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Lecture 12

Lecture 12. 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 D ealers outcome completely random (no strategy)

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Lecture 12

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  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 • 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 friendsselect 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

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