Intuitive Criterion.

Also, the Nelson effect is confirmed by this result. Since a higher amount of reward implies a lower effective price, this proposition suggests that a high-quality seller chooses a lower price, because it can generate more repeat purchases.

This result has some implication on social welfare. Although it is well known that a higher level of education than the Riley outcome in the job market signaling model of Spence (1973) is socially inefficient, it is not the case in our model, because a refund, which is a monetary transfer between a seller and a consumer, is not socially wasteful although it may be wasteful to the seller. If we assume that c = 0, as a high-type seller pays a larger refund amount to the first-period consumer, the effective price gets lower and this increases the first-period sale, and as a result, the second-period sale is also likely to be increased. This clearly improves social efficiency.

4.2 Pooling Equilibria

We now consider pooling equilibria. Let r _λ be a pooling equilibrium cashback amount. Under the most pessimistic belief, r _λ must satisfy the following two incentive compatibility conditions:

The first inequality is the L type’s incentive compatibility condition which requires that an L type prefers being pooled by choosing r _λ rather than giving up looking better by deviating to r ^f (L). That is, r _λ should not be too far from r ^f (L). On the other hand, the second inequality is the H type’s incentive compatibility condition requiring that an H type also prefers being pooled by choosing r _λ rather than deviating from it and being perceived to be a low type, implying that r _λ should not be too far from r*(H, L).

The set of r _λ satisfying the two inequalities, [ICL2] and [ICH2], is illustrated in Figure 3, as [ r ̲ L λ , r ̄ L λ ] and [ r ̲ H λ , r ̄ H λ ] , respectively. Let E p = [ r ̲ L λ , r ̄ L λ ] ∩ [ r ̲ H λ , r ̄ H λ ] be the set of pooling equilibrium cashback amounts. It is characterized in the following proposition.

Figure 3:

Pooling equilibria.

In pooling equilibria described above, the cashback amount cannot convey any information about the quality of the product and thus the review outcome is the only source from which consumers can get information about the quality.

We may resort to a more refined equilibrium concept such as Intuitive Criterion just as in the case of separating equilibria, if our equilibrium concept wPBE based on the most pessimistic belief appears to be too weak. However, it is not difficult to see that C–K Intuitive Criterion does not eliminate all the pooling equilibria, although it can eliminate some.

For example, in Figure 3, r _λ  = r ^f (L) can be eliminated by CK Intuitive Criterion. If one takes an off-the-equilibrium message r ″ = r ̂ + ϵ , it clearly satisfies (CK-i), because Π(r ^f (L), L, M) > Π(r″, L, q ^e ) for any q ^e  = H, L. It is also clear that (CK-ii) is satisfied because Π(r ^f (L), H, M) < Π(r″, H, H). The intuitive reason why some pooling equilibria can be eliminated by Intuitive Criterion is that there exists a deviant message that cannot be profitable to a low-type seller but increases only the profit of a high-type seller.

On the other hand, Figure 3 also shows the possibility of a pooling equilibrium that survives the C–K Intuitive Criterion. Eliminating an equilibrium message r λ = r ̄ L λ requires that there exist an off-the-equilibrium message r′ that is equilibrium dominated for an L type, which would be possible if r ′ > r ̄ L . However, for any r ′ > r ̄ L , Π ( r ̄ L λ , H , M ) > Π ( r ′ , H , H ) . This means that none of such an off-the-equilibrium message r ′ > r ̄ L satisfies (CK-ii), implying that the original equilibrium passes the Intuitive Criterion. Intuitively, an equilibrium passes the Intuitive Criterion if there exists no deviant message that is not profitable to a low type but profitable only to a high type. When we consider the pooling equilibrium message r ̄ L λ in Figure 3, a deviation of a low type cannot be profitable if the deviant refund rate is too large, i.e. r ≥ r ̄ L , but those messages are not profitable to a high-type seller, either.

5.1 Rational Consumer

While a naive second-period consumer updates his posterior belief only based on the review outcome, a consumer could be more rational by using more information. Since a high-quality product is sold more than a low-quality product and a consumer who purchases always submits a review in equilibrium, observing a review by a previous consumer can signal that it is more likely to be a high quality.

If the second-period consumer can utilize this information in addition to the review outcome, he will update his posterior belief twice. Let y ∈ Y = {0, 1} indicate the purchasing decision of the first-period consumer (or availability of a review by the first-period consumer) where y = 1 indicates “a review is available” and y = 0 indicates “a review is not available”. Also, let λ ̂ 2 , b ( y ) be the posterior belief of the second-period consumer updated from the prior belief λ based on his observation of y, and λ ̂ 2 , s ( s ) be his posterior belief updated from λ ̂ 2 , b ( y ) based on his observation of s. Note that λ ̂ 2 , b ( y ) is not updated from λ ̂ 1 which was updated by the first-period consumer based on his observation of r, because the second-period consumer is assumed that he cannot observe r. Then, the posterior beliefs are calculated as follows: If a review is available,

and if a review is not available,

It is easy to see that λ ̂ 2 , b ( 1 ) > λ > λ ̂ 2 , b ( 0 ) because D(r _H , H) > D(r _L , L), assuming that r _H  > r _L which will be shortly proved.

If a signal s is observed from a review, he updates the posterior belief one more time to

This is mainly because λ ̂ 2 , s ( H ) > λ ̂ 2 , b ( 0 ) > λ ̂ 2 , s ( L ) if α is accurate enough. The intuition goes as follows. It is obvious that the posterior belief that q = H is higher when the second-period rational consumer receives the review report s = H than when he receives no report. On the other hand, if the consumer receives the review s = L, it is both better news and worse news than when he receives no review. First, the information that the first-period consumer purchased the good and wrote a review updates the belief upwards to λ ̂ 2 , b ( 1 ) ( > λ ̂ 2 ( 0 ) ) . Second, the information that s = L updates the belief downwards to λ ̂ 2 , s ( L ) ( < λ ̂ 2 , b ( 1 ) ) . Therefore, it is unclear whether λ ̂ 2 , s ( L ) > λ ̂ 2 , b ( 0 ) or λ ̂ 2 , s ( L ) < λ ̂ 2 , b ( 0 ) . However, if the signal s is accurate enough, the second effect dominates the first effect; hence, λ ̂ 2 , s ( L ) < λ ̂ 2 , b ( 0 ) < λ ̂ 2 , s ( H ) . The counterpart of Proposition 1 for a rational consumer follows directly from this lemma.

Since Lemma 1 and Lemma 2 still hold with a minor modification of replacing λ ₂(s) by λ ̂ 2 , s ( s ) , it is straightforward that Proposition 1 remains unaffected qualitatively in the case of a rational consumer. The counterpart of Proposition 2 for a rational consumer can be similarly derived from Lemma 1, so we omit it.

5.2 Endogenizing Price

To highlight the role of the refund rate as a quality signal, we have, so far, assumed that price is exogenously given. In this section, we assume that in the first period, the seller chooses the price that will prevail in both periods together with the refund rate.

In this modified game, there may be a separating equilibrium in which the seller uses a separating price strategy as well as a separating refund policy. In the equilibrium, however, the review of the first-period consumer does not convey better information to the second-period consumer than the separating price, because the second-period consumer can infer the true quality of the product from observing the price. Naturally, it is our main interest to see whether there is a separating equilibrium in which the type of a seller is revealed only by the refund rate in the first period. Since our previous analysis was based on the implicit assumption that a pooling equilibrium price exists, the analysis of this section can provide a justification for it.

We can rewrite the profit function of a high-type seller by using Equation (5) as

where D = 1 − p + k − r q e and Q H = D α E ( q | H ) + 1 − α E ( q | L ) + 1 − D M . Here, D is the first-period demand, Q _H is, roughly speaking, the inverse of expected quality given the information available in the second period when the true quality is H, and thus 1 − pQ _H is the second-period demand. We can denote the price and the refund rate maximizing (16) by p*(H, q ^e ) and r*(H, q ^e ).

On the other hand, the profit function of a low-type seller can be rewritten by using Equation (6) as

where Q L = D 1 − α E ( q | H ) + α E ( q | L ) + 1 − D M . We can denote the price and the refund rate maximizing (17) p*(L, q ^e ) and r*(L, q ^e ).

By abusing notation, let (p _λ , r _H ) and (p _λ , r _L ) be the separating strategy pairs of types H and L. Under the most pessimistic belief, the incentive compatibility conditions are as follows.

There can be many equilibrium strategies that satisfy both (18) and (19). Figure 4 illustrates the set of the equilibrium refund rates of the high type r _H with the pooling price p _λ . In the blue region, the low type cannot imitate the high type; i.e. the high-type equilibrium pairs in the blue region satisfy (19). Similarly, in the green circular region, the high type cannot deviate from its equilibrium strategies. The set of (r, p) indicated in red, which is the intersection of the blue region and the green region that satisfy both incentive compatibility conditions, is the high-type’s separating equilibrium pairs of the price and refund rate. The intersection of the red set and the horizontal line at p = p _λ are the set of separating equilibria (p _λ , r _H ) we are interested in.

Figure 4:

Separating equilibria with a pooling price and separating refund rates.

We can also show that point B is the only separating equilibrium that passes the Intuitive Criterion. For example, consider point B′ inside the equilibrium set. Take any point B″ between B and B′. Then, it is not difficult to see that it is equilibrium dominated for a low type. Note that the iso-profit curve (of true type L and perceived type H) passing through its equilibrium point A = (p _λ , r _L ) yields a higher profit than the iso-profit curve possing through B″. Then, it is obvious that if the low type is perceived to be a low type, its profit is even lower when it chooses point B″. On the other hand, it is clear from the figure that the iso-profit curve of the high type passing through point B″ yields a higher profit than point B′. Therefore, such a deviation must be believed to come from a high-type seller, implying that the original equilibrium point B′ fails to pass the Intuitive Criterion. Similar arguments can show that point B is the unique separating equilibrium that survives the Intuitive Criterion.

Although our main interest is in separating equilibria, there can be also pooling equilibria. In a pooling equilibrium, a low type successfully imitates a high type, while the high type gives up preventing the low type from imitating itself because it is too costly. Thus, we will consider the undistorted price and refund rate of the high-type seller when he is perceived to be M = λH + (1 − λ)L.

Under incomplete information, both types of seller use the same price and refund rate. Let (p _λ , r _λ ) be the pooling equilibrium pairs of the price and refund rates of types H and L where p _λ  = p*(H, M) and r _λ  = r*(H, M). Under the most pessimistic belief, i.e. λ ̂ ( p , r ) = 0 for any (p, r) ≠ (p _λ , r _λ ), it must satisfy the following incentive compatibility conditions.

Both inequalities require that neither type of seller has an incentive to deviate from the pooling price and refund rate.

It is clear that Inequality (20) is always satisfied, that is, the high-type seller has no incentive to deviate from (p _λ , r _λ ), because it is the optimal strategy of a high type when q ^e  = M and any deviation from it simply leads into the belief that the deviant seller is a low quality type.

Let us consider the incentive compatibility of the low type. We start from the case that δ = 0. Then, it is also clear that (21) is trivially satisfied, because (p*(H, M), r*(H, M)) maximizes Π ( p , r , H , M ) = Π ( p , r , L , M ) = ( p − c − r ) 1 − p + k − r M . In fact, any pair (p, r) such that p − r = p ̃ = M − k + c 2 is optimal. Now, we increase δ from zero. If δ > 0, the optimal pair of (p, r) is no longer indeterminate. Now, Π of either type depends both on p and r, not a function of the effective price p ̃ solely. We will assume that the optimal pair of (p, r) is unique for any (q, q ^e ) when δ > 0. If δ > 0, we cannot generally compare the relative size of Π(p, r, L, M) and Π(p, r, L, L), because the first-period profit of a low type is clearly higher when the perceived quality is M but the second-period profit of the low-type seller is higher when the perceived quality is L. Intuitively, this is because if the first-period sales volume is high due to its high perceived quality, its second-period demand can be low due to bad reviews. However, if δ is very low, the first-period effect outweighs the second-period effect, implying that Π(p, r, L, M) > Π(p, r, L, L). Also, since Π(p, r, L, M) is continuous with respect to δ, there exists δ ̄ ( > 0 ) such that for any δ ≤ δ ̄ , Π(p*(H, M), r*(H, M), L, M) > Π(p*(L, L), r*(L, L), L, L).

5.3 Conditional Cashback Policy

In this section, we consider an extended model in which the seller can choose a conditional cashback policy of paying only a consumer who writes a positive review on its product.

Under the conditional cashback policy, the first-period consumer’s decisions are significantly affected. Although we argued that a consumer never makes a dishonest review under the unconditional cashback policy due to a positive lying cost, it is possible that if he can get a refund only for a positive review, he may choose “H” message in his review after receiving a signal s = L even with the lying cost. We denote the lying cost by η( > 0).

Consider the first-period consumer’s reviewing decision given a conditional refund policy. It is clear that if he receives s = H, he has no reason to lie so he will review honestly if the benefit exceeds the cost, i.e. r ≥ k. If he receives s = L, he can get paid only when he lies, so he will review by choosing a dishonest message “H” to get a positive amount of cashback. Thus, he will review if r ≥ k + η. The reviewing decision of the first-period consumer given a conditional refund rate can be summarized as follows; (i) if r ≥ k + η, a consumer who purchases reviews for any s = H or L, (ii) if k ≤ r < k + η, only the consumer who receives s = H after purchasing reviews, and (iii) if r < k, a consumer who purchases never reviews for any s = H or L. In the first case that r ≥ k + η, the review message conveys no useful information as in a babbling equilibrium in a cheap talk game, although the review message is not cheap talk in the sense that it has a negative cost of r and the cost differs across messages, i.e. it is payoff-relevant.

Now, we consider the seller’s policy choice. The main advantage of using a conditional refund policy is to induce more positive reviews. It is clear that the advantage is larger for the low-type sellers than for the high-type seller because the low-quality product is more likely to generate a low signal s = L. Thus, it is our first guess that an unconditional refund policy signals high product quality while a conditional refund policy signals low quality. In other words, our main interest is whether there exists a separating equilibrium in which a high-quality seller chooses an unconditional refund policy and a low-quality seller chooses a conditional refund policy in equilibrium.

If a high-type seller can signal its high quality by using an unconditional refund policy itself (rather than a conditional policy), it does not need to involve a costly distortion in its price and refund rate. So, suppose that a high-quality seller charges p ^f (H) and offers r ^f (H) unconditionally while a low-quality seller charges p ^f (L) and offers r ^f (L) conditionally in equilibrium. In this equilibrium, the low-type seller’s total profit is

If the low-type seller deviates by imitating a high-type seller, its profit is

Intuitively, if a low type imitates a high type by using the unconditional refund policy with (p ^f (H), r ^f (H)), it can increase the first-period demand by being perceived to be a high type. So, if δ = 0, the low type will always imitate the high type. As δ gets larger, the cost of imitation gets larger in the second period, because a low message is more costly when the first-period sales volume is larger. The following proposition captures this intuition.

The proof of this proposition in the Appendix shows that it is more difficult for a low type to imitate a high type as the signal gets more accurate (i.e. α gets larger), δ gets larger, and the quality difference gets larger (i.e. H − L gets larger). For example, (38) shows that as δ gets larger, the imitation’s positive effect of increasing the first-period sales is more likely to be outweighed by the second-period negative effect. It is also clear that as the signal gets more accurate, the second-period negative effect gets larger. If the quality difference gets larger, the low type’s second-period negative effect gets much larger.

In this equilibrium, the first-period consumer infers the quality by using the signal of the refund policy and the second-period consumer infers the quality by using the signal of the first-period consumer’s message. Note that a negative review may be more helpful to a seller than a positive review in this separating equilibrium, because it is a signal that the seller is a high type.