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Hobbes, Harsanyi and the Edge of the Abyss*

Published online by Cambridge University Press:  10 November 2009

Gabriella Slomp
University of Strathclyde
Manfredi M. A. La Manna
University of St. Andrews


The authors present a new game-theoretic interpretation of Hobbes's state of nature that, unlike existing rational-choice models, questions the possibility of individually rational decision making. They provide a general formulation of the two-player two-strategy game applied to the state of nature and derive existing models as special cases. A nonstandard version of Chicken under incomplete information, that interprets “death” as infinitely bad, provides an explanation for important and hitherto unaccounted for claims by Hobbes. The authors suggest that rational choice in Hobbesian political philosophy ought to examine not so much the mechanics of rational action in natural conditions, but rather the means whereby citizens already living in civil associations can be persuaded of the irrationality of civil war.


Les auteurs proposent une nouvelle interprétation du concept hobbésien d'état de nature. Leur approche, reprise de la théorie des jeux, remet en cause le principe de la rationalité individuelle dans la prise de décision, pourtant admis traditionnellement par les modèles du choix rationnel. Ils donnent une définition générale du jeu à deux joueurs et à deux stratégies dans l'état de nature dont les modèles existants s'avèrent être des cas particuliers. Ils analysent d'une manière originale certains des principaux arguments de Hobbes à l'aide d'une variante du jeu du « Chicken » en environnement incertain, où la « mort » est considérée infiniment mauvaise. Ils suggérent que l'analyse des choix politiques rationnels dans une perspective hobbésienne se doit avant tout de considérer les moyens par lesquels des citoyens vivant déjà dans des associations politiques peuvent être convaincus de l'irrationalité d'une guerre civile, plutôt que de décrire le mécanisme qui conduit à une action rationnelle dans l'état de nature.

Research Article
Copyright © Canadian Political Science Association (l'Association canadienne de science politique) and/et la Société québécoise de science politique 1996

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1 Hobbes, Thomas, De Cive, The Clarendon Edition of the Philosophical Works of Thomas Hobbes, Vol. 2, ed. by Warrender, H. (Oxford: Clarendon Press, 1983), 130.Google Scholar

2 Kavka, Gregory, Hobbesian Moral and Political Theory (Princeton: Princeton University Press, 1986)Google Scholar; Hampton, Jean, Hobbes and the Social Contract Tradition (Cambridge: Cambridge University Press, 1986)Google Scholar; Taylor, Michael, The Possibility of Cooperation (Cambridge: Cambridge University Press, 1987)Google Scholar; and McLean, Iain, “The Social Contract in Leviathan and the Prisoner's Dilemma Supergame,” Political Studies 29 (1981), 339–51.CrossRefGoogle Scholar

3 Neal, Patrick, “Hobbes and Rational Choice Theory,” Western Political Quarterly 41 (1988), 635–52.CrossRefGoogle Scholar

4 Neal argues that “rational choice theory reaps a good less than Hobbes attempted to sow and serves to obscure more than illuminate his teaching” (ibid., 635).

5 Hampton, Hobbes and the Social Contract Tradition.

6 Hobbes compares the state of nature to a state of civil war: “it may be perceived what matter of life there would be, where there were no common power to fear, by the manner of life, which men that have formerly lived under a peaceful government, use to degenerate into, in a civil war” (Hobbes, Thomas, Leviathan, Vol. 3 of The English Work of Thomas Hobbes, ed. by Molesworth, W. [London: Scientia Aalen, 1962], 114–15)Google Scholar. We should stress that our claim to novelty refers to the way in which we use rational-choice theory to show the irrationality of descent into the state of nature. This reading of Hobbes's theory goes back to some of his contemporaries and two recent examples are Hardin, Russell, “Hobbesian Political Order,” Political Theory 19 (1991), 156–80CrossRefGoogle Scholar, and Hampsher-Monk, Iain, A History of Modern Political Thought (Oxford: Blackwell, 1992)Google Scholar, who calls it the “virtual contract” theory.

7 Hobbes, De Cive, 49.

8 Harsanyi, John C., “Games with Randomly Disturbed Payoffs: A New Rationale for Mixed Strategy Equilibrium Points,” International Journal of Game Theory 2 (1973), 123.CrossRefGoogle Scholar

9 The substantive qualitative points we establish in section 4 are unaffected by extending the analysis to N (>2) players. The same does not apply to some of the games examined in section 2, which rely on interpreting Hobbes's model as a compound two-player game.

10 See, for example, Hobbes, De Cive, 177; see also Hobbes, Leviathan, chap. 5, and Hobbes, Thomas, The Elements of Law Natural and Politic, ed. by Tönnies, F. (2nd ed.; London: F. Cass, 1969)Google Scholar, chap. 1.

11 For a formal definition of common knowledge, see Aumann, Robert, “Agreeing to Disagree,” Annals of Statistics 4 (1976), 1236–39CrossRefGoogle Scholar; see also Sugden, Robert, “Rational Choice: A Survey of Contributions from Economics and Philosophy,” Economic Journal 101 (1991), 751–85CrossRefGoogle Scholar. Assumptions IR and CK are part of what Neal (in “Hobbes and Rational Choice Theory”) calls “E-rationality.”

12 According to Hobbes, the content of the right of nature and of the natural laws can be understood by everybody, and everybody can be assumed to understand it (see Hobbes, Leviathan, chap. 5, especially 144).

13 Hobbes, Leviathan, 110; Hobbes, Elements of Law, 70; and Hobbes, De Cive, 45.

14 We ignore the trivial case of total indifference where both P = D and S = W.

15 On the relationship between the CK assumption and Nash equilibrium, see Bacharach, M., “A Theory of Rational Decision in Games,” Erkenntnis 27 (1987), 1755CrossRefGoogle Scholar, and Sugden, “Rational Choice.”

16 Taylor, The Possibility of Cooperation, and McLean, “The Social Contract,” are two loci classici.

17 See, for example, Hobbes, Leviathan, chap. 6, at 44.

18 Ibid., chap. 13, at 88, emphasis added.

19 That is, a state that, once entered into, cannot be escaped from.

20 As explained in Appendix A, the dotted lines in Figures 4 and 6 refer to a weakly dominated strategy and can be ignored.

21 “[B]ees and ants live sociably one with another (which are therefore by Aristotle numbred amongst Politicall creatures), and yet have no other direction, than their particular judgements and appetites; nor speech, whereby one of them can signifie to another, what he thinks expedient for the common benefit: and therefore some man may perhaps desire to know why mankind cannot do the same” (Hobbes, Leviathan, 156; see also Hobbes, Elements of Law, 102, and Hobbes, De Cive, 87).

22 The first applications of the Prisoner's Dilemma to Hobbes's theory can be found in Gauthier, David, The Logic of Leviathan (Oxford: Clarendon Press, 1969)Google Scholar, and Watkins, John, “Imperfect Rationality,” in Borger, R. and Cioffi, F., eds., Explanation in the Behavioural Sciences (Cambridge: Cambridge University Press, 1970)Google Scholar. In the literature on PD applications to Hobbes's theory, usually only strict inequalities are considered; however, the same substantive outcome obtains even with weak inequalities.

23 Neal, “Hobbes and Rational Choice Theory,” 642.

24 Kavka, Hobbesian Moral and Political Theory, especially 182–88; Hampton, Hobbes and the Social Contract Tradition, 150–60; and Taylor, The Possibility of Cooperation, chaps. 2 and 3.

25 Exceptions are McLean, , “The Social Contract,” and Sugden, Robert, The Economics of Rights, Co-operation and Welfare (Oxford: Blackwell, 1986)Google Scholar, who apply the hawk-dove game originally developed by Smith, John Maynard and Price, G. R. (in “The Logic of Animal Conflict,” Nature 246 [1973], 1518)CrossRefGoogle Scholar to Hobbes's state of nature.

26 Hobbes, Leviathan, chap. 24, at 233, emphasis added; see also chap. 13.

27 Hobbes, De Cive, 33, emphasis added.

28 Hobbes, Elements of Law, 29; Hobbes, De Cive, 74–75, 177; and Hobbes, Leviathan, 28–29, 146.

29 “All men [are] not alike affected with the same thing, nor the same man at all times” (Hobbes, Leviathan, 146; see also Hobbes, De Cive, 74, 177).

30 In other words, player i knows their own type ti and has some beliefs regarding the other players’ types, embodied in a probability distribution pi(t-i), where t-i ≡ (ti,…, ti−1, ti+1,…. tN) are the types of all other players. More generally one would write pi (t-i|ti) to allow for the possibility that player i'S belief regarding the other players’ types can depend on their own type, ti. However, in the state of nature as described by Hobbes, it is more likely that players’ types are independent.

31 Harsanyi, “Games with Randomly Disturbed Payoffs.”

32 See ibid, for a formal statement and proofs.

33 Kavka, Hobbesian Moral and Political Theory; Hampton, Hobbes and the Social Contract Tradition; and Taylor, The Possibility of Cooperation.

34 Hobbes, Elements of Law, 188.

35 See McNeilly, F. S., The Anatomy of Leviathan (London: Macmillan, 1968).Google Scholar

36 Hobbes, Thomas, Thomas White's De Mundo Examined (Anti-White), ed. by Jones, H. Whitmore (Bradford: Bradford University Press, 1976), 408Google Scholar, emphasis added.

37 “[Necessity of nature maketh men… to avoid that which is hurtful; but most of all that terrible enemy of nature, death” (Hobbes, Elements of Law, 71); “[reason] teaches every man to fly a contre-naturall dissolution, as the greatest mischiefe that can arrive to Nature” (Hobbes, De Cive, 27); “for every man is desirous of what is good for him, and shuns what is evill, but chiefly the chiefest of natural evills, which is death” (ibid., 47). See also Hobbes, Elements of Law, 71–72, 94; Hobbes, De Cive, 47, 53; and Hobbes, Leviathan, 329. Only in a few scattered passages of his works does Hobbes put forward a weaker claim on self-preservation by suggesting that a few individuals regard life in dishonour or with scorn or without liberty as not worth living (see Hobbes, Elements of Law, 39, 86; Hobbes, De Cive, 67; and Hobbes, Leviathan, 140).

38 Mountain climbing, for example, is fully compatible with S∞.

39 Attaching an infinitely bad payoff to violent death at the hands of others is equivalent to the following two-stage decision process: first, partition the set of social states into the two disjoint subsets of life-endangering actions (actions that may result in violent death) and non-life-endangering actions; second, define preferences (according to glory) over the latter subset; for details see Gabriella Slomp, “Hobbes's Impossibility Theorem” (mimeographed, University of Wales, Swansea, 1995).

40 See, for example, Hobbes, Elements of Law, 47; Hobbes, De Cive, 43; and Hobbes, Leviathan, 156.

41 Hobbes, Elements of Law, 70–71; see also Hobbes, Leviathan, 112.

42 ibid., 156; see also Hobbes, Elements of Law, 102, and Hobbes, De Cive, 87.

43 Neal, “Hobbes and Rational Choice Theory.”

44 Neal's game (case B2) shares with cases Bl and D2 the property that mutual avoidance is an equilibrium even if W = −∞.

45 Let p be the probability of the other player avoiding fighting. The expected pay-offs from Fight and Avoid are respectively:

To find a Nash equilibrium we find the value of p that solves π(F) = π(A) whence we obtain that each player's optimal probability of avoiding fighting is given by

46 That is, tR and tC are independent draws from a uniform distribution on [0, ξ].

47 Of course, an infinitely bad payoff violates the assumption of bounded utility and thus it could be argued that it is not surprising that no rational decision making is feasible in Hobbes's characterization of the state of nature. Our response to this line of criticism is as follows. Bounded utility is sufficient, but not necessary, to guarantee the existence of equilibria in mixed strategies in non-cooperative games. Nash Theorem (Nash, John, “Equilibrium Points in N-Person Games,” Proceedings of the National Academy of Sciences 36 [1950], 4849)CrossRefGoogle ScholarPubMed states that if mixed strategies are allowed, then any normal-form finite game has at least one equilibrium. This result depends crucially on the payoff functions being bounded (for an elegant proof, see Theorem 3.1 in Friedman, James, Game Theory with Applications to Economic [2nd ed.: Cambridge: MIT Press, 1990])Google Scholar. It is simple to verify that there are cases (Neal's coordination game, B2 and its variant B1, as well as the bees and ants game, D2) that yield mutual avoidance as an equilibrium even if W = W′ = −∞.

48 With reference to the payoff rankings listed in Appendix A, cases A1, C1 and C2 (insofar as they assume that S < W) are incompatible with W = −∞. Case A2, which subsumes our assumption on glory insofar as it posits that D > P, shares with our version of Chicken the non-existence of an equilibrium, as can be easily verified.

49 Interestingly, in his translation of Thucydides’ History, Hobbes came across what is probably the earliest statement of the Prisoner's Dilemma, that he rendered thus: “Everyone supposeth, that his own neglect of the common estate can do little hurt, and that it will be the care of somebody else to look to that for his own good: not observing how by these thoughts of every one in several, the common business is jointly ruined” (Hobbes, Thomas, The History of the Grecian War Written by Thucydides and Translated by Thomas Hobbes, Vol. 8 of English Works of Thomas Hobbes [London: John Bohn, 1839], 147)Google Scholar; for an analysis of Thucydides’ influence on Hobbes, see Slomp, Gabriella, “Hobbes, Thucydides, and the Three Greatest Things,” History of Political Thought 11 (1990), 565–86.Google Scholar

50 Hobbes, Elements of Law, 73; and Hobbes, De Cive, 49.

52 Hobbes, Elements of Law, 34.

53 Hobbes, Leviathan, 113.

54 In a follow-up to this article, one of us provides a new interpretation of Hobbes's political construct from a fully fledged external-observer perspective (see Slomp, “Hobbes's Impossibility Theorem”).

55 We ignore the trivial case P = D, S = W.

56 Gender neutrality is a substantial component of Hobbes's theory; on this, see Slomp, Gabriella, “Hobbes and the Equality of Women,” Political Studies 42 (1994), 441–52.CrossRefGoogle Scholar

57 For a defence of the elimination of weakly dominated strategies, see, for example, Myerson, Roger, Game Theory: Analysis of Conflict (Cambridge: Harvard University Press, 1991)Google Scholar, Theorem 1.7, 30. The potential problem of eliminating weakly dominated strategies, namely, that the order in which the elimination takes place may affect the resulting equilibrium, simply does not arise in two-strategy games such as those considered here.

58 Pursuing the box-crossing analogy sketched a few lines above, we may say that what matters is going from one side of the box to the opposite side, and that wandering around on the zebra crossings on either side of the box is a waste of time.

59 The static Bayesian game in normal form is Ω. = {AA, AF; TC, TR; prC, prR.; uC, uR), the action spaces are AA = AF = {Avoid, Fight}, the type spaces are TC = TR = [0,ξ], the beliefs are prC(tR) = prR(tC) = 1 / ξ, ∀ tR, tC and the payoffs uC, uR are as described in Figure 10.

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