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Beating Sportsbooks Odds part 3
Alpha Docs #3
Break down:
Sports betting is a game where you allocate a fraction of your money to bet on the possible outcomes of a match. Each outcome has associated odds set by the bookmaker. If the outcome you bet on happens, you receive a payout based on the odds and the fraction you allocated. If the outcome doesn't happen, you lose the fraction you bet. The goal of the player and the bookmaker is to maximize their profits in the long-term. The player wants to allocate their money to target a high total expectation of profit. A betting strategy is a function that maps estimates of probabilities and bookmaker's odds onto fractions of current wealth to be wagered over game outcomes. In a fixed-odds betting setup, odds distribution is always known in advance of the game for the player's strategy to calculate with.
Key Idea Terms:
Player and Bankroll: The player (denoted as "p") participates in sports betting and has a bankroll (denoted as "W") representing their available funds for betting.
Time Step: The betting process occurs over discrete time steps (denoted as "t").
Fractional Allocation: At each time step, the player allocates a distribution of fractions (denoted as "fi") from their current bankroll. These fractions represent the portion of the bankroll allocated to different possible outcomes of a match.
Stochastic Results: The outcomes of a match (denoted as "ri") are stochastic or random variables. The distribution Pr(ri) describes the likelihood of each possible outcome.
Odds and Bookmaker: The bookmaker (denoted as "b") sets the odds (denoted as "oi") for each possible match outcome. The odds represent the potential payoff for the player if that outcome occurs.
Payoff and Profit: If a specific outcome (ri) is realized, the player receives a payoff equal to the odds (oi) multiplied by the allocated fraction (fi) and the current bankroll (W). If the outcome is different, the player loses the allocated portion (fi) of their bankroll to the bookmaker.
Net Profit and Expectation: The potential net profit (wi) from betting on the i-th outcome is calculated based on the odds, allocated fraction, and bankroll. The expectation (EPr[wi]) represents the average profit over all possible outcomes, taking into account the probabilities (Pr(ri)) of each outcome occurring.
Zero-Sum Game: In a closed system with only bettors and bookmakers, the profits of the player and the bookmaker are directly opposite, making it a zero-sum game. Both the player and the bookmaker aim to maximize their long-term profits.
Goal and Utility: The goal for both the player and the bookmaker is to maximize their long-term profits, measured by their respective utilities. The specific utilities are not mentioned in this passage.
Fraction Allocation Strategy: The player aims to allocate fractions (f1,...,fn) in a way that maximizes the total expectation of profit (EPr[W]) over all possible outcomes. This involves choosing the allocation policy (p: ri 7→ fi) based on the current wealth (Wt).
Market Maker and Market Taker: In this context, the bookmaker assumes the role of the market maker, setting the initial odds, while the player takes on the role of the market taker, reacting with their betting strategy based on the odds provided.
Multiple Outcome Game: In real sports betting situations, there are usually more than two possible outcomes. For example, in soccer, the outcomes could be Win, Draw, and Loss, or in a horse race, the outcomes could be the winners among multiple horses.
Matrix Representation: To represent the game in a more compact way, a generic matrix O (outcome matrix) is proposed. The columns of O represent the possible outcome assets, and the rows represent the joint realizations of all the outcomes. Each element in O represents the specific odds for a particular outcome realization.
Cash Asset: An artificial risk-free "cash" asset (denoted as c) is introduced to the game. It allows the player to put money aside safely and can model situations where leaving money aside incurs a small cost (e.g., due to inflation) or offers a possibility of increasing wealth through interest rates (e.g., in a savings account).
Betting Strategy: The betting strategy (denoted as g) now involves allocating the full amount of current wealth (W) among the available outcome assets. Out of the n available assets, n-1 are risky and stochastic, while 1 is the risk-free cash asset. The allocation fractions (f) satisfy the constraint that the sum of all fractions equals
Probabilistic Game: The game is probabilistic, with k possible joint outcome realizations (possible worlds). The probabilities (p1, p2, ..., pk) represent the likelihood of each joint outcome realization. The odds for each outcome asset in each of the k world realizations are specified in the columns of the matrix O.
Odds Matrix: The odds matrix O is constructed with columns representing the odds for each outcome asset and a column for the cash asset. The odds for each outcome asset in each world realization are specified in the respective rows of the matrix.
Excess Odds Matrix: To simplify further calculations, a modified odds matrix ρ is defined. It represents the excess odds, which removes the return amount of the placed wager itself, resulting in the net profit W.
Exclusive Outcomes: In the provided example of a football game with three outcomes (W, D, L), the outcomes are exclusive, meaning that only one outcome can occur. This exclusive nature simplifies the computation of optimal strategies. However, in the broader context, the review acknowledges that outcomes can have varying multiplicities, which adds complexity to the problem.
Maximum Sharpe Strategy Explained:
Modern Portfolio Theory (MPT) involves selecting a portfolio of investments based on their risk and expected return. One aspect of MPT is choosing a risk measure, which determines how risk is quantified. Another important decision is how to select a specific portfolio from the set of all possible portfolios.
One popular approach is to find a portfolio on the efficient frontier (a set of portfolios with the highest return for a given level of risk) that maximizes expected profits relative to risk. This measure is known as the Sharpe ratio. The Sharpe ratio is calculated by subtracting the risk-free rate (a hypothetical risk-free investment) from the expected return of the portfolio and dividing it by the standard deviation of the return.
In the context of sports betting, where there is no risk-free investment, the risk-free rate can be neglected. Therefore, the optimization problem is reformulated as maximizing the expected return multiplied by a risk measure, which we'll call the "MSharpe" strategy.
It's worth noting that the variance-based risk measures used in MPT have received criticism because they penalize both excessive losses and excessive returns, which may not be desirable. Additionally, calculating the MaxSharpe solution is sensitive to errors in probabilistic estimates and can lead to extreme solutions. To address these concerns, additional controls may be necessary. However, despite these limitations, the MaxSharpe strategy remains a popular investment practice in sports betting and is often used in experiments and analysis.
Quadratic Approximation:
The Kelly strategy is a method for determining how much to bet in gambling or investments. However, calculating the exact bet amount can take a long time, especially when dealing with a lot of data.
To make the calculation quicker and easier, a simpler approach called the Quadratic Kelly strategy was proposed. It approximates the Kelly strategy using a mathematical technique called the Taylor series expansion.
The idea is to approximate the logarithm of the Kelly criterion using a simple formula. By assuming that the profits are close to zero, we can transform the problem into a new form that is easier to solve. This new form is called the Quadratic Kelly strategy.
Interestingly, the Quadratic Kelly strategy can also be expressed in terms of Modern Portfolio Theory (MPT), which is a popular investment strategy. This connection shows that the Kelly strategy and MPT are related in some ways.
In simpler terms, the Quadratic Kelly strategy is a quicker and simpler way to calculate the optimal bet amount. It is based on approximations and has a connection to a popular investment strategy called Modern Portfolio Theory.
Kelly criterion:
is a mathematical formula that seeks to maximize the long-term growth of continuously reinvested wealth. It does so by finding the optimal portfolio allocation based on expected multiplicative growth. The criterion can be formulated as an optimization problem, and it does not explicitly include a risk term, as risk is considered implicitly in the growth-based perspective.
Results:
the results of experiments conducted to compare informal betting strategies with formal strategies, specifically focusing on the performance of Modern Portfolio Theory (MPT) and the Kelly Criterion. Here's an explanation of the key points:
Informal vs. Formal Strategies: The experiments confirm that the informal betting strategies commonly used by bettors are clearly inferior to formal strategies. This finding aligns with previous reports and supports the notion that relying on informal strategies can lead to unfavorable outcomes. In fact, the informal strategies often result in ruin, even when there is a statistical advantage provided by the model. Due to their poor performance, the informal strategies were not further considered in the analysis.
Performance of MPT and Kelly Criterion: The formal strategies based on Modern Portfolio Theory (MPT) and the Kelly Criterion showed reasonable performance in scenarios where there was a statistical advantage, as indicated by AKL (a measure of precision of the model). However, it is noted that these strategies are based on mathematical assumptions that may not accurately reflect real-world risk profiles. Therefore, their actual risk characteristics in practice may be different from what is expected based on the mathematical models alone.
Additional Risk Management: The experiments also explored the impact of incorporating additional risk management practices (discussed in Section 5). It was found that using these practices generally reduced volatility while maintaining reasonably high wealth progression for a typical bettor in terms of both mean and median outcomes.
Ruin in Disadvantageous Scenarios: The passage mentions that the pure form of both MPT and the Kelly Criterion can lead to ruin in scenarios where there is no statistical advantage provided by the model. This observation is consistent with the findings presented in Table 3 and Table 4. However, it is also mentioned that with smart strategy modifications, it is possible to generate profits even in statistically disadvantageous scenarios. It is important to consider the specific properties of the underlying models when assessing the potential for profits, as there may be scenarios where no strategy can yield profits.
In summary, the experiments support the superiority of formal strategies over informal betting strategies. While MPT and the Kelly Criterion perform reasonably well in scenarios with statistical advantages, their actual risk profiles in practice may differ. Incorporating additional risk management practices can help reduce volatility while maintaining favorable wealth progression. However, it is crucial to consider the specific properties of the models and recognize that there may be scenarios where no strategy can generate profits.