Optimal Strategy

The Math of Good Decisions

Game theory is not just about interactions between players. It also provides frameworks for making decisions under uncertainty when you do not control the outcome. How much to bet when the odds are in your favor. When to cut losses. How to size positions so that a string of bad luck cannot wipe you out. These are not intuitions. They are solved problems with mathematical answers. Most people invest by feel. The frameworks in this lesson replace gut feeling with calculation.

Expected Value

Expected value is the probability-weighted average of all possible outcomes. If a coin flip pays you $200 on heads and costs you $100 on tails, the expected value is: (0.5 x $200) + (0.5 x -$100) = $50. Positive expected value means the bet is worth taking over many repetitions. Negative expected value means the house wins over time. Every casino game has negative expected value for the player. Every insurance policy has negative expected value for the policyholder (otherwise insurance companies would go bankrupt). You buy insurance anyway because the utility of avoiding catastrophe exceeds the expected monetary loss. This distinction between expected value and expected utility is why rich people self-insure and why poor people need insurance most.

• EV = sum of (probability x outcome) for all possible results
• Positive EV: worth taking repeatedly. Negative EV: the house wins long-term.
• A $1 lottery ticket with a 1-in-10-million chance at $5M has EV of -$0.50
• A rental property with 70% chance of 12% return and 30% chance of -5% has EV of +7%
• All investing decisions are expected value calculations with estimated probabilities

The Kelly Criterion

Knowing a bet has positive expected value is not enough. You also need to know how much to bet. The Kelly Criterion, developed by John Kelly at Bell Labs in 1956, gives the mathematically optimal bet size to maximize long-term wealth growth. The formula: f = (bp - q) / b, where f is the fraction of your bankroll to bet, b is the odds received (net profit per dollar wagered), p is the probability of winning, and q is the probability of losing (1 - p). The Kelly fraction maximizes the geometric growth rate of your capital. Betting more than Kelly is reckless (increases variance without increasing long-term growth). Betting less than Kelly is conservative (reduces variance at the cost of slower growth). Most professional investors use fractional Kelly (half or quarter Kelly) to reduce volatility while maintaining positive trajectory.

Position Sizing in Practice

You find a rental property deal. You estimate 70% probability it generates a 15% annual return and 30% probability it loses 10% (vacancy, repairs, market downturn). Expected value: (0.7 x 15%) + (0.3 x -10%) = 7.5%. Positive. The Kelly formula says: f = (1.5 x 0.7 - 0.3) / 1.5 = 0.5. Kelly says you could allocate up to 50% of your capital to this deal. Half-Kelly says 25%. Quarter-Kelly says 12.5%. A portfolio of quarter-Kelly positions across multiple uncorrelated deals gives you strong expected growth with manageable downside risk. This is why diversification works mathematically, not just intuitively. Each position is sized so that even total loss on one deal cannot cripple the portfolio.

• Full Kelly: Maximum growth but high variance. One bad streak feels terrible.
• Half Kelly: 75% of max growth with significantly lower variance. Most common among professionals.
• Quarter Kelly: 50% of max growth, very smooth ride. Good for sleep quality.
• Over Kelly: Mathematically worse than full Kelly. More risk, less long-term growth. Pure recklessness.

Minimax: Preparing for the Worst

Minimax strategy minimizes your maximum possible loss. Instead of asking "what's the best case?", you ask "what's the worst thing that can happen, and how do I limit that?" In investing, this manifests as: never invest money you cannot afford to lose. Keep 3-6 months of expenses in cash before investing a dollar. Never put more than X% of your portfolio in a single asset. Use stop-losses or position sizing to cap downside. The minimax investor gives up some upside to eliminate catastrophic downside. This is the mathematical reason behind emergency funds (Lesson 1-4), diversification (Track 2), and entity protection (Track 4). Each of those strategies is a minimax move: accepting slightly lower expected returns in exchange for removing tail risk.

Expected value tells you whether a decision is worth making. The Kelly Criterion tells you how much to commit. Minimax strategy tells you how to survive being wrong. Together, these frameworks replace intuition with calculation. Every lesson that preceded this one, from budgeting to entity structures, can be understood as an application of these principles. Investing is applied game theory.