I NTRODUCTI ON: IS MOB RULE OUR I DEAL OF REPRESENTATI VE DEMOCRACY?

Their platforms reflect concern with enhancing the probability of being elected, but some candidates often, or all candidates occasionally, voluntarily adept stances that reduce that probability. Governments care about their popularity, but sometimes they choose, even before an election, to announce or implement policies that are unpopular. For most people, the phenomenon is no news and is not altogether a unhappy one. Deprecating words such as demagogy or "mob rule" and praising ones such as leadership or statesmanship express a deeply-rooted, widelyshared concern about the possibility that democratic politicians could be too subservient to public opinion.

But that concern is not reflected in the economic theory of democracy, which implicitly takes as its benchmark a world in which every wish or whim of a majority of voters would be perfectly reflected in the platforms and actions of politicians. In other words, put somewhat bluntly, democracy ought to mean mob rule of the purest kind. Then, in most of the literature, it is argued that the benchmark, viewed as an ideal, cannot be reached and therefore that the mechanism of democracy is necessarily "imperfect" or "inefficient". For instance, the wishes of the voters are such that a stable majority cannot exist (the cycling problem). Or politicians exploit voters' rational ignorance to further their own interests and those of the pressure groups they transact with (politicians "cheat" or "shirk"). The possibility that politicians might follow too closely the policy preferences of a majority of the electorate is not considered. In a recent article, Donald Wittman (1989) argues that democracy is efficient, but, his aim being only to show that a stable majority obtains and that politicians act according to its preferences -the presence of politicians introducing no significant distance or distortion -his message comes down, put somewhat bluntly again, to saying that the pessimists are wrong because, in the real world, thanks heavens, the mob does rule.

Let me formulate three disclaimers. First, when referring to mob rule I do not want to suggest that any distinction should be made between "masses" and "elites". There is ample evidence that the views expressed by "elites" (whoever they are) on most issues and particularly on the really important ones cannot be singled out for superior quality (whatever that means).

Second, I do not deny the relevance of citizen preferences (in some sense to be considered below), and the need, for a political system to deserve being called democratic, that they be reflected (in some way) in actual or proposed policy-making. Third, contrary to Wittman, I think that "shirking" by politicians can be the source of an important "failure" of representative democracy.

Under uncertainty and information costs and asymmetry, agents such as politicians (or doctors) have considerable discretion. That there will be some misuse (from the point of view of principals) of that discretion is unavoidable and potentially a very serious problem. 1 But the fact that discretion has a cost (the possibility of cheating) does not imply that we should always deplore its existence. 2 In other words there are two sides to discretion. I am concerned here with the second one, without attempting to relate it to the first Why discretion may be a condition for democracy to work is not a question that the economic theory of democracy, particularly the spatial theory of elections, can address. But that means that it loses touch with a part of reality which almost everybody else considers as essential.

The politicians we have in mind are not concerned with becoming members of assemblies.

They run in popular elections for functions such as President of the United States, France or Peru, Prime Minister of Britain or Ontario, Chancellor of Germany, Mayor of New York City, Athens (Greece) or Dijon (France), Governor of California. Focussing on this kind of actors and elections does not mean that others are less significant For instance political parties are clearly essential in some countries, such as Italy, and important everywhere. But taking them into account would complicate matters too much. The aspect of politics we concentrate on is one among others. It is enough that it is not unimportant Section I discusses voters' preferences over policies. The main argument there is that voters know that their policy preferences are tentative and wish them to be interpreted as such by politicians. In other words, the voter-politician relation is in some respects analogous to the patient-doctor one. 3 The second section considers the objectives of politicians. That these objectives do not always consist exclusively in maximizing popularity in the short run or winning the next election is increasingly aknowledged, but I remain unconvinced by the currently dominant alternative which consists in endowing politicians with policy preferences. Assuming that politicians have policy preferences has proved productive but it has troublesome implications. I shall argue that a better way to qualify their objectives is to add some reputation concern to their short-run, purely electoral objectives. A common assumption in these two sections is that voters cannot distinguish between popular and unpopular policies. Under that assumption, the feasibility of unpopular policies is straightforward.But in section III voters can assess, at least to some extent, popularity, and that ability is common knowledge (politicians know that voters have it, and voters know that politicians know that voters have it). Things then may become more complicated.

Policies which are unpopular (in the sense that they do not coincide as much as possible, given the constraints, with policies preferred by a majority of voters) can be advocated by politicians whose horizon and objectives are limited to winning the next election. There is an obvious need at this stage for a formal model but that task is left for future research.

Section I: The nature of the policy preferences expressed by voters

In it simplest form, the theory of multidimensional voting applied to elections assumes that each voter has an ideal policy position and votes for the candidate whose position is the nearest to that 4 The ideal position is sometimes called a "bliss point". But the dimensions of the space considered are policies while the variables individuals are really concerned with, the arguments in their utility functions, are effects of policies.5 Although some dimensions of the effect space and of the policy space may be the same (when policies as such enter the utility functions of voters independently of their consequences), the two sets are normally very different Policies have effects which cannot be limited to their objectives or purposes. [START_REF] Fiorina | Retrospective voting in American national elections[END_REF] misses that point when he argues that busing in the United States constitutes an exception to his thesis that voters care about policy outcomes rather than policy instruments. He writes (p. 13):

"Ostensibly busing is a means to an end -integrated education and ultimately an integrated society. currently in the Western democracies, for most matters, and with the exception of extremists (of both sides), that is not too much the case. For that reason, as well as for the more methodological one underlying rational expectations thinking, I think we can assume that voters arc fully aware of the uncertainty that affects their policy preferences. That extends to the policies, already referred to, which affect utility directly. Thus, taxing the rich may be pleasant perse, but the voters who enjoy that policy are aware that it may have all sots of consequences or side-effects that are difficult to predict and may change general equilibrium in such a way that its net effect on their utility becomes negative. Now, for voters to consider their policy preferences as tentative or corrigible, uncertainty is a necessary but not a sufficient condition. There must also be some distance between the expression of preferences and the actual choice. The latter must not be directly and mechanically the product of the former. It seems clear that answering the question in an opinion survey "Do you prefer the governement to reduce taxation or to provide more social policies?" has little to do with answering one's dealer "I prefer a rebate in price to a free radio set". But why exactly? Strictly speaking, the buyer cannot be sure that getting a given sum of money instead of a radio set will increase his or her utility. Buyer uncertainty can be substantial. Yet, in most cases, no distinction is made between the space of decisions a consumer faces when buying and the space of outcomes which is relevant for his or her utility function. The reason for that is that expressing one's preferences and buying, that is, deciding, are assumed to be one and the same. But things can be otherwise. If our buyer's answer had been "I think I should choose the rebate", it would have revealed not only some uncertainty but also some acceptance that the choice was yet to come and could still be influenced by collective deliberation. Our buyer's preferences could then have been considered as tentative.

Yet

In collective decision-making, there is always at least some distance between the preferences of individuals over alternatives and the actual or final decision. But the distance is greatest in the case of representative democracy. Voters do not directly choose policies but politicians, and all constitutions ensure that elected politicians will have some freedom in choosing policies. There is an old and abundant literature on the subject but we shall invoke here the perhaps unexpected authority of Condorcet While better known fa* his famous theorem on the outcome of conflicting preferences, Condorcet was mainly concerned with the way a democratic institution could enhance its probability of coming to the "most correct" decision -his less well-known jury theorem (see, e.g., [START_REF] Giesbert | Rousseau's general will: a Condorcetian perspective[END_REF][START_REF] Estlund | Symposium under the title: "Democratic theory and the public interest Condorcet and Rousseau revisited[END_REF]. He may deserve thus to be considered as the father of what has been called the epistemic conception of democracy (cf. [START_REF] Cohen | An epistemic conception of democracy[END_REF][START_REF] Coleman | Democracy and Social Choice[END_REF], although different approaches, such as Popper's "piecemeal engineering", would also deserve that name. Contrary to Rousseau, Condorcet saw in representative democracy a solution to the ignorance which he thought likely to prevail in the people. He thought that "in the majority of matters submitted to the decision of an assembly, the same voters whose opinions have such a small probability of being true can be enlightened enough -certainly not to pronounce with some probability of truth as to which man among a great number has the most merit -but to choose, as the most enlightened, one of those whose opinions will have a large enough probability of being true. Thus a numerous assembly who are not very enlightened could be usefully employed only to choose the members of a less numerous assembly to whom the decisions on other matters would then be entrusted" (from [START_REF] Estlund | Symposium under the title: "Democratic theory and the public interest Condorcet and Rousseau revisited[END_REF]Estlund et al., , p. 1325). I do not think that the mass-elite dichotomy is involved here but, to make sure that it is not, it is sufficient at this stage of the argument to assume that, whathever their level of enlightement prior to be elected, representatives, especially those who constitute the executive, have more knowledge than voters because they have an interest in collecting information and can get it more cheaply (with the help of the bureaucracy).

Again, I think we can safely assume that voters are aware of the difference in knowledge between them and governments, and this goes some way, we can note in passing, in explaining that the latter feel free to engage in unpopular policies. In fact, there is some empirical evidence on this awareness. According to a survey of English workers conducted in 1966 by Eric Nordlinger, the majority approved of a government gang ahead "with policies ("like raising taxes") which it believes to be for the good of the country even though the majority of the people do not agree with them". There was a majority even when the formulation of the question implied that the party supported by the respondents was in the opposition. Reasons for the dominant answer were typically of the kind: "The government should do what it thinks right"; "We put them in power to get on with the job"; "The government is there to lead." (see Nordlinger 1981, p.79). I do not know how robust that empirical evidence is, but if sufficiently so, it confirms that citizens, first, do agree or disagree with policies and, second, accept that their opinions are shaky and should be bypassed when the government thinks it is necessary.

In the last sentence, I used the word "opinion", instead of preference (Condorcet did also).

In the context, it came naturally. It is not accidental that we speak of "public opinion" rather than of "public preferences". In many cases, respondents to surveys, or even voters, do not really and exactly "prefer" a particular policy to another one. They think that the policy ought to be chosen depending on some information which they are more or less aware of not having. We are in a world of opinions, beliefs, judgements, feelings, inclinations more than in one of preferences as the term is used in economics (but as we know beliefs, inclinations, etc. can be quite strong). 7

Two obvious problems remain to be dealt with before moving on to politician motivation.

Section II: The objectives of politicians

The simplest hypothesis about politicians' behaviour is that their sole objective is to maximize the probability of winning the next election and that, for this purpose, candidates adopt policies that are likely to be favoured by a majority of voters (we consider only the two-candidate case). Under that assumption, and given sufficient uncertainty, the platforms of the two candidates converge to a common position (which is not necessarily the median voter's "bliss point" under the assumption of probabilistic voting). The assumption I shall make is that, to various extents, politicians are interested in their reputation, not in the eyes of their "competitors" (as in Alesina), but in the eyes of voter's and of future generations. Consequently politicians will not endorse any policy for the sake of winning the next election. Policies proposed or implemented by politician A's competitors will be of some concern only inasmuch as they affect electoral outcomes and/or A's reputation.

That view has been challenged in various ways

In this section, we assume that voters do not know whether a policy is popular or not In other words voters do not know how other voters will react to policies. Although very crude, the following possible scenario should be sufficient for illustrating the point I want to make at this stage.

(1) Politician A expects a particular policy to be currently, i.e. at time t, popular (unpopular), and to remain so until t+n, but to have a high probability of proving in retrospect (say, from time t+n) to have been detrimental (favourable) to the interest of the country or to the interest of the particular constituency (say, the working class) she is associated with.

(2) A expects advocating a implementing the policy to result in a higher (lower) probability of winning elections, or increased (decreased) support from the particular constituency, until t+n, but in a lower (higher) reputation (in some sense to be discussed) after t+n.

That reputation has sane value to A may result simply from it being a form of political capital, in the sense that a higher reputation means, ceteris paribus, an increased probability of winning elections a gaining the support of a particular constituency. But if the only value attached to reputation is of that kind, our scenario comes down to the assertion that some politicians may look beyond the next election(s) and adopt a long-term view of their political career. In cases in which A knows that she will not run after time t+n, the assumption that reputation is purely instrumental implies that popular (unpopular) policies whose detrimental (unfavourable) effects will be felt only after t+n, will (not) be advocated or implemented. This is disturbing. Thus I prefer to assume that politicians may also be concerned with their reputation in the eyes of future generations (or historians), which seems plausible in the case of the politicians we have in mind.

The assumption then allows some policies that bear fruit only in the very long-term and are unpopular in the meanwhile to be advocated or implemented even by dd politicians, which seems to me not only more pleasant, but also more realistic.

Whether a politician will adept or not a policy will thai depend on the expectations he or she forms about its effect on the next election(s), on more distant ones, on reputation after death or retirement; it will depend also on age and on preferences (including preferences towards risk). It is clear that there will be differences among individual politicians placed in the same conditions as to the way they will perceive and value the costs and benefits of a policy. This has important consequences on how voters behave in elections. Before we turn to that, let us note that, in a less simplified analysis, we should consider the interaction between the strategies of politicians. For instance, reputation is largely a positional good, which may explain to some extent why rivalry is an essential characteristic of the political g^me even when, for some reason, there is no competition for office. 11

Section III. Voter assessments of candidates

According to Give Bean and Anthony Mughan, the perceived personal quality of leaders which coitributed most to explaining the results of recent parliamentary elections in Australia and Britain

was "effectiveness" (followed in Britain, at substantial distance, by "caring" and "decisiveness"), while, in both countries, "likability as a person" had no impact (e.g., Margaret Thatcher was perceived as much less "likable as a person" than her opponent but that had a negligible impact on the vote, while being perceived as much less "caring" had a significant negative one). They conclude that "prime ministerial candidates are judged against some kind of well-defined schema in public mind" and "will have a positive electoral impact to the extent that they conform to this But, first, as we saw in the last section, that does not mean that candidates are not looking beyond the next election and thus do not take into account the effect of their present policy position on the future perception of their personal characteristics, that is, on their reputation. Most candidates, consequently, will not adopt the policy position that would maximize their probability of winning the next election, which means that they will tolerate in their platforms some policies that reduce their popularity in the short run.

To qualify that statement, we should make a distinction here between premises and actions.

Engaging in actions, i.e. implementing policies, that may be resented in retrospect undoubtedly affects future reputation. Premises are another matter. We argued that voters are aware of the difference in knowledge that exists between government and outsiders. Since candidates who are not in office may also be perceived as outsiders, especially if they have not been in office for a long time, their reneging on their promises may have no serious consequences to their personal reputation, which means that they can feel freer than incumbents to make excessive promises. But even the violation of promises (e.g., not raising taxes) by reelected incumbents may not affect their reputation durably. Lying or deceiving might be regarded as a lesser sin than mismanaging. The matter becomes more interesting and at the same time more difficult to handle when voters have at least some perception of the popularity of policies. To see the nature of the problem, let us first consider what would happen if we could assume that voters know only retrospectively whether a policy was popular or not The mechanism presented in the last section gets reinforced Politician A expects a policy to be unpopular now, i.e. to have an electoral cost in the short run, and to be considered favourably in retrospect at a later time (after t+n). If she proposes or adopts that policy, A can expect a positive effect on her reputation after t+n. Now, that positive effect will be higher if unpopularity at time t is widely known at time t+n. The known fact that A did not choose a popular but mistaken policy provides additional favourable information on her personal characteristics. The presumption will be stronger that at least she was not exclusively concerned with success in the short-run. The admiration we have for some politicians, or for that matter for some painters or economists, is enhanced by our knowledge that they ran against the tide. I argued at the beginning of the last section that various factors (pressure groups, party activists, ideology, etc.) constrain the policy choice of candidates seeking to maximize the probability of winning the next election (or some other short-run electoral or popularity objective).

As Nordlinger's

The optimal policy position of a candidate is thus a "constrained optimal position". We must take that into account in our analysis. A finer assessment of the past behaviour of politicians will Of course, the story is completely unacceptable. Whatever it is that voters know, in this context politicians know more. Consequently, even a pure short-run "office-oriented" or "votemaximizing" politician might want to mimic politicians who are not, and fa* that purpose advocate an unpopular policy. In fact, something of the like certainly obtains in reality. To simplify, let us disregard the constraints on policy positions referred to above. Voters know that public opinion about policies is P (this is another simplification of course). Suppose that the space in which P is defined has a large number of dimensions. And suppose that candidate A adopts platform P. We have seen in the first section that voters know both that they know little and that politicians know more. Thus, they would not expect a "high quality" (not exclusively office-oriented) politician to follow public opinion on all matters. They would then conclude from the perfect coincidence between public opinion and A's platform that A is somewhat "unprincipled" or "low quality", that is, in our framework, cares exclusively about short-run approval. 14 Thus, I think that even exclusively office-oriented politicians will typically try to avoid excessively obvious coincidence and consequently adopt original or idiosyncratic stances at least on some issues, preferably issues that are both miner (do not affect much the vote) and very noticeable (because, for instance, they attract the attention of the media). In a dynamic context, the concern could explain that no politician follows the vagaries of public opinion in all matters, even when the pressure of support groups would allow for it In terms of pooling and separating equilibria, candidates foolish enough to follow public opinion in all details would separate themselves, that is, signal their "low quality". Such candidates probably do not exist In a context such as this one -of voters assessing public opinion, politicians knowing that they do, and so forth -can "high quality" candidates signal their high quality? A credible signal is one too costly, in terms of utility, for "low-quality" candidates to undertake. Although this remains to be proved formally, I am sure that, under some conditions, such signalling is possible.

If that is confirmed, we have a somewhat modified mechanism for current policy positions to influence the perception of personal characteristics and for unpopular but useful policies to be feasible. As a consequence, choosing between motions is always in part dedding whom, among those who express an opinion on the competing proposals, to trust more (or distrust less). As we have seen, complications result from the fact that advocating an alternative affects trust even in the short run. I claim no originality ft?r these ideas, which could well be regarded, especially by noneconomists, as platitudes.But the question whether unpopular but necessary or useful policies are feasible in democracies should be seen as a very important one. That democracy is a system which precludes their occurrence is the main argument of its critics. Demagogy is their favorite word.

Section IV. Conclusion

The problem is not that in fact we do not know the answer to the question. It is rather that our models are not adapted to the issue, or worse, support the wrong answer. In other words, the problem is a theoretical one. As happens frequently in economics (perhaps increasingly frequently), it consists in finding a way to fit into the theory important but perfectly unoriginal observations on or about the real world. A precondition for progress in doing this is some awareness that a problem exists. I hope at least to have contributed to that NOTES 11 argue elsewhere (Salmon 1987b[START_REF] Salmon | The logic of pressure groups and the structure of the public sector[END_REF], that the problem is less serious whenever international or interjurisdictional comparisons of performance are available.

2 This is also one of the main themes in Breton and Wintrobe's analysis of bureaucracy -see, e.g., their 1982 book.

3 Information asymmetry here is largely of the hidden knowledge kind: the principal (i.e., the patient or the voter) may observe the actions of the agent (the doctor or the politician) but cannot say if they are appropriate. 6 One implication of the assumption is that campaigns can only aim at persuading voters that seme policies will be implemented, and at providing more details about these policies, not at informing voters about their effects. This is rather unfortunate since we all know that political rhetoric tends to concentrate on some possible consequences of policies (especially ominous ones according to Riker). 7 See the discussion of opinions and of Mill's view on them in [START_REF] Galeotti | An economic theory of political parties[END_REF]. 8 Cf. note 1. 9 Although it is one of the traditional approaches to representative democracy, the theory of retrospective theory in its extreme form is associated nowadays with the work of V.O. Key. For that author, the electorate is mainly "an appraiser of past events, past performance and past actions. It judges retrospectively; it commands prospectively only insofar as it expresses either approval or disapproval of that which has happened before". And: "Voters may reject what they have known; or they may approve what they have known. They are not likely to be attracted in great numbers by promises of the unknown. Once innovation has occurred they may embrace it, even though they would have, earlier, hesitated to venture forth to welcome it" (from Fiorina 1981, pp. 7 and 9). Let me also quote Max Weber, although his views are somewhat ambiguous: "Concepts like the 'will of the people', the true will of the people, no longer exist for me -They are fictions. It is just as if one were to speak of the will of the purchaser of a pair of boots as being authoritative for how the cobbler ought to pursue his craft. The buyer may know how the shoe pinches -but never how to make a better shoe" (from Rune Slagstad 1988, p. 125). l^To defend extreme forms of retrospective voting, one would have to suppose either that voters are not rational (e.g., driven by emotions) or that they adopt a strategy of reward and punishment But the rational adoption of such a strategy by atomistic voters would require that they be coordinated (see footnote 9 of [START_REF] Alesina | Credibility and policy-convergence in a two-party system with rational voters[END_REF], which is implausible.

11 For instance if the question "Who will be the greatest president of the 20th century, and who will be the worse?" is considered relevant It is common observation that retired persons do not typically derive utility from the successes of their successors But I must mention that, a few days after President Bush decided to pour troups in Saudia Arabia, former President Carter wrote in the Washington Post "I wish him well as he wrestles with this recurrent threat..He has our prayers and our support" (from the International Herald Tribune, 13 August 1990).

12 And Margaret Thatcher is reported (by Arthur Seldon, 1987, p. 125)

  the political debate usually simplifies to one in which busing is both policy instrument and policy outcome. To millions of citizens the policy outcome is simply whether or not their children are bused". But what we should say here is that busing has more than one effect voters are aware of. As seen by voter X, the effects of busing are a more integrated society (admittedly a possible argument in her utility function), but also the time her children spend in a bus, the benefits and costs for them of studying in an integrated school, the change in the taxes paid by the familly, every member of it having to wake up half an hour earlier, the extreme right becoming more powerful, and many others. If one takes into account all effects of policies, including idiosyncratic and side effects, it is clear that models of voting cannot handle voter preferences in the space of effects or consequences, even if we assume a world of perfect certainty and information. Preferences have to be defined over policies. But in such a world, the distinction does not matter because each voter makes perfects forecasts about all the consequences of each possible set of policies and thus operates a straightforward translation or mapping between his or her preferences over consequences and his or her preferences over policies. That uncertainty and imperfect information are important in the real world has of course been aknowledged by the literature. But uncertainty has not been introduced in the models where we would have wished it to be introduced for the purpose of this paper. In the literature surveyed by Enelow and Hinich (1990), translation, mapping, derived preferences, even the distinction between an "outcome space" and a "policy space" (Austen-Smith and Riker 1987), are discussed as ways to deal with voter uncertainty about what exactly the proposed policies are, or will be after the vote or the election. In the probabilistic voting literature, candidates do not know with certainty how each voter will vote (see Lafay 1990). In both cases, the implicit assumption remains that each voter is at least certain about his or her own ideal point in the policy space (to my knowledge, only Riker 1990 departs from that assumption). 6 My point is not that voters do not derive preferences over policies from their preferences over effects but that they should feel extremely uncertain about the result. The reason for that should be obvious to economists, whose discipline has thrived on the notion that very complex interdependence is the essential characteristic of society. But are voters really aware of the complexity of the linkage between policies and their effects? Two opposite answers are offered us on this point. Some thinkers (e.g. Karl Popper) have argued that people tend to underestimate ignorance and complexity, and thus ascribe what they do not like in the circumstances they observe to hidden intentional actions of some person or group -they subscribe to a "conspiracy theory". On the other hand, rational expectations theory tells us that we should not assume in our models that agents are more stupid or ignorant than we, as model-builders, are. Thus, if our view is that public decisions have far-flung or distant consequences, we should assume that the agents in our models also know that they have such consequences. There are examples, in the past, or currently in sane societies, of public opinion falling victim of the conspiracy theory. But,

First, voters have

  different interests and values. Policies, for instance have a redistributive impact After a decade of leftist government in France, over which the stock market index, property prices, inequality, unemployment, and deprivation ("new poverty") increased substantially, I would tend to consider ex ante predictions of the redistributive impact of policies as highly uncertain and to suspect that voters are well aware of the fact More important, redistributive considerations are not as dominant in the political process as sometimes assumed, nor is the choice of objectives. Most citizens want peace, prosperity, price stability, efficient public services, honest officials, moderate taxation, etc., and they want simultaneously more of each. In other words, there is a large area of common interests and also of non rivalry between objectives. Economists usually transpose to collective choice the famous "there is no such thing as a free lunch" dictum, illustrated by the equally famous gun versus butter trade-off. At the same time, they think that there is ample room for Pareto improvements, which means that collective decision-making is not just choosing a point on the production-possibility curve. Good or bad government is often the underlying main issue. Still, conflicting interests and values exist Thus, we cannot reason in terms of truth as Condorcet did. We are not in a purely epistemic context Voters are not impartial spectators or members of juries. If we could reduce uncertainty, some differences across voters in their socalled policy preferences would disappear, but others would be increased, which means that these differences in policy preferences reflect differences both in beliefs and in "genuine" policy preferences. Taking full account of this dual nature complicates the exposition but does not change the substance of our argument Within any homogeneous category of citizens, sharing the same interests and values, the perceived distance remains between what policies they think are in their common interest and the policy choice which would be made by their better informed professional representative if enabled to choose. The second problem is how genuine preferences of voters, in terms of effects of policies, get respected in the political process if voters have the attitude which has been ascribed to them in the foregoing discussion. If they are ready to consider that they may be wrong and politicians right, what limits the discretionary power the latter consequently enjoy? Here, I think retrospective assessments are essential. As argued elsewhere, their feasibility is enhanced by the availability of international comparisons of the kind which obtains in Western Europe. 8 That does not mean that we should follow retrospective voting theory completely. 9 If they are rational, voters must be forward-looking, which implies that they use retrospective assessments of past actions, outcomes or performance as a means to form expectations about future actions, results and performance (as argued already by Downs, 1957). 10 And if we want politicians to engage in the kind of unpopular policies which may be needed in some circumstances, we cannot ascribe to them the simple motivation assumed by retrospective theory, as we shall see now.

  in the theoretical literature. A distinction can be made between two categories of proposed changes in the basic assumptions. First, a number of factors constrain the choice of an optimal position by a candidate in such a way that convergence will not obtain. Convergence may be prevented by the possibility that it would result in extremist voters being alienated and thus abstain from voting. Pressure groups, different ones for the two candidates, may impose, as a counterpart to their financial support to campaigns, some policies that have an electoral cost. Poor information of voters about policies may justify that policies be homogeneized or made consonant under the umbrella of a perceptible ideology. Support from party activists may be necessary for a candidate first to be nominated and then for his or her campaign to gather strength, and this may entail that activists' preferences, not exclusively focussed on electoral victory, be given more weight in the determination of the candidate's optimal positioa Taken individually or together, these additional considerations constitute constraints that may well explain non-convergence of the positions of two candidates in the policy-space. I shall assume that the foregoing factors imply that, for the purpose of maximizing the probability of winning the next election, there is still a single optimal position for each of the two candidates in the policy-space and that these two positions do not coincide.Second, the motivation assumption itself can be changed. Let us note that it has two components: the exclusive objective of getting in office and the short-term character of the horizon. I shall argue that the two components must be changed together. But there are a number of models in the literature in which the first component only is changed. The models of Wittman (1983) and Randall Calvert (1985) are such. They assume that candidates have policy preferences in addition to the objective of winning elections. Alberto Alesina (1988) shows that their model is likely to meet a time-inconsistency problem: if the premises of the candidates are not binding, and they can hardly be binding under the assumptions, once elected a candidate will renege on them and apply his or her most preferred policies. If there is common knowledge, voters expect the elected candidate's preferred policies to be implemented, candidates know that voters expect this, and so on, and the time-consistent equilibrium of that one shot-game is that platforms reflect exclusively candidate preferences. But, if, contrary to Alesina's assumptions, voters understand the reneging mechanism but do not know the candidate true preferences, presumably anything can happen (for instance voters may as well toss a coin). The implication of this, in my view, is that no plausible model of voting for candidates (or political parties) can exist that does not assume a longer horizon. Alesina (e.g., 1988) offers us such a model. He refers to political parties instead of individual candidates and assumes that they have policy preferences as well as a preference for being in office. When the two parties have an infinite horizon, Alesina demonstrates that, assuming that announced platforms are implemented, the parties' first-best policies consist in choosing the same platform, which will be followed with certainty. That platform is on the portion of the efficient frontier in which the two parties are better off than in the one-shot equilibrium, its exact position resulting from a bargaining process whose Nash solution is spelled out in the paper. To give the flavour of the argument, let me quote in full footnote 6 (p.799) -after explaining that in the text itself the problem solved by the two candidates, i.e. "problem (8)", is formulated under the assumption that the utility of being in office (k) is zero. Footnote 6 reads: "These results generalize to any value of k. If k * 0 the two parties would still solve problem (8) to find the "first-best" policies. Then they would split k, namely the benefits of being in office: for example, they would agree to share the term of office. For a formal treatment of this problem see Alesina-Spear, 1987" [now, 1988]. When parties cannot make binding agreements, one may fear that, once in office, a party returns to his most preferred policy, that is, that cooperation is not sustainable. What may prevent that behaviour is each party's fear of losing one's reputation (in the eyes of the other party), which would entail being brought back for ever, starting from the next election, to the inferior equilibrium of the one-shot game. That cooperation is the first best for each player is hardly surprising. It always pays to collude. Still, there is something verging on innocence or naivety, I cannot help feeling, in the image of political behaviour suggested by Alesina's model, even if we allow fa* the fact that the model is concerned with political parties rather than individual candidates. It implies that (in a world of two parties) what party A resents, and retaliates against, is a move by party B, in office, to policies that are more distant to the ones party A prefers and what party A welcomes is a move by party B towards policies that party A prefers. But, typically, what we observe is the reverse. Neither parties nor individual politicians truly like their preferred platform or policy-mix to be ransacked by their competitors. Of course, part of the dislike stems from their concern with winning future elections and this is taken into account to some extent by the variable k (being in office) referred to above. But I do not think the explanation is sufficient Even when probabilities of winning future elections are not affected, I would think parties and politicians typically resent rather than welcome others moving their platform closer to theirs. As Alesina suggests himself when saying that "each party represents the interests of a different constituency" (1987, p. 651), his model applies to transparent parties, in fact proxies for different categories of pressure groups or social classes. Social classes and interest groups may cooperate but competition or rivalry seems to me an essential characteristic of the party system we know in most countries (especially two-party systems). The empirical evidence which apparently supports the model (cf . Alesina and Sachs, 1988; Alesina 1989; see also Raldam, forthcoming) has no bearing on the part of the model which entails the superiority or attractiveness of a cooperative solution, that is, on what we should consider the implausible part It supports non-convergence but we have already conceded that under the assumption of pure vote-maximization. It also supports the form erf* the political business cycle predicted by the author but that form can be derived from non-convergence plus rational expectations. Since Alesina's theorems are unassailable, the responsibility for the implication I consider to be counter-intuitive lies in the assumption, shared with Wittman and Calvert, that policies as such are independent arguments in the utility function of candidates a* parties. That assumption necessarily means that politicians or parties increase their utility, ceteris paribus, from the implementation of the policies they prefer whether or not they are in office, which seems to me implausible as a general proposition. Yet, I do not think that all politicians or parties are interested in being elected to the extent that they would do anything for that purpose. In other words, there are policies which they will not propose if in opposition or implement if in office. How can one model that behaviour without introducing policy preferences in the utility function, which almost inevitably leads, as shown by Alesina, to replacing competition by collusion?

  Since voters know that they have to rely on politicians and since politicians are different, it is clear that the expected behaviour of candidates under a variety of circumstances is an important factor in each voter's choice. That remains true whether we focus on common interest or consider political bias, explained by conflicting interests or values, to be important also. On what basis can a voter try to form expectations about the future behaviour of a candidate? Clearly, perceived personal characteristics are important Many journalists, political commentators, communication specialists and politicians put great weight on the most trivial ones. That these count is certain. But there is at least sane statistical evidence (cf . Miller, Wattenberg and Malanchuk, 1986; Bean and Mughan, 1989) that voters give priority to those characteristics which can be associated with what they would wish the candidate's behaviour to be once in office.

  mental image of what a leader should be like" (p. 1176). 12 But if the analysis of the last section is correct, voters should also be influenced by their perception of the opportunity costs and constraints that each candidate faces. For instance, a wellknown politician, secure about his or her position and fame even if not elected, should not, ceteris paribus, be expected to behave as somebody who could expect to fall into oblivion were he or she to lose the next election. That type of consideration was important in the case of De Gaulle in the late fifties and early sixties. Although his opponents raised the possibility that he would use the new constitution in an undemocratic way, many voters apparently felt that he would remain careful not to jeopardize his already well-established reputation, and thus, for instance, would not try to remain in office at any cost (an expectation which proved correct). The literature tends to separate the policy positions of candidates and the perception by voters of their personal characteristics.The latter are introduced as a second, independent, explanatory variable -what Endow and Hinich (1984, ch. 5) call the "non-policy" component of voters' evaluations. I think that this approach is mistaken, even if we assume as we shall keep doing for a while that voters have no information on the popularity of policies. At one point of time the past performance, actions, positions, and so on, of candidates are fixed and exogenous assets or liabilities (see also Fiorina 1981, chapter 7) -used by voters (with other kinds of information) to build up the conjectures about personal characteristics that will enable them to form (again, with other kinds of information) expectations about future behaviour and performance.

  English blue-collar workers were saying, "the government should do what it thinks right". Reneging on one's promises might even be interpreted by some voters as a display of courage, or statesmanship. Thus, at this stage of the argument, we cannot expect platforms (sets of promises), contrary to implemented policies, to be very different from what they would be under the assumption that candidates maximize their probability erf* winning the next election. But there is a second objection to a complete separation between personal characteristics and policy positions of candidates. In fact, Enelow and Hinich implicitly accept the point. They argue that candidates cannot change their policy position in the short run without incurring the high electoral cost of a blurred image in the eyes of the electorate. We can interpret that as meaning that personal characteristics and policy position are independent so to say at equilibrium only. It is because candidates know that they are not independent that they act in such a way that the two sets of variables can be considered as independent But I agree with Charles Rowley (in his review of their book) that Enelow and Hinich's assumption that candidates cannot change their position in the short run is not plausible, nor verified empirically. Even if moving one's position has, perse, an electoral cost, the electoral benefit involved in occupying a better policy position may be greater. Thus we should accept only that part of Enelow and Hinich's message which implies the recognition of a link between the current policy position of candidates and the current perception by voters of their personal characteristics. In the course of campaigns, candidates will move in the policy space with due recognition that they affect votes through these two channels simultaneously.

  aknowledge the existence of these constraining factors. In other words, what should and, I submit, does count is the perception of a discrepancy between the actual position and the "constrained optimal" one. In 1983, the leftist government in France (elected in 1981) started a policy consisting in reducing real wages, restoring profits, accepting lay-offs in public entreprises, and using very strict monetary means to reduce inflation. Such a policy, which was adopted only after seme intense internal debate had taken place within the government and among the president's collaborators and friends (see Giesbert 1990, pp. 169-83), was contrary to the government's avowed ideology, to the demands of the pressure groups with which it was most closely associated (the unions), to the programme of the parties which staffed it, and it could be expected to be detrimental to the government's popularity and to the supporting o lition's probability of winning the next election (which, actually, it lost). But, in retrospect, such a policy reversal increased considerably the political capital or reputation erf* the president I am not concerned here with his exact motivation. The point here is that the policy reversal was perceived as especially courageous or bold given the constraining factors, the perception of these factors being thus an essential ingredient in the enhanced reputation (De Gaulle's reversal of policy towards Algeria in the early sixties would be another case in point). Unfortunately for the simplicity of the reasoning, to some extent voters perceive popularity or unpopularity of policies, as well as constraining factors or forces, right away. And that extent, whatever it is, may be enough to influence electoral tactics, even when exclusively geared to the short run. Let us start with the first part of the proposition, that is, with what voters know. Although the point is seldom made in the public choice literature, voters may know rather well where public opinion stands at any point of time or what the direction and order of magnitude of its movements are. In recent times, an obvious reason for that is the results of opinion surveys being so widely publicized by the media But awareness of the state and dynamics of public opinion is apparently a more general phenomenon (at least, in democracies, the matter being completely different in unfree regimes, see Kuran 1989). Although the way they get informed is not so clear, it seems that individuals have an amazing capacity to assess directly (without consulting survey reports) the state of public opinion on many issues and to perceive variations. Some impressive evidence, whose robustness I cannot assess, is provided by Elisabeth Noelle-Neumann (1980). Let mc just quote one of her conclusions: "Through the methods of survey research today we recognize the highly sensitive human capacity of a quasistatistical sense organ to perceive -without using statistical techniques -frequency distributions and changes of opinion in the environment" (1984 ed., p. 115). If that is so, voters can also perceive discrepancies between the stance adopted by a candidate and the state of public opinion, or between movements in the candidate's position and movements taking place concurrently in public opinion. To some degree, voters can also perceive without delay the factors that constrain the choice of policy positions by politicians. They know that agricultural policy is heavily constrained by the pressure exerted on the government by the farmer unions as well as by foreign policy commitments within EC or GAIT, or that teacher unions necessarily have a strong influence on the education policy of a leftist government They are aware of the influence of party activists or of ideological requirements. They even have some way to assess the intensity of pressure and its variation. These are widely discussed in the media and can also be inferred from statements, demonstrations, strikes, parliamentary debates and votes, or from the information circulated within the various interest groups voters belong to. Thus, in Britain over a substantial period of time, it was easy for voters to know that the position of the leaders of the left was strongly constrained by ideology, party activists and trade unions. The same was true in France in the late seventies and early eighties (the party activists there being mainly the Comnu *nists and the leftwing factions within the Socialists). 13 It is, I think, common knowledge that the pressure is now much less in both countries. We do not need to assume that assessments are precise or even unbiased for their existence or possibility to influence the behaviour of politicians. Let us call P the policy position that candidate A would adopt to maximise the probability of electoral success in the short run if voters had no information on popularity and constraining factors. Let us call P the position, known to the voters, that A does adopt. The distance (D) between P and P has an electoral or popularity cost of X (in the short run). Let us assume now that voters have such information, that politician A does not know that voters have it, and that voters know that A does not know that voters have it Then, voters can derive from D the presumption that politician A does not exclusively seek electoral success or popularity in the short run. Under our assumptions, that perception has a positive effect R on A's reputation, hence a positive effect Y on the vote. The net electoral effect of D will be the sum of X and Y. It may be positive or negative.

I

  have explored in this paper a mechanism that may account for the feasibility of advocating or, mere importantly, implementing unpopular policies in a representative system. It is only one among others that are at work to produce the result I do not overestimate its reliability. But the others also have shortcomings. Political parties as such, abstracting from the real persons who lead them, are usually long-lived. To seme degree and in some respects, they are clearly concerned with their long-term reputation But they are large organisations and subject to the opportunistic behaviour of their members. The hallmark of statemanship often consists in a leader bypassing his or her party, that is, in supporting or implementing policies that are disliked both by it and by public opinion. Pressure groups are better equipped than individual voters, sometimes even than politicians, to predict some of the outcomes of particular policies, but they usually disregard the overall effects their members are concerned with as voters or citizens (and as members of many other groups). And they have other limitations. Not all concerns give rise to mobilized groups. Most pressure groups are not independent actors and their leaders are more or less suspected by members (as argued elsewhere 1 5). I can only mention, despite their obvious relevance, other actors such as the media and the bureaucracy (which in France tends to consider itself as the natural guardian of public interest, all the others assumed to misbehave). As noted in the case of political parties, statemanship involves (almost by definition) bypassing these various actors in some circumstances. That does not mean that they may not play also a role in the solution of our problem. Anyhow, the existence of other mechanisms does not affect an underlying theme of the paper, namely, that voting is mainly a matter of conjectures about actors (individual ones in the paper, certainly also collective actors in a more general perspective) rather than a choice of policies, although what voters are really concerned with ultimately is policies (or their effects). The discussion has been limited to representative democracy but the theme is relevant in a broader context It has been argued recently (Endow and Hinich 1984, Mueller 1989) that the theory of elections and the theory of committee voting should be completely separated because uncertainty affects the first much more than the second. But uncertainty of the kind stressed here, that is, uncertainty about the effects of actions, is also a major feature of decision-making in committees (e.g., boards of directors). The one man-one vote principle notwithstanding, members of committees have a very unequal capacity to reduce that uncertainty and are well aware of the fact.

4

  The problems related to the existence of an equilibrium are disregarded in this paper (but see Salmon and Wolfelsperger 1990).5 Our discussion here has some relation with Albert Breton's defense of what he calls the "goods are goods" assumption, although, for the purpose of this paper I do not consider as final goods the policies he treats as such (see[START_REF] Breton | The growth of competitive government[END_REF]).

  , to have said: "If we heed the grumblers, we would be bending and turning with every twist in the opinion polls... our style is to decide what is right, not temporally convenient". And: "Applause dies with the day. Belief lives on..." 13 In 1977-78, the French Communist Party, whose support was necessary for the Socialist candidates to win the next legislative election, put much pressure on them and made it widely known. As analysed by Franks Mitterand (the Socialist leader) himself, the pressure and its visibility proved helpful. He was seen as being able both to resist the Communists and to remain the champion of unity (see Giesbert, p. 38).

Although one usually likes other persons to express agreement, one may become suspicious of someone who would on every point


Mariano Rivera

Half-quadratic transportation problems

Keywords: quadratic transportation problems. 2017 General transportation problem, Half-quadratic potentials, Sequential quadratic programming

come

Half-quadratic transportation problems

Introduction

The general transportation problem (GTP) deals with the distribution of goods from m suppliers with production capacities p = {p i } i=1,...,m to n destinations with demands q = {q i } i=1,...,n . Without loss of generality, we assume balanced production and demand: i p i = j q j . A classical approach to this problem assumes that the cost of transport remains constant, independently of the the quantity to be transported. In real problems, this is not the case. The cost may increase or decrease according the volume of the transported good. We can write the general transportation problem as follows:

min x i,j f ij (x ij ) (1)
s.t.

j x ij = p i (2)
i x ij = q j (3)

x ij ≥ 0. ( 4
)
where x ij denotes the quantity of goods to be transported from the ith supplier to the jth destination, f : [m × n] × R + → R + is a continuous cost function that depends on the supplier i, the destination j and the volume x ij .

The first case of a transportation problem was formulated by Hitchcock assuming that the cost functions are linear: [START_REF] Hitchcock | The distribution of a product from several sources to numerous localities[END_REF]. Another popular model is the quadratic one:

f ij (x ij ) = c ij x ij
f ij (x ij ) = a ij x 2 ij + b ij x ij .
According to Ref. [START_REF] Adlakha | On the quadratic transportation problem[END_REF], the quadratic model is popular because it can approximate other cost functions. Despite such flexibility, the limitation of quadratic models has been well documented in the context of robust statistics, and its applications to image processing and computer vision [START_REF] Geman | Nonlinear image recovery with half-quadratic regularization[END_REF][START_REF] Charbonnier | Deterministic edgepreserving regularization in computed imaging[END_REF]. In our opinion, the main limitations of quadratic models are the following:

1. The impossibility of limiting the cost for large values; i.e., one has lim

x→+∞ |q ij (x ij )| = ∞.
2. The limitation of promoting sparse solutions; i.e., solutions that use a reduced number of routes.

In this work, we present an approximation scheme that allows us to define new cost functions that overcome the aforementioned limitations. We also present a primal-dual algorithm with limited memory requirements.

The GTP is relevant in modern computer science applications such as computer vision [START_REF] Rubner | The earth mover's distance as a metric for image retrieval[END_REF], machine learning [START_REF] Baccianella | Feature selection for ordinal text classification[END_REF] and data analysis [START_REF] Levina | The earth mover's distance is the mallows distance: some insights from statistics[END_REF][START_REF] Kusner | From word embeddings to document distances[END_REF]. The Earth Mover Distance (EMD) is an interesting application of the transportation problem where the optimum cost is used as a metric between the histograms, vectors, p and q. In Ref. [START_REF] Rubner | The earth mover's distance as a metric for image retrieval[END_REF], EMD is used as a metric for image retrieval in the context of computer vision. Recently, the EMD was proposed as a measure of reconstruction error for non-negative matrix factorisation [START_REF] Zen | Simultaneous ground metric learning and matrix factorization with earth mover's distance[END_REF]. The Word Moving Distance is the metric version for comparing documents based on the transportation problem [START_REF] Kusner | From word embeddings to document distances[END_REF]. Recent EMD applications include the quantification of biological differences in flow cytometry samples [START_REF] Orlova | [END_REF]. In addition, there is current interest in the learning of metrics for particular problems [START_REF] Cuturi | Ground metric learning[END_REF]; since our proposal is parametrised, the parameters involved can be learned.

Preliminaries

Before presenting our transportation formulation, we review an important result reported in the context of robust statistics and continuous optimisation applied to image processing [START_REF] Charbonnier | Deterministic edgepreserving regularization in computed imaging[END_REF]. The purpose of such work was to transform some non-linear cost functions to halfquadratic functions [START_REF] Geman | Nonlinear image recovery with half-quadratic regularization[END_REF][START_REF] Charbonnier | Deterministic edgepreserving regularization in computed imaging[END_REF]. A half-quadratic function is quadratic in the original variable and convex in a new auxiliary variable, where the minima of the auxiliar variable can be computed with a closed formula. The next proposition resumes the conditions imposed on the cost function f and the transformed half-quadratic function.

Proposition 1. Let f : R + → R + be a function that fulfils the following conditions: 3. f (t) ≥ 0.

1. f (t) ≥ m with f (0) = m, for t ≥ 0 and m > -∞. 2. f is continuously differentiable.
4. lim t→+∞ f (t)/(2t) = 0.

lim t→+0

+ f (t)/(2t) = M , 0 < M < +∞.
Then,

1. there exists a strictly convex and decreasing function ψ : (0, M ] → [0, β), where

β = lim t→+∞ f (t) -t 2 f (t)/(2t) such that f (t) = inf 0<ω≤M ωt 2 + ψ(ω) ; (5)
2. the solution to inf 0<ω≤M {ωt 2 + ψ(ω)} is unique and given by

ω * = f (t)/(2t). (6)
Proof. The proof is presented in Ref. [START_REF] Charbonnier | Deterministic edgepreserving regularization in computed imaging[END_REF].

Observe that our version of the half-quadratic Proposition assumes a non-negativity constraint on the primal variables.

Half-quadratic transportation problem

In this section we present a memory efficient primal-dual algorithm for solving GTP which cost functions satisfy Proposition 1.

Proposition 2. Let f ij be a cost function in [START_REF] Hitchcock | The distribution of a product from several sources to numerous localities[END_REF], such that each f satisfies Proposition 1; then, a solution to the transportation problem can be computed with Algorithm 1.

Algorithm 1 Half-quadratic transportation solver.

Require: i q i = j p j and f, p, q ≥ 0; 1: function hqTP(f, p, q) 2:

Initialise λ = γ = 1, ω ij = c ij and use ωi,j = 1/ω ij 3: repeat 4: repeat 5:
Update s with (20); Update ω with (10); 1. On the half quadratic transportation problem. By [START_REF] Rubner | The earth mover's distance as a metric for image retrieval[END_REF], the cost (1) can be rewritten as

min x ij f ij (x ij ) = min x,w ij {ω ij x 2 ij + ψ(ω ij )}. ( 7
)
2. On the algorithm convergence. Let L denotes the Lagrangian of the half-quadratic transportation problem, then one can interchange the order of the minimisations; i.e., min x,ω max y L(x, ω, y) = min x min ω max y L(•) = min ω min x max y L(•); where we denote with y the Lagrange's multiplies vectors. This suggests an alternating minimisation scheme w.r.t. ω and (x, y). Let x k , ω k and y k be the current feasible values, then we define

ω k+1 = argmin ω L(x k , ω, y k ) ( 8
)
to be the updated ω value. Thus, x and y are updated by solving the quadratic transportation problem:

x k+1 , y k+1 = argmin x argmax y L(x, ω k+1 , y). (9)
We define

F (x) = ij f ij (x ij ) and F (x, ω) = ij {ω ij x 2 ij + ψ(ω ij )} and observe that F (x k ) = F (x k , ω k ) ≥ F (x k , ω k+1 ) ≥ F (x k+1 , ω k+1 ) = F (x k+1
). Then, the alternated minimisations w.r.t. ω and x produce a feasible convergent sequence {x k , x k+1 , x k+2 , . . .} that reduces the cost of the GTP:

F (x k ) ≥ F (x k+1 ) ≥ F (x k+2 ) ≥ . . ..

3.

On the alternated minimisations. From ( 6), the optimum ω in ( 8) is computed as

ω ij = f ij (x ij )/(2x ij ). ( 10
)
for a given x. We define y equal to (λ, γ, s) where λ and γ are the Lagrange's multipliers for the equality constraints ( 12) and ( 13), respectively; and s are the Lagrange's multipliers for the non-negativity constraint. Then, the minimisation (9) corresponds to finding the vectors (x, λ, γ, s) that solve the Karush-Kuhn-Tucker conditions (KKTs) with ω fixed:

ω ij x ij -λ i -γ j -s ij = 0, (11)
j x ij -p i = 0, ( 12
) i x ij -q j = 0, (13)
s ij x ij = 0, (14)
s ij , x ij ≥ 0, (15)
A strategy for solving the KKTs is to use an iterative Projected Gauss-Seidel scheme [START_REF] Morales | An algorithm for the fast solution of symmetric linear complementarity problems[END_REF]. Thus, from [START_REF] Cuturi | Ground metric learning[END_REF]:

x ij = ωij (λ i + γ j + s ij ) , (16)
where we define ωij = 1/ω ij . Substituting x ij in (12), we have

j ωij (λ i + γ j + s ij ) = p i . (17)
We solve for λ i and obtain

λ i = p i -j (γ j + s ij )ω ij j ωij . (18)
Similarly, we substitute x ij in (13) and solve for γ j :

γ j = q j -i (λ i + s ij )ω ij i ωij . ( 19
)
From ( 11), ( 14) and ( 15), we see two cases:

x ij = 0 and s ij = -(λ i + γ j ) ≥ 0; or x ij = ωij (λ i + γ j ) ≥ 0 and s ij = 0. Thus s ij = max{0, -(λ i + γ j )}. ( 20
)
The complete procedure is shown in Algorithm 1.

In practice, we observed that the internal loop in Algorithm 1 requires only a few iterarations to approximate the dual variables and is not necessary to achieve convergence; in our experiments we used five iterations. The solution is the global minima or a local one by depending if the cost is or is not convex. Following, we present two particular and interesting cases of half-quadratic transportation problems.

Proposition 3 (Approximation for Linear Transportation). The linear cost can be approximated with the differential function [START_REF] Charbonnier | Deterministic edgepreserving regularization in computed imaging[END_REF]:

f ij (ij) = c ij (x 2 ij + β 2 ) 1 2 , with β ≈ 0. Thus, ω is computed with ω ij = 1 2 c ij /(x 2 ij + β 2 ) 1 2 in Algorithm 1.
Proof. It follows from

lim β→0 c ij x 2 ij + β 2 1 2 = c ij x ij ; c ij , x ij ≥ 0. ( 21
)
Hence, the formula for ω follows directly from [START_REF] Orlova | [END_REF].

Proposition 4 (Approximation for L 0 Transportation). The cost function of the form

f ij (x ij ) = c ij [1 -δ(x ij )], for c ij , x ij > 0 (where δ(t)
is the Kroneker's delta) can be approximated with the half-quadratic function:

fij (x ij ) = c ij x 2 ij /(β 2 + x 2 ij ) (22)
with β ≈ 0. Thus,

ω ij = c ij β 2 /(β 2 + x 2 ij ) 2 .
Proof. It follows from lim

β→0 t 2 β 2 + t 2 = 1 -δ(t) = 0 for t = 0, 1 for t ∈ R\{0}. (23)
The formula for ω follows directly from [START_REF] Orlova | [END_REF].

Figure 1 plots the half-quadratic functions that approximate the L 1 and L 0 norms.

Relationship with the quadratic transportation problem

Proposition 5 (Simple Quadratic Transport (SQT)). The SQT problem is defined by the cost function

f ij (ij) = c ij x 2 ij ; thus, ω ij = c ij .
Proof. It follows directly from [START_REF] Orlova | [END_REF].

Proposition 6 (Simple Quadratic Transportation Algorithm). In the case of the QT problem, the Algorithm 1 is reduced to the dual Algorithm 2.

Algorithm 2 Simple quadratic transportation solver.

Require:

i q i = j p j and c, p, q ≥ 0; 1: function qTP(c, p, q) 2:

Initialise λ = γ = 1 and use ωi,j = 1/c ij 3:

repeat 4:
Update s with (20); return [x, λ, γ]; 10: end function Proof. From Proposition 5, we note that ω is constant, and that the computations of λ, γ and s are independent of x.

Dorigo and Tobler discussed the relationship between the QTP and the push-pull migration laws implemented in Algorithm 2 [START_REF] Dorigo | Push-pull migration laws[END_REF].

Proposition 7 (Quadratic Transportation (QT)). The QT is defined by a cost function of the form

f ij (ij) = a ij x 2 ij + b ij x ij .
Thus, the dual algorithm is derived with ω = c ij and using the condition

ω ij x ij -λ i -γ j -s ij = b ij (24)
instead of [START_REF] Cuturi | Ground metric learning[END_REF].

Proof. It follows directly from the KKTs.

Remark. An alternative to Proposition 7 is given by the half-quadratic approximation

a ij x 2 ij + b ij x ij ≈ a ij x 2 ij + 2b ij (x 2 ij + β 2 ) 1 2 , with β ≈ 0; thus, ω ij = a ij + 1 2 b ij /(x 2 ij + β 2 ) 1 2
. This approximation is presented with the sole aim of illustrating the potential of our approach. It is clear that the dual algorithm derived according to Proposition 7 is more accurate, faster and requires less memory to be implemented.

Discussion and Conclusions

The transportation problem is the base of the Earth Mover Distance which has become a relevant metric to for compare distributions in applications to data analysis and computer vision. The presented technique can motivate the design of new algorithms in those areas.

In order to demonstrate the versatility of our proposal, we generate two random vectors p and q (depicted in Figure 2); we compute the optimum transported volumes x with three cost function models: the quadratic (c ij x 2 ij ), the approximation L 1 (c ij x 2 ij + β 2 , with Figure 2: Randomly generated levels of production, p, and demand, q.

β 2 = 1 × 10 -3 ) and the approximation L 0 (c ij x 2 ij /(β 2 + x 2 ij ), with β 2 = 1 × 10 -1 ). In all the cases, we use c ij = |i -j| + 1. Figure 3 depicts the computed x values. One can observe that the quadratic cost function promotes dense solutions; i.e., there are many x's with small values. On the other hand, one can observe the sparseness of the solution is induced with the use of the approximated L 1 -norm. Such sparsity is emphasised with the approximated L 0 -norm.

We have presented a model to approximate solutions to the general transportation problems by approximating the transportation cost functions with half-quadratic functions. The approach guarantees convergence using an alternated minimisation scheme. In the case of a non-convex cost function f the convergence is guaranteed to a local minimum. Although we present a minimisation algorithm with reduced memory requirements, our scheme accepts other efficient solvers for the quadratic transportation subproblem; such as those reported in Refs. [START_REF] Adlakha | On the quadratic transportation problem[END_REF][START_REF] Megiddo | Linear time algorithms for some separable quadratic programming problems[END_REF][START_REF] Cosares | Strongly polynomial algorithms for the quadratic transportation problem with a fixed number of sources[END_REF].

Figure 1 :

 1 Figure 1: Plots of half-quadratic cost functions (for x ≥ 0): the L1-norm approximation x 2 + β 2 and the L0-norm approximation x 2 /(x 2 + β 2 ).

  return [x, λ, γ]; 13: end function Proof.

Figure 3 :

 3 Figure 3: Computed transport values x (in grayscale) corresponding to (a) quadratic cost, (b) half-quadratic L 1 -approximation and (c) half-quadratic L 0 -approximation. The solution computed with the non-convex cost function (L 0 -approximation) depicted in panel (c) is sparser.