Sports Betting/Wagering/Gambling
ObjectiveFor any sports market, a generalizable framework is as follows. For a payout odds of implied probability \(p_{\text{market}}\), by using a set of historical features and signals of truth, \(\vec{X}_{\text{training}}\), \(y_{\text{training, truth}}\), model \(M\) and modeling parameters \(\vec{A}\), then the objective is to try to minimize a cost function \(C\left(M\left(\vec{A}, \vec{X}_{\text{training}}\right),y_{\text{training, truth}}\right)\) to arrive at an optimal set of parameters, \(\vec{A}^{*}\), \[ \begin{equation} \vec{A}^{*} = \underset{\vec{A}}{\text{argmin }} C\left(M\left(\vec{A},\vec{X}_{\text{training}}\right),y_{\text{training, truth}}\right)\label{eq:1} \end{equation} \] and thus, try to ensure \(M\left(\vec{A}^{*},\vec{X}_{\text{test}}\right) = p_{\text{predict}} > p_{\text{market}}\) where \(y_{\text{test, truth}}=1\). Additionally, to maximize wagering effectiveness, optimal wagers \(\vec{W}^{*}\) are found with an expectation maximization for the expected payout \(E\left[P_{\text{payout}}\left(\tilde{p}_{\text{market}}, \vec{W}\right)\right]\), for wagers \(\vec{W}\), where \(\tilde{p}_{\text{market}}=\tilde{p}_{\text{market}}\left(p_{\text{predict}}\right)\) is the predicted \(p_{\text{market}}\): \[ \begin{equation} \vec{W}^{*} = \underset{\vec{W}}{\text{argmax }} E\left[P_{\text{payout}}\left(\tilde{p}_{\text{market}}, \vec{W}\right)\right].\label{eq:2} \end{equation} \] Thus, a prediction and wagering strategy results in maximizing overall expectated return; using \(\eqref{eq:1}\) and \(\eqref{eq:2}\), across \(n\) wagering opportunities, \[ \text{Expected return} = \displaystyle\sum_{i}^{n} p_{\text{predict}}\left(\vec{A}^{*},\vec{X}_{\text{test}}\right)_{i}E\left[P_{\text{payout}}\left(\tilde{p}_{\text{market}}, \vec{W}\right)\right]_{i}. \; \llap{\mathrel{\boxed{\phantom{\text{Expected return} = \displaystyle\sum_{i}^{n} \tilde{p}_{\text{predict}}\left(\vec{A}^{*},\vec{X}_{\text{test}}\right)_{i}E\left[P_{\text{payout}}\left(p_{\text{market}}, \vec{W}\right)\right]_{i}…}}}} \] \[ \text{Realized return} = \displaystyle\sum_{i}^{n} y_{\text{test, truth}, i}P_{\text{payout}}\left(p_{\text{market}}, \vec{W}\right)_{i}. \; \llap{\mathrel{\boxed{\phantom{\text{Realized return} = \displaystyle\sum_{i}^{n} y_{\text{test, truth},i}P_{\text{payout}}\left(p_{\text{market}}, \vec{W}\right)_{i}…}}}} \] We end up having to use numerical optimization techniques and high performance computing to solve such functions, where the objective functions are tailored to the sport of interest. This is generalizable to any (nonsports) market, as well. ResultsI have personal experience in systematically / algorithmically wagering on a sport, using techniques from domains of mathematical modeling, Data Science, computation, numerical optimization, to result in the following revenue made:
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