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Expected Value and the Kelly Criterion

Activate when: user asks 'how much should I bet/invest on this?', 'what's the expected value here?', 'Kelly criterion', 'optimal bet size', 'fractional Kelly... Skill: Expected Value and the Kelly Criterion Owner: deciqai Summary: Activate when: user asks 'how much should I bet/invest on this?', 'what's the expected value here?', 'Kelly criterion', 'optimal bet size', 'fractional Kelly... Tags: latest:1.0.4 Version history: v1.0.4 | 2026-07-16T17:59:26.940Z | user Description tail link + agents machine-readable metadata line (deciqai.com/s/expected-value-and-kelly.json) v1.0

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959

Updated

Oct 12, 2026

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1.0.4

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Capability contract not published. No trust telemetry is available yet. 959 downloads reported by the source. Last updated 10/12/2026.

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Vendor: Clawhubvendor · observed Oct 12, 2026
Protocol compatibility: OpenClawcompatibility · observed Oct 12, 2026
Adoption signal: 959 downloadsadoption · observed Oct 12, 2026
Latest release: 1.0.4release · observed Jul 16, 2026
Handshake status: UNKNOWNsecurity

What changed

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Release 1.0.4release
Description tail link + agents machine-readable metadata line (deciqai.com/s/expected-value-and-kelly.json)
Observed Jul 16, 2026 · source

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Setup complexity: low.

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SKILL.md

---
name: expected-value-and-kelly
description: "Activate when: user asks 'how much should I bet/invest on this?', 'what's the expected value here?', 'Kelly criterion', 'optimal bet size', 'fractional Kelly', 'how big a position should I take?', or is allocating capital across repeated decisions (ad spend by segment, VC portfolio construction, position sizing, A/B test ramp).
  Do NOT activate when: the decision is one-shot and non-repeating (career change, marriage) — use regret-minimization instead; or when the user cannot estimate probabilities or payoffs even roughly. More: deciqai.com/c/expected-value-and-kelly"
---

# 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

_meta.json

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  "slug": "expected-value-and-kelly",
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references/sources.md

# Sources — expected-value-and-kelly

> *Primary sources for the [expected-value-and-kelly](../SKILL.md) skill.*

- **Kelly, J. L., Jr.** "A New Interpretation of Information Rate." *Bell System Technical Journal*, Vol. 35, No. 4 (July 1956), pp. 917–926. **The primary source for the Kelly formula and its proof.** https://www.princeton.edu/~wbialek/rome/refs/kelly_56.pdf
- **Thorp, Edward O.** *Beat the Dealer: A Winning Strategy for the Game of Twenty-One*. Random House, 1962 (2nd ed. Vintage, 1966). **Primary source** for the first practical application of Kelly to a positive-edge real-world bet (blackjack card-counting).
- **Thorp, Edward O.** "The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market." Paper for the 10th International Conference on Gambling and Risk Taking, Montreal, 1997. The clearest concise summary by Thorp himself; covers the fractional-Kelly trade-off. https://www.eecs.harvard.edu/cs286r/courses/fall12/papers/Thorpe_KellyCriterion2007.pdf
- **Thorp, Edward O.** *A Man for All Markets: From Las Vegas to Wall Street, How I Beat the Dealer and the Market*. Random House, 2017. Autobiography; primary source on the Princeton-Newport years and Thorp's evolution from blackjack to markets.
- **Bernoulli, Daniel.** "Exposition of a New Theory on the Measurement of Risk." Commentarii Academiae Scientiarum Imperialis Petropolitanae, Tomus V (St. Petersburg, 1738). English translation by Louise Sommer in *Econometrica*, Vol. 22, No. 1 (January 1954), pp. 23–36. **Primary source** for the utility-versus-EV distinction.
- **Benter, William.** "Computer Based Horse Race Handicapping and Wagering Systems: A Report." In *Efficiency of Racetrack Betting Markets*, edited by Donald B. Hausch, Victor S. Y. Lo, and William T. Ziemba, pp. 183–198. Academic Press, 1994 (reprinted World Scientific, 2008). **Primary source** for combining a fitted probability model with public odds and sizing pari-mutuel bets by fractional Kelly under estimated inputs and market impact.
- **Poundstone, William.** *Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street*. Hill & Wang, 2005. The popular history; useful as a secondary reference and for the Bell Labs / Shannon / Kelly intellectual lineage.
- **U.S. Securities and Exchange Commission filings and quarterly earnings materials of Microsoft, Alphabet, Amazon, Meta, and Nvidia (2023–2025).** Durable **primary sources** for AI-driven data-center capital-expenditure guidance and Nvidia's data-center-segment revenue growth during the 2023–2026 AI investment boom — used in preference to fast-moving press figures, which should be verified against these filings. https://www.sec.gov/cgi-bin/browse-edgar
- **Contemporaneous business-press coverage of frontier-AI-lab financing rounds and valuations (e.g., The Wall Street Journal, Financial Times, The Information, 2023–2026).** **Secondary** reference for reported startup valuations (OpenAI, Anthropic, a

examples/bill-benter-hong-kong-horse-racing-1985-2001.md

# Method in Action: Bill Benter and Computer-Model Horse Betting in Hong Kong (1985 → 2001)

> *Example for the [expected-value-and-kelly](../SKILL.md) skill.*

A worked example in a domain far from a card table: **pari-mutuel horse racing**, where the odds are set by the crowd's own money and your bet moves the payout against you. Primary-source documented in Benter's own 1994 technical paper.

In the mid-1980s **William "Bill" Benter** — who had learned card-counting from Thorp's *Beat the Dealer* and been barred from casinos — turned to the Hong Kong Jockey Club, then the largest pari-mutuel racing pool in the world. His insight was statistical, not tipster's intuition: build a **multinomial logit model** that estimates each horse's true win probability from fundamental factors (past times, weight, jockey, barrier, going), then combine that estimate with the **public's implied probability** encoded in the tote odds. The crowd is a strong predictor; a model that ignores it loses, and a model that only mimics it has no edge. Benter's edge came from the *residual* — the systematic gap between his model's probabilities and the crowd's.

That residual defined the bet. On any race, a horse is a positive-EV wager only when Benter's estimated win probability *p* exceeds the break-even probability implied by its tote odds after the Jockey Club's ~17–19% takeout. Most horses in most races failed that gate. The discipline was to bet only the small fraction that cleared it — and to size each qualifying bet by Kelly.

Pari-mutuel racing punishes overbetting twice. First, Kelly's ordinary lesson: a positive-EV bet staked too large courts ruin through variance. Second, a mechanical trap absent from casino play — **your own money enters the pool and lowers the odds you receive.** A full-Kelly stake computed against the pre-bet odds is, after your bet lands, an *overbet* against the true post-bet payout. Benter's paper is explicit that they used **fractional Kelly**, both to absorb estimation error in the probability model and to blunt this self-impact on the tote. Run over hundreds of thousands of bets across many seasons, the syndicate is widely reported to have won on the order of a billion dollars — the integral effect of a small per-race edge, correctly sized and relentlessly repeated.

Walk the EV-Kelly Sizing through Benter's case:

- **Repeatability (Step 1):** Hundreds of races per season, many seasons, thousands of qualifying bets. Emphatically repeated. ✓
- **Bet map (Step 2):** For a candidate horse, model win probability *p*; payoff *W* set by the tote (dividend minus stake) after ~17–19% takeout; loss *L* = the stake.
- **EV (Step 3):** EV = p·W − q·L per unit staked. Computed per horse, per race.
- **Edge gate (Step 4):** Bet only when *p* exceeds the takeout-adjusted break-even implied by the odds. The overwhelming majority of horses fail this gate and are not bet.
- **Estimation uncertainty (Step 5):** High and irreducible — *p* comes from a f

examples/ed-thorp-blackjack-and-princeton-newport-1961-1988.md

# Method in Action: Ed Thorp, Blackjack, and Princeton-Newport (1961 → 1988)

> *Example for the [expected-value-and-kelly](../SKILL.md) skill.*

A worked example. Not a casino fable — primary-source documented, including the Bell Labs origin.

In 1960, **Ed Thorp**, a young mathematics professor at MIT, was reading Baldwin et al.'s 1956 paper on basic-strategy blackjack and realized something the casino-gambling literature had missed: because cards in blackjack are drawn *without replacement* from a finite deck, the composition of the remaining deck shifts the player's edge from hand to hand. A player who **tracks** the cards already played (a "counter") sees a small but real positive edge on certain deck compositions — typically when the remaining deck is rich in tens and aces.

Thorp's contribution had two parts. First, he proved the math: a counting system (his "Ten-Count," later refined to the "High-Low" by Harvey Dubner) yields an average edge of roughly **+1% over the house** when applied disciplinedly. Second — and this is what most blackjack popularizers miss — he applied **Kelly sizing** to the bet: vary the wager up and down with the count, betting more when the count favored the player and the minimum when it didn't.

To prove it in the field, Thorp went to Las Vegas and Reno casinos in spring 1961, funded by **Manny Kimmel**, a New York businessman with $10,000 in seed capital. Over a series of trips, they confirmed the math at the tables. The result was published in **Thorp's 1962 book *Beat the Dealer*** (Random House) — a primary source for everything in this Method-in-Action section. The casinos' second-order response was rapid: they changed rules, introduced multiple decks and shoe shuffles, and trained dealers to spot counters. Within a few years, Thorp's specific edge in single-deck games had been engineered out of most casinos.

But the *framework* — combine a measurable edge with disciplined Kelly sizing across a repeated series of independent bets — translated. In 1969, Thorp co-founded **Princeton-Newport Partners**, one of the first quantitative hedge funds. The fund's strategy was convertible-bond arbitrage, statistical arbitrage, and other measurable-edge market trades, sized using Kelly principles (Thorp typically used a fractional Kelly given the higher estimation uncertainty in markets vs. casinos). **From 1969 to 1988, the fund returned 19.1% annually before fees with no down year** (Thorp, *A Man for All Markets*, 2017, ch. 14). The Princeton-Newport partnership ended for unrelated legal reasons in 1988.

Walk through the EV-Kelly Sizing on Thorp's blackjack case:

- **Repeatability (Step 1):** Tens of thousands of hands per casino trip. Strongly repeated. ✓
- **Bet map (Step 2):** Per hand at a favorable count, win probability *p ≈ 0.505*, loss *q ≈ 0.495* (the +1% edge); payoff approximately 1:1 (with some variance from doubles, splits, blackjack 3:2).
- **EV (Step 3):** ≈ +0.01 per unit staked. *Positive but sma

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