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