Bundling Customers: How to Exploit Trust Among Customers to Maximize Seller Profit
Abstract
We consider an auction of identical digital goods to customers whose valuations are drawn independently from known distributions. Myerson’s classic result identifies the truthful mechanism that maximizes the seller’s expected profit.
Under the assumption that in small groups customers can learn each others’ valuations, we show how Myerson’s result can be improved to yield a higher payoff to the seller using a mechanism that offers groups of customers to buy bundles of items.
1 Introduction
1.1 Second version note
After posting the first version of this paper we learned that much of its mathematical content already appears in the literature, for example in an article of Armstrong [3], although in a slightly different context of bundling products, rather than customers.
1.2 Bundling items
Bundling is the practice of joining together a number of products into a “bundle”, so that customers may not buy each product separately, but must choose to either buy the entire bundle or have non of the included items. Alternatively, customers may be allowed to purchase a single item, but at a higher cost; that is, the price of the bundle is set to below the sum of the prices of the individual items that comprise it. Examples range from McDonald’s happy meals to enormous defense contracts [1] (see also the recent attention to bundling of scientific journal subscriptions [8]). Bundling has also received much attention from theorists (cf. [2, 11, 9] and many more).
However, consider a population of consumers who are potential customers for some mass produced product (i.e., the number of available items is unlimited). Assume also that customers generally have no need for more than one item. For example, the product might be an upgrade to an operating system, a cellphone data package or removal of tax offenses record. This class of products is sometimes referred to as digital goods.
Since each customer has no need for more than one item, bundling items
does not seem to offer an advantage to the seller. Indeed,
Myerson [12] shows that in a Bayesian setting the
best strategy available to the seller is to offer a fixed peritem
price
1.3 Bundling customers
We consider a different kind of bundling, which, although also widespread, seems (to our knowledge) to have been largely overlooked by theorists. We propose that the seller may increase its profit beyond Myerson’s bound by bundling customers: here customers are arbitrarily grouped into pairs and are offered to buy two items for a price that is lower than the sum of the prices of the individual items. The same can of course be done for larger groups of customers, so that a group of customers are jointly offered to buy items for a discount.
Our key assumption is what we call group rationality: namely that a bundle of customers will accept the group offer if there is a way for them to share the cost so that all of them benefit. For example, consider two customers who are each interested in buying a copy of a book whose (single item) price is set to Gold Dinars. Let customer be willing to pay at most Dinars, and let customer be willing to pay at most Dinars. Let the cost of a single book be set to Dinars. Group rationality implies that if and were offered to jointly buy two books for Dinars then they would accept and find a way to split the cost, since both can benefit; customer can contribute Dinars and customer can contribute Dinars, and then has paid two Dinars less than she would have paid on her own, and was able to buy the book, which he wouldn’t been able to do on his own. Note that assuming that the cost of printing a book is small, then the seller is also strictly better off.
Our group rationality assumption is novel in the context of Myerson auctions, and is in fact what allows us to increase the seller’s profit past Myerson’s bound on truthful auctions. We note that indeed there is no truthful mechanism for two customers to agree on a division of costs when a feasible one exists; this is nothing but the well known “splitting the dollar” game. In the example above, if customer manages to convince that he is not willing to pay more than Dinars, then might settle for paying herself, which still leaves her better off than buying a single book for Dinars.
However, we argue that it is important to consider group rationality; it is in fact a phenomenon that, in other contexts, has been widely studied theoretically and experimentally, and falls under the general titles of cooperation and altruism (cf. [4, 13, 6, 14]).
Specifically, families and tribes are often group rational (for obvious evolutionary reasons, cf. [10]), as are other groups of people who expect to have to rely on each other in the future. A further argument to support group rationality in our setting is the observation that when the stakes (i.e., the savings) are high, one could expect that in any small group people would be sufficiently incentivized to find a way to compromise, trust and share, even if there is a danger of being shortchanged; in reality, prisoners do sometimes choose to “cooperate” even when facing the risk of “defection” by cellmates, and the tragedy of the commons can be averted (cf. [7]).
1.4 Results
We consider a Bayesian setting with independent customer valuation distributions and group rational customers. Our main result is that under mild smoothness conditions of the customers’ valuation distributions, the seller can expect a strictly higher profit when bundling customers into pairs, as compared to selling single items.
We also show that when valuations are uniformly bounded then, as the size of the bundle increases, the seller’s expected profit from the customers approaches the sum of their expected valuations for the product, which is an upper bound on the seller’s profit. This bound is achieved in single customer auctions only when the customer reveals its valuation to the seller.
Approaching this limit by bundling ever larger groups of customers would require ever more trust among them. Note that assuming group rationality for larger groups is a stronger assumption than group rationality for smaller groups. Indeed, as the size of the group grows, the believability of group rationality diminishes; all else being equal, it seems harder to expect honesty and trust among a hundred people than among a couple.
Our results can therefore be interpreted to show that the seller can
exploit trust among customers
2 Model
Let be the set of customers. Each customer
has a private valuation , which is the maximum price that it
would be willing to pay for the product. These valuations are not
known to the seller, who however has some knowledge of what they might
be. We model the seller’s uncertainty by assuming that each valuation
is picked independently
We make a number of mild smoothness conditions on the distributions of
valuations: We assume that is nonatomic and differentiable with
bounded density (PDF) . We assume all valuations are in
for some , so that is zero outside this interval for
all . We further assume that for some it holds for all
that in the interval
Let be an auction mechanism or sales strategy. We assume that it can result in each of the customers either receiving or not receiving an item, and parting with some sum of money. In the context of , we denote by the event that customer receives an item. We denote by the price, or the amount of money customer paid the seller for the item. We denote customer ’s utility by , where
(1) 
and denote the customer’s expected utility by .
Let denote the seller’s utility from selling to customer . We assume that the cost of an item is zero, and so define
(2) 
We denote the seller’s expected utility by . We denote the seller’s total expected utility by .
We assume throughout that given a seller’s strategy, the customer will pick a strategy that will maximize its expected utility. Given that, a seller will pick a strategy that will maximize its own total expected utility. We largely ignore the possibility of ties (i.e., two strategies that result in the same expected utility, for either the customer or the seller), since, as we assume the distribution of the valuations is nonatomic, it will be the case for the strategies that we consider that ties will occur with probability zero.
2.1 Sales strategies
Single customer one time offer
We assume that the seller wishes to maximize the sum of the expected revenues it extracts from the customers. A possible strategy would be to give each customer a one time offer to buy the product at price . Myerson [12] shows that this sales strategy, of all the truthful strategies, maximizes the profit of the seller, for the appropriate choice of .
The customer’s utility in this case is . Therefore, assuming the customer wishes to maximize its utility, it would buy iff (or equivalently , since ). Hence , the gain by the seller is , and the seller’s expected utility is
(3) 
with
(4) 
If we assume that is nonatomic, differentiable and only supported on , then is continuous and differentiable and must have a maximum in . By solving we can show that any which maximizes satisfies
(5) 
Furthermore, under these assumptions , whereas clearly is positive for some . Hence this maximum does not occur at or .
Bundling customers
We next consider the strategy of bundling the customers. Let be a set of customers. The bundling strategy here is parametrized by a vector of single item prices and the bundle price .
The customers are given the option to buy a bundle of items (i.e., each gets an item) for the total price of . Additionally, each customer may buy a single item for the price of .
We assume group rationality, so that the customers choose to buy the bundle if the cost can be shared in a way that is profitable for all. That is, the customers buy the bundle if there exist such that the following holds:

.

for all . That is, each customer’s utility for buying the bundle is positive, or better than the utility for not buying.

for all . That is, each customer’s utility for buying the bundle is better than the utility for buying individually.
Hence, we assume that if the cost of the bundle can be shared in a way that, for each customer, improves the utility over the other alternatives, then the customers will find a way to share the cost and will choose to buy the bundle. When this is not the case then each customer , independently, decides to either buy or to buy, depending on whether , as in the single customer case. Formally, iff the condition above holds or . Note that “” can be replaced by “” throughout, since ties occur with probability zero.
Note that when the conditions above apply  i.e., accepting the bundle is group rational for some prices  then accepting the offer and paying is a Nash Equilibrium: it is better for customer to accept the offer for rather than shop alone, since then it would have to pay more.
3 Results
3.1 Smoothness and boundedness conditions
We make the following assumptions on the distribution of customer valuations . Recall that we denote by and the CDF and PDF of the distribution of .

Customers valuations are independent and nonatomic.

There exists such that, for all , is in .

The distribution of has a density (PDF) .

There exists such that, for all , for .
3.2 Theorem statements
In the statement of the following theorem we mark quantities related to the single customer strategy by , and quantities related to the bundling strategy by . E.g., is the seller’s utility from customer using the single customer strategy, and is the seller’s expected utility from customer using the pair bundling strategy.
Theorem 3.1.
Let be a pair of customers with valuation distributions satisfying the smoothness and boundedness conditions in 3.1. Let
be the seller’s total expected utility when using the single customer strategy with prices and . Let
be the seller’s total expected utility when using the pair bundling strategy with prices , and . Then
(6) 
That is, the best bundling strategy is strictly better than the best single customer strategy.
The next theorem shows that when valuations are bounded then, as the size of the bundle grows, the expected utility of the seller from the customers approaches the sum of their expected valuations.
Theorem 3.2.
Consider a set of customers with valuation distributions satisfying the smoothness and boundedness conditions in 3.1.
Let be customer ’s expected valuation, and let be the sum of the customers’ expected valuations. Let
be the seller’s total expected utility when bundling all customers with prices and .
Then the seller’s total expected utility satisfies
(7) 
Note that since a customer will never pay more than its valuation then
4 Proofs
Proof of Theorem 3.1.
Let be the prices that maximize the seller’s total expected utility for the single customer strategy. We will prove the theorem by showing that there exists such that
(8) 
Note that since is optimal for the single customer strategy, then
Hence there exist a constant such that for all small enough it holds that
(9) 
Let denote the event that the customers buy the bundle. Recall that in the bundling strategy with prices occurs if and only if there exist and such that
(10)  
Using Eq. (10) we can substitute and arrive at the following equivalent condition: occurs if and only if there exists a such that
In this form it is apparent that occurs if and only if
We now partition our probability space into the disjoint events (see Fig. 1), where the in the superscripts denotes the fact that these events depend on . We compare and in each event.

Let be the event that . Then in , in the single customer strategy both customers buy an item for a total of , and in the bundling strategy the customers buy the bundle for . Hence
and
(11) 
Let be the event that . In this region the bundle is not bought, and neither does customer 2 buy an item on their own, in either strategies. Hence in this region customer 1 buys the item iff in the bundling strategy. Since and are independent then the (expected) utility for the seller in the bundling strategy is identical to what it would be when offering a single item to customer 1 for . Since, by Eq. (9), this expected utility is maximized when the price is (as is done in the single item strategy), then
and
(12) 
Let be the event that . In this region the bundle is not bought, and neither does customer 1 buy an item on their own, in either strategies. Customer 2, however, buys in both strategies. Therefore the seller’s utility is identical in this region:
and
(13) 
Let be the event that . In this case we note that
and
Now, since we assumed that the distribution of is nonatomic and since both and are bounded then there exists a constant such that for small enough it holds that , and so there exists a constant such that
(14) for small enough.

Finally, let be the event that . Here in the single customer strategy customer 1 buys an item for and customer 2 does not buy. In the bundling strategy the customers purchase a bundle for . Hence
and
Since the distribution of is supported on , and since (see note at the end of Section 2.1.1), then there exists a constant such that for small enough . Hence
(15) for small enough.
Since the events are disjoint and since then
with a similar expression for . Therefore, as a conclusion of Eqs. (11), (12), (13), (14) and (15) we have that for small enough
and therefore for small enough
∎
Proof of Theorem 3.2.
Consider the bundling strategy with individual prices for all (i.e., no single item sales) and . In this case the customers will either buy the bundle if its cost is less than the sum of their valuations, and buy nothing at all otherwise. Denote the sum of their valuations by .
Since the customers buy the bundle when then the seller’s expected utility equals and it holds that
Since in the interval then and , and so it holds that
where the second inequality follows from the fact that .
Since the optimal strategy yields at least as much utility to the seller as this one, then
∎
5 Conclusion
We showed how sellers may maximize profits by offering bundles of items to rational groups of customers. Our work suggest a number of future research directions we wish to mention.
5.1 Optimal auctions and optimal profit
Our results show that it suffices to bundle pairs of customers to increase profits under mild conditions, and that if the customers are bundled in large groups it is possible to extract profit which approaches the theoretical bound, as the group size increase. For example  assume that there are individuals  are paired into rational groups, another are partitioned into rational groups of size , and the rest are partitioned into rational groups of size . Assuming that all valuations are drawn i.i.d. from the same distribution  what is the optimal auction and by how much is it better than the single item auction?
5.2 Overlapping Rational Groups and Social Networks
The problem presented in the previous subsection can be generalized further to a situation where an individual may belong to more than one rational group: for example an individual may belong to a family and to a small startup company. The two groups are rational and are offered different bundles. Understanding the optimal auction in this setup and its relationship to the social network structure is, in our opinion, an interesting open problem.
Footnotes
 Weizmann Institute and U.C. Berkeley. Email: mossel@stat.berkeley.edu. Supported by a Sloan fellowship in Mathematics, by BSF grant 2004105, by NSF Career Award (DMS 054829) by ONR award N000140710506 and by ISF grant 1300/08
 Weizmann Institute. Supported by ISF grant 1300/08. Omer Tamuz is a recipient of the Google Europe Fellowship in Social Computing, and this research is supported in part by this Google Fellowship.
 Different fixed prices may be offered to different customers in a practice called price differentiation.
 Note that the customers are not required to trust the seller!
 Despite some recent progress [15], it seems that Myerson auctions are generally difficult to analyze when valuations are not independent. We conjecture that our results hold also for the case of correlated valuations.
 These assumptions can be significantly relaxed at the price of a significantly more technical and difficult to read paper.
 We use the following version of Bernstein’s inequality: Let be independent random variables such that and for all . Then for any it holds that
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