Chapter 25 Key Takeaways: Optimization Methods for Betting

Key Concepts

Linear Programming for Betting: Betting allocation can be formulated as a linear program that maximizes expected profit subject to bet limits, exposure constraints, and group constraints. LP provides globally optimal solutions, efficient computation (even for thousands of bets), and valuable sensitivity analysis through shadow prices that reveal the marginal value of relaxing each constraint.
Sensitivity Analysis and Shadow Prices: The shadow price of a constraint tells you exactly how much the optimal profit would increase if that constraint were relaxed by one unit. For a bettor, this means knowing precisely how much expected profit is lost due to each sportsbook's bet limit. Non-binding constraints have zero shadow price and are not currently limiting profitability.
Mean-Variance Portfolio Optimization: Markowitz portfolio theory adapted to betting constructs the efficient frontier --- the set of portfolios achieving maximum expected return for each level of risk. The key insight is that diversification across uncorrelated or negatively correlated bets reduces portfolio risk without sacrificing expected return. Bet correlations (same-game, same-sport, model-based) must be estimated and incorporated.
The Efficient Frontier: The curve in return-risk space that represents all Pareto-optimal portfolios. Points below the frontier are suboptimal (dominated), and no feasible portfolio exists above it. The maximum Sharpe ratio portfolio lies at the tangent point. Moving along the frontier involves a trade-off: more return requires accepting more risk.
Arbitrage Detection: Arbitrage exists when the best available odds across sportsbooks produce a total implied probability below 100%. For a two-outcome event, the condition is $1/d_A + 1/d_B < 1$. Algorithmic scanning across books enables detection, but execution requires sub-second speed, pre-funded accounts, and strategies to avoid sportsbook detection and limitation.
Constrained Kelly Criterion: The multi-bet Kelly criterion maximizes expected log-growth of the bankroll: $\max_{\mathbf{f}} E[\log(1 + \sum f_i R_i)]$ subject to individual and total exposure constraints. For small numbers of bets ($n \leq 15$), exact enumeration over all $2^n$ outcome combinations is feasible; for larger problems, Monte Carlo approximation is required.
Fractional Kelly: Full Kelly is almost always too aggressive in practice because probability estimates contain errors. Using a fraction $k$ of the full Kelly bet (typically $k = 0.25$ to $0.50$) sacrifices a small amount of expected long-run growth rate in exchange for dramatically lower variance and drawdown risk. This is the standard professional practice.
Multi-Objective Optimization: Betting decisions involve multiple competing objectives: profit, risk, account sustainability, and liquidity. The Pareto frontier reveals the set of strategies where no objective can be improved without worsening another. The weighted-sum method and epsilon-constraint method are the two primary approaches for tracing the frontier.

Key Formulas

Formula	Expression	Application
LP Standard Form	$\max \mathbf{c}^T\mathbf{x}$ s.t. $\mathbf{Ax} \leq \mathbf{b}$, $\mathbf{x} \geq 0$	Betting allocation with linear constraints
Expected Return	$\mu_i = p_i \cdot d_i - 1$	Per-dollar expected return on bet $i$
Return Variance	$\sigma_i^2 = p_i(1-p_i) \cdot d_i^2$	Variance of per-dollar return on bet $i$
Portfolio Return	$E[R_p] = \mathbf{w}^T\boldsymbol{\mu}$	Expected return of weighted portfolio
Portfolio Variance	$\text{Var}(R_p) = \mathbf{w}^T\boldsymbol{\Sigma}\mathbf{w}$	Risk of weighted portfolio
Two-Way Arb Condition	$1/d_A + 1/d_B < 1$	Arbitrage exists
Arb Profit	$\Pi = B(1/S - 1)$ where $S = \sum 1/d_i$	Guaranteed profit on investment $B$
Arb Stake Allocation	$x_i = B / (d_i \cdot S)$	Equal-profit staking across outcomes
Single-Bet Kelly	$f^* = (p \cdot d - 1) / (d - 1)$	Optimal fraction for one bet
Multi-Bet Kelly	$\max_\mathbf{f} E[\log(1 + \sum f_i R_i)]$	Log-growth maximization for $n$ bets
Sharpe Ratio	$SR = E[R_p] / \sigma_p$	Risk-adjusted return measure
HHI Concentration	$HHI = \sum (w_i / \sum w_j)^2$	Portfolio concentration measure

Quick-Reference Optimization Workflow

When allocating a bankroll across a set of positive-expected-value bets, follow this six-step workflow:

Step 1 --- Identify the opportunity set. List all available bets with estimated true probabilities, decimal odds, and sportsbook limits. Compute expected returns $\mu_i = p_i d_i - 1$ and filter to positive-EV bets only. Record which bets share a game, sport, and sportsbook.

Step 2 --- Estimate correlations and build the covariance matrix. Assign correlations based on structural relationships: same-game bets (0.15-0.30), same-sport-different-game bets (0.03-0.08), cross-sport bets (near zero). Convert to a covariance matrix using $\Sigma_{ij} = \rho_{ij} \sigma_i \sigma_j$. Verify positive semi-definiteness.

Step 3 --- Define constraints. Specify the maximum bet per opportunity, maximum total exposure (as a fraction of bankroll), per-sport or per-sportsbook exposure limits, and any minimum bet sizes. More constraints reduce maximum return but improve robustness.

Step 4 --- Choose the optimization method. Use LP (Section 25.1) when correlations can be ignored and the objective is purely expected profit. Use mean-variance (Section 25.2) when correlations matter and you want the risk-return trade-off. Use constrained Kelly (Section 25.4) when maximizing long-run growth rate with correlated outcomes. Use multi-objective (Section 25.5) when balancing profit, risk, and operational constraints simultaneously.

Step 5 --- Solve and analyze. Run the optimization, inspect the solution for reasonableness (are any weights suspiciously large or small?), and perform sensitivity analysis. For mean-variance, compute the efficient frontier. For Kelly, compare full, half, and quarter Kelly. For LP, examine shadow prices and binding constraints.

Step 6 --- Execute with discipline. Convert optimal weights to dollar amounts. Place bets at the sportsbooks specified by the optimizer. Track actual returns and compare against the portfolio's expected performance. Re-optimize daily as new opportunities arise and the bankroll changes.

The core principle: Optimization makes trade-offs explicit. Instead of guessing how much to bet on each opportunity, define the objective, constraints, and parameters, then let the solver find the mathematically optimal allocation. Even approximate optimization (with imperfect inputs) systematically outperforms ad hoc allocation.

Ready for Chapter 26? Self-Assessment Checklist

Before moving on to Chapter 26, confirm that you can do the following:

[ ] Formulate a betting allocation problem as a linear program with objective function, decision variables, and constraints
[ ] Solve a linear program using PuLP or cvxpy and interpret the solution, including shadow prices and binding constraints
[ ] Compute the expected return vector and covariance matrix for a set of bets with known probabilities and correlations
[ ] Construct the efficient frontier using mean-variance optimization and identify the maximum Sharpe ratio portfolio
[ ] Explain why bet correlations matter and estimate correlations from structural features of the betting slate
[ ] Detect two-way and three-way arbitrage opportunities across multiple sportsbooks
[ ] Compute optimal stake allocations for arbitrage to equalize guaranteed profit across outcomes
[ ] Implement the constrained multi-bet Kelly criterion for simultaneous bets with correlations
[ ] Explain why fractional Kelly dominates full Kelly in practice and choose an appropriate Kelly fraction
[ ] Design multi-objective optimization problems with competing objectives and compute the Pareto frontier

If you can check every box with confidence, you are well prepared for Chapter 26, where the focus shifts from mathematical optimization to the practical implementation challenges of deploying these methods in real betting markets.