Alain Marciano

Sado-Masochism in Buchanan's Samaritan's Dilemma. A Constitutional Perspective $

Keywords: Altruism, Samaritan's dilemma, Buchanan, Exploitation, Sadism, Masochism, Cooperation JEL codes -C72

In this paper, we study the impact of altruism on an interaction between a samaritan and a recipient/parasite in the frame of Buchanan's samaritan 's dilemma (1975). We show that, as soon as altruism reaches a certain threshold, the equilibrium of the game corresponds to the situation Buchanan called a samaritan's dilemma. We also show that the Nash equilibrium reached for these levels of altruism is a Pareto-efficient outcome. Thus, the situation Buchanan characterized as a samaritan's dilemma is not a dilemma at all. Both players are satisfied with the situation as it is and need each other, up to the point of giving birth to a sado-masochistic equilibrium. We also show that this result holds if and only if the constitutional rules are given-either the ethical rules followed by the individuals, or the form of the game. This equilibrium could be avoided if the players adopted a constitutional perspective on the situation.

Introduction

Samaritanism, benevolence, helping people in need, trying to relieve their grief through aid and donations, may have negative rather than positive consequences. First, on the beneficiaries of such help. Indeed, as it has been put forward in a rather large literature, helping people may contribute to maintaining them in a state of poverty, dependent on aid and assistance. It reduces the recipient's willingness to work-as Burns put this rather old concern, "aid may discourage self-reliance" (2009, p. 1)-as well as the incentives to invest in productive behaviors (Williamson, 2010). In addition, these negative effects tend to spread through strategic behavior and imitation-"[g]iving aid to those who make impoverishing choices will encourage others to do the same, thereby worsening the problem" (Wagner, 2005), or "there is a strong incentive in these situations for an endogenous increase in the supply of those who will receive the benefits from our policy." (Boettke and Martin, 2010) Helping creates a situation of "tragic commons" (Schmidtz, 2000, 686). Why? The reason seems straightforward. Help transforms the incentives the recipients face and creates a situation of "moral hazard"-also called a "charity hazard" (see Andor, Osberghaus and Simora, 2017;

Raschky and Weck-Hannemann, 2007; Browne and Hoyt, 2000): because they feel insured against problems by the presence of samaritans, recipients for instance work or save less or increase risky behaviors. Even if the evidence is not always clear, the phenomenon seems frequent.

Part of this literature was influenced by James Buchanan's essay "The samaritan's dilemma" (1975)-that seems to be the first time the expression was used. In his work, Buchanan pointed to another problem: samaritans can be durably hurt as well. According, to Buchanan, samaritans are caught in a situation in which they help-via, for instance, some transfer of money-someone who takes the money but nonetheless does not "return the favor" by "cooperating" with the donor's wishes. The recipient-to Buchanan, a "parasite" (1975)-"refuse[s] to work" (Buchanan, 1975, 76). [START_REF] Andreoni | Impure Altruism and Donations to Public Goods: A Theory of Warm-Glow Giving[END_REF] He "exploits"-still Buchanan's term-the samaritan. The latter is dissatisfied with "the absence of work on the part of the recipient" (Buchanan, 1975, 76) because she precisely hoped and wished that her charity would have led the recipient to change his behavior and work. This asymmetrical situation-the samaritan helps, the beneficiary does not work-could be ended if the samaritan decided not to behave charitably [START_REF] Basu | The moral basis of prosperity and oppression: altruism, other-regarding behaviour and identity[END_REF] . Choosing this course of action would suppress the samaritan's dissatisfaction with seeing her altruism not reciprocated and, in Buchanan's eyes, would incite the recipients to change their behavior towards working. Yet, according to Buchanan, this process is costly: the samaritan would suffer from seeing the recipient suffer, and also from a loss of utility because of the impossibility of satisfying her altruistic preferences. Hence, the dilemma the samaritan faces: either she helps someone who by that help is induced to exploit the helper, or she does not help but suffers from the cost of not helping.

Buchanan did not explain the origins of the game. He assumed its structure and payoffs, because his objective was to analyze the consequences of the situation and explain how to escape from it. He did not even make specific assumptions regarding the origins of the samaritan's action. In particular, he did not assume that samaritans are altruists. [START_REF] Battigalli | Dynamic Psychological Games[END_REF] Yet, altruism is most of the time, if not always, associated with Buchanan's samaritan's dilemma. This is why we also assume that samaritans are altruist. But, then, a question arises: does the samaritan's dilemma still exist if it is assumed that the samaritan is an altruist? This is what we discuss in the present paper, in a more illustrative than demonstrative way. [START_REF] Battigalli | Psychological Game Theory[END_REF] We show that the asymmetric situation that corresponds to Buchanan's samaritan's exists if-and as soon as-altruism reaches a certain level. However, we also show that the Nash equilibrium reached for these levels of altruism is a Pareto-efficient outcome. This means that the situation Buchanan characterized as a dilemma is not a dilemma at all. Indeed, neither the samaritan nor the recipient would want to play otherwise.

To be more precise, and to be consistent with in Buchanan's distinction between within rule choices and choices of rules of a game, what we actually show is that there is no dilemma as long as, and because, the samaritan and the recipient play a (given) game by following their own (given) subjective preference terms. Our result holds, and the samaritan is satisfied with the situation in which she helps, and the recipient is satisfied with making no effort, because the two individuals choose within given rules-which technically comprise the subjective preferences of the actors. This was the situation Buchanan found problematic in the US in the early 1970s.

Samaritans were (myopically) enjoying their altruism. We then suggest that the situation can be interpreted (rather provocatively) the situation as a sado-masochistic equilibrium: both the samaritan needs people to help and the beneficiary needs helpers. It is even worse (or less desirable) than what Buchanan diagnosed. But, precisely because we reason in terms of game-forms, of within rule choices and choices of rules, we can also suggest that instead of being trapped in this sadomasochistic equilibrium, the players could also envisage to change the rules. Put differently, the Samaritan and the parasite would have reason to wish that the rules of the game were different.

They could try to find a constitutional solution to the situation. From this perspective, "The Samaritan's Dilemma" must not (only) be interpreted as a message of despair but also as a plea for a change in the rules of the game.

Sympathy for strangers and the conditions of cooperation

To be as general as possible, we simply consider a large population of individuals, is and js, paired to interact in a one-shot prisoner's dilemma game. Let us pause here to make two important remarks about this setting. First, we assume that individuals are non-embedded in an institutional structure. The reason is that we want to isolate the impact of altruism and separate it from the role institutions may play. Stated differently, our objective is to explain exploitation induced by individual preferences and not to link it to, or explain it by, a specific institutional structure (as done by Kaushik Basu, 2010). Second, we study only interactions that are not repeated. [START_REF] Becker | A theory of social interactions[END_REF] Changing from one-shot to repeated interactions would necessarily modify the dominance properties of the repeated game strategies and therefore the strategic nature of the interaction (backward induction being an additional assumption anyway). We want to focus on "interactions" such as the one taking place between an individual and, for instance, an unknown beggar or the unknown victim of a natural catastrophe.

Our starting point is the game depicted in Table 1. Here, the row-and the column-player both (privately) have two strategies. They can choose a generous, G, or a selfish, S, action. The numbers in each cell are the material payoffs the two players receive in their interaction. At this stage, Table 1 is the objective description of the interaction between the two individuals. It is assumed initially that their subjective perceptions of the game and their subjective preferences coincide with the objective payoffs in their 'natural ranking'. [START_REF] Becker | Altruism, Egoism, and Genetic Fitness: Economics and Sociobiology[END_REF] Table 1. The initial game, a prisoners' dilemma

Player j's options Player i's options G j S j G i (3, 3) (0, 5) S i (5, 0) (1, 1)
As is familiar, in such a prisoner's dilemma game mutual defection is the unique Nash equilibrium in strictly dominant strategies. This is the only pure strategy combination which is not Paretoefficient and strictly payoff-dominated by mutual cooperation. [START_REF] Becker | A Treatise on the Family[END_REF] Remaining with an interaction characterized by this game-form, we move to a different game if we introduce other-regardingness and assume that individual i is characterized by an altruism parameter that represents her concern for the welfare of the other person, j, with whom she interacts. This parameter is α ij and is ≥ 0.9 The degree of sympathy, altruism or other-regarding concern is represented as the weight with which i values the objective payoffs earned by j relative to her own in her subjective ranking of results. In the emerging game, it is no longer assumed that the self-regarding objective or substantive payoffs alone represent the preference ranking. If Π i and Πj denote the material and objective payoffs received by i and j, then the induced subjective payoff of i is given by:

Π i = [1 -α ij ].Πi + α ij .Πj. 10 (1)
This means that, although the game-form-in our case a prisoner's dilemma game-form-does not change, the game individuals play is different because they play it as they perceive it. The game is depicted in Table 2, where α ij corresponds to the degree of altruism i feels towards j and α ji is the degree of altruism j feels towards i. It is the game altruistic individuals play when the game is the one given in Table 1.

Table 2

Player j's options Player i's options

G i S i G j (3; 3) (5α ij ; 5[1 -α ji ]) S j (5[1 -α ij ]; 5α ji ) (1; 1)
The outcome of the game obviously depends on the values of α.11

Let us start by assuming that the society is perfectly homogenous, at least in terms of altruism and concern towards others. This can be viewed as a benchmark situation in which

α ij = α ji = α.
Then according to equation (??), this leads to the symmetric "psychological game" depicted in

G i S i G j (3; 3) (5α; 5[1 -α]) S j (5[1 -α]; 5α) (1; 1)
One immediately sees that the outcome of the game depends on two threshold values, 1/5 and 2/5, that determine three possible situations. [START_REF] Boettke | Exchange, production, and samaritan dilemmas[END_REF] This is straightforward but useful for the discussion we propose in the next sections. First, if α < 1/5, the game remains a prisoner's dilemma. Defection is a strictly dominant strategy. Individuals with a degree of altruism towards others α < 1/5 are unconditional defectors, even though they feel a certain concern for others. Their degree of altruism is too low or insufficient for the 'psychological' game to be of a different strategic nature than the 'material' game. Second, at the other end of the spectrum, if α > 2/5, then cooperation is strictly-dominant; individuals are unconditional cooperators. Third, for intermediary values of α-between 1/5 and 2/5, individuals are conditional cooperators: cooperation is the unique best response to cooperation and defection is the unique best response to defection. In addition, such individuals cooperate if they expect their opponent to cooperate with probability greater than 3-5α 1+α . The prisoner's dilemma is turned into a chicken game in which there are two asymmetric strict Nash equilibria-(G i , S j ) and (S i , G j )-and one symmetric mixed equilibrium-where both individuals choose G with probability 5α -1.

The paradox of non-reciprocal altruism: from cooperation to exploitation

Let us now turn to the situation that we want to analyze more specifically, namely when an interaction takes place between a moral-or altruist or benevolent-and a non-moral-or egoist and selfish-individual randomly paired with each other. [START_REF] Brennan | Voter Choice and the Evaluation of Political Alternatives[END_REF] The latter, j, cares for his own payoffs only-that is, α ji = 0, ∀i-and thus Πj = Πj. By contrast, the moral individual, the samaritan, i, is such that α ij > 0. She is affected by j's welfare and takes his payoffs into account. Therefore,

Πi = [1 -α ij ].
Πi + α ij .Πj. She will not base her choice on the material payoffs she will receive but on her subjective rankings of results of outcomes of play.

To simplify the presentation and, since α ji = 0, we denote α ij = α.

i's options G j S j G i (3; 3) (5α; 5) S i (5[1 -α]; 0) (1; 1)
In this situation, the non-moral individual j still has a dominant strategy to defect whatever the moral individual i does. The strategy the latter is going to choose depends on how much she cares about the other player, i.e. depends on the value of α. The threshold values are the same as above, 1/5 and 2/5. When α < 1/5, the game remains a prisoner's dilemma with a strictly dominant defection strategy for both players. The second-when 1/5 < α < 2/5-and third situations-in which α > 2/5-represent particularly interesting and problematic outcomes that result from an interaction between a moral and a non-moral individual. When 1/5 < α < 2/5, a moral player not anticipating that the non-moral co-player has a dominant strategy of defecting could in principle choose either to cooperate or to defect. Anticipating the behavior of the co-player the moral player will, however, for 1/5 < α < 2/5 choose to cooperate. When α > 2/5 cooperation is a strictly dominant strategy for the moral actor who then need only know her own payoffs. 15 [START_REF] Brennan | Voter Choice and the Evaluation of Political Alternatives[END_REF] We analyze situations in which an individual characterized by a degree of altruism, α, different from 0, face another individual characterized by an α equal to 0. This is a simplification used to analyze situations in which individuals are characterized by positive but different α. It does not change the nature of the reasoning. [START_REF] Burns | Solving samaritan's dilemmas in Irrigation Investment[END_REF] As a corollary, one may note that cooperation does not require to go as far as to follow the Biblical teaching "love your neighbor as yourself" (Leviticus, 19:18). "[E]nlightened self-interest" (Simon, 1983) or "imperfect altruism" (Stark, 1989) are sufficient to lead individuals to cooperate with others.

Whenever α > 1/5, there is only one asymmetric strict Nash equilibrium-(G i , S j ). It corresponds to a situation in which the moral individual cooperates while the non-moral individual defects and the game is no longer a standard prisoner's dilemma but rather corresponds to a samaritan's dilemma. Or, to be more precise, the equilibrium of the game corresponds to the asymmetrical situation in which the samaritan cooperates and the parasitic recipient defects. Thus, this equilibrium is reached whenever a sufficiently altruistic samaritan interacts with an egoist. Indeed, the exploitation equilibrium is the consequence of an altruistic preference of the samaritan. Does this equilibrium correspond to Buchanan's samaritan's dilemma? To answer the question, one must recall that, to Buchanan, the dilemma did not result from the existence of this asymmetric situation but from the impossibility to escape from it. In the next section, we show that indeed the situation is stable. Neither the samaritan nor the recipient want it to change.

Exploitation as mutual advantage, and the end of the samaritan's dilemma

Let us start by noting that, as soon as α > 1/5 then Π i (G, S) > Π i (S, S), and the samaritan's utility is greater than it would be if she chose to behave selfishly. Moreover, the game depicted in Table 3 shows that, when α > 3/5, then i's gains if j plays S are greater than her gains if j plays G. This means that the samaritan prefers (G i , S j ) to (G i , G j ). And this can, also, be easily be explained. If obliged to cooperate, the recipient would get 3 instead of 5. But he would not be the only one to be affected. The samaritan would be too: (G i , G j ) corresponds to a situation in which the samaritan also suffers from the loss of utility of the recipient. This means that samaritans with a relatively high degree of altruism-α > 3/5-prefer to interact with a selfish recipient who behaves selfishly than with a selfish beneficiary who would react to her altruism and behave generously as it would be the case if Becker's Rotten Kid theorem applied. One may note that the more altruistic the samaritan is-the purer her altruism-the less she is interested in any form of cooperation or reciprocation from the recipient. Again, the stability that was required by Buchanan to characterize a samaritan's dilemma exists.

More broadly, when α > 3/5, the outcome (G i , S j ) appears to be preferred by both the moral and the non-moral individual, the samaritan and the recipient. Exploitation is thus the samaritan's and the parasite's preferred outcome. Therefore, exploitation is Pareto-efficient. Both the samaritan and the recipient would be made worse off by any attempt to oblige the recipient to choose strategy G, that is to behave non-selfishly. The same stability of the asymmetrical equilibrium is verified.

But, a particularly important corollary has to be added: here, with an altruist samaritan, the stability of the off-diagonal outcome has not the same origin as in Buchanan's model. To Buchanan, exploitation implied a dissatisfaction for the samaritan. This is not what we have obtained. When α reaches a certain threshold, the samaritan does not want to change her behavior and does not want the recipient to behave differently within the rules of the given game. The samaritan has no desire to end a situation that she perceives as favorable. In other words, by contrast with what Buchanan wrote, there is no dilemma. This arises from the fact that the samaritan receives some subjective, symbolic benefits from being benevolent and these gains can be really important for people with a high degree of concern for others. This is the warm glow effect, already emphasized in the literature on altruism (see Andreoni, 1990, for a seminal work on this issue). This means that altruism cannot be viewed only for its incentive dimension (to some extent, this is a problem that comes from the first economic studies, in particular Becker's about altruism that envisage altruism as conditional). Benevolence has also a consumption value. The samaritan is perfectly satisfied with the asymmetric situation in which she finds herself. Indeed, there is no dilemma. Or, rather, what Buchanan interpreted as a dilemma is just a stable equilibrium that satisfies both players. This does not however mean that they should be satisfied. Although the exploitation equilibrium is not a dilemma, it is nonetheless problematic.

Masochism and sadism in the samaritan's dilemma

If we assume that both individuals follow their own subjective preferences-do not try to be a different person-in a set of given rules, it appears that exploitation in objective terms is needed to increase both player's subjective satisfaction. More precisely, samaritans need parasites as much as parasites need samaritans to realize mutual advantage. That reciprocal and mutual need creates a very specific sado-masochistic relationship. Indeed, it can obviously be said that the samaritan is willing to, and even 'enjoy', reducing her objective game form payoff to allow an increase in the objective payoff and utility of the recipient. Such a behavior could be said to reveal a form of masochistic preferences. This echoes a point emphasized by Elias Khalil about Becker's altruism and its 'unavoidable' masochistic dimension (2001, 2004a; see also Nida-Rümelin, 1991). Khalil argued that "Becker cannot analytically distinguish between the actions of altruists and masochists. Both actions involve the reduction of one's welfare for the purpose of enjoying vicariously the welfare of others." (2004b, 143) As he explained more precisely, "Becker's theory is properly a good theory to explain why the masochist (the altruist) is ready to suffer some loss or some pain so that the sadist (the beneficiary) can have some enjoyment: as the sadist is having some enjoyment, the masochist feels pleasure via mirroring or reflexivity. If we think that this mechanism is what takes place in the sadist-masochist relation, Becker's theory is saying that the altruist-beneficiary relation is no different." (Khalil, 2004a, 102) Viewed this way the samaritan cares sufficiently for the recipient to sacrifice her objective payoff.

The utility ('pleasure') she derivess from sacrificing her payoff is compensated by the pleasure obtained by the parasite. [START_REF] Browne | The Demand for Flood Insurance: Empirical Evidence[END_REF] This masochism turns into or is complemented by a form of sadism. And this is the second result of the game-illustrated in Table 3. Since the samaritan needs parasites to exploit her to reach a subjectively preferred situation, this means that the samaritan prefers that recipients remain in the situation of being helped. Her masochism, that leads her to sacrifice objective payoff to see others enjoying more utility (respectively higher objective payoffs), also leads her to be a sadist-in the sense that she may well "welcome tragedies" (Khalil, 2004a, 102). The failure to distinguish between objective and subjective payoffs -respectively, the within rule and the choice of rules constitutional perspective -leads Khalil to diagnose the emergence of "absurd results": "[t]he altruist qua masochist may not abhor natural disasters befalling others. While such an altruist may refrain from expediting disasters, he would celebrate the opportunities such disasters afford him." (2004, 102) Or, in other words, "Becker's model entails that altruists should feel joyful over the prospect of the miseries of others because such miseries occasion for them the opportunity to be aroused." (Khalil, 2001, 431) That is not all. There is sadism from the parasite too. Since exploitation can be said to be "mutually advantageous"-preferred by both the samaritan and the parasite-, this then means that the parasite wants the samaritan to be exploited in terms of a loss of objective payoffs she receives. This obviously makes the parasite a sadist too.

The sado-masochist equilibrium in a constitutional perspective

Buchanan made two points of importance in his essay that complement each other, and can be seen in constitutional terms. First, he insisted on the importance of strategic courage, of the need to assume one's responsibilities as a Samaritan-and this includes not being an altruist in the short term to secure long term benefits. Now, ethical rules are constitutional rules, at least if one follows Buchanan. Then, one way to avoid or prevent the samaritan's dilemma or rather the sadomasochistic equilibrium is that individuals adopt new ethical rules. That they become "better" individuals (see Buchanan, 1979, 100). Or, as Buchanan wrote it: "the argument suggests the appropriateness of adopting rules for personal choice behavior as opposed to retaining individual flexibility of action." (1975, 75) That was the problem Buchanan wanted to address. It was because individuals were no longer able to refrain from these short-term satisfactions that the dilemma existed-which is what we have demonstrated. To Buchanan, in the Western societies of the early 1970s, there no longer existed the individual ethic of responsibility, very similar to the "Puritan ethic" that consists "of inhibiting personal behavior that was aimed solely at gratification of instant desires, whether these be charitable or selfish." (1975, 80) Or, in the less positive terms Buchanan used, the "modern man has 'gone soft.'" (1975, 75) Much to the contrary. Buchanan wrote: "The quasi-revolutionary shift in modern behavioral standards that widespread adherence to the responsibility ethic would represent does not seem likely to occur. Indeed all signs point in the opposing direction, and we shall probably witness a continuing erosion in strategic courage at all levels of decision." (1975, 82) [START_REF] Bruce | The Rotten-Kid Theorem Meets the samaritan's dilemma[END_REF] . Individuals, in particular samaritans, are not ready to assume the responsibility that necessarily goes with being a samaritan. To paraphrase what Buchanan would write later (2005), they are afraid to be responsible. [START_REF] Bruce | Transfers in Kind: Why They Can Be Efficient and Nonpaternalistic[END_REF] This fits with Buchanan's perhaps rather irritating diagnosis that refusing to spank a child out of a myopic interest in avoiding the displeasure of that act may not be in the best interest of the child. Independently of the precarious empirical assumption that spanking is an effective educational instrument of punishment, there is the very general problem that punishing may be an altruistic act of overcoming the short run subjective resistance of the punisher against such acts. This indicates a subjective dimension of altruism in so-called 'altruistic punishment' that has recently become popular in behavioral and experimental economics as a term referring exclusively to objective game-form payoffs.

Second, he also claimed that, lacking such a courage, that is assuming that they could not choose the subjective preferences that would become operative in within rule choices directly, the samaritan and the parasite would have reasons to alter the game in a constitutional choice of different rules of the game-form. He thus wrote that "There may be no escape from the generalized samaritan's dilemma, in its public form, except through the collective adoption and enforcement of rules that will govern individual situational responses." (1975, 81) Although "such rules must be coercive", individuals "may... freely agree to their adoption at some prechoice, or constitutional stage of deliberation. Indeed, if the public form of the dilemma is a genuine one, it will be in the potential interest of most members of the community to adopt some such rules." (1975, 81-82) This is precisely the point we put forward in this article. If 'rules of the game' are understood this way, two classes of rules of games can be distinguished. On the one hand, there are those belonging to the objective game-form and on the other, there are the subjective rules (basically the given 'beliefs and desires') or subjective preferences as represented in the subjective utility functions. The game arises from 'combining' these two separate aspects of what can not be changed by within rule choices in plays of the game. From a constitutional perspective in which players imagine that they can make choices of the rules of the games they play this raises evaluative issues in which Buchanan is mainly interested.

Concretely, the samaritan is "exploited" in terms of the objective payoffs of the game-form but in terms of the game she plays with her given subjective preferences and the given game-form she is also subjectively better off than if she would choose against her subjective preferences within the constraints of the given game-form. Assuming that she cannot choose her own utility function (Frank 1987), the samaritan has reason to wish that the game-form be altered in a Buchanan type constitutional choice. Adopting Buchanan's constitutional point of view, she can understand that she has reasons to wish that the situation ends once she understands the logic of the model Buchanan lays out in his samaritan's dilemma. Indeed, the intuitions Buchanan expressed in his discussion of what he called the samaritan's dilemma were guided by his fundamental distinction between a constitutional choice of the rules of a game and choices within the rules of a game.

Conclusion

In this paper we have studied the consequence of altruism on an interaction between a samaritan and a recipient/parasite in the frame of Buchanan's samaritan's dilemma (1975). It appears that, as soon as altruism reaches a certain threshold, the equilibrium of the game corresponds to the equilibrium of what Buchanan called a samaritan's dilemma. It is the situation in which the samaritan cooperates and the recipient defects in objective game-form terms. For Buchanan, the dilemma arises from the inability of the samaritan to stop being a samaritan as long as she is playing a pd game-form according to her own other-regarding subjective preferences. This is the consequence of the 'psychological' benefits the samaritan derives from being an altruist. Without the tensions between those subjective benefits which guide within rule choices and the objective payoffs that accrue according to the game-form and could be realized only by a constitutional choice of changing the rules of the game, there would be no dilemma: both players prefer exploitation to any other situation; they need each other, which gives birth to a sado-masochistic relationship. This confrontation between a myopic perspective of within rule choices in a game and the constitutional perspective of choosing the rules of a game was what Buchanan clearly had in mind.

In this sense we confirm Buchanan's results. It seems also clear that there is a fundamental difficulty as long as we assume that preferences on the level of within rule choices and the level of the choice of rules are the same. An easy constitutional solution to the dilemma only emerges if the samaritan's subjective preferences and interests on the constitutional level diverge from the direct interests that guide the samaritan in her within rule choices (a topic Buchanan sought to address later in indirect ways jointly with Geoffrey Brennan and Viktor Vanberg respectively). Once we distinguish between subjective rankings of outcomes in a game and outcomes in a game-form and assume that on the constitutional level objective interests matter the samaritan's form of altruism can be diagnosed as encouraging 'unhealthy' tendencies in the giver as Buchanan clearly hinted. This is, again, perfectly consistent with how Buchanan analyzed the concept of Pareto optimality.

In "The Relevance of Pareto Optimality" (1962), Buchanan explained that economists used the concept of Pareto optimality independently from the "'constitutional' limits" (1962, 343) that should be placed on the region in which optima were located, as if all Pareto optima were acceptable or, put differently, as if the concept could be used as absolute criterion. He disagreed with that view, wondering if it made sense to use the criterion to qualify a situation in which "one man could make all his fellow citizens his slaves?" (1962, 343) Obviously, it could be optimal, but this did not mean that it was acceptable. In his view, a "[m]eaningful use of the Pareto criterion, for

Table 2 -

 2 bis.[START_REF] Buchanan | Natural and Artifactual Man[END_REF]

Table 2

 2

-bis Player j's options Player i's options

Table 3 .

 3 Non

-reciprocal altruism Player j's options Player

[START_REF] Buchanan | The samaritan's dilemma[END_REF], The parasite has two strategies, "work" and "refuse to work".

Letho and Meadowcroft (2021) insist on how important were these off-diagonal outcomes, or asymmetrical equilibria, for Buchanan after a while.[START_REF] Marciano | The Political Economy of Buchanan's Samaritan's dilemma[END_REF] shows that the samaritan's dilemma was present in Buchanan's work since the late 1950s.

[START_REF] Battigalli | Dynamic Psychological Games[END_REF] Although "The samaritan's dilemma" was published in a collective volume on altruism, Buchanan did not write it with altruism in mind. When he wrote his article, in the early 1970s, Buchanan was concerned with other issues that had not much to do with altruism (see[START_REF] Fleury | The Making of a Constitutionalist: James Buchanan on Education[END_REF]

).[START_REF] Battigalli | Psychological Game Theory[END_REF] For a more detailed analysis of how it is possible to make sense of Buchanan's dilemma with altruist samaritans, see[START_REF] Dughera | Altruism and Strategic Courage. Inside Buchanan's samaritan's dilemma[END_REF]

For an analysis of the samaritan's dilemma with repeated interactions, see for instanceSchmidtchen, 1989.

Using different numerical values would not change the logic of the results. And, since we are only concerned with players' ordinal preferences over outcomes, the value of an individual's degree of altruism is allowing us to generate alternative outcome rankings in a comprehensive way.

As Buchanan suggested many times, in a standard prisoner's dilemma game-form, a purely subjective re-ranking of the moves that renders the co-operative choices subjectively dominant would yield also the result that makes both actors objectively better off in terms of the game-form.

Edgeworth was the first to explicitly consider sympathetic utility functions through the use of an "effective coefficient of sympathy". He wrote that "between the frozen pole of egoism and the tropical expanse of utilitarianism [there is] the position of one for whom in a calm moment his neighbour's utility compared with his own neither counts for nothing, nor 'counts for one', but counts for a fraction" (1881, p.102). See Collard (1975) for a detailed account of Edgeworth's treatment of altruism. The use of such linear forms of altruistic utility functions is frequent: see[START_REF] Becker | A theory of social interactions[END_REF], Bernheim and Stark (1988), Bruce and Waldman (1990), Montgomery (1994), and Mulligan (1993, 1997), Rescher (1975), Rotemberg (1994). Levine (1998) or Sally (2001) use a different form of altruistic utility function.

Sally (2001, pp. 7-8), criticizes this "usual linear form" because "an altruistic person is subject to the samaritan's dilemma in a multi-period setting, as demonstrated by Bruce and Waldman". Our point is precisely that there is no need to envisage a "multi-period setting" to end up with a samaritan's dilemma. This is why we adopt this form.

The degree of moral concern plays no role when the two individuals gain the same payoffs. When payoffs differ, then altruism affects the objective function and, hence, the behavior of the moral individual.

On "psychological game theory" see Geanakoplos, Pearce and Stacchetti (1989), andBattigalli and Dufwenberg (2009, 2019). An important difference must be noted here between these works, as well as with what we do in this article. Battigalli and Dufwenberg assume that "players' utility depends on endogenous beliefs" (2019, 4). As they note, Geanakoplos, Pearce, and Stacchetti's analysis "only encompasses utilities that depend on player's initial hierarchical beliefs." (2019, 7; italics in original). Indeed, Geanakoplos, Pearce, and Stacchetti noted: "The principal distinguishing characteristic of what we call a psychological game is that the players' payoffs depend not only on what everybody does but also on what everybody thinks." (1989, 61; italics in original) Our approach differs and is less sophisticated than the one adopted by these authors. We assume that neither the Samaritan nor the recipients modify their beliefs, when they play the game. As we will see it, this is precisely what is problematic.

Threshold values can easily be calculated: for i, G is preferred to S if 3 > 5[1 -α] and 5α > 1. This gives 1/5 and 2/5.

Let us note that this contrasts whatBuchanan and what the literature on the subject after him argued, namely that, even if it leads to exploitation, asymmetrical or unilateral altruism is beneficial to both parties (see also[START_REF] Singh | Unilateral altruism may be beneficial: A game-theoretic illustration[END_REF]. This is also different from Nicholas Resher's approach, in which altruism is distinguished from self-sacrifice(1975, p. 11).

On the importance of strategic courage in Buchanan's samaritan's dilemma, see Skarbek (2016); Goodman and Hertzberg (2020);[START_REF] Dughera | Altruism and Strategic Courage. Inside Buchanan's samaritan's dilemma[END_REF]

In "Afraid to Be Free: Dependency as Desideratum" (2005), Buchanan referred to the individuals' incapacity to assume the responsibility for their actions or choices that necessarily goes with freedom. The similarity with the incapacity to assume the responsibility that goes with benevolence, or with samaritanism, is strinking. I thank John Meadowcroft for pointed this out to me.

classifying positions or states, must, therefore, be limited to 'regions' that allow for some but not any and all changes in an existing set of social rules." (1962, 343). This is exactly the point we make here. The sado-masochistic equilibrium makes sense if one accepts this kind of equilibria.


Alain Marciano

The Political Economy of Buchanan's Samaritan's dilemma

Keywords: Buchanan, samaritan's dilemma, cooperation, exploitation, numbers, public goods, externalities

Written by James Buchanan in the early 1970s, "The Samaritan's Dilemma" is a pessimistic essay, marked by his author's negative views about the situation in Western societies at that time. Yet, the situation described in this essay also fits into Buchanan's approach of cooperation and free-riding. Put differently, it is perfectly with Buchanan's views in public economics. This is what we aim at showing in this short article. Our demonstration develops in two parts. First, we show that Buchanan's main argument about cooperation in the provision of public goods or removal of externalities necessarily leads to situations such as the one described in the Samaritan's dilemma. Second, we show that Buchanan did not take the situation seriously until the end of the 1960s, a few years before he wrote his essay on the dilemma.

Introduction

The samaritan's dilemma is a situation that has largely been studied in economics and social sciences. It has been shown that it occurs a large number of situations [START_REF] Boettke | Exchange, production, and Samaritan dilemmas[END_REF][START_REF] Boone | Politics and the effectiveness of foreign aid[END_REF][START_REF] Bovard | Feeding Everybody: How federal food programs grew and grew[END_REF][START_REF] Burnside | Aid, policies, and growth[END_REF][START_REF] Coyne | After war: The political economy of exporting democracy[END_REF]Coyne, , 2013;;[START_REF] Futagami | Government transfers and the Samaritan's dilemma in the family[END_REF][START_REF] Rajan | Aid and growth: What does the cross-country evidence really show?[END_REF][START_REF] Skarbek | Aid, ethics, and the Samaritan's dilemma: strategic courage in constitutional entrepreneurship[END_REF][START_REF] Stone | The Samaritan's dilemma: should government help your neighbor?[END_REF]Wilson, Andersson, Ostrom and Shivakumar, 2005;[START_REF] Wagner | Redistribution, poor relief, and the welfare state[END_REF][START_REF] Williamson | Exploring the failure of foreign aid: The role of incentives and information[END_REF]. It is therefore an important phenomenon. However, the reference to the first economist who-as it seemsused the term for the first time, James Buchanan, is frequently lost. This paper focuses on the role the Samaritan's dilemma played in Buchanan's work.

It was in an essay published in 1975 in a collective volume edited by Edmund Phelps, entitled Altruism, Morality, and Economic Theory (1975) that Buchanan put forward and analyzed the problem Samaritans could face when they help people in need, for instance, by transferring them money. The dilemma is a trap in which Samaritans are caught when and because the persons in need they help do not react to the help received, make no effort to reciprocate the help from the Samaritan. Now, Buchanan believed or started from the assumption that a Samaritan expects that her help, gift or transfer will lead the recipient to change his behavior, make some additional efforts and, in particular, "work". 3 Therefore, the Samaritan cannot but be dissatisfied by the lack of effect the transfer has on the recipients' behavior. To avoid this asymmetrical, unequal-and unpleasant-situation in which she helps and the recipient does not make any effort in return, the Samaritan could decide to stop behaving charitably, stop helping the potential recipient. Choosing this course of action would suppress the Samaritan's dissatisfaction with seeing her charity not reciprocated and also, in Buchanan's mind, would incite the recipient to change his behavior and work.

Symmetry and equality would thus be restored. But, according to Buchanan, not helping people in need is costly. The Samaritan would suffer from seeing the recipient starve and from the loss of utility that stems from the impossibility of behaving benevolently. An unpleasant situation replaces another one. Hence, the dilemma the Samaritan faces: either she helps someone who does not return her help or she does not help but suffers from the cost of not helping.

3

To Buchanan, Samaritans could prevent being caught in the dilemma if and only if they were able to follow an ethic of individual responsibility, exhibit what he called "strategic courage"-that is the courage not to help others even despite the costs it implies for the Samaritan. That was the responsibility Samaritans had and should face. But Buchanan doubted they would. The "cliche that modern man has 'gone soft'" (Buchanan 1975, 75) was only too true. Samaritans could not resist to their own benevolence and to the demands of recipients Buchanan "labelled... the parasite" (1975, 75, e.g.). They were "incapable of making the choices that are required to prevent [their] exploitation by predators of [their] own species." (74) Buchanan insisted: "What we may call 'strategic courage' may be a markedly inferior economic good, and what we may call 'pragmatic compassion' may be markedly superior. " (1975, 75) To Buchanan, individuals were afraid to behave as responsible Samaritans, to assume their responsibilities.

These particularly negative statements can be explained by what was happening in the U.S.A. in the late 1960s and early 1970s, when Buchanan started to draft his essay. He was extremely pessimistic about the situation in his country and in Western countries in general. It was this pessimism that had lead him to write "The Samaritan's dilemma" (see Fleury and Marciano, 2018). Yet, a product of its times, "The Samaritan's dilemma" also perfectly fits Buchanan's view in public economics and in public choice or, put differently, his views on political economy. This is what we argue in this essay. More precisely, our goal is to show why situations like the ones described in the 1975 essay are perfectly consistent, and even the consequence of Buchanan's analysis on cooperation in the provision of public goods, and the removal of externalities. In other words, Samaritan's dilemmas are inherent to how Buchanan envisaged cooperation and defection around public goods and externalities. More globally, the Samaritan's dilemma is part of Buchanan's objective of developing a form of non-Samuelsonian welfare economics.

Our demonstration rests on Buchanan's claim that individuals cooperate in small groups-they spontaneously internalize the external consequences of their actions-but do not cooperate in large groups. What we show, and this is original, is that Samaritan's dilemmas arise in what can be called "intermediary" situations, that is when individuals behave as if they were in a small group while they are in a large one, or the reverse. This is the first element we put forward. Then, this is the second aspect of our analysis, Buchanan was perfectly aware that this kind of situation could occur. He acknowledged it at the end of the 1950s but he did not took the situation seriously before the end of the 1960s. This is also what we show in this paper.

The originality of our analysis lies in the fact that it has never been shown that the Samaritan's dilemma is consistent with Buchanan's analysis of cooperation. This finding is interesting to shed new light on Buchanan's work. That is not all. Our analysis is useful also because of the role of "numbers" in the occurrence of the dilemma. Indeed, if it is true that our analysis means that the scope for cooperation is narrower than what Buchanan himself acknowledged, it also reinforces the importance of relying on small groups to guarantee cooperation among individuals.

Market failures, cooperation and numbers

One of the most frequently admitted claims in economics is that, as soon as interdependencies exist between individuals, markets fail to allocate resources efficiently. This is therefore the case when there are public goods or spillover-external-effects. Indeed, individuals being rational and self-interested have no incentive to reveal their preferences-their willingness to pay-for the public good or to internalize the costs of their action on others. This was, in particular, the view held by Richard [START_REF] Musgrave | The voluntary exchange theory of public economy[END_REF][START_REF] Musgrave | The theory of public finance[END_REF]who spoke of "the absence of a general willingness to comply with the obligation to contribute." (1939,220). This was also how Paul Samuelson concluded his article on "The Pure Theory of Public Expenditure": "It is in the selfish interest of each person to give false signals, to pretend to have less interest in a given collective consumption activity than he really has." (1954,(388)(389) Buchanan for hist part wanted to develop a "non Samuelsonian" perspective in economics. He himself used the expression in a comment he sent to Samuelson in 1955 about the famous paper Samuelson had published in 1954(cf. Boettke et Marciano, 2020;[START_REF] Marciano | James Buchanan: clubs and alternative welfare economics[END_REF]. Buchanan meant that he refused the use of a social welfare function to The suggestion made sense only because Buchanan-and, beyond, also the economists from the Virginia school of Political Economy-was convinced that individuals reveal their true preferences and therefore accept to contribute to the provision of the public goods they consume or internalize the effects their actions have on others even if they are rational and motivated by their self-interest [START_REF] Marciano | Buchanan's non-coercive economics for self-interested individuals: Ethics, small groups, and the social contract[END_REF]. A conviction Buchanan held for a long time (Marciano, 2013). And, when he started to admit that individuals could not contribute to the provision of public goods, Buchanan only partially abandoned his confidence in individuals.

Voluntary contribution remained the rule. A condition had nonetheless to be satisfied to guarantee that individuals would cooperate: that the group of individuals involved in the interaction be rather small. Thus, one may note, Buchanan put explicitly forward his first arguments on the role of the size of groups in the early sixties, that is before Mancur Olson (1965).

Thus, as the size of group increases, individuals more and more follow their selfinterest in the narrowest sense of the word, that is: they do not take into account the welfare of others. As Buchanan noted in a rather early article, "the individual's interest in the welfare of his fellow citizen falls off sharply as the group is enlarged." (1961a, 340) Therefore, "as the size of the group increases, any tacit adherence to moral or ethical principles that might inhibit individual utility-maximizing behaviour becomes more difficult to secure" (ibid.; see also, among others, 1967, 1968a, 1968b[START_REF] Buchanan | Markets, states, and the extent of morals[END_REF][START_REF] Buchanan | The calculus of consent: logical foundations of constitutional democracy[END_REF][START_REF] Buchanan | A note on public goods supply[END_REF]. Individuals cooperate less as the size of group increases. This is what Buchanan called a "probabilistic" theory of free-riding (1965b; 1968a, 85-88). How does cooperation, and free riding, relate to the size of groups?

Different strategies in different types of groups

To Buchanan, individuals' choice to cooperate or not to cooperate unsurprisingly depends on the gains, and costs, they expect to receive and incur from their behavior. Those gains then obviously depend on how others behave. More precisely, since no one cannot know in advance how others will behave they depend on each individual's "own predictions about the behavior of others" (1965b, 3). Indeed, it is a matter of probability-"[t]he expected values depend, of course, on the probabilities that the individual assigns to the various patterns of behavior for 'others' than himself" [START_REF]Meadowcroft for pointed this out to me. Buchanan from the archives[END_REF] or, Buchanan repeated almost word for word, depends on the "probabilities assigned [by the individual] to each of the possible behavior patterns of others. " (1968a, 85) These probabilities in turn depend on the impact or influence each individual believes his or her behavior will have on others. Individuals thus cooperate after having contemplated wether or not their behavior will "exert some influence on the behavior of others in the group." (1968a, 86) However, the effect such a belief will have on cooperation can be positive or negative. An individual "may behave cooperatively hoping that his... "rivals" will emulate his... action." (Buchanan, 1967, 121) Or, by contrast, he may decide not to cooperate because he expects that "his own action in contribution will lower, not increase, the probability of others' making contributions of their own." (Buchanan, 1968a, 86) To have a definite answer, we need to take into account the size of the group.

Buchanan distinguished between two types of environments-large and small number environments. In large groups, individuals do not anticipate that their own behavior will influence the behavior of others-"only when the size of the group is critically large. Only in such large groups will the individual consider his own action to exert substantially no effect on the actions of others." (1965b, 9) As a consequence, individuals behave non-strategically, "simply react [ing] or adjust [ing] to the behavior of "others'' in a manner similar to his reaction to natural environment." (1967,113) "Natural" refers to the fact that individuals treat others as part of nature, taking their behavior as given and assuming that how others behave is independent from their own behavior-"The behavior of the other is embodied as data in the choice calculus, but the other person is not considered to be subject to influence or control, positively or negatively." (Buchanan, 1967, 111) Therefore, individuals behave as if they were independent from others. They ignore the interdependencies that link them to others. They simply maximize their own, private, utility. Or, as Buchanan put it, they follow "the private maxim" or "the expediency criterion" (1965b, 2). Now, since the decision to maximize one's utility, to follow the expediency criterion or the private maxim comes from the belief that each individual is independent from others, one understands that each individual makes this choice by ignoring what others do. Whether or not others cooperate, each individual privately prefers not to cooperate-"[i]n a group of critically large size, the individual will tend to adopt the rule of following the expediency criterion even if he thinks that all of his fellow citizens are saints" (1965b, 7; italics in original). Even if they understand that unconditional defection is not the best strategy and that would be better off by cooperating with others. Convinced that they are not able to influence the behavior of others, no one will change his own behavior. No reason can lead individuals to choose to behave differently-"Rationally, he cannot adopt the moral law as a principle for his own behavior." (1965b, 7) Or, to put in terms that will be useful for the rest of the analysis, individuals have a dominant strategy. They are unconditional defectors. Buchanan was rather clear about that: "The individual in a large-group, public-goods interaction... face[s]... no pressure or incentive to behave cooperatively", because he behaves independently from others (1967,121). As a consequence, the individuals who are in large groups are trapped in what Buchanan called the "large number dilemma" (1965b), which is a form of prisoner's dilemma: "The dilemma is a real one, and it is similar to, although not identical with, that which is commonly discussed in game theory as "the prisoners' dilemma." (Buchanan, 1965b, 8). This situation corresponds to what Samuelson, and other economists, viewed as the standard case of market failure.

Let us note here that, to Buchanan, "The individual is caught in a dilemma by the nature of his situation; he has no sensation of securing benefits at the expense of others in any personal manner." (1968a, 83) Not cooperating, following the expediency criterion, means that the individuals maximize their own utility without taking into account or acknowledging the presence of others. Each of them behaves as if others were not there. In other words, choosing not to cooperate in a large group was not interpreted by Buchanan as meaning that individuals would free ride. From this perspective, Buchanan found the terminology used in public economics about free riding "misleading" (1968a, 83). Indeed, free riding "suggests some deliberate effort on the part of the choosing individual to secure benefits at the expense of his fellows." (Buchanan, 1965b, 9) It implies that individuals try "to shift a major share of the burden onto the other while himself securing a share of the benefits." (1967,114) They really adopt anti-social behaviors [START_REF] Marciano | Buchanan on pro-social behaviors: Why is ethics necessary[END_REF]. Or, to use another of Buchanan's words, they try to "exploit" others. In other words, a free rider acknowledges and takes into account others, which is incompatible to how individuals are supposed to behave in large groups. Free riding is more a behavior that could arise, but don't, in small groups.

In small groups such as an "isolated setting" (Buchanan 1965b, 6) of 3 persons or "a desert island " (1965b, 6) or when "personal interaction is recognized" (Buchanan, 1968a, 86), the situation is different. Indeed, "utility maximization in a small number setting will not exhibit the observable properties of utility maximization in a large number setting" (Buchanan, 1978, 366). More precisely, individuals no longer follows their "narrowly defined self-interest" (Buchanan, 1978, 366) but rather adopt "moral or ethical principles" (Buchanan, 1961a, 340). Individuals adapt their behavior to the behavior of others or, to use Buchanan's words, they behave "strategically" (1968a, 91). They no longer ignore the interdependencies that link them to others. Indeed, by contrast with what happens in large number environments, individuals are aware that their behavior matter and may influence others: each individual "will tend to recognize that his own choice of a rule, and subsequent adherence to it, will to some considerable extent influence the similar choices to be made and followed by the other two members" (1965b, 6). Or, "So long as the interaction is limited to small groups, [the individual] will recognize that his own action can exert some influence on the behavior of others in the group." (1967,115) In particular, an individual who "contributes nothing... may assess the probability of noncooperation on the part of others higher than if he contributes some share. This change alone may be sufficient, on rational grounds, to cause him to contribute." (Buchanan, 1968a, 86) In other words, individuals cooperate because they know and anticipate that this behavior will lead others to cooperate too (and that defection would probably lead them to defect too).

It thus seems reasonable to claim that, to Buchanan, the individuals who are in a small group or who are not narrow utility maximizers are unconditional cooperators, as long as they are in a small group or among other ethical individuals. They have a dominant strategy that consists in cooperating to the provision of the public goods or internalize the externalities their behavior can generate. Therefore, the large number dilemma disappears. Individuals cooperate, contributing to the provision of public goods or internalizing the external effects of their actions on others. Even free riding does not exist.

When is there a Samaritan's dilemma? Thus, in Buchanan's analysis of free riding and cooperation, two main possibilities are put forward: the generalized prisoners' dilemma of large groups and the cooperative situation of the small groups. What about intermediary situations? What if in a group, whether large or small, some individuals follow the expediency criterion and others adopt the ethical rule of conduct?

Before examining how Buchanan came to acknowledge this possibility, let us see where the analysis discussed in the previous section leads us. The outcome of an interaction between individuals who do not follow the same rule of behavior is rather complex. One would have understood that individual behaviors depends on whether or not they behave independently from the others or strategically. And, in addition, adopting one behavior or the other depends on the size of the group. Thus, the outcome of an asymmetric interaction depends on whether a narrow maximizers finds himself in a small group or if an ethical behavior finds herself in a large group.

A narrow maximizer by definition ignores the interdependencies with others. As indicated above, this kind of individual does not change his behavior, even if others cooperate, because he does not expect that not cooperating (or cooperating, for that matter) could change the behavior of others. When he enters a small group and interacts cooperators, and when he recognizes that there are cooperators around him, such an individual could be expected to cooperate. The fear of being sanctioned and excluded from the group should lead him to change his behavior. Then, the outcome of the game would be the cooperative situation, where both players cooperate. But there are cases in which the narrow maximizer could decide not to change his behavior: if or as long as he ignores others or believes that he cannot influence them or, and this is very important, if he realizes that the other members of the group might not be able to sanction and exclude him. After all, the costs of exclusion could be so high that narrow maximizers could not be excluded and remain in small groups.

In that case, the narrow maximizer defects. It can even be said that he free rides and exploits the cooperator. Therefore, the outcome is the same as the asymmetric one described in Buchanan's "The Samaritan's dilemma" where the Samaritan of Buchanan's article is the individual who follows the small number environment rule of behavior and the recipient is the narrow maximizer.

The ethical individual entering a group of narrow maximizers faces a similar situation.

He cooperates as long as he does not realize that the individual with whom he interacts does not behave strategically. The other, being a member of a group of narrow maximizers, defects. Therefore, the outcome corresponds to a Samaritan's dilemma. However, the dimension of free riding and exploitation does not exist. The narrow maximizers do not behave strategically. They do not try to shift their share of the burden of the public good on the ethical individual. The latter is nonetheless the only one who cooperates in a world of defectors. This may not last. When the ethical individual finally understands the new situation, his behavior changes and he switches to the non cooperative strategy. The interaction between the two individuals becomes that of the prisoner's dilemma that already prevailed in the group. But, again, the outcome of the interaction depends on the rapidity with which the individual adjusts to the new environment and on the costs of adjustment too. One would remind that, as said in the introduction of this paper, an important assumption made by Buchanan in the 1975 paper was that Samaritans were incapable of changing their behavior and of not behaving charitably because of their "softness".

Therefore, a Samaritan's dilemma exists because or as long as neither the ethical individual nor the narrow maximizer realizes that the nature of the interaction has changed and that they no longer interact with someone who adopts the same behavior as they do. The outcome of the interaction remains that of a Samaritan's dilemma as long as the "confusion" lasts or as long as individuals find it not advantageous enough or too costly to change their behavior. In other words, Samaritan dilemmas are one of the three possible outcomes that can result from individual interactions in Buchanan's theory of cooperation based on numbers and the size of groups or, put differently, from his probabilistic theory of free-riding.

Towards the Samaritan's dilemma

That such dilemmas could arise was perfectly clear in Buchanan's own writing before he started to write explicitly on cooperation and the size of groups and, accordingly, before he wrote on the Samaritan's dilemma. Yet, it took him some time before admitting that such a situation was possible and durable.

What seems to be the first instance of a description of a Samaritan's dilemma can be found in the comment Buchanan made on the papers presented by Musgrave and, more interestingly, by Charles Tiebout at a 1959 conference in public finance he organized (see Marciano, 2013;Boettke, Marciano and Stein, 2021). Tiebout had discussed a situation that corresponds to the one described above. Newcomers arrive in a group in which a public good is already provided. They thus benefit from the good without having to contribute to its provision. Tiebout then put forward an institutional solution to deal with the problem. To him, zoning laws could be used to exclude these potential users from the consumption of the public good they did not have paid for (1961,94; see also Tiebout, 1956, 420).

Buchanan did not disagree on the principle. Such laws or restrictions aiming at "[p]rohibition on entry" (1961b, 129) could indeed allow early settlers "to create a structure of property rights in 'taxpayers' surplus'" (129). In other words, the individuals who were financing the provision public goods could exclude newcomers-who were supposed not to contribute to the provision of the goods in question-to guarantee their surplus. But, he added, exclusion was costly. Indeed, to exclude the newcomers, the early settlers are forced to "forego capital gains in order to prevent entry of "undesirables" into the community." (128) And, as Buchanan explained, "this sacrifice of capital gains on possible land holdings may be more than offset by the retention of a greater share of taxpayers' surplus." (128). The costs of creating these excluding devices could thus be too large compared to the benefits. This meant that the early settlers will not enact, and then enforce, the laws that would be necessary to exclude the potential free riders arriving in the group although it would be collectively rational to do so. They were indeed in the situation of a Samaritan's dilemma.

After having acknowledged the possibility of such dilemmas, Buchanan did not spend more time analyzing them in the 1960s. Much to the contrary. He tended to ignore them. One of the reasons can be found in the phenomenon he described in his 1959 comment: free riders are difficult to exclude once they are in the group; it is therefore crucial to exclude them before they enter the group. Unsurprisingly, thus, Buchanan studied and proposed mechanisms aimed at excluding potential free riders. This is what "clubs" are-"the theory of clubs is, in one sense, a theory of optimal exclusion, as well as one of inclusion." (Buchanan, 1965a, 13) Provided that property rights are correctly defined. Indeed, Buchanan wrote, property rights should "be adjusted to allow for optimal exclusion." (13) What if property rights could not be adjusted? Then, Buchanan concludes, "the " free rider " problem arises." (13) Which means that free riders would benefit from the good provided by the club members. Or, in other words, that a Samaritan's dilemma would take place. Did Buchanan acknowledge the problem? No. He had unlocked a door that he would not immediately push open. In "An Economic Theory of Clubs" (1965a), Buchanan insisted on the importance for "allowing for more flexible property arrangements and for introducing excluding devices" (14), as if he did not want to focus too much on the risks of letting free riders enter clubs. In "Ethical Rules, Expected Values, and Large Numbers" (1965b), he also stressed that the large-number dilemma could be avoided by "reducing group size, or at least modifying the rules so that something similar to small-group results emerges." (10) In other words, he put the emphasis on the possibility to change property rights and re-design groups to avoid free riding and the occurrence of the Samaritan's dilemma. This is what he repeated a few years later, in The Demand and Supply of Public Goods (1968a), noting that in "small groups,... the possibility of excluding genuine non-conformists will normally be present." (87; italics in original).

At the end of the 1960s, Buchanan admitted the existence of the problem more frequently. Two papers from 1967 and 1968 are, from this perspective, particularly importante. In "Cooperation and Conflict in Public-Goods Interactions" (1967), Buchanan demonstrated again how the size of groups affect the individuals' decision to cooperate and contribute optimally to the provision of a public good. Thus, in small groups, did he claim, "there will remain some motivation for the individual to behave strategically" and "some incentive for... tacit cooperation..., but as the group size grows this incentive becomes increasingly faint while the pressures toward "anti-social" behavior become increasingly strong." (115) Also, he demonstrated that strategic behaviors would produce different results depending on whether individuals choose between two or three possible strategies. With two strategies-to share the provision of the public good and to behave independently from others-, the outcome of the game could either be the optimal cooperative solution in which both players cooperate or the worst possible situation in which no one cooperates. Both were possible, even if the second one was more probable. Then, when a third strategy was introduced-exploitation or free riding-, the outcome of the game would be unstable: "Once this additional opportunity for genuine "exploitation" of the other person is recognized, dominance no longer characterizes the utility payoff matrix." (114) He thus admitted the possibility of asymmetric outcomes as in the Samaritan's dilemma. However, and this is important to note, Buchanan built his game matrix to show that such outcomes would not last.

And that was also the point he made in "A Behavioral Theory of Pollution" (1968b), that Buchanan came even closer to describing a Samaritan's dilemma. Buchanan demonstrated that free riders in "a cooperating depollution club" could achieve a "higher utility level than that achieved by club members." (355) In other words, Buchanan had demonstrated the existence of a Samaritan's dilemma-an asymmetric situation in which some individuals cooperate and others do not. He even noted that this equilibrium was "suboptimal in the Pareto sense." (355) However, at this stage, he had not yet admitted that this situation could be a trap and that the cooperating club members could be facing a dilemma. Buchanan then noted that the cooperating members of the club could accept the situation, "acquiese in the free-riders' gains, if they observe the latter to be reinforcing, even if feebly, their own "depollution" efforts." (355) Free-riding could be tolerated if it was not too important and if free-riders evidenced a disposition to change their behavior. Or, put differently, Buchanan did not (at least, seem to) believe yet that the asymmetric situation of a Samaritan's dilemma could and would last.

These articles were the last Buchanan wrote before acknowledging a problem that he had identified in the late 1950s and that was implicit in his approach of cooperation and of free riding. About two years later, in 1970, he started to write "The Samaritan's dilemma", admitting that exploitation was a possible and stable (durable) outcome of an interaction between two individuals adopting a different rule of conduct.

Conclusion

There is no doubt that "The Samaritan's dilemma" was, as it is usually said, written in, and in reaction to, a specific context. Buchanan was pessimistic, very much affected by the situation in the Academia and, more broadly, in the American society when he wrote his essay. The latter however perfectly fits into the frame Buchanan had built to analyze cooperation, and conflict, in public economics. His probabilistic approach of free riding, his analysis of how the size of groups influence cooperation, does not only imply that individuals will cooperate if they are in a small group and behave strategically, or that they will not cooperate in large groups. As we have shown in this paper, Buchanan's insistence on small number environments also suggests that asymmetric and conflictual outcomes-under the form of exploitation-are possible and indeed inevitable. It suggests the possibility of situations that can be depicted by a game like the Samaritan's dilemma. 4 Buchanan was aware of that very early, but admitted it only at the turning of the 1960s to the 1970s. Later, Buchanan will insist more frequently on the risks that such games, and their off-diagonal outcomes, could represent (for instance, [START_REF] Buchanan | Law and the Invisible Hand[END_REF][START_REF] Buchanan | How can constitutions be designed so politicians who seek to serve 'public interest' can survive and prosper?[END_REF][START_REF] Buchanan | Majoritarian logic[END_REF]Buchanan and Lomasky 1984). As Otto Lehto and John Meadowcroft note, Buchanan was then very much concerned that "the off-diagonal outcomes...could be relatively stable." (2021,149) In other words, he was concerned that Samaritan dilemmas could take place. Now, Samaritan's dilemmas are much more problematic than Prisoner's dilemmas. While the latter corresponds to an anarchic situation, a situation of war of all against all that obviously requires a specific action. The former is not an exceptional situation. It corresponds to an equilibrium in a stable political-legal order that all the individuals accept, because they directly benefit from it or because it would be too costly or too difficult to get out of it.

That was one of the most important aspects and lessons of Buchanan 1975 essay. Individuals were not ready to accept the responsibilities that go with helping others-which includes not being charitable in the short term to secure long term benefits. They do not accept, put differently, to become "better" individuals (see Buchanan, 1979, 100). This anticipated the individuals' incapacity to assume the responsibility for their actions or choices that necessarily goes with freedom that Buchanan put forward forty years later in "Afraid to Be Free: Dependency as Desideratum" (2005). 5

  determine the price individuals should pay for public goods (or external effects). To him, no external observer could know what were the individual preferences for public goods and therefore could say what were their marginal willingness to pay. As a consequence, such prices as those based on the use of a social welfare function would in all likelihood differ from what individuals were ready to pay. To avoid that coercive move, Buchanan suggested to rely on what individuals want to pay for the public goods they consume, or for the external effects their behaviors generate.

I thank John Meadowcroft for comments on a previous version of the paper.

Université de Montpellier, Faculté d'Économie, Avenue Raymond Dugrand, CS 79606, F-34960 Montpellier cedex 2 France; alain.marciano@umontpellier.fr

In his description of the interaction between the Samaritan and the recipient, Buchanan wrote that the recipient had two strategies: "the potential parasite... may work ...Or... refuse work" (1975, 76). Then, the many examples he gave of Samaritan dilemmas evidence that it was not only a matter of "work". For instance, he used the case of a mother who hesitates to "spank a misbehaving child". Indeed, it hurts her to have to punish her child but "spanking may be necessary to instill in the child the fear of punishment that will inhibit future misbehavior." (76) Here, it is no longer a matter of "work" strictly speaking. it seems to be more of a matter of changing the behavior of the child.