Benson’s Algorithm and Its Application to Sports Betting: A Deep Dive

Introduction

In the realm of optimization and game theory, Benson's Algorithm is a well-known method used for solving multi-objective linear programming problems (MOLP). Originally developed by Harold P. Benson in the 1970s, the algorithm provides an efficient way to find the entire set of Pareto optimal (efficient) solutions for a given MOLP.

While Benson’s algorithm was initially conceived for abstract mathematical optimization problems, its versatility and robustness have led to a wide range of applications. In recent years, one particularly intriguing application has emerged: sports betting. The inherently uncertain and probabilistic nature of sports events, combined with multi-objective considerations like maximizing expected return while minimizing risk, makes sports betting a fertile ground for advanced optimization techniques like Benson’s algorithm.

This article explores Benson’s algorithm in detail, explains its underlying mathematics, and shows how it can be adapted for strategic decision-making in sports betting.

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Part 1: Understanding Benson’s Algorithm

1.1 Multi-Objective Linear Programming (MOLP)

Most optimization problems focus on a single objective (e.g., maximize profit, minimize cost). However, many real-world problems involve multiple conflicting objectives. In such cases, instead of a single optimal solution, there exists a Pareto frontier—a set of non-dominated solutions, where improving one objective worsens another.

A general form of a MOLP is:

$$\text{maximize } \mathbf{C}x \quad \text{subject to } Ax \leq b,\ x \in \mathbb{R}^n$$

Where:

• $x \in \mathbb{R}^n$ is the decision vector.

• $A \in \mathbb{R}^{m \times n}$, $b \in \mathbb{R}^m$

• $C \in \mathbb{R}^{q \times n}$ maps the decision space to the objective space (with $q$ objectives).

The Pareto frontier consists of solutions for which no objective can be improved without degrading at least one other.

1.2 Benson’s Algorithm Overview

Benson’s Algorithm computes all efficient extreme points of the feasible region in the objective space. It works primarily in the objective space rather than decision space, which makes it efficient for generating the entire Pareto frontier.

Basic Steps of Benson’s Algorithm:

1. Initialization: Define a bounding box in the objective space that is guaranteed to contain the Pareto frontier.

2. Iterative Refinement: At each step:

   • Generate a linear programming problem to find the boundary point of the feasible region.

   • Check whether the point lies on the Pareto frontier.

   • If not, refine the bounding region.

3. Termination: The algorithm stops when the approximation of the Pareto frontier is within a defined accuracy.

1.3 Advantages

• Computational Efficiency: Works in objective space, reducing dimensionality.

• Global Insight: Offers full Pareto set rather than a single solution.

• Adaptability: Can incorporate constraints and weightings dynamically.

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Part 2: Sports Betting as a Multi-Objective Problem

Sports betting is an excellent candidate for MOLP due to competing objectives:

• Maximizing Expected Return: Choosing bets with the highest expected value.

• Minimizing Risk: Avoiding high-variance bets that could lead to large losses.

• Diversifying Portfolio: Spreading risk across different games, leagues, or bet types.

• Beating the Market (Odds): Exploiting inefficiencies in bookmaker odds.

2.1 Modeling Sports Bets

Let’s define the decision variable $x_i$ as the amount to bet on outcome $i$. Each outcome has:

• A probability $p_i$ (estimated by the bettor).

• A payout $o_i$ (offered by the bookmaker).

• A potential return $r_i = x_i \cdot o_i \cdot p_i$

Then, the expected return is:

$$\text{E}[R] = \sum_i x_i \cdot o_i \cdot p_i$$

Variance (risk) can be modeled as:

$$\text{Var}[R] = \sum_i x_i^2 \cdot o_i^2 \cdot p_i \cdot (1 - p_i)$$

This leads to a natural MOLP, where we aim to maximize expected return and minimize variance, subject to:

• Total capital constraint: $\sum x_i \leq B$

• Non-negativity: $x_i \geq 0$

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Part 3: Applying Benson’s Algorithm to Sports Betting

3.1 Step-by-Step Framework

Step 1: Define Objectives

• Objective 1: Maximize Expected Return $f_1(x) = \sum x_i o_i p_i$

• Objective 2: Minimize Risk $f_2(x) = \sum x_i^2 o_i^2 p_i (1 - p_i)$

Note: Since MOLP is typically about maximization, we can model -f₂(x) to treat it as a maximization problem.

Step 2: Construct Feasible Region

• Betting constraints (budget, min/max bet sizes, market exposure).

• Add realistic constraints such as limits on the number of concurrent bets.

Step 3: Apply Benson’s Algorithm

• Use Benson’s method to generate the Pareto frontier in the return-risk space.

• Each point on the frontier represents a different trade-off between reward and risk.

Step 4: Select Betting Portfolio

• From the frontier, select the point that aligns with your betting style:

  • Risk-averse → select low-variance portfolio.

  • Risk-tolerant → choose higher expected return.

Step 5: Real-Time Adjustment

• Update probability estimates with new data (team news, weather, odds changes).

• Recalculate frontier as needed—Benson’s algorithm allows incremental updates.

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Part 4: Practical Considerations

4.1 Estimating Probabilities

• Use machine learning models, historical data, and expert analysis.

• Maintain a database of outcomes and features to dynamically improve estimates.

4.2 Computational Tools

• Use Python packages like PyMOLP, cvxpy, or custom linear solvers.

• For Benson’s algorithm specifically, libraries like Bensolve (MATLAB) are well-established.

4.3 Real-World Challenges

• Market Efficiency: Bookmakers adjust odds to limit arbitrage.

• Liquidity Constraints: Not all markets accept large bets.

• Psychological Biases: Bettors may deviate from optimal strategy due to emotion or overconfidence.

• Data Quality: Garbage in, garbage out—accurate input is vital.

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Part 5: Case Study (Hypothetical Example)

Imagine you have a bankroll of $1,000 and face 10 betting opportunities. You estimate probabilities using your model and take odds from a major sportsbook.

You want to:

1. Maximize profit (Expected Value)

2. Minimize risk (Variance)

You run Benson’s Algorithm and get a Pareto frontier with 12 points. After analyzing these trade-offs, you pick a point with:

• Expected profit: $120

• Standard deviation: $50

You distribute your bets accordingly, e.g.:

Match	Outcome	Bet Size ($)	Odds	p (your estimate)
Match A	Home Win	150	2.0	0.55
Match B	Away Win	100	3.5	0.32
Match C	Draw	50	3.2	0.29
…	…	…	…	…

Over time, you repeat this process and adjust based on outcomes, refining your model.

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Conclusion

Benson’s Algorithm, a classic tool from multi-objective optimization, offers a powerful way to structure betting strategies in the complex and uncertain world of sports. By enabling a bettor to see the full range of efficient options, it supports better-informed decisions that balance the eternal trade-off between risk and reward.

While implementing Benson’s Algorithm may require a solid grasp of linear programming and some computational effort, its potential to transform betting from a gamble into a structured, data-driven investment strategy makes it a tool worth considering for serious bettors.