Auctions for Procuring Options James Schummer∗ Rakesh V. Vohra† MEDS Department, Kellogg Graduate School of Management Northwestern University, Evanston, IL 60208–2009
March 12, 2001
Abstract We examine the mechanism design problem for a single buyer to procure purchase-options for a homogeneous good when that buyer is required to satisfy an unknown future demand. Suppliers have 2dimensional types in the form of commitment costs and production costs. The efficient schedule of options depends on the distribution of demand. To implement an efficient outcome, we introduce a class of mechanisms which are essentially pivotal mechanisms (Vickrey– Clarke–Groves) with respect to the expected costs of the suppliers. We show that the computational task of running such mechanisms is not burdensome. Our discussion uses electricity markets as an example.
Keywords: auctions, options, multi-dimensional types. ∗ †
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1
Introduction
When firms face an uncertain future demand for a good—especially when that demand is inelastic—they often buy options: they pay now for the right to purchase units in the future, at a predetermined price (or price schedule). There are many examples of markets involving this type of transaction between buyers and suppliers, including markets for electricity, clothing (Eppen and Iyer (1997)), automotive parts, etc. One simple description of such an option involves the specification of an upfront fixed fee (option price), which obligates the supplier to make a specified quantity available for future purchase, and a production fee (strike price), which is charged to the buyer contingent on the future purchase quantity.1 Depending on the environment, the terms of such options may be settled in negotiation or in a market setting such as with an auction. In this paper, we address the issue of designing an auction to arrange for the efficient purchase of such options. One setting for which such auctions have been the focus of much recent attention is the electricity market. System operators in the electricity industry purchase options for future capacity in an auction where suppliers make 2-part bids, corresponding to option and strike prices. In this particular market, those prices have a direct correspondence to costs: suppliers incur both a cost to reserve capacity and a cost to actually supply the electricity.2 In this paper, we examine the issue of efficiently procuring options from capacity-constrained suppliers with these two types of costs—a stand-by (commitment) cost and a production cost—both of which are functions of the committed quantity. First, we observe that, in order to choose suppliers 1
Barnes-Schuster et al. (1998) describe agreements of this kind in a 2-period supply model. 2 Chao and Wilson (1999) provide a more detailed description centered around the California electricity market.
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efficiently, it is necessary to know the probability distribution of demand (see the example in Section 3.1). This observation, emphasized with an example below, contrasts with an auction design that has been used in various electricity procurement auctions (and has recently been discontinued by the California Power Exchange). With this observation in hand, we examine the computational issue of calculating an efficient options schedule. We show how the problem of deciding how many options to procure from each supplier can be formulated as a linear program. In fact, these linear programs belong to the class of network flow problems, which admit particularly fast solution algorithms. Finally, we address the issue of implementing such a minimum-expectedcost options schedule. To do this, we propose a class of incentive-compatible, efficient auctions for procuring options. We show that auctions in this class can be described as ex ante Clark–Groves schemes. Different auctions within this class, though, provide different levels of risk to suppliers. We advocate the use of the one we show to minimize this risk. It should be noted that we are not explicitly addressing the issue of minimizing the buyer’s expected expenditures (i.e., we are not addressing the issue of optimal auctions in the sense of Myerson (1981)). However, results from Ausubel and Cramton (1999) suggest that when there can be resale among bidders after an auction, the auctioneer’s objective of expenditure minimization (or revenue maximization) is aligned with the objective of social efficiency. We leave the issue of expenditure minimization (when there are restrictions on post-auction trading) to future research. 1.1
Electricity Markets
One motivation for this paper is the “hour ahead” auctions prevalent in many electricity markets. However, the ideas presented here apply to any setting where firms or agents purchase options for future contingent demand. 3
The last few years have been witness to the deregulation of electricity markets in both the U.K. and the U.S. Many of these markets have been reorganized so that suppliers provide electricity via an auction. These mechanisms have typically resembled a “uniform price” auction in which suppliers area all paid the same per-unit price for power consumed. For example, under a mechanism recently used in California’s market,3 after suppliers submit 2-part bids (assumed to represent their costs as described above), the set of winning bids is accepted solely on the basis of stand-by costs. Our discussion above and the example in Section 3.1 explain why this is not efficient. It is intuitively clear that if a supplier has a very low stand-by cost and a very high production cost, it may not be optimal to secure an option from him. Second, all winning bids are paid the same amount, equal to the highest winning bid. Such a mechanism is manipulable: it does not induce suppliers to truthfully reveal their costs. Arguments demonstrating this are provided by Ausubel and Cramton (1998). They argue that one way for suppliers to artificially inflate the highest winning bid is by withholding capacity. A similar way to manipulate such an auction is to inflate the reported costs (i.e., submit very high bids) for the units of supply that have a low likelihood of being accepted by the auction process. This observation is not just a theoretical possibility, as the following report from The Wall Street Journal (August 4, 2000, p. A1) indicates. “What PECO and PPL did was offer much of their output at low prices so that the majority of their plants would be called into service. But knowing demand was so high, they offered power from their tiniest plants at vastly higher bids, in a way that often set the peak price for a number of hours. Consumers that day ended up paying millions of extra dollars for power.” 3
Again, see Chao and Wilson (1999) for a more detailed description.
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Chao and Wilson (1999) demonstrate that if a market is perfectly competitive, then a supplier cannot profit by misreporting his costs. The manipulation reported above by the Wall Street Journal, and the fact that the California electricity market has fewer than thirty suppliers, may not make perfect competition arguments relevant for these applications. 1.2
Related Literature
Applicable to our model is the general notion of combinatorial auctions. Bidders have certain preferences over subsets of objects which, in our case, correspond to ordered units of potential supply (i.e., the first unit to be supplied if at least one unit is demanded, the second unit, etc.). The problem of determining the winning set of bids in the general case of such environments is typically very hard; see de Vries and Vohra (2000) for a survey). It is therefore interesting that we show, in our specific setting, that our auction can be implemented in a computationally efficient way. Computational considerations are important, for example, in the electricity context, where many auctions take place with short turn-around times. With regard to the particular application of electricity markets, Elmaghraby (1998) argues that efficiency gains could be made by altering the structure of the way electricity is supplied. Rather than running separate auctions for each hour of supply, she argues that bidders should be allowed to bid for the right to supply over consecutive hours. Finally, Gallien and Wein (2000) examine a (deterministic) procurement model in which bidders use bidding devices to participate in a dynamic auction. They derive bounds on the bids, and convergence results. In the next section we introduce the model. In Section 3, we describe the efficient selection of suppliers, and computability. In Section 4 we provide results on mechanisms, and in Section 5 we discuss how our results easily extend to generalizations of our model. We close with a discussion comparing 5
the proposed auction to that adopted in the California electricity market.
2
Model
There is a single buyer (consumer) of a homogeneous good. The buyer has uncertain demand d ∈ [0, D] for the good, distributed according to a cumulative distribution F (with density f ). Most parts of the paper concern the case in which d is a discrete random variable. When d is continuous, we assume that f is integrable. The buyer must arrange for (potential) supply from a set of suppliers N = {1, 2, . . . , n} by buying (call) options from them. Each supplier i ∈ N has two types of costs. First, there is a stand-by cost, ci ∈ R, which is a cost (per unit) of writing an option on a single unit (i.e., a commitment cost).4 Second, there is a cost (per unit) of producing a unit, pi , called a production cost.5 For example, if a supplier were to sell a standard call option on one unit at a price of ci , with a strike price of pi , the supplier would be guaranteed a profit of 0 regardless of whether the option is exercised. Finally, each supplier i ∈ N has a maximum production capacity of Ki . Our purpose is to examine situations in which no supplier exercises any type P of monopoly power, so we assume that for all i ∈ N, N \{i} Kj ≥ D. In Section 4, we investigate the buyer’s task of designing a mechanism to procure options on D units of the good. We begin by assuming the buyer must have options for D units. Implicitly, we are imposing an arbitrarily high cost for falling short of realized demand. It is straightforward to extend the results to the case in which the buyer incurs a cost for failing to meet demand (Section 5.2). 4
Our results also apply to the case in which stand-by costs are not constant per unit. In Section 5, we discuss more general cost structures. 5 In Chao and Wilson’s (1999) terminology of the electricity market, ci is the capacity cost and pi is the energy cost.
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Since our focus is on dominant strategy mechanisms, it is essentially without loss of generality to restrict attention to revelation mechanisms, i.e., mechanisms that ask agents to simply reveal their costs. This is due to the Revelation Principle, which concerns the issue of “What is theoretically implementable?” In practice, direct revelation mechanisms may not be desirable. For example, in repeated situations, or in situations where this same cost information could be used against suppliers in unrelated negotiations, suppliers may be reluctant to reveal their costs. For this reason, there has been much interest in dynamic, ascending-bid implementations of these mechanisms (Ausubel (1997,2000)). Such schemes may be more transparent to bidders, and help to mask the costs of winning bidders. A (direct) mechanism accepts as input the reported (pairs of) costs from suppliers. As output, the mechanism specifies (i) a quantity of options to purchase from each supplier, along with individual (ii) options prices and (iii) strike prices. Formally, a mechanism is a function ci , p˜i , qi ))i∈N ϕ: (R2 )n → (R3 )n : ((ci , pi ))i∈N → ((˜ where qi ≤ Ki unit-options are purchased from supplier i at a per-unit price of c˜i with a strike price of p˜i . Notice that different suppliers may receive different prices. For the moment, though, we require that the strike price p˜i be fixed for each of the qi units potentially supplied by supplier i. This may appear at first to be less general than a mechanism which prescribes a more general supply curve, or strike prices contingent on d. For risk-neutral agents, however, our restriction is without loss of generality. More general mechanisms of the class we consider would be equivalent to ours in terms of expectation.
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3 3.1
Optimal Selection of Suppliers The Importance of the Distribution
The objective of minimizing the expected cost of supplying the buyer’s demand requires knowing the probability distribution of that demand. This is illustrated by the following example. Suppose the buyer’s demand can take on one of three values d ∈ {0, 1, 2}. Let f (1), f (2) be the probabilities that d = 1, 2 respectively. There are two potential suppliers, each capable of producing up to two units. The first has a per-unit stand-by cost of c1 = 1 and a production cost of p1 = 1. The costs for the second supplier are c2 = 1/2 and p2 = 2. For now, denote the expected cost of securing an option for two units from supplier i by Ci = 2ci + pi (f (1) + f (2)) + pi f (2). Therefore C1 = 2+f (1)+2f (2), and C2 = 1+2f (1)+4f (2). Similarly, the cost of securing an option for one unit from each of the two suppliers is C12 = 1.5 + f (1) + 3f (2) (knowing it is best to exercise the option from supplier 1 if d = 1). The optimal choice clearly depends on f (1) and f (2). Each of the three option schedules is optimal for some choice of probabilities. In fact, when f (1) = 0 and f (2) = .5, we have c1 = C2 = C12 . Intuitively, when f (2) is relatively high, supplier 1 is best. When both probabilities are low, supplier 2 is best. For some intermediate values, it is best to obtain a single-unit option from each. Finding the optimal solution must involve using this probability information. This observation is in contrast to auction procedures used in various electricity markets, including the California Power Exchange. 3.2
The Objective
In this section, we discuss the optimal selection of suppliers in order to minimize expected incurred costs. Since, in Section 4, we discuss mechanisms 8
that induce suppliers to truthfully reveal their costs, these results also apply to the efficient selection of winning bidders in such auctions. The social optimum is to minimize the expected cost of meeting the buyer’s demand d, given costs (ci , pi )N and capacities (Ki )N , by choosing quantities (qi )N for the suppliers to (conditionally) supply, such that P qi = D. The expected cost incurred by a supplier i ∈ N depends both on the distribution F , and on the order in which the options are executed. For the remainder of this section, suppose that suppliers are ordered by production costs, i.e., i < j ∈ N implies pi ≤ pj . It is clear that in an optimal solution, upon realization of any demand d, supply should be obtained from low-cost suppliers first. This means that if i < j ∈ N, supply should be obtained from supplier j only after all qi units of supply from supplier i have been obtained.6 The expected cost for supplier i ∈ N, given quantities (qi )N , is (in the continuous case) Ci (q, c, p) = ci qi + pi (E(quantity supplied by i)) Z qi X = ci qi + pi (1 − F (( qj ) + x))dx 0
(1)
j i supplies anything.
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X
qi ≥ D
N
If a list of quantities q ∗ = (qi∗ )N solves this minimization exercise, we say q ∗ is efficient (with respect to c and p). 3.3
Computation with Continuous Demand
We show that the problem of finding efficient quantities in the continuous case can be expressed as the problem of minimizing an additively separable, convex function subject to linear constraints. This is perhaps one of the simplest classes of convex programming problems since, for many convex functions, such problems can be solved by a simple greedy type algorithm.7 The problem of finding quantities q can be rewritten as the problem of P finding cumulative quantities: For all i ∈ N, let Qi = j≤i qj , and let Q0 = 0. Then the cost minimization problem becomes min Q
subject to
X
Z ci (Qi − Qi−1 ) + pi
i∈N
Qi Qi−1
(1 − F (x))dx
(3)
∀i ∈ N, Qi−1 ≤ Qi ≤ Qi−1 + Ki , and Qn ≥ D
The objective function is additively separable in the Qi ’s. In particular, the terms in eqn. (3) that involve Qi have the form H(Qi ) = ci Qi − ci+1 Qi + pi (Qi − Fˆ (Qi )) − pi+1 (Qi − Fˆ (Qi ))
(4)
where Fˆ is the integral of F . It is sufficient to show that H(Qi ) is convex, which is done by showing 7
See Tang (1990), and software references at http://www-fp.mcs.anl.gov/otc/Guide/SoftwareGuide/Categories/constropt.html.
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that its second derivative is non-negative. The first derivative is ci − ci+1 + pi (1 − F (Qi )) − pi+1 (1 − F (Qi )) The second derivative is −pi (f (Qi )) + pi+1 (f (Qi )) ≥ 0 where the inequality follows from the fact that pi ≤ pi+1 . Hence the objective function is also convex. In Section 5.1, we observe that this result also holds when stand-by costs are not necessarily linear. 3.4
Computation with Discrete Demand
In this section we address the problem of finding an efficient algorithm to determine the optimal list of quantities when demand d is discrete. We do this in two different ways. The first is by writing the cost minimization problem as a transportation problem; the second, as a shortest-path problem. We provide both methods because the first should be a transparent representation of our basic model, while the shortest-path setup permits various generalizations of the suppliers’ cost structure. 3.5
Transportation Problem
Our transportation problem has n ‘supply’ nodes, D ‘demand’ nodes, and n(D + 1) edges. Thus using a polynomial time minimum cost flow algorithm, one can solve this transportation problem in time polynomial in n and D. The dependence on D is unavoidable; to describe a discrete distribution over the integers in [0, D] requires D + 1 numbers. The actual complexity will depend on which minimum cost flow algorithm is used. For example, the enhanced capacity scaling algorithm would have, on our problem, a complexity 11
of O([nD log(n + D)] · [nD + (n + D) log(n + D)]); see Ahuja et al. (1993). To describe the transportation problem, we introduce one supply node for each supplier. The capacity of the supply node will be Ki . We also introduce one demand node for each integer in [1, D], with a demand of 1 unit. From each supply node we insert directed edges to each demand node in [1, D]. Thus each directed edge is identified by its supplier node and demand node. An edge from supplier node i to demand node j ∈ [1, D] is assigned a weight equal to the expected cost of supplier i supplying the jth unit: wij = ci + pi · P (d ≥ j) = ci + pi (1 − F (j − 1)). The goal is to find a minimum weight flow from the supply nodes such that both (i) the demand of each demand node is satisfied, and (ii) no supply node sends more flow than its capacity. The problem can be formulated as a linear program in the usual way: min
X X
wij xij
i∈N j∈[1,D]
subject to
X
xij ≤ Ki ∀i ∈ N
j∈[1,D]
X
xij = 1 ∀j ∈ [1, D]
i∈N
0 ≤ xij ≤ 1 ∀i, j where xij is the amount of flow sent from supplier i to demand node j. 3.6
Shortest Path Problem
Our optimization problem can also be formulated as one of finding a shortest path through a network that has O(nD 2 ) vertices. Thus, using standard shortest-path algorithms (Ahuja et al. (1993)) the winning set of bids can again be found in time polynomial in n and D. P Given a list of quantities q, let Qi = j≤i qj denote the cumulative quan12
tity potentially supplied by suppliers 1 through i. The discrete analog of eqn. (1) for the expected cost for supplier i as a function of quantities becomes Ci (q, c, p) = ci qi + pi
qi X
P(d ≥ x + Qi−1 )
x=1 Qi X
= ci (Qi − Qi−1 ) + pi
P(d ≥ x)
x=Qi−1 +1 Qi −1
X
= ci (Qi − Qi−1 ) + pi
(1 − F (x))
x=Qi−1
Though we write i’s cost as a function of q, i’s cost only depends on Qi and Qi−1 . When qi > 0, this cost is equal to i’s stand-by cost for those units, plus his expected production cost from potentially supplying units Qi−1 + 1 through Qi . In a slight abuse of our previous notation, it is convenient to denote i’s expected cost from potentially supplying units j + 1 through k (where 0 ≤ j < k ≤ D) as Ci (j, k) = ci (k − j) + pi
k X
P(d ≥ x)
(5)
x=j+1
where Ci (j, j) ≡ 0. Recall that suppliers in N are in increasing order of production cost. We construct a network whose paths assign suppliers intervals [j, k], such that consecutive suppliers have consecutive intervals. The network consists of one source node, one sink node, and n layers of nodes—one per supplier. The ith layer contains a node for each interval [j, k] ⊂ [0, D] such that j ≤ k and k − j ≤ Ki + 1. Such a node in the ith layer is denoted [j, k](i) , and represents supplier i potentially supplying 13
units (j + 1) through k. Some nodes are connected by edges as follows. • From the source node to each node in the first layer of the form [0, j](1) , there is an edge of length zero. • For i = 1, . . . , n − 1, from each node [j, k](i) to each node of the form [k, `](i+1) , there is an edge of length Ci (j, k). • From each node in the nth layer of the form [j, D](n) , there is an edge to the sink node of length Cn (j, D). A shortest path through this network finds an optimal list of quantities. Each node on the shortest path determines each supplier’s quantity, and in what order its supply is demanded. The network has O(nD 2 ) nodes and O(nD 2) edges. Using a version of the label correcting algorithm for shortest paths the shortest path in our network can be found in O(n2 D 4 ) steps (see Ahuja loc. cit.).
4
The Mechanisms
The social optimization problem we have discussed concerns minimizing the sum of (expected) suppliers’ costs. If demand d were known (i.e., there were no uncertainty about d), then there would be no meaningful distinction between stand-by costs and production costs. It would be straightforward to implement, in dominant strategies, efficient quantities q ∗ by using a Clarke– Groves mechanism8 in the usual deterministic sense. Such a mechanism would specify lump-sum payments to be made to the suppliers in exchange for the (known) quantities they need to supply. There would be no need to distinguish between an option price and a strike price when there is no uncertainty. 8
See Clarke (1971) and Groves (1973).
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In our setting, however, demand is uncertain. Supply commitments must be made ex ante, so a mechanism needs to pay each supplier an option price up front, and a strike price contingent on supply being obtained from that supplier. Our approach is to extend the notion of Clarke–Groves payments to the stochastic case. The idea of Clarke–Groves mechanisms is essentially to prescribe payments in direct proportion to a supplier’s contribution to social efficiency. This has the effect of aligning a supplier’s incentive to reveal his costs with the social objective of maximizing efficiency. Since, in our setting, the social objective concerns expected costs, such a mechanism should align expected payments in relation to the effect an agent’s presence has on the optimal choice of quantities. Under any of these mechanisms, risk-neutral suppliers will find it a dominant strategy to truthfully reveal their costs. We now define our class of mechanisms for the case in which d is a continuous variable. For each supplier i ∈ N, let q ∗\i be efficient quantities subject to the additional constraint that qi = 0, that is, X
q ∗\i ∈ arg min
5
q K,qi=0
Cj (q, c, p)
(6)
j∈N
These quantities would be efficient if supplier i did not exist. A mechanism ϕ is an Expected Groves Mechanism if for all reported costs c, pˆ) = (˜ c, p˜, q ∗ ), then both (i) q ∗ is efficient cˆ, pˆ ∈ Rn+ , it is the case that if ϕ(ˆ for (ˆ c, pˆ), and (ii) for all i ∈ N, c˜i qi∗ =
Z + p˜i X j6=i
0
qi∗
(1 − F (
X
qj∗ + x))dx
j 0. II Suppose supplier 1’s actual costs are (c1 , p1 ) = (3, 1), and that he believes P (d = 1) = 1. As above, by reporting his costs truthfully, supplier 1 would not supply an option, and would receive a net profit of 0. By mis-reporting his costs to be (2, 1), however, the mechanism buys an option from him, again, as described above. Under his beliefs in this case, supplier 1 perceives his expected payoff p1 − p1 ) = c˜1 − 3 + p˜ − 1. This from misreporting to be c˜1 − c1 + P (d = 1)(˜ is a profitable misrepresentation if c˜1 + p˜ − 4 > 0. In order for supplier 1 to have the incentive to report his costs truthfully in both cases I and II, we need both c˜1 ≤ 1 and c˜1 + p˜ ≤ 4. These are, together, incompatible with eqn. (8).
5
Extensions
In this section, we discuss natural extensions of the basic model, and their implications on computation. First, we discuss more general (non-linear) 12
A similar example can be constructed with P (d = 1) = .
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cost structures. Second, we allow the buyer to secure potential supply for fewer than D units of demand, when there is a per-unit cost of doing so. This models situations in which the buyer may find it efficient to procure, say, the D th unit of demand in a spot market when stand-by costs are high, relative to the probability of actually requiring a D th unit of demand. In each case, there is a straightforward generalization of the class of EVCG mechanisms (and, more generally, the class of Expected Groves Mechanisms). We omit the obvious formalization of such generalizations. 5.1
Non-linear costs
Stand-by costs. A more general setting than ours would specify non-linear stand-by costs. For instance, it may be the case that a supplier has decreasing (or increasing, or arbitrary) marginal stand-by costs involved with making various quantities available for production. In the case of discrete demand, our results hold for arbitrary stand-by costs. The shortest-path network described in Section 3.6 should be altered merely by changing the length of each edge in accordance with the cost of supplier i potentially supplying units (j + 1) through k. That is, eqn. (5) would be modified to be Ci (j, k) = γi (k − j) + pi
k X
P(d ≥ x)
x=j+1
where γi is a function of the quantity k − j, equal to the sum of marginal stand-by costs of supplier i’s first through (k − j)th units. In the continuous case, if stand-by costs are convex rather than linear, it is simple to see that we still have a well-behaved minimization exercise. That is, the corresponding version of eqn. (3) still yields an additively separable, convex function.
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Production costs. Another obvious way our original setting should be generalized is by considering production costs to be non-linear. In the discrete case, for example, suppose that the cost to supplier i of supplying the k th unit is pik , and that the supplier has non-decreasing marginal production costs, so pi1 ≤ pi2 · · · ≤ piKi . In this case, the shortest-path network described in Section 3.4 can be modified to compute an efficient solution by ordering individual units in increasing order of cost, rather than ordering only suppliers. The description of the network is as follows. The network consists of one source node, one sink node, and
P
Ki
layers of nodes—one per potential unit of supply. The ith k layer contains a node for each interval [j, j + `] ⊂ [0, D] such that ` ∈ {0, 1}. (ik ) , and represents the Such a node in the ith k layer is denoted [j, j + `]
kth unit of supply from supplier i either being not supplied (if ` = 0), or being used to potentially fill the j + 1st unit of demand (if ` = 1). The ik th layer precedes the i0k0 th layer only if pik ≤ pi0 k0 . Edges are constructed in the same way as in the previous shortest-path network.
This network is referred to as “S2” in Table 1 below. It is constructed with no significant increase in complexity: Such a network would still have O(nD 2 ) vertices. The astute reader might observe that this modified shortest-path network allows for infeasible solutions. That is, some paths correspond to a solution in which a supplier’s second unit of supply is used, but his first unit is not. However, since production costs are increasing, such an (infeasible) solution cannot be optimal. Both costs increasing. Finally, and most generally, if both stand-by costs and production costs are weakly increasing, the discrete computation problem can be formulated as an assignment problem. On one side of the problem, 24
there is a set of nodes, each node corresponding to a unit of demand in [1, D]. On the other side, there is, for each agent, a node for each unit of potential supply. Each demand-node is connected to a supply-node by an edge weighted by the expected cost of supply (in a way similar to that of the transportation weighting in Section 3.5).
P One side of the market has D nodes, while the other has Ki . This is P P solvable in time polynomial in n and D, i.e. O((D + Ki )(D Ki )) (Ahuja et al. (1993)).
ci ’s constant increasing other
constant T,S1,S2 S1,S2 S1,S2
pi ’s increasing S2 S2
other
Table 1: Networks representing various cost structures. T is the transportation network of Section 3.5; S1 is the shortest-path network of Section 3.6; S2 is the shortest-path network of Section 5.1.
5.2
Spot Markets
In certain situations (including the electricity market), a firm may not only rely on supply via options, but also may choose to rely on supply from a spot market, internal production, inventory, etc. In particular, at the time that demand is realized, the firm may purchase supply at a spot-price s. The spot-price may either be known in advance or a random variable. In the former case, the spot market may be viewed as a supplier that has a stand-by cost of zero, and unlimited capacity. Furthermore, this “supplier,” if used, must be paid a production price of s. When s is deterministic, it is straightforward to extend the model by treating the spot market as a supplier. Once EVCG payments are computed, 25
we know that the spot market must be paid at its actual “production cost” s, and need not be paid a stand-by fee. Since the spot market is not a strategic player, this does not destroy the incentive-compatibility of any Expected Groves mechanism. When s is not known until after options have been purchased, however, the generalization is not as straightforward, though the convexity result of Section 3.3 carries through. For instance, the expected cost to supplier i in eqn. (1) must be modified in order to account for the probability that s will exceed his production cost. That is, even after d is realized, we cannot determine which suppliers should be used until we know s. Precisely, suppose that first, options are purchased, second, d and s are realized, and third, demand is filled via the exercise of options and/or the spot market. It is (socially) efficient to exercise an option from supplier i only if pi ≤ s. If pi > s, it is cheaper to use the spot market without exercising that option. Thus, given quantities (qi )N , and assuming s and d are independent, the expected cost for supplier i is P(s < pi )ci qi + P(s ≥ pi )C(q, i) Total expected cost equals the sum of suppliers’ costs plus expected expenditure on the spot market. To show that this is convex, it suffices to show the convexity of expected expenditure on the spot market, which is X i∈N
Z sP(pi−1 < s ≤ pi )
D
Qi−1
where p0 = −∞.
(x − Qi−1 )f (x)dx
RD The term sP(pi−1 < s ≤ pi )[ Qi−1 (x − Qi−1 )f (x)dx] can, via integration
26
by parts, be written as
Z
Qi−1 )F (x)]D Qi−1
−
Z < s ≤ pi ) (D − Qi−1 ) −
D
sP(pi−1 < s ≤ pi ) [(x −
D
F (x)dx
Qi−1
which is equal to sP(pi−1
F (x)dx
Qi−1
The first derivative is −1+F (Qi−1), and the second derivative is f (Qi−1 ) ≥ 0, so we have convexity. Unfortunately, we do not necessarily have convexity when the variables are dependent.
6
Relation to Walrasian Prices
In certain auction environments, Vickrey–Clarke–Groves payments correspond to Walrasian prices—prices at which each bidder is consuming his most-preferred bundle of “goods.” For example, consider a special case of our model, with two suppliers, D = 1, and a 50% probability of realizing d = 1. Suppose the costs of the suppliers are (c1 , p1 ) = (1, 2) and (c2 , p2 ) = (2, 1). In this case, the efficient quantities are those that secure the single option from supplier 1. An EVCG mechanism prescribes payments to that supplier (˜ c1 , p˜1 ) that satisfy p = c2 + .5p2 = 2.5 c˜1 + .5˜
(9)
while supplier 2 receives zero payment. A Walrasian price vector for these efficient quantities is a price pair (ˆ c, pˆ) ∈ R2 , corresponding to the single option, such that supplier 2 does not (strictly) wish to offer an option at those prices. I.e., the expected rev-
27
enue to supplier 2 would not exceed the expected cost: cˆ + .5ˆ p ≤ c2 + .5p2 = 2.5
(10)
Note that by (9), payments (˜ c1 , p˜1 ) prescribed by any EVCG mechanism must be Walrasian prices. Furthermore, they are maximal : there is no Walrasian price vector that vector-dominates any EVCG payments. Crawford and Knoer (1981) consider a dynamic auction process that finds such Walrasian prices13 in a model where firms have additive valuations for heterogeneous workers. Their model can be thought of as a deterministic version of ours. As the example above suggests, when D = 1, the correspondence between EVCG payments and maximal Walrasian prices holds with any number of suppliers, with any costs. This equivalence also holds for the case D > 1 when suppliers do not have capacity constraints.14 Gul and Stacchetti (2000) show that this equivalence also holds when generalized versions of the economies defined by Crawford and Knoer (1981) are replicated. In other situations, though, it may be the case that EVCG payments cannot be decomposed into (maximal) Walrasian prices. Gul and Stacchetti show that (in the language of our procurement setting) in a model with more general valuations, expected EVCG payments are always weakly greater than the sum of the corresponding maximal Walrasian prices. If this inequality is strict, then for any decomposition of EVCG payments into per-object prices, at least one agent would prefer another consumption set (at those prices) to the one he is prescribed at the efficient outcome; in other words, EVCG payments cannot be decomposed into anonymous prices that support the 13
Since they are not in a procurement setting, they obtain minimal Walrasian prices. Their discussion is in terms of core outcomes. 14 Without capacity constraints, efficiency can be obtained by holding a separate secondprice auction for each individual option, equating it to the case D = 1.
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Unit of potential demand Supplier 1 2 3 1 c1 + g1 p1 c1 + g2 p1 c1 + g3 p1 2 c2 + g1 p2 c2 + g2 p2 c2 + g3 p2 3 c3 + g 1 p 3 c3 + g 2 p 3 c3 + g 3 p 3 Probability g1 g2 g3 Table 2: Expected costs for each unit of potential demand. The efficient quantities are bold-faced, assuming c1 < c2 < c3 and p1 < p2 < p3 . efficient outcome in the Walrasian sense.15 We demonstrate this with the following example. Let D = 3, N = {1, 2, 3}, and capacities be Ki = 2 for each i ∈ N. Let the suppliers’ costs be such that supplier 1 has unambiguously lower costs than supplier 2, who has lower costs than supplier 3. I.e., let c1 < c2 < c3 , and p1 < p2 < p3 .16 In this case, efficiency dictates that (as long as the support of F contains {1, 2, 3}), supplier 1 should be contracted to supply the first two potential units of demand, and supplier 2 should contract to supply the third. In the absence of supplier 1, supplier 2 moves up to supply the first two units, and supplier 3 supplies the third. Let g1 be the probability that d ≥ 1; let g2 and g3 be defined similarly. Then from eqn. (7), the expected value of supplier 1’s EVCG payment must be equal to (see Table 2) (c2 + g1 p2 ) + (c2 + g2 p2 ) + (c3 + g3 p3 ) − (c2 + g3 p2 )
(11)
A Walrasian price vector for the efficient quantities is a triple of price pairs (ˆ cj , pˆj )j=1,2,3 ∈ R6 , one for each potential unit of demand, such that 15
Bikhchandani and Ostroy (2000) consider deriving Vickrey payments via nonanonymous pricing. See also Bikhchandani et al. (2001). 16 Similar examples can be obtained without this ordering of costs.
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at those prices, each supplier maximizes expected profits by supplying his prescribed quantities. For example, since supplier 2 is prescribed to supply only the third unit, he has an additional unit of capacity available. Therefore, at Walrasian prices, supplier 2 must be unwilling to supply the first two units, i.e., cˆ1 + g1 pˆ1 ≤ c2 + g1 p2
(12)
cˆ2 + g2 pˆ2 ≤ c2 + g2 p2
(13)
Since supplier 1 is supplying the first two units of potential demand, the total of his expected payments from Walrasian prices is cˆ1 + g1 pˆ1 + cˆ2 + g2 pˆ2 From the inequalities (12) and (13), and since c2 < c3 and p2 < p3 , this expected payment must be strictly less than supplier 1’s EVCG payment given in (11). Therefore, this example shows that with capacity constraints, the correspondence between Walrasian prices and EVCG payments does not hold, even in our model of additive valuations.
7
Conclusion
We have examined a setting in which a buyer wishes to secure, in advance, a source of supply for unknown demand. The cost to obtaining this supply involves a commitment cost, called the stand-by cost, realized for all units committed to meet potential demand, plus a production cost, realized only for those units demanded ex-post. In order to achieve an ex ante efficient solution to the problem of securing options in this environment, one needs to examine the tradeoff between 30
the price of making units available, and the expected cost of actually producing them—a tradeoff which depends on the distribution of demand (see Section 3.1). This observation contrasts with the way the corresponding bid information is used in practice in various electricity markets, where bids are accepted only on the basis of the portion of the bid that corresponds to stand-by costs. In a world with no capacity constraints, our optimization problem can be easily solved in a decentralized way by running a standard (single-object) second-price auction on each unit of the good to potentially be supplied (e.g., first run an auction on the right to potentially supply the first unit of demand, then on the second unit, etc.). With linear costs, the outcome of one of these mini-auctions has no effect on the relevant parameters (and, hence, incentives) of the next one. With capacity constraints, however, this simple method does not work. Therefore, we examine centralized mechanisms that solve the constrained optimization problem discussed in Section 3, involving the probability information discussed above. We show that making use of this information does not lead to a computationally difficult problem. Furthermore, a class of mechanisms exists which implements this solution while making it a dominant strategy for risk-neutral suppliers to truthfully reveal their costs. From this class, we recommend a particular mechanism (Proposition 1) which both minimizes variance of suppliers’ profit and guarantees non-negative profits ex post. Our analysis differs from previous work by emphasizing the role played by the distribution of demand.
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