Expected Value and the Kelly Criterion

Overview

Two questions decide most repeated bets: is this bet good? (EV) and how big? (Kelly). Most professional ruin comes from positive-EV bets sized wrong. EV = p · W − q · L. If EV ≤ 0, do not bet. Kelly f* = (bp − q) / b maximizes long-term geometric growth (Kelly, Bell Labs, 1956). Full Kelly requires casino-grade certainty; default to half- or quarter-Kelly for estimated edges.

Neighbors: first-principles · occams-razor · second-order-thinking · inversion · regret-minimization (for non-repeating life decisions).

When to Use

• Decision repeats many times — capital allocation, position sizing, VC portfolio, ad spend, A/B test budget
• How big to bet matters as much as whether to bet; you have a measurable or estimable edge
• Someone says: "expected value," "EV," "Kelly," "optimal bet size," "how much should we put on this?"
• Sizing bets in a boom with power-law payoffs and possible ruin — how much to allocate to AI startups / GPU-compute capex / AI-exposed equities given frothy AI valuations, uncertain AI adoption, and correlated bets

When NOT to use: one-shot life decisions → regret-minimization; negative-EV bets (don't bet); unestimable probabilities; correlated bets without portfolio adjustment.

Coaching Novices (Adaptive Front Door)

Engine mode: user has a concrete repeated bet → run The Process directly. Coach mode: user is unfamiliar → guide step by step.

In Coach mode, respond one step at a time. Each [WAIT] is a hard stop — output only that step's question, then stop.

1. One-line what-it-is: EV tells you whether the bet is worth taking; Kelly tells you what fraction of bankroll to stake — sized to maximize long-term growth without ruin.
2. Check fit against When to Use / When NOT to use. If one-shot life decision, redirect to regret-minimization. If EV is negative, say "don't bet" and stop.
3. Elicit their real bet. Ask for a concrete repeated decision with measurable inputs. > [WAIT — do not advance until user responds]
4. Walk The Process one step per turn: outcomes → probabilities → payoffs → EV → Kelly → fractional Kelly. > [WAIT — do not advance until user responds]
5. Close by naming their sizing rule: "bet f* × bankroll, use half-Kelly given estimation uncertainty" — and the trigger that would change it. > [WAIT — do not advance until user responds]

The Process

Run the EV-Kelly Sizing (EV → Kelly → fractional Kelly → stop trigger):

1. Confirm the decision is repeated. If "once," stop → use regret-minimization.
2. Map the bet. Win prob p, loss prob q = 1−p, payoff on win W, loss L, odds b = W/L.
3. Compute EV. EV = Σ(pᵢ · payoffᵢ). State per unit staked.
4. Edge gate. EV > 0? If no, stop — do not bet.
5. Estimate input uncertainty. Are p and W measured or estimated? Write 80% CI on edge.
6. Compute Kelly fraction. f* = (bp − q) / b. For continuous: f* ≈ μ/σ².
7. Apply fractional Kelly. Half-Kelly under modest uncertainty; quarter-Kelly under serious uncertainty.
8. Set stop trigger. "I will re-estimate if: (a) drawdown > X%, (b) outcomes diverge N σ over Y trials, (c) regime change invalidates edge model."

Output: EV-Kelly Sizing

# EV-Kelly Sizing: <bet/decision>
## Repeatability: <count or "ongoing"> — if one-shot, STOP.
## Bet map: p=<>, q=<>, W=<>, L=<>, b=<>
## EV: p·W − q·L = <number> — Edge: <positive/negative/zero>
## Estimation uncertainty: <measured/estimated>; 80% CI on edge: <range>
## Kelly fraction: f* = <full Kelly> → practitioner: <half/quarter-Kelly> = <number>
## Stop trigger: "I will re-estimate if <condition>."
## Correlation check: bets independent? <yes/no — adjustment>

→ Method in Action: Ed Thorp, Blackjack, and Princeton-Newport (1961 → 1988) · Bill Benter, Hong Kong Horse Racing (1985 → 2001) → 2026 lens: Sizing Hyperscaler GPU/Compute Capex (2023–2026) — one CFO's repeated capex bet: wide CI, stranded-capital ruin tail, and fractional Kelly as staged, survivable capex.

Sizing Packs

Domain	Fractional-Kelly	Stop trigger
Active equity	quarter-Kelly or less	drawdown > 2× expected annual vol
Venture capital	portfolio-level quarter-Kelly	hit rate diverges from model by vintage
Ad spend by segment	half- to full Kelly	ROAS falls >2σ over N conversions
A/B test ramp	fractional Kelly on traffic %	regression in primary metric

Applying It Well

• Positive EV is necessary but not sufficient. A positive-EV bet sized too large still wrecks you.
• Default to fractional Kelly. Full Kelly is for measured edges. Half- or quarter-Kelly for estimated edges — growth cost is small, protection is large.
• Correlation eats Kelly fast. All positions long tech, all VC bets in one vintage — adjust portfolio Kelly down.
• Kelly is for bankroll, not for life. Use regret-minimization for career, marriage, time.
• Update continuously. Static Kelly on a stale edge is how winning strategies ride into ruin.

→ Primary sources: references/sources.md

Common Rationalizations

[D] = designed upfront | [O] = observed in real use. [O] entries are more valuable.

Fake move	Reality
[D] Applying Kelly to one-shot decisions	Kelly maximizes long-run geometric growth rate across many bets. For a one-time bet, EV is the right concern; for life decisions, use regret-minimization.
[D] Treating estimated p as known p	Kelly is brutal when probability estimates are wrong. Use fractional Kelly to compensate for estimated inputs.
[D] Using full Kelly with subjective probabilities	Full Kelly is for casino-grade certainty. Half-Kelly costs ~25% of growth and cuts ruin risk by far more (Thorp 1997).
[D] Confusing positive EV with "should bet"	EV ignores variance, bankroll, path-dependency. A +1% EV bet that ruins you 1% of the time is not equivalent to one that ruins you 0%.
[D] Ignoring correlation across bets	Kelly assumes independent bets. Correlated bets (same sector, same vintage) require a lower portfolio-level Kelly.
[D] Using Kelly on negative-EV bets	Kelly returns zero or negative fraction when EV ≤ 0. No sizing rescues a -EV bet. Don't bet.
[D] Treating EV as the only number that matters	Bernoulli 1738: utility of money is non-linear. A +$1000 EV bet that risks your rent ≠ one that risks your rounding error.
[D] Forgetting Kelly's brutal variance	Even correct full-Kelly expects 50%+ drawdowns. Most professionals use half/quarter-Kelly to survive psychologically.
[D] Applying Kelly to non-financial "bets"	Kelly assumes a compoundable bankroll. Relationships, careers, time, attention don't compound across independent trials.
[D] Computing Kelly once, ignoring updates	A static fraction on a now-stale edge is the textbook path to ruin. Pre-commit to a re-estimation trigger.
→ Add [O] entries here after each real use — paste the actual failure pattern	What went wrong and why

Red Flags

• Kelly applied to a decision that happens only once
• Full Kelly used with subjective probabilities and no calibration check
• EV is negative but a positive fraction is being computed
• No stop-and-re-estimate trigger is named
• Bets are obviously correlated but Kelly is computed per-bet without portfolio adjustment
• The "bankroll" is not liquid or fungible (career time, relationship capital)
• A historical drawdown wiped out the strategy; same fraction still applied without re-examining the edge

Verification

• Decision confirmed repeated, not one-shot
• Outcomes, probabilities, and payoffs stated explicitly in correct units
• EV computed per unit staked; edge gate (positive/zero/negative) named
• Estimation uncertainty named (measured/estimated/mixed); 80% CI on edge given
• Kelly fraction computed with correct formula (binary vs. continuous)
• Fractional-Kelly adjustment applied with explicit justification
• Stop-and-re-estimate trigger named with concrete conditions
• Correlation across bets considered; per-bet Kelly adjusted if needed

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Part of deciqAI Knowledge Skills — 227 open-source thinking skills that make rigor executable for AI agents. The same skills power every deciqAI agent, which runs them autonomously to operate your company. See it run → https://www.deciqai.com/c/expected-value-and-kelly · ⭐ Star the repo → https://github.com/deciqAI/knowledge-skills · Contributions welcome.

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