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I am trying to learn about expected value calculations in sports betting... As an example, say I am a sports bettor, and there is a race with 3 horses, $a$, $b$ and $c$ and I know their respective true probabilities, $\Pr(a)$, $\Pr(b)$ and $\Pr(c)$. Assume that a bookkeeper is offering the implied probabilities (consider these the probabilities decided by the market), $\hat{\Pr(a)}$, $\hat{\Pr(b)}$ and $\hat{\Pr(c)}$. I am trying to find the maximum possible expected value, $EV_{max}$, by deciding the wagers, $w_a$, $w_b$ and $w_c$ that I should place on the respective horses. I framed this as a maximisation function (essentially return weighted by probability of event), but am unsure how to select the optimal wagers.

$EV_{max}=\\max(\Pr(a)(\frac{w_a}{\hat{\Pr(a)}}-w_a-w_b-w_c)+\\\Pr(b)(\frac{w_b}{\hat{Pr(b)}}-w_a-w_b-w_c)+\\\Pr(c)(\frac{w_c}{\hat{Pr(c)}}-w_a-w_b-w_c))$

Harry Stuart
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  • This is hard to follow. First of all, "odds" and "probability" are not the same thing, at least not numerically. Which do you mean? More broadly, if you magically know the probabilities for each outcome, then you can easily compute the expected value of each bet. Why not maximize that? If your utility function is something other than "maximize expected gain" then you need to specify. – lulu Sep 25 '22 at 14:47
  • Should say, I see no reason why you should think that the horse with the highest true probability of winning is always the one to bet on. Maybe the market thinks the probability of that horse winning is even higher than you do. In that situation, betting on that horse will have a negative expected result (from your point of view). – lulu Sep 25 '22 at 14:49
  • I understand, my calculation uses odds defined at the beginning (the way they're defiend by bookkeepers), hence the divisor is an odd. I want to know what wagers to place to maximise return. The implied odds are relevant because they determine what the bookkeeper pays out (e.g. 4.6 implied odds with a 100 wager is a payout of $4.6*100-100$) – Harry Stuart Sep 25 '22 at 14:51
  • It does not make sense to quote odds as single numbers. Odds must quoted, e.g., at $2:1$. In any case, I told you how to pick the best bet (for the utility function I guessed at). Just evaluate each possible bet (including combination bets if you like). – lulu Sep 25 '22 at 14:55
  • @lulu I understand, it's just the vernacular of bookkeepers, perhaps I should have used $\frac{1}{Pr}$... Is the function you allude to not the one I articulated? I believe we agree semantically, but I suspect I have not implemented it correctly – Harry Stuart Sep 25 '22 at 14:59
  • Not quite, I am not looking for arbitrage. While I do wish to calculate a positive expected profit, it need not be an arbitrage. An arbitrage guarantees profit after one event, a positive expected value suggests a net profit after many events. – Harry Stuart Sep 25 '22 at 15:24
  • As I say, you need to think about your utility function. "Maximize expectation" is a lazy shortcut, and you probably don't mean it. A very low probability bet with high expected return is probably not an optimal use of capital. Most people would want to weight return by volatility or something like that. But this is a choice you need to make...nobody can pick it for you. – lulu Sep 25 '22 at 15:33
  • Yes, I want the latter. But I believe that in my function, the true odds in fact act as those weights, right? By maximising my function, you are looking for bets that have high return and have high probability. – Harry Stuart Sep 25 '22 at 15:46
  • Again, you only reference "maximise return" which is almost certainly not what you mean. Which would you prefer: a bet which is guaranteed to return $100$ or a bet which only succeeds with probability $.01$ but which has an expected payout of $101$? One common thing people do is to consider the Sharpe Ratio or some natural variant of that. – lulu Sep 25 '22 at 15:50
  • And I still don't understand how you are quoting your odds, so I am not looking at your function. I have never heard of anyone quoting them as the reciprocal of probability. The usual thing would be to quote odds like $x:y$ from which one deduces a win probability of $\frac x{x+y}$. Thus a $1:1$ bet has a win probability of $\frac 12$. For clarity, and as this is a math site, not a gambling site, I would just quote everything in probabilities. – lulu Sep 25 '22 at 15:52
  • I have rephrased with probabilities. I am referring to maximum expected return, not maximum return. I would of course prefer a $100 guaranteed win. The formula I have provided aims to maximise each return weighted by the respective probabilities. I am just unsure about how to choose the wagers to maximise it. Or get the formula in terms of the wagers. – Harry Stuart Sep 25 '22 at 16:18

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First, this is not betting advice. Second, you should have some limit on the total amount bet, or you may be able to increase your expected winnings simply by betting more. Let W be the total amount you will wager. $w_a$ the wager on horse $a$, $w_b$ the wager on horse $b$, and I a simplifying this to a two-horse race. Let $r_a$ be the return for betting one dollar on horse $a$. If you think odds, if the odds are 2-1, betting 1 wins you 2, 3:2, betting 1 gets you 1.5. Let $p_a$ be the probability of horse $a$ winning.Then your expected winnings are $$p_a(w_ar_a)-(1-p_a)w_a +p_b(w_br_b)-(1-p_b)w_b.$$ $$w_a[p_a(r_a+1)-1]+w_b[p_a(r_a+1)-1].$$ So the linearity shows that which ever bet has the higher expected return, put all your money on that. Of course we can simplify this by noting that $w_b=W-w_a$ and $p_b=1-p_a.$

Trurl
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