Philippe Poinsot
email: philippe.poinsot@univ-paris1.fr.

Alain Béraud

Shirine Sabéran

Gilbert

The Foundations of Justice in Jules Dupuit's Thought

Keywords: Jules Dupuit, justice, general interest, welfare, the School of Paris Classification J.E.L.: B19, D63, H11, H21

The ongoing interpretation of Dupuit's conflict with the French liberal school on the question of justice consists of a simple opposition between public utility and natural rights. This paper aims to show that Dupuit's position is far more complex: justice is based on welfare -and thus on public utility -although natural rights are not excluded from his scheme. An understanding of Dupuit's concept requires a clarification of the differences between three notions -public interest, welfare and public utility -and, by the same token, of the factors linking each notion to the others.

Introduction

The question of justice was a much-debated issue in 19 th century France. Against the backdrop of the 1848 revolution, the development of social doctrines and, more broadly, the political context of the time (Sagot-Duvauroux, 2002a, 2002b), these debates focused on the possible foundations of property rights.

Dupuit and French liberal economists 1 do not share the same position. Many members of the liberal school -such as Bastiat, Frederic Passy or Baudrillart -assert that the law must conform to natural rights whereas Dupuit, following the example of Turgot and Condorcet 2 , feels that property rights do not stem from natural law but from public utility. This position isolated him from the French liberals and led some commentators to consider him as a utilitarian. Hence, Vatin (2002, p. 106) states that, "the utilitarianism of Dupuit is without ambiguity". 3 Contrary to what is becoming the usual interpretation, I argue that Dupuit's opposition to the majority of the members of the French liberal school does not consist of a simple opposition between public utility and natural rights. As I have already mentioned, most French liberal economists believe that legislation must conform to natural rights in order to maximize the welfare of society: hence justice is based on natural rights and is therefore referred to as "natural justice" (Baudrillart, 1867, p. 15). On the contrary, Dupuit believe that natural rights may be in conflict with each other; consequently, legislation cannot be entirely founded on these rights but on another criterion that he calls "general interest" or "public interest" 4 (section 2). However, natural rights are not excluded from Dupuit's scheme (section 3). The law may have to restrict natural liberty -which is a natural right -whenever this is necessary to increase the wealth and thus the welfare of the nation.

The law must nevertheless protect the other natural rights -self-ownership, self-defense and the respect of contracts. We will show that both arguments arise from public interest while the former is also based on a different concept -that of public utility.

Dupuit's analytical framework is therefore based on three specific notions:

1) The goal of society -or of the social contract -is to maximize the welfare of the nation by restricting natural liberty as little as possible and protecting the other natural rights.

As we will see, the welfare of the nation is measured by average life expectancy and, 2 According to Turgot and Condorcet, even if the right of property is a natural right it can be regulated by public utility. See Faccarello (2006, pp. 10-12). 3 Ekelund and Hébert are more cautious. They claim that "Dupuit was a utilitarian in the broad sense, and in a manner that reflected the tradition of Bentham, Pascal, Montesquieu, and Mirabeau" (1999, p. 307). Sagot-Duvauroux (2002a, 2002b) only refers to utilitarianism when dealing with Dupuit's position concerning property rights. 4 Echoing Dupuit, I will consider the terms 'general interest' and 'public interest' to be synonymous. Dupuit's reference to general interest surely stems from his education at the Polytechnique and the Ecole des Ponts et Chaussées. The notion certainly existed in engineering concepts taught at the Polytechnique and the Ecole des Ponts et Chaussées, which constantly made reference to public interest. On this point, see [START_REF] Etner | Histoire du calcul économique en France[END_REF]; Grall (2004), p. 31;[START_REF] Mosca | Jules Dupuit, the French 'ingénieurs économistes' and the Société d'Economie Politique[END_REF] and [START_REF] Smith | The Longest Run: Public Engineers and Planning in France[END_REF].

according to Dupuit, is determined by two factors: moral responsibility and the ratio of the quantity of wealth to the population level;

2) The general or public interest consists of achieving the goal of the social contract;

3) Public utility is a quantitative notion that political economy indicates to the legislator in order to increase wealth.

These three notions are central to an understanding of Dupuit's thought and his specificity compared to his contemporaries.

From natural rights to public interest

All the authors of the French liberal school -Courcelle-Seneuil excluded [START_REF] Sigot | Utility and justice: French liberal economists in the 19 th Century[END_REF] -consider that, before entering the state of society, individuals have rights, referred to as natural rights. These rights are either a "gift" from God (Bastiat, 1848(Bastiat, /1854-5, pp. 277-8) -5, pp. 277-8) or from Providence and they are located in the consciousness of human beings (Sigot, 2010, pp. 12-13). Humans have an innate knowledge of these natural rights (Ibid), and as it is innate, the liberals assert that the experience and the observation of human nature prove their existence: "killing, beating, stealing or fraudulously failing to deliver what has been promised to be returned are unjust actions. Does your consciousness not cry out, even if there are no written laws? Well! This is what we call the natural right" (Baudrillart, 1867, p. 14; see also 1857, pp. 33-34). 5 In the same way, Dupuit states that natural rights "are located in consciousness" (Société d'économie politique (SEP), 1862, p. 153). He proves their existence through the observation of human behavior: "The essence of the natural right is to be the same everywhere and at any time. If I am attacked, if somebody tries to kill me, I do not need to open the [civil] code to know if I have the right to defend by myself. Wherever I am, in France, in England, in Spain, in China, I am certain not to be violating the law by repelling an aggression that I did not provoke, or even by killing an individual who wanted to kill me, if I cannot defend my life by another means, because the right of selfdefense is a natural right that every reasonable human being recognizes" (Dupuit, 1861a, p. 330).

In this consensus, however, one can find two important divergences between Dupuit and the French liberal school: the first concerns the range of these rights (2.1) while the second is a matter of their role with regard to justice (2.2).

2.1. The range of natural rights: opposition on the question of property rights According to the French liberals, natural rights can be grouped into five types: 1) selfownership (body, faculties and labor), 2) self-defense, 3) natural liberty, 4) the respect of free contracts and conventions and 5) ownership of things.

The liberals all accept the idea of a hierarchical and logical link among these rights: the most important natural right is the self-ownership, since every other natural right stems from this. Indeed, as long as we accept that a man is the owner of his own person (item 1 above), we must accept his right to defend his body against external attacks (item 2), his freedom to enter into contracts which is a part of natural liberty (item 3) and to respect them (item 4).

As for property rights, the French liberals put forward an argument that can now be found in libertarian thought. According to (1) we know that individuals are the owners of their own faculties and their own labor; thus objects made through an individual's work are the property of that individual (item 5) (Bastiat, 1847(Bastiat, /2002, p. 39 & p. 468;, p. 39 & p. 468;Faucher, 1852, p. 464;Garnier, 1875, p. 97). For example, Baudrillart assumes that before human labor, land was inappropriate to humans, and he affirms that the first individual to cultivate it through his own labor thus becomes its owner.

"Bare land! But do you know what bare land is? It is blackberry bushes and reptiles, it is pestilent swampland, it is wrestling, it is pain… It is often death that occurs after horrible deprivations and illnesses that slowly consume the epic pioneer of culture and civilization…". "In cultivating land, it is necessary to…[make] many thorough preparations which become inseparable from the land. Landed property is less of an instrument within the state of nature, which is not worth anything without some form of work, than an advanced tool… Everything that the owner adds… [to the land] by his labour and capital belongs to him…" (1867, p. 34 & p. 39). on justice in land distribution. Indeed, the latter is the result of the differences of humans in nature, since individuals are born with unequal faculties (Garnier 1873, p. 62 & p. 491) 7 .

Dupuit opposes the French liberals with his conception of property rights, even though he accepts their idea of a hierarchy among the remaining natural rights and their logical connection: he claims that the right of property is not a natural right but a human invention. 8 He rejects property rights as natural for two reasons.

First, he contests the way in which Baudrillart and the French liberal school depict the natural state of land: "human beings do not create anything…[T]here are trees and plants whose roots, stems, flowers or fruit could immediately serve as food [in the natural state of nature]. We can affirm this [point]…, because if this was not so, the animal species that we possess, humans included, would have perished…" (Dupuit, 1861a, p. 325). He considers that the link between self-ownership and the individual right of property over land is inappropriate. He even states that, from his own standpoint, the reference to natural rights and to a state of nature reinforces the socialist position. If, in the natural state, land was the property of all human beings and supplied products in abundance independent of the will and the work of these humans, the products would not therefore be the result of individual work as asserted by the French liberals. 9 Thus the individual property right over land would, as the socialists claim, merely be the result of violence (Ibid, pp. 326-28 & pp. 338-339;SEP, 1855, pp. 146-7). Moreover Dupuit adds that, even if we admitted that they were true, the arguments of the school of Paris could only legitimize primitive land owners and not the current ones (1861a, pp. 329-30).

The second argument deals with the existence of different modes of appropriation of wealth, depending on the nation and the timeframe. Consequently, individual property rights cannot be natural, since natural rights are essentially the same everywhere and at any time (Ibid, p. 330) while property rights are not.

7 On this point, see Sigot (2010, p. 14). 8 In his position on property rights, Dupuit cites the influence of many authors such as Pascal, Montesquieu and Mirabeau. On this point, see [START_REF] Ekelund | Secret Origins of Modern Microeconomics. Dupuit and the Engineers[END_REF]Hébert (1999, p. 307), Vatin (2002) and Sagot-Duvauroux (2002b). 9 The same argument is used for intellectual property rights because of the community chest (see Dupuit, 1861b). This argument is relevant for land property and intellectual property, but not for all property rights. For instance, property rights in industrial production only arise from human labour. However, in support of Dupuit, it should be noted that his argument is linked with debates at that time in France which entirely concerned land and intellectual works. "M. Dupuit absolutely admits that there are natural rights, but he denies that the property right is one of them. The principles of natural right are more limited than we generally think: one may consider only what makes part of the human consciousness and what is recognized in all nations. Humans have made and can make many conventions about property rights without violating the natural right" (SEP, 1862, p. 153).

The existence of various means of appropriating wealth -either private or collective -leads Dupuit to distinguish between "natural rights" and "natural sentiments" 10 , and to consider that property rights are a natural sentiment. He specifies that:

"Natural sentiment leads to good or bad, just or unjust actions; whereas the natural right only has justice as its purpose and motive. If we say that natural sentiments, which motivate man to appropriate all things that can procure his happiness, has motivated and accelerated the establishment of property ownership in society, I will not deny this assertion; but if we say that this sentiment creates rights, we are mistaken." (Dupuit, 1861a, p. 328-329).

This distinction between natural right and natural sentiment is not easy to understand. Dupuit's position could be understood in the following way: the property right varies according to time and space. Thus, it cannot be a natural right. However, it is a natural sentiment: i.

It is a sentiment because any sentiment is changeable, depending on the culture of individuals (see for example SEP, 1855, p. 145);

ii.

It is natural because, in order to satisfy their needs, individuals spontaneously appropriate things (in either an individual or collective way);

iii.

Consequently, this natural sentiment does not create rights.

In order to demonstrate that the property right is not a natural right, Dupuit argues the existence of different means of appropriating wealth from one country to another. I find this argument controversial because he actually combines two different subjects: facts and rights. Even if facts do not prove a natural right, this does not mean that a right is not natural -for instance even if one does not use self-defense, it is still a natural right. 11 Furthermore, this approach may even be used against the right to self-ownership. As long as different forms of slavery legislation exist, it may be concluded that self-ownership is not a natural 10 The French liberals do not distinguish between natural sentiments and natural rights and use these words synonymously. See Garnier, for example (1875, p. 96).

11 Nevertheless, as soon as the School of Paris uses an observation as proof of the existence of natural rights, the existence of differences in the means of appropriating wealth between countries constitutes a criticism of French liberals' conception of natural rights.

right. This is a crucial issue: if self-ownership is not a natural right, there would be no natural rights at all, yet Dupuit provides no argument to justify self-ownership as a natural right. He simply states that "slavery violates the principles of moral and of natural right..." (SEP, 1863b, p. 473; see also SEP, 1862, p. 149).

For Dupuit, the suitable means of appropriating wealth is only defined according to public utility: it is this criterion that enables the legislator to judge if private appropriation is more suitable in a particular case than collective appropriation. In fact, public utility provides the basis for calculating which mode of appropriation increases the volume of production more quickly; this is the sole condition that matters. Dupuit claims that individual appropriation of land is better than collective appropriation in terms of increasing agricultural output, as it stimulates self-interest. Moreover, in increasing the intellectual satisfaction of the property owner and his sense of responsibility, this mode increases the welfare of the nation. 12 Dupuit asserts that this mode of "appropriation is [...] excellent for two reasons: excellent for the landowners, because they find particular pleasures in it which encourage them to prefer this kind of appropriation over others; excellent for those who are not landowners, because this [mode of appropriation] gives them the greatest amount of product for the cheapest price." (Dupuit, 1861a, p. 347).

Potential incompatibility among natural rights

As mentioned above, the French liberal school believes that natural rights are the only source of justice. These rights are unquestionable; they cannot be limited by the legislator.

Dupuit criticizes this position because there might be an incompatibility among different natural rights. More precisely, there is an incompatibility between self-ownership and natural liberty. Consequently, they cannot be the only source of justice. Imagine for instance that you "need an amount of money in order to save your honor…, you go to find a rich man and you tell him: Give me this amount of money, and you will have over me all the rights that you have over your horse; in short, I will be your slave" (Dupuit, 1861b, p. 42).

Natural liberty means that you are free to undertake a contract. This implies that you can sell your person in any circumstances whatsoever and nobody could prevent you from doing so. However, doing this means that you will forego your right of self-ownership. This is the only example that Dupuit gives to prove the possibility of an incompatibility among natural rights.

It is interesting that Dupuit implicitly states that there is no criterion for choice in this conflict among natural rights -in this case the self-ownership and natural liberty. This is why However, Dupuit's criticism is not really relevant, since he uses a different conception of the boundaries of natural liberty from that of the French liberal school. They both share the idea that natural liberty means free will and the absence of any interference or constraint in terms of individuals' actions and the freedom of individuals to enter into contracts -the "negative liberty" of Isaac Berlin. However, they differ with respect to the boundaries of natural liberty. The French liberal school posits that natural liberty is limited by the rights of others (Bastiat, 1848(Bastiat, /1854-5, p. 282;-5, p. 282;Garnier, 1864, p. 274;Molinari, 1861, p. VI-VII), while Dupuit implicitly distinguishes two kinds of liberty: i. Natural liberty, which means that the freedom of individuals is unlimited (Dupuit, 1861b, p. 51) and only concerns the state of nature;

ii.

The liberty of individuals in society, in other words the liberty approved by laws

(see 3.2).
Consequently, the French liberals' definition of natural liberty renders the question of incompatibility moot because humans can only appropriate things, but not other humans.

On the contrary, this problem may arise when liberty is defined as unlimited. 13 Dupuit's position on public expropriation is explained in 3.2.

The introduction of the notion of public interest thus becomes necessary in order to eliminate this conflict among natural rights. I will now clarify this notion and its relation to Dupuit's conception of justice. This will lead me to explain the status of the notion of welfare in Dupuit's thought.

The dual sources of ethical justice in Dupuit's thought: welfare and natural rights

Natural rights, according to Dupuit, cannot be the only decision-making tool for the legislator. As he states, the legislator aims to achieve the objective of the social contract of increasing the welfare of the country and the protection of certain natural rights. This objective results from an agreement or a contract between all members of society. As there is a positive correlation both between welfare and wealth and between wealth and public utility, public utility is a priority decision-making tool to increase welfare. Dupuit continues by assuming that public utility is a norm for public decisions: it indicates whether or not laws are just. Another concept of justice therefore appears which is ethical rather than legal in nature (3.1). In addition to public utility, some natural rights also constitute a source of ethical justice (3.2). This dual source of ethical justice is relevant to a better understanding of Dupuit's economic views and in particular his position regarding the problem of financing public goods (3.3).

Justice and public utility

The objective of society is to maximize the welfare of the nation (Dupuit, 1861a, p. 334). In this framework, general interest is a formal notion without any substance in itself; it consists only of achieving the goal of the social contract. 14 Laws are enforced in order to attain this goal. A clarification of the notion of welfare is therefore essential.

14 Since general interest is an exclusively formal notion, trying to define it without any reference to the objective of society is absurd.

The objective of society: the welfare of the population

The "purpose of society is the welfare of its members" (Dupuit, 1861b, p. 53). 15 But what is welfare 16 ? Dupuit does not provide any clear definition of this notion of welfare.17 He nevertheless clearly explains its causes (economic and moral) and its consequences for the population.

Dupuit identifies two causes of welfare, one economic and the other moral. On the one hand, increasing wealth in society enhances the welfare of the population (Dupuit, 1865, p. 28). 18 It seems that the wealth of a nation is defined by its public wealth -"the harbors, canals, roads, rivers, museums, prisons, hospitals, etc." (Dupuit, 1853, p. 17 The relationship between these two causes of welfare (wealth and responsibility) is a cumulative process: the more responsible the individual is, the richer he will be and the 15 In this article, the question of what Dupuit's expression "its members" refers to is not raised, since it does not fall within the framework of our problem relating to the clarification of the sources of ethical justice. Moreover, in this sentence Dupuit implicitly supposes that, other than natural liberty, natural rights cannot be in conflict with increasing welfare. The protection of certain natural rights is also part of the objective of society. See 3.2. 16 In this article, I will not explain why the objective of the social contract is the welfare of the country. This would involve summarising the emergence of such an objective and hence the notion of human nature in Dupuit's thought. However, that would extend beyond the framework of this article. In the same way, I will not explain whether or not the welfare of the nation is the summation of individual welfare.

richer he is, the more responsible he will be (Dupuit, 1865;SEP, 1862, pp. 472-5;SEP, 1863a, pp. 344-8). 19 According to Dupuit, this process justifies measuring the level of welfare of a country by means of the average life expectancy: "in places where humans live for a long time, they are certainly more comfortable and their needs are better satisfied" (Dupuit, 1865, p. 6).

"The average life expectancy", he adds, "may be considered as the true thermometer of the welfare of populations" (Ibid, p. 7).

An increase in the welfare of a country thus supposes either an improvement in the moral state of the population and/or an increase in wealth. In the case of the former, there is no need for the legislator to intervene directly. More precisely, Dupuit considers that the mission of the law is not to force individuals to adopt adequate moral behavior -only incentive mechanisms can be implemented to this end.

These mechanisms are either spontaneous -in which case they are the result of individual responsibility -or artificial. In the first case, for instance, behavior that is not in accordance with morality will result in the death of individuals who are incapable of satisfying their needs. In the second case, the law encourages individuals to modify their behavior, and this is how Dupuit justifies the system of taxes for private goods. 20 Dupuit justifies state intervention in order to increase wealth and specifies the principles that must guide this intervention: the notion of "public utility" is therefore introduced into his analysis, referring here to an ethical conception of justice.21

19 On this point, Dupuit uses the same argument as the French liberals (Sigot, 2010, p. 3). 20 For private goods, taxes should be a function of the welfare of the nation. For example, in order to improve the health of a nation, taxes must incite individuals to consume 'good' instead of 'bad' products. Actually, at SEP, Dupuit claims that "taxes on sugar, coffee, and wine, have negative effects on the consumption of commodities which are useful to the welfare [and] the health of a population. In diminishing its consumption, the tobacco tax is more favourable than harmful to public health [and so to its welfare]" (1859, p. 470).

Utility conforms to moral justice

Dupuit claims that the legislator is not free to decide the means of increasing wealth.

This leads him to introduce another conception of justice into his analysis, not in its legal sense, but as a norm that enables public policies to be judged.

Since political economy is the science of wealth, the legislator must follow its recommendations with regard to public utility 22 (Dupuit, 1844(Dupuit, /1995, p. 128), p. 128). This point has been examined in depth in secondary literature (see for example [START_REF] Ekelund | Secret Origins of Modern Microeconomics. Dupuit and the Engineers[END_REF][START_REF] Etner | Histoire du calcul économique en France[END_REF][START_REF] Mosca | Jules Dupuit, the French 'ingénieurs économistes' and the Société d'Economie Politique[END_REF]. However, in contrast to these commentators, I believe that welfare, general interest and public utility are different notions.

First, general interest is more global than welfare. Public interest is the conformity of public decisions with the objective of society to maximize welfare by restricting natural liberty as little as possible and protecting the other natural rights (see 3.2). Every law which increases welfare and protects all natural rights other than natural liberty therefore complies with the general interest.

Second, welfare differs from public utility. As we have seen, public utility is linked to wealth, itself linked to welfare; public utility is therefore a criterion for assessing whether or not the objective of increasing welfare is correctly pursued.

Besides being "the only solution to every problem of wealth appropriation", as I have already explained, the principle of public utility is the solution to "many other economic questions" (Dupuit, 1861b, p. 53). These questions include "the question of taxes, which is much better established and allocated if it does not harm the wealth [of the nation] […] [or moreover], the question of the coalition or the association of certain interests which should only be authorized on the condition that it does not harm the public interest" (Ibid: footnote 1). Therefore, the means of making choices in order to increase wealth -public utility -must be distinguished from the objective of society -welfare. 22 Dupuit defines public utility or the "Utilité absolue 'collective'" by the sum of the "utilité relative" of all consumers and the profits of the producer. An individual's utilité relative is represented by the difference between the maximum price that an individual is willing to pay for the service (what I call the 'individual' utilité absolue in order to differentiate it from 'collective' utilité absolue) and the asking price set by the producer (Dupuit, 1844[START_REF] Dupuit | De la mesure de l'utilité des travaux publics[END_REF](Dupuit, , 1849(Dupuit, /1933(Dupuit, , 1853)). It should be noted that the concept of individual utilité absolue is different from the cardinality of utility. Indeed, Dupuit understands individual utilité absolue in the sense of an individual's preference for buying or not buying an object, rather than as a measure of his satisfaction (1844/1995, p. 65; see also Allais, 1989, p. 192 andPoinsot, 2007, p. 43). I do not use the term of present day micro-economics -"consumers' surplus" -because this could lead to confusion with respect to the concept developed by Alfred Marshall. The categories used by Dupuit are very different from Marshall's categories (see [START_REF] Béraud | Richesse et Valeur: la contribution des économistes français du début du 19éme siècle[END_REF][START_REF] Poinsot | De l'Intérêt Général à l'Utilité Collective, étude comparative des précurseurs de deux traditions: Jules Dupuit (1804-1866) et Alfred Marshall[END_REF].

Public utility is, of course, correlated to welfare, although the former is only indirectly linked to the welfare of the population through the wealth of the country.

The most remarkable consequence of Dupuit's statement is that public utility conforms to what may be termed "ethical" or "moral" justice.

He claims that members of society have to respect laws because when "[positive] laws are established, it is unjust to disobey them" (Pascal quoted by Dupuit, 1861a, p. 333). The French liberals strongly oppose this statement, arguing that unjust laws exist. Dupuit recognizes this, but claims that unjust laws must also be respected by members of society, and laws which are not based on public utility are unjust (Dupuit, 1861b, pp. 49-50). Public utility thus becomes a norm for public decision-making. In my opinion, we can understand Dupuit as meaning that the welfare of the nation is the objective of society and every law that increases welfare conforms to ethical justice. As welfare increases with wealth and public utility is a criterion for increasing wealth, so public utility conforms to ethical justice.

Hence, any laws that do not comply with public utility are unjust. This can be observed in Dupuit's criticism of the analysis of the French liberal school which, according to Dupuit, only defends private interests:

"In order to find out that [public] utility [conforms to ethical] justice, one only needs to adopt an alternative viewpoint to that of my honorable opponents, to depart from private interests and consider public interest. When one listens to certain commentators, it would appear that only the individual has rights and that society has one single right which is to defend them" (Ibid, p. 37).

Public utility is therefore a source of ethical justice. But are natural rights totally excluded from Dupuit's definition of ethical justice?

3.2. Pure consequentialism? The persistence of natural rights At first glance, Dupuit seems to adopt a contradictory position. On the one hand, he claims that limiting natural rights in order to increase welfare conforms to moral justice:

"[Society] can only exist if some [positive] laws or conventions adjust the interaction of its members and the use of some objects, and restrict the liberty and the natural rights of individuals.

The general principle of these conventions is to provide the maximum possible overall welfare for all of the individual members of society, by respecting -insofar as this goal allows -their liberties and their natural rights, which are in themselves among the most precious forms of wealth" (Ibid, p. 53).

On the other hand, he also states that the legislator has to respect natural rights. This is his position, for instance, on public expropriation. While he considers that property is not based on natural rights, he asserts that the legislator has to pay compensation in the event of expropriation. This is a direct consequence of Dupuit's conception of the respect of a contract as a natural right (item 4): "any owner of any form of property by virtue of a [precedent of positive] law, may only be deprived of said property on the condition that

[s]he is granted an equivalent [indemnity]. This is a form of natural right" (Ibid, p. 48). In the same article, Dupuit claims that:

"It is a great mistake to believe that a law may destroy what another law has established. If it has the character of a contract, and this is the case of property laws, it may only be changed with the consent of the contracting parties […] The worker who earns 100,000 francs and buys either a plot of land, or a house […] has an honorable property in his possession […] It is equally true that it cannot be taken away from him without violating natural right […] that it may only be taken away through provision of an indemnity." (Dupuit, 1861a, pp. 333-4). 23This argument concerning public expropriation raises an important issue: by basing his analysis on natural rights, without even referring to pains and pleasures, Dupuit clearly departs from Bentham's utilitarianism. 24But how can these two positions be compatible? Dupuit resolves this contradiction by stating that only natural liberty can be restricted: "we can demand no more liberty than what is suitable to public interest" (ibid, p. 51).

This relationship between natural rights (in particular natural liberty) and welfare in Dupuit's analysis is highly original in comparison to the French liberals' views. The latter believe that welfare is positively correlated to the defense of natural rights: the greater the protection of individuals' liberty and property, the more welfare increases (Garnier, 1875, p. 111;Molinari, 1861, p. VI). Natural rights therefore oppose neither welfare nor utility: the French liberals feel that an action that conforms to utility increases wealth, and consequently welfare, and thus that the government must defend natural rights (Sigot, 2010, p. 1).

On the contrary, Dupuit claims that natural liberty should be limited in society by the legislator with a view to increasing welfare through wealth since it is not possible for the legislator to act directly on the other source of welfare, i.e. moral responsibility. More precisely, government is designed to maximize the welfare of the population, while preserving natural liberty as far as possible: "The best society is that in which the legislator gives each individual the maximum social advantages while imposing as few constraints on his/her natural liberty as possible" (1861b, p. 53). His position is interesting with respect to welfare comparisons: given two situations with the same level of welfare, the situation with greater respect for natural liberty should be preferred.

Dupuit's position nevertheless raises at least two problems.

First, Dupuit has to explain the absence of restrictions on natural rights other than natural liberty -self-ownership, self defense and the respect of contracts. His argument lies in the essence of the social contract, obliging the members of society to consent only to giving up their natural liberty: "to ask for an absolute, radical [and] complete liberty is to ask for the state of nature. [We must] accept the need for laws for the sake of public interest" (Ibid, p. 51).

Second, conflict may arise between the two aspects of the objective of societywelfare and the protection of all natural rights other than natural liberty that the legislator cannot restrict. In this case, however, Dupuit does not specify how the legislator should make his decision and even implicitly supposes that it is impossible for him to do so.

In brief, Dupuit asserts that:

i. Ownership of things is not a natural right but a natural sentiment;

ii.

Natural liberty can be restricted by the legislator with a view to maximizing welfare; and

iii. Three other natural rights remain that must be protected by the legislator: selfownership, self defense and the respect of contracts. This illustrates Dupuit's position with respect to ethical justice whereby ethical justice stems from both public utility and natural rights and this position applies to his dealing with the problem of financing public goods.

The problem of financing public goods

Dupuit distinguishes between private and public goods. The production of goods serves to increase the wealth of the country and the State is more efficient than a private company in producing and managing public goods: "If the State [alienates] the monopoly of a means of communication into the hands of a private company, this means immediately loses its economic characteristic as an inexpensive instrument of transport to become a machine for major profits" (Dupuit, 1854d, p. 853). Dupuit asserts that "we recognize that it falls within the government's functions, or departments of the government, to construct and to maintain certain means of communication" (Dupuit, 1854b, p. 380).

As far as public goods are concerned, two questions arise, the first concerning the financing principle to be adopted and the second concerning the rate of tax. In other words, the first question is "who has to pay for public goods?" while the second is "what rate of tax must they pay?" I will only examine the first question, since this is the area in which Dupuit's dual sources of ethical justice are the most clear-cut.

With regard to the principle of financing public goods, it seems that Dupuit distinguishes between public goods that are excludable (bridges, ships, canals, etc.) and those that are non-excludable (streets, courthouses, etc.). The former must be financed by their users while the latter must be financed by all members of society. In my opinion25 , this distinction can be understood through the issue of restricting natural liberty. As we have already seen, for a given level of welfare, a state where natural liberty is less restricted is preferable to a more restrictive state.

Dupuit treats excludable public goods in the same way as private goods. It conforms to ethical justice in that only users pay for such services. 26 Indeed, the user "is perfectly free to buy it or not to buy it" (Dupuit, 1844(Dupuit, /1995, p. 59, p. 59).27 Hence, he states: "nobody can dispute the just character of the tax [that charges those who benefit from it], not even those who do not pay for it. In fact, it is obvious that if they did not find a certain advantage in using it, in spite of the toll, then they would not use it. Nothing justifies charging those who do not use this means of communication […]" (Dupuit, 1854c, p. 556;see also SEP, 1865, p. 132).

In the case of non-excludable public goods, no contract is possible as nobody can be excluded from the service. In order to finance public goods, the legislator therefore has to diminish the natural liberty of the members of society by forcing them to contribute (even if they do not take advantage of such goods) and to make use of them: "No form of society is possible without a certain dose of communism; there are forms of wealth which must be personally appropriated, and there are the others that must remain collective, because of their intrinsic nature or destined usage" (Dupuit, 1863, p. 118). The government can force members of society to contribute either by means taxes or loans. Indeed, "there are […] plenty of things that are of common and sustainable use, and for which the government, provincial or municipal councils can […] legitimately borrow so that these things may be established. Public highways, streets, squares, markets, harbors, courts, prisons and so on require expenditure, whose burden may be transferred on to future generations without their having the right to complain"(SEP, 1860, p. 484).

Thus, according to Dupuit, all public goods from which some individuals can be excluded must be financed by their users, since this mode of taxation does not restrict natural liberty. However, when it is impossible to exclude individuals, the legislator may restrict natural liberty in order to finance this form of public good.

  ) -in addition to individual wealth as measured by the ratio of the quantity of wealth to the total population(Dupuit, 1865, p. 28). On the other hand, welfare increases if, and only if, individuals adopt good moral behavior. As welfare depends on the population level, Dupuit makes a value judgment concerning the good and bad behavior of individuals. Good behavior equals moral responsibility -restricting the number of children in the family in accordance with individual wealth and limiting the consumption of tobacco or alcohol(SEP, 1859, p. 470). This moral responsibility derives from the capacity of individuals to act according to their own will. Indeed, if individuals are free to act, they are responsible for their acts and their consequences.

  Dupuit introduces a higher criterion: public interest. The legislator has to protect selfownership by restricting natural liberty because, as I have already stated, self-ownership is the most important natural right (cf. 2.1): "whenever public interest appears, the law mediates in order to restrict the individual [natural] right [...] The law will not ratify [a contract to sell your person]. You can sell your field, but you cannot sell your person; you have over your field rights that you do not have over yourself"(Ibid, p. 42). By extension, Dupuit refuses any compensation to slave owners (SEP, 1863b, p. 473) even though public expropriation compensations must in principle be provided by the legislator.13

All translations throughout the paper are mine.

When the French liberals claim that the natural state of lands is not adapted to human life, they do not assume that the quantity of land is insufficient(Bastiat, 1847(Bastiat, /2002, p. 41), p. 41), only that the quality of land is very poor. This is an essential point in understanding their position6 See alsoBaudrillart (1857, pp. 34-5) andGarnier (1846, p. 179 & p. 318).

The definition of welfare will be given in section 3.1.

Dupuit only uses imprecise terms such as "aisance"(Dupuit,

1865, p. 6) or "jouissance" (1861b, p. 49) without defining them.18 On this point, Dupuit agrees with the French liberal school. This affirmation nevertheless supposes that the effects of the distribution of wealth between members of society on welfare are not considered. This problematic question will not be tackled here, as it diverges from the subject of this paper.

The question of the principles guiding the legislator in his/her decisions is important, as it responds to the French liberals' attack on the notion of general interest. They state that the legislator must make his/her decisions in accordance with natural rights, in order to prevent arbitrary decisions from being taken in relation to the general interest (seeSigot, 2010, p. 12).

On the question of international free trade, Dupuit makes the same claim. Indeed, in La liberté commerciale (1861c), Dupuit pronounces in favour of free trade as this will increase the quantity of wealth and hence the welfare of the nation. Free trade will nevertheless destroy certain economic interests. Dupuit therefore claims that a period of transition is necessary for considerations of justice. Indeed, "everything that prevents free trade condemns the population to work more to obtain less […] decreasing the industrial power and also the political power of the country. Does this mean that after having existed for many years under a restrictive regime, a nation is supposed to achieve complete freedom without any form of transition? Probably not; this transition […] [is necessary for] considerations of justice or equity with respect to certain interests […]" (1861c, p. 2).

As opposed to Dupuit, Bentham's argument lies in the pain caused in the absence of compensation. Indeed, this absence is perceived by individuals as reconsidering the positive right of property. A fear of this arises in society (which Bentham named an "alarm") implying a "pain of disappointed expectation".

This way of dealing with the question of financing public goods was suggested to me by one of the referees, whom I would sincerely like to thank.

Dupuit only recommends taxing all members of society (users and non-users) for some excludable public services such as roads because "the State only charges directly for services that it provides when this can be achieved fairly and economically"(SEP, 1865, p. 132; see also 1854b, p. 556). He nevertheless adds that "theoretically, a traffic toll applied to roads would be fair and just"(SEP, 1865, p. 132).

This remark leads to the conclusion that non-users who benefit from excludable public goods should not pay for such goods. In actual fact, even if an individual profits from the existence of such goods, this benefit is independent of his or her will.

Conclusion

In this article, I have attempted to demonstrate that a more general conception of ethical justice may be found in Dupuit's work than may be found in the secondary literature.

Contrary to his commentators, Dupuit's originality does not stem from public utility; rather it results from the coexistence of two sources of ethical justice -welfare and natural rights.

Through this conception of ethical justice, Dupuit differs from the French liberal school without for all that being bound by Bentham's utilitarianism. Indeed, even if his analysis leads to utilitarian conclusions, the absence of an a priori definition of rights that characterizes Bentham's utilitarianism contradicts Dupuit's position on natural rights.

Moreover, Dupuit's ethical justice is essential to an understanding of his economic analysis. In this paper, I have chosen to tackle the question of financing public goods, but many other subjects are concerned, such as Dupuit's justification of intellectual property rights or of the distribution of wealth.


Patrick Loiseau
email: patrick.loiseau@eurecom.fr

Galina Schwartz
email: schwartz@eecs.berkeley.edu

John Musacchio
email: johnm@soe.ucsc.edu

Saurabh Amin
email: amins@mit.edu

Shankar S Sastry
email: sastry@eecs.berkeley.edu

Congestion

S Shankar Sastry

Congestion Pricing Using a Raffle-Based Scheme

Keywords: congestion pricing, raffle-based incentive schemes, public good, probabilistic pricing, demand management

come

I. INTRODUCTION

Congestion is becoming increasingly frequent in large infrastructures such as the Internet, transportation and power networks. Congestion occurs when a large number of users share a common resource, and user demand varies with time. Environments with frequent congestion include access points of the Internet (e.g., a base-station of a wireless 3G network or a DSLAM for wired residential networks) or transportation network (e.g., the Bay Bridge in California). In these cases, congestion management becomes necessary when the inefficiencies increase and result in a substantial decrease of user utility. In the case of power networks, high peak demand could cause instability and an increase in the costs of provision.

In all these examples, congestion is driven by incentives misalignment of individual users relative to social optimum. Thus, decongestion can be cast as a public good. Indeed, when a user reduces or moves part of his demand to another time, the benefits of the reduced congestion are shared by all the users. With that perspective, ideas from the economic literature on incentivizing contributions to a public good can be brought to bear on the decongestion problem.

In this paper, we propose a raffle-based scheme for the decongestion of a shared resource. We start by formulating the decongestion problem as a public good provision problem using a game-theoretic model in a setup with a finite number of atomic users. Next, we introduce a setup with an infinite number of non-atomic users, and show that the former converges toward the latter when the number of users goes to This research was supported by NSF grant CNS-0910711 and by TRUST (Team for Research in Ubiquitous Secure Technology), which receives support from the NSF (#CCF-0424422) and the following organizations: AFOSR (#FA9550-06-1-0244), BT, Cisco, DoCoMo USA Labs, EADS, ESCHER, HP, IBM, iCAST, Intel, Microsoft, ORNL, Pirelli, Qualcomm, Sun, Symantec, TCS, Telecom Italia, and United Technologies. infinity (in the following, we will simply call these setup the atomic setup and the non-atomic setup respectively).

Our raffle-based scheme builds on the ideas of Morgan [START_REF] Morgan | Financing public goods by means of lotteries[END_REF] who pioneered the economic analysis of public good provision via lotteries. However, the specifics of the congestion management problem require substantial changes from Morgan's model [START_REF] Morgan | Financing public goods by means of lotteries[END_REF]. The idea of lotteries have also been used in other contexts, e.g., for the raffle scheduling technique applied in computer operating systems [START_REF] Waldspurger | Lottery scheduling: Flexible proportional-share resource management[END_REF]. Recent interest in application of lotteries to congestion management was facilitated by Mergu, Prabhakar, and Rama who demonstrated with a field study that lottery-based mechanisms can be used to decongest transportation systems [START_REF] Merugu | An incentive mechanism for decongesting the roads: A pilot program in bangalore[END_REF]. In contrast, our focus is methodological. We approach lotteries as a technical tool of congestion management. We compare our setup(s) to Morgan's scheme of using lotteries for funding public goods.

II. OVERVIEW OF MORGAN'S MODEL

Let N = {1, . . . , n} be a set of users (players). Each user i ∈ N has a wealth w i , and chooses an amount x i ∈ [0, w i ] to contribute to a public good. With voluntary contributions, the level of public good G = n i=1 x i is the sum of contributions, and the objective of each user i ∈ N is to maximize his utility U i = w i + h i (G) -x i , where h i (•) is an increasing strictly concave function which reflects user i's valuation of the public good.

The raffle-based scheme (fixed-price raffle in [START_REF] Morgan | Financing public goods by means of lotteries[END_REF]) gives a reward R > 0 to one or more users and user's expected reward is proportional to his fraction of the total contribution. The scheme is financed by deducting the reward from the total contribution. Then each user i ∈ N has expected utility

U i (x i , x -i ) = w i + h i (G) -x i + R • x i n i=1 x i , (1)
where the level of public good is now G = n i=1 x i -R, and x -i denotes the vector of contributions of all users but i. Finally, Morgan [START_REF] Morgan | Financing public goods by means of lotteries[END_REF] assumes that if the total contribution is insufficient to cover the prize up to an arbitrarily small amount δ > 0 (i.e., if n i=1 x i < R -δ), then the raffle is canceled and each user's contribution is returned.

At a Nash equilibrium, each user maximizes his utility (1). We denote with a superscript (eq) all quantities at Nash equilibrium. The aggregate welfare W is

W = n i=1 U i . (2)
We define social optimum as the point where the aggregate welfare is maximized. All quantities at social optimum are denoted with a superscript * .

Theorem 1 (Summary of Morgan's results [START_REF] Morgan | Financing public goods by means of lotteries[END_REF]).

(i) For any R > 0, there exists a unique Nash equilibrium, whereas for R = 0 (voluntary contributions), there can exist multiple equilibria, all with the same amount of public good. (ii) There exists a unique level of public good G * , which maximizes the aggregate welfare, and for any R > 0, we have G (eq) (0) < G (eq) (R) < G * . (iii) The equilibrium amount of public good with the rafflescheme G (eq) (R) can be made arbitrarily close to the socially optimal level G * by choosing a sufficiently large reward R. (iv) For any R > 0, G (eq) (R) > 0 if and only if G * > 0.

III. DECONGESTION OF A SHARED RESOURCE

To address the problem of decongestion, we first formulate it as a public good provision problem using a game-theoretic model in an atomic setup (with a finite number of users) as in Morgan [START_REF] Morgan | Financing public goods by means of lotteries[END_REF] (Sec. III-A). Then, we propose a non-atomic setup (Sec. III-A), as it is often done in economics to model games with a large number of players [START_REF] Aumann | Markets with a continuum of traders[END_REF].

A. Atomic setup

Let N = {1, . . . , n} be the set of users. For simplicity, we assume that each user has identical demand d = D/n for the shared resource, where D is the total demand. There are two times of the day: peak and off-peak, and each user decides how to divide their demand between them. Each user i ∈ N is endowed with a type θ i ∈ Θ that characterizes the user's preferences between the two times. Without any incentive scheme, we assume that the utility of a user i ∈ N who chooses to put a fraction x i of his demand in the off-peak time is:

U i (x i , x -i ) = d P θi (1 -x i ) + O θi (x i ) -p- (1 -x i ) • L P D -d j x j -x i • L O d j x j ,
where P θ (•) and O θ (•) are the utilities that the user gets for his demand in the peak and off-peak periods respectively, while L P (•) and L O (•) are the costs of delay in the peak and offpeak periods respectively. These delay costs are per unit of demand, hence they are multiplied by the fraction of demand in each time. The whole expression is scaled by the user's total demand d since this form will later be useful for considering the limit in which each user's fraction of the total demand diminishes. The quantity p is a fixed monthly subscription price and x -i is the vector of choices of all users but i. Finally, we assume that since there are many more off-peak hours than peak hours, congestion at off-peak times never becomes significant. Therefore, L O (•) ≈ 0.

We assume that, in the absence of latency, users always prefer to use the service at peak-time, i.e., P θ (1 -x) is larger than O θ (x) for all user types θ ∈ Θ and x ∈ [0, 1]. We may therefore define the notion of the cost of shifting as the loss of utility that a user of type θ ∈ Θ incurs when shifting a fraction x of his demand from peak to off-peak time:

c θ (x) = ūθ -(P θ (1 -x) + O θ (x)) , ∀θ ∈ Θ, where ūθ = P θ (1) + O θ (0)
is the maximal utility that a user could get without shifting any of his demand if there was no congestion. We assume that c θ (•) is increasing and strictly convex (this assumption may also be derived from the assumption that P θ (•) and O θ (•) are strictly concave). This assumption reflects the fact that for a given x i , users will shift the most easily shiftable parts of their demand before shifting demand that is more costly to shift. We also assume that c θ (•) is twice differentiable, and that it has a slope that is bounded away from zero by a constant independent of θ. To simplify the proofs, we also assume that c θ (•) is bounded on [0, 1] by a constant independent of θ, but this assumption can likely be relaxed. We view each players choice of x i as a contribution to the public good since it represents demand withheld from the peak time. Thus we define

G = n i=1 x i d. (3)
We may now recast the utility of each user as

U i (x i , x -i ) = d [ū θi + (1 -x i )h(G) -c θi (x i ) -p]
where h(G) = -L P (D -G). We assume that L P (•) is increasing and strictly convex and twice differentiable. This assumption results in h(•) being increasing strictly concave. The raffle-based scheme introduces for each user a reward proportional to his fraction of the total shifted demand dx i /G. However, it is financed here by charging an extra price ∆p (per unit of d) to each user. With the raffle-based scheme of parameter R ≥ 0, the utility of user i ∈ N then becomes

U i (x i , x -i ) = d ūθi + (1 -x i )h(G) -c θi (x i ) -p + R • x i G -∆p , (4)
where ∆p = R D , so that

n i=1 d • R • xi G -∆p = 0.
Finally, we assume that if no user shifted any of his demand, i.e., x i = 0 for all i ∈ N (hence G = 0), then the reward is not given. The raffle-based scheme could be implemented as a conventional raffle in which the whole prize R is awarded randomly to one player with a probability proportional to that player's contribution to the public good as in [START_REF] Morgan | Financing public goods by means of lotteries[END_REF]. However the scheme could also be implemented deterministically -i.e., each player i could be given a payment or rebate of the amount R • dxi G . The marginal utility of user i ∈ N is

∂U i ∂x i = d • -h(G) + (1 -x i )dh (G) -c θi (x i ) + R • G -x i d G 2 . ( 5
)
At a Nash equilibrium, each user chooses his contribution x i to maximize his utility (4). Since utility (4) is strictly concave, for a vector of contributions x to be a pure Nash equilibrium, it is necessary and sufficient that is satisfies the first-order conditions (FOCs)

∂U i ∂x i (x i , G)    ≤ 0, ∀i ∈ N : x i = 0, = 0, ∀i ∈ N : x i ∈ (0, 1), ≥ 0, ∀i ∈ N : x i = 1, (6)
and equation [START_REF] Merugu | An incentive mechanism for decongesting the roads: A pilot program in bangalore[END_REF]. In general, a Nash equilibrium does not coincide with the social optimum in which ( 2) is maximized.

B. Non-atomic setup

When the number of users is large, the demand d of each user becomes negligible w.r.t. to the total demand D and a setup with an infinite number of non-atomic users becomes more appropriate [START_REF] Aumann | Markets with a continuum of traders[END_REF], [START_REF] Khan | Non-cooperative games with many players[END_REF]. We assume here that the total demand D remains constant and the individual demand tends to zero. We assume that user types are i.i.d. distributed according to a measure µ on the set of types. Since the utilities are strictly concave, two users of the same type have the same contribution at equilibrium. Therefore, we work with the distribution of types directly instead of working with the distribution of users as in [START_REF] Aumann | Markets with a continuum of traders[END_REF], [START_REF] Khan | Non-cooperative games with many players[END_REF].

Let (Θ, F, µ) be a measured space; where Θ is the set of user types, F is a σ-algebra and µ is a finite measure accounting for the distribution of the user types. We assume that users have identical demand and that each user of type θ ∈ Θ chooses a fraction x θ of his demand to contribute to the public good. This defines a measurable function x : Θ → [0, 1] on (Θ, F, µ). In this framework, the total demand is D = Θ dµ(θ) and the level of public good is

G = Θ x θ dµ(θ). ( 7
)
With the raffle-based scheme, a user of type θ ∈ Θ has utility

u θ (x θ , x -θ ) = ūθ + (1 -x θ )h(G) -c θ (x θ ) -p + R • x θ G -∆p, (8)
which corresponds to utility (4) normalized by the demand d.

Since users are non-atomic, u θ may also be interpreted as a "density of utility" at type θ ∈ Θ. We still assume that if no user contributes, then the reward is not given. However, if a set of users of measure zero contributes, then each contributing user gets an infinite reward distributed so that the integral with respect to the measure of users is R.

Marginal utility (5) becomes:

∂u θ ∂x θ = -h(G) -c θ (x θ ) + R G . (9)
A Nash equilibrium is defined as a function x (eq) such that for all θ ∈ Θ, u θ (x

(eq) θ , x (eq) -θ ) ≥ u θ (a, x (eq) -θ ), ∀a ∈ [0, 1]. It is equivalent to x (eq) satisfying the FOCs ∂u θ ∂x θ (x θ , G)    ≤ 0, ∀θ ∈ Θ : x θ = 0, = 0, ∀θ ∈ Θ : x θ ∈ (0, 1), ≥ 0, ∀θ ∈ Θ : x θ = 1, (10)
and equation [START_REF] Fudenberg | Game Theory[END_REF]. At social optimum, the aggregate welfare

W = Θ u θ dµ(θ) (11)
is maximized.

IV. ANALYSIS AND CONVERGENCE RESULTS

We first analyze the equilibrium and social optimum in the non-atomic setup (Sec. IV-A). Then, in Sec. IV-B, we analyze the equilibrium in the atomic setup and we show that it converges toward equilibrium of the non-atomic setup when the number of users tends to infinity.

A. Analysis of the non-atomic setup

Our first result establishes existence and uniqueness of the Nash equilibrium in the non-atomic setup.

Theorem 2. For any R ≥ 0, there exists a unique Nash equilibrium x (eq) (R) in the non-atomic setup.

Proof: See Appendix A. The discontinuity in utilities at points where G = 0 necessitates some special treatment to prove the existence of a Nash equilibrium. Moreover, the uniqueness property is nontrivial and is derived from the specifics of our model.

The intuition behind the result of Theorem 2 is as follows. For a given value of G, each user of type θ ∈ Θ chooses his best-response contribution x (resp) θ (G) ∈ [0, 1] to maximize his utility, i.e., to solve the FOCs [START_REF] Bodoh-Creed | Approximation of large games with applications to uniform price auctions[END_REF]. Integrating the contribution of each type gives the aggregate best-response G (resp) (G) = Θ x (resp) θ (G)dµ(θ), i.e., the amount of public good that users want to provide in response to a given G. At an equilibrium, both quantities are equal, which corresponds to solving the fixed-point equation

G (resp) (G) = G. (12)
A key feature of our model is that the marginal utilities (9) are decreasing when G increases. Intuitively, if G is higher, users are less willing to contribute, both because they want to enjoy more the resource if it is less congested (the term -h(G) in ( 9)) and because the reward per unit of contribution is lower (the term R G in ( 9)). Therefore, the aggregate best response G (resp) (G) decreases when G increases. It is also a continuous function of G, due to the strict convexity of the cost functions. This leads to a unique fixed-point G (eq) (R) for equation [START_REF] Altman | Routing games in the many players regime[END_REF], and then to a unique Nash equilibrium x (eq) (R).

Our next result concerns the social optimum.

Theorem 3. There exists a function x * , uniquely determined almost-everywhere, maximizing [START_REF] Haurie | On the relationship between Nash-Cournot and Wardrop equilibria[END_REF]. Moreover, x (eq) (R) = x * almost-everywhere, and hence

G (eq) (R) = G * , for R = R * where R * = G * h (G * )(D -G * ). (13)
Proof: See Appendix B. Intuitively, the result holds because the externality faced by a user (-h(G) + R G ) is independent of his type θ. Therefore, by fixing a reward that is also independent of the type θ, it is possible to achieve social optimum by making users effectively pay a Pigovian tax [START_REF] Pigou | The Economics of Welfare[END_REF]. Then, Theorem 3 shows that by tuning the reward parameter R, the raffle-based scheme is able to enforce socially optimal contributions from every users at the Nash equilibrium.

The next two propositions further characterize the variations of the Nash equilibrium with the reward parameter R. Proposition 1. For any R > 0, we have G (eq) (R) > 0.

Proof: If G = 0, each user wants to contribute positively (the unit reward is infinite), hence it is not an equilibrium.

Note that this result is consistent with Theorem 3: if G * = 0, then the only parameter value that permits to achieve social optimum at Nash equilibrium is R * = 0. Proposition 2. For any R > R, G (eq) (R ) ≥ G (eq) (R); and the inequality is strict if G (eq) (R) ∈ (0, D).

Proof: See Appendix C. Proposition 2 finally formalizes the intuition that contributions increase with the reward R, due to the increase of the marginal utility.

B. Analysis of the atomic setup and convergence

In the previous section, we established existence and uniqueness of Nash equilibrium in the non-atomic setup of the decongestion problem model. We now show that there is also a unique Nash equilibrium in the atomic setup, which converges toward the equilibrium of the non-atomic setup.

We start with the existence and uniqueness of the Nash equilibrium for the atomic setup.

Theorem 4. For any number of users n ≥ 2, any sequence of (θ i ) i∈N and any R ≥ 0, there exists a unique Nash equilibrium x (eq,n) (R) in the atomic setup.

Proof: See Appendix D. As for the non-atomic setup, the existence of a Nash equilibrium cannot be derived from continuity and concavity conditions (see e.g., [START_REF] Fudenberg | Game Theory[END_REF]) due to the discontinuity at the point x = (0, • • • , 0) where no user contributes. However, with our assumptions, as long as n ≥ 2 (at least two users), this point cannot be a Nash equilibrium. Indeed if no user contributes, each user has a positive incentive to contribute a small fraction of his demand so as to get the entire reward R. Then, our proof goes through the formal steps to show the existence of a unique Nash equilibrium. The discontinuity at the origin created by the presence of the sum of contribution at the denominator also appears in different contexts, e.g., [START_REF] Johari | Efficiency loss in a network resource allocation game: the case of elastic supply[END_REF], [START_REF] Maheswaran | Efficient signal proportional allocation (espa) mechanisms: decentralized social welfare maximization for divisible resources[END_REF], where it is handled through different means such as the introduction of a reservation price in [START_REF] Maheswaran | Efficient signal proportional allocation (espa) mechanisms: decentralized social welfare maximization for divisible resources[END_REF].

When the number of users is large, it becomes more appropriate to represent the population as a continuum of nonatomic users whose types follow distribution µ. To establish a link between the atomic and non-atomic setups, we assume that in the atomic setup, users have types that are random variables distributed according to µ and let the number of users n go to infinity. Formally, let (Ω, F , P) be a probability space.

Let (θ i ) i≥1 be a sequence of i.i.d. random variables defined on this probability space, taking values in Θ and distributed according to the probability measure µ(•)/D. For any integer n, the atomic setup corresponds to n users whose types are the first n values of the random sequence (θ i ) i≥1 . Then we have the following result.

Theorem 5. For any R, as n goes to infinity, we have P-almost surely:

(i) The equilibrium level of public good in the atomic setup converges toward the equilibrium level of public good in the non-atomic setup:

G (eq,n) = i∈N x (eq,n) i • d P-a.s.
----→ n→∞ G (eq) ;

(ii) The equilibrium contribution of each user in the atomic setup converges toward the equilibrium contribution of the corresponding user type in the non-atomic setup:

x (eq,n) i P-a.s.

----→ n→∞ x (eq) θi .

Proof: See Appendix E. Theorem 5 establishes a link between the atomic and nonatomic setups: as n goes to infinity and the demand of each user becomes negligible compared to the total demand, the fraction that user i contributes to the public good at equilibrium in the atomic setup approaches the equilibrium contribution of the type θ = θ i in the non-atomic setup. This justifies the use in practical cases of the non-atomic setup which permits easier analyzes and simulations. Our convergence result is closely related to the results of Bodoh-Creed [START_REF] Bodoh-Creed | Approximation of large games with applications to uniform price auctions[END_REF] for Bayesian-Nash equilibrium in games with semianonymous utilities (our model is a particular case of such games). Although it is different in spirit, our result is also related to [START_REF] Haurie | On the relationship between Nash-Cournot and Wardrop equilibria[END_REF], [START_REF] Altman | Routing games in the many players regime[END_REF] in the context of routing games, where the number of user classes remains fixed and each user is replaced by a growing number of identical copies.

V. COMPARISON WITH MORGAN'S MODEL

Finally, let us compare our results with Morgan's results [START_REF] Morgan | Financing public goods by means of lotteries[END_REF]. The heterogeneity in our model is due to cost heterogeneity (c θi (•)) rather than heterogeneity of user valuation of the public good (h(•)), as in [START_REF] Morgan | Financing public goods by means of lotteries[END_REF]. Also, our cost of contribution is strictly convex rather than linear as in [START_REF] Morgan | Financing public goods by means of lotteries[END_REF]. However, the most important difference is that in our model, user valuation of the public good (h(•)) is reduced by (1 -x i ), reflective of reduced demand for the resource. Another modification of [START_REF] Morgan | Financing public goods by means of lotteries[END_REF] is the financing of the incentive scheme. In [START_REF] Morgan | Financing public goods by means of lotteries[END_REF], the reward is subtracted from the total user contribution, that is the amount available for public good financing is equal to total user contribution net of the reward R. In contrast, our rafflebased scheme is financed by an increase ∆p of the subscription price.

The result of Theorem 2 partially contrasts with the Morgan's results ((i) of Theorem 1). In our model, the case R = 0 leads to a unique equilibrium whereas the case of voluntary contributions can lead to multiple equilibria in [START_REF] Morgan | Financing public goods by means of lotteries[END_REF]. Indeed, in Morgan's model, the equilibrium level of public good could be achievable at multiple combinations of user contributions (when the sum of user contributions remains equal to equilibrium level). This multiplicity of equilibria is driven by the linear cost in [START_REF] Morgan | Financing public goods by means of lotteries[END_REF] and does not occur in our model due the convexity of the cost of shifting.

The result of Theorem 3 differs from Morgan's result ((ii) and (iii) of Theorem 1) in two respects. First, in [START_REF] Morgan | Financing public goods by means of lotteries[END_REF], only the level of public good is uniquely determined at social optimum, but it may be induced by multiple combinations of contributions. This is again due to the linear costs in [START_REF] Morgan | Financing public goods by means of lotteries[END_REF]. Second, the socially optimal level of public good in our model is exactly achievable, whereas in [START_REF] Morgan | Financing public goods by means of lotteries[END_REF], it can only be achieved in the limit of infinitely large reward. This difference is driven by the difference in the scheme's financing. The difference between the result of Proposition 1 and Morgan's result ((iv) of Theorem 1) is also driven by the difference in the scheme's financing.

Proposition 2 implies that our raffle-based scheme can lead to overprovision of the public good. Indeed, the equilibrium level of public good exceeds G * (hence welfare is suboptimal) if the reward exceeds R * . However, in reality, this is unlikely to occur because the same welfare could always be achieved with a reward lower than R * (and public good level lower than G * ).

VI. CONCLUDING REMARKS

In this paper, we developed a raffle-based scheme for congestion management, building on the economic literature on public good provision by means of lotteries. Our scheme can be viewed as probabilistic pricing: a user's reward depends not only on his contribution but also on the contribution of the other users. It is implementable via lottery-like mechanism.

The design of our scheme by a provider requires estimations of the user utilities to set the parameter to its optimal value. Imperfect information on user utilities may lead to overshooting of the parameter. However, the effect of this overshooting will be limited due to the close-loop mechanism inherent in our raffle-based scheme: the more users contribute, the smaller the incentive to contribute.

Our analysis is valid only if the price increase required by our scheme does not affect user participation. This clearly holds if each user has a sufficiently high utility before the price increase, so that even with an increased monthly price all users continue to buy the service. This also holds if we assume that users are heterogeneous on the timescale of a day but homogeneous on the timescale of a month; e.g., each day, users have their types drawn independently according to a common stationary distribution. Then, for any R for which the raffle-based scheme is welfare improving (around R * ), each user utility over the timescale of a month is larger than it was before the imposition of the scheme.

Our scheme can be applied in various settings for demand management. An application to decongestion of peak-time in broadband access networks is proposed in [START_REF] Loiseau | Incentive schemes for internet congestion management: Raffles versus time-of-day pricing[END_REF]. A similar scheme could be applied to manage residential demand for electricity. In that case, privacy and security considerations make our scheme advantageous relative to real-time pricing. Indeed, our scheme requires no real-time user-dispatcher communication. In addition, unlike currently suggested realtime pricing schemes (e.g., [START_REF] Boudec | Satisfiability of elastic demand in the smart grid[END_REF]), our scheme requires only aggregate data. Overall, this makes our raffle-based scheme an attractive tool to improve the allocation of a shared resource.

APPENDIX A PROOF OF THEOREM 2

Let R be fixed. For any given demand G ∈ [0, D] shifted by the population, the best response (solving the FOCs ( 10)) defines a measurable function x (resp) (G) : Θ → [0, 1] given for all θ ∈ Θ by

x (resp) θ (G)=    0 if R G -h(G) -c θ (0) ≤ 0, 1 if R G -h(G) -c θ (1) ≥ 0, (c θ ) -1 R G -h(G) otherwise. ( 14
)
Due to the convexity assumption of c θ (•), c θ (•) is strictly increasing, hence invertible and with an increasing inverse function. Therefore, ( 14) uniquely defines x (resp) (G). Let

G (resp) (G) = Θ x (resp) θ (G)dµ(θ) (15)
be the aggregate best response, i.e., the total demand that the population wants to shift in response to G. By definition and by strict concavity of the utility functions, a measurable function x : Θ → [0, 1] is a Nash equilibrium if and only if there exists G ∈ [0, D] satisfying the fixed-point equation ( 12) such that x = x (resp) (G).

To conclude the proof of Theorem 2, we show that (12) admits a unique fixed-point. Lemma 1. There exists a unique solution of [START_REF] Altman | Routing games in the many players regime[END_REF].

Proof: The r.h.s. of ( 12) (G) is clearly a strictly increasing continuous function of G. It goes from 0 to D as G increases from 0 to D.

For the l.h.s. (G (resp) (G)), first note that it is a continuous function of G. Indeed, due to the convexity assumption of

c θ (•), (c θ (•)) -1 (•) is strictly increasing continuous, hence x (resp) θ (G) is continuous in G for all θ ∈ Θ. Moreover, the function x (resp) (G) : Θ → [0, 1]
is dominated by the constant function equal to 1 (i.e., |x (resp) (G)| ≤ 1) which is integrable w.r.t. µ. Therefore, for any G ∈ [0, D] and for any sequence

(G n ) n≥0 which converges to G, we have x (resp) (G n ) ----→ n→∞ x (resp) (G)
lim n→∞ G (resp) (G n ) = lim n→∞ Θ x (resp) θ (G n )µ(dθ) = Θ lim n→∞ x (resp) θ (G n ) µ(dθ) = G (resp) (G).
Clearly, the l.h.s. (G (resp) (G)) is also a non-increasing function of G taking values in [0, D]. Therefore, there is a unique fixed-point of [START_REF] Altman | Routing games in the many players regime[END_REF].

APPENDIX B PROOF OF THEOREM 3

A. Proof of existence and uniqueness of x * Let X be the set of functions x : Θ → R such that Θ |x θ | µ(dθ) < ∞ and let X 0 ⊂ X be the set of functions x : Θ → [0, 1]. Consider the aggregate welfare [START_REF] Haurie | On the relationship between Nash-Cournot and Wardrop equilibria[END_REF] as a functional on X 0 taking values in R:

W (x) = h Θ x θ µ(dθ) • D - Θ x θ µ(dθ) - Θ c θ (x θ )µ(dθ) + Θ ūθ µ(dθ) -pD.
Since X 0 is compact and the functional W is continuous, it has a maximum (see Corollary 38.10 of [15, p. 152]). Let x * ∈ X 0 be such that W is maximal and let

G * = Θ x * θ µ(dθ).
Define the three subsets of Θ: Θ 1 , Θ 2 and Θ 3 where x * = 0, x * ∈ (0, 1) and x * = 1 respectively. We now derive necessary conditions for x * to maximize W in each subset. We start with the subset Θ 2 corresponding to interior points. Let y ∈ X be such that y θ = 0 for all θ ∈ Θ\Θ 2 . We define the directional derivative (also called Gâteaux derivative) of W around x * in the direction y as

dW (x * , y) = lim t→0 W (x * + ty) -W (x * ) t .
Then, we have

dW (x * , y) = Θ2 y θ •[h (G * )(D -G * )-h(G * )-c θ (x * θ )] dµ(θ),
where the exchange between limit and integration in the last term (giving -y θ c θ (x θ )) is justified by Lebesgue's dominated convergence theorem whenever

Θ |y θ • c θ (x θ )| µ(dθ) < ∞.
This holds here due to the assumption that c θ (•) has a bounded slope.

For x * to be optimal, it is necessary that dW (x * , y) = 0, i.e., Θ2

y θ •[h (G * )(D -G * ) -h(G * ) -c θ (x * θ )] dµ(θ) = 0.
For this to hold for any function y such that y θ = 0 for all θ ∈ Θ\Θ 2 , it is necessary that we have

h (G * )(D -G * ) -h(G * ) -c θ (x * θ ) = 0, (16)
for almost-all θ ∈ Θ 2 , i.e., for almost-all θ such that x * θ ∈ (0, 1).

We now treat the case of subset Θ 1 , which corresponds to the points of the lower boundary. Let y ∈ X be such that y θ ≥ 0 for all θ ∈ Θ 1 and y θ = 0 for all θ ∈ Θ\Θ 1 ; that is y is a direction that "pushes up" the values of x * that are at zero. The directional derivative of W around x * in the direction y is defined similarly to the previous case but with a limit t > 0:

dW (x * , y) = lim t→0 + W (x * + ty) -W (x * ) t , which gives dW (x * , y) = Θ1 y θ •[h (G * )(D -G * )-h(G * )-c θ (x * θ )] dµ(θ).
Here, for x * to be optimal, it is necessary that dW (x * , y) ≤ 0, that is

Θ1 y θ •[h (G * )(D -G * ) -h(G * ) -c θ (x * θ )] dµ(θ) ≤ 0.
For this to hold for any function y such that y θ ≥ 0 for all θ ∈ Θ 1 and y θ = 0 for all θ ∈ Θ\Θ 1 , it is necessary that

h (G * )(D -G * ) -h(G * ) -c θ (x * θ ) ≤ 0, (17)
for almost-all θ ∈ Θ 1 , that is for almost all θ such that x * θ = 0. The case of subset Θ 3 is handled similarly and yields the necessary condition:

h (G * )(D -G * ) -h(G * ) -c θ (x * θ ) ≥ 0, (18)
for almost all θ such that x * θ = 1. In summary, ( 16)-(18) show that for function x * to maximize W , it is necessary that x * is solution for almost all θ ∈ Θ of the FOCs [START_REF] Bodoh-Creed | Approximation of large games with applications to uniform price auctions[END_REF] where the unit reward R G is replaced by h (G)(D -G). By the assumption on h(•), this is a decreasing function of G. Therefore the same proof as for Theorem 2 applies to show that x * is uniquely determined almost everywhere.

B. Proof of coincidence with Nash equilibrium

From the previous proof (App. B-A), it is clear that if R = R * , then the FOCs (10) at a Nash equilibrium coincide with the optimality condition, which immediately brings the desired conclusion.

APPENDIX C PROOF OF PROPOSITION 2

Case 1: If G (eq) (R) = 0, then the result is obvious. Case 2: If G (eq) (R) = D, then we have R D -c θ (1) ≥ 0 for almost all θ ∈ Θ, which implies R D -c θ (1) ≥ 0. Hence G (eq) (R ) = D.

Case 3: If G (eq) (R) ∈ (0, D), for a given G, x (resp) (G) of ( 14) is non-decreasing when R increases to R , and strictly increasing for θ's s.t. x (resp) (G (eq) (R)) ∈ (0, D). Since the set of such θ's is of positive measure, the new fixed-point has G (eq) (R ) > G (eq) (R).

APPENDIX D PROOF OF THEOREM 4

A. Existence

For n = 1, a Nash equilibrium may not exist. For example, if the user's utility without the raffle-based scheme is decreasing in his contribution, there is no equilibrium. Indeed, if the user has a positive contribution, he wants to reduce it (and still get the entire reward); whereas if he has a null contribution he wants to increase it to get the reward. Note however that the one-user case is not of interest to us since G is simply the user's contribution which is not a public good.

For n ≥ 2, the utility is discontinuous at x = (0, • • • , 0), so that existence of an equilibrium cannot be derived from a standard existence theorem based on continuity assumptions (see e.g., [START_REF] Fudenberg | Game Theory[END_REF]). Clearly, x = (0, • • • , 0) is not a Nash equilibrium because any user can improve his utility by removing a small fraction of his demand. We now show that there exists a Nash equilibrium where at least one user shifts a positive fraction of his demand.

Let > 0 and consider the convex closed bounded domain D = x ∈ [0, 1] n : i∈N x i ≥ . In D , the utility (4) is continuous in x and concave in x i for each fixed value of x -i . Then, by Theorem 1 of Rosen [START_REF] Rosen | Existence and uniqueness of equilibrium points for concave n-person games[END_REF], there exists an equilibrium point x (eq,n) ∈ D such that for all i ∈ N :

U i (x (eq,n) i , x (eq,n) -i ) = max xi:(xi,x (eq,n) -i )∈D U i (x i , x (eq,n) -i ). ( 19
)
If is small enough and n ≥ 2, this equilibrium point is not on the corner of D , i.e., G (eq,n) = i∈N x (eq,n) i

• d > . Indeed, suppose that we had an equilibrium with G (eq,n) = . Then, for at least one user i ∈ N (e.g., a user such that x (eq,n) i ≤ /2), the marginal utility of that user

∂U i ∂x i = d • -h( ) + (1 -x (eq,n) i )dh ( ) -c θi (x (eq,n) i ) + R -x (eq,n) i • d 2
is positive (for small enough); which contradicts the equilibrium property.

To conclude, we show that an equilibrium x (eq,n) of the concave n-person game defined by D for small enough so that G (eq,n) > is also a Nash equilibrium of the initial game corresponding to the raffle-based scheme (with strategy space [0, 1] for each user). First for i ∈ N such that j∈N \i x (eq,n) j ≥ , we have {x i : (x i , x

(eq,n) -i ) ∈ D } = [0, 1] so that (19) implies U i (x (eq,n) i , x (eq,n) -i ) = max xi∈[0,1] U i (x i , x (eq,n) -i ). (20)
Second for i ∈ N such that j∈N \i x (eq,n) j < , since

G (eq,n) > , x (eq,n) i
is not on the lower boundary of the domain

{x i : (x i , x (eq,n) -i
) ∈ D }. Therefore, for (19) to be satisfied, we must have ∂Ui ∂xi (x

(eq,n) i , x (eq,n) -i ) ≤ 0. Since for x (eq,n) -i fixed, U i (x i , x (eq,n) -i
) is a strictly concave function of x i on [0, 1], this implies that (20) is satisfied. In summary, (20) is satisfied for all i ∈ N which means that x (eq,n) is a Nash equilibrium.

B. Uniqueness

First recall that, as for the non-atomic setup, for a fixed G, the best responses are uniquely determined due to the strict concavity of the the utility (4) in x i .

Contrarily to the non-atomic setup, the marginal utilities may not be decreasing in G due to the"raffle term" of ( 5

): R G-xi•d G 2 , which is increasing for G ∈ [x i • d, 2x i • d] and decreasing for G > 2x i • d.
Suppose that two equilibria exist with contributions x (1) and x (2) levels of public good G 1 and G 2 . Without loss of generality we assume that G 1 < G 2 and we distinguish the two possibles cases (which are exhaustive).

Case 1. Suppose that for all users i ∈ N , x

(1) i ≤ G1

2 . Then, for all users i ∈ N we have ∂Ui ∂xi (x

(1) i , G 2 ) ≤ ∂Ui ∂xi (x (1)
i , G 1 ), so that at the second equilibrium, we must have x

(2) i ≤ x (1) i for all i ∈ N , hence G 2 ≤ G 1 ; which is a contradiction. Case 2. Suppose that for one user i ∈ N , x (1) i > G1
2 . Then, for all other users k ∈ N \i, we have x

(1) k ≤ G1 2 and therefore x (2) k ≤ x (1) k . If x (1) i = 1, then G 2 ≤ G 1 which is a contradiction. If x (1) i < 1, then we have ∂U i ∂x i (x (1) i , G 1 ) = d• -h(G 1 ) + (1 -x (1) i )h (G 1 ) -c θi (x (1
) i ) + R k∈N \i x (1) k • d G 2 1 ≤ 0, which implies ∂Ui ∂xi (x (1) i , G 2 ) < ∂Ui ∂xi (x (1)
i , G 1 ) ≤ 0. Therefore, we must have x

(2) i ≤ x (1) i which leads to G 2 ≤ G 1 , a contradiction.

APPENDIX E PROOF OF THEOREM 5 A. Proof of (i)

For a given G ∈ [0, D], the individual best response for the non-atomic setup is defined by [START_REF] Boudec | Satisfiability of elastic demand in the smart grid[END_REF]. For the atomic setup with n players, it is defined by

x (resp,n) i (G) =        0, if g (n) θi,G (0) ≤ 0, 1, if g (n) θi,G (1) ≥ 0, g (n) θi,G -1 (0), otherwise;
where for all θ ∈ Θ, function g

(n) θ,G is defined for all x ∈ [0, 1] by g (n) θ,G (x) = -h(G) + (1 -x)dh (G) -c θ (x) + R • G -x • d G 2 .
Note that for all G ∈ [0, D], x (resp,n) i (G) is well defined since by our assumptions on h(•) and c θ (•), g

(n) θ,G (•) is strictly decreasing continuous.
Let G (eq) be the equilibrium level of public good in the nonatomic setup. Recall that, by Proposition 1, we have G (eq) > 0. Let η ∈ (0, G (eq) ). Recall that d = D/n where D is a constant. We now provide the proof of Theorem 5 in three steps. In the first step, we prove the uniform convergence of the atomic setup's response function, x (resp,n) i (•), to the nonatomic setup's response function, x (•) of the population of n users sampled from µ/D to the aggregate best response G (resp) (•). Finally, in the third step, we prove that at equilibrium, the level of public good for the sampled population, G (eq,n) , almost surely converges to G (eq) as n → ∞.

Step 1. First we show that for any integer i, we have x θi (G)

≤ 1 -x (resp,n) i (G) • h (G) -R • x (resp,n) i (G) G 2 • D n , ≤ h (η) + R η 2 • D n .
Due to strict convexity, c θ (•) is invertible and due to the assumption that c θ (•) has a slope bounded away from 0, (c θ ) -1 (•) is Lipschitz continuous with a factor independent of θ. We conclude that for all G ∈ [η, D] and all type θ i ∈ Θ,

x (resp,n) i (G) -x (resp) θi (G) ≤ K • 1 n , ( 21
)
where K is a constant independent of G and θ i . ----→ n→∞ 0.

This exactly means that the second part of (22) converges almost surely towards zero, uniformly (in G) on [η, D].

Step 3. We finally show that G (eq,n) a.s.

----→ n→∞ G (eq) .

We work on a set of P-measure 1 where G (resp,n) (•) converges uniformly towards G (resp) (•) on [η, D] (see step 2). Recall that by Theorems 2 and 4, G (eq,n) and G (eq) are uniquely determined if n ≥ 2. The uniform convergence property implies that G (resp,n) (G (eq,n) ) -G (resp) (G (eq,n) ) + G (resp,n) (G (eq) ) -G (resp) (G (eq) ) ----→ n→∞ 0, that is, by definition of G (eq,n) and G (eq) , G (resp) (G (eq,n) ) -G (eq,n) + G (resp,n) (G (eq) ) -G (eq) ----→ n→∞ 0.

(23)

Let φ : [η, D] → R be the strictly decreasing function such that for all G ∈ [η, D], φ(G) = G (resp) (G) -G. Suppose that G (eq,n) does not converge toward G (eq) . Then there exists > 0 and subsequence {n j } such that G (eq,nj ) -G (eq) ≥ ∀j. Therefore, φ(G (eq,nj ) ) ≥ δ > 0 where δ = min φ G (eq) + , φ G (eq)if G (eq) ≤ Dand δ = φ(D -) if G (eq) > D -. Hence, there exists > 0 and subsequence {n j } such that G (resp) (G (eq,nj ) ) -G (eq,nj ) ≥ δ > 0 ∀j which contradicts (23).

B. Proof of (ii)

We have x (eq,n) i -x (eq) θi ≤ x (resp,n) i (G (eq,n) ) -x (resp) θi (G (eq,n) ) + x (resp) θi (G (eq,n) ) -x (resp) θi (G (eq) ) .

From (21), the first part goes to zero and from step 3 and the continuity of x (resp) θi , the second part goes to zero.

  pointwise (by continuity of x (resp) θ (G) w.r.t to G for all θ ∈ Θ) and by Lebesgue dominated convergence theorem,

  The second step establishes the almost sure convergence of the aggregate best responsesG (resp,n) (•) = i∈N x (resp,n) i

(

  resp,n) i (•) ----→ n→∞ x (resp) θi (•), uniformly on [η, D] and uniformly in θ. From the definition of x (resp,n) i and x (resp) θi , we have, for all G ∈ [η, D], c θi x (resp,n) i (G) -c θi x (resp)

Step 2 .

 2 Next, we show that G (resp,n) (•) a.s. ----→ n→∞ G (resp) (•), uniformly on [η, D].

For

  all G ∈ [η, D], we haveG (resp,n) (G) -G (resp) (G) ), it is clear that the first part converges towards zero, uniformly (in G) on [η, D]. For the second part, consider the set of functions FU = {f : Θ → [0, 1] s. t. f (•) = x (resp) • (G) for some G ∈ [η, D]}.By application of a Glivenko-Cantelli theorem (e.g., Theorem 1 in[17, p. 837] whose conditions can be verified with the result (21) of step 1), we obtainsup f ∈F U 1 n i∈N f (θ i ) -E µ D f (θ)a.s.