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/( ) =.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 (approximately) So, the probability of having the defect given a negative test result is about 1 in 10,000

II. Updating beliefs in deterrence How does Bayes’ Rule help us to understand how beliefs change in IR? Consider deterrence Three 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 Challenges Hegemon Acquiesces Punishes 0, a b, 0 b-1, -x t

OPEC game Assume: 0<b<1 (ally benefits from acquiescence) a>1 (hegemon prefers that ally obeys) x t = {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