This paper investigates how a buyer can design a competition to maximize the quality-of-service she receives from a set of heterogeneous suppliers. Our focus is on competitions that use an allocation function to award demand shares to suppliers based on the service level each supplier promises to provide. We develop a general framework for characterizing optimal allocation functions for any qualityof-service measure the buyer wishes to maximize. In particular, we introduce the notions of servicemaximizing and objective-consistent allocations and show that together these properties provide a sufficient condition for an allocation function to be optimal. We also identify two families of allocation functions that are service-maximizing and examine how they can be used to develop optimal allocation functions for specific definitions of quality-of-service. To illustrate the impact of the quality-of-service definition, we consider two examples. The first defines quality-of-service as the suppliers’ demandweighted average service level, while the second defines it as the suppliers’ minimum service level.
Numerical results illustrate the impact of supplier heterogeneity and the definition of quality-of-service on the performance of optimal allocation functions relative to other commonly used competition schemes.
Keywords: supplier competition, service quality, economic contests, design of incentives, Nash equilibrium

Although procurement decisions have been traditionally driven by price, firms are placing increased importance on the service levels suppliers provide. A recent survey of firms that have outsourced significant parts of their business reveals that commitment to service quality ranks as the number one factor in supplier selection (Outsourcing Institute 1998). Large retailers, such as Wal-Mart, and manufacturers, such as Dell, are known to put a premium on the service levels their suppliers provide and have developed sophisticated supplier rating systems for tracking and rewarding supplier performance.
Service quality features prominently in the supplier rating systems used by other firms as well. For example, Raytheon, a major aerospace and defense systems supplier, places the largest weight (35%) in its supplier selection score
1
on delivery time performance while assigning a relatively small weight to price (10%). Saturn Electronics, a global supplier to original equipment manufacturers (OEMs), also uses a rating system
2
that weighs more significantly on-time delivery (20%) and quality compliance (30%) than cost (15%).
This increased focus on supplier service level
3
is driven in part by the availability of a large number of low cost suppliers. The relatively weak bargaining position of these suppliers allows the buying firms to set the price, with service level becoming a primary factor in differentiating between suppliers.
Concern about service level is also driven by the operational policies adopted by many firms, such as justin-time and lean , which emphasize on-demand production and on-time delivery. These policies make firms particularly vulnerable to poor supplier performance because of the limited safety stocks and safety leadtimes the firms keep. High supplier service level is of course critical to firms that have, themselves, chosen to compete on the basis of customer service.
While the importance of having suppliers provide high service levels is clear, it is less clear how firms should go about inducing their suppliers to do so. One common approach is to simply select those suppliers that promise to offer the highest service level. This amounts to carrying out a reverse auction where bids are specified in terms of service level. However, a reverse auction does not always guarantee
1 https://raysrs.raytheon.com/srsrc/rc/web/srsrcMain.jsp
2 http://www.saturnee.com/supplierinformation.htm
3 Service level is typically measured in terms of the availability of the demanded good or service at the time it is requested. For physical goods, typical measures of service quality include fill rate, expected order delay, the probability that order delay does not exceed a quoted lead-time, and the percentage of orders fulfilled within specification. For services, service level measures include expected customer waiting time, the probability that the customer receives service within the specified time window, and the probability that a customer does not renege before being served.

that suppliers would provide the highest feasible service level (as we show in this paper). Another alternative is to negotiate an incentive contract with a subset of suppliers where, for example, suppliers are financially penalized for deviating from a negotiated service level. However, the outcome of such negotiation can be unpredictable. More importantly, the potential of negative financial flows can limit the ability of suppliers, or dampen their incentive, to offer the highest possible service levels (Elahi 2006).
In this paper, we explore a third alternative where a firm (the buyer) stages a competition among a set of potential suppliers and allocates a fraction of demand to one or more suppliers based on the service level each promises to provide. The fraction of demand allocated to each supplier is calculated using a scoring function, which we call the allocation function . The buyer chooses this allocation function and announces it to the suppliers. The suppliers respond by quoting a service level to which they promise to commit should they be allocated any fraction of the buyer’s demand. In determining a service level to quote, each supplier maximizes his own profit, which is a function of the fraction of demand he receives and the cost of service level he provides. The buyer’s objective is to design the allocation function so as to maximize an overall measure of service levels she receives from the suppliers. We call this overall measure quality-of-service .
We provide a general framework for characterizing optimal allocation functions for a general definition of quality-of-service. We introduce the notions of service-maximizing and objective-consistent allocations and show that together these properties provide a sufficient condition for an allocation function to be optimal. We then identify two families of allocation functions that are service-maximizing
(a service-maximizing allocation function is one that induces suppliers to provide the maximum feasible service level for the amount of demand allocated) and show how a buyer can use these to design optimal allocation functions for specific definitions of quality-of-service. We compare the performance of optimal allocations to alternative allocation schemes, including auctions, and show that the difference in qualityof-service to the buyer can be significant. We also show how a buyer can customize the proposed allocation to enable outcomes for the competition other than maximizing quality-of-service.
A related version of this problem is treated in Benjaafar et al. (2007). The supplier competition we describe here is similar to the SA competition discussed in Benjaafar et al. (2007). However, that paper is not concerned with determining optimal allocation functions. Instead, the focus is on studying the behavior of suppliers under an exogenously-determined allocation function, where the function is assumed to have a simple proportional form. In that paper, the analysis is limited to identical suppliers

with identical costs and revenue structures and the definition of quality-of-service is limited to the demand-weighted average service level. These assumptions greatly simplify the analysis. To our knowledge, our paper is the first to characterize optimal allocation functions for a system with nonidentical suppliers and to show how such allocation functions can be designed for a general definition of quality-of-service.
A general review of the literature on supplier service competition is included in Benjaafar et al.
(2007). For the sake of brevity, we will not reproduce it here. However, we should note that much of the existing literature has focused on either competition among identical suppliers or competition involving specific allocation functions, typically proportional allocation functions. Few results exist for settings with heterogeneous suppliers. Also, few results exist on how to construct optimal allocation in such settings.
There is related literature in economics on optimal mechanism design (reviews can be found for example in Klemperer 1999 and Karla and Mengze 2001). Papers in this area focus on finding the optimal way for a principal to orchestrate a competition in order to elicit maximum effort or minimum price from a set of agents. This literature spans diverse application areas, including designing employee incentives, incentives for sales agent, contest rules among independent agents to elicit maximum effort, and auctions.
The emphasis is on situations where there is information asymmetry regarding agent characteristics or effort. Our paper can be viewed as involving mechanism design. However, in our case, we assume complete information among all parties. Our objective is to design incentives via the allocation functions so as to induce suppliers to exert maximum effort (in the form of service levels) for the amount of demand allocated and to allocate demand in a way that maximizes overall quality-of-service.
There is also related literature in economics on rent-seeking contest. In a rent-seeking contest, there are N contestants who compete for a prize. The probability that a contestant wins the prize (the rent) increases with his expenditures and decreases in the expenditures of other contestants, where the probability with which a contestant wins the prize is generally assumed to be exogenous. A review of important results from this literature can be found in Nitzan (1994). A focus of this literature is on documenting the so-called inefficiency of rent-seeking contests. Rent-seeking is viewed as wasteful since the total expenditures by the contestants can equal the value of the prize itself, a phenomenon called rent dissipation . However, in systems with non-identical contestants, it has been shown that there may not be complete rent dissipation; see for example Hilman and Reily (1989), Paul and Wilhite (1990), Nti (1999),

and Dixit (1987). That is, heterogeneity in the contestants’ characteristics can diminish the intensity of the competition. In this paper, we show that by carefully designing an allocation function (the equivalent of a selection probability in the rent seeking context), one may induce the contestants to exert the maximum feasible effort even when they have non-identical characteristics.
In addition to Benjaafar et al. (2007), the other paper that is most related to ours is Cachon and Zhang
(2007). They consider a specific context where suppliers are modeled as single server queues and compete in terms of investment in service rates. Higher service rates translate into higher service levels in the form of lower queueing delays for the buyer. Similar to Benjaafar et al. (2007), they treat the case of homogeneous suppliers with identical revenue and cost structures. They compare different demand allocations and show that a so-called linear allocation function leads suppliers to invest in the maximum feasible service rates for the fraction of demand they are allocated (see section 4 for further discussion).
However, because they consider only symmetric allocations (suppliers that provide the same service rates are allocated the same amount of demand), the proposed allocation does not necessarily maximize overall quality-of-service.
In summary, this paper makes the following contributions relative to the existing literature: (1) it introduces sufficient conditions for an allocation function to be optimal (service-maximization and objective consistency); (2) it expands the common view of homogenous suppliers to include suppliers with heterogeneous production and service costs; (3) it identifies families of allocation functions that are service-maximizing while also providing the unique property of inducing any targeted allocation of demand shares among suppliers; and (4) shows how these allocation functions can be used to construct optimal allocation functions for different definitions of the buyer’s optimality objective.
The rest of the paper is organized as follows. In Section 2, we formulate the problem and characterize the properties of an optimal allocation. In Section 3, we present two families of allocation functions that can induce the maximum feasible service levels for any set of desired allocations. In Section 4, we compare optimal allocation functions to other allocation schemes. Finally, in Section 5, we offer concluding remarks.

We consider a system that consists of a single buyer who seeks to allocate an expected demand quantity λ to N potential suppliers. Suppliers compete for demand based on the service level they choose to offer.
The objective of the buyer is to maximize via demand allocation a measure of service quality determined by the service levels offered by the suppliers. We assume the buyer’s acquisition cost is constant across suppliers, and so not a factor in the buyer’s allocation decision. Let s i
≥ 0 denote the service level offered by supplier i , λ i
the amount of demand allocated to supplier i , 0 ≤ λ i
≤ λ and
∑ N i = 1
λ λ .
The competition proceeds as follows. First, the buyer announces a criterion for allocating demand across suppliers with the property that the fraction of demand a supplier receives is increasing in the service level she offers. More specifically, the buyer carries out the allocation through a demand allocation function vector α = ( α
1
, …, α
N
) where α i
α i s s
− i
) specifying the fraction of demand allocated to supplier i given the supplier’s service level s i
and the service levels s
− i
= s s i − 1
, s i + 1
,..., s
N
) offered by his competitors with 0 ≤ α
− i
) ≤ 1. Therefore, the demand received by each supplier is i
( ,
− i
) λ . We assume the function α i s s
− i
) is non-decreasing in s i and equal to zero when s i
= 0, for i = 1, …, N .
Each supplier i responds to the announced α by choosing a service level that maximizes his expected
π
− i
), subject to the behavior of other suppliers. In choosing their service levels, suppliers trade off the potential revenue benefit of a higher service level against the cost of providing this service. r i earns for each unit demand received. Since the buyer’s acquisition cost is fixed, any difference in unit revenue across suppliers is due solely to differences in unit operating costs (e.g., r i
= − i
, where p c i ). Let denote the service cost supplier i incurs in providing s i
. Examples of service related costs include investments in capacity, inventory, and/or continuous improvement efforts. We assume is continuous and strictly increasing in s i
with (0) 0 .
4
This cost model differs from previous studies
(e.g., Benjaafar et al. 2007, Cachon and Zhang 2007) in that we allow for a non-convex service cost function, and supplier heterogeneity in both unit revenue and service cost.
4 Our setup is similar to the setup described in Benjaafar et al. (2007) for the case of demand-independent service costs, where the cost of providing service is independent of the quantity of demand received. This cost structure arises naturally in many different manufacturing and service applications. For example, Benjaafar et al. (2007) show that this reflects the structure of service cost for both make-to-order and make-to-stock suppliers when process times and demand inter-arrival rates are exponentially distributed. Cachon and Zhang (2007) also use a similar cost structure in their analysis of a make-to-order supplier competition when the suppliers compete based on their effort levels.

The supplier’s profit can now be stated as
π i s s
− i
) = α i
( ,
− i
) λ − i
( )
.
(1)
Note that profit depends on a supplier’s own service level s i as well as the service level profile s
-i
of his competitors. In keeping with previous studies, we assume all parties have complete information, costs are incurred only if the supplier receives a positive portion of demand (i.e., λ i
> 0 ), service levels are enforceable, and all suppliers are risk neutral. We also assume that the reservation profits of the suppliers are zero.
Let θ i s λ = v s i
λ r denote supplier i ’s ratio of service cost to revenue received for a given service level and demand allocation. Note that the profit of supplier i θ s λ = . Let s i max λ denote the service level of supplier i which results in ( , ) 1 . This represents the maximum service level supplier i is willing to provide given demand allocation λ i
. Since s i max λ is clearly increasing in λ i
, s i max λ provides an upper bound on the decision space of supplier i . This upper bound is also a measure of the efficiency of supplier i . Without loss of generality, we assume throughout that suppliers are labeled in descending order of their efficiencies so that s
1 max ensure that s i max
≥ s max
2
λ s max
N
. To
λ is finite, we restrict our analysis to the non-trivial case where the supremum of i
( ), > 0 is strictly greater than λ r i
.
As the first mover in this two-stage game, the buyer selects a demand allocation function α that maximizes the quality-of-service she receives from these suppliers. Let q ( , ) denote the buyer’s qualityof-service function when the suppliers provide service levels s = s s
N
) and receive demand shares
δ = δ δ
N
)
e assume q is nondecreasing in s i for all i = 1,…, N . This property reflects most reasonable interpretations of quality-of-service across a supply base, including average service level and minimum service level.
In a competition where the suppliers’ demand shares are set by an allocation function α ( ),
= ( )
he buyer’s problem can be stated as
( α
1 max s α
N s
) ∈ F
N q s α s (2) subject to: where F N and
∑ i
N
= 1
α i
=
{ ( α
1 x α
N x
) } s i
= arg max x
(
is the set of all
α
N i
( ,
− i
) λ − ) i = 1,..., N , (3)
-dimensional vectors of functions with α i
: R N a [0,1]
. The optimization is carried out over all vectors of functions in F N .
The constraints in
(3) reflect the suppliers’ subgame, where each supplier chooses a service level to maximize his own profit

for any given set of service levels chosen by his competitors. We refer to an allocation function as optimal, for a given quality-of-service definition, if it solves the buyer’s problem (2)-(3) and results in a unique Nash equilibrium
5
.
Since the buyer’s problem involves searching for optimal functions (not optimal quantities), it is difficult to solve directly. Instead, we take an indirect approach by first introducing two properties which together are sufficient for an allocation function to be optimal. We refer to these properties as servicemaximizing and objective-consistent . These conditions will be used later, in sections 3 and 4, to evaluate candidate allocation functions.
We define an allocation function to be service-maximizing if it intensifies the competition to a level where at Nash equilibrium all suppliers choose the maximum feasible service level for their level of demand allocation. In other words, an allocation function is service-maximizing if it induces an equilibrium service level vector s * = ( ,..., s *
N
) where π i the equilibrium allocation resulting from α, that is we also know that θ i
( , δ λ = i
, for i =
and s i
* = s i max ( δ λ ) for i = N
To define objective-consistency, let δ * = for i = N Let δ = δ δ
N
) denote
If α is service-maximizing,
(
δ δ *
N
) max
δ
1
,..., δ
N
) q
(
be the solution to the following problem: s max ( δ λ ), δ
)
s max ( δ λ ) =
( s
1 max
∑ i
N
= 1
δ i
( δ λ ),..., s max
N
( δ λ ) .
)
= 1
δ i
≥ 0, i = 1,..., ,
An allocation function α is objective-consistent if its
δ i
* = α i
( ) , for i = Note that the solution to (4)-(5) varies with the given quality-of-service function, q ( , ( )) .
We are now ready to state our main theorem for this section which characterizes the optimal solution to the buyer’s problem (2)-(3).
Theorem 1.
If an allocation function α( s ) is both service-maximizing and objective-consistent , for a given q ( , ( )) , then it is necessarily optimal.
The proof of Theorem 1 and of all subsequent results can be found in the Appendix. Theorem 1 states that an optimal allocation induces the maximum feasible service levels that each supplier can afford, s i max ( δ λ ) , while the induced demand shares guarantee the maximum quality-of-service,
5 An allocation function which results in multiple Nash equilibria could be optimal if the smallest Nash equilibrium quality-ofservice is greater than or equal to the values of Nash equilibrium quality-of-service resulting from all other allocation functions.

q
( s max ( δ * λ ), δ *
)
. This also implies that we can test whether or not an allocation function is optimal by verifying whether or not it leads to a quality-of-service equal to q
( s max ( δ * λ ), δ *
)
. Note that Theorem 1 gives a sufficient condition for optimality but does not guarantee the existence of such an allocation function for a given quality-of-service function.
The two conditions for an optimal allocation function are independent, in the sense that a servicemaximizing allocation function may not be objective-consistent (or vice versa). This fact will become more concrete in section 4 when we analyze the impact of different quality-of-service functions. For a quick illustration, consider the case of identical suppliers where
( ) = ( ) and r i
= r for i = 1,..., N .
In this case, it is easy to show that when the buyer wishes to maximize the minimum service level, q s α s = s s
N
) , the allocation function must induce identical Nash equilibrium service levels, s
1
*
α i
...
s *
N
= s * . We know from previous research that a simple service proportional allocation, s = s i
/
∑ N j = 1 s j
, induces identical service levels and identical demand shares at the Nash equilibrium.
Therefore, a service proportional allocation function is objective-consistent for q s α s = s s
N
) .
However, it is also easy to show that this allocation function does not induce zero profit for the identical suppliers, so it is not service-maximizing.
Another example is provided by the allocation function α i
= s i
/( − 1) /
∑ N j = 1 s j
/( − 1) which is analyzed in Benjaafar et al. (2007), where they assume identical suppliers and a more restrictive service cost function. They show that it is possible to use this allocation function to induce zero profits for the suppliers. In other words, the allocation function is service-maximizing. Because this is a symmetric allocation function, it also induces equal demand shares at the Nash Equilibrium. So, if
δ = is not the optimal division of demand shares for the given quality-of-service measure, this allocation function will be service-maximizing but not objective-consistent. This is true, for example, if quality-of-service is defined as the demand weighted average service level, q ( , ( )) = ∑ N j = 1
α i s s j
; see section 4 for further detail.
We continue in this section by identifying and analyzing two families of service-maximizing allocation functions. These families draw from allocation functions previously studied in the literature. The first allocates demand in a proportional manner, while the second allocates demand in a linear fashion.

Proportional allocation is a commonly cited method for assigning demand across competing suppliers.
Much of the existing literature has focused on proportional allocation functions that are symmetric (see for example Benjaafar et al. 2007, Cachon and Zhang 2007, Allon and Federgruen 2005, and the references therein). Symmetric functions imply that if two or more suppliers choose the same service investment, they will receive the same proportion of demand. This type of allocation function is known to be service-maximizing in some special cases when suppliers are identical (Benjaafar et al. 2007, Cachon and Zhang 2007). However, a symmetric function is not service-maximizing when suppliers are heterogeneous. In searching for a service-maximizing allocation function, we must consider the following more general characterization that allows for heterogeneity across suppliers:
α i
( ,
− i
) =
∑ N j i
= 1
( )
, (6) for i = 1,..., N , where i
( ) is a non-decreasing function of s i
with (0) 0.
Note that symmetric g s = g s i
for i =
We know from prior research that a unique Nash equilibrium solution is not guaranteed for all allocation functions defined by (6). For example, Benjaafar et al (2007) show that a symmetric proportional allocation function in a system with identical suppliers does not guarantee uniqueness of the
Nash equilibrium when ( ) is non-concave (see also Cachon and Zhang (2007) for a similar result). It follows that in our more general case of heterogeneous allocation functions and heterogeneous suppliers, a unique Nash equilibrium will not be possible for all ( ).
The following theorem defines a specific form for ( ) that not only guarantees a Nash equilibrium, but also results in a service-maximizing allocation. The function is parameterized for each supplier i θ s λ , and the proportion of demand, δ i
, the buyer wishes to allocate to that supplier.
Theorem 2.
If the buyer uses a proportional allocation function as defined in (6) with
= δ θ s δ λ ) 1/(1 − δ i
) , 0 < δ i
< 1 , and
∑ N i = 1
δ i
= 1 , then the following holds.
(a) A Nash equilibrium exists, with the suppliers’ service levels and profits given by s i
* = s i max ( δ λ ) and
π i s * for i = Furthermore, allocation levels at this equilibrium are given by α i s * = δ i for i =

(b) This Nash equilibrium is unique if suppliers are constrained to provide a strictly positive service level
( i.e., s i
> 0 for i = 1,..., N ) .
Unlike previous research involving symmetric allocation functions, Theorem 2 guarantees a Nash equilibrium without the assumption that the resulting allocation function is concave and the service cost function is convex. A relatively mild assumption, namely that all suppliers who want to participate in the competition must provide a positive service level, is required to ensure uniqueness. Without this requirement, other equilibria may arise where one or more suppliers provide zero service level and receive zero allocation (see Elahi (2007) for details).
The implications of Theorem 2 are rather remarkable. By simply manipulating the parameters of the allocation function, the buyer can orchestrate the competition so that each supplier, regardless of his efficiency, has enough incentive to spend all his revenue to provide the maximum feasible service level.
The parameter vector δ serves a special purpose by allowing the buyer to induce a specific equilibrium allocation for each supplier. We will leverage this feature of the allocation function later, in section 4, to identify functions that are both service-maximizing and objective-consistent for specific definitions of q ( , ( )).
Intuitively, by varying δ i
, the buyer affects the incentive for supplier i to offer a higher/lower service level via the rate at which supplier i increases its demand share with increases in his service level.
However, this must be done while maintaining enough competitive pressure on all suppliers so that each continues to offer the maximum feasible service level for the amount of demand allocated.
Cachon and Zhang (2007) introduced the concept of a linear allocation function in their study of competitions among identical make-to-order suppliers. This functional form is motivated, in part, by the earlier results of Bell and Stidham (1983). The allocation function is linear in the sense that the demand allocated to a supplier is a linear function of the suppliers’ service levels. In this subsection, we expand the formulation of a linear allocation function to include non-symmetric functions that are linear in a set of functions of service levels. We then show how these functions can be designed to be servicemaximizing.
To begin, let us define φ i n s s
− i
)
φ i n
as a linear combination of functions ( ) . That is, s s
− i
) ≡ g s i
−
1 n
(
∑ n j = 1
−
)
(7)

for , i = 1, …, N , and g s is a non-decreasing function of s i
with g i
(0) 0.
We also define for each vector of possible service levels ( s
1
, …, s
N
), an index vector ( m
1
, …, m
N
), where m k
∈ N
= 1, …, N and m k
≠ m l if , such that φ N m
1
( s m
1
, s
− m
1
) ≥ φ N m
2
( s m
2
, s
− m
2
≥ ≥ φ N m
N
( s m
N
, s
− m
N
for k
) .
A general linear allocation function can be stated as follows where
α m k
( s m k
, s
− m k
)
⎧
= ⎨
⎪⎩
φ
0
N m k
( s m k
, s
− m k
) if k ≤ othe rwise,
(8)
ˆ ≤ N is the largest integer such that α i s s
− i
> if s m
> 0 . Note that the symmetric linear allocation function proposed by Cachon and Zhang (2007) is a subset of this general family which can be obtained by choosing = θ s i
ρ
, or equivalently φ i n ( ,
− i
) ≡ θ s i
ρ −
(
θ ∑ n j = 1 s j
ρ − 1 /
) n , gives us a symmetric linear allocation function similar to Cachon and Zhang’s (2007).
The following theorem shows that by defining ( ) in a particular way, we can devise a class of service-maximizing allocation functions that result in a unique Nash equilibrium.
Theorem 3.
If the buyer uses a linear allocation function as defined in (7) (8) with g s = θ i s i
δ λ ω i
,
ω i
=
N
N
− 1
1
γ i
δ i
, 0 δ 1, i = N
∑ N i = 1
δ i
= 1, (9) and
γ i
=
N +
1
(
N
N
δ γ
−
1
1)( −
1
)
, i = γ min i N δ
1
+
N
(
δ
δ
1 i
− δ
1
)
, (10) then the allocation is service-maximizing and leads to a unique Nash equilibrium. That is, at the Nash equilibrium the suppliers’ service levels and profits are s i
* = s i max ( δ λ ) and π i
Furthermore, the equilibrium allocation levels are given by α i
= δ i for i = N for i =
Although a linear allocation function, as specified in Theorem 3, has a more complicated functional form than a proportional allocation function, it has the advantage of resulting in a unique Nash equilibrium without requiring a positive service level from all suppliers. To see why this condition is not needed, let s * = s s *
N
) be the Nash equilibrium of a competition based on a service-maximizing linear allocation function. Now suppose one of the suppliers, say supplier j , chooses to deviate from this Nash equilibrium and set s j
= 0 . This removes the impact of supplier j on the remaining competitors and effectively softens the competition. The remaining suppliers will then decrease their service levels to settle on a new equilibrium (see the proof of theorem 3 for details). However, this cannot be a Nash

equilibrium since supplier j could increase his service level to its initial value, s j
= s * j
, and earn a positive profit. The allocation function thus guards against any deviation from this unique equilibrium.
We have now laid out two families of allocation functions that are service-maximizing for competitions involving heterogeneous suppliers. If we can show that some subset of these allocation functions are also objective-consistent then, by virtue of Theorem 1, we have identified an optimal set of allocation functions. Proving that an allocation function is objective-consistent requires further specifying the quality-of-service function that the buyer wishes to maximize. In this section, we consider two common quality-of-service measures, average service level and minimum service level, and derive the associated vector of optimal demand shares, δ * = δ δ *
N
) , for each (i.e., solve problem (4)-(5)). We then compare the performance of various allocation functions, including those outlined in the previous section, for these problem instances.
Before proceeding, we should note that our formulation of the buyer’s problem (2)-(3) assumes that the buyer is unconstrained in the way demand shares are set. However, in some settings the buyer may wish to constrain the demand share for a given supplier to a range or even a specific number. For example, the buyer may be contractually obligated to provide a minimum share to a long-term supplier, or may wish to limit her risk exposure by limiting all shares to less than some specified maximum value
(where the value is some fraction greater than 1/ N ). These additional constraints are easily handled in our framework, and can actually make the problem of identifying an optimal allocation function easier in some instances.
One of the most common aggregate service objectives is to maximize the demand-weighted average of the service levels provided by the suppliers, i.e. q ( , ( )) = ∑ i
N
= 1
α i s s i
. This is the measure of choice, for example, when the buyer is interested in maximizing the average fill-rate, length of delay, or order lead time across her supply base.
To define the demand shares that optimize this objective, we must introduce the following notation.
Let k be the number of suppliers who achieve the highest possible efficiency rating (i.e., s
1 max λ ).
Because we assume suppliers are ordered according to their efficiency rating, we have

s
1 max = s max
2
λ s max k
> s max k + 1
λ s max
N
Note that k = 1 represents the situation where one supplier dominates all others in efficiency. The following result defines the set of demand shares an allocation function must induce to optimize the demand weighted average service level.
Proposition 1.
Suppose quality-of-service is defined by q ( , ( )) = ∑ N i = 1
α i
α ( ) s where δ * j
= 1 for only one j ∈ {1, ..., }, and δ i
* = 0 for i ≠ j .
An allocation function
α ( ) = δ * ,
Proposition 1 implies that when the buyer is interested in optimizing demand-weighted average service level, problem (4)-(5) is optimized by allocating all demand to one supplier in the group of the k most efficient suppliers. The chosen supplier must equal (if not dominate) all other suppliers in terms of the maximum service level he can offer. It is easy to see that assigning demand to less efficient suppliers would only degrade this demand-weighted average service measure.
Proposition 1 uncovers a specific requirement for an allocation to be objective-consistent but it does not guarantee that such an allocation function actually exists. In this sense, the result provides an upper bound on the buyer’s problem (2)-(3) for this definition of service quality. More precisely, if an allocation
α s results in Nash equilibrium service levels s * = ( ,..., s *
N
) , we know that q s α s ≡ ∑ i
N
= 1
α i
* s i
* ≤ s
1 max (11)
Since our service-maximizing allocation functions in section 3 require the condition δ i
> 0, i = 1,..., N , we cannot identify an allocation function within these families that are objective-consistent for this definition of quality-of-service. However, it is possible to construct an allocation function from these families that induces demand shares approaching the solution outlined in Proposition 1.
For example, suppose we define all demand shares in terms of the proportion of demand given to supplier 1,
δ i
= − δ
1
) /( N − 1), i = N (12) and use these parameter settings as input to the service-maximizing allocation functions outlined in
Theorems 2 and 3. It is easy to show that by choosing δ
1
sufficiently close to 1, the buyer can get arbitrarily close to the objective-consistent solution outlined in Proposition 1. Since these allocation functions are service-maximizing, we can also get arbitrarily close to the maximum feasible quality-ofservice, s
1 max λ More precisely, δ
1
→ 1 implies s
1
* → s
1 max i = 1, …, N . Therefore, δ
1
→ 1 also implies q s α s → s
1 max
, α
1
( ) → 1, and s i
* → 0, α i s * = for
. Altogether, this implies a service-

maximizing allocation function that is asymptotically optimal since for any ε > 0, the buyer can choose δ
1 such that s
1 max λ – q s * α s < ε .
The optimal allocation has similarities with an auction that awards all demand to the supplier offering the highest service level (or randomly, in the case of a tie, selects from the highest service level suppliers). The demand shares resulting from such an auction are consistent with the outcome outlined in
Proposition 1, so an auction is clearly objective-consistent. However, an auction is not always servicemaximizing since in some cases the chosen supplier does not have an incentive to fully invest in service
(i.e., choose s
1 max λ
Whether or not the auction is service-maximizing depends on the value of k . When k = 1, it is easy to show that under an auction supplier 1 will offer the smallest possible service level that is still greater than s max
2
λ Strictly speaking, a Nash equilibrium does not exist in this case since service levels can be varied continuously. However, if we assume service levels are varied in ε increments where ε is arbitrarily small, then a unique Nash equilibrium does exist and the associated solution for supplier 1 is to offer s
1
* = s max
2
λ + ε .
This is clearly not service-maximizing. The difference in service quality for the buyer relative to the upper bound (11) depends on the difference s
1 max − s max
2
.
On the other hand, when k > 1, it is straightforward to show that a Nash equilibrium exists with each of the k most efficient suppliers providing their maximum feasible service level. The buyer can then respond by randomly selecting one of these suppliers. This results in quality-of-service q ≡ ∑ i
N
= 1
α i
* s i
* = s
1 max which is clearly optimal. In this case, an auction dominates the service-maximizing allocation functions outlined in section 3. However, the difference in performance can be made arbitrarily small by setting the allocation function parameter δ
1
(or more generally, δ j
for j ∈ {1, ..., } ) arbitrarily close to 1.
Another common quality-of-service objective is to maximize the minimum service level received across the supply base, q s α s = s s
N
) . This may be critical, for example, when the buyer assigns products or services directly from specific suppliers to specific customers. In this setting, customers do not receive the average service level over all suppliers but rather the actual service level of a given supplier. By maximizing the minimum service level, the buyer ensures that the service level experienced by all customers is reasonable in this setting.

The set of demand shares an allocation function must induce to optimize this objective is different from the one we saw in the previous subsection. In this case, the definition of demand shares depends on
θ λ , or more precisely the structure of and r i
, for i =
Proposition 2.
Suppose quality-of-service is defined by q s α s
α ( )
α i
= δ i
* , i = 1, ..., N , where δ i
* = θ i s ˆ λ
= s s
N
) . An allocation function and ˆ
∑ i
N
= 1
θ i s ˆ λ = .
Proposition 2 reveals that in equilibrium an objective-consistent allocation function, with quality-ofservice defined by minimum service level, will induce demand shares δ i
* = θ i s ˆ λ for supplier i. Since
θ i s ˆ λ = v s ˆ λ r i
) , this results in less efficient suppliers receiving higher demand shares than more efficient suppliers. In other words, δ
1
* ...
δ *
N
. In this way, the suppliers with lower efficiencies are given an incentive to increase their service levels to equal that of suppliers with higher efficiencies. This, in turn, results in a higher minimum service level for the buyer.
Proposition 2 outlines specific requirements for an allocation to be objective-consistent, but (similar to Proposition 1) does not guarantee its existence. However, since we know from Proposition 2 that
δ i
* = θ i
λ > for i = 1 , …, N , the two families of service-maximizing allocation functions from section
3 can be used to create a pair of optimal allocation functions. This is accomplished by simply setting the parameters δ i
= equal to δ i
* as specified by Proposition 2.
It is interesting to note that the resulting optimal allocation functions induce equal service levels for all suppliers. To confirm this result, note that because an optimal allocation function is service-maximizing
(as well as objective-consistent), we know it induces θ i
( , δ λ = . Rearranging terms, we have i
( ) / r i
λ δ i
* , or δ i
* = θ i i
* λ . We also know from Proposition 2 that δ i
* = θ i s ˆ λ . Comparing these two equalities, we see that s i
* = s ˆ, i = 1, …, N . That is, to maximize the minimum service level, an optimal allocation has to induce identical equilibrium service levels.
In the previous subsection, where the buyer’s objective was to maximize average service level, the maximum quality-of-service was determined by the efficiency of the most efficient supplier. However, we now see that when the buyer’s objective is to maximize the minimum service level, the maximum quality-of-service is ˆ which depends on the efficiencies of all suppliers through equation
∑ i
N
= 1
θ i s ˆ λ = . In this case, if the efficiency of any supplier improves, the buyer will receive a higher quality-of-service.

To further investigate the performance of these optimal allocation functions for different quality-ofservice measures, we consider a numerical example. The example is from an application described in
Benjaafar et al. (2007). Suppose a buyer wishes to outsource demand for a product or service to two suppliers ( N =2). Supplier produce items one unit at a time in a make-to-order fashion, which we model as two single server queues. A supplier’s service level is defined by his probability of meeting a given lead-time target τ , i.e., s i
= Pr( W i
≤ τ ) where W i is a random variable representing lead-time for supplier i .
We assume demand arrives according to a Poisson process with mean λ and the processing time at supplier i is exponential with mean 1/ μ i
.
Suppliers have the opportunity to increase their service level by investing in capacity at an amortized capacity cost of k i
per unit of service rate. In addition, supplier i incurs production cost c i for each item produced. In terms of our general framework, this can be shown to translates into a unit revenue of r i p c k i and a service cost function of where s i
= Pr( W i
≤ τ ) 1 v s = k i ln[1/(1 − s i
τ e
− ( μ λ τ
, see Benjaafar et al. (2007) for details.
We will use this application to illustrate how our optimal allocation compares with other common allocation schemes including the auction scheme alluded to earlier. For our example, let λ = 50 units per day, τ = 1 day, p = $1000 , c
1
= $450 , and k
1
= $500 . To investigate the impact of supplier heterogeneity, c k
2
. In particular, we allow the ratio
2
/
2
= k
2
/( −
2
− k
2
) to vary between 0.1 and 1.0.
Figure 1 illustrates the impact of supplier heterogeneity when quality-of-service is defined in terms of average service level. Supplier heterogeneity is measured by the ratio β = k r
1 k r
2
The optimal average service level is 99% irrespective of the level of heterogeneity. This is because optimal qualityof-service in this case is determined entirely by the most efficient supplier (supplier 1), whose parameters are fixed. The auction performs poorly, as expected, when the cost efficiencies of the two suppliers diverge. The gap between an auction and optimal allocation converges as the ratio of the suppliers’ efficiencies converges to 1. Figure 1 also displays the relative performance of a simple serviceproportional allocation function where ( ) = s i
, for i = N a commonly cited allocation function in symmetric competitions. Not surprising, this simple function performs quite poorly for heterogeneous

suppliers and its gap in performance remains large even when the efficiencies of the two suppliers converge.
Figure 2 reports similar information for the case where quality-of-service is defined as the minimum service level. Here we see that, unlike the previous case, the optimal quality-of-service now increases with the efficiency of the least efficient supplier (supplier 2). The performance of the serviceproportional allocation function continues to significantly lag the optimal. However, the gap between the two is slightly smaller than what was observed for the average service measure, particularly for higher levels of supplier heterogeneity. Note that we do not graph the results of an auction here since an auction by definition results in a single supplier and minimum service level is meaningless with one supplier.
Figure 3 illustrates how the allocation of demand shares between the two suppliers changes with the level of heterogeneity. As expected, the proportion of demand given to the least efficient supplier
(supplier 2) under the optimal allocation decreases as this supplier’s efficiency improves to match his competitor. On the other hand, we see the opposite behavior under a service proportional allocation, where an improvement in supplier 2’s efficiency now increases his demand share. This occurs because the optimal allocation assigns a larger demand share to the less efficient supplier to help him maintain the same service level as his more efficient competitor. In contrast, a service proportional allocation allocates demand in proportion to the service level each supplier provides. Since the service proportional allocation is symmetric, the buyer has no way to discriminate between suppliers based on their relative efficiency.
Because supplier 1 can provide the same level of service as supplier 2 at a lower cost, supplier 1 offers a higher service level in equilibrium and thus captures a larger portion of demand.
In this research we have studied an outsourcing problem in which suppliers compete for the demand of a single buyer based on the service levels they provide. The buyer can control the intensity of this competition through the way she allocates her demand among the suppliers. Therefore, the buyer’s objective is to design an optimal allocation function which is capable of eliciting high service levels from her pool of suppliers and maximizing an aggregate measure of service quality. We introduced two conditions for an allocation function to be optimal: service-maximization and objective consistency. An allocation function is service-maximizing if it intensifies the competition to a level where all suppliers

1.0
Optimal allocation
0.8
Auction
0.6
0.4
Service proportional allocation
0.2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Supplier heterogeneity ratio, β
Figure 1: Impact of supplier heterogeneity under an average service level objective
1.0
0.8
0.6
0.4
0.2
Optimal allocation
Service proportional allocation
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Supplier heterogeneity ratio, β
Figure 2: Impact of supplier heterogeneity under a minimum service level objective

1.0
0.8
0.6
Optimal allocation
0.4
Service proportional allocation
0.2
0.0
0.1 0.2
0.3 0.4
0.5 0.6
0.7 0.8
0.9 1.0
Supplier heterogeneity ratio, β
Figure 3: Impact of supplier heterogeneity on demand shares under a minimum service level objective

provide maximum feasible service levels for a given set of demand shares. The allocation function is objective-consistent if it induces a set of demand shares that is able to maximize a given quality-ofservice while suppliers provide their maximum feasible service levels.
We presented two examples of service-maximizing allocation functions. We showed that using these allocation functions, the buyer can induce any desired set of shares while keeping the intensity of the competition at this maximum level. We also characterized the sets of demand shares which make an allocation function objective-consistent with respect to two different optimality criteria: maximizing the average service level and maximizing the minimum of service levels. Finally, we compared the performance of the optimal allocation function with the performance of two other common allocation schemes and showed that the benefit from using the optimal allocation can be significant.
There are several potential avenues for future research. For example, our current model assumes complete information among all parties. By releasing this assumption, we can model situations where the cost structure of each supplier is private information. In this case the buyer and other suppliers may possibly know only a probability distribution of each supplier’s service cost. This information asymmetry could affect both the form of the optimal allocation function and the intensity of the competition. We expect incomplete information to decrease the intensity of the competition, with buyers possibly no longer able to induce suppliers to provide the maximum feasible service level. That is, information asymmetry could leave the suppliers with a surplus as a result of their private information.
Another interesting research avenue is to study a setting where service levels offered by the suppliers affect the demand of the buyer. This would be the case, for example, when higher service levels stimulate demand. In such a setting, by increasing the service level he offers, a supplier not only increases the fraction of demand he receives but also the size of total demand. Consequently, a higher service level would benefit all suppliers, rather than just the individual supplier that offers it. Clearly the notions of service-maximizing and objective consistency would have to be redefined to capture how demand is affected by service levels. Moreover, we expect the functional characteristics of service costs and demand to play an important role in the performance of different allocation functions.

Allon, G. and A. Federgruen, “Outsourcing Service Processes to a Common Service Provider under Price and Time Competition,” working paper, Columbia University, 2005.
Bell, C. E. and S. Stidham , “ Individual Versus Social Optimization in the Allocation of Customers to
Alternative Servers,” Management Science , 29 , 831-839, 1983
Benjaafar, S., E. Elahi, and K. Donohue, “Outsourcing via Service Competition,” Management Science ,
53 , 241-259, 2007.
Cachon, G. P. and F. Zhang, “Obtaining Fast Service in a Queueing System via Performance-Based
Allocation of Demand,” Management Science , 53 , 408-420, 2007.
Elahi, E., “Incentive Contracts versus Allocation Functions in Supplier Competitions.” Working paper,
University of Minnesota, 2006.
Dixit, A., “Strategic Behavior in Contests,” American Economic Review , 77 , 891-898, 1987.
Hillman A. L. and J. G. Riley, “Politically Contestable Rents and Transfers,” Economics and Politics , 1 ,
17-39, 1989.
Kalra, A. and M. Shi, “Designing Optimal Sales Contests: A Theoretical Perspective,” Marketing
Science , 20 , 170-193, 2001.
Klemperer, P., “Auction Theory: A Guide to the Literature,” Journal of Economics Surveys , 13 , 227-286,
1999.
Nitzan, S., “Modelling Rent-Seeking Contests,” European Journal of Political Economy , 10 , 41-60, 1994.
Nti, K. O., “Rent-Seeking with Asymmetric Valuation,” Public Choice , 98 , 415-430, 1999.
Paul C. and L. Wilhite, “Efficient Rend-Seeking under Varying Cost Structure,” Public Choice , 64 , 279-
290, 1990.

Let δ * = δ δ *
N
) be the solution to problem (4)-(5). If an allocation function is both servicemaximizing and objective-consistent then the resulting quality-of-service is q
( s max ( δ * λ ), δ *
)
We will show that
and
ˆ = s
) q q
( s ˆ δ ≤ q
( s max (
ˆ λ ),
ˆ s max ( δ * λ ), δ *
)
q
( s max ( δ * λ ), δ
q
δ * is the solution to problem (4)-(5),
*
) q
( s max (
ˆ λ ),
ˆ ≤
)
We first prove part ( b ) of the theorem. Let us write the profit functions of the suppliers in terms of y i
= i
( ) = δ θ s δ λ ) 1/(1 − δ i
) . That is,
π i
( ,
− i
) = y i
G
λ r i
− λ δ δ i y i
(1 − δ i
) (A1)
Where G = ∑ N k = 1 y k
. The constraint that all suppliers provide positive service levels is equivalent to y i
> 0, i = 1, ..., N . Since y i
> 0, a Nash equilibrium point must satisfy the first order optimality condition. That is, in order to prove part ( b ) of the theorem, we need to show that the following system of
( N +1) equations with unknowns
∂ π i
( ,
∂ y i
− i
)
=
− i
G 2 i
,
λ
= r i
1,...,
−
N
λ δ
and
δ i
(1 −
G
δ
has a unique solution that maximizes A1. i
) y i
− δ i
= 0, i = 1,..., N , and G = ∑ N k = 1 y k
, (A2) or, equivalently,
Y i
δ i
(1 − Y i
) − δ i
δ i
(1 − δ i
) G 1 − δ i
∑ N k = 1
Y k
= 0, i = 1,..., N , and (A3)
= 1, (A4) where Y i
= i
/ . We can rewrite the first order optimality conditions as
D Y = δ i
δ i
(1 − δ i
) G 1 − δ i
, i = 1, ..., N , (A5)
D Y ≡ Y i
δ i
(1 − Y i
) D = D = ,
Y i
> 1 . Also,
for 0
( ) δ + δ i
) which is equal to δ i
δ i
Y i
1 D Y i
< for
/(1 + δ i
) 1 + δ i
. Hence, for any

max =
(
1/(1 − δ i
)(1 + δ i
) 1 + δ i
) 1/1 − δ i , equation A5 has two solutions 0 < Y i ,1
< Y i ,2
< 1 . We want to argue that Y i ,1
corresponds to a local minimum for supplier i ’s profit function, A1. We observe that the sign of the derivative of the profit function of supplier i , equation A3, changes from negative to positive when we increase Y i
from values smaller than Y i ,1
to values bigger than Y i ,1
. We also observe that, for fixed decisions of the other suppliers, y i
Y i
= y i
/( y i
+ ∑ y j
) is increasing in . Therefore, when we increase
, the sign of equation A3 changes from negative to positive at y i
=
(
Y i ,1
/(1 − Y i ,1
)
) ∑ y j
, which in turn means that Y i ,1
corresponds to a local minimum of supplier i 's profit function. Thus, Y i ,1
cannot be a feasible solution. Similarly, we can show that Y i ,2
corresponds to a local maximum of supplier i 's profit function. Therefore, Y i ,2
, i = 1,..., N is the unique solution to the system of equations A5 which illustrates the above argument.
Lemma A1 below shows that G max > 1 for any 0 < δ i
< 1 . For G = 1 , the only solution for equation A5 that maximizes profit function A1 is Y i ,2
Therefore, Y i
= δ i
. It is easy to see that Y i ,2 is decreasing in G (see figure A1).
Therefore, for G < 1 , we have Y i ,2
Also, for G > 1 , we have Y i ,2
> δ i
or equivalently
< δ i
or equivalently
∑ i
N
= 1
Y i ,2
∑ i
N
= 1
Y i ,2
> 1 , which is not a feasible solution.
< 1 , which is not a feasible solution as well.
= δ i
and G = 1 (or equivalently y i
* = δ i
) is the unique solution of the system of equations
A3-A4 which maximizes each profit function in A1, given all suppliers provide a positive service level.
δ i
δ i
(1 − δ i
) G 1 − δ i
Y i ,1
1
δ
+ i
δ i
Y i ,2
Y i
Figure A1 – The solution to first order optimality condition (equation A5)
We now prove part ( a ) of the theorem. We showed that y i
* = δ i
satisfies the first order optimality condition A2. If we release the positive service level constraint, y i
* = δ i
, i = 1,..., N is still a Nash equilibrium since any supplier j cannot increase his profit by choosing y j
= 0 while other suppliers choose y i
* = δ i
, i ≠ j . We can easily verify that y i
* = δ i
results in s i
* = s i max ( δ λ ) , π i
( ) 0 , and
α i
= δ i

Lemma A1: For any 0 x 1 , we have
(
1/(1 − x )(1 + x ) 1 + x
) 1/(1 − x )
> 1 .
Proof: It is enough to prove that
Z =
= − x )(1 + x ) 1 +
. Therefore, it is enough to show that x < 1 for any 0
( ) x 1
is decreasing in the interval (
Z
0, 1)
=
. The derivative of where
( ) can be written as dx
= + x ) 1 + x ⎡
⎣
(1 − x
( ) ⎤
⎦
(1 x ) 1 + x
= − x
( + x + )
. Noticing that (0) 1 , (1) 0 , and dx
= − ( ln(1 x ) 1
) 1
1
−
+ x x
< 0 ,
[ we can conclude that ( ) is decreasing and less than 1 in (0,1) . Hence,
]
( ) is decreasing and less than 1 in ( 0,1) . This completes the proof of the lemma.
To prove that s i
* = s i max ( δ λ ), i = 1,..., N is a Nash equilibrium of the described game, we have to show that π i
( , *
− i
) = α i s s *
− i
) λ − i s is maximized at s i
* . Returning to the definition of our linear demand allocation (7)-(8), it is easy to verify that choosing s * = ( ,..., s *
N
) results in ˆ = N and
α i s = δ i
, i = 1,..., N . We also know that 0 < δ i
< 1, for i = N Therefore, there exist a and b , a s i
* < b such that for any s i
∈ ( , ) we have ˆ = N when the service level vector is s = ( , *
− i
) . We choose a and b in a way that for any s i
≤ a we have α i s s *
− i
= and for any s i
≥ b we have
α j s s *
− i
= s i
* as long as means
for some j i . We will show that the maximum of π i
( , *
− i
) will be zero and happens at
ˆ = N
π i s s *
− i
≤
or, equivalently, as long as s i
∈ ( , ) . For any s i
≤ a we have α i s s *
− i
= , which
. Therefore, π i
( , *
− i
) cannot attain a positive value for 0 ≤ ≤ a . That is,
π i s s *
− i
) ≤ π i s s *
− i
) for 0 ≤ ≤ a . For s i
≥ b we have α j s s *
− j
= for some j i . Therefore,
ˆ < N . In this case, one can easily verify that π i
( , *
− i
) has a maximum at s i
< s i
* (see lemma A2).
Since π i
( , *
− i
) has a unique maximum for any ˆ < N , π i s s *
− i
) will be decreasing for any s i
≥ b .
Knowing that π i
( , *
− i
) is continuous, we conclude that π i
( , *
− i
) cannot attain any maximum for s i
≥ b . Therefore, if we show π i
( , *
− i
) has a maximum at s i
* then it will be the only maximum. The first order optimality condition for π i
( , *
− i
) , when there are N participating suppliers, can be written as
∂ π i
( ,
∂ s s s i
*
− i
)
=
N
N
− 1
⎝
⎜
⎛
1
δ λ r i
⎟
⎞
⎠
γ i
γ γ i
− 1
∂ v s
∂ i
( ) s i
ω λ r i
−
∂ v s
∂ i
( ) s i
= 0, (A6)

which leads to
= δ λ r i
⇒ s i
= s i
* = s i max ( δ i
λ ) .
Since (A6) depends only on s i
, we conclude that s * = ( ,..., s *
N
) is the only solution that satisfies the system of equations resulting from the first optimality conditions. It is straightforward to verify that
α i
= δ i
and π i s * = .
We now proceed with the proof of uniqueness. In any Nash equilibrium s ˆ = ( ,..., s ˆ
N
) in which all suppliers provide positive service levels, we have 0 < α i
ˆ ˆ
− i
) 1 , i = 1,..., N satisfies the first order optimality condition. Hence, ˆ = * is the only possible Nash equilibrium with positive service levels for all suppliers. To complete the proof of uniqueness, we need to rule out the possibility of an equilibrium in which one or more suppliers provide zero service levels. To do so, we follow an approach similar to the one in Cachon and Zhang (2007; see proof of their theorem 3). We first notice that the case where all suppliers provide zero service levels cannot be a Nash equilibrium since any one of the suppliers can provide a positive service level and receive the buyer’s entire demand. The case where only one supplier provides positive service level cannot be a Nash equilibrium either since in that case the supplier with positive service level would provide an arbitrarily small service level, creating the opportunity for another supplier to provide a positive service level and earn a positive expected profit.
Now, consider the case in which M suppliers, 2 < M < N , provide positive service levels and ( N − M ) suppliers provide zero service levels. In this case, a new Nash equilibrium, ˆ s = ( ,..., s ˆ
N
) , would form as a result of the competition between the M suppliers who provide positive service levels. By virtue of lemma A2, we have s ˆ i
= v i
− 1
(
η i
( )
)
< s i
* , where η =
⎣
⎡ (
( M − 1) / M
) (
N − 1) / N
)
⎦
⎤
1 − γ i < 1 and v i
− 1 (.) is the inverse of v i
(.) ( v i
− 1 (.) is well defined since we assumed ∂ i
( ) / ∂ > 0 ). If any supplier j who we have assumed decided not to participate in the competition increases his service level to s * j
, then he would receive a demand share greater than δ j
. This is due to the fact that the number of participating suppliers is less than or equal to N and any participating supplier i provides a service level less then s i
* . This means that supplier j can increase his expected profit to a positive value by increasing his service level to s * j
, which in turn implies that this cannot be a Nash equilibrium. This completes the proof.
Lemma A2: In the competition between suppliers as stated in theorem 3, if only M < N suppliers participate in the competition, the Nash equilibrium service levels of the competing suppliers satisfy s i
< s i
* .

Proof: For any supplier i who participates in the competition, we can write the first order optimality condition as or equivalently,
∂ π i
( , *
− i
)
∂ s i
=
M
M
− 1 ⎛ 1
⎝
⎜
δ λ r i
⎟
⎞
⎠
γ i
γ γ i
− 1
∂ i
( )
∂ s i
N
N
− 1
δ i
γ i
M
M
− 1
( δ λ r i
) 1 − γ i
=
N
N
− 1 v i s 1 − γ i .
λ r i
−
∂ i
( )
∂ s i
= 0,
Knowing that i i
* = δ λ r i
, we can simplify the above equation to:
M
M
− 1 i i
* 1 − γ i
=
N
N
− 1
1 − γ i
Therefore, when M < N , the maximum of π i
( , *
− i
) happens at s i
< s i
* .
We first prove the result for j = 1 . If an allocation function α ( ) induces Nash equilibrium service level vector s * such that α s * q
( s max ( δ * λ )
)
= s
1 max ( λ )
then for δ * = α s
s
1 max
the objective function of problem (4)-(5) is
δ * = (1,0,...,0) is a solution to this problem, which in turn implies that if an allocation function induces α ( ) (1,0,...,0) , then it is objective-consistent. For any δ = δ δ
N
) , the objective function of problem (4)-(5) can be written as q
( s max ( δ λ )
)
= ∑ N i = 1
α i
( s max ( δ λ )
) s i max ( δ i
λ )
Since 0 ≤ δ i
≤ 1 and s i max (.) is an increasing function, we have q
( s max ( δ λ )
)
≤
The last inequality is due to the fact that
∑ i
N
= 1
α i
∑ i
N
= 1
α i
(
( s max ( δ λ ) s max ( δ λ )
)
) s i max s i max
≤ s
1 max
λ is the weighted average of s i max λ , i = 1,..., N . We know that the weighted average of a set of numbers is less than or equal to the maximum of that set. Therefore, we have q
( s max ( δ λ )
)
≤ s
1 max . This completes the proof for j = 1 . Since s max j
= s
1 max for any j ∈ k , we can replace supplier 1 with any other supplier in { k and obtain the same result.

Let δ i
* = θ i s ˆ λ = i
( ) / λ r i
ˆ is the unique solution of
∑ i
N
= 1
θ i s ˆ λ s
1 max ( δ λ ) ...
= . It is easy to verify that s max
N
( δ λ ) = s ˆ
ˆ is an upper bound for the solution of problem (4)-
(5).
et ( ) be an allocation function which induces a Nash equilibrium service level vector s = s s
N
)
s i max (.)
α s = δ i
, i = 1,..., N . If ≠ *
δ j
< δ * j
j
s max j
( δ λ ) < s max j
( δ * j
λ )
Knowing that s
1 max ( δ λ ) ... = s max
N
( δ λ ) = s ˆ
min
{ s
1 max ( δ λ ),..., s max
N
( δ λ )
}
< min
{ s
1 max ( δ λ ),..., s max
N
( δ λ )
}
= s ˆ .
Noticing that the objective function of problem (4)-(5) can be written as q
( s max ( δ λ ), δ
)
= min
{ s
1 max ( δ λ ),..., s max
N
( δ λ )
}
s
1 max ( δ λ ) ...
s max
N
( δ λ ) = s ˆ
the solution of problem
(4)-(5). Therefore, any allocation which induces δ * = δ δ *
N
)