SelfEnforcing Social Contracts

1
Self-Enforcing Social Contracts

• Lecture 5

2
Outline

• Hobbes on Anarchy
• Locke on Anarchy
• Anarchy as a one-shot PD
• Anarchy as a repeated PD
• Cooperation Under Anarchy
• Thucycides on the Plague in Athens

3
Hobbes on Anarchy
In such condition, there is no place for
Industry because the fruit thereof is uncertain
and consequently no Culture of the Earth no
Navigation, nor use of the commodities that may
be imported by see no commodious Building no
Instruments of moving and removing such things as
require much force no Knowledge of the face of
the Earth no account of Time no Arts no
Letters no Society and which is worse of all
continuall feare, and danger of violent death
And the live of man, solitary, poore, nasty,
brutish, and short. Hobbes, Leviathan,
p.186
4
Hobbes on International Relations
But though there had never been any
time, particular men were in a condition of
warre one against another yet in all times,
Kings, and Persons of Soveraigne authority,
because of their Independency, are in continuall
jealousies , and in the state and posture of
Gladiators having their weapons pointing, and
their eyes fixed on one another that is their
Forts, Garrisons, and Guns upon the Frontiers of
their Kingdomes and continuall spyes upon their
neighbours which is a posture of War.
5
One-shot PD
Player B Player A

• If VHgt VC gt VN gt VL then
• Nash (DD)
• Pareto Eff. (CC)
• In one-shot PD, the Nash equilibrium is not
  Pareto Efficient.

6
Locke on Anarchy
And there we have the plain difference between
the state of nature and the state of war, which
however some men have confounded, are as
distant, as the state of peace, good will, mutual
assistance and preservation, and a state of
enmity malice, violence and mutual destruction,
are one from another. Men living together
according to reason, without a common superior
on earth, with authority to judge between them,
is properly the state of nature. But force, or a
declared design of force, upon the person of
another, where there is no common superior on
earth to appeal to for relief is the state of
war. Locke. Second Treatise. Chap III 19.
7
Locke on International Relations
It is often asked as a mighty objection, where or
ever were there any men in such a state of
nature? To which it may suffice as an answer at
present, that since all princes and rulers of
independent governments all through the
world, are in a state of nature, it is plain the
world never was not ever will be, without numbers
of men in that state. Locke Chap II 14.
8
Locke on Enforcement
And that all men may be restrained from invading
others rights, and from doing hurt to one
another, and the law of nature be observed, which
willeth the peace and preservation of all
mankind, the execution of the law of nature is,
in that state, put into every mans
hands, whereby every one has a right to punish
the transgressors of that law to such a degree,
as many hinder its violation. For the law of
nature would, as all other laws that concern men
in this world, be in vain if there were no body
that in the state of nature had the power to
execute that law. Locke Chap II 7
9
Locke on Enforcement (continued)
but where there is no such appeal to a common
superior , as in the state of nature, for want
of positive laws, and judges with authority to
appeal to, the state of war once begun,
continues with the right of the innocent party
to destroy the other whenever he can, until the
aggressor offers peace and desires reconciliation
on such terms as may repair any wrongs he has
already done, and secure the innocent for the
future. Locke Chap III 20.
10
Repeated PD

• Let t 0,1,2, denote a period of time
• In each period, players A and B play a single
  round of prisoners dilemma.
• There is a constant probability 0 lt k lt 1 that a
  player will survive until the next period.
• Assume that if a player dies, another will step
  into her place at the next round of play.

11
Action set, history, strategy

• Let Xi C,D be the action set for player i
  (iA,B).
• Let ht (CC)1, (DC)2, (DD)t-1 be a history
  of the game up to but not including period t.
• A strategy si for player i specifies an action
  from Xi at every period t after every possible
  history of the game ht.

12
Examples of strategy

• Define sA as the following strategy
• Cooperate all the time
• Defect all the time
• Grim trigger
• in the first period play C.
• In all other periods, play C if the other player
  played C in all previous periods, and play D if
  ever the other player played D.
• sA C if ht(CC)1(CC)t-1 and D otherwise.

13
Is Grim Trigger a Nash?

• sA in the first period play C. In all other
  periods, play C if player B played C in all
  previous periods, and play D if player B ever
  played D.
• sB in the first period play C. In all other
  periods, play C if player A played C in all
  previous periods, and play D if player A ever
  played D.
• Is the strategy profile s (sA, sB) a Nash
  equilibrium?

14
Incentives to Deviate

• What are the possible deviations?
• Deviations following a history of cooperation
• ht (CC)1, (CC)2, (CC)t-1
• 2) Deviations following a deviation from
  cooperation
• ht (CC)1, (CC)2, (CC)t-2, (CD)t-1

15
Deviations following a history of cooperationht
(CC)1, (CC)2, (CC)t-1
Proposed Nash

• VC k(VC k(VC k(VC )))
• VC/(1-k).
• EuA(sA,sB)

• EuA(sA,sB)

VH k(VN k(VN k(VN ))) VH
kVN/(1-k).
One-time Deviation by A
Note we are assuming that there are more than
two players so that a player does not have to
worry about his current partner survival rate.
16
1) Deviations following a history of cooperation
(continued)

• A does not deviate if
• EuA(sA,sB) ? EuA(sA,sB)
• VC/(1-k) ? VH kVN/(1-k)
• ?
• By symmetry, the same is true for player B.

17
2) Deviation following a deviation from
cooperation ht (CC)1, (CC)2,
(CC)t-2,(CD)t-1

• EuA(sA,sB)
• VN k(VN k(VN k(VN)))
• VN /(1-k)
• - EuA(sA, sB)
• VL k(VN k(VN k(VN )))
• VL kVN/(1-k).

One-time Deviation by A
18
2) Deviation following a deviation from
cooperation (continued)

• A does does not deviate if
• EuA(sA,sB) EuA(sA,sB)
• VN /(1-k) VL VN k/(1-k)
• ?VN VN k/(1-k) VL VN k/(1-k)
• ?VN VL,
• which is true by assumption.
• By symmetry, the same is true of player B.

19
Results

• For k (VH-VC)/(VH-VN), no player has an
  incentive to deviate following either a history
  of cooperation or one of defection.
• Consequently, s (sA, sB) is a Nash equilibrium.

20
Implications

• High temptation payoff (VH-VC ) makes cooperation
  harder.
• Harsh punishment threat (VH-VN ) makes
  cooperation easier.
• Low k makes cooperation harder
• second-term administration
• graduating senior
• the plague!

21
Thucydides and the Plague
For the catastrophe was so overwhelming that
men, not knowing what would happen to them next,
became indifferent to every rule of religion and
law. Thus they resolved to spend their money
quickly and to spend it on pleasure, since money
and life alike seem equally ephemeral. As for
what is called honour, no one showed himself
willing to abide by its laws, so doubtful was
it whether one would survive to enjoy the name of
it. It was generally agreed that what was
honourable was the pleasure of the moment. No
fear of god or law of man had a restraining
influence. Thucycides, Book II 52.