Optimal auctions for networked markets with externalities

TL;DR

This paper extends optimal auction mechanisms to complex networked markets with externalities, improving efficiency and providing benchmarks for practical mechanisms in electricity markets.

Contribution

It generalizes previous models to markets with multiple agents and piecewise linear costs, offering a new algorithm and broader applicability.

Findings

Mechanism effectively reduces market power in complex networks.

Algorithm efficiently solves the principal allocation problem.

Provides a benchmark for evaluating practical auction mechanisms.

Abstract

Motivated by the problem of market power in electricity markets, we introduced in previous works a mechanism for simplified markets of two agents with linear cost. In standard procurement auctions, the market power resulting from the quadratic transmission losses allows the producers to bid above their true values, which are their production cost. The mechanism proposed in the previous paper optimally reduces the producers' margin to the society's benefit. In this paper, we extend those results to a more general market made of a finite number of agents with piecewise linear cost functions, which makes the problem more difficult, but simultaneously more realistic. We show that the methodology works for a large class of externalities. We also provide an algorithm to solve the principal allocation problem. Our contribution provides a benchmark to assess the sub-optimality of the mechanisms used in practice.

Figures

Tables

Table 1. Table 1 : The gap Δ Δ \Delta for some standard probabilities

Name	$\propto f\left(x\right)$	$\propto F\left(x\right)$	$K\left(x\right)$	$\Delta$
Uniform	$1$	$x$	$x$	$c^{+}$
Power Function	$\lambda {\left(\frac{x}{c^{+}}\right)}^{\lambda -1}$	$c^{+}{\left(\frac{x}{c^{+}}\right)}^{\lambda }$	$\frac{x}{\lambda }$	$\frac{c^{+}}{\lambda }$
Weibull	$\lambda {\left(\frac{x}{c^{+}}\right)}^{\lambda -1}{e}^{{\left(-\frac{x}{c^{+}}\right)}^{\lambda }}$	$c^{+}\left(1-e^{-{\left(\frac{x}{c^{+}}\right)}^{\lambda }}\right)$	$\frac{c^{+}}{\lambda }{\left(\frac{c^{+}}{x}\right)}^{\lambda -1}\left(e^{{\left(\frac{x}{c^{+}}\right)}^{\lambda }}-1\right)$	$c^{+}\frac{e-1}{\lambda }$
Laplace	$\frac{1}{2}{e}^{-\lambda \left|x-\frac{c^{+}}{2}\right|}$	$x>\frac{c^{+}}{2}$, $\frac{2-e^{-\lambda \frac{c^{+}}{2}}{e}^{-\lambda \left(x-\frac{c^{+}}{2}\right)}}{2\lambda }$		$\frac{2}{\lambda }\left(e^{\frac{c^{+}}{2}\lambda }-1\right)$
Exponential (reversed)	$\lambda e^{-\left(c^{+}-x\right)\lambda }$	$e^{-c^{+}\lambda }\left(e^{x\lambda }-1\right)$	$\frac{1-e^{-x\lambda }}{\lambda }$	$\frac{1-e^{-c^{+}\lambda }}{\lambda }$

Table 2. Table 2 : Results for a linear (a) and piecewise linear (b) instances of the problem solved with the fixed point algorithm and CVX.

	Fixed Point	CVX
cost	83.2	83.195
time (s)	2.03	30.23

	Fixed Point	CVX
cost	83.2	83.195
time (s)	2.03	30.23

	Fixed Point	CVX
cost	4971.4	4971.4
time (s)	28.39	35.23

Equations

$$C^c(q)=\sum _{j=1}^N{c}_j\min ((q-(j-1)\bar{q}{)}^{+},\bar{q}),$$

$$u_i=x-C^c(q).$$

$$x_i-C^{c_i}(q_i)\ge 0.$$

$$f_{-i}(c_{-i})=\prod _{i^{\prime }\in I\backslash i}{f}_{i^{\prime }}(c_{i^{\prime }})\text{mbox}andf(c_1,..,c_n)=\prod _{i\in I}{f}_{i^{\prime }}(c_i),$$

$$U_i(c_i,c_i^{\prime })={\mathbb{E}}_{-i}{u}_i=X_i(c_i^{\prime })-\sum _{j\in J}{c}_i^j{Q}_i^j(c_i^{\prime }).$$

$$Q_i^j(c_i)={\mathbb{E}}_{-i}\min ((q_i(c_i,c_{-i})-(j-1)\bar{q}{)}^{+},\bar{q})\text{mbox}and{X}_i(c_i)={\mathbb{E}}_{-i}{x}_i(c_i,c_{-i})$$

$$V_i(c_i)=U_i(c_i,c_i).$$

$${\mathbf{C}}_i=[c_i^{1-},c_i^{1+}]\times \dots \times [c_i^{N-},c_i^{N+}]$$

$$U_i(c_i,c_i)\ge U_i(c_i,c_i^{\prime }).(IC)$$

$$U_i(c_i,c_i)\ge 0.(PC)$$

$$q_i(c)+\sum _{i^{\prime }\in V(i)}{h}_{i^{\prime },i}(c)-h_{i,i^{\prime }}(c)-\frac{h_{i,i^{\prime }}^2(c)+h_{i^{\prime },i}^2(c)}{2}{r}_{i,i^{\prime }}\ge d_i.(SD)$$

$$\underset{(q,x,h)\in (\mathcal{Q},\mathcal{X},\mathbb{H})}{\mathrm{minimize}}\sum _{i\in I}\mathbb{E}{x}_i(c)$$

$$\forall c\in {\mathbf{C}}^n,\forall i\in I:q_i(c)+\sum _{i^{\prime }\in V(i)}{h}_{i^{\prime },i}(c)-h_{i,i^{\prime }}(c)-\frac{h_{i,i^{\prime }}^2(c)+h_{i^{\prime },i}^2(c)}{2}{r}_{i,i^{\prime }}\ge d_i(SD)$$

$$\forall c\in {\mathbf{C}}^n,\forall (i,i^{\prime })\in E:h_{i,i^{\prime }}(c)\ge 0$$

$$\forall i\in I,\forall (c_i^{\prime },c_i)\in {\mathbf{C}}_i^2:U_i(c_i,c_i)\ge U_i(c_i,c_i^{\prime })(IC)$$

$$\forall i\in I,\forall c_i\in {\mathbf{C}}_i:U_i(c_i,c_i)\ge 0(PC).$$

$$\underset{(q,x,h)\in (\mathcal{Q},\mathcal{X},\mathbb{H})}{\mathrm{minimize}}\sum _{i\in I}\mathbb{E}{x}_i(c)$$

$$\forall c\in {\mathbf{C}}^n,\forall i\in I:q_i(c)+\sum _{i^{\prime }\in V(i)}{h}_{i^{\prime },i}(c)-h_{i,i^{\prime }}(c)-\frac{h_{i,i^{\prime }}^2(c)+h_{i^{\prime },i}^2(c)}{2}{r}_{i,i^{\prime }}\ge d_i(SD)$$

$$\forall c\in {\mathbf{C}}^n,\forall (i,i^{\prime })\in E:h_{i,i^{\prime }}(c)\ge 0$$

$$\forall i\in I,\forall j\in J,(c^{-j},t_1,t_2),(c^1,\dots ,t_k,\dots ,c^N)\in {\mathbf{C}}_i,:V_i(c^1,..,c^{j-1},t_1,c^{j+1}..,c^N)$$

$$-V_i(c^1,..,c^{j-1},t_2,c^{j+1}..,c^N)={\int }_{t_1}^{t_2}{Q}_i^j(c^1,..,c^{j-1},s,c^{j+1}..,c^N)\mathrm{d}s(H1)$$

$$\forall i\in I,\forall (c,c^{\prime })\in c^2:(c-c^{\prime }).(Q_i(c)-Q_i(c^{\prime }))\le 0,(H2)$$

$$\forall i\in I,\forall c_i\in {\mathbf{C}}_i:V_i(c_i)\ge 0(PC),$$

$$\underset{(q,h)\in (\mathcal{Q},\mathbb{H})}{\mathrm{minimize}}\mathbb{E}\sum _{i\in I}\sum _{j\in J}{q}_i^j(c)(c_i^j+K_i^j(c_i^j))$$

$$\forall (c,i)\in {\mathbf{C}}^n\times I:q_i(c)+\sum _{i^{\prime }\in V(i)}{h}_{i^{\prime },i}(c)-h_{i,i^{\prime }}(c)-\frac{h_{i,i^{\prime }}^2(c)+h_{i^{\prime },i}^2(c)}{2}{r}_{i,i^{\prime }}\ge d_i(SD)$$

$$\forall c\in {\mathbf{C}}^n,\forall (i,i^{\prime })\in E:h_{i,i^{\prime }}(c)\ge 0.$$

$$\forall c\in {\mathbf{C}}_i,\forall i\in I:x_i(c)=\sum _{j\in J}{q}_i^j(c)c_i^j+{\int }_{c_i^j}^{c_i^{j+}}{q}_i^j(c_i^1\dots c_i^{j-1},t,c_1^{(j+1)+}\dots c_i^{N+};c_{-i})\mathrm{d}t.$$

$$(c_i-c_i^{\prime }).(Q_i(c_i)-Q_i(c_i^{\prime }))\le 0$$

$$U(c,c)\ge U(c,c^{\prime })\text{mbox}andU(c^{\prime },c^{\prime })\ge U(c^{\prime },c)$$

$$X(c)-\sum _{j\in J}{c}^j{Q}^j(c)\ge X(c^{\prime })-\sum _{j\in J}{c}^j{Q}^j(c^{\prime })$$

$$X(c^{\prime })-\sum _{j\in J}{c}^{j^{\prime }}{Q}^j(c^{\prime })\ge X(c)-\sum _{j\in J}{c}^{j^{\prime }}{Q}^j(c).$$

$$V(c^1,\dots ,c^{j-1},t_1,c^{j+1},\dots ,c^N)=$$

$$-{\int }_{t_2}^{t_1}{Q}^j(c^1,\dots ,c^{j-1},s,c^{j+1},\dots ,c^N)\mathrm{d}s$$

$$t\to U((c^1,..,c^{j-1},t,c^{j+1}..,c^N),c)=X(c)-\sum _{k\in J\backslash \{j\}}{c}^k{Q}^k(c)-t{Q}^j(c)$$

$$V(c)=\sum _{j\in J}{\int }_{c^j}^{c^{j+}}{Q}^j(c^1\dots c^{j-1},t,c^{(j+1)+},\dots ,c^{N+})\mathrm{d}t.$$

$$\sum _{j\in J}{\int }_{c_j}^{c^{j+}}{Q}^j(c^1\dots c^{j-1},t,c^{(j+1)+},\dots ,c^{N+})\mathrm{d}t=$$

$$\sum _{j\in J}V(c^1,..,c^{j-1},c^j,c^{(j+1)+},\dots ,c^{N+})-V(c^1,..,c^{j-1},c^{(j)+},\dots ,c^{N+})$$

$$=V(c)-V(c^{1+},\dots ,c^{N+}).$$

$$V_i(c_i^{1+},\dots ,c_i^{N+})=0.$$

Full text

JEL classification: D44, D62, D82

††journal: Journal of Economic Theory

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 [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 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 [3], 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 [1]. 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 [4] for a non-divisible good and no externalities.

We build on an electricity market model introduced by the second author in two previous papers [2] and [1]. The authors wrote a brief presentation of this model in [5]. Other models were proposed for example in [6], [7], and [8], 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 [9]. Distributed markets were also studied in [10, 11], 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 [12], [13], [14], [15], the chapter 45 of [16] and [17] for general introductions on principal-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 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 bid-taker. 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 [2]) would be realization-wise 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 sub-optimality 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 ($q$ and $Q$), a decreasing rate estimation for the fixed point iterations, some results of numerical experiments with the fixed point algorithm.

We describe the market in the next section. In §3 we introduce and solve the mechanism design problem. In §5, we study the standard allocation problem and propose an algorithm to solve it. In §6 we sum up and comment on our main results and propose possible continuations of this work.

2 Market description

The production cost of each agent is assumed to be piecewise linear, non-decreasing and convex in the quantity produced. This class of functions is sufficiently rich to represent real-life 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 $\bar{q}$ such that the changes of slope only occur at the multiples of $\bar{q}$. Thus, the authors find it practical to write the production cost functions in the form

[TABLE]

where $N\in\mathbb{N}$ and the $c_{j}$ are some slopes coefficients specific to the agent, while $q$ is the quantity produced. We will sometimes refer to the vector of the $c_{j}$ as the cost vector (of the agent). If we denote by $q_{i}^{j}$ the quantity produced by agent $i$ at marginal cost $c_{i}^{j}$, then $q_{i}^{j}=\min((q_{i}-(j-1)\bar{q})^{+},\bar{q})$, where $q_{i}$ is the total quantity produced by this agent. Let $c_{*}<c^{*}\in\mathbb{R}^{*+}$ and $\boldsymbol{C}$ a set of non-decreasing $N$-tuples of $[c_{*},c^{*}]$. To each element $c$ of $\boldsymbol{C}$ we associate the piecewise linear cost function $q\rightarrow C^{c}(q)$. Throughout the paper we set, for any $c\in\boldsymbol{C}$, $c^{N+1}=c^{*}$ to simplify the notations in some proofs. Note that in practice a capacity constraint of the type $q\leq j\bar{q}$ for a given agent can be implemented by setting its $(j+1)^{th}$ slope $c_{j+1}$ equal to a big positive number. If an agent of cost vector $c$ produces a quantity $q$ and receives a transfer $x$, then its profit is

[TABLE]

There are $n$ agents numbered from $1$ to $n$ in the market. We denote $I=[1\ldots n]$ and use generically the letter $i$ to refer to a specific agent, and $-i$ to refer to $I\backslash\{i\}$. We denote $J=[1\ldots N]$ 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 node $i$ we find the corresponding agent $i$ and a local demand $d_{i}$. The nodes are connected by undirected edges. We write $V(i)$ the set of nodes different from $i$ connected to $i$. Obviously if $i_{1}\in V(i_{2})$ then $i_{2}\in V(i_{1})$. We denote $E=\{(i_{1},i_{2}):i_{1}\in V(i_{2})\}$ the set of undirected edges. For each $(i_{1},i_{2})\in E$, we introduce a quadratic loss coefficient $r_{i_{1},i_{2}}$ such that $r_{i_{1},i_{2}}=r_{i_{2},i_{1}}$. In the context of electricity markets, this quadratic coefficient corresponds to the Joule effect within the lines. We make the non restricting assumption that $N$ is big enough so that in what follows production at each node is smaller than $\bar{q}N$.

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 $x_{i}$ to agent $i$ to make her produce a quantity $q_{i}$ - this agent being free to accept or decline the offer- and if the agent $i$ has a production cost defined by $c_{i}$, then he accepts the offer if

[TABLE]

Then for agent $i$, either $x_{i}\geq C^{c_{i}}(q_{i})$ or $q_{i}=0$. Thus, if the principal knew the cost vectors $c_{i}$, he would solve an allocation problem with those $c_{i}$, 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 $c_{i}$ (not necessarily their real cost vectors), and then the principal decides based on those values. In this case, previous works [1] showed that the agents could receive non-zero 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 $f_{i}$, which is a probability density of support included in $\boldsymbol{C}$ and an element $c_{i}$ of $\boldsymbol{C}$ drawn according to $f_{i}$. Only agent $i$ knows $c_{i}$, which is private information. The other agents and the principal only know the probability $f_{i}$ with which it was drawn. The density $f_{i}$ corresponds to the public knowledge on agent $i$’s production costs so the principal won’t accept any bid $c_{i}$ that is not in the support of $f_{i}$. We assume that the cost slopes are not correlated for a given agent and between agents, i.e. their laws $f_{i}^{j}$ are independent. In particular $f_{i}(c_{i})=\prod_{j\in J}f_{i}^{j}(c_{i}^{j})$. In such situation, it makes sense to define

[TABLE]

and $\mathbb{E}$ (respectively $\mathbb{E}_{c_{-i}}$) the mean operator with respect to $f$ (respectively $f_{-i}$). The density $f$ (resp. $f_{-i}$) represents the uncertainty from the principal’s (resp. agent $i$) perspective. To simplify notations we will use the symbol $\boldsymbol{C}^{n}$ to denote the product of the supports of the $f_{i}$s. We denote by $\mathcal{Q}$ the set of allocation functions - which are the applications from $\boldsymbol{C}^{n}$ to $\mathbb{R}_{+}^{n}$, by $\mathcal{X}$ the set of payments functions -which are the applications from $\boldsymbol{C}^{n}$ to $\mathbb{R}^{n}$, and by $\mathbb{H}$ the set of flow functions - which are the applications from $\boldsymbol{C}^{n}$ to $\mathbb{R}^{E}$ -. A direct mechanism is a triple $(q,x,h)\in(\mathcal{Q},\mathcal{X},\mathbb{H})$. Let $(q,x)\in(\mathcal{Q},\mathcal{X})$. For this payment function and this allocation function, the expected profit of agent $i$ of type $c_{i}$ and bid $c_{i}^{\prime}$ is

[TABLE]

where the capitalized quantities

[TABLE]

correspond to the average of their non capitalized counterpart. We also denote by

[TABLE]

the expected profit of agent $i$ if he is of type $c_{i}$ and bids her true production cost.

For $i\in I$, $j\in J$ and $c_{i}\in\boldsymbol{C}_{i}$ let $K^{j}_{i}(c_{i})=\int_{c_{i}^{j-}}^{c_{i}^{j}}f_{i}(c^{-j}_{i},s)ds/f_{i}(c_{i}).$ We point out that by independence of the laws of the $c_{i}^{j}$, $K^{j}_{i}(c_{i})=\int_{c_{i}^{j-}}^{c_{i}^{j}}f_{i}^{j}(s)ds/f_{i}^{j}(c_{i}^{j})=K_{i}^{j}(c_{i}^{j})$. Thus $K_{i}^{j}$ is simply the ratio of the cumulative distribution and the probability density for $c_{i}^{j}$. Our main assumption is the discernability assumption: for all $i\in I$ and $c_{i}\in\boldsymbol{C}_{i}$, the virtual cost $J_{i,j}(c_{i}^{j})=c_{i}^{j}+K_{i}^{j}(c_{i}^{j})$ is increasing in $j$. 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 $\boldsymbol{C}_{i}$ the support of $f_{i}$, then $\boldsymbol{C}_{i}$ should be of the form:

[TABLE]

with $c_{i}^{1-}<c_{i}^{1+}<\ldots<c_{i}^{N-}<c_{i}^{N+}$. 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 $c_{i}^{j}\rightarrow c_{i}^{j}+K_{j}^{i}(c_{i}^{j})$ is increasing in $c_{i}^{j}$. 333This 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 $\boldsymbol{C}^{n}=\prod_{i\in I}\boldsymbol{C}^{i^{\prime}}$ and $\boldsymbol{C}^{-i}=\prod_{i^{\prime}\in I\backslash\{i\}}\boldsymbol{C}^{i^{\prime}}$.

3 Mechanism Design

We begin with the revelation principle as expressed in [18].

Theorem 1** (Revelation Principle).**

To any Bayesian Nash equilibrium of a game of incomplete information, there exists a payoff-equivalent 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 $i$ of type $c_{i}$ should be maximal when agent $i$ bids her true type $c_{i}$ i.e. for all $(c^{\prime}_{i},c_{i})$

[TABLE]

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 $c_{i}$

[TABLE]

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 $h_{i,i^{\prime}}$ of the divisible good is sent from one node $i$ to another $i^{\prime}$. This loss is equal to $r_{i,i^{\prime}}h_{i,i^{\prime}}^{2}$, where $r_{i,i^{\prime}}$ 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 [1]). 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 $i\in I$ and $c\in\boldsymbol{C}^{n}$,

[TABLE]

We point out that for an optimal allocation (see §5) , $h_{i,i^{\prime}}h_{i^{\prime},i}=0$.

The principal decision is a triple $(q,x,h)\in(\mathcal{Q},\mathcal{X},\mathbb{H})$. 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**.**

[TABLE]

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**.**

[TABLE]

and

Problem 3**.**

[TABLE]

The inequality on the scalar product in (H2) is the piecewise linear equivalent of a monotonicity condition already encountered in [3]. The first two problems are very similar, 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 3 is simpler, as the optimization part can be solved pointwise (and $x$ 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 $(IC)$ to deduce the two next results. The first lemma indicates that any solution of the first problem should be such that $Q$ is monotonous. This is a classic result already introduced in [4] and [3], 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** ($Q$ monotonicity).**

If $(q,x,h)$ is admissible for Problem 1, then for all agent $i\in I$ and all $(c_{i},c^{\prime}_{i})\in\boldsymbol{C}_{i}^{2}$

[TABLE]

where . is the scalar product in $\mathbb{R}^{N}$.

Proof.

We omit the $i$ in the proof, as it plays no role. First, let $(c,c^{\prime})\in\boldsymbol{C}_{i}^{2}$ by the (IC) constraint,

[TABLE]

i.e.

[TABLE]

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 $i$th working 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 $i$) for any $c$, $t_{1}$ and $t_{2}$

[TABLE]

Proof.

The inequality $U(c,c)\leq U(c,c^{\prime})$ implies that $c^{\prime}\rightarrow U(c,c^{\prime})$ is maximal at $c$ for any $c\in\mathcal{C}_{i}$. Moreover,

[TABLE]

is absolutely continuous, differentiable with respect to $t$ for all $c$, and its derivative is $-Q^{j}(c)$. By definition of $q^{j}$, $Q^{j}\leq\bar{q}$. 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 $(q,x,h)$ is an optimal solution to Problem 2 then (omitting $i$) for all $c\in\boldsymbol{C}_{i}$

[TABLE]

Proof.

According to (H1)

[TABLE]

This is an expression for $V(c)$ as a sum of a positive function of $c$ and a constant $V(c^{1+},\ldots,c^{N+})$. It is clear that to optimize the criteria, this constant should be as small as possible. The participation contraint (PC) imposes that $V(c^{1+},\ldots,c^{N+})\geq 0$, therefore $V(c^{1+},\ldots,c^{N+})=0$. ∎

A consequence of this is:

Corollary 1**.**

If $(q,x,h)$ is an optimal solution of Problem 2 then for all $i\in I$,

[TABLE]

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 $(q,x,h)$ is an optimal solution of Problem 2, the expected profit of agent $i$ (over his type) is

[TABLE]

Proof.

By Lemma 3 and Fubini’s lemma, $\mathbb{E}V_{i}(c)$ is equal to

[TABLE]

Our task is now to compute the inner term. Applying again Fubini’s lemma, this term is equal to

[TABLE]

We get the lemma by summing all the inner terms. ∎

Lemma 5**.**

If (H1) is satisfied, then for any $(a,b)\in\boldsymbol{C}_{i}^{2}$ (omitting $i$)

[TABLE]

Proof.

Because of its length the proof is detailed in Appendix A ∎

Lemma 6**.**

If $(q,x,h)$ verifies (H1) and (H2) and $Q_{i}^{j}$ is independent of $c_{i}^{j^{\prime}}$ for $j^{\prime}>j$, then for all $(c,\tilde{c})\in\boldsymbol{C}^{2}$

[TABLE]

Proof.

Since (H1) is satisfied, equation (20) of Lemma 5 applies. We combine this relation with the definition of the expected profit $U$ from (5). We obtain:

[TABLE]

where we used the independence hypothesis for the second equality. By (H2), which implies the decreasingness of $Q^{j}$ with respect to $c_{i}^{j}$ when all other quantities are fixed, if $c^{j}<\tilde{c}^{j}$ then for any $t\in[c^{j},\tilde{c}^{j}]$, $Q^{j}(t)-Q^{j}(\tilde{c}^{j})\geq 0$. Otherwise, we use the formula $\int_{a}^{b}=-\int_{b}^{a}$ and the fact that any $t\in[\tilde{c}^{j},c^{j}]$ verifies $Q^{j}(t)-Q^{j}(\tilde{c}^{j})\leq 0$. Therefore $U(c,c)-U(c,\tilde{c})$ is non negative. ∎

3.3 Necessary conditions for Problem 3

We derive some properties for Problem 3.

Lemma 7**.**

There is an optimal solution $(q,x,h)$ for Problem 3 such that $q_{i}^{j}$ (and $Q_{i}^{j}$) is independent of $c_{i}^{k}$ for $k\neq j$.

Proof.

First note that $x$ is not taking any role in the optimization problem: it is defined afterward. The only real optimization variables are then $q$ and $h$. Remember that $q_{i}^{j}$ is defined as a function of $q$ by $q_{i}^{j}=\min((q_{i}-(j-1)\bar{q})^{+},\bar{q})$. The constraints are defined for each $c\in\boldsymbol{C}^{n}$ and the integral criterion is in fact a sum of independent criteria depending on $q(c)$ for $c\in\boldsymbol{C}^{n}$. Therefore we can solve Problem 3 with a pointwise optimization. By the discernability assumption, for any $c\in\boldsymbol{C}^{n}$ and $i\in I$, $c_{i}^{j}+K_{i}^{j}(c_{i}^{j})$ is increasing in $j$. Therefore for all $c\in\boldsymbol{C}^{n}$, $i\in I$, $\sum_{j\in J}q_{i}^{j}(c)(c_{i}^{j}+K^{j}_{i}(c_{i}^{j}))$ is a convex criteria in $q_{i}$ and therefore the pointwise problem corresponds to Problem 4 of §5. In particular, we can apply Lemma 10 from the next section. Thus $q_{i}^{j}$ only depends on $c_{i}^{j}$ and $c_{-i}$. This property is preserved by integration over the $c_{-i}$: $Q_{i}^{j}$ only depends on $c_{i}^{j}$. ∎

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 $(q,x,h)$ is the pointwise optimal solution of Problem 3 and $K_{i}^{j}$ is smooth in $c_{i}^{j}$ for $(i,j)\in I\times J$ and $c\in\boldsymbol{C}_{i}$, then for all $i\in I$, $Q_{i}$ is $C^{\infty}$ over $\boldsymbol{C}_{i}$.

Proof.

Remember that $c_{i}^{j}\rightarrow c_{i}^{j}+K_{i}^{j}(c_{i}^{j})$ is increasing, so by composition with smooth bijection, we can do the proof as if the costs involved were $c_{i}^{j}$ instead of $c_{i}^{j}+K_{i}^{j}(c_{i}^{j})$. According to Lemma 11, $q_{i}$ is continuous. Since $q_{i}$ is bounded, we can apply the dominated convergence theorem to show that $Q_{i}$ is continuous. We can then we proceed by mathematical induction. Assume that $Q_{i}$ is $C^{l}$, then take $c_{i}^{0}\in\boldsymbol{C}_{i}$ and $c_{i}^{k}$ a sequence in $\boldsymbol{C}_{i}$ that converges to $c_{i}^{0}$. Since $\hat{\mathcal{S}}=\cup_{k\in\mathbb{N}}\mathcal{S}(c_{i}^{k})$ 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 $\boldsymbol{C}^{-i}\backslash\hat{\mathcal{S}}$ instead of $\boldsymbol{C}^{-i}$. Since $q_{i}$ and its derivatives are bounded, we can apply the dominated convergence theorem to compute the limit of $\frac{Q_{i}^{(l)}(c_{i}^{0})-Q_{i}^{(l)}(c_{i}^{k})}{c_{i}^{0}-c_{i}^{k}}$ as $k$ goes to $+\infty$ as the integral of a limit. Since we removed the point over which this limit was not defined, we get that $\frac{Q_{i}^{(l)}(c_{i}^{0})-Q_{i}^{(l)}(c_{i}^{k})}{c_{i}^{0}-c_{i}^{k}}$ has a limit, and this limit does not depend on the sequence $c_{i}^{k}$. So $Q_{i}$ is $l+1$ times derivable at $c_{i}$, for all $c_{i}$. 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 $(q_{i}^{j},h)$ be defined such that for any $c\in\boldsymbol{C}^{n}$, $(q_{i}^{j}(c_{i}^{j},c_{-i}),h(c))$ solves

[TABLE]

and set

[TABLE]

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 optimal for Problem 3, moreover, by construction $(q,h,x)$ satisfies (SD) and $h\geq 0$.

• 2.

Then note that by Lemma 4, $(q,h,x)$ solves a relaxation of Problem 2, but is it admissible for Problem 2 ?

• 3.

By definition of $V$ (omitting $i$),

[TABLE]

where we used the definition of $x$, the definition of $Q$ and Fubini lemma’s for the second, third and fourth equalities. Threfore $(q,h,x)$ satisfies (H1).

• 4.

By construction, $q_{i}^{j}$ is non-increasing in $c_{i}^{j}+K_{i}^{j}(c_{i}^{j})$ then using the third assumption, $q_{i}^{j}$ is non-increasing in $c_{i}^{j}$ so for any $(a,b,c_{-i})\in\boldsymbol{C}^{2}\times\boldsymbol{C}^{-i}$, $(a_{i}^{j}-b_{i}^{j})(q_{i}^{j}(a_{i}^{j},c_{-i})-q_{i}^{j}(b_{i}^{j},c_{-i}))\leq 0$, so by integration with respect to $c_{-i}$, $(a_{i}^{j}-b_{i}^{j})(Q_{i}^{j}(a_{i}^{j})-Q_{i}^{j}(b_{i}^{j})\leq 0$ and then by summation over $j$, $(c-c^{\prime}).(Q_{i}(c)-Q_{i}(c^{\prime}))\leq 0$, i.e. (H2) is satisfied.

• 5.

Since (H1) is satisfied, $V_{i}(c_{i})\geq V_{i}(c_{i}^{+})$. Moreover, $V_{i}(c_{i}^{+})=0$ by construction of $x$. Therefore the participation constraint (PC) is satisfied.

• 6.

Therefore $(q,h,x)$ is admissible for Problem 2. So it solves Problem 2.

• 7.

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 admissible for Problem 1, but is it optimal ?

• 8.

By Lemmas 1 and 2, any optimal solution of Problem 1 should be admissible for Problem 2. Since the criteria are the same, we conclude that $(q,h,x)$ is an optimal solution of Problem 1.

∎

3.5 Comments

In the optimal mechanism, the agents are paid at a marginal price which is equal to their bid augmented by an information rent. This information rent depends on the problem structure since it is built from a collection of allocation problems, and it depends on the available information by the fact that, in these optimization problems, the marginal prices are replaced by the virtual marginal prices $c_{i}^{j}+K_{i}^{j}(c_{i}^{j})$. We point out that, as already noted in [19], the computation of such rent may pose a practical difficulty for large problems.

Notice that, by construction, the optimal mechanism is incentive compatible no matter 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 producers’ cost parameters.

The model extends to the case in which some nodes do not have a producer and where for others, the demand is null. In particular, we can consider the buyer/suppliers setting where there is demand only at one node.

One may argue that one limit of the current result is that it does not take into account any network constraints. Nonetheless, the structure of the proof makes it clear that we exploited only some properties of the allocation problem. Therefore, the optimal mechanism construction is valid for any market for which the allocation problem satisfies these 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 the solution 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 large variety of network constraints. This extension is of particular importance for power market 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 is known to bring its lot of technical challenges. As argued in [20] , the allocation problem can be written:

[TABLE]

With $g_{i}$ being concave functions, $A$ and $B$ are $p\times m$ and $p\times n$ real matrices; $b\in\mathbb{R}^{p}$, and $H$ and $Q$ are (convex) products of segments in $\mathbb{R}^{m}$ and $\mathbb{R}^{n}$ respectively.

They observe that whenever the multipliers of the first set of constraints in (23) are positive (*), then the optimal solution of (23) is also a solution of

[TABLE]

where

[TABLE]

and

[TABLE]

We assume that (*) is satisfied. Otherwise said, we require the sub-gradient of the value of (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 $\delta_{C}$ the support function of $C$ and set $U=\left\{u=\left(u_{1},\ldots,u_{n}\right)|u_{i}\leq 0\right\}$ Applying Theorem 10.1 from [21], we get that a necessary and sufficient condition for an allocation to be optimal is that:

[TABLE]

Now observe that

[TABLE]

The last equation requires the qualification constraint (Q) from [22] to be satisfied, so one can use Theorem 4.3 from [22]. Still, note that no matter Q being satisfied, $N_{C}$ does not depends on $c$.

Theorem 4**.**

If () is satisfied, then $q_{i}^{j}$ is independent of $c_{i}^{k}$ for $k\neq j$, moreover, the mechanism proposed ib the previous section can be applied to (23).*

Proof.

Fix $c$ and consider the associated optimal allocation $\mathbf{q}$. Observe that either $\mathbf{q}_{i}^{j}\in]0,\bar{q}_{i}^{j}[$ or $\mathbf{q}_{i}^{j}\in\{0,\bar{q}_{i}^{j}\}$.

First case

If $\mathbf{q}_{i}^{j}\in]0,\bar{q}_{i}^{j}[$, take $k\neq j$ then $c_{i}^{k}$ does not appear in the first order condition (27). By Berge’s Maximum Principle [23], the optimal allocation is upper hemicontinuous with respect to the parameter $c_{i}^{k}$, by unicity of the solution of (24), we get that $q_{i}$ is continuous with respect to $c_{i}^{k}$. Thus there is a neighbourhood of $c_{i}^{k}$ such that $q_{i}^{j}$ is still in $]0,\bar{q}_{i}^{j}[$. In this neighbourhood, condition (27) is satisfied for $q_{i}^{j}=\mathbf{q}_{i}^{j}$, by unicity of the solution, $q_{i}^{j}$ is constant with respect to $c_{i}^{k}$ on this neighbourhood.

Second case

$\mathbf{q}_{i}^{j}\in\{0,\bar{q}_{i}^{j}\}$. Without loss of generality, let us assume that $\mathbf{q}_{i}^{j}=\bar{q}_{i}^{j}$. Here, we need to observe that the sub-differential of the criteria with respect to $q_{i}$ is $[c_{i}^{j},c_{i}^{j+1}]$, thus the reasoning of the first case can be reproduce whenever $k\neq j+1$. So we only need to deal with the situation where $k=j+1$. Moreover, since $q_{i}$ is non-increasing in $c_{i}^{j+1}$, only an increase of $c_{i}^{j+1}$ can potentially trigger a change in $q_{i}^{j}$. Observe that by Berge’s Maximum Principle, $q_{i}^{j}$ is continuous with respect to the parameter of interest $c_{i}^{j+1}$. If it happens to take a value different than $\bar{q}_{i}^{j}$, then this value is also a solution to (27) for the initial parameters, which is in contradiction with the unicity of the solution of (27). ∎

4.1 Discussion on the non-overlaping zone assumption

We show here how the non-overlaping zone structure naturally emerges if we envision slightly 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 have a density $f$ over $C$ 444we still keep non decreasing marginal costs, otherwise we would loose the continuity of $q$ , denoting by $f_{j}$ the marginals, (4) it will prove to be convenient to use $q*c:=C^{c}(q)$ and denote by $j(q)$ the integer part of $q/\bar{q}$.

Let $q$ and $x$ be some optimal allocation and payment rules for the producer. We assume $q$ and $x$ are continuous and have derivatives almost everywhere. The producer’s profit is $x(s)-q(s)*c$ whenever his type is $c$ and he signals himself as of type $s$. When the competition is known, we pinpoint that for any value $q=q(c)$ of the allocation function, there should be a unique payment that we denote by $x_{q}$. If it was not the case, then the producer would be better off revealing what is required to get the highest possible 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 particular that $c_{j(q)}\in\partial_{q}x(q_{c})$. Hence, almost everywhere and whenever $q$ is not a multiple of $\bar{q}$, $c_{j(q)}$ is uniquely defined. Setting $c_{q}=c_{j(q)}$, we observe that $x_{q}=\int_{0}^{q}c_{t}{\rm d}t$($\star$).

If we take $s\in C$ such that $q(s)$ is not a multiple of $\bar{q}$, then by (IC) we know that $c\rightarrow x(c)-q(c)*s$ should be maximal at $s$. Take $t$ such that $t_{j(q(c))}=s_{j(q(c))}$ and $q(t)\neq q(s)$, then we should have $j(q(t))\neq j(q(s))$ which would imply a discontinuity of $q$ somewhere. Hence: whenever $s_{j(q(s))}=t_{j(q(s))}$, $q(s)=q(t)$ ($\star\star$). Let $C_{k}=\{c\in C,j(q(c))=k\}$, $c_{k}^{-}=\inf\{c_{k},c\in C_{k}\}$, and $c_{k}^{+}=\sup\{c_{k},c\in C_{k}\}$. By combining ($\star\star$) with the monotony of $q$, we get that the intersection of the $[c_{j}^{-},c_{j}^{+}]$ has an empty interior.

4.2 Examples with log-concave functions

We point out that a sufficient condition to check the monotone likelihood ratio property 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 $f/F$ is decreasing iff $\ln F^{\prime}$ 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 literature are log-concave: the uniform, the normal, the exponential and the power function and 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 by monotonic transformation and truncation. Moreover, it happens that if a probability density distribution is log-concave, then the corresponding cumulative distribution is log-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 assumption imposes a gap $\Delta$ equals to $K_{i}^{j}(c_{i}^{j+})$ between $c_{i}^{j+}$ and $c_{i}^{(j+1)-}$. We compute this gap for some standard cases. To simplify the notations and the computation, we assume without loss of generality that $c^{j-}=0$ and write $c^{j+}=c^{+}$. This results in the following table:

We truncate the probabilities so that they have support in $[0,c^{+}]$. The symbole $\propto$ means that we express $f$ and $F$ modulo the multiplication by a common constant (due to the truncation) and $\lambda$ is a positive parameter that should be greater than $1$ for the Power function and the Weibull probability. For the uniform distribution, we see that the intervals should be of non-decreasing sizes. For instance, one could take $c^{1}\in[\bar{c},2\bar{c}]$, $c^{2}\in[3\bar{c},4\bar{c}]$, $c^{3}\in[5\bar{c},6\bar{c}]$, 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

5.1 The standard auction problem

The previous section motivates the study of the allocation problem for different reasons. Firstly, as we have already pointed out in the proofs, the results of §3 rely on some properties of the solution of the standard allocation problem. In addition to those properties, we derive in this section an algorithm to compute the solution of the standard allocation problem. According to 3, these algorithms can be used for both the original auction problem and the optimal mechanism design. To benchmark the mechanism design equilibrium against an equilibrium of the Bayesian game related to the standard auction, numerical efficiency is pivotal: indeed, the Bayesian equilibrium estimation requires a large number of allocations computations.

Let us first introduce the standard allocation problem. In a standard mechanism, the principal solves an allocation problem based on the bids he receives. Those bids will be denoted by $c_{i}^{j}$, where as before $i\in I$ corresponds to the ith agent and $j\in J$ corresponds to the jth working zone with constant marginal price. To model the fact that the production costs are piecewise linear, we use some positive variables $q_{i}^{j}$ so that $q_{i}^{j}\leq\bar{q}$, for any $i\in I$, the quantity produced by agent $i$ is $q_{i}=\sum_{j\in J}q_{i}^{j}$ and the related production cost is $\sum_{j\in J}c_{i}^{j}q_{i}^{j}$. As before, an allocation should satisfy the constraint that production exceeds demand. We end up with Problem 4:

Problem 4**.**

[TABLE]

The notations for the dual the variables associated with each constraint are indicated in parentheses. Those variables are in $\mathbb{R}_{+}$.

For any node $i\in I$, we define the function $F_{i}$ for $\lambda\in[\min_{i}c_{i}^{1},\max_{i}c^{N}_{i}]^{n}$

[TABLE]

Later on we justify that this function could be interpreted as the production of agent $i$ when the multipliers are $\lambda_{i}$ and $\lambda_{-i}$. Its partial derivative with respect to $\lambda_{i}$ is

[TABLE]

The derivative is negative: when $i$ increases its price it is assigned smaller production quantities. The partial derivative of $F_{i}$ for $i^{\prime}\in I\backslash\{i\}$ is

[TABLE]

When another agent becomes less competitive, $i$ is assigned more production. Let $k\in J\cup\{0\}$. The limit at $+\infty$ and [math] of $F_{i}(x,\lambda_{-i})-k\bar{q}$ are

[TABLE]

and

[TABLE]

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 the intermediate value theorem, $F_{i}-k\bar{q}$ has a zero. Since $F_{i}-k\bar{q}$ is decreasing in $\lambda_{i}$, this solution is unique. Now we define for $i\in I$ and $k\in J\cup\{0\}$, $g^{k}_{i}$ as the function that associates any $\lambda_{-i}\in[\min_{i}c_{i}^{1},\max_{i}c^{N}_{i}]^{n-1}$ with the unique $x$ such that and $F_{i}(x,\lambda_{-i})=k\bar{q}$ and $x>0$:

[TABLE]

Lemma 8**.**

For any $i\in I$, $k\in J\cup\{0\}$, $\lambda^{-i}\in[\min_{i}c_{i}^{1},\max_{i}c^{N}_{i}]^{n-1}$ and $i^{\prime}\in V(i)$

[TABLE]

In particular, $g_{i}^{k}$ is increasing in $\lambda_{i^{\prime}}$ for $i^{\prime}\in V(i)$.

Proof.

According to the implicit function theorem

[TABLE]

∎

It is clear that $g_{i}^{k}(\lambda_{-i})$ is decreasing in $k$. We proceed with the computation of the dual of Problem 4. If a strong duality theorem applies, then we should have

[TABLE]

so that for any $(i,i^{\prime})\in E$, by necessary and sufficient first order condition

[TABLE]

By replacing $h$ by its expression in the dual variables we get something equivalent to

[TABLE]

The expression of $\gamma$ with respect to $\lambda$ follows. For any $(i,i^{\prime})\in E$

[TABLE]

thus the dual problem is equivalent to

[TABLE]

because $\mu$ does not play any role in the admissibility of the other variables nor in the objective, this is equivalent to

[TABLE]

The expression of $\nu$ follows. For any $(i,j)\in I\times J$

[TABLE]

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:

[TABLE]

where

[TABLE]

For $i\in I$ we maximize the criteria

[TABLE]

which is strictly concave for any $\lambda_{-i}$ (sum of concave and strictly concave functions). We denote by $\Lambda_{i}(\lambda_{-i})$ its maximizer. The first order necessary and sufficient condition on $\Lambda_{i}$ is:

[TABLE]

where

[TABLE]

We conclude

Lemma 9**.**

For any $i\in I$ and any $\lambda^{-i}\in[\min_{i}c_{i}^{1},\max_{i}c^{N}_{i}]^{n-1}$, $\Lambda_{i}(\lambda_{-i})$ is the unique solution of

[TABLE]

We point out that the primal (and dual) solution unicity is a desirable property that is not systematic for the allocation problems of centralized market models. The expression of $h$ with respect to $\lambda$ (39) together with the fact the fact the supply constraint should be binding at optimality justify the interpretation of $F_{i}$ proposed at the beginning of this subsection. In the following sequel we use this property many times.

5.2 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 $c\in\boldsymbol{C}^{n}$. We denote by $q(c)$ the solution of Problem 4 with the cost vector $c$. Similarly, we define $q_{i}(c)$, $q_{i}^{j}(c)$, $\lambda(c)$ and $\lambda_{i}(c)$. We give here two properties of the allocation problem solution. By integration, we showed in the previous section that the solution of the mechanism design inherits those properties.

Lemma 10**.**

Let $(q(c),h(c))$ be a solution of Problem 4, then $q_{i}^{j}(c)$ does not depend on $c_{i}^{l}$ for $l\neq j$:

[TABLE]

Proof.

Let $i\in I$, $j\in J$, $c_{-i}\in\boldsymbol{C}^{n-1}$, $c=(c^{1},\ldots,c^{N})\in\boldsymbol{C}$ and $s=(s^{1},\ldots,s^{N})\in\boldsymbol{C}$ such that $s^{j}=c^{j}$. We shall prove that $q_{i}^{j}(s,c^{-i})=q_{i}^{j}(c,c^{-i})$. We denote by $\lambda^{c}$ (resp. $\lambda^{s}$) the dual variables associated with the nodal contraints for the allocation problem parametrized with $c$ (resp. $s$). First if

[TABLE]

then by lemma 9 $\lambda^{c}_{i}=c_{i}^{j}$ and using Lemma 9 again, $\lambda^{s}_{i}=c_{i}^{j}$. Therefore $\lambda^{s}=\lambda^{c}$, from which we deduce that $q_{i}^{j}(c,c^{-i})=q_{i}^{j}(s,c^{-i})$.

Therefore without loss of generality, we can assume that

[TABLE]

Then using Lemma 9 we get

[TABLE]

so that $\lambda^{c}_{i}\geq\lambda^{s}_{i}$. If $\lambda^{c}_{i}>\lambda^{s}_{i}$, then $\lambda^{c}_{-i}\geq\lambda^{s}_{-i}$ by non-decreasingness of $\Lambda_{i^{\prime}}$, $i^{\prime}\in I\backslash\{i\}$ (explained in §5.3) Therefore all the other agents are producing less, which is absurd since $i$ is already producing less.

∎

We extend the notations by setting for all $i\in I$, $c_{i}^{0}=c_{*}$. We consider the subset $\mathcal{S}$ of $\boldsymbol{C}$ for which at some nodes $i$, the multiplicator $\lambda_{i}$ is equal to the marginal cost and the production is a multiple of $\bar{q}$ (i.e. stuck in an angle):

[TABLE]

The set $\mathcal{S}$ corresponds to the points of transition between the two possibilities defined by the first order condition (48). Because of the angle, it is natural to think that this is where irregularities may happen (see the proof of the next lemma). We introduce this set to show some regularity properties of $q$ and $Q$. We detail the proof in the Appendix. The approach consists in showing that $\mathcal{S}$ is a finit union of sets of zero measure. This is also true for the projection of $\mathcal{S}$ on the $\{c_{i}\}\times\mathcal{C}^{-i}$. Then we observe that on $\boldsymbol{C}\backslash\mathcal{S}$, the relations between the primal and dual variables are smooth.

Lemma 11**.**

The function $q$ is $C^{\infty}$ on $\boldsymbol{C}^{n}\backslash\mathcal{S}$ and $C^{0}$ on $\boldsymbol{C}^{n}$.

Proof.

We postpone the proof to Appendix B ∎

5.3 Fixed point

In this subsection we show that the solution of the dual problem is the unique fixed point of a monotone operator. We define

[TABLE]

Lemma 12**.**

For any $i\in I$, $\Lambda_{i}$ is non-decreasing.

Proof.

Let $\lambda_{-i}<\lambda_{-i}^{\prime}$ and the corresponding $\Lambda_{i}$ and $\Lambda_{i}^{\prime}$. Assume $\Lambda_{i}>\Lambda_{i}^{\prime}$. Since $F_{i}$ is decreasing in the first variable and increasing in the second

[TABLE]

Moreover for any $x\in K(\Lambda_{i}^{\prime})$ and $y\in K(\Lambda_{i})$, $x\leq y$ and $F_{i}(\Lambda_{i},\lambda_{-i})\in K(\Lambda_{i})$, $F_{i}(\Lambda_{i}^{\prime},\lambda_{-i}^{\prime})\in K(\Lambda_{i}^{\prime})$. Therefore $F_{i}(\Lambda_{i}^{\prime},\lambda_{-i}^{\prime})\leq F_{i}(\Lambda_{i},\lambda_{-i})$ which is absurd. ∎

We will use the following classical result (see [17] for a proof and definition of complete lattice).

Theorem 5** (Knaster-Tarski fixed point).**

Let $L$ be a complete lattice and let $f$ an application from $L$ to $L$ and order preserving. Then the set of fixed points of $f$ in $L$ is a complete lattice.

In particular, the set of fixed points is non empty. Since $\Lambda$ is order preserving and $[c_{*},c^{*}]^{n}$ is a lattice when we consider the natural order, there is a fixed point, and the set of fixed points is a lattice.

Lemma 13**.**

$\lambda$* is optimal for the dual $\Leftrightarrow$ $\lambda$ is a fixed point of $\Lambda$.*

Proof.

If $\lambda$ is optimal for the dual, then each component $i$ maximizes the criteria (47), thus $\lambda$ is a fixed point of $\Lambda$.

• 2.

If $\lambda$ is a fixed point of $\Lambda$, then by definition, each component $i$ maximizes the criteria (47). Hence since the problem is (strictly) concave, $\lambda$ is optimal.

∎

A consequence of the previous lemma is that

Lemma 14**.**

The set of fixed points of $\Lambda$ is a singleton.

Definition 1** (Continuous for monotone sequence).**

We consider the natural partial order on $\mathbb{R}^{n}$. 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 continuous for increasing and decreasing sequences.

Lemma 15**.**

The operator $\Lambda$ is continuous for monotone sequences.

The intuition of the proof is that we can use the monotony of the sequence and Lemma 9 to characterize the behaviour of $\Lambda$ on the neighborhood. We find that $\Lambda$ is either constant or characterized by the implicit function theorem.

Proof.

Let $\bar{\lambda}_{-i}$, $j\in[1\ldots N]$, we first deal with the ’nice’ case, that corresponds to $F_{i}(\Lambda(\bar{\lambda}_{-i}),\bar{\lambda}_{-i})\in]j-1,j[\bar{q}$

If $\Lambda_{i}(\bar{\lambda}_{-i})\in]c_{i}^{j},c_{i}^{j+1}[$ (we do not treat the case $j=N$, which is very similar to what follows) then since $F_{i}$ is $C^{\infty}$ and of invertible derivative (non zero) in $\lambda_{i}$, the implicit function theorem tells us that the solution $\psi$ of $F_{i}(\psi(\bar{\lambda}_{-i}),\bar{\lambda}_{-i})=j\bar{q}$ is continous in a neighborhood $B$ of $\bar{\lambda}_{-i}$. Thus we can make $B$ small enough so that for $\lambda_{-i}\in B$, $\psi(\lambda_{-i})\in]c_{i}^{j},c_{i}^{j+1}[$. On this neighborhood, $\psi$ satisfies the first order conditions and so by unicity of the solution of the optimization problem, since those conditions are sufficient, $\psi=\Lambda_{i}$ on $B$. Therefore $\Lambda_{i}$ is continous at $\bar{\lambda}_{-i}$.

• 2.

If $\Lambda_{i}(\bar{\lambda}_{-i})=c_{i}^{j}$ (as before, we do not treat the case $j=N$), then by Lemma 9 $F_{i}(\Lambda_{i}(\bar{\lambda}_{-i}),\bar{\lambda}_{-i})=[j-1,j]\bar{q}$, if $F_{i}\in]j-1,j[\bar{q}$ (we deal with the border case in the next point) then since $F_{i}$ is continuous, there is a neighborhood $B$ of $\bar{\lambda}_{-i}$ such that $F_{i}(\Lambda_{i}(\bar{\lambda}_{-i}),\lambda_{-i})\in]j-1,j[\bar{q}$, so on $B$ $\Lambda_{i}$ is constant and therefore continuous.

• 3.

We proceed with the borders. If $F_{i}(\Lambda_{i}(\bar{\lambda}_{-i}),\bar{\lambda}_{-i})=(j-1)\bar{q}$ and $\Lambda_{i}(\bar{\lambda}_{-i})=c_{i}^{j}$.

• (a)

Decreasing case: Let us take $\epsilon\in\mathbb{R}^{n-1}_{+}$ such that $F_{i}(\Lambda_{i}(\bar{\lambda}_{-i}),\bar{\lambda}_{-i}+\epsilon)\in[j-1,j]\bar{q}$ ($F_{i}$ is continuous and increasing in $\lambda_{-i}$). Then $\Lambda_{i}(\bar{\lambda}_{-i}+\epsilon)=\Lambda_{i}(\bar{\lambda}_{-i})$ checks the first order condition so $\Lambda$ is constant, and we get the continuity for decreasing sequences.

• (b)

Increasing case: $F_{i}(\Lambda_{i}(\bar{\lambda}_{-i}),\bar{\lambda}_{-i})=(j-1)\bar{q}$ hence there exists a ball $B$ such that the implicit function theorem applies and there exists $\psi$ such that $F_{i}(\psi(\bar{\lambda}_{-i}-\epsilon),\bar{\lambda}_{-i}-\epsilon)=(j-1)\bar{q}$ and $\psi(\bar{\lambda}_{-i})=\Lambda_{i}(\bar{\lambda}_{-i})=c_{i}^{j}$ (remember that $\Lambda_{i}(\bar{\lambda}_{-i})=c_{i}^{j}$ by hypothesis) . Since $F_{i}$ is increasing in the second variable and decreasing in the first, $\psi$ is increasing. For $\epsilon$ of positive components and sufficiently small, $\psi(\bar{\lambda}_{-i}-\epsilon)\in]c_{i}^{j-1},c_{i}^{j}[$ (since $\psi(\bar{\lambda}_{-i})=\Lambda_{i}(\bar{\lambda}_{-i})=c_{i}^{j}$) and check the first order condition. Therefore for $\epsilon$ of positive components and sufficiently small, $\psi=\Lambda_{i}$ by uniqueness of the solution. Thus $\Lambda_{i}$ is continuous for increasing sequence.

• 4.

We do the same analysis if $F_{i}(\Lambda_{i}(\bar{\lambda}_{-i}),\bar{\lambda}_{-i})=j\bar{q}$ and $\Lambda_{i}(\bar{\lambda}_{-i})=c_{i}^{j}$.

The conclusion follows. ∎

We could have alternatively used the Berge Maximum theorem for strictly concave criterion to get the continuity of $\Lambda$. Yet, we chose to present this proof for pedagogical reasons because it contains some key ideas (see appendix).

Theorem 6**.**

The sequence $(\Lambda^{k}(c_{1}^{N}...c_{n}^{N}))_{k\in\mathbb{N}}$ converges to the solution of the dual.

Proof.

Since $\Lambda(c_{1}^{N}...c_{n}^{N})\leq(c_{1}^{N}...c_{n}^{N})$, and since $\Lambda$ is order preserving, the sequence $\Lambda^{k}(c_{1}^{N}...c_{n}^{N})=\lambda^{k}$ is non increasing and bounded, therefore it converge to a point $x$. Since $\Lambda$ is continuous for monotone sequence, $x$ is a fixed point. ∎

Theorem 7**.**

For any $i\in I$, $\lambda_{-i}\in[c_{*},c^{*}]^{n-1}$, $\Lambda_{i}(\lambda_{-i})$ has the following explicite expression:

[TABLE]

Proof.

We denote by $G_{i}$ the RHS of (58) and show that for any $i$

[TABLE]

and then we conclude with a uniqueness argument.

If there is $j\in J$ such that $G_{i}(\lambda_{-i})=c_{i}^{j}$, then either $F_{i}(c_{i}^{j},\lambda_{-i})<j\bar{q}$ or $g_{i}^{j}(\lambda_{-i})=c_{i}^{j}$. This last possibility implies by definition of $g_{i}^{j}$ that $F_{i}(c_{i}^{j},\lambda_{-i})=j\bar{q}$. Thus $F_{i}(c_{i}^{j},\lambda_{-i})\leq j\bar{q}$. Remember that $K(G(\lambda_{-i}))=[j-1,j]\bar{q}$. So we need to prove that $F_{i}(c_{i}^{j},\lambda_{-i})\geq(j-1)\bar{q}$. Suppose the contrary, i.e. $F_{i}(c_{i}^{j},\lambda_{-i})<(j-1)\bar{q}$. Then since $G_{i}(\lambda_{-i})=c_{i}^{j}$, $F(c_{i}^{j},\lambda_{-i})<(j-1)\bar{q}$, which in turn implies that

[TABLE]

Now observe that since $G_{i}(\lambda_{-i})=c_{i}^{j}$, $F(c_{i}^{j-1},\lambda_{-i})>(j-1)\bar{q}$, which implies that

[TABLE]

Combining (60) and (61) with the definition of $G$, we see that $G(\lambda_{-i})\leq g_{i}^{j}(\lambda_{-i})$. But $G(\lambda_{-i})=c_{i}^{j}$ and $g_{i}^{j}(\lambda_{-i})<c_{i}^{j}$, which is absurd. Therefore $F_{i}(c_{i}^{j},\lambda_{-i})\geq(j-1)\bar{q}$.

Otherwise, let us assume that there is not such $j$. Then there is $k\in[0\ldots N-1]$ such that $G_{i}(\lambda_{-i})=g_{i}^{k}(\lambda_{-i})$. By definition of $g_{i}^{k}$, $F_{i}(G_{i}(\lambda_{-i}),\lambda_{-i}))=k\bar{q}$ and by definition of $G$, $G_{i}(\lambda_{-i})\in[c_{i}^{k},c_{i}^{k+1}]$. So again $F_{i}(G_{i}(\lambda_{-i}),\lambda_{-i}))\in K(G_{i}(\lambda_{-i}))$. We can now conclude that $\Lambda=G$. ∎

We can interpret the fixed point algorithm as if some benevolent agents situated at each node of the network were exchanging information. They collectively try to minimize 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 of importation from the adjacent nodes. At each iteration, the agents compute how much they are going to produce based on their current marginal cost. They then update their marginal cost based on the information they just received and transmit this marginal cost to the adjacent nodes. We point out that the information used by each agent is local.

5.4 Decreasing Rate

We derive in this section an estimate for the decreasing rate. We denote $\alpha=\max_{(e,e^{\prime})\in E^{2}}r_{e}/r_{e^{\prime}}$. We have the following bound:

Lemma 16**.**

For any $(i,i^{\prime},k,\lambda_{-i})\in E\times[0,N]\times[c_{*},c^{*}]^{n-1}$,

[TABLE]

Proof.

We combine (38) with (32) and (33). ∎

Lemma 17**.**

Since $(\lambda^{k}_{i})_{k\in\mathbb{N}}$ is non-increasing for all $i\in I$, there is a finite number of $k$ for which at least one coordinate $\lambda_{i}^{k}$ satisfies

[TABLE]

or

[TABLE]

We denote by $\mathcal{K}$ this set. Let $(k_{1},k_{2})\in\mathbb{N}^{2}$ such that $[k_{1}-1,k_{2}+1]\cap\mathcal{K}=\emptyset$. Then for $k\in[k_{1},k_{2}]$ and $i\in I$ such that $\lambda^{k-1}_{i}\neq\lambda^{k}_{i}$

[TABLE]

Proof.

By definition of $\lambda^{k}$, $\lambda^{k}_{i}-\lambda^{k+1}_{i}=\Lambda^{i}(\lambda^{k-1}_{-i})-\Lambda^{i}(\lambda^{k}_{-i}).$ By construction, there exists $j\in[0,N-1]$ such that $\Lambda^{i}(\lambda^{k-1}_{-i})=g_{i}^{j}(\lambda_{-i}^{k-1})$ and $\Lambda^{i}(\lambda^{k}_{-i})=g_{i}^{j}(\lambda_{-i}^{k})$. Then by monotony of $g$, $g_{i}^{j}(\lambda_{-i}^{k})-g_{i}^{j}(\lambda_{-i}^{k-1})$ is lower bounded by

[TABLE]

for $i^{\prime}\in V(i)$. We then take the $i^{\prime}\in V(i)$ that maximizes $(\lambda^{k-1}_{i^{\prime}}-\lambda^{k}_{i^{\prime}})$ and use the previous lemma to get the result. ∎

5.5 Algorithm Implementation

We implemented this algorithm in Matlab. We used a dichotomy to compute the $g_{i}^{k}$. Note that for linear cost the analysis is similar. We define $g_{i}(\lambda_{-i})$ as the unique $x$ such that $f_{i}(x,\lambda_{-i})=0$ and $x\geq 0$ and define $\Lambda$ such that

[TABLE]

We performed some numerical comparisons with CVX, a package for specifying and solving convex programs [26, 27] for both linear and piecewise linear production cost functions. We generated a graph with 100 nodes connected randomly. To generate the graph, we used a Barabasi-Albert model [28] to ensure some scaling properties. The experiment was performed on a personal laptop (OSX, 4 Go,1.3 GHz Intel Core i5). The networks randomly generated to test the implementations are displayed in Figures 1(a) and 1(b), and the results are summarized in Table 2.

Both CVX and the fixed point algorithm converges to an estimate of the optimal value. We did not try to optimize the numerical algorithm, but some trick could be used to avoid the costly estimation of the $g$s. Still, the linear version of the fixed point algorithm was about ten times faster than the CVX resolution. Note that the algorithm could be distributed, since at each iteration, the computation at each node only depends on the values of the previous iteration.

6 Conclusion

In this paper we have shown how to characterize and compute the optimal mechanism for a network market. We observed in particular that the allocation problem for the optimal and the standard mechanism are the same. We have proposed an algorithm based on a fixed point to solve the allocation problem and derived regularity properties of the solution. Our contribution provides a direction to benchmark mechanism proposals.

Appendix A Proof of Lemma 5

Proof.

By definition

[TABLE]

We use (H1) for the last equality. Then we apply a telescopic formula

[TABLE]

Reordering the last three terms, we get

[TABLE]

We end up with

[TABLE]

∎

Appendix B On $\mathcal{S}$ and the regularity of $q$

Remember that the set $\mathcal{S}$ corresponds to the points of transition between the two possibilities defined by the first order condition (48):

[TABLE]

Our first goal is to show that $\mathcal{S}$ is a finite union of sets of zero measure (Lemmas 18 and 20). To do so, we apply the implicit functions theorem. From this we deduce the regularity of $q$ (proof of Lemma 11). For any $I_{A}$, $I_{B}$ partition of $I$, and $I_{C}\subset I_{B}$ not empty, $j\in J^{I}$ and $j^{\prime}\in J^{I}$ such that for all $i$, $j^{\prime}\in\{j_{i},j_{i}+1\}$, we denote by $S(I_{A},I_{B},I_{C},j,j^{\prime})$ the set

[TABLE]

For an element $c$ of such set, we denote by $M$ the matrix

[TABLE]

We need to study the invertibility of $M$ to apply the implicit functions theorem (Lemma 19). Note that the function $S$ is defined on a finite set. We use the image of $S$ to show that the measure of $\mathcal{S}$ with respect to the Lebesgue measure is zero. We first show in the next lemma that $\mathcal{S}$ is included in the finite union of the $S(I_{A},I_{B},I_{C},j,j^{\prime})$ family. Then we will show that each element of this family has a measure equal to zero.

Lemma 18**.**

$\mathcal{S}\subseteq\cup S(I_{A},I_{B},I_{C},j,j^{\prime})$**

Proof.

Take $c\in\mathcal{S}$, then by definition of $\mathcal{S}$, there exist $i\in I$, $j\in J$ and $j^{\prime}\in\{j,j+1\}$ such that $q_{i}(c)=j\bar{q}$ and $\lambda_{i}(c)=c_{j^{\prime}}$, therefore $I_{C}$ is not empty. By Lemma 9, for all $i\in I$, $i$ is in $I_{A}$ or $I_{B}$. Hence we have a set $S(I_{A},I_{B},I_{C},j,j^{\prime})$ such that $c$ is in this set, so $\mathcal{S}$ is included in the union of those sets. ∎

Lemma 19**.**

For any $c\in\mathcal{C}^{n}$ the matrix $M(c)$ is invertible.

Proof.

Assume that there are some coefficients $\alpha_{i}$ such that $\sum_{i}\alpha_{i}M_{i}=0$ where $M_{i}$ is the ith column of $M$. Then by (32) and (33), the ith row of this relation writes:

[TABLE]

We denote $b_{i,j}=\frac{\lambda_{j}^{2}\lambda_{i}}{r_{i,j}(\lambda_{i}+\lambda_{j})^{3}}$ and $a_{i}=\frac{\alpha_{i}}{\lambda_{i}}$. Then (72) is equivalent to

[TABLE]

Insofar as we can slightly perturb the demand, we assume without loss of generality that t it is not possible to produce a multiple of $\bar{q}$ at each node and satisfy exactly the nodal constraints ($\star$).

Considering the biggest $a_{i}$, we get that all $a_{i}$ are equal by convexity, thus either all are equal to zero or

[TABLE]

which is not the case since $I_{A}$ is not empty by ($\star$). ∎

Next we show that $S(I_{A},I_{B},I_{C},j,j^{\prime})$ has a zero Lebesgue measure.

Lemma 20**.**

For any $I_{A}$, $I_{B}$ partition of $I$, and $I_{C}\subset I_{B}$ not empty, $j\in J^{I}$ and $j^{\prime}\in J^{I}$ such that for all $i$, $j^{\prime}\in\{j,j+1\}$, the measure of the set $S(I_{A},I_{B},I_{C},j,j^{\prime})$ is zero.

Proof.

We assume in the market description that it is not possible to produce a multiple $\bar{q}$ at each node and satisfy exactly the nodal constraints (($\star$). Therefore it is not possible that $I_{B}=I$, therefore $I_{A}$ is not empty. By definition of $S_{I_{A},I_{B},I_{C},j,j^{\prime}}$, for all $i\in I_{B}$,

[TABLE]

which is a system of equations in $\lambda_{I_{B}}$ parametrized by $c_{I_{A}}^{j^{\prime}}$. Let $c\in\boldsymbol{C}$ such that the system is satisfied, by Lemma 19, we can apply the implicit function theorem, hence there is a ball around $c$ in which $S(I_{A},I_{B},I_{C},j,j^{\prime})$ is included in a smooth surface. By compacity of $\boldsymbol{C}$, we can choose a sequence dense in $S(I_{A},I_{B},I_{C},j,j^{\prime})$. We apply the result to each element of this sequence. By density, $S(I_{A},I_{B},I_{C},j,j^{\prime})$ is a countable union of smooth surfaces. Therefore the measure of $S(I_{A},I_{B},I_{C},j,j^{\prime})$ is zero. ∎

A direct consequence of Lemma 20 and Lemma 18 is

Lemma 21**.**

The measure of $\mathcal{S}$ is zero.

We proceed with the proof of Lemma 11.

of lemma 11.

Let $c=(c_{1}\ldots c_{n})\in\boldsymbol{C}^{n}\backslash\mathcal{S}$. Let us show that $q$ is infinitely differentiable at $c$. We consider the two assertions:

[TABLE]

[TABLE]

By Lemma 9 and by defintion of $\mathcal{S}$, for any $i\in I$ 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 $\tilde{c}$ of $V$, $F_{i}(\tilde{c})\in]k_{i}-1,k_{i}[\bar{q}$, therefore on $V$, $\lambda(\tilde{c})=\tilde{c}$.

Else $I_{B}$ is not empty and by definition of $B_{i}$

[TABLE]

which we can see as an equation in $\lambda_{I_{B}}$ parametrized by $\lambda_{I_{A}}$. This equation is satisfied at $\lambda(c)$. If we denote by $M$ the matrix

[TABLE]

then $M$ is invertible (see lemma 19), the implicit function theorem applies and there exists a function $\lambda_{I_{B}}$ so that in a neighborhood $V$ of $c$, for all $i\in I_{B}$, we have $F_{i}(\lambda_{I_{A}},\lambda_{I_{B}}(\lambda_{I_{A}}))=\bar{q}k_{i}$. Moreover, since $F_{i}$ is $C^{\infty}$ on $[c_{*},c^{*}]^{n}$, $\lambda_{I_{B}}$ is $C^{\infty}$ on $V$. Then if $\tilde{c}\in V$, $(\tilde{c},\lambda_{I_{B}}(\tilde{c}))$ checks the first order condition thus by uniqueness $c_{I_{A}},\lambda_{I_{B}}(\tilde{c})$ is the dual solution, and so, $q_{i}=F_{i}(\lambda_{I_{B}}(\tilde{c}),\tilde{c})$ for all $i\in I$ on $V$, so $q_{i}$ is $C^{\infty}$ at $c$. This concludes the proof of the first 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**.**

Let $i\in I$ and $c_{i}\in C_{i}$, then the Lebesgue measure of the set

[TABLE]

is zero.

Proof.

Using Lemma 18, $\mathcal{S}_{i}(c_{i})\subseteq\{c_{-i}\in C_{-i},(c_{i},c_{-i})\in\cup S(I_{A},I_{B},I_{C},j,j^{\prime})\}$. Now let $c_{-i}\in\mathcal{S}_{i}(c_{i})$, $I_{A}$, $I_{B}$ a partition of $I$, and $I_{C}\subseteq I_{B}$ not empty, and $j$, $j^{\prime}$ such that $(c_{i},c_{-i})\in S(I_{A},I_{B},I_{C},j,j^{\prime})$. There are three possible cases:

$i\in I_{A}$ then as explained in the proof of Lemma 20, $S(I_{A},I_{B},I_{C},j,j^{\prime})$ is locally a surface parametrized by $c_{i}$ so by projection over an hyperplane of the type $c_{i}=x$ it also a surface in $\boldsymbol{C}_{-i}$.

• 2.

$i\in I_{B}\backslash I_{C}$ locally, $q$ is independant of $c_{i}$ therefore if $S(I_{A},I_{B},I_{C},j,j^{\prime})\cap(c_{i},\mathcal{S}_{i}(c_{i}))$ is of strictly positive measure, then $S(I_{A},I_{B},I_{C},j,j^{\prime})$ has also a strictly positive measure in $\boldsymbol{C}^{n}$, since this is not true, $S(I_{A},I_{B},I_{C},j,j^{\prime})\cap(c_{i},\mathcal{S}_{i}(c_{i}))$ is of zero measure in the neighborhood.

• 3.

Else $i\in I_{C}$, which is the tricky part. First by definition of $I_{C}$, for any element $c$ of $S(I_{A},I_{B},I_{C},j,j^{\prime})$, $q_{i}(c)=j_{i}\bar{q}$ and $\lambda_{i}(c)=c_{i}^{j_{i}^{\prime}}$. Without loss of generality, we assume $j_{i}^{\prime}=j_{i}$, the other case can be treated similarly. Then we make the observation that we do not modify the $c_{-i}$ of $S(I_{A},I_{B},I_{C},j,j^{\prime})$ if we set $c_{i}^{j+1}=c_{i}^{j}$. Since we are interested in $S(I_{A},I_{B},I_{C},j,j^{\prime})\cap(c_{i},\mathcal{S}_{i}(c_{i}))$, we can assume without loss of generality that $c_{i}^{j+1}=c_{i}^{j}$. Then we have reduced to the case $i\in I_{A}$.

We conclude as in the proof of Lemma 20. ∎