Outcome $o$ is a function of three variables:
$$o = f(\alpha, \beta, \gamma, \epsilon)$$
Where $\alpha \in [0,1]$, $\beta \in [1.5,2.5]$, $\gamma \in {0, 1}$, and $\epsilon \in [0,\infty]$
Note that $\alpha, \beta, \gamma$ are given in a dataset (below), and $\epsilon$ can vary.
It is important to note that $\alpha, \beta, \gamma$ are pairwise dependent (of course the sort of their relations can be further clarified by mining the dataset attached below).
The function that defines $o$ has the following form:
$$o = (\gamma = 1 \to \epsilon * (1 - \beta))\land(-\epsilon)$$
That reads as "if gamma = 1, outcome is a product of epsilon and one minus beta, else, minus epsilon".
A series of outcomes of the length $n$ is $O$:
$$O = \sum_{i=1}^n o$$
Our goal:
$$\max_{\epsilon_i = f(\alpha_i, \beta_i)}(O)$$
That reads as "maximize sum of the outcomes by optimizing epsilon as a function of alpha and beta".
With restrictions (!):
$$\sum_{i=1}^n epsilon = \sum_{i=1}^n 1$$
Motivation
This question can be of use when one tries to optimize the bet amount in sports betting, when the probability of a game outcome is better than the one a book house provides.
Suppose there is a history of bets an investor (me) made while trying to make money via sports betting. The full history can be downloaded from this github account.
The investor is able to calculate a probability of team winning for each game he is in. He also gets a coefficient for this win, ranging from $1.5$ to $2.5$.
His strategy is to bet $1$ unit of capital all the time. He does not consider reinvesting strategies (e.g., the Kelly system), and he does not want to increase the total amount of investments.
He observes that his way of calculating the probability of win is quite good (using R below):
glm(formula = win_bet ~ probs, family = binomial(link = "logit"),
data = bet_stats)
Deviance Residuals:
Min 1Q Median 3Q Max
-1.8425 -1.4128 0.8111 0.8808 1.0534
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -3.689 1.146 -3.219 0.00129 **
probs 6.135 1.573 3.900 9.6e-05 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 1516.6 on 1219 degrees of freedom
Residual deviance: 1501.0 on 1218 degrees of freedom
AIC: 1505
Number of Fisher Scoring iterations: 4
Note that $probs$ are biased upwards.
Question: Is there a mathematically grounded way to distribute his investments differently, using the probabilities (and, possibly, the coefficients)?
My blind approach was as follows:
$$V = \frac{\text{base}}{1 - \text{probs}}$$
Where $V$ is the betting volume, and $base$ is a naive volume. In my case, $base$ = 0.262, so that the sum of investments would be equal to that when using 1. $$\sum_{i=0}^n V_i = \sum_{i=0}^n 1$$
With this approach the sum of winnings increased by $11.9\%$ using the history on hand.