Abstract
The Samaritan’s Dilemma has largely been investigated, frequently by assuming that Samaritans help recipients out of altruism. Yet, Buchanan did not make any behavioral assumption regarding the Samaritan’s motives. In this paper, we explicitly introduce this assumption in Buchanan’s original model and analyze how it changes the nature of the game. We show that altruism alone does not explain the Dilemma. A parameter that captures the disutility the Samaritan feels when helping someone who does not reciprocate her benevolence must be introduced to make sense of the different version of Buchanan’s Samaritan’s Dilemma. We also show that the Samaritan’s Dilemma is an evolutionary stable outcome, which confirms Buchanan’s intuitions. Finally, a third important point put forward in the paper is that the more altruistic are the Samaritans, the less likely it is that they will show the kind of strategic courage envisaged by Buchanan, which is one of the most important traits Samaritans should display to avoid being trapped in a Dilemma.
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30 March 2023
A Correction to this paper has been published: https://doi.org/10.1007/s10101023002945
Notes
See for instance, Thompson (1980), Veall (1986), Charlton (1987), Kotlikoff (1987), Lindbeck and Weibull (1988), Hansson and Stuart (1989), Wagner (1989), Bruce and Waldman (1990, 1991), Coate (1995), Easterly (2003), Lagerlof (2004), Poulsen and Svendsen (2005), Blouin and Pallage (2008), Leeson (2008), Burns (2009), John and Storr (2009) and Skarbek (2016).
Throughout his 1975 text, Buchanan (1975a: 75) identifies as “pragmatic” those behaviors that are implemented without any form of strategic thinking, that arise automatically (even unconsciously) in response to certain type of situations. Interestingly, he saw this type of behavioral automatisms as one of the main causes behind the existence of the Dilemma.
See Fontaine (2007) for a detailed historical account of Buchanan’s role and the place “The Samaritan’s Dilemma” in the history of economic analyses of altruism. Fontaine convincingly argues, and provides evidence, that Buchanan was interested in altruism and philanthropy. Yet, when he wrote his article, Buchanan was concerned by other issues that had not much to do with altruism (see Fleury and Marciano 2018).
To be more precise, we show that this form of altruism is not sufficient to make sense of both forms of Buchanan’s Samaritan’s Dilemma. With other forms of altruism, it is possible to show that altruism generates a Samaritan’s Dilemma (see Marciano 2022). This does not however invalidate our result: if the Samaritan’s Dilemma was the consequence of altruism, then it should exist with any form of altruistic concern.
In particular, the game is a Passive Samaritan’s Dilemma as long as \(2\alpha > 4\alpha  h  c > 3\alpha  h > \alpha\); and an Active Samaritan’s Dilemma as long as \(3\alpha  h > 4\alpha  h  c > 2\alpha > \alpha\). Clearly, there are multiple set of parameters satisfying these conditions.
Of course, one can think of many reallife situations where helpers may condition their assistance on the recipients’ future behavior, as well as others where the recipients’ intentions can be signalled or inferred during the matching procedure. In addition, allowing for repeated interactions would ultimately change the model’s predictions. While each of these scenarios is by no means irrelevant and generate a number of problems that can be addressed through other gametheoretic techniques, our focus here is on the multitude of everyday contingencies that fits with the above assumptions—e.g., see the examples of the passerby giving a coin to a panhandler or helping a driver change a flat tire in the Introduction.
The last assumption, \(\rho \left( {0,0} \right) = \rho \left( {1,1} \right) = \rho \in \left( {0,1} \right)\) is made to avoid notational clutter. While all our results would be slightly different in quantitative terms by allowing for the possibility that \(\rho \left( {0,0} \right) \ne \rho \left( {1,1} \right)\), none would be altered qualitatively.
Meaning that all trajectories starting from any initial pair \(\left( {x_{0} ,y_{0} } \right) = \left( {1,\hat{y}} \right)\), \(\left( {x_{0} ,y_{0} } \right) = \left( {0,\hat{y}} \right)\), \(\left( {x_{0} ,y_{0} } \right) = \left( {\hat{x},0} \right)\) and \(\left( {x_{0} ,y_{0} } \right) = \left( {\hat{x},1} \right)\) will lie on the side with \(x = 1\), \(x = 0\), \(y = 0\) and \(y = 1\) respectively, where \(0 \le \hat{x} \le 1\) and \(0 \le \hat{y} \le 1\).
When there are noown population effects, the players’ choices do not depend on the behavior of the other members of their population. In our framework, this can be seen from the fact that \(\frac{{\partial \left[ {U\left( 1 \right)  U\left( 0 \right)} \right]}}{\partial x} = 0\) and \(\frac{{\partial \left[ {V\left( 1 \right)  V\left( 0 \right)} \right]}}{\partial y} = 0\).
Moreover, Buchanan makes an additional remark concerning the influence of group size on individual behavior. In small groups, individuals anticipate that their behavior will affect what others do, so they adopt an ethical rule of conduct—they cooperate. In large groups, they believe that what they do will have no consequence and therefore behave egoistically—they do not cooperate.
The situation is surely not new in game theory, and resembles a classic “Battle of the Sexes” where neither of the two Nash equilibria simultaneously maximizes the payoff of the two players.
Interestingly, this strategic property of the Samaritan’s Dilemma is never mentioned by Buchanan in his original contribution, probably, because it sounded like a sort of contradiction to the very point he was trying to make about the need for recipients to make some effort and work.
In The Limits of Liberty (1975b), written at about the same period “The Samaritan’s Dilemma” was published, Buchanan used the same game to defend a social contract and a contractualist view of institutions.
We thank a reviewer for having insisted on the importance of such situations and having suggested us that recipients too could display courage or adhere to an ethic of responsibility that leads them to work more when helped, instead of not working more when helped. Modeling the situation would nonetheless lead us too far away from the objective of the paper.
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The original online version of this article was revised: Due to a production error, the affiliation details for the author Alain Marciano were incorrect. Now, it has been corrected.
Previous versions of the paper were presented at the Adam Smith Seminar (Center of Conflict of Resolution, Munich, 29 March 2021), CEPN (Predatory state, conflict and resistance, 18 June 2021), Annual conference of the Association Française d'Économie du Droit (14–15 October 2021), NiceNancyMontpellier online workshop (2 December 2021), and at the Séminaire d’Économie Théorique et Appliquée (THEMA, Cergy Université, 25 octobre 2022). We thank all the participants for their comments. In particular, Benoît Le Maux, Antoine Pietri, and Mehrdad Vahabi.
Appendix 1
Appendix 1
Proof of Lemma 2
The \(\left( {x^{*} ,y^{*} } \right)\) equilibrium exists when \(x^{*}\) and \(y^{*}\) both \(\in \left( {0,1} \right)\). This, in turn, requires that two of the following conditions are simultaneously satisfied:
Rearranging condition (4′) yields condition (1); rearranging conditions (6′) and (7′) yields, respectively, conditions (8) and (9). The concluding statement in Lemma 1 follows from the fact that conditions (1) and (9) are simultaneously satisfied when the game is a Passive Samaritan’s dilemma, since both are implied by condition (3), while the parametrization defined in condition (1) is inconsistent with any of the possible combinations across conditions (4′)–(7′), so that \(\left( {x^{*} ,y^{*} } \right)\) never exists when the game is an Active Samaritan’s dilemma.□
Proof of Proposition 1
The Jacobian matrix of the system is given by:
As it is wellknown, a stationary point is attractive if both the eigenvalues of the Jacobian matrix evaluated at that equilibrium have negative real parts. In addition, when either of the partial derivative \({\text{d}}\dot{x}/{\text{d}}y\) and/or \({\text{d}}\dot{y}/{\text{d}}x\) is = 0—which happens whenever \(x = 0\) or \(x = 1\) and/or \(y = 0\) or \(y = 1\)—the eigenvalues exactly correspond to the partial derivatives \({\text{d}}\dot{x}/{\text{d}}x\) and \({\text{d}}\dot{y}/{\text{d}}y\). With these facts in mind, the topological properties of the four vertices \(\left( {0, 0} \right)\), \(\left( {0, 1} \right)\), \(\left( {1, 0} \right)\) and \(\left( {1, 1} \right)\) can be straightforwardly checked by evaluating the sign of \({\text{d}}\dot{x}/{\text{d}}x\) and \({\text{d}}\dot{y}/{\text{d}}y\) at each of the four vertices, since the partial derivatives \({\text{d}}\dot{x}/{\text{d}}y\) and \({\text{d}}\dot{y}/{\text{d}}x\) are always = 0 at each of these points.
At \(\left( {0,0} \right)\) we have that:
Hence, we have three cases:

1.
if \(1  e  \rho < 0\) and \(h + c  \rho \alpha < 0\), \(\left( {0,0} \right)\) is a sink (asymptotically stable).

2.
If \(1  e  \rho > 0\) and \(h + c  \rho \alpha > 0\), \(\left( {0,0} \right)\) is source (asymptotically unstable).

3.
If \(1  e  \rho < 0\) and \(h + c  \rho \alpha > 0\), or \(1  e  \rho > 0\) and \(h + c  \rho \alpha < 0\), \(\left( {0,0} \right)\) is a saddle (asymptotically unstable).
Given Lemma 1, when the game is either an Active or a Passive Samaritan’s Dilemma, \(\left( {0,0} \right)\) is always a sink.
At \(\left( {1,1} \right)\) we have that:
Hence, we have three cases:

1.
if \(e  \rho < 0\) and \(\alpha \left( {1  \rho } \right)  h < 0\), \(\left( {1,1} \right)\) is a sink (asymptotically stable).

2.
If \(e  \rho > 0\) and \(\alpha \left( {1  \rho } \right)  h > 0\), \(\left( {1,1} \right)\) is source (asymptotically unstable).

3.
If \(e  \rho < 0\) and \(\alpha \left( {1  \rho } \right)  h > 0\) or \(e  \rho > 0\) and \(\alpha \left( {1  \rho } \right)  h < 0\), \(\left( {1,1} \right)\) is a saddle (asymptotically unstable).
Given Lemma 1, when the game is an Active Samaritan’s Dilemma, \(\left( {1,1} \right)\) is always a saddle; when the game is a Passive Samaritan’s Dilemma, \(\left( {1,1} \right)\) is always a sink.
At \(\left( {1,0} \right)\) we have that
Hence, we have three cases:

1.
if \(\rho + e  1 < 0\) and \(h  \left( {1  \rho } \right)\alpha < 0\), \(\left( {1,0} \right)\) is a sink (asymptotically stable).

2.
If \(\rho + e  1 > 0\) and \(h  \left( {1  \rho } \right)\alpha > 0\), \(\left( {1,0} \right)\) is source (asymptotically unstable).

3.
If \(\rho + e  1 < 0\) and \(h  \left( {1  \rho } \right)\alpha > 0\), or \(\rho + e  1 > 0\) and \(h  \left( {1  \rho } \right)\alpha < 0\), \(\left( {0,1} \right)\) is a saddle (asymptotically unstable).
Given Lemma 1, when the game is an Active Samaritan’s Dilemma, \(\left( {1,0} \right)\) is always a saddle; when the game is a Passive Samaritan’s Dilemma, \(\left( {1,0} \right)\) is always a source.
At \(\left( {0,1} \right)\) we have that
Hence, we have three cases:

1.
if \(\rho  e < 0\) and \(\rho \alpha  h  c < 0\), \(\left( {0,1} \right)\) is a sink (asymptotically stable).

2.
If \(\rho  e > 0\) and \(\rho \alpha  h  c > 0\), \(\left( {0,1} \right)\) is source (asymptotically unstable).

3.
If \(\rho  e < 0\) and \(\rho \alpha  h  c > 0\), or \(\rho  e > 0\) and \(\rho \alpha  h  c < 0\), \(\left( {0,1} \right)\) is a saddle (asymptotically unstable).
Given Lemma 1, when the game is an Active Samaritan’s Dilemma, \(\left( {0,1} \right)\) is always a saddle; when the game is a Passive Samaritan’s Dilemma, \(\left( {0,1} \right)\) is always a source.
To prove the stability properties of the \(\left( {x^{*} ,y^{*} } \right)\) equilibrium, we study its Trace and Determinant and see that:
From Lemma 1, we already know that \(\left( {2\rho  1} \right)\alpha  c > 0\) and \(2\rho  1 > 0\) are simultaneously satisfied when condition (3) is satisfied. In this case, \({\text{Det }}J_{{\left( {x^{*} ,y^{*} } \right)}} < 0\), which proves that \(\left( {x^{*} ,y^{*} } \right)\) is a saddle when the game is a Passive Samaritan’s dilemma. In addition, there exists parametrizations for which \({\text{Det }}J_{{\left( {x^{*} ,y^{*} } \right)}} \ge 0\), which completes the Proof of Proposition 1.□
Proof of Proposition 2
From the results in Proposition 1, it is straightforward to derive the following set of results. First, the existence conditions of the four monostable regimes are as follows:

1.
The stationary point \(\left( {0,0} \right)\) is the unique attractor iff \(e > \max \left\{ {\rho ,1  \rho } \right\}\) and \(\alpha > \max \left\{ {\frac{h}{1  \rho },\frac{h + c}{e}} \right\}\).

2.
The stationary point \(\left( {1,1} \right)\) is the unique attractor iff \(e < \min \left\{ {\rho ,1  \rho } \right\}\) and \(\alpha < \min \left\{ {\frac{h}{1  \rho },\frac{h + c}{e}} \right\}\).

3.
The stationary point \(\left( {1,0} \right)\) is the unique attractor iff \(e < \min \left\{ {\rho ,1  \rho } \right\}\) and \(\alpha > \max \left\{ {\frac{h}{1  \rho },\frac{h + c}{e}} \right\}\).

4.
The stationary point \(\left( {0,1} \right)\) is the unique attractor iff \(e > \max \left\{ {\rho ,1  \rho } \right\}\) and \(\alpha < \min \left\{ {\frac{h}{1  \rho },\frac{h + c}{e}} \right\}\)
Second, the existence conditions of the two bistable regimes are as follows:

1.
The stationary points \(\left( {0,0} \right)\) and \(\left( {1,1} \right)\) simultaneously attract iff conditions (1) and (9) are simultaneously satisfied.

2.
The stationary points \(\left( {1,0} \right)\) and \(\left( {0,1} \right)\) simultaneously attract iff simultaneously attract iff conditions (7) and (8) are simultaneously satisfied.
Third, the existence conditions of the two cyclical regimes are as follows:

1.
The system exhibits cyclical behavior with counterclockwise oscillations around \(\left( {x^{*} ,y^{*} } \right)\) iff conditions (7) and (9) are simultaneously satisfied.

2.
The system exhibits cyclical behavior with clockwise oscillations around \(\left( {x^{*} ,y^{*} } \right)\) iff conditions (1) and (8) are simultaneously satisfied.
Observe that when the game is an Active Samaritan’s Dilemma, the monostable regime featuring \(\left( {0,0} \right)\) as the only attractor results; when the game is a Passive Samaritan’s Dilemma, the bistable regime featuring \(\left( {0,0} \right)\) and \(\left( {1,1} \right)\) as attractors result.□
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Dughera, S., Marciano, A. Inside Buchanan's Samaritan's Dilemma: altruism, strategic courage and ethics of responsibility. Econ Gov 24, 207–233 (2023). https://doi.org/10.1007/s10101023002927
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DOI: https://doi.org/10.1007/s10101023002927