The Fundamental Theorem of Linear Programming and Its Application to Sports Betting

Introduction

Linear programming (LP) is a mathematical method used to optimize an objective function subject to constraints. One of the most important results in LP is the Fundamental Theorem of Linear Programming, which guarantees that if an optimal solution exists for a linear program, it can be found at a vertex (or corner point) of the feasible region.

In this article, we will explore the Fundamental Theorem of Linear Programming in detail and discuss its application to sports betting, a field where decision-making under constraints is crucial for maximizing expected profit.

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Understanding the Fundamental Theorem of Linear Programming

1. Basics of Linear Programming

A linear program consists of:

• Decision variables: The values we want to determine.
• Objective function: A linear function to be maximized or minimized.
• Constraints: Linear inequalities or equalities that restrict the values of decision variables.

A general LP can be formulated as:

$$\text{Maximize (or Minimize) } Z = c_1x_1 + c_2x_2 + ... + c_nx_n$$

subject to:

$$a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n \leq b_1$$ $$a_{21}x_1 + a_{22}x_2 + ... + a_{2n}x_n \leq b_2$$ $$\vdots$$ $$a_{m1}x_1 + a_{m2}x_2 + ... + a_{mn}x_n \leq b_m$$ $$x_1, x_2, ..., x_n \geq 0$$

where:

• $x_1, x_2, ..., x_n$ are decision variables.
• $c_1, c_2, ..., c_n$ are coefficients in the objective function.
• $a_{ij}$ are constraint coefficients.
• $b_1, b_2, ..., b_m$ are resource limits or constraints.

2. Fundamental Theorem of Linear Programming

The Fundamental Theorem of Linear Programming states that:

1. If an LP has an optimal solution, it occurs at a vertex (extreme point) of the feasible region.
2. If there is a feasible region and the objective function is bounded, there exists an optimal solution.
3. If the feasible region is unbounded and the objective function can increase indefinitely, there is no finite optimal solution (the solution is unbounded).
4. If constraints contradict each other, the LP has no feasible solution.

Key insight: Instead of searching the entire feasible region, optimization algorithms like the Simplex Method or Interior-Point Methods focus on the vertices of the feasible region, making the solution process more efficient.

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Application of Linear Programming in Sports Betting

1. The Optimization Problem in Sports Betting

Sports betting involves placing wagers on the outcomes of sporting events to maximize expected profit. The bettor needs to decide how much to bet on different outcomes given the odds offered by bookmakers. This scenario can be formulated as a linear programming problem.

Consider a sport bettor who wants to distribute their capital optimally among multiple bets while ensuring risk control. The decision variables represent the amount bet on each outcome, and the objective function represents the expected return. The constraints arise from limited capital, betting limits, and risk tolerance.

2. Formulating a Linear Program for Betting Optimization

Suppose a bettor has a budget $B$ and wants to distribute it among $n$ different bets with odds $o_i$ and probabilities $p_i$. The bettor's goal is to maximize the expected return while ensuring they do not bet more than their capital.

Decision Variables

Let $x_i$ represent the amount bet on outcome $i$, where $i = 1, 2, ..., n$.

Objective Function

The goal is to maximize expected return:

$$\text{Maximize } Z = \sum_{i=1}^{n} p_i o_i x_i$$

where:

• $p_i$ is the probability of outcome $i$.
• $o_i$ is the odds for outcome $i$.
• $x_i$ is the amount bet on outcome $i$.

Constraints

1. Budget constraint:

$$\sum_{i=1}^{n} x_i \leq B$$

2. Non-negative bets:

$$x_i \geq 0, \quad \forall i$$

3. Risk tolerance constraints (optional):
   If a bettor does not want to risk more than a certain fraction of their bankroll on any single bet:

$$x_i \leq \alpha B, \quad \forall i$$

where $\alpha$ is a fraction of total bankroll $B$ (e.g., Kelly Criterion suggests setting $\alpha$ based on edge and variance).

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3. Arbitrage Betting Using Linear Programming

Arbitrage betting exploits discrepancies in odds offered by different bookmakers to guarantee profit.

Example: Arbitrage in a Two-Outcome Event

Consider a tennis match with two possible outcomes:

• Player A wins (odds = 2.10)
• Player B wins (odds = 1.90)

If different bookmakers offer these odds, the bettor can determine the optimal bet distribution to guarantee a profit.

Formulating the Arbitrage Problem as an LP

Let $x_1$ be the amount bet on Player A and $x_2$ the amount bet on Player B. The goal is to ensure a guaranteed return by solving:

$$\text{Minimize } -\left( x_1 \times 2.10 + x_2 \times 1.90 \right)$$

subject to:

• Total bet constraint: $x_1 + x_2 = B$
• Guaranteed return condition: Ensure profit in either case.

Solving this LP using the Simplex Method yields the optimal bet amounts.

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4. Bankroll Management and Risk Control Using LP

Bankroll management is crucial in professional betting. Bettors want to allocate their funds in a way that maximizes returns while minimizing risk.

Using LP, bettors can:

• Optimize bet sizing based on expected value (EV) and risk constraints.
• Diversify bets to avoid overexposure to a single event.
• Adjust bets dynamically based on changing odds.

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Conclusion

The Fundamental Theorem of Linear Programming provides a powerful foundation for optimizing problems where decisions must be made under constraints. In sports betting, LP helps bettors optimize capital allocation, identify arbitrage opportunities, and manage risk efficiently.

By formulating sports betting strategies as linear programs, bettors can use mathematical optimization techniques like the Simplex Method to maximize long-term profitability while controlling risk.