QR 38

4/10 and 4/12/07

Bayes’ Theorem

I. Bayes’ Rule

II. Updating beliefs in deterrence

III. Hegemonic policy

I. Bayes’ Rule

How to address the potential for learning: using observed actions of others to update beliefs about their type?

Use a mathematical formula, Bayes’ Rule. This provides a way to draw inferences about underlying conditions from actions that we observe.

Updating beliefs

In the U.S.-Japan trade game, US bases its strategy on its beliefs about whether J is a cooperative type or not.

• In that simple game, no opportunity for US to observe J’s behavior and improve its information

• But US may be able to observe something relevant, e.g., J behavior in another trade dispute.

Updating beliefs

• If US can observe J’s behavior, rational to use this information to develop more precise estimates about the probability that Japan is cooperative.

• Bayes’ Rule (or Theorem, or Formula) gives us a way to draw inferences about underlying conditions (type) from actions that we observe.

Conditional probabilities

Remember the concept of conditional probabilities: the probability of something happening given that some other condition holds.

• Here, we are interested in the probability that a player is of a certain type conditional on observed actions.

Bayes’ Rule

Notation:

• O=observation

• C=condition (type)

• |=“given”

• p(C|O) is what we care about: the probability that a player is of a certain type (the condition) given an observation

Bayes’ Rule

• Prior beliefs = p(C) (also called initial beliefs)

• p(C|O) = posterior or updated beliefs

How do we get to these updated beliefs?

Genetic test example

D&S example:

• Test for a genetic condition that exists in 1% of the population.

• The test is 99% accurate. – If you get a negative result, the chance that is

it wrong is 1%– If you get a positive result, the chance that it

is wrong is 1%.

Genetic test example

Assume 10,000 people take the test.

• 100 of these (1%) will have the defect.

• Of these 100, 99 will get a correct positive test result.

• Of the 9,900 without the defect, 99 (1%) will get a false positive.

• So of the positive test results within this group, only 50% are accurate.

Genetic test example

• Using the above notation, write p(C)=.01 (1%)

• p(C|+)=.5 (50%)

Baye’s Rule:

p(C|O)=

p(O|C)p(C)/(p(O|C)p(C)+p(O|~C)p(~C))

• ~C reads “not C”

Genetic test example

In this example, let O=a positive test

• p(C)=.01

• p(~C)=.99

• p(O|C)=.99

• p(O|~C)=.01

Genetic test example

Plug into Bayes’ rule:

p(C|O) = .99(.01)/(.99(.01)+.01(.99))

=.0099/(.0099+.0099)

=.5

Genetic test example

Let O=a negative test• p(O|C)=.01• p(O|~C)=.99• p(C|O)=.01(.01)/(.01(.01)+.99(.99)) =.0001/(.0001+.9801) =.0001 (approximately)• So, the probability of having the defect

given a negative test result is about 1 in 10,000

II. Updating beliefs in deterrenceHow does Bayes’ Rule help us to

understand how beliefs change in IR?• Consider deterrenceThree types of deterrence:1. General: prevent any change to SQ

(India-Pakistan over Kashmir)2. Extended: deter attacks on third parties

(U.S. protection of W. Europe during Cold War)

3. Extended immediate: deter attack on third party during a crisis (Berlin)

Deterrence

In deterrence, the central problem is the credibility of the defender’s threats.

• Determining credibility means determining the defender’s type: tough or weak?– Will threats really be carried out?

• Challenger has some prior beliefs about defender’s type (e.g., 50-50).

• Then uses observations of defender to update– Force structure, other crises

Deterrence

Need to calculate the challenger’s posterior probability (updated belief) in order to determine whether a challenge is likely to lead to a response.

Iraq (I) example

Example of whether Saddam Hussein believed the Bush (senior) would actually carry out an attack against Iraq if Iraq invaded Kuwait.

• Was Bush bluffing?

• Bush could be one of two types: weak or tough

Iraq example

Prior:

• p(w)=0.7

• p(t)=0.3

Bush first had to decide about an air war, then a ground war.

• Decision on the first provided information about the credibility of the second.

Iraq example

• p(A|w)=0.5

• p(~A|w)=0.5

• p(A|t)=1.0

Observe A.

• What is the posterior, p(w|A)?

Iraq example

p(w|A) = p(A|w)p(w)/(p(A|w)p(w)+p(A|t)p(t))

=.5(.7)/(.5(.7)+1(.3))

=.35/(.35+.3)

=.54

Here, the observation=an air attack; the condition=weak; want p(C|O)

Terrorism

BdM also applies this logic, less formally, to terrorism.

• Assume that terrorists are trying to decide whether US is responsive (willing to negotiate) or repressive (not)

• Terrorists observe US unwillingness to negotiate in other crises, or the stated policy of no negotiations

Terrorism

• Then the terrorists’ posterior probability that the U.S. is a repressive type will go up.

• If terrorists in fact prefer negotiations to terror, they will then be discouraged and turn to terror instead.

• Note that in this analysis BdM neglects reputational effects with other terror groups.

III. Hegemonic policy

Can also apply this model of signaling to “hegemonic stability”:

• Hegemonic stability is the idea that stability in IR results from the ability of a hegemon (a single powerful state) to create stability.– May create stability through coercion– Through side-payments– Through creation of institutions

OPEC and hegemony

Apply hegemonic stability to OPEC: • Saudi Arabia is the hegemon, with the largest

share of oil reserves• Stability defined as a stable price for oil• Saudis enforce production limits with threat of

increasing its own production and driving prices down; but this is costly for the Saudis.

• Are Saudi threats to punish in order to enforce the cartel credible?

OPEC game

Consider a game played over two periods where the hegemon has an opportunity in the first period to build a reputation for being tough.

• The hegemon faces a potential challenge from an ally (another OPEC member) in each period.

OPEC game

Ally

Obeys

ChallengesHegemon

Acquiesces

Punishes

0, a

b, 0

b-1, -xt

OPEC game

Assume:

• 0<b<1 (ally benefits from acquiescence)

• a>1 (hegemon prefers that ally obeys)

• xt = {1 with probability w

0 with probability 1-w

• So w is the prior probability that the hegemon is weak and will bear a cost from punishing (x)

OPEC game• The game is played twice• A second ally observes the action of the

hegemon in the first round and updates w.• We want to calculate updated beliefs: p(w|

acquiesce) and p(w|punish).• Use Bayes’ rule to do this; look for a

Bayesian equilibrium.• Beliefs must be updated in a reasonable

way• Beliefs and actions must be consistent

OPEC game results

Four cases (equilibria) result, depending on the value of the temptation facing allies (b):

1. Very low b means little benefit from challenging, so there is no challenge in equilibrium.

-- Even if an out-of-equilibrium challenge did occur, the hegemon would not punish because there is little need for deterrence

OPEC game results

2. Slightly higher b: ally still afraid to challenge.

-- But if an irrational challenge did occur, the hegemon would respond because deterrence is now necessary.

OPEC game results

3. Still higher b: hegemon needs to establish a reputation. – Uses a mixed strategy: responds to any

challenge probabilistically. – Depending on the value of b, Ally 1 may or

may not be deterred.– If Ally 1 is deterred, Ally 2 challenges,

because the hegemon has had no chance to build a reputation by punishing

– If the hegemon punishes a challenge by Ally 1, Ally 2 adopts a mixed strategy.

– So deterrence sometimes works

OPEC game results

4. High b: allies always challenge.– Hegemon never punishes, since there is no

point in building a reputation.

In case 3, why does the hegemon use a mixed strategy?

– It is useful to keep allies guessing– If the hegemon always punished in

round 1, punishment would convey no information