PoliSci30 Week 3 Notes: Pure and Mixed Nash Equilibrium
PoliSci30 Week 3 Notes: Pure and Mixed Nash Equilibrium Poli Sci 30
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This 10 page Class Notes was uploaded by Christine Cheung on Saturday October 17, 2015. The Class Notes belongs to Poli Sci 30 at University of California - Los Angeles taught by Chwe in Summer 2015. Since its upload, it has received 33 views. For similar materials see Politics and Strategy in Political Science at University of California - Los Angeles.
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Date Created: 10/17/15
Review How do we write down a strategic form game First focus on relevant individuals Second Write down alternatives Third Assign payoffs Prediction Nash equilibrium Situation in which no one can gain from deviating Situation of best response Example 2a 2b 2c 1a 81 00 2 NE b c 1b 40 34 lc 19 2 68 Example protesting P2 protests P2 stays home Pl protests m m 5 0 P2 stays home 0 5 Q Q 0 NE protest protest and home home Big Question 3 Why is it difficult to overthrow a regime even if it is widely hated Issue of assurance you have to be pretty positive that everyone else is going to show up P2 protests P2 stays home Pl protests m m 5 0 P2 stays home 0 5 Q Q 0 It s not easy to get everyone to protest all at once to move from the bad NE to the good NE Example Battle of the Sexes Bibi goes to Bibi goes to Prizefight Opera Asha goes to Prizefight 2 l O O 0 NE prize ght prize ght and opera opera 0 Here there are two NE and they disagree about which one is better What about ties 2a 2b 1a 8 l 00 1b i6 0 NE la 2a and lb 2a and lb 2b 0 BUT lb weakly dominates la Thus iterative elimination of weakly dominated strategies con icts with Nash Equilibrium We have 2 methods of making a prediction in a strategic form game 0 Iterative elimination of strongly and weakly dominated strategies 0 Nash equilibrium 0 Can they con ict I Never for strong but possibly for weak I When you iteratively eliminate strongly dominated strategies you never eliminate Nash equilibrium This is because strongly dominated implies that there is an option that is always better the same as Nash equilibrium 0 Just be careful and pay attention Big Question 4 How can a single person create large social change Rosa Parks and the Bus P2 protests P2 stays home Pl protests m m l 0 P2 stays home O 100 O O 0 NE protest protest o By changing one person to make them courageous or reckless her 100 changes to l o This is like the Strictly Ballroom example everyone claps if one person starts it Change it so that person 1 gets sick and tired of staying home Her 0 9 101 P2 protests P2 stays home P1 protests m m 100 0 P2 stays home 101100 1010 0 NE protest protest 0 Again to make protest protest the only NE only one person has to be fed up How many people do you have to repress in order to prevent protest P2 protests P2 stays home P1 protests m m 100 0 P2 stays home O 100 O O 0 NE protest protest and home home 0 Say that person 1 gets a reward of 15 if you stay home P2 protests P2 stays home P1 protests 10 10 100 0 P2 stays home 15 100 150 I NE stay stay I If you want to eliminate protest protest NE you only have to pay off one person 0 Say that person 1 is punished if he protests P2 protests P2 stays home P1 protests 40 m 150 0 P2 stays home Q 100 M I NE home home I To eliminate the protest protest NE you only have to threaten one person 0 Guerrilla Warfare Example I The most important thing is shoes I Don t attack the whole army attack small sections I If you see the enemy walking in a line only re at the rst person If you always kill the rst person then no one will want to be the rst person A Beautiful Mind 0 Note everyone wants to invite the blonde but there are multiple Brunettes 2 invites Blonde 2 invites Brunette 1 invites blonde O O 10 4 1 invites brunette 4 10 5 5 0 Prediction I No strongweak domination I Nash equilibrium 2 invites Blonde 2 invites Brunette 1 invites blonde O O Q Q 1 invites brunette 4 m 5 5 I NE brunette blonde and blonde brunette I Both going for brunette is not a NE even though that s What Nash said would be best for everyone This is because NE is not about group bene t it s about individual bene t I This is like both the Arms Race example and the Chicken example We have 2 methods of making a prediction in a strategic form game 0 Iterative elimination of strongly and weakly dominated strategies 0 Nash equilibrium 0 Is that it Example Penalty Kick Striker East Striker West Keeper East 1 O O l Keeper West 0 l l O 0 Matching Pennies No Nash Equilibrium 0 Expand the game to include random strategies Striker East 5050 Striker West Keeper East 1 O O 1 7 5050 05 05 9 9 739 397 Keeper West 0 1 1 O Interlude Probability and expected value 0 The probability of a random event is the likelihood it will happen I A probability is a number between 01 I Add up the probabilities of all possible events to get 1 0 Example Roulette I Probability of ball landing in 26 is 138 Probability of ball landing in red is 1838 Probability of ball landing in 112 is 1238 I If I put 1 on red What is my average payout Probability of red is 1838 Some other color is 2038 If I win I get 2 If I lose I get 0 On average I get 2 X 1838 0 X 2038 3638 095 o The expected value is 3638 0 The expected value of a gamble is the probability of the event times the payoff from that event summed over all events I The expected value of betting on red in Roulette is 3638 I The expected value of betting a 26 138 x 36 3738 x 0 3638 I The expected value of betting on 112 1238 x 3 2638 x 0 3638 0 Example Rain I Today there is a 20 chance of snow 50 chance of rain and 30 chance of no precipitation I Payoffs Snow 10 rain 20 and no precipitation 0 I The expected value of my payoff is 020 x 10 050 x 20 030 x 0 8 Penalty Kick example Striker East 5050 Striker West Keepelsmiker East1 05050 r3 triker West 1 Keeper East 10 05 05 9 0 1 5050 05 05 5050 05 05 05 05 05 05 Keeper West 0 1 9 1 0 Keeper West 0 1 05 05 39 10 Work Keeper goes east always Keeper s payoff 12 0 12 l 12 Striker s Payoff 12 0 12 l 12 Keeper goes 5050 Keeper s payoff 12 0 12 l 12 Striker s payoff 12 0 12 l 12 Keeper goes west always Keeper s payoff 12 0 12 l 12 Striker s payoff 12 0 12 l 12 Work Striker goes east always Keeper s payoff 12 0 12 l 12 Striker s Payoff 12 0 12 l 12 Striker goes 5050 Keeper s payoff 12 0 12 l 12 Striker s payoff 12 0 12 l 12 Striker goes west always Keeper s payoff 12 0 12 l 12 Striker s payoff 12 0 12 l 12 We have 2 methods of predicting in a strategic form game Iterative elimination of stronglyweakly dominated strategies Nash equilibrium But what about predictions Striker East Keeper East 1 0 5050 05 05 Keeper West 0 1 Nash Equilibrium 5050 5050 Striker West LS 05 01 05 05 05 05 l0 So we predict that both players will go 5050 How do we include all possible randomized strategies p Keeper East lp Keeper West Striker East lq Striker West 0 1 10 Keeper goes east with prob p and west with prob lp The Striker goes east with prob q and west with prob lq game with all poss1ble randomized strategies looks like this Striker East Striker West 1 Keeper goes Q Q 0 East p1 How do we fill in all the payoffs Just fill in best responses amp Keeper West PO What s the keeper s best response Strikers Striker East Switchover Striker West Q1 O 05 Q 0 Keeper East P1 Switchover PO5 96 96 96 gtXltgtXltgtXltgtXltgtXltgtXlt 96 96 96 i Keeper West P20 NE Keeper goes E W prob 05 W Wprob 05 Striker goes E W prob 05 W Wprob 05 221 2b 1a 4 O O 3 No pure strategy NE Mixed Example Switchover for person 1 EU expected utility q lq 2a 2b EU1 1a 4xq 01q 4q p 1a 4 O O 3 lp 1b 17 62 EU21b 1q61qq6 6q65q What s the switchover q for person 1 4q 2 6 5 q Person 2 9q 6 2a Switchover 2b q 69 23 Q1 Q 23 QO 1a p21 Switchover for person 2 EU1 2a Op 71p 77p 96 EU2 2b 3p 2lp 3p 22p 2 13 gtlltgtllt gtlltgtlltgtlltgtlltgtlltgtllt Switchover p 58 7 7p 2 p 5 8p 1b p0 p 58 23 13 2a 2b 58 1a 40 03 38 lb 1 7 6 2 Mixed strategy NE 1 plays 1a w prob 58 lb wprob 38 2 plays 2a wprob 23 2b wprob 13 Example Beautiful Mind 01 lq 2 Blonde 2 Brunette p I approach Blonde 0 O 10 4 1p I approach Brunette 4 10 5 5 EU1 blonde Oq 101q 10 10q EU1 brunette 4q 51q 4q 5 Sq 5 q 10 10q 5 q 5 9q q 59 EU2 blonde Op 101p 10 10p EU2 brunette 4qp 51p 4p 5 5p 5 p 10 10p 5 p 5 9p p 59 Blonde Switchover Brunette Q1 Q59 QO Blonde p1 6 6 6 6 Switchover p59 gtXlt gtXlt gtXlt gtXlt gtilt gtilt NE 6 6 6 Brunette p0 Pure Strategy NE Brunette blonde blonde brunette Mixed strategy NE 1 plays blonde wprob 59 brunette wprob 49 2 plays blonde wprob 59 brunette wprob 49 To nd mixed NE do I always have to write down the box diagram No Once you get used to it you just have to calculate the switchover possibilities Conclusion and preview We know how to write down a strategic form game How do we make a prediction 0 Iterative elimination of strongly or weakly dominated strategies 0 Pure strategy Nash equilibrium 0 Mixed strategy Nash equilibrium That s it 0 John Nash proved in 1950 that these methods can nd predictions for any type of strategic form game However note that in a strategic form game we implicitly assume that each person chooses independently 0 When choosing your actions you can t respond to the actions of others 0 People choose their actions simultaneously 0 But in many situations you respond to the actions of others I People move in sequence I To model these we use an extensive form game