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Suppose we have $n$ independent events. We know their probabilities $p_i,\dotsc,p_n$. We are given betting odds $c_1,\dotsc,c_n$.

We can make bets to any of the events, and also any combination of events. For every bet (be it a single or combined-event), the bookkeepers take a certain percentage $t$ from the winnings amount.

It is obvious that we can make $N=\sum_{i=1}^{n}\binom{n}{i}$ different bets.

We bet a fraction $b_i$ of our "bankroll" for the $i$-th of the N possible bets. If $b_i=0$, it means we don't make the $i$-th bet at all.

Is there a way of determining the optimal set of $b_1,\dotsc,b_N$ so that, in the long run, the "bankroll" increases maximally? (Something like the Kelly criterion)

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    $\begingroup$ (Joke) There are $N+1=\sum_{i=0}^n \binom{n}{i}$ different bets. In most (all?) gambling situations, the empty bet is best. $\endgroup$ – André Nicolas Sep 19 '11 at 19:52
  • $\begingroup$ Is one allowed to bet on the same alternative multiple times? Is the order of bet payoffs specified? Is $b_i$ an amount bet on event $i$, or on some subset of the events? $\endgroup$ – James Waldby - jwpat7 Sep 19 '11 at 20:47
  • $\begingroup$ $b_i$ is the amount bet on a certain set of events. There are N possible sets of events you can bet on. You only make one bet in the amount of $b_i$ for the $i$-the event set. $\endgroup$ – vedran Sep 19 '11 at 21:11
  • $\begingroup$ Ok, let an $i$ be given. What events are elements of bet #$i$ ? Is the amount $b_i$ spread evenly across the elements of bet #$i$ ? Does bet #$i$ get paid off and its result reported to you before you make bet #$i+1$? $\endgroup$ – James Waldby - jwpat7 Sep 19 '11 at 21:23
  • $\begingroup$ Ordering is unimportant, since the point is to FIND the $b_i$. For example, $n=4$, so there are 4 events which we label with numbers $1,\dotsc,4$ Then the possible bets are $\{\{1\},\{2\},\{3\},\{4\},\{1,2\},\{1,3\},\{1,4\},\{2,3\},\{2,4\},\{3,4\},\{1,2,3\},\{1,2,4\},\{1,3,4\},\{2,3,4\},\{1,2,3,4\}\}$ (I hope i didn't miss some). So $b_1$ is the amount bet on $\{1\}$, and so on through all possible $N=15$ sets. The amount bet on a certain set is not paid off immediately - the bets are simultaneous. $\endgroup$ – vedran Sep 19 '11 at 21:50
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Using the strategy described in the Wikipedia article on the Kelly Criterion, there absolutely are situations where you should place different bets on multiple different outcomes, including outcomes with a less-than-maximum Expected Value. There are even situations where you should bet on outcomes with a negative Expected Value.

Strategy:

To summarize the strategy described in the article (for multiple, exclusive, "Many horses" event situations):

1) Order all of the possible events from most to least profitable (highest to lowest Expected Value).

2) For each event, see if the Expected Value for that event exceeds the "Reserve Rate" for your existing set of bets. (Your "Reserve Rate" is initially "1" when your set of planned bets is empty.) If the Expected Value is higher, then add that event to your set of bets.

3) Once you have discovered your optimal set of outcomes to bet on, compute the optimal bet size for each outcome using the "Fraction to Bet" equation below (where "Fraction to Bet" is a fraction of your current bankroll.)

Reserve Rate = (1 – (sum of each probability bet on)) / (1 – (sum of each 1/payoff))

Fraction to Bet = Probability – Reserve Rate/Payoff

Example:

Here is a gambling article which gives a horse-racing example in which your optimal betting set would include a horse with a negative Expected Value. In this example situation, betting on sub-optimal horses allow you to safely wager a much larger percentage of your overall bankroll, which enables you to maximize your long-term expected profit as defined by the Kelly Criterion.

"Independent" vs. "Exclusive" Events:

Your question deals with Independent events, and the Kelly article deals with Exclusive events. But you can still use the strategy by considering all of the Exclusive possibilities. First compute your Reserve Rate for not betting. Then compute your Reserve Rate for the highest EV bet. Then compute the Reserve Rate for the top two highest EV bets where you separately compute the EV for each exclusive outcome (A and B, A and Not B, B and not A, etc...)

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Expected winnings will be maximized by either betting everything on the best event or by refraining from making a bet.

To maximize winnings over a series of bets, see http://en.wikipedia.org/wiki/Kelly_criterion#Many_horses

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  • $\begingroup$ Sure, but we don't want to maximize expected winnings for a single bet round. We need to maximize the growth rate of the bankroll over a long series of such bettings, like in the Kelly criterion. $\endgroup$ – vedran Sep 19 '11 at 20:02
  • $\begingroup$ @vedran: Do you always have the same odds and probabilities? $\endgroup$ – Charles Sep 20 '11 at 3:34
  • $\begingroup$ Yes, always the same. $\endgroup$ – vedran Sep 20 '11 at 6:41
  • $\begingroup$ @vedran: Are the probabilities independent, disjoint, or not known to be either? $\endgroup$ – Charles Sep 20 '11 at 6:49
  • $\begingroup$ The probabilities are independent. Any of the events could happen, or none, or all of them. Some of the probabilities may be 1, some 0. They are totally unrelated events. $\endgroup$ – vedran Sep 20 '11 at 7:00
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Yes, there is a way.

Like in a single bet (Kelly) where the new bankroll $B_n$ will be (the bankroll just before this new one) $B_{n-1}$ multiplied with gain: $$B_n=B_{n-1} (1-u)^{1-s} (1+u(k-1))^s,$$ where $k$ is decimal odds, $u$ is invest (% of bankroll) and $s$ is the probability for win.

Thus, for two bets we have: $$B_n=B_{n-1} (1-u_1-u_2)^{(1-s_1)(1-s_2)} (1+u_1(k_1-1)-u_2)^{s_1(1-s_2)} \\·(1-u_1+u_2(k_2-1))^{(1-s_1)s_2} (1+u_1(k_1-1)+u_2(k_2-1))^{s_1 s_2}.$$

I hope you get the picture of what comes with three and more bets.

So, if we can find maximums - we can find optimum bet sizes. It is hard to do that kind of calculus without a computer, so I have made an algorithm that does that and have put it in a link, but the original post has unfortunately been deleted (it was .exe).

If administrator sees it appropriate - maybe he will return the link back on. Hope this helps :) Cheers!


I have made "multiKelly" calculator so you can find your maximums.. :)

Calculator for "multiple Kelly"

Cheers!

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    $\begingroup$ WARNING: This links to an .exe file. I should remind everyone reading this that downloading executables from untrusted sources on the internet is a very fast way to get your machine infected. $\endgroup$ – tomasz Nov 22 '15 at 11:00
  • $\begingroup$ You may have posted this with the best and most altruistic of reasons, but 1) we generally frown upon link only -answers, 2) posting links to your own site may amount to advertising, which is against the rules, and considered to be spam, when blatant (not the case here perhaps). $\endgroup$ – Jyrki Lahtonen Nov 22 '15 at 11:15

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