## Abstract

The price discrimination literature typically makes the assumption of no consumer arbitrage. This assumption is increasingly violated in the digital economy, where coupons are traded with increased frequency online. In this paper, we analyze the welfare impacts of coupon trading using a modified Hotelling model where firms send coupons to poach each other’s loyal customers. The possibility of coupon trading renders this important instrument for price discrimination less effective. Moreover, coupon distribution has unintended consequences when coupon traders sell coupons back to a firm’s loyal customers. Consequently, coupon trading may reduce firms’ incentive to distribute coupons, leading to higher prices and profits. We find that, an increase in coupon distribution cost lowers promotion frequency but raises promotion depth, and an increase in the fraction of coupon traders lowers both promotion frequency and promotion depth.

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## Notes

These are only listings of coupons auctioned, and not all of them are actually sold. To get a sense of how many are sold, we searched for a specific coupon (Staples coupon), and checked the 10 listings with the earliest expiration time. We found that 6 of them had bids submitted.

This is somewhat similar to “reciprocal dumping” in the trade literature (e.g. Brander and Krugman 1983; Deltas et al. 2012). In both settings, each firm has disadvantage in one market, whether it is due to weaker preferences of consumers in that market (our case) or higher transportation cost to serve consumers in that market (the reciprocal dumping case). Firms poach each other’s strong markets, leading to lower profits for both firms and a

*prisoner’s dilemma*game.For example, some consumers (one of the authors included) may be familiar with eBay and may have various accounts already set up for transactions, so the incremental transaction cost of trading coupons on eBay is minimal.

Promotion frequency is the probability that a random consumer will receive a firm’s coupon. Promotion depth is the dollar off value when a consumer gets and uses a coupon. More details are provided in Sect. 2.

Coupons can also enable firms to reward repeat purchase customers. In particular, firms can issue coupons to consumers buying from them, and these coupons offer discounts when these consumers buy from the same firms later. See Fong and Liu (2011) for details.

In our model, firms do not have information to differentiate between coupon traders and non-traders. It is intuitive that firms would have unilateral incentive to acquire such information which would then allow them to send coupons to non-traders only.

Behavior-based price discrimination is based on past purchases rather than on observable and exogenous consumer characteristics. Results in this literature have somewhat similar flavor as those in the standard third-degree price discrimination but due to its dynamic nature there are other considerations (e.g., the use of long-term contracts). See Fudenberg and Villas-Boas (2006) for a survey of this literature.

Our analysis is more applicable to markets where firms only have crude information on consumers and thus cannot identify and target individual customers. Such limitation on information may due to technology restrictions or it may be too costly to acquire more refined information.

A similar model with \(l=p_1-p_2\) has been used in Shaffer and Zhang (2002) and Liu and Serfes (2006). We define marginal consumer by \(l=p_2-p_1\) so consumers on the left (right) like firm 1’s (2’s) product more. We do not conduct comparative statics with respect to

*L*. In our model, an increase in*L*means higher level of product differentiation and larger market size. Alternatively, one can assume that the density of consumers is \(\frac{1}{2L}\) so the total measure of consumers will be fixed at 1. In this case, an increase in*L*implies higher product differentiation only.One can think of a dynamic model where firms choose uniform price in the first period. Consumers’ purchasing decisions then reveal their preferences and our assumed information structure can be obtained after the first period. Considering such a dynamic game explicitly will complicate our analysis a great deal. For simplicity, we assume that such information structure is exogenous. This information structure is also similar to that in Bester and Petrakis (1996) and Armstrong (2006).

In our setting, firms have only crude information and can identify only two groups of consumers, not individual consumers. If a defensive coupon (say with face value

*d*) is offered, the same coupon will be offered to all customers in the firm’s own turf. Such a promotion strategy is dominated by (i) lowering both regular price and the value of the offensive coupon by*d*and then (ii) getting rid of defensive coupon. A more detailed analysis of defensive couponing is available upon request.Alternatively, coupon distribution cost may be increasing but concave in

*L*, the size of the market. Also, one can introduce a fixed cost of couponing so that firms may have an incentive not to distribute any coupon in equilibrium under certain conditions.In Sect. 5, we introduce coupon non-users, i.e., those who incur prohibitively high cost when using coupons. The results do not change qualitatively.

For tractability, we assume that \(\alpha \) is exogenous. Alternatively, one can endogenize coupon trading choices by introducing a smooth distribution of coupon trading costs among consumers. That is, everything else held the same, consumers with lower trading costs would be more willing to trade coupons. We analyze this setup in an extension, and find that the results are qualitatively the same as in our main model.

Similar to Bester and Petrakis, we model the price and promotion strategies as a simultaneous game. An alternative way of modeling is a sequential-move game where firms chooses one strategy (say price) before they choose the other strategy (say promotion strategy). However, it is unclear to us whether firms should choose price strategy or promotion strategy first. On the one hand, it is often viewed that regular price is a higher level managerial decision and is relatively slow to adjust in practice than promotions. On the other hand, we often observe that regular price changes while promotion strategy (e.g. coupon face value) is relatively stable over time.

We assume that firms do not match each other’s coupons. If coupons are matched, then firms would have no incentive to send poaching coupons unless some consumers do not request coupon-matching.

Notice that, non-traders may make different purchasing decisions depending on whether they receive coupons or not. Traders, on the other hand, will never use offensive coupons. If they receive such coupons from a firm, they will trade them away to others who were the firm’s loyal customers in the first place. Therefore, receiving coupons affects how well-off they will be but not their purchasing decisions. As such, demand functions are the same for traders whether they receive poaching coupons or not.

Alternatively, one can think of a double auction environment where all coupons are sold at once and the price is determined by a linear combination of the bid and ask prices that clears the market. In a buyer’s bid double auction, at the one extreme, the price, depending on the level of participation, will be again between zero and the coupon face value. Assuming that valuations are independent and private (IPV), the coupon distribution problem is nearly efficient (converging to efficiency at a rate of \(O(n/m^2)\) where

*m*is the number of buyers and*n*is the number of sellers. See Zacharias and Williams 2001). In an ask market when the sellers highest ask is determining the uniform price, price would be close to the coupon face value. These different forms vary in terms of the transaction prices of the coupons, but the outcomes are all the same. Poaching coupons received by traders will be traded back to the coupon issuing firms’ loyal customers—who value the coupons the highest (at their face values).Different consumers may value the same coupon differently. For example, suppose that \(p_1=p_2\) and consider the two consumers located on \([-L, 0)\) and (0,

*L*] respectively. If the consumer on \([-L, 0)\) receives a coupon from firm 2 with face value \(r<L\), she will still buy from firm 1 so her valuation of firm 2’s coupon is 0. On the other hand, the consumer located at (0,*L*] will buy from firm 2 and value this coupon at its face value*r*. Therefore, the consumer at \([-L, 0)\) will have an incentive to sell this coupon to the consumer at (0,*L*]. For our purpose, it does not matter at what price this transaction occurs, but rather that*it occurs*.For this to happen it is important that coupon traders have zero hassle cost of trading coupons. Coupon sellers in general value coupons less than face value while coupon buyers value these coupons at their face value. Since the demand of coupons from traders is higher than the supply, together with zero hassle cost, all coupons reaching traders will be traded. This means that for a seller who values the coupon arbitrarily close to the face value, the selling price must be arbitrarily close to the face value as well. We consider the case of positive hassle costs of trading coupons in Sect. 5.

This requires that there is more demand than supply for each firm’s coupons, and the consumers in the neighborhood of \(l_{c}\) will not have coupons. Intuitively this holds if distributing coupons is sufficiently costly (

*k*is large) so that \(\lambda _{ij}\) is significantly less than 1.When

*k*is sufficiently small, firms may have an incentive to deviate. See Proof of Proposition 1 in the Appendix for details.Sending no coupons is equivalent to choosing \(\lambda =0\), and can never be optimal given the quadratic coupon distribution cost and \(\alpha <\frac{1}{2}\) .

Poaching coupons, as a tool for third-degree price discrimination, only intensify competition (best-response asymmetry, Corts 1998) and hurt firms’ profits. Therefore, when an increase in \(\alpha \) or

*k*reduces firms’ couponing intensity, firms become better off. On the other hand, if firms have the ability to target consumers based either on their willingness to pay when the market is not covered or on their unit transport cost*t*which differs across consumers groups (called “choosiness” in Armstrong 2006), then couponing may actually improve firms’ profits.The qualitative results remain the same if \(L\ne 1\) is chosen.

The Maple file which contains all the expressions is available upon request. In the Maple file, we also fix the value of either \(\alpha \) or

*k*and plot the equilibrium price, promotion intensity and profit against the other parameter.More details are provided in a separate online Appendix available upon request.

Alternatively, whether coupons are transferrable or not may be a matter of specificities of the market. In this case, only symmetric configurations may be realistic, i.e., either both firms’ coupons are transferrable or both are non-transferrable.

The case where both firms choose non-transferrable coupons is equivalent to \( \alpha =0\) in our model, i.e., no coupon traders.

A companion Maple file for the proof is available for download at http://faculty-staff.ou.edu/L/Qihong.Liu-1/research.html.

There are three solutions. We pick the one that is real and positive.

If \(l_{b2}\le 0\), then our formula of \(d_{2b}\) would be exaggerated. This is because the relevant demand is capped at

*L*while our formula leads to \( d_{2b}\ge L\). Since we show that firm 2 has no incentive to deviate under the exaggerated demand function, it surely has no incentive to deviate under the correct demand function. Thus we ignore the case of \(l_{b2}<0\). Note that \(l_{b1}<0\) must hold. This is because, as the deviating firm, firm 2 must be able to sell to some of firm 1’s loyal customers, i.e., \(l_{b1}<0\).Details are available in the companion Maple file. For \(L=1\) and \(\alpha =0\) , the threshold value for

*k*is around \(k=0.159\). Technically, there is additional constraint on*k*, namely, it should not be too small so that \( \lambda ^{*}\) does not exceed 1 but this constraint is never binding.

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## Additional information

We would like to thank the Editor Giacomo Corneo and two anonymous referees whose comments allowed us to improve the paper significantly. We would also like to thank Henry Chappell, Patrick Greenlee, Beatriz Maldonado-Bird, Chun-Hui Miao, Catherine Mooney, Brian Piper, Cindy Rogers, Kostas Serfes and seminar and conference participants at University of Oklahoma, University of South Carolina, International Industrial Organization Conference (2008), CRETE (2009), Theory Workshop on Industrial Organization (2010), Hong Kong Economic Association Conference (2010) and Chinese Economists Society Conference (2011) for helpful comments and suggestions. Kosmopoulou gratefully acknowledges the Office of the Vice President for Research at the University of Oklahoma for financial support. Any opinion, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

## Appendix

### Appendix

###
*Proof of Proposition 1*

We divide this proof into two parts.^{Footnote 33} In part 1, we derive the optimal prices and couponing strategies. This is the equilibrium candidate. Then in part 2, we show that neither firm has an incentive to deviate unilaterally.

**Part 1: equilibrium candidate**

Firms’ profit functions are

Taking derivative of \(\pi _{2}\) with respect to \(p_{2}\), \(r_{2}\) and \(\lambda _{2}\) respectively, then imposing the symmetry conditions (\(p_{1}=p_{2},r_{1}=r_{2}\) and \(\lambda _{1}=\lambda _{2}\)), we can obtain

Since the cost of coupon distribution is quadratic in \(\lambda \), and the rest is roughly linear in \(\lambda \), it must be that the optimal \( \lambda _{2}>0\). Then, Eq. (7) implies,

From this expression, we can see that \(p_2>r_{2}\).

Next, we substitute the expression for \(r_{2}\) into Eq. (8) and solve for \(\lambda _{2}\). We obtain

Using \(r_{2}\) and \(\lambda _{2}\) in Eq. (9), we can solve for the equilibrium price^{Footnote 34}

where

We can then substitute \(p_{2}\) back into the expressions for \(r_{2}\) and \( \lambda _{2}\). The final expressions are too lengthy to report.

So far, we have used first-order conditions (FOCs) to solve for the optimal choices of prices and promotion intensities. However, FOCs are necessary but not sufficient. We need to make sure that the solution we obtained indeed constitutes an equilibrium. Instead of checking whether the Hessian matrix is negative semidefinite (which is quite messy), we show that neither firm has an incentive to unilaterally deviate from this pair of strategies (Bester and Petrakis use a similar method). Without loss of generality, we fix firm 1’s price and promotion strategies as given in Proposition 1, and show that firm 2 to has no incentive to deviate.

**Part 2: firm 2 has no incentive to deviate**

Note that, the demand/profit functions depend on the locations of marginal consumers and there are two cases. In the first case, \(p_{2}\ge p_{1}\) still holds and thus \(l_{a}\ge 0\). In the second case, \(p_{2}<p_{1}\). In both cases, we assume that \(l_{b1}<0\) and \(l_{b2}>0\).^{Footnote 35}

Start with case 1 where \(p_{2}\ge p_{1}\). Firm 2’s deviation profit is given by equation (6), with \(p_{1}=p^{*}\), \( r_{1}=r^{*}\) and \(\lambda _{1}=\lambda ^{*}\). We normalize \(L=1\). The optimal choice requires that

Solving the first order conditions, we obtain

The first order conditions are necessary and sufficient. Note that, we do not substitute \(p^{*}\) in these expressions as they would be too lengthy to report. Notice that, firm 2’s deviation profit depends only on \( p_{2}^{dev}\), \(\alpha \) and *k*. We want to check whether firm 2 can increase its profit by choosing \(p_{2}^{dev}\ne p^{*}\), i.e., to have

We tried various combinations of \(\alpha \) and *k*, and we found that firm 2 can never increase its profit by choosing a price different than \(p^{*}\) . Therefore, firm 2 has no incentive to deviate. We then proceeded to the case of \(l_{a}<0\) (i.e. \( p_{2}<p_{1}\)). The steps are similar and we found that firm 2 has no incentive to deviate if *k* is not too small relative to *L*.^{Footnote 36}
\(\square \)

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Kosmopoulou, G., Liu, Q. & Shuai, J. Customer poaching and coupon trading.
*J Econ* **118**, 219–238 (2016). https://doi.org/10.1007/s00712-016-0474-8

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DOI: https://doi.org/10.1007/s00712-016-0474-8