Initial commit

2025-11-30 08:38:26 +08:00
commit 41d9f6b189
304 changed files with 98322 additions and 0 deletions
--- a/skills/market-mechanics-betting/resources/betting-theory.md
+++ b/skills/market-mechanics-betting/resources/betting-theory.md
@@ -0,0 +1,393 @@
+# Betting Theory Fundamentals
+
+This resource explains the core theoretical foundations of rational betting, expected value, variance management, and market efficiency.
+
+**Foundation for:** All betting and forecasting decisions
+
+---
+
+## Why Learn Betting Theory
+
+**Core insight:** Betting theory separates decision quality from outcome quality. Make +EV decisions repeatedly and survive variance.
+
+**Enables:**
+- Think probabilistically (convert beliefs to quantifiable edges)
+- Manage risk rationally (distinguish bad decisions from bad outcomes)
+- Avoid costly mistakes (identify predictable failure modes)
+- Optimize long-term growth (balance aggression with preservation)
+
+**Research foundation:** Kelly (1956), Samuelson (1963), Thorp (1969), behavioral economics (Kahneman & Tversky), market efficiency (Fama).
+
+---
+
+## 1. Expected Value Framework
+
+### Definition and Formula
+
+**Expected Value (EV):** Probability-weighted average of all possible outcomes.
+
+```
+EV = Σ(Probability × Outcome)
+
+Binary bet:
+EV = (P_win × Amount_won) - (P_lose × Amount_lost)
+```
+
+**Example:**
+```
+Bet $100 on 60% event at even odds (+100)
+EV = (0.60 × $100) - (0.40 × $100) = $20
+EV% = +20% per $100 wagered
+```
+
+### Positive vs Negative EV
+
+**Decision Framework:**
+- **EV > +5%:** Strong bet (after fees/uncertainty)
+- **EV = 0% to +5%:** Marginal (consider passing)
+- **EV < 0%:** Never bet (unless hedging)
+
+**Critical Rule:** Judge decisions by EV, not outcomes. Good decisions lose sometimes; bad decisions win sometimes. Process matters in small samples, results matter over 100+ trials.
+
+### Converting Market Odds to EV
+
+**Step 1: Implied probability**
+```
+Decimal odds: P = 1 / Odds
+  Example: 1.67 → 60%
+
+American (+): P = 100 / (Odds + 100)
+  Example: +150 → 40%
+
+American (-): P = |Odds| / (|Odds| + 100)
+  Example: -150 → 60%
+```
+
+**Step 2: Calculate edge**
+```
+Your probability: 70%
+Market probability: 60%
+Edge = 70% - 60% = +10%
+```
+
+**Step 3: Calculate EV**
+```
+Bet $100 at 1.67 odds:
+EV = (0.70 × $67) - (0.30 × $100) = +$16.90 = +16.9%
+```
+
+### Law of Large Numbers
+
+**Key principle:** Observed frequency converges to true probability as sample size increases.
+
+**Practical thresholds:**
+- 10 bets: High variance, might be down despite +EV
+- 100 bets: Convergence starting, likely near EV
+- 1000 bets: Results tightly centered around EV
+
+**Application:** Don't judge strategy on <30 trials. Variance dominates small samples.
+
+---
+
+## 2. Variance and Risk
+
+### Standard Deviation
+
+**Measures outcome dispersion around EV.**
+
+**Formula:**
+```
+σ = √(P_win×(Win-EV)² + P_lose×(Loss-EV)²)
+```
+
+**Example ($100 bet, 60% win, even odds):**
+```
+EV = $20
+σ = √(0.60×(100-20)² + 0.40×(-100-20)²)
+σ = √9600 = $98
+
+Coefficient of Variation: σ/EV = $98/$20 = 4.9
+```
+
+**Interpretation:** Standard deviation ($98) is 5× the EV ($20). Variance dominates signal.
+
+### Volatility Categories
+
+**Coefficient of Variation (CV = σ/EV):**
+- CV < 1: Low volatility (10-30 trials to see EV)
+- CV = 1-3: Moderate (30-50 trials)
+- CV = 3-10: High (50-100 trials)
+- CV > 10: Extreme (100+ trials)
+
+**Higher CV requires:** Larger bankroll, more patience, stronger discipline.
+
+### Risk of Ruin
+
+**Probability of losing entire bankroll before profit.**
+
+**Practical Guidelines:**
+
+| Bet Size | Risk of Ruin | Assessment |
+|----------|--------------|------------|
+| 50% of bankroll | ~40% | Reckless |
+| 25% of bankroll | ~20% | Aggressive |
+| 10% of bankroll | ~5% | Moderate |
+| 5% of bankroll | ~1% | Conservative |
+| 2% of bankroll | ~0.1% | Very conservative |
+
+**Kelly Criterion naturally manages risk of ruin. Never bet >10% of bankroll on single bet.**
+
+### Managing Volatility
+
+**1. Fractional Kelly (Primary Tool):**
+- Full Kelly: 100% variance, 40%+ drawdowns
+- Half Kelly: 25% variance, ~20% drawdowns
+- Quarter Kelly: 6% variance, ~10% drawdowns
+
+**2. Diversification:**
+- Multiple uncorrelated +EV bets
+- Requires independence (correlation < 0.3)
+
+**3. Expected Drawdown:**
+- Even optimal betting experiences 20-40% drawdowns
+- Mentally prepare for temporary losses
+- Don't confuse drawdown with -EV strategy
+
+---
+
+## 3. Bankroll Management
+
+### Defining Your Bankroll
+
+**Valid:** Money you can afford to lose entirely, separate from emergency fund, investment portfolio, daily expenses. **Starting:** $500-$5000 recreational, $10,000+ serious.
+
+**NOT valid:** Money needed for bills, emergency fund, retirement, money you'd be devastated to lose.
+
+### Separation Principle
+
+**Why:** Prevents scared money and revenge betting. Clear accounting, tax clarity, risk containment.
+
+**Implementation:** Separate betting account, never add money mid-downswing, withdraw profits periodically, stop if bankroll → $0.
+
+### Growth vs Preservation
+
+**Preservation (Default):** 1/4 to 1/2 Kelly, for most bettors and bankrolls <$5000
+**Growth (Advanced):** 1/2 to full Kelly, for large bankrolls and high variance tolerance (requires 2+ years track record)
+
+### Dynamic Sizing
+
+Bet size scales with bankroll. Example: $1000 bankroll at 5% = $50. After wins → $1500 → bet $75. After losses → $600 → bet $30.
+
+**Recalculate:** Daily if >20% change, weekly (active), monthly (casual).
+
+### Withdrawal Strategy
+
+**Recommended:** When bankroll doubles, withdraw original amount, continue with profit (break-even if lose profit).
+**Conservative:** 50% profit monthly. **Aggressive:** Never withdraw (full compounding).
+
+---
+
+## 4. Market Efficiency
+
+### Efficient Market Hypothesis
+
+**Core claim:** Prices reflect all available information. **Reality:** Semi-strong efficient in liquid, mature markets.
+
+**Market knows:** Published polls/news, historical base rates, expert commentary, obvious statistical patterns.
+
+### Where Edges Exist
+
+**1. Information Asymmetry:** Local knowledge, domain expertise
+**2. Model Superiority:** Better statistical model, proper extremizing
+**3. Lower Transaction Costs:** Market 5% fee vs your 0-1%
+**4. Behavioral Biases:** Recency bias, base rate neglect, narrative following
+**5. Market Immaturity:** Low liquidity, niche topics, few informed traders
+
+**Before betting, ask:** "What information or model do I have that the market doesn't?"
+- Nothing → Pass | Vague → Pass | Specific → Investigate
+
+### Trust vs Question Market
+
+**Trust:** Liquid, mature, objective outcome, many informed participants, low emotion
+**Question:** Illiquid, new, subjective outcome, few informed participants, high emotion (politics, fandom)
+
+---
+
+## 5. Common Betting Mistakes
+
+### Chasing Losses
+**What:** Increasing bet size after losses. **Why:** Loss aversion, emotional arousal.
+**Fix:** Never increase bet size after loss, use bankroll %, take break after 2+ losses.
+
+### Tilt (Emotional Betting)
+**Triggers:** Bad beat, streaks, external stress. **Symptoms:** No analysis, ignoring Kelly, revenge betting.
+**Fix:** Pre-commit no bets when tilted. Checklist: Calm? Calculate EV? Kelly sizing? Betting for +EV not revenge?
+
+### Overconfidence Bias
+**What:** Overestimating probability accuracy (90% when true is 70%).
+**Fix:** Track calibration, log predictions + outcomes, calculate curve quarterly. Do 70% predictions happen 70%?
+
+### Ignoring Variance
+**What:** Judging strategy on <30 trials. Example: "Down 15% after 20 bets, strategy sucks" (normal variance).
+**Fix:** Require 50+ bets minimum, 100+ preferred, 200+ for high confidence.
+
+### Outcome Bias
+**What:** Judging by results not process. +15% EV lost = good decision (bad outcome). -10% EV won = bad decision (lucky).
+**Fix:** Checklist: EV correct? Edge > threshold? Kelly fraction? Followed system? YES = good decision regardless of outcome.
+
+### Hindsight Bias
+**What:** After outcome, "I knew it would happen."
+**Fix:** Pre-commit logging, write probability before event, don't revise after, accept 40% events happen 40%.
+
+---
+
+## 6. Integration with Kelly Criterion
+
+### EV Drives Kelly
+
+**Kelly derives from:** Expected value (edge), odds received, bankroll optimization (maximize log wealth).
+
+**Key relationship:** `f* = (bp - q) / b`. Edge drives bet size: 10% edge → ~10% Kelly, 5% edge → ~5% Kelly, 0% edge → 0% bet.
+
+### Variance Tolerance
+
+| Fraction | Variance | Growth | Drawdown |
+|----------|----------|--------|----------|
+| Full (1.0) | 100% | 100% | ~40% |
+| Half (0.5) | 25% | 75% | ~20% |
+| Quarter (0.25) | 6% | 50% | ~10% |
+
+### Bankruptcy Protection
+
+Kelly never bets 100%: prevents ruin, keeps capital for next bet, scales down as bankroll shrinks. **Practical:** Stop if bankroll drops 80-90%.
+
+---
+
+## 7. Practical Examples for Forecasters
+
+### Example 1: Election Prediction Market
+
+**Scenario:** Market 55%, your forecast 65%, bankroll $2000
+
+**Step 1: Edge**
+```
+Edge = 65% - 55% = +10%
+Threshold: 5%
+Decision: +10% > 5% → Proceed
+```
+
+**Step 2: EV**
+```
+Bet $100 at 1.82 odds → Win $82
+EV = (0.65 × $82) - (0.35 × $100) = +$18.30 = +18.3%
+```
+
+**Step 3: Kelly**
+```
+Full Kelly: 22.3%
+Half Kelly: 11.2%
+Bet: $2000 × 11.2% = $224
+```
+
+### Example 2: Brier Score Tracking
+
+**50 forecasts, goal: Brier < 0.15**
+
+| Forecast | Your P | Outcome | (P-O)² |
+|----------|--------|---------|--------|
+| Event A | 80% | YES (1) | 0.04 |
+| Event B | 30% | NO (0) | 0.09 |
+| Event C | 90% | YES (1) | 0.01 |
+| Event D | 60% | NO (0) | 0.36 |
+| Event E | 70% | YES (1) | 0.09 |
+
+**Brier:** 0.59 / 5 = 0.118 (Excellent)
+
+**Analysis:** Event D large error normal (40% events happen). Don't game metric by avoiding 60% predictions.
+
+### Example 3: Extremizing
+
+**Forecasts:** You 72%, A 68%, B 75%, C 70%, Market 71%
+**Average:** 71.2%
+
+**Extremize:**
+```
+Factor: 1.3 (moderate)
+Extremized = 50% + (71.2% - 50%) × 1.3 = 77.6% ≈ 78%
+
+Edge: 78% - 71% = +7%
+Half Kelly ≈ 3.5% of $5000 = $175 bet
+```
+
+### Example 4: Correlated Portfolio
+
+**Scenario:** Democrats House (60% yours, 55% market) + Senate (55% yours, 50% market)
+**Correlation:** 0.7 (high)
+
+**Naive (WRONG):**
+```
+Bet A: 5% × $10k = $500
+Bet B: 5% × $10k = $500
+Total: $1000 (10%)
+```
+
+**Correct:**
+```
+Adjust for correlation: 1 - (0.7 × 0.5) = 0.65
+Bet A: $500 × 0.65 = $325
+Bet B: $500 × 0.65 = $325
+Total: $650 (6.5%)
+```
+
+**Reasoning:** Positive correlation amplifies risk. Reduce sizing to maintain tolerance.
+
+---
+
+## Key Takeaways
+
+### The 10 Commandments
+
+1. **Expected Value is King** - Judge decisions by EV, not outcomes
+2. **Variance is Inevitable** - Embrace it; don't fight it
+3. **Bankroll is Sacred** - Protect it above all else
+4. **Kelly is Your Guide** - But use fractional (1/4 to 1/2)
+5. **Market is Usually Right** - You need edge to beat it
+6. **Discipline Over Impulse** - System beats emotion
+7. **Sample Size Matters** - 50+ bets before judgment
+8. **Calibration is Honesty** - Track it religiously
+9. **Correlations Kill** - Adjust for portfolio risk
+10. **Survival Enables Profit** - Can't win if bankrupt
+
+### Mental Models
+
+**Betting = Business**
+- Bankroll = Working capital
+- EV = Profit margin
+- Variance = Market volatility
+- Kelly = Capital allocation
+
+**Decision Quality ≠ Outcome Quality**
+- Good decisions lose sometimes (variance)
+- Bad decisions win sometimes (luck)
+- Process > Results (small samples)
+- Results > Process (large samples 100+)
+
+### Integration Workflow
+
+**Before betting:**
+1. Make forecast (Bayesian, reference class)
+2. Calculate edge vs market
+3. Check edge > threshold (5%+)
+4. Use Kelly for sizing
+5. Execute and log
+
+**After betting:**
+1. Track outcome
+2. Update calibration
+3. Calculate Brier score
+4. Don't judge single bet
+5. Evaluate after 50+ bets
+
+---
+
+**Return to:** [Main Skill](../SKILL.md#interactive-menu)
--- a/skills/market-mechanics-betting/resources/kelly-criterion.md
+++ b/skills/market-mechanics-betting/resources/kelly-criterion.md
@@ -0,0 +1,494 @@
+# Kelly Criterion Deep Dive
+
+Mathematical foundation for optimal bet sizing under uncertainty.
+
+## Table of Contents
+
+1. [Mathematical Derivation](#1-mathematical-derivation)
+2. [Formula Variations](#2-formula-variations)
+3. [Fractional Kelly](#3-fractional-kelly)
+4. [Extensions](#4-extensions)
+5. [Common Mistakes](#5-common-mistakes)
+6. [Practical Implementation](#6-practical-implementation)
+7. [Historical Examples](#7-historical-examples)
+8. [Comparison to Other Methods](#8-comparison-to-other-methods)
+
+---
+
+## 1. Mathematical Derivation
+
+### The Core Question
+
+**Problem**: What fraction of your bankroll maximizes long-term growth?
+
+**Why it matters**: Bet too little → Leave money on the table. Bet too much → Risk ruin, high variance.
+
+### Logarithmic Utility Framework
+
+**Key insight**: Maximize expected logarithm of wealth, not expected wealth.
+
+**Why log utility?**
+- Captures diminishing marginal utility ($1 matters more when you have $100 vs $1M)
+- Makes repeated multiplicative bets additive: log(AB) = log(A) + log(B)
+- Geometric mean emerges naturally (what matters for repeated bets)
+- Prevents betting 100% (avoids ruin)
+
+### Derivation for Binary Bet
+
+**Setup**:
+- Current bankroll: W
+- Bet fraction: f
+- Win probability: p, Loss probability: q = 1 - p
+- Net odds: b (bet $1, win $b net)
+
+**Outcomes**:
+- Win (probability p): New wealth = W(1 + fb)
+- Lose (probability q): New wealth = W(1 - f)
+
+**Expected log utility**:
+```
+E[log(W_new)] = p × log(1 + fb) + q × log(1 - f) + log(W)
+```
+
+**Objective**: Maximize g(f) = p × log(1 + fb) + q × log(1 - f)
+
+### Finding the Optimum
+
+**Take derivative**:
+```
+dg/df = pb/(1 + fb) - q/(1 - f)
+```
+
+**Set equal to zero and solve**:
+```
+pb/(1 + fb) = q/(1 - f)
+pb(1 - f) = q(1 + fb)
+pb - pbf = q + qfb
+pb - q = f(pb + qb) = fb(p + q) = fb
+
+f* = (pb - q) / b = (bp - q) / b
+```
+
+**The Kelly Criterion**:
+```
+f* = (bp - q) / b
+
+Where:
+f* = Optimal fraction to bet
+b  = Net odds received
+p  = Win probability
+q  = 1 - p
+```
+
+### Alternative Form
+
+**Edge** = Expected return per dollar bet = bp - q
+
+**Kelly formula**: f* = Edge / Odds = (bp - q) / b
+
+**Example**: p = 60%, b = 1.0 (even money)
+- Edge = 0.6 × 1 - 0.4 = 0.2
+- f* = 0.2 / 1 = 20%
+
+### Optimality
+
+**Second derivative**: d²g/df² < 0 at f = f* → Maximum confirmed
+
+**Growth rate**: G(f*) maximizes long-run geometric growth
+
+**Comparison**:
+- f < f*: Lower growth (too conservative)
+- f > f*: Lower growth (too aggressive, variance dominates)
+- f > 2f*: Negative growth (eventual ruin)
+
+---
+
+## 2. Formula Variations
+
+### Converting Market Odds
+
+**Decimal odds** (e.g., 2.50): b = Decimal - 1 = 1.50
+
+**American odds**:
+- Positive (+150): b = 150/100 = 1.50
+- Negative (-150): b = 100/150 = 0.667
+
+**Fractional odds** (3/1): b = 3.0
+
+**Implied probability**: Market p = 1/(b + 1)
+
+### Multi-Outcome Bet
+
+**Horse race**: Multiple options, bet on any with positive Kelly
+
+**Formula for outcome i**:
+```
+f_i* = (p_i(b_i + 1) - 1) / b_i
+
+If f_i* > 0: Bet f_i* on outcome i
+If f_i* ≤ 0: Don't bet
+```
+
+### Continuous Outcomes (Merton's Formula)
+
+**Stock market application**:
+```
+f* = μ / σ²
+
+Where:
+μ = Expected return (drift)
+σ² = Variance of returns
+```
+
+**Example**: μ = 8%, σ = 20% → f* = 0.08/0.04 = 2.0 (200%, use leverage)
+
+**Reality**: Too aggressive, use fractional Kelly → 50-100% more reasonable
+
+---
+
+## 3. Fractional Kelly
+
+### Why Fractional Kelly?
+
+**Problems with full Kelly**:
+1. **Extreme volatility**: Wild swings, can lose 50%+ in bad runs
+2. **Model error**: If probability estimate wrong, full Kelly overbets dramatically
+3. **Practical ruin**: 20% chance of 50% drawdown before doubling
+4. **Non-ergodic**: Most can't bet infinitely many times
+
+### Formula
+
+```
+Fractional Kelly = f* × Fraction
+
+Common choices:
+- Half Kelly: f*/2
+- Quarter Kelly: f*/4 (recommended)
+- Third Kelly: f*/3
+```
+
+### Growth vs. Variance Trade-off
+
+| Strategy | Growth Rate | Volatility | Max Drawdown |
+|----------|-------------|------------|--------------|
+| Full Kelly | 100% | 100% | -50% |
+| Half Kelly | ~75% | 50% | -25% |
+| Quarter Kelly | ~50% | 25% | -12% |
+
+**Key**: Half Kelly gives 75% of growth with 25% of variance → Better risk-adjusted return
+
+### Robustness to Error
+
+**Example**: You think p = 0.60, true p = 0.55, even money bet
+
+**Full Kelly** (f = 20%):
+- Growth rate = 0.55×log(1.20) + 0.45×log(0.80) ≈ 0 (breakeven!)
+
+**Half Kelly** (f = 10%):
+- Growth rate = 0.55×log(1.10) + 0.45×log(0.90) ≈ 0.005 (still positive)
+
+**Lesson**: Overbetting much worse than underbetting. Fractional Kelly provides buffer.
+
+### Recommended Fractions
+
+| Situation | Fraction | Reasoning |
+|-----------|----------|-----------|
+| Professional gambler | 1/4 to 1/3 | Reduces career risk |
+| High model uncertainty | 1/4 or less | Error buffer crucial |
+| High confidence | 1/2 to 2/3 | Can use more aggression |
+| Institutional | 1/4 to 1/3 | Drawdown = career risk |
+
+**Default**: Quarter Kelly (1/4) for most real-world situations
+
+---
+
+## 4. Extensions
+
+### Multiple Simultaneous Bets
+
+**Matrix form** (N assets):
+```
+f* = Σ⁻¹ × μ
+
+Where:
+Σ = Covariance matrix
+μ = Expected returns vector
+```
+
+**Key insight**: Correlated bets reduce optimal sizing
+
+**Heuristic**: Adjusted Kelly = Individual Kelly × (1 - ρ/2), where ρ = correlation
+
+**Example**: ρ = 0.6, Individual Kelly = 15%
+- Adjusted: 15% × (1 - 0.3) = 10.5%
+
+### Correlated Outcomes
+
+**Common correlations**:
+- Political: Presidential + Senate races
+- Sports: Team championship + Player MVP
+- Markets: Tech stock A + Tech stock B
+
+**Extreme cases**:
+- ρ = 1 (perfect correlation): Only bet on one
+- ρ = -1 (negative correlation): Bets hedge, can bet more
+- ρ = 0 (independent): No adjustment
+
+### Dynamic Kelly
+
+**Problem**: Probability changes over time (new information)
+
+**Process**:
+1. Start with p₀, bet f₀*
+2. New information → Update to p₁ (Bayesian)
+3. Recalculate f₁*
+4. Rebalance (adjust bet size)
+
+**Consideration**: Transaction costs limit rebalancing frequency
+
+---
+
+## 5. Common Mistakes
+
+### Mistake 1: Full Kelly Overbet
+
+**The error**: Using full Kelly in practice
+
+**Why wrong**: Assumes perfect probability estimate (never true)
+
+**Impact**: Bet 2×f* → Negative growth rate
+
+**Fix**: Always use fractional Kelly (1/4 to 1/2)
+
+### Mistake 2: Ignoring Model Error
+
+**The error**: Treating probability as certain
+
+**Adjustment**:
+```
+Uncertain Kelly = f* × Confidence
+
+Example: f* = 20%, 80% confident → Bet 16%
+```
+
+**Better**: Use fractional Kelly (implicitly adjusts)
+
+### Mistake 3: Neglecting Bankruptcy
+
+**Reality**: Finite games + estimation error → real ruin risk
+
+**Drawdown stats** (full Kelly, p=0.55):
+- 25% chance of -40% before recovery
+- 10% chance of -50% before recovery
+
+**Practical bankruptcy**: Client fires you, forced liquidation, can't maintain discipline
+
+**Fix**: Use fractional Kelly, set stop-loss (if down 25%, pause)
+
+### Mistake 4: Ignoring Correlation
+
+**Example disaster**:
+- 10 bets, each Kelly 10%
+- All highly correlated (same theme)
+- Bet 100% total → Single adverse event → Large loss
+
+**Fix**: Measure correlations, use portfolio Kelly, diversify themes
+
+### Mistake 5: Misestimating Odds
+
+**Common confusion**:
+- Decimal 2.0: b = 1.0 (not 2.0)
+- "3-to-1": b = 3.0 ✓
+- American +200: b = 2.0 (not 200)
+
+**Fix**: Always convert to NET payout (b = total return - 1)
+
+### Mistake 6: Static Bankroll
+
+**Problem**: Calculate once, never update
+
+**Fix**: Recalculate before each bet using current bankroll
+
+---
+
+## 6. Practical Implementation
+
+### Step-by-Step Process
+
+**1. Convert odds to decimal**:
+```python
+# Decimal odds: b = decimal - 1
+# American +150: b = 150/100 = 1.50
+# American -150: b = 100/150 = 0.667
+# Fractional 3/1: b = 3.0
+```
+
+**2. Determine probability**: Use forecasting process (base rates, Bayesian updating, etc.)
+
+**3. Calculate edge**:
+```python
+edge = net_odds * probability - (1 - probability)
+```
+
+**4. Calculate Kelly**:
+```python
+kelly_fraction = edge / net_odds
+```
+
+**5. Apply fractional Kelly**:
+```python
+fraction = 0.25  # Quarter Kelly recommended
+adjusted_kelly = kelly_fraction * fraction
+```
+
+**6. Calculate bet size**:
+```python
+bet_size = current_bankroll * adjusted_kelly
+```
+
+**7. Execute and track**:
+- Record: Date, event, probability, odds, edge, Kelly%, bet
+- Set reminder for resolution
+- Note new information
+
+### Position Tracking Template
+
+```
+Date: 2024-01-15
+Event: Candidate A wins
+Your probability: 65%
+Market odds: 2.20 (implied 45%)
+Net odds (b): 1.20
+Edge: 0.43 (43%)
+Full Kelly: 35.8%
+Fractional (1/4): 8.9%
+Bankroll: $10,000
+Bet size: $890
+Resolution: 2024-11-05
+```
+
+---
+
+## 7. Historical Examples
+
+### Ed Thorp - Blackjack (1960s)
+
+**Application**: Card counting edge varies with count → Dynamic Kelly
+
+**Implementation**:
+- True count +1: Edge ~0.5%, bet ~0.5% of bankroll
+- True count +5: Edge ~2.5%, bet ~2.5% of bankroll
+
+**Results**: Turned $10k into $100k+, proved Kelly works in practice
+
+**Lessons**: Used fractional Kelly (~1/2), dynamic sizing, managed "heat" (detection risk)
+
+### Princeton-Newport Partners (1970s-1980s)
+
+**Strategy**: Statistical arbitrage, convertible bonds
+
+**Kelly application**: 1-3% per position, 50-100 positions (diversification)
+
+**Results**: 19.1% annual (1969-1988), only 4 down months in 19 years, <5% max drawdown
+
+**Lessons**: Fractional Kelly + diversification = low volatility, dominant strategy
+
+### Renaissance Technologies / Medallion Fund
+
+**Strategy**: Thousands of small edges, high frequency
+
+**Kelly application**:
+- Each signal: 0.1-0.5% (tiny fractional Kelly)
+- Portfolio: 10,000+ positions
+- Leverage: 2-4× (portfolio Kelly supports with diversification)
+
+**Results**: 66% annual (gross) over 30+ years, never down year
+
+**Lessons**: Kelly optimal for repeated bets with edge. Diversification enables leverage. Discipline crucial.
+
+### Warren Buffett (Implicit Kelly)
+
+**Concentrated bets**: American Express (40%), Coca-Cola (25%), Apple (40%)
+
+**Why Kelly-like**: High conviction → High p → Large Kelly → Large position
+
+**Quote**: "Diversification is protection against ignorance."
+
+**Lessons**: Kelly justifies concentration with edge. Still uses fractional (~40% max, not 100%).
+
+---
+
+## 8. Comparison to Other Methods
+
+### Fixed Fraction
+
+**Method**: Always bet same percentage
+
+**Pros**: Simple, prevents ruin
+
+**Cons**: Ignores edge, suboptimal growth
+
+**When to use**: Don't trust probability estimates, want simplicity
+
+### Martingale (Double After Loss)
+
+**Method**: Double bet after each loss
+
+**Fatal flaws**:
+- Requires infinite bankroll
+- Exponential growth (10 losses → need $10,240)
+- Negative edge → lose faster
+- Betting limits prevent recovery
+
+**Conclusion**: **NEVER use**. Mathematically certain to fail.
+
+### Fixed Amount
+
+**Method**: Always bet same dollar amount
+
+**Cons**: As bankroll changes, fraction changes inappropriately
+
+**When to use**: Very small recreational betting
+
+### Constant Proportion
+
+**Method**: Fixed percentage, not optimized for edge
+
+**Difference from Kelly**: Doesn't adjust for edge/odds
+
+**Conclusion**: Better than fixed dollar, worse than Kelly
+
+### Risk Parity
+
+**Method**: Allocate to equalize risk contribution
+
+**Difference from Kelly**: Doesn't use expected returns (ignores edge)
+
+**When better**: Don't have reliable return estimates, defensive portfolio
+
+**When Kelly better**: Have edge estimates, goal is growth
+
+### Summary Comparison
+
+| Method | Growth | Ruin Risk | When to Use |
+|--------|--------|-----------|-------------|
+| **Kelly** | Highest | None* | Active betting with edge |
+| **Fractional Kelly** | High | Very low | **Real-world (recommended)** |
+| **Fixed Fraction** | Medium | Low | Simple discipline |
+| **Fixed Amount** | Low | Medium | Recreational only |
+| **Martingale** | Negative | Certain | **NEVER** |
+| **Risk Parity** | Low-Med | Low | Defensive portfolios |
+
+*Kelly theoretically no ruin risk, but model error creates practical risk → use fractional Kelly
+
+**Final Recommendation**: **Quarter Kelly (f*/4)** for nearly all real-world scenarios.
+
+---
+
+## Return to Main Skill
+
+[← Back to Market Mechanics & Betting](../skill.md)
+
+**Related resources**:
+- [Betting Theory Fundamentals](betting-theory.md)
+- [Scoring Rules and Calibration](scoring-rules.md)
--- a/skills/market-mechanics-betting/resources/scoring-rules.md
+++ b/skills/market-mechanics-betting/resources/scoring-rules.md
@@ -0,0 +1,494 @@
+# Scoring Rules and Calibration
+
+Comprehensive guide to proper scoring rules, calibration measurement, and forecast accuracy improvement.
+
+## Table of Contents
+
+1. [Proper Scoring Rules](#1-proper-scoring-rules)
+2. [Brier Score Deep Dive](#2-brier-score-deep-dive)
+3. [Log Score](#3-log-score-logarithmic-scoring-rule)
+4. [Calibration Curves](#4-calibration-curves)
+5. [Resolution Analysis](#5-resolution-analysis)
+6. [Sharpness](#6-sharpness)
+7. [Practical Calibration Training](#7-practical-calibration-training)
+8. [Comparison Table](#8-comparison-table-of-scoring-rules)
+
+---
+
+## 1. Proper Scoring Rules
+
+### What is a Scoring Rule?
+
+A **scoring rule** assigns a numerical score to a probabilistic forecast based on the forecast and actual outcome.
+
+**Purpose:** Measure accuracy, incentivize honesty, enable comparison, track calibration over time.
+
+### Strictly Proper vs Quasi-Proper
+
+**Strictly Proper:** Reporting your true belief uniquely maximizes your expected score. No other probability gives better expected score.
+
+**Why it matters:** Incentivizes honesty, eliminates gaming, optimizes for accurate beliefs.
+
+**Quasi-Proper:** True belief maximizes score, but other probabilities might tie. Less desirable for forecasting.
+
+### Common Proper Scoring Rules
+
+**1. Brier Score** (strictly proper)
+```
+Score = -(p - o)²
+p = Your probability (0 to 1)
+o = Outcome (0 or 1)
+```
+
+**2. Logarithmic Score** (strictly proper)
+```
+Score = log(p) if outcome occurs
+Score = log(1-p) if outcome doesn't occur
+```
+
+**3. Spherical Score** (strictly proper)
+```
+Score = p / √(p² + (1-p)²) if outcome occurs
+```
+
+### Common IMPROPER Scoring Rules (Avoid)
+
+**Absolute Error:** `Score = -|p - o|` → Incentivizes extremes (NOT proper)
+
+**Threshold Accuracy:** Binary right/wrong → Ignores calibration (NOT proper)
+
+**Example of gaming improper rules:**
+```
+Using absolute error (improper):
+True belief: 60% → Optimal report: 100% (dishonest)
+
+Using Brier score (proper):
+True belief: 60% → Optimal report: 60% (honest)
+```
+
+**Key Principle:** Only use strictly proper scoring rules for forecast evaluation.
+
+---
+
+## 2. Brier Score Deep Dive
+
+### Formula
+
+**Single forecast:** `Brier = (p - o)²`
+
+**Multiple forecasts:** `Brier = (1/N) × Σ(pi - oi)²`
+
+**Range:** 0.00 (perfect) to 1.00 (worst). Lower is better.
+
+### Calculation Examples
+
+```
+90% Yes → (0.90-1)² = 0.01 (good) | 90% No → (0.90-0)² = 0.81 (bad)
+60% Yes → (0.60-1)² = 0.16 (medium) | 50% Any → 0.25 (baseline)
+```
+
+### Brier Score Decomposition
+
+**Murphy Decomposition:**
+```
+Brier Score = Reliability - Resolution + Uncertainty
+```
+
+**Reliability (Calibration Error):** Are your probabilities correct on average? (Lower is better)
+
+**Resolution:** Do you assign different probabilities to different outcomes? (Higher is better)
+
+**Uncertainty:** Base rate variance (uncontrollable, depends on problem)
+
+**Improving Brier:**
+1. Minimize reliability (fix calibration)
+2. Maximize resolution (differentiate forecasts)
+
+### Brier Score Interpretation
+
+| Brier Score | Quality | Description |
+|-------------|---------|-------------|
+| 0.00 - 0.05 | Exceptional | Near-perfect |
+| 0.05 - 0.10 | Excellent | Top tier |
+| 0.10 - 0.15 | Good | Skilled |
+| 0.15 - 0.20 | Average | Better than random |
+| 0.20 - 0.25 | Below Average | Approaching random |
+| 0.25+ | Poor | At or worse than random |
+
+**Context matters:** Easy questions expect lower scores. Compare to baseline (0.25) and other forecasters.
+
+### Improving Your Brier Score
+
+**Path 1: Fix Calibration**
+
+**If overconfident:** 80% predictions happen 60% → Be less extreme, widen intervals
+
+**If underconfident:** 60% predictions happen 80% → Be more extreme when you have evidence
+
+**Path 2: Improve Resolution**
+
+**Problem:** All forecasts near 50% → Differentiate easy vs hard questions, research more, be bold when warranted
+
+**Balance:** `Good Forecaster = Well-Calibrated + High Resolution`
+
+### Brier Skill Score
+
+```
+BSS = 1 - (Your Brier / Baseline Brier)
+
+Example:
+Your Brier: 0.12, Baseline: 0.25
+BSS = 1 - 0.48 = 0.52 (52% improvement over baseline)
+```
+
+**Interpretation:** BSS = 1.00 (perfect), 0.00 (same as baseline), <0 (worse than baseline)
+
+---
+
+## 3. Log Score (Logarithmic Scoring Rule)
+
+### Formula
+
+```
+Log Score = log₂(p) if outcome occurs
+Log Score = log₂(1-p) if outcome doesn't occur
+
+Range: -∞ (worst) to 0 (perfect)
+Higher (less negative) is better
+```
+
+### Calculation Examples
+
+```
+90% Yes → -0.15 | 90% No → -3.32 (severe) | 50% Yes → -1.00
+99% No → -6.64 (catastrophic penalty for overconfidence)
+```
+
+### Relationship to Information Theory
+
+**Log score measures bits of surprise:**
+```
+Surprise = -log₂(p)
+
+p = 50% → 1 bit surprise
+p = 25% → 2 bits surprise
+p = 12.5% → 3 bits surprise
+```
+
+**Connection to entropy:** Log score equals cross-entropy between forecast distribution and true outcome.
+
+### When to Use Log Score vs Brier
+
+**Use Log Score when:**
+- Severe penalty for overconfidence desired
+- Tail risk matters (rare events important)
+- Information-theoretic interpretation useful
+- Comparing probabilistic models
+
+**Use Brier Score when:**
+- Human forecasters (less punishing)
+- Easier interpretation (squared error)
+- Standard benchmark (more common)
+- Avoiding extreme penalties
+
+**Key Difference:**
+
+Brier: Quadratic penalty (grows with square)
+```
+Error: 10% → 0.01, 20% → 0.04, 30% → 0.09, 40% → 0.16
+```
+
+Log: Logarithmic penalty (grows faster for extremes)
+```
+Forecast: 90% wrong → -3.3, 95% wrong → -4.3, 99% wrong → -6.6
+```
+
+**Recommendation:** Default to Brier. Add Log for high-stakes or to penalize overconfidence. Track both for complete picture.
+
+---
+
+## 4. Calibration Curves
+
+### What is a Calibration Curve?
+
+**Visualization of forecast accuracy:**
+```
+Y-axis: Actual frequency (how often outcome occurred)
+X-axis: Stated probability (your forecasts)
+Perfect calibration: Diagonal line (y = x)
+```
+
+**Example:**
+```
+Actual %
+   100 ┤                             ╱
+    80 ┤                       ●
+    60 ┤               ●
+    40 ┤       ●               ← Perfect calibration line
+    20 ┤ ●
+     0 └───────────────────────
+       0    20   40   60   80  100
+            Stated probability %
+```
+
+### How to Create
+
+**Step 1:** Collect 50+ forecasts and outcomes
+
+**Step 2:** Bin by probability (0-10%, 10-20%, ..., 90-100%)
+
+**Step 3:** For each bin, calculate actual frequency
+```
+Example: 60-70% bin
+Forecasts: 15 total, Outcomes: 9 Yes, 6 No
+Actual frequency: 9/15 = 60%
+Plot point: (65, 60)
+```
+
+**Step 4:** Draw perfect calibration line (diagonal from (0,0) to (100,100))
+
+**Step 5:** Compare points to line
+
+### Over/Under Confidence Detection
+
+**Overconfidence:** Points below diagonal (said 90%, happened 70%). Fix: Be less extreme, widen intervals.
+
+**Underconfidence:** Points above diagonal (said 90%, happened 95%). Fix: Be more extreme when evidence is strong.
+
+**Sample size:** <10/bin unreliable, 10-20 weak, 20-50 moderate, 50+ strong evidence
+
+---
+
+## 5. Resolution Analysis
+
+### What is Resolution?
+
+**Resolution** measures ability to assign different probabilities to outcomes that actually differ.
+
+**High resolution:** Events you call 90% happen much more than events you call 10% (good)
+
+**Low resolution:** All forecasts near 50%, can't discriminate (bad)
+
+### Formula
+
+```
+Resolution = (1/N) × Σ nk(ok - ō)²
+
+nk = Forecasts in bin k
+ok = Actual frequency in bin k
+ō = Overall base rate
+
+Higher is better
+```
+
+### How to Improve Resolution
+
+**Problem: Stuck at 50%**
+
+Bad pattern: All forecasts 48-52% → Low resolution
+
+Good pattern: Range from 20% to 90% → High resolution
+
+**Strategies:**
+
+1. **Gather discriminating information** - Find features that distinguish outcomes
+2. **Use decomposition** - Fermi, causal models, scenarios
+3. **Be bold when warranted** - If evidence strong → Say 85% not 65%
+4. **Update with evidence** - Start with base rate, update with Bayesian reasoning
+
+### Calibration vs Resolution Tradeoff
+
+```
+Perfect Calibration Only: Say 60% for everything when base rate is 60%
+  → Calibration: Perfect
+  → Resolution: Zero
+  → Brier: 0.24 (bad)
+
+High Resolution Only: Say 10% or 90% (extremes) incorrectly
+  → Calibration: Poor
+  → Resolution: High
+  → Brier: Terrible
+
+Optimal Balance: Well-calibrated AND high resolution
+  → Calibration: Good
+  → Resolution: High
+  → Brier: Minimized
+```
+
+**Best forecasters:** Well-calibrated (low reliability error) + High resolution (discriminate events) = Low Brier
+
+**Recommendation:** Don't sacrifice resolution for perfect calibration. Be bold when evidence warrants.
+
+---
+
+## 6. Sharpness
+
+### What is Sharpness?
+
+**Sharpness** = Tendency to make extreme predictions (away from 50%) when appropriate.
+
+**Sharp:** Predicts 5% or 95% when evidence supports it (decisive)
+
+**Unsharp:** Stays near 50% (plays it safe, indecisive)
+
+### Why Sharpness Matters
+
+```
+Scenario: Base rate 60%
+
+Unsharp forecaster: 50% for every event → Brier: 0.24, Usefulness: Low
+Sharp forecaster: Range 20-90% → Brier: 0.12 (if calibrated), Usefulness: High
+```
+
+**Insight:** Extreme predictions (when accurate) improve Brier significantly. When wrong, hurt badly. Solution: Be sharp when you have evidence.
+
+### Measuring Sharpness
+
+```
+Sharpness = Variance of forecast probabilities
+
+Forecaster A: [0.45, 0.50, 0.48, 0.52, 0.49] → Var = 0.0007 (unsharp)
+Forecaster B: [0.15, 0.85, 0.30, 0.90, 0.20] → Var = 0.1150 (sharp)
+```
+
+### When to Be Sharp
+
+**Be sharp (extreme probabilities) when:**
+- Strong discriminating evidence (multiple independent pieces align)
+- Easy questions (outcome nearly certain)
+- You have expertise (domain knowledge, track record)
+
+**Stay moderate (near 50%) when:**
+- High uncertainty (limited information, conflicting evidence)
+- Hard questions (true probability near 50%)
+- No expertise (unfamiliar domain)
+
+**Goal:** Sharp AND well-calibrated (extreme when warranted, accurate probabilities)
+
+---
+
+## 7. Practical Calibration Training
+
+### Calibration Exercises
+
+**Exercise Set 1:** Make 10 forecasts on verifiable questions (fair coin 50%, Paris capital 99%, two heads 25%, die shows 6 at 16.67%). Check: Did 99% come true 9-10 times? Did 50% come true ~5 times?
+
+**Exercise Set 2:** Make 20 "80% confident" predictions. Expected: 16/20 correct. Common: 12-14/20 (overconfident). What feels "80%" should be reported as "65%".
+
+### Tracking Methods
+
+**Method 1: Spreadsheet**
+```
+| Date | Question | Prob | Outcome | Brier | Notes |
+Monthly: Calculate mean Brier
+Quarterly: Generate calibration curve
+```
+
+**Method 2: Apps**
+- PredictionBook.com (free, tracks calibration)
+- Metaculus.com (forecasting platform)
+- Good Judgment Open (tournament)
+
+**Method 3: Focused Practice**
+- Week 1: Make 20 predictions (focus on honesty)
+- Week 2: Check calibration curve (identify bias)
+- Week 3: Increase resolution (be bold)
+- Week 4: Balance calibration + resolution
+
+### Training Drills
+
+**Drill 1:** Generate 10 "90% CIs" for unknowns. Target: 9/10 contain true value. Common mistake: Only 5-7 (overconfident). Fix: Widen by 1.5×.
+
+**Drill 2:** Bayesian practice - State prior, observe evidence, update posterior, check calibration.
+
+**Drill 3:** Make 10 predictions >80% or <20%. Force extremes when "pretty sure". Track: Are >80% happening >80%?
+
+---
+
+## 8. Comparison Table of Scoring Rules
+
+### Summary
+
+| Feature | Brier | Log | Spherical | Threshold |
+|---------|-------|-----|-----------|-----------|
+| **Proper** | Strictly | Strictly | Strictly | NO |
+| **Range** | 0 to 1 | -∞ to 0 | 0 to 1 | 0 to 1 |
+| **Penalty** | Quadratic | Logarithmic | Moderate | None |
+| **Interpretation** | Squared error | Bits surprise | Geometric | Binary |
+| **Usage** | Default | High-stakes | Rare | Avoid |
+| **Human-friendly** | Yes | Somewhat | No | Yes (misleading) |
+
+### Detailed Comparison
+
+**Brier Score**
+
+Pros: Easy to interpret, standard in competitions, moderate penalty, good for humans
+
+Cons: Less severe penalty for overconfidence
+
+Best for: General forecasting, calibration training, standard benchmarking
+
+**Log Score**
+
+Pros: Severe penalty for overconfidence, information-theoretic, strongly incentivizes honesty
+
+Cons: Too punishing for humans, infinite at 0%/100%, less intuitive
+
+Best for: High-stakes forecasting, penalizing overconfidence, ML models, tail risk
+
+**Spherical Score**
+
+Pros: Strictly proper, bounded, geometric interpretation
+
+Cons: Uncommon, complex formula, rarely used
+
+Best for: Theoretical analysis only
+
+**Threshold / Binary Accuracy**
+
+Pros: Very intuitive, easy to explain
+
+Cons: NOT proper (incentivizes extremes), ignores calibration, can be gamed
+
+Best for: Nothing (don't use for forecasting)
+
+### When to Use Each
+
+| Your Situation | Recommended |
+|----------------|-------------|
+| Starting out | **Brier** |
+| Experienced forecaster | **Brier** or **Log** |
+| High-stakes decisions | **Log** |
+| Comparing to benchmarks | **Brier** |
+| Building ML model | **Log** |
+| Personal tracking | **Brier** |
+| Teaching others | **Brier** |
+
+**Recommendation:** Use **Brier** as default. Add **Log** for high-stakes or to penalize overconfidence.
+
+### Conversion Example
+
+**Forecast: 80%, Outcome: Yes**
+```
+Brier: (0.80-1)² = 0.04
+Log (base 2): log₂(0.80) = -0.322
+Spherical: 0.80/√(0.80²+0.20²) = 0.971
+```
+
+**Forecast: 80%, Outcome: No**
+```
+Brier: (0.80-0)² = 0.64
+Log (base 2): log₂(0.20) = -2.322 (much worse penalty)
+Spherical: 0.20/√(0.80²+0.20²) = 0.243
+```
+
+---
+
+## Return to Main Skill
+
+[← Back to Market Mechanics & Betting](../SKILL.md)
+
+**Related Resources:**
+- [Betting Theory Fundamentals](betting-theory.md)
+- [Kelly Criterion Deep Dive](kelly-criterion.md)
+