Betting rules and information theory

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1 Betting rules and information theory Giulio Bottazzi LEM and CAFED Scuola Superiore Sant Anna September, 2013

2 Outline Simple betting in favorable games The Central Limit Theorem Optimal rules

3 The Game Consider a simple game of chance, like tossing a coin or rolling some dice, in which you bet an amount x and if you win you get 2x and if you loose you loose the entire bet. Let W 0 be the initial wealth so that if a fraction α is bet after the game you have { (1 α)w 0 + 2αW 0 = (1 + α)w 0 if you win W 1 = (1 α)w 0 if you loss How much would you like to bet?

4 Classification of games A game is favorable if the expected value of what you get is higher than what you bet. A game is fair if the expected value of what you get is equal to what you bet. A game is unfair if the expected value of what you get is lower than what you bet.

5 wealth maximization Let the probability to win be p. The expected wealth after participation into the game can be computed E[W 1 ] = (1 α)w 0 + p2αw 0 = (1 α + 2αp)W 0 If p = 2/3 then E[W 1 ] = (1 + α/3)w 0 and to maximize the expected wealth one must chose α = 1, so that E[W 1 ] = 4/3W 0. The choice α = 1 is made each time 2 p 1 > 0, that is for any favorable game.

6 Now suppose that the game is repeated T times. If one plays with α = 1 after T games it will have an expected wealth of E[W T ] = ( ) 4 T W 0 3

7 Now suppose that the game is repeated T times. If one plays with α = 1 after T games it will have an expected wealth of E[W T ] = ( ) 4 T W 0 3 but the probability to have nothing is equal to Prob{W t = 0} = 1 ( ) 2 T 3

8 Now suppose that the game is repeated T times. If one plays with α = 1 after T games it will have an expected wealth of E[W T ] = ( ) 4 T W 0 3 but the probability to have nothing is equal to Prob{W t = 0} = 1 ( ) 2 T 3 With T = 10 the expected wealth is E[W T ] 17 W 0 but the probability to have nothing is around 98%.

9 Consider the evolution of wealth in a sequence of repeated games { (1 + α)w t if ω t = 1 W t+1 = (1 α)w t if ω t = 0 where ω t is a random variable taking value 1 for a win and 0 for a loss. The sequence of random variables ω = (ω 1,..., ω T ) is a random process.

10 Consider the evolution of wealth in a sequence of repeated games { (1 + α)w t if ω t = 1 W t+1 = (1 α)w t if ω t = 0 where ω t is a random variable taking value 1 for a win and 0 for a loss. The sequence of random variables ω = (ω 1,..., ω T ) is a random process. The final level of wealth depends on the number of 1 s and 0 s in the sequence ω.

11 The probability to have t wins in a sequence of length T is distributed according to a binomial ( ) T Prob{wins = t} = p t (1 p) T t t

12 The probability to have t wins in a sequence of length T is distributed according to a binomial ( ) T Prob{wins = t} = p t (1 p) T t t Example: t = 2 and T = 3. The number of sequence having 2 wins is 3 (the ways or combinations of selecting 2 elements in a set of 3) and the probability to obtain each sequence is p 2 (1 p).

13 The probability to have t wins in a sequence of length T is distributed according to a binomial ( ) T Prob{wins = t} = p t (1 p) T t t Example: t = 2 and T = 3. The number of sequence having 2 wins is 3 (the ways or combinations of selecting 2 elements in a set of 3) and the probability to obtain each sequence is p 2 (1 p). In general how does the distribution of t looks like?

14 probability density -> binomial with T = 3, p = 2/ For T = 3 the possible values of t range from 0 to 3. Notice that Prob[wins = 3] is larger than Prob[wins = 0] because the probability of a win is larger than 1/2. t ->

15 probability density -> binomial with T = 9, p = 2/ For T = 9 the possible values are from 0 to 9. The distribution is peaked around the mean and median value p T = 2/3 9 = 6. t ->

16 probability density -> binomial with T = 30, p = 2/ For T = 30 the distribution has already assumed a bell shape around p T = 2/3 30 = 20. t ->

17 probability density -> binomial and normal with T = 30, p = 2/ The distribution is well approximated by a normal distribution with mean µ = pt and standard deviation σ = p(1 p)t. In this case the mean is 20 and the standard deviation This is the Central Limit Theorem at work. t ->

18 probability density -> binomial and normal with T = 30, p = 2/ Moreover, due to the normal approximations, we can conclude that in the 99.73% of sequences the number of wins t will be in the interval [µ 3σ, µ + 3σ]. t ->

19 probability density -> binomial and normal with T = 30, p = 2/ Different betting strategies work best in different cases: α = 0 is the best if t = 0 and α = 1 is the best if t = T. These are however unlikely events. It is more effective to have a strategy that works better around the modal value of the distribution of t. t ->

20 The most probable level of wealth after T bets for a given α is WT modal = (1 + α) pt (1 α) (1 p)t W 0. It grows exponentially with T.

21 The most probable level of wealth after T bets for a given α is WT modal = (1 + α) pt (1 α) (1 p)t W 0. It grows exponentially with T. Substituting the minimal and maximal value of t for the interval derived before, in the 99.73% of sequences the wealth at time T satisfies 1 T log W T WT modal [ 3 p(1 p) log 1 α T 1 + α, 3 ] p(1 p) log 1 + α T 1 α.

22 The most probable level of wealth after T bets for a given α is WT modal = (1 + α) pt (1 α) (1 p)t W 0. It grows exponentially with T. Substituting the minimal and maximal value of t for the interval derived before, in the 99.73% of sequences the wealth at time T satisfies 1 T log W T WT modal [ 3 p(1 p) log 1 α T 1 + α, 3 ] p(1 p) log 1 + α T 1 α. When T becomes large the interval is reduced to 0 so that, in probability lim W t = WT modal T which is just the Law of Large Numbers.

23 The best betting rule in the long run is the rule which better performs when the number of wins is equal to the modal value α = arg max {(1 + α) pt (1 α) (1 p)t} removing the unnecessary T and taking the log α = arg max {p log(1 + α) + (1 p) log(1 α)}

24 The best betting rule in the long run is the rule which better performs when the number of wins is equal to the modal value α = arg max {(1 + α) pt (1 α) (1 p)t} removing the unnecessary T and taking the log α = arg max {p log(1 + α) + (1 p) log(1 α)} Notice that the argument in the previous expression is just the expected log growth rate E[log(W t+1 /W t )], thus α is the log-optimal rule.

25 In our case taking the derivative w.r.t. α one has p 1 α 1 p 1 α = 0 whose solution reads α = 2p 1.

26 In our case taking the derivative w.r.t. α one has p 1 α 1 p 1 α = 0 whose solution reads α = 2p 1. According to the log-optimal rule: if the game is favorable (p>1/2) bet a finite amount of money; if the game is fair (p=1/2) bet nothing; if the game is unfair one wants to bet a negative amount, that is take the bet and commit herself to pay the possible payoff.

27 Wealth trajectories associated to different α s will diverge. Set w 0 = 1 and imagine to play different sequences of games of length T with a given α and report the wealth levels log(w t ) on a graph. Then repeat the experiment with a different α. 6 wealth trajectories for α =.5,.6 and p = 2/

28 When the length of the sequence increases, the performances of the two strategies becomes more differentiated wealth trajectories for α =.5,.6 and p = 2/

29 On a longer time horizon the divergence is clear. 160 wealth trajectories for α =.5,.6 and p = 2/

30 expected wealth ratios Starting with the same wealth at time t, consider the wealth at time t + 1 obtained with two different rules α and α W α t+1 = { (1 + α)w t if win (1 α)w t if loss W α t+1 = { (1 + α )W t if win (1 α )W t if loss. then the expected ratio at time t + 1 is [ ] W α E t+1 W α t+1 = p 1 + α 1 α + (1 p) 1 + α 1 α

31 ratio-optimal rule The question is whether there exists a betting rule α such that for any α it is [ ] W α E t+1 1. W α t+1

32 ratio-optimal rule The question is whether there exists a betting rule α such that for any α it is [ ] W α E t+1 1. W α t+1 By rearranging terms [ ] W α E t+1 W α t+1 = p 1 + α + 1 p ( p 1 α + α 1 + α 1 p ) 1 α

33 ratio-optimal rule The question is whether there exists a betting rule α such that for any α it is [ ] W α E t+1 1. W α t+1 By rearranging terms [ ] W α E t+1 W α t+1 = p 1 + α + 1 p ( p 1 α + α 1 + α 1 p ) 1 α This can be made greater than 1 for some α if the expression inside parenthesis is different from zero. Thus the only solution is α = 2p 1 = α. ratio-optimal = log-optimal

34 absolute vs. relative The log-optimality is related to the betting rules that beats all other rules: the probability to have a lower wealth than anyone else using a different rule tend to zero when the number of games increases.

35 absolute vs. relative The log-optimality is related to the betting rules that beats all other rules: the probability to have a lower wealth than anyone else using a different rule tend to zero when the number of games increases. This idea is NOT related to the maximization of some function of the individual wealth, but is instead based on a notion of relative performance of one rule with respect to other rules.

36 Entropy Given a random variable X taking N values with probabilities (p 1,..., p N ) its entropy is defined as S(X) = i p i log p i

37 Entropy Given a random variable X taking N values with probabilities (p 1,..., p N ) its entropy is defined as S(X) = i p i log p i The entropy (or negative information, or information loss) does no depend on the value taken by the random variable, but only on its probability distribution. is non-negative is maximal when the information of the variable is minimal, p i = 1/N it is minimal when the random variable is maximally informative, i.e. there is a j such that p j = 1

38 Relative entropy Given two random variables X and Y taking values on a set of N events with probabilities (x 1,..., x N ) and (y 1,..., y N ), the relative entropy or Kullback-Leibler divergence (sometimes distance) is D(X Y) = x i log x i y i i and measure the information missing for the knowledge of X when one knows Y.

39 Relative entropy Given two random variables X and Y taking values on a set of N events with probabilities (x 1,..., x N ) and (y 1,..., y N ), the relative entropy or Kullback-Leibler divergence (sometimes distance) is D(X Y) = x i log x i y i i and measure the information missing for the knowledge of X when one knows Y. The relative entropy is non-negative is asymmetric is minimal when X and Y coincides

40 entropy-optimal The expression for the expected log growth rate can be rewritten as (1 α)/2 + p 1 p E[log(W t+1 /W t )] = p log (1+α)/2 (1 p) log p log p + (1 p) log(1 p) + log(2)

41 entropy-optimal The expression for the expected log growth rate can be rewritten as (1 α)/2 + p 1 p E[log(W t+1 /W t )] = p log (1+α)/2 (1 p) log p log p + (1 p) log(1 p) + log(2) The betting rule can be seen as a random variable taking values on the two events win and loss with probabilities proportional to 1 + α and 1 α respectively. Then E[log(W t+1 /W t )] D({p, 1 p} betting rule) the relative entropy of the occurrences (win/loss) given the betting rule.

42 entropy-optimal The expression for the expected log growth rate can be rewritten as (1 α)/2 + p 1 p E[log(W t+1 /W t )] = p log (1+α)/2 (1 p) log p log p + (1 p) log(1 p) + log(2) The betting rule can be seen as a random variable taking values on the two events win and loss with probabilities proportional to 1 + α and 1 α respectively. Then E[log(W t+1 /W t )] D({p, 1 p} betting rule) the relative entropy of the occurrences (win/loss) given the betting rule. Maximizing E[log(W t+1 /W t )] is equivalent to minimize the relative entropy. entropy-optimal = log-optimal.

43 Summary The dominant betting rule is the log-optimal rule, the Kelly rule

44 Summary The dominant betting rule is the log-optimal rule, the Kelly rule On average, no other rules can provide an higher level of wealth in the long run

45 Summary The dominant betting rule is the log-optimal rule, the Kelly rule On average, no other rules can provide an higher level of wealth in the long run Under some well specified conditions, the log-optimal rule is the rule that minimizes the conditional entropy

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