Kelly when splitting bet into multiple multis

[iambasil]

Hi,

Please forgive any bad forum etiquette - I'm a newbie!

I'm trying to figure out a staking system for when I group different bets together into multis, but I can't get one that makes mathematical sense to me.

I'll illustrate an example. There are 4 bets to consider:
A - Odds: 1.85, True odds: 1.65 (i.e. they pay out as if it is 54.1% likelihood of success, when true likelihood is 60.6% - hence I have an edge of ~6.5%)
B - Odds: 1.4 (same true odds)
C - Odds: 1.5 (same true odds)
D - Odds: 2 (same true odds)

Bet A is obviously the bet that I'm most interested in.

Let's say I have a bank of \$10,000:
- The Kelly criterion would suggest that the optimal stake to place on Bet A is \$1,426
- Win profit is \$1,212
- The expected return on the bet is 12.1% profit
- Average profit on a bet would be \$173 (i.e. if you were to repeat the same bet over and over)

However, instead of placing this single bet, I need to combine bets into multiples. Bets B, C & D all have 0% edge - i.e. there are equal odds paid out compared to probability of success, hence you would break even on those bets in the long term.

So, let's say I create a double with bet B:
- A + B: Odds 2.59, true odds: 2.31 (4.7% edge)
- Kelly suggests optimal stake of \$762
- Win profit is \$1,212 (same as the single)
- Expected return remains 12.1% profit
- Average profit on a bet would be \$92 (i.e. if you were to repeat the same bet over and over)

Question 1
I would assume that the optimal stake for the double would give the same expected profit over time compared to the single: You would bet less on the double, because you have higher returns when it wins but lower chance of success.

However, the way that Kelly operates, it seems to match the win return rather than expected return. Using Kelly to guide staking, I would be more profitable placing singles rather than doubles. This doesn't make sense to me! Surely when doubling with a 0% edge bet, it is the average (or 'expected' as you might more formally say) profit that should match the single rather than the win profit - because you'd obviously win on far fewer occasions.

Can anybody shed some light on this or suggest an alternative system?

Question 2
Rather than just doubling bet A with bet B, I want to have three separate doubles:
1. A + B: Odds 2.59, true odds: 2.31 (4.7% edge)
2. A + C: Odds 2.78, true odds: 2.48 (4.4% edge)
3. A + D: Odds 3.3, true odds: 3.7 (3.3% edge)

These bets are not independent, since they all have bet A within them, hence the independent Kelly stakes ($762, $683 and £449 respectively) are too high. I'm trying to figure out the best statistical way to evaluate the optimal stake across these three doubles to somehow replicate the expected profit I could make out of the single whilst managing the increased risk to the bank.

I've found this old post which touches the Kelly stats in simultaneous events, but that doesn't quite answer my question.

Really appreciate any help!

Basil

 

[Jameson]

Hi iambasil, (Wave)

Welcome to MHB! I'm sorry I won't be able to help you with your question but I want to welcome you to our forum and point out one issue that I've fixed in your post. If you want to have a dollar sign appear you should put a backslash in front of the dollar sign, otherwise the site thinks you are trying to use Latex code :)

So if I write "backslash, dollar sign, 10,000" I get \$10,000, the desired result.

Again, welcome to MHB and I'm sure someone will help you soon enough.

Regards,
Jameson

 

[iambasil]

Thanks for the welcome and for the tip Jameson!

Since writing the post, I've got a slightly better understanding. There are two options:
1. Accept that going through Kelly to guide staking for multiples, I'll be reducing my ROI by the likelihood of the secondary bets losing
2. I can use Kelly to guide how best to split any bets between multiple multis, however the only way that I can keep the same ROI is by keeping the same stake as I would for the single - I would obviously never do this when doubling with a long shot.

I'm not really satisfied with either. Does anybody have any alternative ideas that meet somewhere in the middle?

Thanks,

Basil