Week 7 - Chap 7A

State prob addition and multiply rules, X and P meaning

1. X1 and X2 disjoint elements, probability that EITHER X1 or X2 occur = addition

2. Probability that BOTH X1 and X2 occur = multiply

3. Random variable = {possible values xi​ and their probabilities pi​}

• X = possible outcomes (must be numbers)

• P = probability of each outcome happening

State expected value equation

• Average payoff you would get in the long run if the game were repeated many times

• Possible outcomes x1, x2, …, xn 

• Each with a probability of happening p1, p2, …, pn 

State NE of mixed

List of mixed strategies, one for each player, such that choice yields highest expected payoff, given the mixed strategies of the other players

Explain general way to find mixed strategy equilibria

1. Strategies not mixed MUST give worse payoffs 

2. Player must be indifferent between all strategies being mixed 

3. Find expected payoffs 

4. Equate E for different strategies 

5. For graphs, draw table and consider inequalities

Describe pure strategy

1. Pure strategy = choose one action with certainty (probability = 1) 

• They are degenerate mixed strategies

• Pure strategy = special case of mixed strategy where probability = 1 for one action (normal NE)

Describe mixed strategy

1. Mixed strategy = randomise between pure strategies using probabilities

• Frequency at which such an event has occurred over a large number of observations 

• Support for mixed strategy = pure strategies that are played with positive probability

• Eg. Play A with 0.7, Play B with 0.3 -> Support = {A, B}

• Eg. Play A with 1, Play B with 0 -> Support = {A}

State conditions for mixed strategy equilibria

1. All strategies in the support (mixing bunch) give equal payoff = indifferent condition 

2. Payoff of unused strategies ≤ Expected payoff of used strategies

Compare asymmetric and symmetric equilibria

1. Pure strategy Nash equilibrium (asymmetric equilibrium): One man chooses blonde, everyone else chooses brunette

2. Symmetric mixed equilibrium (same for each player)

Prove mixed strategy condition: All strategies in the support (mixing bunch) give equal payoff = indifferent condition is fufilled for P1

Prove mixed strategy condition: All strategies in the support (mixing bunch) give equal payoff = indifferent condition is fufilled for P2

Prove mixed strategy condition: Payoff of unused strategies ≤ Expected payoff of used strategies