courses › Thinking

Decision math

level 49 course

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.

Builds on: Risk & the long run (not open yet) · Exponentials & logarithms (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

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

A two-move zero-sum game where no single move is safe.

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

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

A go or no-go choice, and a way to learn the outcome before you decide.

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

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

Tips by skill

TipInformation in bits: An outcome with chance 1/2ᵏ carries k bits. Multiply each outcome's bits by its chance, then add them up.
TipHow much to bet: Stake p − q/b: the win chance minus the loss chance divided by the odds. Write it as a percent.
TipMixed strategies: Your mix makes their two moves pay you the same; their mix makes your two moves pay you the same. Set equal, solve.
TipWhat information is worth: Chance × gain, acting only in the good case, minus what you would expect deciding now (never below $0).

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.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

Information in bits par 20 s not tried yet
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
1. An outcome with chance 1/2ᵏ carries k bits. Entropy = Σ p × bits.
2. 1/2 × 1 + 1/4 × 2 + 2 × 1/8 × 3 = 0.5 + 0.5 + 0.75 = 1.75 bits.
How much to bet par 30 s not tried yet
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%
1. Kelly: f = p − q/b = 0.46 − 0.54/4 = 0.325, so stake 32.5%.
2. (Many people bet half of that to cut the swings.)
Mixed strategies par 45 s not tried yet
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
1. Choose p (your chance of High) so your opponent can't exploit you.
2. If they play High, you average 4p + 3(1 − p) = p + 3; if Low, −p + 4(1 − p) = −5p + 4.
3. Setting them equal: 6p = 1, so p = 1/6.
What information is worth par 45 s not tried yet
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
1. Without it, opening expects 0.2 × $130,000 − 0.8 × $460,000 = −$342,000, so you wouldn't: $0.
2. With it you go ahead only when traffic is strong: 0.2 × $130,000 = $26,000.
3. The information is worth $26,000 − $0 = $26,000.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

1 day
3 days
7 days
14 days
30 days
60 days
mastered · every 90 days

your rounds

No rounds yet.