In this work we deploy a mechanism design approach for allocating a divisible commodity (electricity in our example) among consumers. We consider to have an energy availability function and for each consumer an associated personal valuation function of the energy resource during a certain time interval. We aim to select the optimal consumption pro le for every user avoiding consumption peaks when the total required energy could exceed the energy production. Initially, we apply a Vickrey-Clarke-Groves (VCG) mechanism, we show its properties and we discuss its weakness. Then, we describe our future works, developing a mechanism with veri cation. Our aim is to guarantee the maximization of social welfare (e ciency) and satisfy the budget balance property (the distributor's income must remain stable). So, we evaluate to de ne a cost-sharing mechanism in which for each tick of time if we exceed the amount of available energy the cost of exceeding energy provided by the distributor will be shared among users, deploying the Shapley value.
In this work, we face a problem that involves a community of users and a set
of resources: the energy. The issue deals with the management of users'
consumption according to the amount of available energy. There are several reasons
for managing energy, rst of all avoiding blackouts. A blackout could occur in
situations in which the produced energy is not enough to satisfy users' energy
demand and it causes a huge loss of money and security risks of the
electricitybased systems [
De nition 1 (Player's Valuation Function) Let us consider a set of players N = f1; : : : ; ng and a set of alternatives or outcomes A. Every player i has a preference over alternatives that is described by a valuation function: vi : A ! R, where vi(a) denotes the valuation that player i assigns to outcome a. Furthermore, vi 2 Vi where Vi RjAj is a set of possible valuation functions for player i.
De nition 2 (Social Choice Function) The social choice function selects an alternative (or outcome) from the set of alternatives A according to the vector of users' valuation functions: f : V1 Vn ! A. So, an outcome a, from the set A of alternatives, depends on each possible pro le v = (v1; v2; : : : ; vn): a = f (v). This outcome is called social choice for that pro le.
When considering a mechanism with money, that is a mechanism where there are money transfers between the mechanism and players, a payment function computes money transfers for every players.
De nition 3 (VCG Mechanism with Clarke Pivot Rule) A VCG mechanism determines f (v) that is the social choice function with A as the possible outcomes, such that: f (v) 2 argmaxb2A Pn j=1 vj (b); 3
An approach close to our work is[
Suppose that the total users' consumption is less or equal than produced energy, we have that the social choice function will select the outcome maximizing the sum of valuations.
Case 2 In this second case, we propose a mechanism that assigns to users all the available energy till reaching the total amount of produced energy. Considering a scenario with n players, we have always a set of possible outcomes A [0; xi]n where xi is the desired consumption of user i. As in the case in Section 3, we do not consider all the possible elements of A but A = fa : Pjn a[j] P g and P is the energy available. In this way, the distributor can provide an arbitrary amount of energy to user i where the maximum xi is the i's optimal consumption. The valuation function of user i is: vi(a) = a[i]. Case 3 We start from the mechanism described in Section 3. But, the main di erence is that in this fourth case we take into account also the time t. Formally, we introduce a time variable: t 2 [0; T ], where T is the maximum length. Considering a scenario with n players, we have that a possible outcome is a = (a1(t); : : : ; an(t)) where ai(t) 2 R+ for every t 2 [0; T ] and where xi : [0; T ] ! R is the desired consumption function of user i over time t. The function ai(t) represents the assigned power to user i for every tick t, in the same way the function P (t) is the function of available power for every tick t, where the energy is the power per unit time.
The valuation function is: vi(a) = ( ai(t) xi(t) if ai(t) xi(t) if ai(t) > xi(t) representing the total amount of energy received by the i-th customer once the excess energy has been discarded. 4
In the last period, we applied a VCG mechanism for the DSM problem. We
propose several con gurations of the mechanism, starting from the simplest one
to a more complicated con guration which has the property of assigning the
available energy to users according to their desired energy minimizing the
energy wasting while maximizing the aggregate utility of all users. A mechanism
is composed of a social choice function and a payment scheme. In this case we
use the VCG payment scheme but in this way users will not pay according to
the amount of energy assigned to them but they pay a di erent amount with
respect to their energy consumption. For this reason, we decided to study
different solutions where the payment will be more related to the energy used.
Considering that this social choice process depends on the information collected
from agents, they may nd it convenient to misreport their preferences. For this
reason, the VCG mechanism is used in a context in which the \truthfulness"
is a basic aspect, in fact the dominant strategy for a user is to declare his real
preferences. On our speci c case, we do not have to deal with truth or lies but
the concept of truth is represented by the concept of \consume according to
the available energy" and the lie by the concept of \consume when there is no
available energy". So, the punishment comes when a user consume too much
and the consumption is greater than the available energy. Thus, we want to
develop a mechanism with veri cation. In fact, in this context of fair allocations of
goods with monetary compensation is possible to focus on agents' declarations
on allocated goods that can be veri ed before payments are performed. But, the
veri er could evaluate only what it has already happened. First of all we allocate
the goods then the mechanism control if users have behaved as they had said.
Then, we deploy the Compensation and Bonus Mechanism [
Our aim is to guide users to a energy-aware behaviour not to encourage users to tell the truth. Furthermore, we must de ne a fair allocation of resources with a di erent payment scheme that makes it convenient to consume where the energy is su cient. The nal objectives of our mechanism will rst of all to guarantees the maximization of social welfare (e ciency) and satis es the budget balance property (the distributor's income must remain stable).
These characteristics could be reached by the de nition of a cost-sharing
mechanism in which for each tick of time if we exceed the amount of available energy
the cost of exceeding energy provided by the distributor will be shared among
users. This cost division could be calculated by the Shapley value [
The mechanism will be subsequently tested by a multi-agent system, this framework will take into account also users' types (riches and poors, waster and environmentalist,...) starting from a set of real consumption data.
After the simulation phase, we will evaluate the existence of di erent kind of
equilibria. It was showed that determining whether a game has a pure Nash
Equilibrium is NP-hard [
Until now, we assume to deal with fully rational agents but sometimes agents
decide their strategy according to their moral or ethical beliefs. Considering this
concept, the mechanism becomes quite di cult regarding the incentive
compatibility. In the literature, we can nd several papers that considering this aspect
explain the concept of robustness of an incentive mechanism. As in [
So, the main di culties are that we want to handle a community of heterogeneous users and according to their belief also irrational. Irrational in the sense that for instance a rich user is not interested in saving a small amount of money, he prefers to have a slightly greater level of comfort.