1
The state of nature is a game without enforcement
Two people each choose peace or attack, with no one to punish an attack. Whatever the numbers are, they are the payoffs of an interaction in which only the players' own interests apply.
Lesson · Games and the Social Contract · Many players
Hobbes and the State of Nature
Hobbes's state of nature can be written as a game between two people who each choose peace or attack. Where attacking pays whatever the other does, war is the only equilibrium; where attacking is only protection against an attacker, war and peace are both equilibria. A sovereign who punishes attack changes the payoffs, and with a large enough penalty peace is the only equilibrium.
anarchy: attack pays against either move
so both attack, and each does worse
fear alone gives two equilibria
a sovereign changes payoffs, not people
A game whose payoffs a sovereign changes
Core ideas
What this lesson establishes
Hobbes's argument is a claim about incentives, and the three ideas below put it into a table.
2
Attack can pay from greed or from fear
If attacking a peaceful neighbour pays most, attack dominates and the game is the prisoner's dilemma. If it pays no more than defending against an attacker, the game is the stag hunt and the trouble is only that each fears the other.
3
A sovereign changes the payoffs
A penalty on attacking is subtracted from every payoff in which someone attacks. It is a change to the table, and a large enough change moves the equilibrium without changing what anyone wants.
The material
War of all against all, and what ends it
Three tables: Hobbes's competition, the same people under a sovereign, and Hobbes's diffidence.
Hobbes finds three causes of quarrel in human nature: competition, diffidence and glory. Competition makes men invade for gain, diffidence for safety, and glory for reputation. Without a common power to keep them all in awe, he writes, they are in a condition of war — “of every man against every man” — and life is “solitary, poor, nasty, brutish, and short.” Competition and diffidence can be tabulated; glory cannot, and the lab leaves it out.
Take competition first. Two people each choose peace or attack. If both keep the peace each gets 3. An attacker who finds the other at peace takes 5, and the victim gets 0. If both attack, each gets 1.
other: peace other: attack
peace 3, 3 0, 5
attack 5, 0 1, 1
This is the table of “The Prisoner's Dilemma” with new labels. Attack pays 5 against peace and 1 against attack, and peace pays 3 and 0, so attack dominates. Both attack, each collects 1, and both would have collected 3 at peace. The model needs no wicked people to get there: only that each prefers more to less and cannot count on the other.
The sovereign
Now let a sovereign fine whoever attacks, by 4, whatever the target does. Subtract 4 from every payoff in which the person attacks. An attacker who finds the other at peace now gets 1, and two attackers each get −3. The victim's payoffs do not change.
other: peace other: attack
peace 3, 3 0, 1
attack 1, 0 −3, −3
Against peace, peace pays 3 and attack pays 1. Against attack, peace pays 0 and attack pays −3. Peace now dominates, the only equilibrium is peace with peace, and each collects 3. Nobody is fined there, so the penalty is never collected: a threat that works costs nothing to carry out. Hobbes's remark that covenants without the sword are but words is, in the table, the observation that a promise changes no payoff: only the sword does, and it has to change them by enough. A penalty of 2 or less does not make peace dominate, since at 1 attacking a peaceful neighbour still pays 4 against 3, and at 2 the two tie. The lab's dominance tile reports strict dominance only, so it turns at a penalty of 3.
Fear without greed
The third table has no gain from attacking at all. Peace with peace pays 4 each. Attacking pays 3 whatever the other does, because the attacker gets only the safety of not being caught unprepared, and a peaceful victim gets 0. This is the stag hunt of “Coordination, Conventions and the Stag Hunt”, and it has two equilibria: peace with peace and attack with attack.
Hobbes's point about diffidence is that the second equilibrium can be reached by people who want nothing the other has. Peace is worth keeping only if the neighbour keeps it with probability at least 3/4; short of that, attacking first is the better bet. What the sovereign supplies in this version is assurance. The remedy is a common power that makes each person's expectation of the other safe.
The lab does not say who watches the sovereign, who pays him, or why he would fine anyone: the penalty is an assumption of the model, and Hobbes's own argument for an authority that can be trusted to apply it is the part the table leaves out.
Interactive
A game in a table
best responses, equilibria, dominance and efficiency, computed from the payoffs
Method
Writing the state of nature as a game
The same steps for each of Hobbes's causes.
1
List what each person can do
Peace or attack. Hobbes's conclusions follow from this much, so a third option has to be argued for.
2
Order the four outcomes by what each would rather have
Ask what an attacker gains from a peaceful victim. If the gain is more than peace, the cause is competition; if it is only safety, the cause is diffidence.
3
Find the equilibria
Check dominance first, then the cells where both are best-responding. Anarchy by competition has one, and anarchy by diffidence has two.
4
Subtract the sovereign's penalty
Take it from every payoff in which the person attacks, and leave the victim's payoff alone.
5
Find the equilibria again
If peace now dominates, the sovereign's penalty is large enough. If attack still pays against peace, it is not.
Anarchy and the penalty
Score 0 / 0Choose the answer you can justify.
Competition, then a penalty of four
Payoffs for peace with peace 3, 3; an attacker on a peaceful victim 5, 0; two attackers 1, 1.
Anarchy. Other keeps peace: attack 5, peace 3 -> attack
Other attacks: attack 1, peace 0 -> attack
Only equilibrium: (attack, attack), 1 each
Penalty 4 off every attack payoff:
Other keeps peace: attack 1, peace 3 -> peace
Other attacks: attack -3, peace 0 -> peace
Only equilibrium: (peace, peace), 3 each, no fine paid
Nothing in either table says the people have changed. The sovereign has changed what an attack costs, and with it what each does in reply to the other.
Common mistakes
1
Thinking Hobbes took people to be wickedThe diffidence table has no gain from attacking a peaceful neighbour: attack pays 3 whether the other attacks or not, and peace with peace pays 4. Yet attack with attack is still an equilibrium, because anyone who doubts the neighbour's peace does better to strike first. The model needs only that people are rational and unsure of each other.
2
Expecting the sovereign to be the cause of peaceThe sovereign changes payoffs, and peace follows from the new table. In the penalty table the fine is never paid. If it is too small, as 1 is, attacking still pays and the war continues under a ruler.
3
Reading the model as a history of how states beganThe tables say what follows from a given set of payoffs and say nothing about whether any people were ever in that condition. Hobbes himself treats the state of nature as what would follow without a common power, and the model makes that claim checkable.
Check your understanding
✓
Finish when you can show how a penalty moves an equilibrium.
Write the state of nature as a two-by-two table, find its equilibria, subtract a stated penalty from the attack payoffs, and say whether peace now dominates and, if it does not, how large the penalty must be.
Note: Hobbes's glory, which makes men fight for reputation, has no payoff to attach to it here, and the lab leaves it out rather than invent one. The two people stand for any pair; the next lessons ask what changes when ten take part.