Decision math
How much to bet, how to mix strategies, and what information is worth.
Pen and paper is fine · no calculator needed why?
Opens at level 49. You're level 1. You can read and practice here now.
the idea
Entropy measures uncertainty in bits. An outcome with chance 1/2 carries 1 bit, 1/4 carries 2 and 1/8 carries 3. Entropy is each outcome's bits times its chance, added up, so n equally likely outcomes give log₂ n bits.
The Kelly rule sizes a bet with an edge: stake p − q/b of your bankroll, where p and q are the win and loss chances and b is what $1 staked wins. At 2-to-1 with a 40% win chance, 0.4 − 0.6 ÷ 2 = 0.1, so stake 10%.
The best mixed strategy randomizes in proportions a rival can't exploit. The value of perfect information is what knowing the outcome first is worth.
techniques
Make your rival indifferent
- Let p be your chance of playing your first move.
- Write your average payoff if they play each of their moves.
- Set the two equal and solve for p.
- For their chance q, make your two moves pay you the same instead.
worked example
In a zero-sum game, you and a rival each pick Heads or Tails. Your payoffs (your move first): Heads/Heads +2, Heads/Tails −1, Tails/Heads −1, Tails/Tails +1. At equilibrium, how often should you play Heads? Give a fraction.
- They play Heads: 2p − (1 − p) = 3p − 1.
- They play Tails: −p + (1 − p) = 1 − 2p.
- 3p − 1 = 1 − 2p, so 5p = 2 and p = 2/5.
Answer: 2/5
Price perfect information
- Without it: the expected value of going ahead, or $0 if that is negative.
- With it: go ahead only in the good case, so chance × gain.
- The information is worth the difference.
worked example
Launching a product pays +$50,000 if demand is high (50% chance) and −$30,000 if it is low. A perfect survey would reveal demand first. What is the most the survey is worth?
- Without it: 0.5 × $50,000 − 0.5 × $30,000 = $10,000.
- With it, launch only if demand is high: 0.5 × $50,000 = $25,000.
- Worth: $25,000 − $10,000 = $15,000.
Answer: $15,000
watch out for
- Staking your win chance itself, which risks ruin. Kelly subtracts the loss chance divided by the odds.
- Solving for your rival's mix when the question asks for yours, or the other way round.
- Forgetting what you would expect without the information. Subtract that, or $0 if you would not go ahead.
- Treating uneven outcomes as equally likely. Four outcomes carry 2 bits only when each has chance 1/4.
practice
Information in bits
worked example
Four outcomes have probabilities 1/2, 1/4, 1/8 and 1/8. What is the entropy in bits?
Answer: 1.75 bits
- An outcome with chance 1/2ᵏ carries k bits. Entropy = Σ p × bits.
- 1/2 × 1 + 1/4 × 2 + 2 × 1/8 × 3 = 0.5 + 0.5 + 0.75 = 1.75 bits.
How much to bet
worked example
A bet pays 4-to-1 (win $4 for each $1 staked; lose the $1 otherwise), and you win 46% of the time. What fraction of your bankroll does the Kelly rule say to stake? Give a percent.
Answer: 32.5%
- Kelly: f = p − q/b = 0.46 − 0.54/4 = 0.325, so stake 32.5%.
- (Many people bet half of that to cut the swings.)
Mixed strategies
worked example
A zero-sum game: you pick High or Low, and so does your opponent. Your payoffs: you High, they High: +4; you High, they Low: −1; you Low, they High: +3; you Low, they Low: +4. Your opponent gets the negative of your payoff. At equilibrium, with what probability should you play High? Give a fraction.
Answer: 1/6
- Choose p (your chance of High) so your opponent can't exploit you.
- If they play High, you average 4p + 3(1 − p) = p + 3; if Low, −p + 4(1 − p) = −5p + 4.
- Setting them equal: 6p = 1, so p = 1/6.
What information is worth
worked example
Opening a second store earns +$130,000 if foot traffic is strong (20% chance) and −$460,000 if it's weak. A perfect traffic study would tell you which before you decide. What is the most you should pay for the study?
Answer: $26,000.00
- Without it, opening expects 0.2 × $130,000 − 0.8 × $460,000 = −$342,000, so you wouldn't: $0.
- With it you go ahead only when traffic is strong: 0.2 × $130,000 = $26,000.
- The information is worth $26,000 − $0 = $26,000.