On Hobbes’s state of nature and game theory

Abstract

Hobbes’s state of nature is often analyzed in two-person two-action non-cooperative games. By definition, this literature only focuses on duels. Yet, if we consider general games, i.e., with more than two agents, analyzing Hobbes’s state of nature in terms of duel is not completely satisfactory, since it is a very specific interpretation of the war of all against all. Therefore, we propose a definition of the state of nature for games with an arbitrary number of players. We show that this definition coincides with the strategy profile considered as the state of nature in two-person games. Furthermore, we study what we call rational states of nature (that is, strategy profiles which are both states of nature and Nash equilibria). We show that in rational states of nature, the utility level of any agent is equal to his maximin payoff. We also show that rational states of nature always exist in inessential games. Finally, we prove the existence of states of nature in a class of (not necessarily inessential) symmetric games.

Appendix A: Proofs of the propositions

1.1 Proof of Proposition 1

Proof

All the games gathered in Table 1 satisfy the following necessary and sufficient condition for (Attack, Attack) to be a state of nature according to our definition:

$$\begin{aligned} U_{1}(A, A)\le & {} U_{1}(\overline{A}, A), \end{aligned}$$

(21)

$$\begin{aligned} U_{2}(A, A)\le & {} U_{2}(A, \overline{A}). \end{aligned}$$

(22)

$\square$

1.2 Proof of Proposition 2

Proof

To obtain Eq. (3), we can follow the arguments used by Binmore (1982) to prove his Theorems 6.2.2 and 6.3.1. By definition of the state of nature and the maximin payoff, we have for all agents i:

$$\begin{aligned} U_{i} ( \overline{s}_{i}, \overline{{\mathbf {s}}}_{-i} )= & {} \min _{{\mathbf {s}}_{-i} \in S_{-i}} U_{i} ( \overline{s}_{i}, {\mathbf {s}}_{-i}), \end{aligned}$$

(23)

$$\begin{aligned}\le & {} \max _{s_{i} \in S_{i}} \min _{{\mathbf {s}}_{-i} \in S_{-i}} U_{i} ( \overline{s}_{i}, {\mathbf {s}}_{-i} ), \end{aligned}$$

(24)

$$\begin{aligned}= & {} \alpha _{i}. \end{aligned}$$

(25)

Since the state of nature is also a Nash equilibrium, we also have for all $s_{i} \in S_{i}$

$$\begin{aligned} U_{i}(s_{i}, \overline{{\mathbf {s}}}_{-i})\le & {} U_{i}(\overline{s}_{i}, \overline{{\mathbf {s}}}_{-i}), \end{aligned}$$

(26)

$$\begin{aligned} \Rightarrow \min _{{\mathbf {s}}_{-i} } U_{i}(s_{i}, {\mathbf {s}}_{-i})\le & {} U_{i}(\overline{s}_{i}, \overline{{\mathbf {s}}}_{-i}). \end{aligned}$$

(27)

However, then

$$\begin{aligned} \max _{s_{i} \in S_{i}} \min _{{\mathbf {s}}_{-i} } U_{i}(s_{i}, {\mathbf {s}}_{-i})&\le U_{i}(\overline{s}_{i}, \overline{{\mathbf {s}}}_{-i}). \end{aligned}$$

(28)

This proves that $\alpha _{i} \le U_{i}(\overline{s}_{i}, \overline{{\mathbf {s}}}_{-i})$ and thus that $U_{i}(\overline{s}_{i}, \overline{{\mathbf {s}}}_{-i}) = \alpha _{i}$. $\square$

1.3 Proof of Proposition 3

Proof

Let $\overline{{\mathbf {s}}}$ be a vector of individual strategies, such that for all i $\min _{{\mathbf {s}}_{-i}} U_{i}(\overline{s}_{i}, {\mathbf {s}}_{-i}) = \alpha _{i}$. Then, from Theorem 2.1 of Moulin (1986), we have $U_{i}(\overline{{\mathbf {s}}}) = \alpha _{i}$ for all i.¹⁹ By definition of the maximin payoff, this implies that $\overline{{\mathbf {s}}}$ is a state of nature. Now, from Theorem 2.2 of Moulin (1986), this state of nature is a Nash equilibrium, and therefore, it is also rational.²⁰ $\square$

1.4 Proof of Proposition 4

Proof

Consider individual 1. Let $s_{1}$ be given in S. By Assumption H4, there is a unique solution $(s_{2}(s_{1}), s_{3}(s_{1}), \ldots , s_{n}(s_{1}))$ in $S^{n-1}$ to the problem: $\min _{{\mathbf {y}} \in S^{n-1}} U(s_{1}, {\mathbf {y}})$. By Assumption H3, this solution necessary satisfies: $s_{2}(s_{1}) = s_{3}(s_{1}) = \cdots = s_{n}(s_{1})$. Let then $\phi (s_{1}) \equiv s_{i}(s_{1})$, $i \not = 1$. According to Assumption H4, $\,\phi (.)$ is a continuous application from a compact convex set S into itself. Therefore, by Brouwer’s Theorem (see, e.g., Carter 2001, page 248), this application has a fixed-point $s^{*}$: $s^{*}= \phi (s^*)$. Finally, using assumptions H1–H4, we can check that the strategy profile $(s_{1}, \ldots , s_{n}) = (\overline{s}, \ldots , \overline{s})= (s^{*}, \ldots , s^{*})$ is a state of nature. That is, if any individual plays strategy $s^*$, then the other individuals minimize his utility function by each choosing $s^*$. $\square$