## Abstract

The question of what a group of rational agents would agree on were they to deliberate on how to organise society is central to all hypothetical social contract theories. If morality is to be based on a social contract, we need to know the terms of this contract. One type of social contract theory, contractarianism, aims to derive morality from rationality alone. Contractarians need to show, amongst other things, that rational and self-interested individuals would agree on an impartial division of a cooperative surplus. But it is often claimed that contractarians cannot show this without introducing moral assumptions. This paper argues that on the right understanding of the question contractarians are asking, these worries can be answered. Without relying on moral assumptions, the paper offers a novel derivation of symmetry, which is the axiom responsible for the impartiality of the most famous economic bargaining solutions appealed to by contractarians.

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## Notes

- 1.
See Gauthier (1969) for a reading of Hobbes’

*Leviathan*in game theoretic terms. - 2.
Lensberg (1988) showed that the Nash bargaining solution can also be derived when we substitute an axiom called ‘stability’ for the independence of irrelevant alternatives. This axiom is often considered more plausible than independence of irrelevant alternatives, and claims that, in a multiple person bargaining game, the bargaining solution must be such that, if we hold the utility allocation to some players fixed and apply the bargaining solution to the subgame involving the remaining players, those players must be allocated the same amount as when the bargaining solution was applied to the original bargaining game.

- 3.
This is especially apt given the fact that Gauthier later gave up his original bargaining solution and endorsed the Nash bargaining solution (see Gauthier 1993).

- 4.
There are also non-cooperative implementations of the KS solution, but they involve much more complicated procedures for bargaining. See, for instance, Moulin (1984).

- 5.
An unpublished manuscript by Joseph Heath made me aware of this point.

- 6.
- 7.
Sugden provides a convincing argument that what he calls the ‘principle of rational determinacy’ is not generally true. That is, it is not true in all games that the rationality of the agents guarantees that there is a single determinate outcome to the game. However, this is still consistent with it being the case that in the bargaining game, in particular, the principle holds.

- 8.
Arguably, Gauthier’s

*Morals by Agreement*leaves the bargaining procedure open to the agents. He does describe bargaining as a 2-stage process, whereby agents first claim their ideal utilities, and then offer concessions until a feasible division has been reached (p.131). The process of making concessions is not well described, however. It is not clear, for instance, whether the agents see what concessions everybody else made, whether they make concessions simultaneously, whether they have to concede anything each round, whether their concessions have to become larger every round of bargaining, etc. Later on, Gauthier claims that bargaining will follow Zeuthen’s principle (proposed in Zeuthen 1930), which claims that the person with the smallest relative concession has to concede and make a larger concession each round. However, this is not a further description of the structure of the bargaining game, but rather a principle that Gauthier takes to be implied by the bargainers’ equal rationality. Most of what Gauthier says about bargaining in fact follows the axiomatic approach in leaving the process of bargaining unspecified. He starts off much like Nash, by imposing some conditions on the final bargaining solution (p.130). It must be a point on the Pareto frontier, and it cannot require any interpersonal comparisons of utility. And his more formal derivation of the principle of minimax relative concession focuses on the properties of various outcomes, namely on whether some division of the cooperative surplus is one that all bargainers are rationally willing to entertain. - 9.
Gauthier introduces this assumption, almost as an afterthought, on p. 156 of

*Morals by Agreement*. This feature of Gauthier’s set-up is ignored by Sugden, whose argument that uniqueness cannot be presupposed by Gauthier relies on a reading on which the agents play a game of pure coordination with a predetermined end-point (in which case there are infinitely many Nash equilibria). - 10.
Note the similarity to the Coase Theorem (Coase 1960).

- 11.
We only characterised what the agents’ beliefs have to be like when agreement is reached, but not how the convergence of beliefs on what agreement is feasible is supposed to be achieved. Uniqueness would require that beliefs can only converge on one particular utility allocation, which we have not shown.

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## Acknowledgments

I would like to thank Arthur Ripstein, Olivier Roy, Sergio Tenenbaum, and Joseph Heath for helpful feedback on drafts of this paper. An audience at LMU Munich also provided helpful comments, and I benefitted from conversation with David Gauthier. I am grateful for travel funding from the International Balzan Prize Foundation as part of Ian Hacking’s ‘Styles of Reasoning’ project.

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Thoma, J. Bargaining and the impartiality of the social contract.
*Philos Stud* **172, **3335–3355 (2015). https://doi.org/10.1007/s11098-015-0472-7

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### Keywords

- Contractarianism
- Game theory
- Bargaining
- Symmetry