1. Lecture 10

2. Simplified roulette European roulette has numbers 0,1,…36. https://www.youtube.com/watch?v=4OeYU3_ xD0s https://www.youtube.com/watch?v=4OeYU3_ xD0s If you bet x on a number you get 35x if win on –x if loss. Simplification: no minimum or maximum bets

3. Dubins & Savage Problem You are given initial wealth $100 You can place any bets on the simplified roullette (European) Play until either bust or reach $500 What strategy would you use? R experiments continued

4. Variance and risk Variance measures how far can random variable stray from its mean If mean positive (return on stocks, bonds,…) big variance is bad – risk In our game – we want strategy that maximizes variance at each step, but does not overshoot the target

5. Greedy strategy for our game If all bets have the same mean (roullete is like this) at every step select the most variable bet (single number) Bet the amount that will take you exactly to the amount you need in one step (or everything you have if that is smaller). One can mathematically prove that this is the best strategy. (See handout).

6. Game of chance Roulette is a game of chance where consecutive results are INDEPENDENT – The chance of any outcome is unaffected by past results (no number is due…) The house has an average advantage of 1/37 (European) per dollar bet – Through clever betting we were able to make the advantage come down but not disappear. – Optional stopping theorem proves this mathematically – Casinos disallow ``clever’’ betting through minimum/maximum bets.

7. Blackjack The game depends on the status of the deck that changes through time While on average the casino has an edge. At some moments this edge might disappear – To use this one would need to count cards and use clever betting strategies – Casinos disallow card counting, mix several decks together and bring new card decks frequently

8. Goals The aim of the game is to accumulate a total close to 21. Players play against a dealer not agains each other. – Player who has a higher point than the dealer not exceeding 21 wins. The best possible blackjack hand is an opening deal of an ace with any ten-point card. This is called a "blackjack". If the player and dealer have the same point, this is called a "push” (draw). Scoring higher than 21 (called "bust") results in a loss.

9. Card Values Aces are valued as either 1 or 11 according to the player's choice. 2 -- 10-value cards have its numerical values. The jack, queen, and king (also known as "face cards" or "pictures") count as 10.

10. Basic Rules Betting is done before the players receive the cards. The players and the dealer receive two cards each. The dealer deals cards from her left to her far right. – The players' cards are dealt face up. – The dealer has one face down and one face up. Only the dealer can see the hole card (face-down card) at this point. – If a player and the dealer each have a blackjack, the result is a push for that player. – If the dealer has a blackjack, all players not holding a blackjack lose and the game is over.

11. Rules of play If the dealer doesn't have a blackjack, the game goes on. If the player has a blackjack, the player wins 1.5 times of the bet. Otherwise the game goes on with each remaining player taking action - in clockwise order starting to dealer's left. The player can keep his hand as it is (stand) or take more cards from the deck (hit), one at a time, until either the player judges that the hand is strong enough to go up against the dealer's hand and stands, or until it goes over 21, in which case the player immediately loses.

12. Rules for the dealer The dealer must hit if the value of the hand is lower than 17, otherwise the dealer will stand. If the dealer goes bust, all players who are left in the game win and the game is over. If the dealer and all players don't go bust, players or dealer with higher point totals close to 21 win and the game is over.

13. Insurance If the dealer's up-card is an ace, the player is offered the option of taking "insurance". – Insurance is a side bet that the dealer has blackjack and is treated independently of the main wager. The player may add up to half the value of their original bet to the insurance. After insurance, the dealer checks the hole card. If the hole card has a blackjack, players who are choosing an insurance receive twice the insurance bet but lose original bet. If not blackjack, players lose insurance bet and the game goes on.

14. Split When players get two starting cards of the same face value, they have the option to split the hand in two. You place another bet of the same size as the original bet and play on with two hands. When you've decided to split a hand, the dealer immediately deals a second card to each hand and the game goes on. If players split aces, they are dealt a second card to each hand as usual, but you are not allowed to take any further cards. You can split again but aces can not be split again.

15. Double-down Players are allowed to double down on the initial two-card hand. When players have chosen to double down, they make the initial bet double and they'll only get one more card from the dealer.

16. Surrender Surrender is only available as the first decision of a hand. – When the player surrenders, the house takes half the player's bet and returns the other half to the player. – Surrenders will not be allowed if the dealer has a blackjack. – Surrenders are not allowed in most casinos