4-24-2012Lecture 4:1I. Uncertainty about OutcomesA. Nature NodesB. Probabilities Sum to OneC. Expected ValueII. Sledding DecisionA. Decisions vs. GamesB. Ex post regretsIII. Asymmetric Uncertainty Rollback is due to strategic anticipation.I. Aid No(0, 0) Edu Limos(2, 2) prob = 0.5 0.5Riot Not (-2, -2) (-2, 5)o Solve the Nature Node by calculating Expected Values of the payoffs.o Nature is not a strategic player, so there are no payoffs, but probabilities.o If X (random variable) takes value X1 with probability P, and value X2 with probability 1-P, then EX = PX1 + (1-P)X2EUp(Limos) = .5 (-2) + (1 - .5)(5) =.5 (-2) + 0.5 (5) = 3/2[Expected Utility]- Take uncertain situations and make decisions by comparing probabilities and payoffs (Expected Value)- Every part of uncertainty is included in Exo 3 uncertainties that can be figured out via ExNature This is another node not controlled by players, but controlled by nature.You may want to add population as one strategic player but he is not acting rationally.PoorRich how good is the good? how bad is the bad? how likely is the good to happen?Aid No(0, 0)Pareto Optical Outcome Edu Limos (-2, 3/2)(2, 2) prob = 0.5 0.5Riot Not (-2, -2) (-2, 5)EUp (Limos) = .5 (-2) + .5 (5) = 3/2- After you caluculate Ex, use it like a real number- NEVER highlight choice in the nature node- Rollback Equilibrium: Rich: Aid Poor: Edu if Rich Aid- don't need to say anything about natureWhat if the probability is not 0.5 | 0.5 ?Aid No(0, 0) Edu Limos(2, 2) p 1-pNaturePoorRichNaturePoorRichRiot Not (-2, -2) (-2, 5)EUp (Limos) = -2p + 5(1-p) = 5 - 7pEUR (Limos) = p (-2) + (1-p) (-2) = -2p + (-2) +2p = -2This is not really necessary if both payoffs are the same.At node 2, Poor chooses Limos if... At node 2, Poor chooses Edu if...Case 1: 5 - 7p > 2 Case 2: 5 - 7p < 2p < 3/7 p > 3/7RE: Rich: No Aid RE: Rich: Aid Poor: Limos if Rich Aid Poor: Edu if Rich AidAid No(-2, 5-7p) (2, 2) (0, 0) Edu Limos(-2, 5-7p)(2, 2) p 1-p Riot Not (-2, -2) (-2, 5) Aid No(0, 0) Edu LimosNaturePoorRichNaturePoorRich(2, 2) = 0.66 0.33Riot Not (-2, -2) (-2, 5)0.33 is pretty high!you could've gotten away with itII. Decision Theory ≠ Game TheoryDecision Trees: There are no politics. Decisions are simpler than games.Ex post regrets: "after the fact"Ex ante regrets: "before the fact"Allow Not 0p 1-p Crash Not-10 5How high can p be to make "Allow" the right choice?Ex ante regretsEUM(Allow) = p(-10) + (1-p)5 = -15p + 5Allow if -15p + 5 > 0 p < 1/3Don't allow if p > 1/3The kids are not included because this is not a game.Mom's probability = 1/10 so you did the right thing. Allowed. but he crashed so ex post regrets. This is the best use of information you have but still comes out wrong and the son