1. Lecture 12

2. Game theory So far we discussed: roulette and blackjack Roulette: – Outcomes completely independent and random – Very little strategy (even that is curbed by minimums and maximum bet rules) Blackjack – Dealers outcome completely random (no strategy) – There is some dependence between draws – Strategy can be useful (counting cards and adjusting play accordingly)

3. Game theory Poker – All the participant are following a strategy that uses the information based of other people What is a rational (best?) strategy? MAXIMIN – Select a strategy S – If the opponent new my strategy, what is the worst they can do to me? – Select the strategy that maximizes this worst case scenario

4. Understanding role of strategy Coin matching game: You and your friend each have a coin. You each secretly choose to show “heads” or “tails”. You simultaneously show your choice. The payoff: – If the coins are both “heads”, your friend pays you $1. – If the coins are both “tails”, your friend pays you $9. – If the coins do not match, you pay your friend $5. Is this a fair game?

5. Coin matching game you\friendHT H1-5 T 9 If both of us flip a coin: Expected gain = 1*0.5*0.5+(-5)*0.5*0.5+(-5)*0.5*0.5+9*0.5*0.5=0 (Looks fair but is it?) We do not need to flip a fair coin – we can choose any proportion of H/T we want! Consider our proportion of H p and friends proportion of H q. For now assume independence between us and friend Expected gain = 1*p*q+(-5)*p*(1-q)+(-5)*(1-p)*q+9*(1-p)*(1-q)=9-14p-14q+20pq How to select p?

6. Maximin Main idea: – We want select p to maximize expected win if positive we might also want to minimize variance – What is best depends on the friends strategy: (Depends on friends choices unknown to me) If I select p – if my friends selects all H gain =9-14p all T gain=-5+6p

7. Maximin Notice that smaller of the two lines is less than -0.8 The best option is p=0.7 – gain =-0.8 

8. Minimax (maximin for friend) Friends choices – What is best for the friend depends on my strategy (unknown to him) My friends select q – if I selects all H gain =9-14q all T gain=-5+6q Notice that larger (my gain his loss!) of the two less than -0.8 The best option is q=0.7 – expected gain =-0.8 (friend gains 0.8 on average) Nash equilibrium – if the other player is playing optimally there is nothing to gain from changing strategies