Optimal auctions for networked markets with externalities
OOptimal auctions for networked markets with externalities
Benjamin Heymann , Alejandro Jofr´e Abstract
Motivated by the problem of market power in electricity markets, we introduced in previ-ous works a mechanism for simplified markets of two agents with linear cost. In standardprocurement auctions, the market power resulting from the quadratic transmission lossesallows the producers to bid above their true values, which are their production cost. Themechanism proposed in the previous paper optimally reduces the producers’ margin tothe society’s benefit. In this paper, we extend those results to a more general marketmade of a finite number of agents with piecewise linear cost functions, which makes theproblem more difficult, but simultaneously more realistic. We show that the method-ology works for a large class of externalities. We also provide an algorithm to solvethe principal allocation problem. Our contribution provides a benchmark to assess thesub-optimality of the mechanisms used in practice.
Keywords:
Optimal auctions, mechanism design, allocation algorithm, electricitymarkets, fixed point.JEL classification: D44, D62, D82
1. Introduction
Our purpose in this paper is to show how oligopolistic behaviors in network marketscan be tackled using mechanism design. We point out that the optimal mechanism weobtain has a surprisingly simple expression. We complete this work with algorithmictools for the computation of this mechanism. Following a model already discussed in[1, 2, 3], we consider a geographically extended market where a divisible good is traded.In this proposal, each market participant is located on a node of a graph, and the nodesare connected by edges. The good can travel from one node to another through thoseedges at the cost of a loss. Since our initial motivation was the electricity market, we willdo the presentation with quadratic loss, but as explained thereafter, our results extendto a broad class of externalities. We are considering the usual transmission networkconstraints with the DC approximation (active power) for the losses.We will use the word principal to designate what could also be called in the centralizedmarket literature a central operator, or in the context of electricity markets, an ISO. Theprincipal, who aggregates the (inelastic) demand side, has to locally match -i.e. at eachnode - production and demand at the lowest expense through a procurement auction. As CMAP, Inria, Ecole polytechnique, CNRS, Universit´e Paris-Saclay, 91128, Palaiseau, France.CMM, Universidad de Chile, Santiago, Chile. CMM and DIM, Universidad de Chile, Santiago, Chile
Preprint submitted to Journal of Economic Theory July 25, 2019 a r X i v : . [ ec on . T H ] J u l rgued in [3], this setting can be applied to describe real electricity markets, but it couldalso be used in other markets where a good is being transported. Either way, there is aclear antagonism between the market participants: the operator wants to minimize hisexpected cost while the producers want to maximize their expected profits. Therefore, atthe same time that there is a transaction and a commitment between each agent and theprincipal, there also exists competition among the agents. In a standard procurementauction, the market power resulting from the quadratic line losses allows the producers tobid above their true values, or production cost [1]. The mechanism reduces the producers’margin and decreases the social cost represented in this case by the optimal value of theprincipal. This optimal auction design was introduced by Myerson in 1981 [4] for anon-divisible good and no externalities.We build on an electricity market model introduced by the second author in two previ-ous papers [2] and [1]. The authors wrote a brief presentation of this model in [5]. Othermodels were proposed for example in [6], [7], and [8], with a focus on the existence of amarket equilibrium. We pinpoint that if our initial motivation was electricity markets,network markets are used in other setting such as telecommunication [9]. Distributedmarkets were also studied in [10, 11], with a focus on efficiency and linear cost for trans-missions. For more information on the techniques we use in this paper the reader canrefer to [12], [13], [14], [15], the chapter 45 of [16] and [17] for general introductions onprincipal-agent theory, mechanism design, game theory and lattices theory respectively.In the sequel we consider, as we did in [3], that every participant knows the demandat each node before the interactions begin and that the production cost of each agentis private information. In a standard setting, the agents are the first to bid their costs,after which the principal, knowing the bids, minimizes his cost. In a standard setting, theprincipal is, therefore, a bid-taker. The producers know they influence the allocation andcompete with each other to maximize their individual profit. Since the demand is knownby everyone, everyone can guess the principal reaction once the bids have been announced:we can therefore virtually remove the principal from the interaction in the standardsetting and consider that the agents are players of a game with incomplete information(since the agents do not know their fellow agents’ preferences). This equivalence istrue provided that the agents are not communicating with each other. The mechanismchanges the payoff function of this game -subject to constraints we detail in this article-so as to minimize the principal’s expected cost before the bids are announced. Allowingthe principal to act first by revealing a committing rule gives him a strategic advantagein the negotiation.We restrict our discussion to deterministic demand, but the reasoning extends natu-rally to random demand as long as any possible realization of the demand satisfies themodel assumptions. Indeed, since the optimal mechanism constructed in this article isincentive compatible, then a random version (where the demand is revealed after theproducers’ bidding phase, as in [2]) would be realization-wise incentive compatible, andso incentive compatible. Observe that the mechanism we propose in the sequel could beadapted to elastic, piecewise linear demand.Our first main result is actually the mechanism design characterization. The resultis valid for a very general class of externalities as explained in the generalization section.This characterization of the optimal mechanism could be used to assess the sub-optimalityof the mechanisms used in practice.Interestingly, the allocation procedures for the optimal and the standard mechanism2re the same (one just needs to modify the input of the allocation procedure of thestandard mechanism to get the allocation of the optimal mechanism). Our second mainresult is a principal allocation algorithm based on a fixed point. The fixed point couldbe interpreted as cooperating agents trying to minimize a global criterion by sharingrelevant information. Our implementation of the algorithm gives good results againststandard methods. We point out that the numerical computation of the Nash equilibriumfor the procurement auction (important to compare the optimal mechanism and thestandard auction setting) requires an efficient algorithm to compute the allocation. Someother additional facts are presented within the paper: the smoothness of the allocationfunctions ( q and Q ), a decreasing rate estimation for the fixed point iterations, someresults of numerical experiments with the fixed point algorithm.We describe the market in the next section. In § §
5, we study the standard allocation problem and propose analgorithm to solve it. In §
2. Market description
The production cost of each agent is assumed to be piecewise linear, non-decreasingand convex in the quantity produced. This class of functions is sufficiently rich to rep-resent real-life problems and is sufficiently simple for theoretical study. In this work weneed to assume that the production levels at which there is a slope change are known inadvance and are exogenous - that is the agents cannot choose them-. Then, without lossof generality, we assume that there is a quantity ¯ q such that the changes of slope onlyoccur at the multiples of ¯ q . Thus, the authors find it practical to write the productioncost functions in the form C c ( q ) = N (cid:88) j =1 c j min(( q − ( j − q ) + , ¯ q ) , (1)where N ∈ N and the c j are some slopes coefficients specific to the agent, while q is thequantity produced. We will sometimes refer to the vector of the c j as the cost vector(of the agent). If we denote by q ji the quantity produced by agent i at marginal cost c ji ,then q ji = min(( q i − ( j − q ) + , ¯ q ), where q i is the total quantity produced by this agent.Let c ∗ < c ∗ ∈ R ∗ + and C a set of non-decreasing N -tuples of [ c ∗ , c ∗ ]. To each element c of C we associate the piecewise linear cost function q → C c ( q ). Throughout the paperwe set, for any c ∈ C , c N +1 = c ∗ to simplify the notations in some proofs. Note that inpractice a capacity constraint of the type q ≤ j ¯ q for a given agent can be implementedby setting its ( j + 1) th slope c j +1 equal to a big positive number. If an agent of costvector c produces a quantity q and receives a transfer x , then its profit is u i = x − C c ( q ) . (2)There are n agents numbered from 1 to n in the market. We denote I = [1 . . . n ] anduse generically the letter i to refer to a specific agent, and − i to refer to I \{ i } . We denote J = [1 . . . N ] and we will use generically j for the cost coefficients of the jth segment(starting from 1). The agents are dispatched on the n nodes of a graph. At each node3 we find the corresponding agent i and a local demand d i . The nodes are connectedby undirected edges. We write V ( i ) the set of nodes different from i connected to i .Obviously if i ∈ V ( i ) then i ∈ V ( i ). We denote E = { ( i , i ) : i ∈ V ( i ) } the setof undirected edges. For each ( i , i ) ∈ E , we introduce a quadratic loss coefficient r i ,i such that r i ,i = r i ,i . In the context of electricity markets, this quadratic coefficientcorresponds to the Joule effect within the lines. We make the non restricting assumptionthat N is big enough so that in what follows production at each node is smaller than¯ qN .We assume that both the agents and the principal are risk neutral: they maximizetheir expected profit. If the principal proposes to pay a price x i to agent i to make herproduce a quantity q i - this agent being free to accept or decline the offer- and if theagent i has a production cost defined by c i , then he accepts the offer if x i − C c i ( q i ) ≥ . (3)Then for agent i , either x i ≥ C c i ( q i ) or q i = 0. Thus, if the principal knew the costvectors c i , he would solve an allocation problem with those c i , and then bid to the agentsthe quantity and the payments corresponding to the solution of the allocation problem.But the principal does not know the cost vectors, and instead what happens is thatthe agents tell him some values for the c i (not necessarily their real cost vectors), andthen the principal decides based on those values. In this case, previous works [1] showedthat the agents could receive non-zero profits and bid above their production costs. Thequestion we now address is how to reduce their margins.To do so, we need to consider an intermediate scenario between the one in which theagent knows nothing (and is a price taker), and the one in which he knows everything(and therefore directly optimizes the whole system as a global optimizer). Each agent ischaracterized by an element f i , which is a probability density of support included in C and an element c i of C drawn according to f i . Only agent i knows c i , which is privateinformation. The other agents and the principal only know the probability f i with whichit was drawn. The density f i corresponds to the public knowledge on agent i ’s productioncosts so the principal won’t accept any bid c i that is not in the support of f i . We assumethat the cost slopes are not correlated for a given agent and between agents, i.e. theirlaws f ji are independent. In particular f i ( c i ) = (cid:81) j ∈ J f ji ( c ji ). In such situation, it makessense to define f − i ( c − i ) = (cid:89) i (cid:48) ∈ I \ i f i (cid:48) ( c i (cid:48) ) and f ( c , .., c n ) = (cid:89) i ∈ I f i (cid:48) ( c i ) , (4)and E (respectively E c − i ) the mean operator with respect to f (respectively f − i ). Thedensity f (resp. f − i ) represents the uncertainty from the principal’s (resp. agent i )perspective. To simplify notations we will use the symbol C n to denote the productof the supports of the f i s. We denote by Q the set of allocation functions - whichare the applications from C n to R n + , by X the set of payments functions -which arethe applications from C n to R n , and by H the set of flow functions - which are theapplications from C n to R E -. A direct mechanism is a triple ( q, x, h ) ∈ ( Q , X , H ). Let( q, x ) ∈ ( Q , X ). For this payment function and this allocation function, the expected4rofit of agent i of type c i and bid c (cid:48) i is U i ( c i , c (cid:48) i ) = E − i u i = X i ( c (cid:48) i ) − (cid:88) j ∈ J c ji Q ji ( c (cid:48) i ) . (5)where the capitalized quantities Q ji ( c i ) = E − i min(( q i ( c i , c − i ) − ( j − q ) + , ¯ q ) and X i ( c i ) = E − i x i ( c i , c − i ) (6)correspond to the average of their non capitalized counterpart. We also denote by V i ( c i ) = U i ( c i , c i ) . (7)the expected profit of agent i if he is of type c i and bids her true production cost.For i ∈ I , j ∈ J and c i ∈ C i let K ji ( c i ) = (cid:82) c ji c j − i f i ( c − ji , s ) ds/f i ( c i ) . We point out thatby independence of the laws of the c ji , K ji ( c i ) = (cid:82) c ji c j − i f ji ( s ) ds/f ji ( c ji ) = K ji ( c ji ). Thus K ji is simply the ratio of the cumulative distribution and the probability density for c ji .Our main assumption is the discernability assumption : for all i ∈ I and c i ∈ C i , thevirtual cost J i,j ( c ji ) = c ji + K ji ( c ji ) is increasing in j . As demonstrated in the next section,the virtual cost could be interpreted as the real marginal cost augmented by a marginalinformation rent. The assumption imposes the marginal information rent to be such thatfor any bid, the virtual marginal prices are increasing, i.e. the virtual production costfunction is convex. The assumption is necessary to show the independence property ofthe reformulation in Lemmas 6 and 7.This assumption implies the non overlapping working zones assumption : if we denoteby C i the support of f i , then C i should be of the form: C i = [ c − i , c i ] × . . . × [ c N − i , c N + i ] (8)with c − i < c i < . . . < c N − i < c N + i . We could interpret each segment over which theagent has a constant marginal cost as a working zone with identified productive assets.The expertise of the market participants should allow them to, based on the working zone,assess the marginal cost of the agent. This makes senses for instance if the setting isrepeated over time. This estimation need to be precise enough so that there is no chancethat it corresponds to another working zone. We use this assumption in particular in theproof of lemma 4. For simplicity we assume that c ji → c ji + K ij ( c ji ) is increasing in c ji . .This assumption can be withdrawn using the ironing technique introduced by Myersonwithout difficulty. To finish with the market presentation, we introduce the products ofthe type sets C n = (cid:81) i ∈ I C i (cid:48) and C − i = (cid:81) i (cid:48) ∈ I \{ i } C i (cid:48) .
3. Mechanism Design
We begin with the revelation principle as expressed in [18]. This is the piecewise linear adaptation of the classic monotone likelihood ratio property assumption heorem 1 (Revelation Principle) . To any Bayesian Nash equilibrium of a game ofincomplete information, there exists a payoff-equivalent direct revelation mechanism thathas an equilibrium where the players truthfully report their types.
According to the revelation principle, we can look for direct truthful mechanisms.Because, there is no reason why the agents should willingly report their types we needto add a constraint on the design to enforce truthfulness. This means that the profit ofany agent i of type c i should be maximal when agent i bids her true type c i i.e. for all( c (cid:48) i , c i ) U i ( c i , c i ) ≥ U i ( c i , c (cid:48) i ) . ( IC ) (9)This is the incentive compatibility (IC) constraint. In addition, since we want all agentsto participate in the market, we need the participation constraint imposing that for all c i U i ( c i , c i ) ≥ . ( P C ) (10)Without this constraint, the principal would optimize as if the agents would accept anydeal (even deals where they would make a negative profit). The last constraint is thatthe supply should be at least equal to the demand at every node. The supply availableat a given node is equal to the production augmented by the imports minus the exportsand the line losses. As explained earlier, there is a loss when some quantity h i,i (cid:48) ofthe divisible good is sent from one node i to another i (cid:48) . This loss is equal to r i,i (cid:48) h i,i (cid:48) ,where r i,i (cid:48) is a multiplicative constant. In order to obtain symmetric expressions, wewill proceed as if half of this quantity was lost by the sender, and the other half by thereceiver (see for instance [1]). Note that we could have equivalently used signed flows,but we would have lost some symmetry in the formulation. Then the supply and demandconstraint writes, for all i ∈ I and c ∈ C n , q i ( c ) + (cid:88) i (cid:48) ∈ V ( i ) h i (cid:48) ,i ( c ) − h i,i (cid:48) ( c ) − h i,i (cid:48) ( c ) + h i (cid:48) ,i ( c )2 r i,i (cid:48) ≥ d i . ( SD ) (11)We point out that for an optimal allocation (see §
5) , h i,i (cid:48) h i (cid:48) ,i = 0.The principal decision is a triple ( q, x, h ) ∈ ( Q , X , H ). This decision is made underthe constraints (IC), (PC) and (SD). Since we assume that the principal is risk neutral,his goal is to minimize his average cost, which translates mathematically by his criterionbeing equal to the expected sum of payments. Finally the optimal mechanism is thesolution of Problem 1. minimize ( q,x,h ) ∈ ( Q , X , H ) (cid:88) i ∈ I E x i ( c ) subject to ∀ c ∈ C n , ∀ i ∈ I : q i ( c ) + (cid:88) i (cid:48) ∈ V ( i ) h i (cid:48) ,i ( c ) − h i,i (cid:48) ( c ) − h i,i (cid:48) ( c ) + h i (cid:48) ,i ( c )2 r i,i (cid:48) ≥ d i ( SD ) ∀ c ∈ C n , ∀ ( i, i (cid:48) ) ∈ E : h i,i (cid:48) ( c ) ≥ ∀ i ∈ I, ∀ ( c (cid:48) i , c i ) ∈ C i : U i ( c i , c i ) ≥ U i ( c i , c (cid:48) i ) ( IC ) ∀ i ∈ I, ∀ c i ∈ C i : U i ( c i , c i ) ≥ P C ) .
6e now proceed to solve the optimal mechanism design problem, which is a functionaloptimization problem with an infinity of constraints, some of which are expressed withintegrals. The essential observation is that this complicated problem is equivalent to amuch simpler one. The proof relies on the comparison with two intermediate problems:
Problem 2. minimize ( q,x,h ) ∈ ( Q , X , H ) (cid:88) i ∈ I E x i ( c ) subject to. ∀ c ∈ C n , ∀ i ∈ I : q i ( c ) + (cid:88) i (cid:48) ∈ V ( i ) h i (cid:48) ,i ( c ) − h i,i (cid:48) ( c ) − h i,i (cid:48) ( c ) + h i (cid:48) ,i ( c )2 r i,i (cid:48) ≥ d i ( SD ) ∀ c ∈ C n , ∀ ( i, i (cid:48) ) ∈ E : h i,i (cid:48) ( c ) ≥ ∀ i ∈ I, ∀ j ∈ J, ( c − j , t , t ) , ( c , . . . , t k , . . . , c N ) ∈ C i , : V i ( c , .., c j − , t , c j +1 .., c N ) − V i ( c , .., c j − , t , c j +1 .., c N ) = (cid:90) t t Q ji ( c , .., c j − , s, c j +1 .., c N )d s ( H ∀ i ∈ I, ∀ ( c, c (cid:48) ) ∈ c : ( c − c (cid:48) ) . ( Q i ( c ) − Q i ( c (cid:48) )) ≤ , ( H ∀ i ∈ I, ∀ c i ∈ C i : V i ( c i ) ≥ P C ) , and Problem 3. minimize ( q,h ) ∈ ( Q , H ) E (cid:88) i ∈ I (cid:88) j ∈ J q ji ( c )( c ji + K ji ( c ji )) subject to ∀ ( c, i ) ∈ C n × I : q i ( c ) + (cid:88) i (cid:48) ∈ V ( i ) h i (cid:48) ,i ( c ) − h i,i (cid:48) ( c ) − h i,i (cid:48) ( c ) + h i (cid:48) ,i ( c )2 r i,i (cid:48) ≥ d i ( SD ) ∀ c ∈ C n , ∀ ( i, i (cid:48) ) ∈ E : h i,i (cid:48) ( c ) ≥ . ∀ c ∈ C i , ∀ i ∈ I : x i ( c ) = (cid:88) j ∈ J q ji ( c ) c ji + (cid:90) c j + i c ji q ji ( c i . . . c j − i , t, c ( j +1)+1 . . . c N + i ; c − i )d t. The inequality on the scalar product in (H2) is the piecewise linear equivalent ofa monotonicity condition already encountered in [3]. The first two problems are verysimilar, but (IC) has been replaced by (H1) and (H2) and (PC) is expressed in terms of V instead of U . This replacement is a trick introduced by Myerson in his 1981 paper.We will show later on how we can compare Problems 2 and 3, but note that Problem 3is simpler, as the optimization part can be solved pointwise (and x can be deduced fromthis pointwise optimization). The main result of this paper is that the three problemshave the same solution. We derive some necessary conditions for a solution of Problem 1. In fact, we onlyuse constraint ( IC ) to deduce the two next results. The first lemma indicates that any7olution of the first problem should be such that Q is monotonous. This is a classic resultalready introduced in [4] and [3], for instance. The novelty here is that in the contextof piecewise linear production cost functions, this monotonicity result is expressed in avectorial sense. Lemma 1 ( Q monotonicity) . If ( q, x, h ) is admissible for Problem 1, then for all agent i ∈ I and all ( c i , c (cid:48) i ) ∈ C i ( c i − c (cid:48) i ) . ( Q i ( c i ) − Q i ( c (cid:48) i )) ≤ where . is the scalar product in R N .Proof. We omit the i in the proof, as it plays no role. First, let ( c, c (cid:48) ) ∈ C i by the (IC)constraint, U ( c, c ) ≥ U ( c, c (cid:48) ) and U ( c (cid:48) , c (cid:48) ) ≥ U ( c (cid:48) , c ) (13)i.e. X ( c ) − (cid:88) j ∈ J c j Q j ( c ) ≥ X ( c (cid:48) ) − (cid:88) j ∈ J c j Q j ( c (cid:48) ) X ( c (cid:48) ) − (cid:88) j ∈ J c j (cid:48) Q j ( c (cid:48) ) ≥ X ( c ) − (cid:88) j ∈ J c j (cid:48) Q j ( c ) . (14)We get the lemma after the summation of the two inequalities and simplification.Lemma 1 indicates that an agent should be producing less on average in his i thworking zone if he is bidding a higher marginal cost for this working zone. Lemma 2. If ( q, x, h ) is admissible for Problem 1 then for any agent (omitting i ) forany c , t and t V ( c , . . . , c j − , t , c j +1 , . . . , c N ) = V ( c , . . . , c j − , t , c j +1 , . . . , c N ) − (cid:90) t t Q j ( c , . . . , c j − , s, c j +1 , . . . , c N )d s (15) Proof.
The inequality U ( c, c ) ≤ U ( c, c (cid:48) ) implies that c (cid:48) → U ( c, c (cid:48) ) is maximal at c forany c ∈ C i . Moreover, t → U (( c , .., c j − , t, c j +1 .., c N ) , c ) = X ( c ) − (cid:88) k ∈ J \{ j } c k Q k ( c ) − tQ j ( c ) (16)is absolutely continuous, differentiable with respect to t for all c , and its derivative is − Q j ( c ). By definition of q j , Q j ≤ ¯ q . The envelope theorem yield the result. We derive some necessary conditions for a solution of Problem 2.
Lemma 3. If ( q, x, h ) is an optimal solution to Problem 2 then (omitting i ) for all c ∈ C i V ( c ) = (cid:88) j ∈ J (cid:90) c j + c j Q j ( c . . . c j − , t, c ( j +1)+ , . . . , c N + )d t. (17)8 roof. According to (H1) (cid:88) j ∈ J (cid:90) c j + c j Q j ( c . . . c j − , t, c ( j +1)+ , . . . , c N + )d t = (cid:88) j ∈ J V ( c , .., c j − , c j , c ( j +1)+ , . . . , c N + ) − V ( c , .., c j − , c ( j )+ , . . . , c N + )= V ( c ) − V ( c , . . . , c N + ) . This is an expression for V ( c ) as a sum of a positive function of c and a constant V ( c , . . . , c N + ). It is clear that to optimize the criteria, this constant should be assmall as possible. The participation contraint (PC) imposes that V ( c , . . . , c N + ) ≥ V ( c , . . . , c N + ) = 0.A consequence of this is: Corollary 1. If ( q, x, h ) is an optimal solution of Problem 2 then for all i ∈ I , V i ( c i , . . . , c N + i ) = 0 . (18) Proof.
See the proof of Lemma 3.Corollary 1 means that if an agent bids a production cost function that is the max-imum of what he could bid, he should not make any profit, which is why he should bepaid exactly his production cost. We see with this lemma that if the public informationis inaccurate and the real cost of an agent is higher than what could be expected, thenthere is a risk that the participation constraint is not satisfied. On the other hand, itshould not be surprising that an agent can have a zero profit: remember that in theextreme case in which the principal knows everything (discussed in § Lemma 4. If ( q, x, h ) is an optimal solution of Problem 2, the expected profit of agent i (over his type) is E V i ( c ) = (cid:88) j ∈ J (cid:90) ( c ..c n ) ∈ C i Q ji ( c , . . . , c j , c ( j +1)+ , . . . c N + ) K ji ( c ) f i ( c )d c. (19) Proof.
By Lemma 3 and Fubini’s lemma, E V i ( c ) is equal to E (cid:88) j ∈ J (cid:90) c j + c j Q ji ( c , . . . , c j − , t, c ( j +1)+ , . . . c N + )d t = (cid:88) j ∈ J (cid:90) c − j ∈ C − j (cid:90) c j + c j = c j − (cid:90) c j + t = c j Q ji ( c , . . . , c j − , t, c ( j +1)+ i , . . . c N + i ) f i ( c )d t d c j d c − j . (cid:90) c j + c j = c j − (cid:90) c j + t = c j Q ji ( c , . . . , c j − , t, c ( j +1)+ , . . . c N + ) f i ( c )d t d c j = (cid:90) c j + t = c j − (cid:90) tc j = c j − Q ji ( c , . . . , c j − , t, c ( j +1)+ , . . . c N + ) f i ( c )d c j d t = (cid:90) c j + t = c j − Q ji ( c , . . . , c j − , t, c ( j +1)+ , . . . c N + )( (cid:90) tc j = c j − f i ( c )d c j )d t = (cid:90) c j + t = c j − Q ji ( c , . . . , c j − , t, c ( j +1)+ , . . . c N + )( (cid:90) tc j = c j − f i ( c ) f i ( c − j , t ) d c j ) f i ( c − j , t )d t = (cid:90) c j + t = c j − Q ji ( c , . . . , c j − , t, c ( j +1)+ , . . . c N + ) K ji ( t ) f i ( c − j , t )d t = (cid:90) c j + c j = c j − Q ji ( c , . . . , c j − , c j , c ( j +1)+ , . . . c N + ) K ji ( c j ) f i ( c i )d c j We get the lemma by summing all the inner terms.
Lemma 5.
If (H1) is satisfied, then for any ( a, b ) ∈ C i (omitting i ) X ( a ) − X ( b ) = (cid:88) j ∈ J [ a j Q j ( a ) − b j Q j ( b ) + (cid:90) b j a j Q j ( b . . . b j − , t, a j +1 . . . a N )d t ] (20) Proof.
Because of its length the proof is detailed in Appendix Appendix A
Lemma 6. If ( q, x, h ) verifies (H1) and (H2) and Q ji is independent of c j (cid:48) i for j (cid:48) > j ,then for all ( c, ˜ c ) ∈ C U ( c, c ) ≥ U ( c, ˜ c ) . (21) Proof.
Since (H1) is satisfied, equation (20) of Lemma 5 applies. We combine this relationwith the definition of the expected profit U from (5). We obtain: U ( c, c ) − U ( c, ˜ c ) = (cid:88) j ∈ J c j Q j ( c ) − ˜ c j Q j (˜ c )+ (cid:90) ˜ c j c j Q j (˜ c , ..., ˜ c j − , t, c j +1 , ...c N )d t + c j Q j (˜ c ) − c j Q j ( c )= (cid:88) j ∈ J ( c j − ˜ c j ) Q j (˜ c , ..., ˜ c j − , ˜ c j )) + (cid:90) ˜ c j c j Q j (˜ c , ..., ˜ c j − , t )d t = (cid:88) j ∈ J (cid:90) ˜ c j c j Q j (˜ c , ..., ˜ c j − , t ) − Q j (˜ c , ..., ˜ c j − , ˜ c j )d t, where we used the independence hypothesis for the second equality. By (H2), whichimplies the decreasingness of Q j with respect to c ji when all other quantities are fixed,10f c j < ˜ c j then for any t ∈ [ c j , ˜ c j ], Q j ( t ) − Q j (˜ c j ) ≥
0. Otherwise, we use the formula (cid:82) ba = − (cid:82) ab and the fact that any t ∈ [˜ c j , c j ] verifies Q j ( t ) − Q j (˜ c j ) ≤
0. Therefore U ( c, c ) − U ( c, ˜ c ) is non negative. We derive some properties for Problem 3.
Lemma 7.
There is an optimal solution ( q, x, h ) for Problem 3 such that q ji (and Q ji )is independent of c ki for k (cid:54) = j .Proof. First note that x is not taking any role in the optimization problem: it is definedafterward. The only real optimization variables are then q and h . Remember that q ji isdefined as a function of q by q ji = min(( q i − ( j − q ) + , ¯ q ). The constraints are defined foreach c ∈ C n and the integral criterion is in fact a sum of independent criteria dependingon q ( c ) for c ∈ C n . Therefore we can solve Problem 3 with a pointwise optimization.By the discernability assumption , for any c ∈ C n and i ∈ I , c ji + K ji ( c ji ) is increasingin j . Therefore for all c ∈ C n , i ∈ I , (cid:80) j ∈ J q ji ( c )( c ji + K ji ( c ji )) is a convex criteria in q i and therefore the pointwise problem corresponds to Problem 4 of §
5. In particular, wecan apply Lemma 10 from the next section. Thus q ji only depends on c ji and c − i . Thisproperty is preserved by integration over the c − i : Q ji only depends on c ji .We point out that, since the pointwise problem has a unique solution, the pointwiseoptimal solution introduced in the proof is uniquely defined. Theorem 2. If ( q, x, h ) is the pointwise optimal solution of Problem 3 and K ji is smoothin c ji for ( i, j ) ∈ I × J and c ∈ C i , then for all i ∈ I , Q i is C ∞ over C i .Proof. Remember that c ji → c ji + K ji ( c ji ) is increasing, so by composition with smoothbijection, we can do the proof as if the costs involved were c ji instead of c ji + K ji ( c ji ).According to Lemma 11, q i is continuous. Since q i is bounded, we can apply the domi-nated convergence theorem to show that Q i is continuous. We can then we proceed bymathematical induction. Assume that Q i is C l , then take c i ∈ C i and c ki a sequencein C i that converges to c i . Since ˆ S = ∪ k ∈ N S ( c ki ) is a countable union of null measuredset (by Lemma 22), its measure is zero. Without changing the results, we can computethe integrals on C − i \ ˆ S instead of C − i . Since q i and its derivatives are bounded, wecan apply the dominated convergence theorem to compute the limit of Q ( l ) i ( c i ) − Q ( l ) i ( c ki ) c i − c ki as k goes to + ∞ as the integral of a limit. Since we removed the point over which thislimit was not defined, we get that Q ( l ) i ( c i ) − Q ( l ) i ( c ki ) c i − c ki has a limit, and this limit does notdepend on the sequence c ki . So Q i is l + 1 times derivable at c i , for all c i . We concludeby induction. Last but not least, we state the main result of the Section.11 heorem 3.
Let ( q ji , h ) be defined such that for any c ∈ C n , ( q ji ( c ji , c − i ) , h ( c )) solvesminimize q ji ,h (cid:88) i ∈ I (cid:88) j ∈ J q ji ( c ji , c − i )( c ji + K ji ( c ji )) subject to ≤ q ji ≤ ¯ q (cid:88) j ∈ J q ii ( c ji , c − i ) + (cid:88) i (cid:48) ∈ V ( i ) h i (cid:48) ,i ( c ) − h i,i (cid:48) ( c ) − h i,i (cid:48) ( c ) + h i (cid:48) ,i ( c )2 r i,i (cid:48) ≥ d i h i,i (cid:48) ( c ) ≥ , and set q i ( c ) = (cid:88) j ∈ J q ji ( c ji , c − i ) and x i ( c ) = (cid:88) j ∈ J q ji ( c ji , c − i ) c ji + (cid:90) c j + i c ji q ji ( t, c − i )d t, (22) then ( q, h, x ) solves the optimal mechanism design problem (Problem 1).Proof. • First note that ( q, h, x ) is the pointwise solution of Problem 3 so it is optimalfor Problem 3, moreover, by construction ( q, h, x ) satisfies (SD) and h ≥ • Then note that by Lemma 4, ( q, h, x ) solves a relaxation of Problem 2, but is itadmissible for Problem 2 ? • By definition of V (omitting i ), V ( c . . . a j . . . c N ) − V ( c . . . b j . . . c N ) = E x ( c . . . a j . . . c N ) − x ( c . . . a j . . . c N ) − [ Q j ( a j ) a j − Q j ( b j ) b j ] = E q ji ( a j , c − i ) a j + (cid:90) c j + i a j q ji ( t, c − i )d t − E q ji ( b j , c − i ) b j − (cid:90) c j + i b j q ji ( t, c − i )d t − [ Q j ( a j ) a j − Q j ( b j ) b j ] = E (cid:90) b j a j q ji ( t, c − i )d t = (cid:90) b j a j Q ji ( t )d t where we used the definition of x , the definition of Q and Fubini lemma’s for thesecond, third and fourth equalities. Threfore ( q, h, x ) satisfies (H1). • By construction, q ji is non-increasing in c ji + K ji ( c ji ) then using the third assumption, q ji is non-increasing in c ji so for any ( a, b, c − i ) ∈ C × C − i , ( a ji − b ji )( q ji ( a ji , c − i ) − q ji ( b ji , c − i )) ≤
0, so by integration with respect to c − i , ( a ji − b ji )( Q ji ( a ji ) − Q ji ( b ji ) ≤ j , ( c − c (cid:48) ) . ( Q i ( c ) − Q i ( c (cid:48) )) ≤
0, i.e. (H2) is satisfied. • Since (H1) is satisfied, V i ( c i ) ≥ V i ( c + i ). Moreover, V i ( c + i ) = 0 by construction of x .Therefore the participation constraint (PC) is satisfied. • Therefore ( q, h, x ) is admissible for Problem 2. So it solves Problem 2. • Since ( q, h, x ) solves Problem 2, by Lemma 6 the incentive compatibility constraint(IC) is satisfied. Moreover, by Lemma 3, (PC) is satisfied. Thus ( q, h, x ) is admis-sible for Problem 1, but is it optimal ?12
By Lemmas 1 and 2, any optimal solution of Problem 1 should be admissible forProblem 2. Since the criteria are the same, we conclude that ( q, h, x ) is an optimalsolution of Problem 1.
In the optimal mechanism, the agents are paid at a marginal price which is equal totheir bid augmented by an information rent. This information rent depends on the prob-lem structure since it is built from a collection of allocation problems, and it depends onthe available information by the fact that, in these optimization problems, the marginalprices are replaced by the virtual marginal prices c ji + K ji ( c ji ). We point out that, asalready noted in [19], the computation of such rent may pose a practical difficulty forlarge problems.Notice that, by construction, the optimal mechanism is incentive compatible no mat-ter the value of K since (H1) is verified anyway as long as the hypotheses are satisfied.If this market is repeated over time, the principal can learn the distribution of the pro-ducers’ cost parameters.The model extends to the case in which some nodes do not have a producer and wherefor others, the demand is null. In particular, we can consider the buyer/suppliers settingwhere there is demand only at one node.One may argue that one limit of the current result is that it does not take intoaccount any network constraints. Nonetheless, the structure of the proof makes it clearthat we exploited only some properties of the allocation problem. Therefore, the optimalmechanism construction is valid for any market for which the allocation problem satisfiesthese properties. We detail this argument in the next section.In addition, the optimal mechanism construction is valid for limiting case in which r = 0 at some edges. In this case, one needs to specify the definition of q since thesolution of the allocation problem may not be a singleton.
4. Extension to General Network Constraints
We now explain why the optimal mechanism proposal can be extended to a largevariety of network constraints. This extension is of particular importance for power mar-ket networks, since the admissibility of an allocation is subject to its physical feasibility.While this difficulty can be avoided as long as the network is radial, the general case isknown to bring its lot of technical challenges. As argued in [20] , the allocation problemcan be written: min (cid:80) i J i ( q i )s.t. g i ( h ) + q i = d i , i = 1 , , . . . , nAh + Bq = bh ∈ H, q ∈ Q (23)With g i being concave functions, A and B are p × m and p × n real matrices; b ∈ R p ,and H and Q are (convex) products of segments in R m and R n respectively.13hey observe that whenever the multipliers of the first set of constraints in (23) arepositive (*), then the optimal solution of (23) is also a solution ofmin (cid:80) i J i ( q i ) s.t. ( h, q ) ∈ C (24)where C = { ( h, q ) ∈ X s.t. g i ( h ) + q i ≥ d i , i = 1 , , . . . , n } (25)and X = { ( h, q ) s.t. Ah + Bq = bh ∈ H, q ∈ Q } . (26)We assume that (*) is satisfied. Otherwise said, we require the sub-gradient of the valueof (24) to be positive. We could equivalently require the uniqueness of the solution of(24).We can then focus on the study of (24). We denote by δ C the support function of C and set U = { u = ( u , . . . , u n ) | u i ≤ } Applying Theorem 10.1 from [21], we get that anecessary and sufficient condition for an allocation to be optimal is that:0 ∈ ∂ (cid:88) i J i ( q i ) + δ C ( h, q ) , (27)Now observe that ∂δ C ( h, q ) = N C ( h, q ) = (28) (cid:40) z − (cid:88) i y i ∇ ( g i ( h ) + q i )( h, q i ) | y ∈ N U ([ g i ( h ) + q i ] i ) , z ∈ N X ( h, q ) (cid:41) (29)The last equation requires the qualification constraint (Q) from [22] to be satisfied, soone can use Theorem 4.3 from [22]. Still, note that no matter Q being satisfied, N C doesnot depends on c . Theorem 4.
If (*) is satisfied, then q ji is independent of c ki for k (cid:54) = j , moreover, themechanism proposed ib the previous section can be applied to (23) .Proof. Fix c and consider the associated optimal allocation q . Observe that either q ji ∈ ]0 , ¯ q ji [ or q ji ∈ { , ¯ q ji } . First case. If q ji ∈ ]0 , ¯ q ji [, take k (cid:54) = j then c ki does not appear in the first order condition(27). By Berge’s Maximum Principle [23], the optimal allocation is upper hemicontinuouswith respect to the parameter c ki , by unicity of the solution of (24), we get that q i iscontinuous with respect to c ki . Thus there is a neighbourhood of c ki such that q ji is stillin ]0 , ¯ q ji [. In this neighbourhood, condition (27) is satisfied for q ji = q ji , by unicity of thesolution, q ji is constant with respect to c ki on this neighbourhood. Second case. q ji ∈ { , ¯ q ji } . Without loss of generality, let us assume that q ji = ¯ q ji . Here,we need to observe that the sub-differential of the criteria with respect to q i is [ c ji , c j +1 i ],thus the reasoning of the first case can be reproduce whenever k (cid:54) = j + 1. So we onlyneed to deal with the situation where k = j + 1. Moreover, since q i is non-increasing in c j +1 i , only an increase of c j +1 i can potentially trigger a change in q ji . Observe that by14erge’s Maximum Principle, q ji is continuous with respect to the parameter of interest c j +1 i . If it happens to take a value different than ¯ q ji , then this value is also a solution to(27) for the initial parameters, which is in contradiction with the unicity of the solutionof (27). We show here how the non-overlaping zone structure naturally emerges if we envisionslightly different context and adapt the notations accordingly:(1) we focus on a specific producer, and refer to him implicitly in this paragraph, (2)we assume the competition is known, (3) we only suppose the types distribution to havea density f over C , denoting by f j the marginals, (4) it will prove to be convenient touse q ∗ c := C c ( q ) and denote by j ( q ) the integer part of q/ ¯ q .Let q and x be some optimal allocation and payment rules for the producer. Weassume q and x are continuous and have derivatives almost everywhere. The producer’sprofit is x ( s ) − q ( s ) ∗ c whenever his type is c and he signals himself as of type s . Whenthe competition is known , we pinpoint that for any value q = q ( c ) of the allocationfunction, there should be a unique payment that we denote by x q . If it was not thecase, then the producer would be better off revealing what is required to get the highestpossible payment at q , which violates the incentive compatibility constraints (IC).We observe that, by (IC), x ( q ) − q ∗ c is maximal at q c , which implies in particularthat c j ( q ) ∈ ∂ q x ( q c ). Hence, almost everywhere and whenever q is not a multiple of ¯ q , c j ( q ) is uniquely defined. Setting c q = c j ( q ) , we observe that x q = (cid:82) q c t d t ( (cid:63) ).If we take s ∈ C such that q ( s ) is not a multiple of ¯ q , then by (IC) we know that c → x ( c ) − q ( c ) ∗ s should be maximal at s . Take t such that t j ( q ( c )) = s j ( q ( c )) and q ( t ) (cid:54) = q ( s ),then we should have j ( q ( t )) (cid:54) = j ( q ( s )) which would imply a discontinuity of q somewhere.Hence: whenever s j ( q ( s )) = t j ( q ( s )) , q ( s ) = q ( t ) ( (cid:63)(cid:63) ). Let C k = { c ∈ C, j ( q ( c )) = k } , c − k = inf { c k , c ∈ C k } , and c + k = sup { c k , c ∈ C k } . By combining ( (cid:63)(cid:63) ) with the monotonyof q , we get that the intersection of the [ c − j , c + j ] has an empty interior. We point out that a sufficient condition to check the monotone likelihood ratio prop-erty is that
F/f is increasing. If F is a smooth cumulative distribution function with f being the corresponding smooth and positive density, then F/f is increasing iff f /F isdecreasing iff ln F (cid:48) is decreasing iff ln F is concave. A function f is said to be log-concave if ln f is concave. Many density functions encountered in economic and engineering liter-ature are log-concave : the uniform, the normal, the exponential and the power functionand the Laplace distribution all have log-concave density functions. We refer to [24]for the results we use on this class of functions. The class of log-concave is stable bymonotonic transformation and truncation. Moreover, it happens that if a probabilitydensity distribution is log-concave, then the corresponding cumulative distribution islog-concave. In mechanism design theory, it is standard to assume F is log-concave [25].We want to see the implication of the discernability assumption . This assumptionimposes a gap ∆ equals to K ji ( c j + i ) between c j + i and c ( j +1) − i . We compute this gap for we still keep non decreasing marginal costs, otherwise we would loose the continuity of q c j − = 0 and write c j + = c + . This results in the following table: Table 1: The gap ∆ for some standard probabilities
Name ∝ f ( x ) ∝ F ( x ) K ( x ) ∆Uniform 1 x x c + Power Function λ ( xc + ) λ − c + ( xc + ) λ xλ c + λ Weibull λ ( xc + ) λ − e ( − xc + ) λ c + (1 − e − ( xc + ) λ ) c + λ ( c + x ) λ − ( e ( xc + ) λ − c + e − λ Laplace e − λ | x − c +2 | x > c + , − e − λc +2 e − λ ( x − c +2 ) λ λ ( e c +2 λ − λe − ( c + − x ) λ e − c + λ ( e xλ − − e − xλ λ − e − c + λ λ We truncate the probabilities so that they have support in [0 , c + ]. The symbole ∝ means that we express f and F modulo the multiplication by a common constant (dueto the truncation) and λ is a positive parameter that should be greater than 1 for thePower function and the Weibull probability. For the uniform distribution, we see thatthe intervals should be of non-decreasing sizes. For instance, one could take c ∈ [¯ c, c ], c ∈ [3¯ c, c ], c ∈ [5¯ c, c ], etc. For the Power, the Weibull and the exponential functions,we see that the gap could be made smaller.
5. Study of the allocation problem
The previous section motivates the study of the allocation problem for different rea-sons. Firstly, as we have already pointed out in the proofs, the results of § c ji , where as before i ∈ I corresponds to the ith agent and j ∈ J corresponds to the jth working zone with constant marginal price. To model the factthat the production costs are piecewise linear, we use some positive variables q ji so that q ji ≤ ¯ q , for any i ∈ I , the quantity produced by agent i is q i = (cid:80) j ∈ J q ji and the relatedproduction cost is (cid:80) j ∈ J c ji q ji . As before, an allocation should satisfy the constraint thatproduction exceeds demand. We end up with Problem 4:16 roblem 4. minimize ( q,h ) (cid:88) i ∈ I (cid:88) j ∈ J q ji c ji subject to ∀ i ∈ I : (cid:88) j ∈ J q ji + (cid:88) i (cid:48) ∈ V ( i ) h i (cid:48) ,i − h i,i (cid:48) − h i,i (cid:48) + h i (cid:48) ,i r i,i (cid:48) ≥ d i ( λ i ) ∀ ( i, i (cid:48) ) ∈ E : h i,i (cid:48) ≥ γ i,i (cid:48) ) ∀ ( i, j ) ∈ I × J : q ji ≥ µ i,j ) ∀ ( i, j ) ∈ I × J : q ji ≤ ¯ q ( ν i,j ) . (30)The notations for the dual the variables associated with each constraint are indicatedin parentheses. Those variables are in R + .For any node i ∈ I , we define the function F i for λ ∈ [min i c i , max i c Ni ] n F i ( λ i , λ − i ) = d i + (cid:88) i (cid:48) ∈ V ( i ) λ i (cid:48) − λ i r i,i (cid:48) ( λ i + λ i (cid:48) ) + ( λ i (cid:48) − λ i ) r i,i (cid:48) ( λ i + λ i (cid:48) ) . (31)Later on we justify that this function could be interpreted as the production of agent i when the multipliers are λ i and λ − i . Its partial derivative with respect to λ i is ∂ λ i F i ( λ i , λ − i ) = − (cid:88) i (cid:48) ∈ V ( i ) r i,i (cid:48) λ i (cid:48) ( λ i + λ i (cid:48) ) < . (32)The derivative is negative: when i increases its price it is assigned smaller productionquantities. The partial derivative of F i for i (cid:48) ∈ I \{ i } is ∂ λ i (cid:48) F i ( λ i , λ − i ) = (cid:40) r i,i (cid:48) λ i (cid:48) λ i ( λ i + λ i (cid:48) ) > i (cid:48) ∈ V ( i )0 else. (33)When another agent becomes less competitive, i is assigned more production. Let k ∈ J ∪ { } . The limit at + ∞ and 0 of F i ( x, λ − i ) − k ¯ q arelim x → + ∞ F i ( x, λ − i ) − k ¯ q = d i − k ¯ q − (cid:88) j ∈ V ( i ) r i,j (34)and lim x → + ∞ F i ( x, λ − i ) − k ¯ q = d i − k ¯ q + (cid:88) j ∈ V ( i ) r i,j . (35)Without loss of generality (otherwise we could impose capacity constraints), we assume,the first term to be strictly negative and the second to be strictly positive, hence by theintermediate value theorem, F i − k ¯ q has a zero. Since F i − k ¯ q is decreasing in λ i , thissolution is unique. Now we define for i ∈ I and k ∈ J ∪ { } , g ki as the function thatassociates any λ − i ∈ [min i c i , max i c Ni ] n − with the unique x such that and F i ( x, λ − i ) = k ¯ q and x > F i ( g ki ( λ − i ) , λ − i ) = k ¯ qg ki ( λ − i ) > . (36)17 emma 8. For any i ∈ I , k ∈ J ∪ { } , λ − i ∈ [min i c i , max i c Ni ] n − and i (cid:48) ∈ V ( i ) ∂ λ i (cid:48) g ki ( λ − i ) > . (37) In particular, g ki is increasing in λ i (cid:48) for i (cid:48) ∈ V ( i ) .Proof. According to the implicit function theorem ∂g ki ( λ − i ) ∂λ i (cid:48) = − ∂F i ∂λ i (cid:48) / ∂F i ∂λ i , (38)It is clear that g ki ( λ − i ) is decreasing in k . We proceed with the computation of thedual of Problem 4. If a strong duality theorem applies, then we should havemin q,h max λ,γ,ν,µ (cid:88) i ∈ I,j ∈ J q ji c ji + (cid:88) i ∈ I λ i { d i − ( (cid:88) j ∈ J q ji + (cid:88) i (cid:48) ∈ V ( i ) h i (cid:48) ,i − h i,i (cid:48) − h i,i (cid:48) + h i (cid:48) ,i r i,i (cid:48) ) }− (cid:88) i ∈ I,j ∈ J γ i,j h i,j + (cid:88) i ∈ I,j ∈ J ν i,j ( q ji − ¯ q ) − µ i,j q ji = max λ,γ,νµ min q,h (cid:88) i ∈ I λ i d i − (cid:88) i ∈ I,j ∈ J ν i,j ¯ q + q ji ( c ji + ν i,j − λ i − µ i,j )+ (cid:88) ( i,i (cid:48) ) ∈ E h i,i (cid:48) { λ i − λ i (cid:48) − γ i,j } + h i,i (cid:48) r i,i (cid:48) λ i + λ i (cid:48) , so that for any ( i, i (cid:48) ) ∈ E , by necessary and sufficient first order condition h i,i (cid:48) = γ i,i (cid:48) + λ i (cid:48) − λ i r i,i (cid:48) ( λ i (cid:48) + λ i ) . (39)By replacing h by its expression in the dual variables we get something equivalent tomaximize ( λ,γ,µ,ν ) (cid:88) i ∈ I { λ i d i − (cid:88) j ∈ J ν i,k ¯ q − (cid:88) i (cid:48) ∈ V ( i ) ( λ i − λ i (cid:48) − γ i,j ) r i,i (cid:48) ( λ i + λ i (cid:48) ) } subject to ∀ ( i, j ) ∈ I × J c ji + ν i,j ≥ λ i + µ i,j . (40)The expression of γ with respect to λ follows. For any ( i, i (cid:48) ) ∈ Eγ i,i (cid:48) = (cid:40) λ i ≤ λ i (cid:48) λ i − λ i (cid:48) else (41)thus the dual problem is equivalent tomaximize ( λ,µ,ν ) (cid:88) i ∈ I { λ i d i − (cid:88) j ∈ J ν i,j ¯ q − (cid:88) i (cid:48) ∈ V ( i ) ( λ i − λ i (cid:48) ) r i,i (cid:48) ( λ i + λ i (cid:48) ) } subject to ∀ ( i, j ) ∈ I × J c ji + ν i,j ≥ λ i + µ i,j , (42)18ecause µ does not play any role in the admissibility of the other variables nor in theobjective, this is equivalent tomaximize ( λ,ν ) (cid:88) i ∈ I { λ i d i − (cid:88) j ∈ J ν i,j ¯ q − (cid:88) i (cid:48) ∈ V ( i ) ( λ i − λ i (cid:48) ) r i,i (cid:48) ( λ i + λ i (cid:48) ) } subject to ∀ ( i, j ) ∈ I × J c ji + ν i,j ≥ λ i , (43)The expression of ν follows. For any ( i, j ) ∈ I × Jν i,j = (cid:40) λ i ≤ c ji λ i − c ji else. (44)We can now justify that we have strong duality: the operator is continuous, convex-concave and the dual variables are restricted to be in a bounded set.The dual of the allocation problem is therefore written:maximize λ ≥ (cid:88) i ∈ I { λ i d i − ¯ q (cid:88) j ∈ J ( λ i − c ji ) δ λ i ≥ c ji − (cid:88) i (cid:48) ∈ V ( i ) ( λ i − λ i (cid:48) ) r i,i (cid:48) ( λ i + λ i (cid:48) ) } , (45)where δ x ≥ y = (cid:40) x ≥ y i ∈ I we maximize the criteria λ i d i − ¯ q (cid:88) j ∈ J ( λ i − c ji ) δ λ i ≥ c ji − (cid:88) i (cid:48) ∈ V ( i ) ( λ i − λ i (cid:48) ) r i,i (cid:48) ( λ i + λ i (cid:48) ) , (47)which is strictly concave for any λ − i (sum of concave and strictly concave functions).We denote by Λ i ( λ − i ) its maximizer. The first order necessary and sufficient conditionon Λ i is: 0 ∈ F i (Λ i , λ − i ) − K i (Λ i ) , (48)where K i ( λ i ) = λ i < c i [ j − , j ]¯ q if λ i = c ji j ¯ q if λ i ∈ ] c ji , c j +1 i [ , j (cid:54) = NN ¯ q if λ i ∈ λ i ∈ ] c Ni , ¯ c [ , (49)We conclude Lemma 9.
For any i ∈ I and any λ − i ∈ [min i c i , max i c Ni ] n − , Λ i ( λ − i ) is the uniquesolution of F i (Λ i , λ − i ) ∈ K i (Λ i ) . (50)We point out that the primal (and dual) solution unicity is a desirable property that isnot systematic for the allocation problems of centralized market models. The expressionof h with respect to λ (39) together with the fact the fact the supply constraint shouldbe binding at optimality justify the interpretation of F i proposed at the beginning of thissubsection. In the following sequel we use this property many times.19 .2. Some properties of the solution If r and d are set, we can see the solution of Problem 4 as a function of the vector c ∈ C n . We denote by q ( c ) the solution of Problem 4 with the cost vector c . Similarly,we define q i ( c ), q ji ( c ), λ ( c ) and λ i ( c ). We give here two properties of the allocationproblem solution. By integration, we showed in the previous section that the solution ofthe mechanism design inherits those properties. Lemma 10.
Let ( q ( c ) , h ( c )) be a solution of Problem 4, then q ji ( c ) does not depend on c li for l (cid:54) = j : q ji ( c , . . . c j − , c j , c j +1 . . . , c N ; c − i ) = q ji ( s , . . . s j − , c j , s j +1 . . . , s N ; c − i ) (51) Proof.
Let i ∈ I , j ∈ J , c − i ∈ C n − , c = ( c , . . . , c N ) ∈ C and s = ( s , . . . , s N ) ∈ C such that s j = c j . We shall prove that q ji ( s, c − i ) = q ji ( c, c − i ). We denote by λ c (resp. λ s ) the dual variables associated with the nodal contraints for the allocation problemparametrized with c (resp. s ). First if q ji ( c, c − i ) ∈ ]0 , ¯ q [ , (52)then by lemma 9 λ ci = c ji and using Lemma 9 again, λ si = c ji . Therefore λ s = λ c , fromwhich we deduce that q ji ( c, c − i ) = q ji ( s, c − i ).Therefore without loss of generality, we can assume that q ji ( c, c − i ) = ¯ q and q ji ( s, c − i ) = 0 . (53)Then using Lemma 9 we get λ ci ≥ c k and λ si ≤ c k , (54)so that λ ci ≥ λ si . If λ ci > λ si , then λ c − i ≥ λ s − i by non-decreasingness of Λ i (cid:48) , i (cid:48) ∈ I \{ i } (explained in § i is already producing less.We extend the notations by setting for all i ∈ I , c i = c ∗ . We consider the subset S of C for which at some nodes i , the multiplicator λ i is equal to the marginal cost and the production is a multiple of ¯ q (i.e. stuck in an angle): S = { c ∈ C n , q i ( c ) = j ¯ q and λ i ( c ) = c j (cid:48) i for some i ∈ I, j ∈ J ∪ { } , j (cid:48) ∈ { j, j + 1 }} . (55)The set S corresponds to the points of transition between the two possibilities definedby the first order condition (48). Because of the angle, it is natural to think that this iswhere irregularities may happen (see the proof of the next lemma). We introduce thisset to show some regularity properties of q and Q . We detail the proof in the Appendix.The approach consists in showing that S is a finit union of sets of zero measure. This isalso true for the projection of S on the { c i } × C − i . Then we observe that on C \S , therelations between the primal and dual variables are smooth. Lemma 11.
The function q is C ∞ on C n \S and C on C n .Proof. We postpone the proof to Appendix Appendix B20 .3. Fixed point
In this subsection we show that the solution of the dual problem is the unique fixedpoint of a monotone operator. We defineΛ( λ , ..., λ n ) = (Λ ( λ − ) , ..., Λ n ( λ − n )) . (56) Lemma 12.
For any i ∈ I , Λ i is non-decreasing.Proof. Let λ − i < λ (cid:48)− i and the corresponding Λ i and Λ (cid:48) i . Assume Λ i > Λ (cid:48) i . Since F i isdecreasing in the first variable and increasing in the second F i (Λ i , λ − i ) < F i (Λ (cid:48) i , λ (cid:48)− i ) (57)Moreover for any x ∈ K (Λ (cid:48) i ) and y ∈ K (Λ i ), x ≤ y and F i (Λ i , λ − i ) ∈ K (Λ i ), F i (Λ (cid:48) i , λ (cid:48)− i ) ∈ K (Λ (cid:48) i ). Therefore F i (Λ (cid:48) i , λ (cid:48)− i ) ≤ F i (Λ i , λ − i ) which is absurd.We will use the following classical result (see [17] for a proof and definition of completelattice). Theorem 5 (Knaster-Tarski fixed point) . Let L be a complete lattice and let f anapplication from L to L and order preserving. Then the set of fixed points of f in L is acomplete lattice. In particular, the set of fixed points is non empty. Since Λ is order preserving and[ c ∗ , c ∗ ] n is a lattice when we consider the natural order, there is a fixed point, and theset of fixed points is a lattice. Lemma 13. λ is optimal for the dual ⇔ λ is a fixed point of Λ .Proof. • If λ is optimal for the dual, then each component i maximizes the criteria(47), thus λ is a fixed point of Λ. • If λ is a fixed point of Λ, then by definition, each component i maximizes thecriteria (47). Hence since the problem is (strictly) concave, λ is optimal.A consequence of the previous lemma is that Lemma 14.
The set of fixed points of Λ is a singleton. Definition 1 (Continuous for monotone sequence) . We consider the natural partialorder on R n . We say that a function G is continuous for monotone (resp. increasing,decreasing) sequences if for any monotone (resp. increasing, decreasing) sequence x n converging to a point x in the domain of G , G ( x n ) goes to G ( x ) as n goes to infinity. Clearly, a function is continuous for monotone sequences if and only if it is continuousfor increasing and decreasing sequences.
Lemma 15.
The operator Λ is continuous for monotone sequences. The intuition of the proof is that we can use the monotony of the sequence andLemma 9 to characterize the behaviour of Λ on the neighborhood. We find that Λ iseither constant or characterized by the implicit function theorem.21 roof.
Let ¯ λ − i , j ∈ [1 . . . N ], we first deal with the ’nice’ case, that corresponds to F i (Λ(¯ λ − i ) , ¯ λ − i ) ∈ ] j − , j [¯ q • If Λ i (¯ λ − i ) ∈ ] c ji , c j +1 i [ (we do not treat the case j = N , which is very similar towhat follows) then since F i is C ∞ and of invertible derivative (non zero) in λ i , theimplicit function theorem tells us that the solution ψ of F i ( ψ (¯ λ − i ) , ¯ λ − i ) = j ¯ q iscontinous in a neighborhood B of ¯ λ − i . Thus we can make B small enough so thatfor λ − i ∈ B , ψ ( λ − i ) ∈ ] c ji , c j +1 i [. On this neighborhood, ψ satisfies the first orderconditions and so by unicity of the solution of the optimization problem, since thoseconditions are sufficient, ψ = Λ i on B . Therefore Λ i is continous at ¯ λ − i . • If Λ i (¯ λ − i ) = c ji (as before, we do not treat the case j = N ), then by Lemma 9 F i (Λ i (¯ λ − i ) , ¯ λ − i ) = [ j − , j ]¯ q , if F i ∈ ] j − , j [¯ q (we deal with the border case in thenext point) then since F i is continuous, there is a neighborhood B of ¯ λ − i such that F i (Λ i (¯ λ − i ) , λ − i ) ∈ ] j − , j [¯ q , so on B Λ i is constant and therefore continuous. • We proceed with the borders. If F i (Λ i (¯ λ − i ) , ¯ λ − i ) = ( j − q and Λ i (¯ λ − i ) = c ji . – Decreasing case: Let us take (cid:15) ∈ R n − such that F i (Λ i (¯ λ − i ) , ¯ λ − i + (cid:15) ) ∈ [ j − , j ]¯ q ( F i is continuous and increasing in λ − i ). Then Λ i (¯ λ − i + (cid:15) ) = Λ i (¯ λ − i )checks the first order condition so Λ is constant, and we get the continuity fordecreasing sequences. – Increasing case: F i (Λ i (¯ λ − i ) , ¯ λ − i ) = ( j − q hence there exists a ball B such that the implicit function theorem applies and there exists ψ such that F i ( ψ (¯ λ − i − (cid:15) ) , ¯ λ − i − (cid:15) ) = ( j − q and ψ (¯ λ − i ) = Λ i (¯ λ − i ) = c ji (remember thatΛ i (¯ λ − i ) = c ji by hypothesis) . Since F i is increasing in the second variableand decreasing in the first, ψ is increasing. For (cid:15) of positive components andsufficiently small, ψ (¯ λ − i − (cid:15) ) ∈ ] c j − i , c ji [ (since ψ (¯ λ − i ) = Λ i (¯ λ − i ) = c ji ) andcheck the first order condition. Therefore for (cid:15) of positive components andsufficiently small, ψ = Λ i by uniqueness of the solution. Thus Λ i is continuousfor increasing sequence. • We do the same analysis if F i (Λ i (¯ λ − i ) , ¯ λ − i ) = j ¯ q and Λ i (¯ λ − i ) = c ji .The conclusion follows.We could have alternatively used the Berge Maximum theorem for strictly concavecriterion to get the continuity of Λ. Yet, we chose to present this proof for pedagogicalreasons because it contains some key ideas (see appendix). Theorem 6.
The sequence (Λ k ( c N ...c Nn )) k ∈ N converges to the solution of the dual.Proof. Since Λ( c N ...c Nn ) ≤ ( c N ...c Nn ), and since Λ is order preserving, the sequenceΛ k ( c N ...c Nn ) = λ k is non increasing and bounded, therefore it converge to a point x .Since Λ is continuous for monotone sequence, x is a fixed point. Theorem 7.
For any i ∈ I , λ − i ∈ [ c ∗ , c ∗ ] n − , Λ i ( λ − i ) has the following explicite ex-pression: Λ i ( λ − i ) = min { c Ni , min j ∈ J { c ji F i ( c ji ,λ − i )
1] suchthat G i ( λ − i ) = g ki ( λ − i ). By definition of g ki , F i ( G i ( λ − i ) , λ − i )) = k ¯ q and by definition of G , G i ( λ − i ) ∈ [ c ki , c k +1 i ]. So again F i ( G i ( λ − i ) , λ − i )) ∈ K ( G i ( λ − i )). We can now concludethat Λ = G .We can interpret the fixed point algorithm as if some benevolent agents situatedat each node of the network were exchanging information. They collectively try tominimize the total cost and, to do so, they communicate their current marginal costs.This marginal cost is the minimum of their local marginal cost and the marginal cost ofimportation from the adjacent nodes. At each iteration, the agents compute how muchthey are going to produce based on their current marginal cost. They then update theirmarginal cost based on the information they just received and transmit this marginalcost to the adjacent nodes. We point out that the information used by each agent islocal. We derive in this section an estimate for the decreasing rate. We denote α =max ( e,e (cid:48) ) ∈ E r e /r e (cid:48) . We have the following bound: Lemma 16.
For any ( i, i (cid:48) , k, λ − i ) ∈ E × [0 , N ] × [ c ∗ , c ∗ ] n − , ∂ λ i g ki (cid:48) ( λ − i ) ≥ N α ( c ∗ c ∗ ) . (62) Proof.
We combine (38) with (32) and (33).
Lemma 17.
Since ( λ ki ) k ∈ N is non-increasing for all i ∈ I , there is a finite number of k for which at least one coordinate λ ki satisfies λ ki > c qi and λ k +1 i ≤ c qi (63)23 ixed Point CVXcost time (s) Fixed Point CVXcost time (s)
Table 2: Results for a linear (a) and piecewise linear (b) instances of the problem solved with the fixedpoint algorithm and CVX. or λ ki = c qi and λ k +1 i < c qi . (64) We denote by K this set. Let ( k , k ) ∈ N such that [ k − , k + 1] ∩ K = ∅ . Then for k ∈ [ k , k ] and i ∈ I such that λ k − i (cid:54) = λ ki λ ki − λ k +1 i ≥ N α ( c ∗ c ∗ ) max i (cid:48) ∈ V ( i ) ( λ k − i (cid:48) − λ ki (cid:48) ) (65) Proof.
By definition of λ k , λ ki − λ k +1 i = Λ i ( λ k − − i ) − Λ i ( λ k − i ) . By construction, there exists j ∈ [0 , N −
1] such that Λ i ( λ k − − i ) = g ji ( λ k − − i ) and Λ i ( λ k − i ) = g ji ( λ k − i ). Then by monotonyof g , g ji ( λ k − i ) − g ji ( λ k − − i ) is lower bounded by | ∂ λ i (cid:48) g ji | ∞ ( λ k − i (cid:48) − λ ki (cid:48) ) , (66)for i (cid:48) ∈ V ( i ). We then take the i (cid:48) ∈ V ( i ) that maximizes ( λ k − i (cid:48) − λ ki (cid:48) ) and use the previouslemma to get the result. We implemented this algorithm in Matlab. We used a dichotomy to compute the g ki .Note that for linear cost the analysis is similar. We define g i ( λ − i ) as the unique x suchthat f i ( x, λ − i ) = 0 and x ≥ i ( λ ) = min( c i , g i ( λ − i )) (67)We performed some numerical comparisons with CVX, a package for specifying andsolving convex programs [26, 27] for both linear and piecewise linear production costfunctions. We generated a graph with 100 nodes connected randomly. To generate thegraph, we used a Barabasi-Albert model [28] to ensure some scaling properties. Theexperiment was performed on a personal laptop (OSX, 4 Go,1.3 GHz Intel Core i5). Thenetworks randomly generated to test the implementations are displayed in Figures 1aand 1b, and the results are summarized in Table 2.Both CVX and the fixed point algorithm converges to an estimate of the optimalvalue. We did not try to optimize the numerical algorithm, but some trick could beused to avoid the costly estimation of the g s. Still, the linear version of the fixed pointalgorithm was about ten times faster than the CVX resolution. Note that the algorithmcould be distributed, since at each iteration, the computation at each node only dependson the values of the previous iteration. 24 a) The network generated to test the linearimplementation of the algorithm (b) The network generated to test thegeneric implementation of the algorithm
6. Conclusion
In this paper we have shown how to characterize and compute the optimal mechanismfor a network market. We observed in particular that the allocation problem for theoptimal and the standard mechanism are the same. We have proposed an algorithmbased on a fixed point to solve the allocation problem and derived regularity properties ofthe solution. Our contribution provides a direction to benchmark mechanism proposals.
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Appendix A. Proof of Lemma 5
Proof.
By definition X ( a . . . a k − , b, a k +1 . . . a N ) − X ( a . . . a k − , c, a k +1 . . . a N ) = V ( a . . . b . . . a N ) − V ( a . . . c . . . a N ) + (cid:88) j (cid:54) = k a j [ Q j ( a . . . b . . . a N ) − Q j ( a . . . c . . . a N )]+ bQ k ( a . . . b . . . a N ) − cQ k ( a . . . c . . . a N )= (cid:90) cb Q k ( a . . . s . . . a N )d s + (cid:88) j (cid:54) = k a j [ Q j ( a . . . b . . . a N ) − Q j ( a . . . c . . . a N )]+ bQ k ( a . . . b . . . a N ) − cQ k ( a . . . c . . . a N ) .
26e use (H1) for the last equality. Then we apply a telescopic formula X ( a ) − X ( b ) = X ( a . . . a N ) − X ( b , a . . . a N ) + X ( b , a . . . a N ) − X ( b , b . . . a N ) + . . . + X ( b . . . b N , a N ) − X ( b . . . b N )= N (cid:88) k =1 ( (cid:90) b k a k Q k ( b . . . s . . . a N )d s ) + N (cid:88) k =1 (cid:88) j S ⊆ ∪ S ( I A , I B , I C , j, j (cid:48) ) Proof. Take c ∈ S , then by definition of S , there exist i ∈ I , j ∈ J and j (cid:48) ∈ { j, j + 1 } such that q i ( c ) = j ¯ q and λ i ( c ) = c j (cid:48) , therefore I C is not empty. By Lemma 9, for all i ∈ I , i is in I A or I B . Hence we have a set S ( I A , I B , I C , j, j (cid:48) ) such that c is in this set,so S is included in the union of those sets. Lemma 19. For any c ∈ C n the matrix M ( c ) is invertible.Proof. Assume that there are some coefficients α i such that (cid:80) i α i M i = 0 where M i isthe ith column of M . Then by (32) and (33), the ith row of this relation writes: α i (cid:88) j ∈ V ( i ) λ j r i,j ( λ i + λ j ) = (cid:88) j ∈ V ( i ) ,j ∈ I B α j λ i λ j r i,j ( λ i + λ j ) . (B.4)We denote b i,j = λ j λ i r i,j ( λ i + λ j ) and a i = α i λ i . Then (B.4) is equivalent to a i = (cid:88) j ∈ V ( i ) ,j ∈ I B a j b i,j (cid:80) k ∈ V ( i ) b i,k (B.5)28nsofar as we can slightly perturb the demand, we assume without loss of generalitythat t it is not possible to produce a multiple of ¯ q at each node and satisfy exactly thenodal constraints ( (cid:63) ).Considering the biggest a i , we get that all a i are equal by convexity, thus either allare equal to zero or (cid:88) j ∈ V ( i ) b i,j = (cid:88) j ∈ V ( i ) ,j ∈ I B b i,j (B.6)which is not the case since I A is not empty by ( (cid:63) ).Next we show that S ( I A , I B , I C , j, j (cid:48) ) has a zero Lebesgue measure. Lemma 20. For any I A , I B partition of I , and I C ⊂ I B not empty, j ∈ J I and j (cid:48) ∈ J I such that for all i , j (cid:48) ∈ { j, j + 1 } , the measure of the set S ( I A , I B , I C , j, j (cid:48) ) is zero.Proof. We assume in the market description that it is not possible to produce a multiple¯ q at each node and satisfy exactly the nodal constraints (( (cid:63) ). Therefore it is not possiblethat I B = I , therefore I A is not empty. By definition of S I A ,I B ,I C ,j,j (cid:48) , for all i ∈ I B , F i ( c j (cid:48) I A , λ I B ( c )) = q i ( c ) = j i ¯ q, (B.7)which is a system of equations in λ I B parametrized by c j (cid:48) I A . Let c ∈ C such that thesystem is satisfied, by Lemma 19, we can apply the implicit function theorem, hencethere is a ball around c in which S ( I A , I B , I C , j, j (cid:48) ) is included in a smooth surface. Bycompacity of C , we can choose a sequence dense in S ( I A , I B , I C , j, j (cid:48) ). We apply theresult to each element of this sequence. By density, S ( I A , I B , I C , j, j (cid:48) ) is a countableunion of smooth surfaces. Therefore the measure of S ( I A , I B , I C , j, j (cid:48) ) is zero.A direct consequence of Lemma 20 and Lemma 18 is Lemma 21. The measure of S is zero. We proceed with the proof of Lemma 11. of lemma 11. Let c = ( c . . . c n ) ∈ C n \S . Let us show that q is infinitely differentiableat c . We consider the two assertions: A i = ” ∃ k i , F i ( λ ( c )) ∈ ] k i − , k i [¯ q and λ i = c ki ” B i = ” ∃ k i , F i ( λ ( c )) = k i ¯ q and λ i ∈ ] c ki , c k +1 i [”By Lemma 9 and by defintion of S , for any i ∈ I either A i or B i is true, but never both.We denote by I A (resp. I B ) the set of elements of I for which A i (resp. I B ) is true. If A i is true for all i then there is a neighborhood V of c such that for any element ˜ c of V , F i (˜ c ) ∈ ] k i − , k i [¯ q , therefore on V , λ (˜ c ) = ˜ c .Else I B is not empty and by definition of B i ∀ i ∈ I B F i ( λ I A , λ I B ) = ¯ qj i , (B.8)29hich we can see as an equation in λ I B parametrized by λ I A . This equation is satisfiedat λ ( c ). If we denote by M the matrix M = (cid:18) ∂F i ( λ ( c )) ∂λ j (cid:19) ( i,j ) ∈ I B , (B.9)then M is invertible (see lemma 19), the implicit function theorem applies and there existsa function λ I B so that in a neighborhood V of c , for all i ∈ I B , we have F i ( λ I A , λ I B ( λ I A )) =¯ qk i . Moreover, since F i is C ∞ on [ c ∗ , c ∗ ] n , λ I B is C ∞ on V . Then if ˜ c ∈ V , (˜ c, λ I B (˜ c ))checks the first order condition thus by uniqueness c I A , λ I B (˜ c ) is the dual solution, andso, q i = F i ( λ I B (˜ c ) , ˜ c ) for all i ∈ I on V , so q i is C ∞ at c . This concludes the proof of thefirst part of the lemma.The continuity of q comes from Berge maximum principle (see Theorem 9.17 in [29])in a convex setting.The next lemma is an important component for the proof of Theorem 2. Lemma 22.