Single sourcing versus multiple sourcing.

We show that in contrast to results in the extant literature, single sourcing may not be the optimal strategy of a buyer facing suppliers with strictly convex costs. As we argue, previous findings relied crucially on the joint assumption that, first, there is only a single buyer and that, second, procurement takes place in an auction organized by the buyer. Relaxing these restrictions, we obtain a richer set of results. In particular, we show that even in the original setting, where suppliers bid, committing to single sourcing is only optimal if the respective buyer controls a sufficiently large fraction of the whole procurement market.

1. Introduction

* Over the last two decades, a growing literature in economics has emerged that studies, both theoretically and empirically, optimal procurement practices. This article is primarily concerned with a particular element of a firm's or public agency's procurement strategy: the choice between single and multiple sourcing.

One of the early seminal contributions to this literature is that by Anton and Yao (1989), who derive the following simple but powerful result: under complete information and if suppliers have strictly convex costs, a single buyer that conducts an auction to procure a fixed volume is strictly better off when committing to single sourcing. (1) Anton and Yao (1989), as well as a number of subsequent papers, then go on to develop arguments for why multiple sourcing (or, likewise, split-award contracts) could still be beneficial, for instance as this encourages more bidder participation or ensures more competition in the long run. (2)

This article revisits the original idea of Anton and Yao (1989) in order to derive a richer set of results. Our main result is the following. We show that if there is more than one buyer, which seems to be characteristic of many though clearly not all settings, then single sourcing may no longer be optimal. In fact, if all buyers demand the same volume, it is now most profitable for each buyer to spread his purchases evenly over suppliers. Single sourcing remains, however, optimal for a buyer who controls a sufficiently large share of the respective procurement market.

That several buyers procure at the same market may often be more realistic than assuming that there is only a single buyer. The key distinction between the two cases is the following. If there are more buyers and if one buyer makes more or even all of his purchases at one supplier, then in equilibrium the whole pattern of purchases and supplies will readjust. As a consequence, although one supplier will then account for most or all of the sales to the particular buyer, other suppliers will sell more to different buyers. In contrast, such a reshuffling of purchases and sales is simply not possible if the buyer is a monopsonist. As we bring out in detail below, this simple difference has profound implications for the optimality of single sourcing.

We also extend the analysis by considering the case where it is now suppliers who run an auction to sell their capacity. In marked contrast to the results obtained if suppliers bid at auctions that are run by buyers, single sourcing is now only optimal if a buyer is sufficiently small. What drives this stark reversal of results is the insight that the (strategic) choice of single sourcing plays an entirely different role in the two scenarios. If suppliers bid and have thus all "contracting power," single sourcing is chosen if it enhances the value of a buyer's outside option, that is, the value of the buyer' second-best alternative, namely to procure instead more from other suppliers. In contrast, if buyers bid, then single sourcing is chosen if it erodes the value of suppliers' outside option.

Overall, we thus obtain a richer set of results, relating the optimality of single versus multiple sourcing to both the size of the respective buyer, relative to that of the total procurement market, and to how procurement is organized. This may help to shed more light on observed variations in the organization of procurement. As documented in Tunca and Wu (2005), companies such as Sun or HP that use online auctions to procure products worth hundreds of millions of dollars frequently opt for multiple sourcing. Moreover, recent studies in operations research and logistics document cases of both single sourcing and multiple sourcing when firms procure via web-based auctions and business-to-business (B2B) platforms (see, e.g., Elmaghraby, 2000). Below we also comment on the implications of our findings for the optimal procurement strategy of public agencies, which are often special in that they account for a substantial fraction of the respective procurement market (e.g., in health care, road construction, or defense).

The rest of this article is organized as follows. Section 2 introduces the basic model where suppliers post bids. Section 3 analyzes two benchmark cases, namely that of a single buyer and that of two symmetric buyers, whereas Section 4 allows for buyers of different size. Section 5 discusses and further extends the results when suppliers bid. In Section 6, we analyze the opposite case where buyers now post bids. Section 7 concludes.

2. The model

* The basic model considers a setting with two symmetric suppliers, indexed by m = S1, S2. Suppliers produce a homogeneous good with a strictly convex and twice continuously differentiable cost function C(x), where C(0) = 0. At the downstream level, there are at most two buyers, n = B1, B2. In our basic model, we further specify that each buyer wants to purchase a fixed volume [X.sub.n]. The total size of the procurement market is thus given by X := [[summation of].sub.n=B1,B2] [X.sub.n].

Buyers' demands and suppliers' costs are common knowledge. We show how even the most simple framework, namely that of a one-shot interaction under complete information, can already give rise to a rich set of predictions. Note also that buyers do not compete at a downstream market. This is realistic for many procurement markets, in particular those for services such as computer programming or consulting, where [X.sub.n] may denote the man days required for a project.

In the procurement game that we study, first each supplier m submits to each buyer n a menu [t.sup.m.sub.n](x), specifying the total transfer that the buyer has to make when purchasing the quantity x. We analyze both the case where buyers do not restrict their purchasing strategies and the case where at least one buyer chooses single sourcing. There may be different ways how a commitment to single sourcing can be sustained, depending on the particular application. For instance, a retailer may dedicate only a limited amount of shelf space to a product category, which limits the number of listed goods. Also, if the purchased good represents some intermediate input, the buyer's choice of the production process (or, likewise of its warehousing and inventory management) may commit to only purchase from a single supplier.

3. Analysis of two benchmark cases

* First benchmark: the case with a single buyer. Suppose first that there is only a single buyer as [X.sub.B1] = X and, consequently, [X.sub.B1] = 0. The analysis is then analogous to that in Anton and Yao (1989).

We know that there exists a plethora of (Nash) equilibria in which the single buyer ends up purchasing different fractions of X from the two suppliers. Although we refer to Section 5 for a more formal analysis, it is straightforward to see already now how different equilibria can be supported. For this purpose, suppose that each supplier submits a "quantity-forcing" contract, namely to, first, supply some fixed quantity [[??].sup.m] for some total payment [??] = [[??].sup.m]([[??].sup.m]) and to, second, make the purchase of any quantity other than [[??].sup.m] prohibitively expensive by demanding a sufficiently high payment [t.sup.m](x) > > [??] for all x [not equal to] [[??].sup.m]. If supplier m follows this strategy, then this makes it optimal for supplier m' [not equal to] m to follow a similar strategy with [[??].sup.m'] = X - [[??].sup.m].

In Our main analysis, we want to abstract from this multiplicity. We do so by applying a common refinement to the set of Nash equilibria, which is owing to Bernheim and Whinston (1986). The key notion of a refinement in the present context is to pin down the parts of the schedules tm that are only of importance off equilibrium. Bernheim and Whinston (1986) do so by essentially fixing the slope of the payment schedules, requiting that they truthfully reflect the respective supplier's marginal costs. Put somewhat differently, under the "truthfulness requirement," each schedule of payments must then satisfy

[t.sup.m](x') - [t.sup.m](x) = C(x') - C(x), (1)

where we have dropped the subscript n = B1 for the supply to the single buyer (given that currently [X.sub.B1] = X and, consequently, [X.sub.B2] = 0). Section 5 shows that our main results still hold if we do not impose the truthfulness requirement (1).