4-17-2012Lecture 3:1I. Variable PayoffsA. Interpreting cases and subcasesB. Mix of variables and numbers***C. Knife Edge casesII. Evaluating Outcomes: Pareto CriteriaA. Pareto ImprovementB. Pareto InefficientC. Pareto Efficient***MOST IMPORTANTThe hardest part is figuring out where to use variables.From now on, we'll be using a mix of variables and numbers.Assume every variable is positive (+).I. Variable PayoffsRF Not RF Not RF NotInc wins Inc wins Chal wins Inc wins(VI-CI , -Cc) (VI-CI , 0) (0 , Vc-Cc) (VI , 0) idea of baseline is needed → using 0 instead of another variableVariables: VI : value of winning to IncVc : value of winning to ChalCI : cost of fundraising to IncCc : cost of fundraising to Chal Assuming: VI , Vc , CI , Cc > 0IncChal Chalcan be differentcost = general & inclusive.$, time ; everything they don't likethink of it as a (+) that we can subtract0 = baseline payoff (not winning & not RF)We get more from solving the game with variables because then the game represents an independent number of games.IA. Interpreting cases and subcasesWe are interested in seeing cases. Be sure to number the different nodes when doing cases.To solve the game, we still use the rollback technique. 1.RF Not(VI-CI , 0) 2. 3. RF Not RF NotInc wins Inc wins Chal wins Inc wins(VI-CI , -Cc) (VI-CI , 0) (0 , Vc-Cc) (VI , 0) Case 1 : Vc > CcChal choose RF at node 3Subcase 1-a : V I > C IRollback Equilibrium: Inc RF Chal N if RF , RF if NSubcase 1-b : V I < C IRollback Equilibrium: Inc N Chal N if RF , RF if N*subcase 1-a & 1-b are different because Inc acts differently. Inc's action activates different parts of Chal's equilibrium Case 2 : Vc < CcChal choose N at node 3Inc choose N at node 1Rollback Equilibrium: Inc N Chal N if RF , N if N*Chal's choice doesn't depend on Inc. This works to Inc's advantage. Chal is always going to N.*doesn't matter whether VI > CI or VI < CIIncChal Chalwhen Vc > Cc AND VI > CICase 1 : (0 , Vc-Cc)Case 2 : (VI , 0)*This is TRICKY. check this with Vc,I = 7 Cc = 10CI = 2 or CI = 10Case 1 and 2 : never going to have a balanced competition between Inc and Chalnever both RF*You won't know the cases until you start solving.IB. Mix of Variables and Numberson HW #2! COMMON.IC. Knife Edge casesVI = CI , Vc = Cco You can ignore the Knife Edge cases before the midterm.o Knife Edge = narrow caseo each option is exactly the same payoffo game theory = not helpful. not that useful in sequential gameo change all < / > to ≤ / ≥ → all in equilibrium o Neither option is wrongex. Case 1 : Vc ≥ Cc Case 2 : Vc ≤ Cc 1-a : VI ≥ CI 1-b : VI ≤ CIII. Evaluating Outcomes: Pareto Criteria RF Not(5 , 0) (0 , 5) RF Not RF NotInc wins Inc wins Chal wins Inc wins(5 , -2) (5 , 0) (0 , 5) (7 , 0) o Predicted outcome: well-funded Inc , anemic Chalo Pareto Criteria says that the predicted outcome is a bad outcome in the world.IncChal Chalo (7 , 0) is a Pareto Improvement over (5 , 0) because one player (Inc) is better off and the other is not worse off.o value judgements. utilities costs We can improve one while not hurting the