1 Introduction
Our purpose in this paper is to show how oligopolistic behaviors in network markets can be tackled using mechanism design. We point out that the optimal mechanism we obtain has a surprisingly simple expression. We complete this work with algorithmic tools for the computation of this mechanism. Following a model already discussed in escobar2010monopolistic ; escobar2008equilibrium ; NicolasFigueroaAlejandroJofrBenjaminHeymann , 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 nodes are connected by edges. The good can travel from one node to another through those edges at the cost of a loss. Since our initial motivation was the electricity market, we will do the presentation with quadratic loss, but as explained thereafter, our results extend to a broad class of externalities. We are considering the usual transmission network constraints with the DC approximation (active power) for the losses.
We will use the word principal to designate what could also be called in the centralized market literature a central operator, or in the context of electricity markets, an ISO. The principal, who aggregates the (inelastic) demand side, has to locally match i.e. at each node  production and demand at the lowest expense through a procurement auction. As argued in NicolasFigueroaAlejandroJofrBenjaminHeymann , this setting can be applied to describe real electricity markets, but it could also be used in other markets where a good is being transported. Either way, there is a clear antagonism between the market participants: the operator wants to minimize his expected cost while the producers want to maximize their expected profits. Therefore, at the same time that there is a transaction and a commitment between each agent and the principal, there also exists competition among the agents. In a standard procurement auction, the market power resulting from the quadratic line losses allows the producers to bid above their true values, or production cost escobar2010monopolistic . The mechanism reduces the producers’ margin and decreases the social cost represented in this case by the optimal value of the principal. This optimal auction design was introduced by Myerson in 1981 myerson1981optimal for a nondivisible good and no externalities.
We build on an electricity market model introduced by the second author in two previous papers escobar2008equilibrium and escobar2010monopolistic . The authors wrote a brief presentation of this model in Heymann:2016aa . Other models were proposed for example in aussel2013electricity , anderson2013mixed , and hu2007using , with a focus on the existence of a market equilibrium. We pinpoint that if our initial motivation was electricity markets, network markets are used in other setting such as telecommunication Altman2006 . Distributed markets were also studied in Babaioff2009 ; Cho2003 , with a focus on efficiency and linear cost for transmissions. For more information on the techniques we use in this paper the reader can refer to laffont2009theory , krishna2009auction , Roughgarden2016 , nisan2007introduction , the chapter 45 of aumann1992handbook and topkis1998supermodularity
for general introductions on principalagent theory, mechanism design, game theory and lattices theory respectively.
In the sequel we consider, as we did in NicolasFigueroaAlejandroJofrBenjaminHeymann , that every participant knows the demand at each node before the interactions begin and that the production cost of each agent is 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, the principal is, therefore, a bidtaker. The producers know they influence the allocation and compete with each other to maximize their individual profit. Since the demand is known by 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 standard setting 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 is true provided that the agents are not communicating with each other. The mechanism changes 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. Allowing the principal to act first by revealing a committing rule gives him a strategic advantage in the negotiation.
We restrict our discussion to deterministic demand, but the reasoning extends naturally to random demand as long as any possible realization of the demand satisfies the model assumptions. Indeed, since the optimal mechanism constructed in this article is incentive compatible, then a random version (where the demand is revealed after the producers’ bidding phase, as in escobar2008equilibrium ) would be realizationwise incentive compatible, and so incentive compatible. Observe that the mechanism we propose in the sequel could be adapted to elastic, piecewise linear demand.
Our first main result is actually the mechanism design characterization. The result is 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 suboptimality of the mechanisms used in practice.
Interestingly, the allocation procedures for the optimal and the standard mechanism are the same (one just needs to modify the input of the allocation procedure of the standard mechanism to get the allocation of the optimal mechanism). Our second main result is a principal allocation algorithm based on a fixed point. The fixed point could be interpreted as cooperating agents trying to minimize a global criterion by sharing relevant information. Our implementation of the algorithm gives good results against standard methods. We point out that the numerical computation of the Nash equilibrium for the procurement auction (important to compare the optimal mechanism and the standard auction setting) requires an efficient algorithm to compute the allocation. Some other additional facts are presented within the paper: the smoothness of the allocation functions ( and
), a decreasing rate estimation for the fixed point iterations, some results of numerical experiments with the fixed point algorithm.
2 Market description
The production cost of each agent is assumed to be piecewise linear, nondecreasing and convex in the quantity produced. This class of functions is sufficiently rich to represent reallife problems and is sufficiently simple for theoretical study. In this work we need to assume that the production levels at which there is a slope change are known in advance and are exogenous  that is the agents cannot choose them. Then, without loss of generality, we assume that there is a quantity such that the changes of slope only occur at the multiples of . Thus, the authors find it practical to write the production cost functions in the form
(1) 
where and the are some slopes coefficients specific to the agent, while
is the quantity produced. We will sometimes refer to the vector of the
as the cost vector (of the agent). If we denote by the quantity produced by agent at marginal cost , then , where is the total quantity produced by this agent. Let and a set of nondecreasing tuples of . To each element of we associate the piecewise linear cost function . Throughout the paper we set, for any , to simplify the notations in some proofs. Note that in practice a capacity constraint of the type for a given agent can be implemented by setting its slope equal to a big positive number. If an agent of cost vector produces a quantity and receives a transfer , then its profit is(2) 
There are agents numbered from to in the market. We denote and use generically the letter to refer to a specific agent, and to refer to . We denote and we will use generically for the cost coefficients of the segment (starting from ). The agents are dispatched on the nodes of a graph. At each node we find the corresponding agent and a local demand . The nodes are connected by undirected edges. We write the set of nodes different from connected to . Obviously if then . We denote the set of undirected edges. For each , we introduce a quadratic loss coefficient such that . In the context of electricity markets, this quadratic coefficient corresponds to the Joule effect within the lines. We make the non restricting assumption that is big enough so that in what follows production at each node is smaller than .
We assume that both the agents and the principal are risk neutral: they maximize their expected profit. If the principal proposes to pay a price to agent to make her produce a quantity  this agent being free to accept or decline the offer and if the agent has a production cost defined by , then he accepts the offer if
(3) 
Then for agent , either or . Thus, if the principal knew the cost vectors , he would solve an allocation problem with those , and then bid to the agents the 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 that the agents tell him some values for the (not necessarily their real cost vectors), and then the principal decides based on those values. In this case, previous works escobar2010monopolistic showed that the agents could receive nonzero profits and bid above their production costs. The question we now address is how to reduce their margins.
To do so, we need to consider an intermediate scenario between the one in which the agent 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 is characterized by an element
, which is a probability density of support included in
and an element of drawn according to . Only agent knows , which is private information. The other agents and the principal only know the probability with which it was drawn. The density corresponds to the public knowledge on agent ’s production costs so the principal won’t accept any bid that is not in the support of . We assume that the cost slopes are not correlated for a given agent and between agents, i.e. their laws are independent. In particular . In such situation, it makes sense to define(4) 
and (respectively ) the mean operator with respect to (respectively ). The density (resp. ) represents the uncertainty from the principal’s (resp. agent ) perspective. To simplify notations we will use the symbol to denote the product of the supports of the s. We denote by the set of allocation functions  which are the applications from to , by the set of payments functions which are the applications from to , and by the set of flow functions  which are the applications from to . A direct mechanism is a triple . Let . For this payment function and this allocation function, the expected profit of agent of type and bid is
(5) 
where the capitalized quantities
(6) 
correspond to the average of their non capitalized counterpart. We also denote by
(7) 
the expected profit of agent if he is of type and bids her true production cost.
For , and let We point out that by independence of the laws of the , . Thus is simply the ratio of the cumulative distribution and the probability density for . Our main assumption is the discernability assumption: for all and , the virtual cost is increasing in . As demonstrated in the next section, the virtual cost could be interpreted as the real marginal cost augmented by a marginal information rent. The assumption imposes the marginal information rent to be such that for any bid, the virtual marginal prices are increasing, i.e. the virtual production cost function is convex. The assumption is necessary to show the independence property of the reformulation in Lemmas 6 and 7.
This assumption implies the non overlapping working zones assumption: if we denote by the support of , then should be of the form:
(8) 
with . We could interpret each segment over which the agent 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 is repeated over time. This estimation need to be precise enough so that there is no chance that it corresponds to another working zone. We use this assumption in particular in the proof of lemma 4. For simplicity we assume that is increasing in . ^{3}^{3}3This is the piecewise linear adaptation of the classic monotone likelihood ratio property assumption. This assumption can be withdrawn using the ironing technique introduced by Myerson without difficulty. To finish with the market presentation, we introduce the products of the type sets and .
3 Mechanism Design
We begin with the revelation principle as expressed in gibbons1992game .
Theorem 1 (Revelation Principle).
To any Bayesian Nash equilibrium of a game of incomplete information, there exists a payoffequivalent direct revelation mechanism that has 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 need to add a constraint on the design to enforce truthfulness. This means that the profit of any agent of type should be maximal when agent bids her true type i.e. for all
(9) 
This is the incentive compatibility (IC) constraint. In addition, since we want all agents to participate in the market, we need the participation constraint imposing that for all
(10) 
Without this constraint, the principal would optimize as if the agents would accept any deal (even deals where they would make a negative profit). The last constraint is that the supply should be at least equal to the demand at every node. The supply available at a given node is equal to the production augmented by the imports minus the exports and the line losses. As explained earlier, there is a loss when some quantity of the divisible good is sent from one node to another . This loss is equal to , where is a multiplicative constant. In order to obtain symmetric expressions, we will proceed as if half of this quantity was lost by the sender, and the other half by the receiver (see for instance escobar2010monopolistic ). Note that we could have equivalently used signed flows, but we would have lost some symmetry in the formulation. Then the supply and demand constraint writes, for all and ,
(11) 
We point out that for an optimal allocation (see §5) , .
The principal decision is a triple . This decision is made under the 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 criterion being equal to the expected sum of payments. Finally the optimal mechanism is the solution of
Problem 1.
subject to  
We now proceed to solve the optimal mechanism design problem, which is a functional optimization problem with an infinity of constraints, some of which are expressed with integrals. The essential observation is that this complicated problem is equivalent to a much simpler one. The proof relies on the comparison with two intermediate problems:
Problem 2.
subject to.  
and
Problem 3.
subject to  
The inequality on the scalar product in (H2) is the piecewise linear equivalent of a monotonicity condition already encountered in NicolasFigueroaAlejandroJofrBenjaminHeymann . The first two problems are very similar, but (IC) has been replaced by (H1) and (H2) and (PC) is expressed in terms of instead of . 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 3 is simpler, as the optimization part can be solved pointwise (and can be deduced from this pointwise optimization). The main result of this paper is that the three problems have the same solution.
3.1 Necessary conditions for Problem 1
We derive some necessary conditions for a solution of Problem 1. In fact, we only use constraint to deduce the two next results. The first lemma indicates that any solution of the first problem should be such that is monotonous. This is a classic result already introduced in myerson1981optimal and NicolasFigueroaAlejandroJofrBenjaminHeymann , for instance. The novelty here is that in the context of piecewise linear production cost functions, this monotonicity result is expressed in a vectorial sense.
Lemma 1 ( monotonicity).
Proof.
We omit the in the proof, as it plays no role. First, let by the (IC) constraint,
(13) 
i.e.
(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 th working zone if he is bidding a higher marginal cost for this working zone.
Lemma 2.
If is admissible for Problem 1 then for any agent (omitting ) for any , and
(15)  
Proof.
The inequality implies that is maximal at for any . Moreover,
(16) 
is absolutely continuous, differentiable with respect to for all , and its derivative is . By definition of , . The envelope theorem yield the result. ∎
3.2 Necessary conditions for Problem 2
We derive some necessary conditions for a solution of Problem 2.
Lemma 3.
If is an optimal solution to Problem 2 then (omitting ) for all
(17) 
Proof.
According to (H1)
This is an expression for as a sum of a positive function of and a constant . It is clear that to optimize the criteria, this constant should be as small as possible. The participation contraint (PC) imposes that , therefore . ∎
A consequence of this is:
Corollary 1.
If is an optimal solution of Problem 2 then for all ,
(18) 
Proof.
See the proof of Lemma 3. ∎
Corollary 1 means that if an agent bids a production cost function that is the maximum of what he could bid, he should not make any profit, which is why he should be paid exactly his production cost. We see with this lemma that if the public information is inaccurate and the real cost of an agent is higher than what could be expected, then there is a risk that the participation constraint is not satisfied. On the other hand, it should not be surprising that an agent can have a zero profit: remember that in the extreme case in which the principal knows everything (discussed in §2), the agents do not make any profit.
Another consequence of lemma 3 is
Lemma 4.
If is an optimal solution of Problem 2, the expected profit of agent (over his type) is
(19) 
Proof.
By Lemma 3 and Fubini’s lemma, is equal to
Our task is now to compute the inner term. Applying again Fubini’s lemma, this term is equal to
We get the lemma by summing all the inner terms. ∎
Lemma 5.
If (H1) is satisfied, then for any (omitting )
(20) 
Proof.
Because of its length the proof is detailed in Appendix A ∎
Lemma 6.
If verifies (H1) and (H2) and is independent of for , then for all
(21) 
Proof.
Since (H1) is satisfied, equation (20) of Lemma 5 applies. We combine this relation with the definition of the expected profit from (5). We obtain:
where we used the independence hypothesis for the second equality. By (H2), which implies the decreasingness of with respect to when all other quantities are fixed, if then for any , . Otherwise, we use the formula and the fact that any verifies . Therefore is non negative. ∎
3.3 Necessary conditions for Problem 3
We derive some properties for Problem 3.
Lemma 7.
There is an optimal solution for Problem 3 such that (and ) is independent of for .
Proof.
First note that is not taking any role in the optimization problem: it is defined afterward. The only real optimization variables are then and . Remember that is defined as a function of by . The constraints are defined for each and the integral criterion is in fact a sum of independent criteria depending on for . Therefore we can solve Problem 3 with a pointwise optimization. By the discernability assumption, for any and , is increasing in . Therefore for all , , is a convex criteria in and therefore the pointwise problem corresponds to Problem 4 of §5. In particular, we can apply Lemma 10 from the next section. Thus only depends on and . This property is preserved by integration over the : only depends on . ∎
We point out that, since the pointwise problem has a unique solution, the pointwise optimal solution introduced in the proof is uniquely defined.
Theorem 2.
If is the pointwise optimal solution of Problem 3 and is smooth in for and , then for all , is over .
Proof.
Remember that is increasing, so by composition with smooth bijection, we can do the proof as if the costs involved were instead of . According to Lemma 11, is continuous. Since is bounded, we can apply the dominated convergence theorem to show that is continuous. We can then we proceed by mathematical induction. Assume that is , then take and a sequence in that converges to . Since is a countable union of null measured set (by Lemma 22), its measure is zero. Without changing the results, we can compute the integrals on instead of . Since and its derivatives are bounded, we can apply the dominated convergence theorem to compute the limit of as goes to as the integral of a limit. Since we removed the point over which this limit was not defined, we get that has a limit, and this limit does not depend on the sequence . So is times derivable at , for all . We conclude by induction. ∎
3.4 Resolution of the mechanism design problem
Last but not least, we state the main result of the Section.
Theorem 3.
Let be defined such that for any , solves
subject to  
and set
(22) 
then solves the optimal mechanism design problem (Problem 1).
Proof.

First note that is the pointwise solution of Problem 3 so it is optimal for Problem 3, moreover, by construction satisfies (SD) and .

Then note that by Lemma 4, solves a relaxation of Problem 2, but is it admissible for Problem 2 ?

By definition of (omitting ),
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