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QR 38 4/10 and 4/12/07 Bayes’ Theorem I. Bayes’ Rule II. Updating beliefs in deterrence III. Hegemonic policy. How to address the potential for learning: using observed actions of others to update beliefs about their type?
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QR 38 4/10 and 4/12/07 Bayes’ Theorem I. Bayes’ Rule II. Updating beliefs in deterrence III. Hegemonic policy
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. I. Bayes’ Rule
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. Updating beliefs
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. Conditional probabilities
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? Bayes’ Rule
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 Genetic test example
How does Bayes’ Rule help us to understand how beliefs change in IR? • Consider deterrence • Three types of deterrence: • General: prevent any change to SQ (India-Pakistan over Kashmir) • Extended: deter attacks on third parties (U.S. protection of W. Europe during Cold War) • Extended immediate: deter attack on third party during a crisis (Berlin) II. Updating beliefs in 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. Deterrence
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 (I) 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) Iraq example
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. Terrorism
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 III. Hegemonic policy
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 and hegemony
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
OPEC game 0, a Obeys b, 0 Ally Acquiesces Challenges Hegemon Punishes b-1, -xt
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
Four cases (equilibria) result, depending on the value of the temptation facing allies (b): • 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 OPEC game results