Political Science 30Study Guide for MidtermKey Concepts1) Sequential Games2) Solving Game Trees3) Rollback Equilibrium4) Strategy5) "Off the Equilibrium Path"6) Pareto Efficient and Inefficient7) Nature Nodes8) Expected Value9) Ex Post Mistake1) Sequential GamesSequential Moves: "The moves in a game are sequential if the rules of the game specify a strict order such that at each action node only one player takes an action, with knowledge of the actions taken (by others or himself) at previous nodes."To be "strategic":1) disagreement2) interdependency3) anticipation Game Theory:1) Preferences2) Strategies3) OutcomesGame Trees are how we solve sequential games.2) Solving Game TreesGame Tree: "Representation of a game in the form of nodes, branches, and terminal nodes and their associated payoffs."decision node: choiceRF Not branch: one option terminal nodes: 4 optionsRF Not RF Not3 decision scenarios(Inc: 1 , Chal: 2)Outcomes connect strategies to preferences; they are the stakes of situations.*Outcomes don't actually appear in the Game Tree. The payoffs are the communication aspect of the game theory; add up the numbers and/or variables that represent the stakes. Payoffs, or utilities, are part of an individualistic framework, so you can never compare one person's payoff to another. You should assume that each player already compared her own payoffs, but be sure to never compare within the paranthesis.Payoff: "The objective, usually numerical, that a player in a game aims to maximize."Counterfactuals are what could happen but does not.RF NotStrategic Equivalents (5, 0) (0, 5)counterfactual RF Not RF NotInc wins Inc wins Chal wins Inc wins(5, -2) (5, 0) (0, 5) (7, 0) payoffs*SET UP THE WHOLE GAME (TOP-DOWN & MAKE PAYOFFFS)BEFORE SOLVING IT (BOTTOM-UP. BECAUSE OF THE ANTICIPATED)3) Rollback EquilibriumRollback: "Analyzing the choices that rational players will make at all nodes of a game, starting at the terminal nodes and working backward to the initial node."aka Backward InductionChalIncChal ChalChalIncRollback Equilibrium: A set of strategies, one for each player, in which each player takes her bestaction at each node given the choices of the others'.4) StrategyStrategies are complex: multiple parts (if ___ RF then ___ )Actions are simple: ordinary language (will not RF)Suppose player has n nodesLet xi be the # of branches at node iPlayer's total # of strategies = xi x x2 x ... xnL M R L R L R L RStrategies:1: L, M, R (3)2: L L L R R R 2 x 2 x 2 = 85) "Off the Equilibrium Path"Equilibrium Path(what happens) RF Not(5, 0) (0, 5) Off the Equilibrium PathRF Not RF Not (but part of RE)IncChal Chal12 22Inc wins Inc wins Chal wins Inc wins(5, -2) (5, 0)(0, 5) (7, 0) Chal's Equilibrium StrategyN if Inc RF , RF if Inc NOff the Equilibrium PathEquilibrium Path (counterfactual usually)(what we expect to play) explains why other player makes her choice*important in understanding / analysis 6) Pareto Efficient and InefficientEfficient Allocation aka Pareto Efficient: "An allocation is called efficient if there is no other allocation that is feasible within the rules of the game, and if it yields a higher payoff to at least one player without giving a lower payoff to any player."RF Not RF Not RF Not(5, 2) (5, 0) (0, 5) (7, 0) Pareto Improvement: (5, 0) to (7, 0) Aid Not(0, 0) Edu Limos(2, 2) (-2, 5)IncChalChalRichPoor (2, 2) and (-2, 5): Pareto Efficient Pareto Improvement: (0, 0) to (2, 2) Rollback Equilibrium: rich: Not poor: Limos if AidIf there is no Pareto Improvement possible, then the situation is Pareto Efficient.It is possible to have all Pareto Efficient.7) Nature NodesAid 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 Ex how good is the good? how bad is the bad?RichPoorNature 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. 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 the 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-pRiot Not (-2, -2) (-2, 5)RichPoorNatureRichPoorNatureEUp (Limos) = -2p + 5(1-p) = 5 - 7pEUR (Limos) = p (-2) + (1-p) (-2) = -2p + (-2) +2p = -2 This 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 Limos(2, 2) = 0.66 0.33Riot Not (-2, -2) (-2, 5)0.33 is pretty high!RichPoorNatureRichPoorNatureyou could've gotten away with it8) Expected ValueExpected Value: "The probability-weighted average of the outcomes of a random variable, that is, its statistical mean or expectation."aka Expected Utility. (see above)9) Ex Post MistakeEx post regrets: "after the fact" Ex ante regrets: "before the fact"Decision trees are much simpler than game trees.Allow Not 0p 1-p Crash Not-10 5EUM(Allow) = p(-10) + (1-p)(5) = 5 - 15pThe expected value allows us to compare to other uncertainty or certainty.Case 1: p < 1/3 Allowoptimal decision: allow when 5-15p > 0p < 1/3crashes happen with probability p sometimes it is easier to assign a number. p = 1/10Case 2: p > 1/3 Not Allowoptimal decision: not