How to bet and win: and why not to trust a winner. Niall MacKay. Department of Mathematics

Transcription

1 How to bet and win: and why not to trust a winner Niall MacKay Department of Mathematics

2 Two ways to win Arbitrage: exploit market imperfections to make a profit, with certainty

3 Two ways to win Arbitrage: exploit market imperfections to make a profit, with certainty Prediction: make more accurate predictions than the market; have an edge

4 Advice for rookie punters Don t bet on the horse you think will win: bet on the horse that is undervalued

5 Advice for rookie punters Don t bet on the horse you think will win: bet on the horse that is undervalued Don t bet the bank on one horse: make small bets in proportion to the undervaluation

6 Our game Split into groups of four. One person is the bookie, and offers the others the following deal: I ll double your stake if the die comes up 1-4, but you lose your stake on 5-6.

7 Our game Split into groups of four. One person is the bookie, and offers the others the following deal: I ll double your stake if the die comes up 1-4, but you lose your stake on 5-6. Each of the other three begins with 1. How should they bet? Punter A: bets 10% of the bank. 1.1 on a win, 0.9 on a loss. Punter B: bets 33% of the bank on a win, 0.67 on a loss. Punter C: bets 50% of the bank. 1.5 on a win, 0.5 on a loss.

8 Our game Split into groups of four. One person is the bookie, and offers the others the following deal: I ll double your stake if the die comes up 1-4, but you lose your stake on 5-6. Each of the other three begins with 1. How should they bet? Punter A: bets 10% of the bank. 1.1 on a win, 0.9 on a loss. Punter B: bets 33% of the bank on a win, 0.67 on a loss. Punter C: bets 50% of the bank. 1.5 on a win, 0.5 on a loss. Now you all play 30 times.

9 Our game: results wins probability

10 Our game: results wins probability

11 Kelly Betting The fraction f of the bank you bet in each round should maximize (1+f) Np (1 f) N(1 p), where p is the chance of success.

12 Kelly Betting The fraction f of the bank you bet in each round should maximize (1+f) Np (1 f) N(1 p), where p is the chance of success. Take the Nth root, differentiate with respect to f and p(1+f) p 1 (1 f) 1 p (1 p)(1 f) p (1+f) p = 0 p 1+f 1 p 1 f f = 2p 1. = 0 Kelly betting: for the greatest long term return, maximize E(log(bank multiplier).

13 Whose tips should you follow? - which tipster should you trust? - trouble is, the bookie has good odds - and the bookie takes a cut (so that, in our game, they wouldn t double your money, they d multiply it by 1.9)

14 Whose tips should you follow? - which tipster should you trust? - trouble is, the bookie has good odds - and the bookie takes a cut (so that, in our game, they wouldn t double your money, they d multiply it by 1.9) - so the really good punter may choose not to bet if (s)he knows (s)he s estimated the odds well, and they do not differ from market odds by more than the margin - whereas even a poor punter may have a short winning streak: but don t think that makes him/her any good!

15 World Cup Niall MacKay BOOKIE Dan M Jason L David H MONKEY Mike B

16 How to choose a tipster Choose the one with the best probability scores. For example, the Brier score:

17 How to choose a tipster Choose the one with the best probability scores. For example, the Brier score: The punter assigns a probability to each outcome;

18 How to choose a tipster Choose the one with the best probability scores. For example, the Brier score: The punter assigns a probability to each outcome; the Brier score is the sum of the squares of the differences between these and the true outcome;

19 How to choose a tipster Choose the one with the best probability scores. For example, the Brier score: The punter assigns a probability to each outcome; the Brier score is the sum of the squares of the differences between these and the true outcome; better predictions lower (average) Brier score.

20 How to choose a tipster Choose the one with the best probability scores. For example, the Brier score: The punter assigns a probability to each outcome; the Brier score is the sum of the squares of the differences between these and the true outcome; better predictions lower (average) Brier score. e.g. You bet on Man U vs Wigan: P(Man U win) = 0.7, P(draw) = 0.2, P(Wigan win) = 0.1.

21 How to choose a tipster Choose the one with the best probability scores. For example, the Brier score: The punter assigns a probability to each outcome; the Brier score is the sum of the squares of the differences between these and the true outcome; better predictions lower (average) Brier score. e.g. You bet on Man U vs Wigan: P(Man U win) = 0.7, P(draw) = 0.2, P(Wigan win) = 0.1. Result is a draw, 0,1,0, : Brier score is (1 0.2) = 1.14 Man U win, 1,0,0, : Brier score is (1 0.7) = 0.14

22 How to choose a tipster Choose the one with the best probability scores. For example, the Brier score: The punter assigns a probability to each outcome; the Brier score is the sum of the squares of the differences between these and the true outcome; better predictions lower (average) Brier score. e.g. You bet on Man U vs Wigan: P(Man U win) = 0.7, P(draw) = 0.2, P(Wigan win) = 0.1. Result is a draw, 0,1,0, : Brier score is (1 0.2) = 1.14 Man U win, 1,0,0, : Brier score is (1 0.7) = 0.14 The MONKEY always chooses P = 1/3 and always scores (1 1/3) 2 +1/3 2 +1/3 2 = 2/3.

23 Probability scores Probability scores can tell you who is best at predicting...

24 Probability scores Probability scores can tell you who is best at predicting... sports results

25 Probability scores Probability scores can tell you who is best at predicting... sports results investment performance

26 Probability scores Probability scores can tell you who is best at predicting... sports results investment performance the weather

27 Probability scores Probability scores can tell you who is best at predicting... sports results investment performance the weather medical diagnoses and outcomes

28 Probability scores Probability scores can tell you who is best at predicting... sports results investment performance the weather medical diagnoses and outcomes...

29 Markowitz Portfolio Theory

30 Markowitz Portfolio Theory Markowitz theory is optimal for parallel investments Kelly betting is optimal for sequential investments

31 Economic Darwinism How should you select an investment adviser?

32 Economic Darwinism How should you select an investment adviser? Do not choose the one who s made most money; (s)he may merely have got lucky. Past performance is not an indicator of future success.

33 Economic Darwinism How should you select an investment adviser? Do not choose the one who s made most money; (s)he may merely have got lucky. Past performance is not an indicator of future success. The adviser whose predictions are most accurate may well have chosen not to invest, if they could not beat the bid/ask spread. D. Johnstone, Economic Darwinism: Who has the best probabilities? Theory and Decision 62 (2007) 47-96

34 The End