Yo guys,
you finds your self in the following situation:
- Event: 4 playesrs competition. only 1 win. (players: A,B,C,D,)
- Position: bought (back) 2 units of player A winning @5 (fractional odds: 4:1).

some thinking:
Why i took this position?
=>Because i think is a +EV position (i.e. i have an edge).
How did arrived to the conclusion that i have an edge?
=> I've assume that there is one mkt that is giving me the efficient price, i.e. the "true" probabilities of this event. i then compare that price with other mkt.
==> the efficient mkt is @3 => therfore i see a miss price here. simply put: i can buy in one mkt @5 (1/5 implied probability) and sell the same thing on another mkt @3 (1/3 implied probability). Here im buying low and selling high (= making money). For the sake of clarity the efficient mkt is an exchange (this is why im saying: you can sell (i.e. lay)).

NOW the question: I found a +EV bet by comparing the (assumed) "true" probabilities of the efficient mkt with another mkt and found a difference between the two. Should i closed the trade and hedge on the efficient mkt and capture all the EV or shall i be naked, not hedge, and run the bet till expiring?

My thinking is the following: since the the true probabilities are given from the efficient mkt, the trade on the efficient mkt is by default a 0EV trade (those are the true probabilities), therefore i shuld hedge and capture all the EV that ive estimated in this bet. Moreover the EV that im estimating is the same if i hedge or not(ofc), the difference is the risk if i hedge ot not. Im reducing my risk (variance of result here) by hedging with a 0EV hedge (i.e. by hedging im assuring to get the EV in all scenarios, actually looks like it will not be an expected value but a sure value => Im gettin a $$ amount equal to the EV in all the 4 scenarios.)

What do you think? I saw a lot of discussion about not heding vs hedging ecc.
Am i doing any logical mistake in my reasoninig? i would love to have a discussion on this.

Cheers,

These are great and important questions you're asking. First off I'll say that even on an exchange that's reliable to use as a model of the efficient market you're likely to pay some amount of vig and not get a truly 0EV hedge. But even if it's slightly -EV, is the hedge correct?

I have an article on this very topic coming out in the February issue of the 2p2 magazine, so check that out in a few days for the answer.

It breaks down to an expected growth calc for multiple wager options, market projections, and game outcomes. (e.g. push frequencies)

My article is up now here: https://www.twoplustwo.com/magazine/...sting-bets.php

Curious to hear what you think.

For anyone who read my article, here's an example problem to try: Say your current bankroll is $10k and you have a futures slip that pays out $1500 if KC wins on Sunday. The lines you can get are +150/-175 (for an implied probability of 38.6% for TB^2). How much should you hedge on the Bucs to optimize your expected growth?

I get 1500 * .382 = 573

When contemplating hedges, I consider their value ex ante. Below is an example.

Suppose one can get +200 future on KC to win SB midseason. An estimate of their chances is 36%, so one concludes this is a +EV bet.

Without considering any possible hedge, one should bet $400 on KC future SB bet. This yields an expected growth of 0.158%.

Let's assume one will be able to get a 0EV hedge on KC's SB opponent. Assume one has $10,000 bankroll. How much should one bet on KC future @ +200?

The answer is uncertain, because we do not know the forward odds of KC winning the SB. This is because we do not know their opponent. (and even if we did, odds change over time) We also don't know what KC's chances are of reaching the SB, which is a required condition to make the SB hedge bet.

To make it simple, let's assume KC is 60% to reach the SB, and their fair value odds will be -150 to win SB. This means their opponent will be +150 fair value.

The optimal bet sizes are $2,000 on KC future, and $3,200 on SB opponent +150. Expected growth is 2.63%. Below are the possible outcomes, and the resulting log of expected growth:

Instead of -150, assume KC is -122.2 (55% win equity) fair value odds to win SB. This means their opponent will be +122.2 fair value.

The optimal bet sizes are $2,000 on KC future, and $4,100 on SB opponent +122.2. Expected growth goes up to 3.84%%. Below are the possible outcomes, and the resulting log of expected growth:

Below is a chart of bet sizes and expected growth for varying KC SB win equities:

Now, these are significantly higher than if you simply placed an optimal EG future bet on KC without considering a hedge. Needless to say, a $2,000 future bet implies one has a 40% chance to lose $2,000. (20% of one's bankroll) So, there is obviously large exposure if KC does not make it to the SB. If KC makes it though, one is essentially in a no-lose situation with the hedge. This fact yields the opportunity to significantly increase EG. Hedging after the fact misses this opportunity.

That's a very interesting theory, but I get a different result than you did. Care to share your math? For example, are you calculating the EG of "KC out of SB" as log(1-0.2) since you bet 20% of your starting bankroll and lost?

I reviewed my calcs and discovered an error calculating the KC lose SB scenario. Below is the chart of the correct growth rates:
Some of the growth rates are significantly lower than what I previously posted. But even those are multiples higher than the unhedged KC play.

To be fair, it is possible to hedge the KC future without initially intending to do so. For example, with KC 60% SB win equity the proper hedge bet on TB is $480, which increases total EG to 0.257%. But again, this is significantly lower than the ex ante hedge EG of 0.404%.

IRT your question, I calculate the logs of bankroll growth for each scenario, multiply by the probabilities of each scenario, and add them up. Using Excel this is probably the simplest method of calculating EG for multiple scenarios. For KC 60% SB win equity, the 3 scenarios are calculated as below:

1. KC out of SB: 0.4 * log((10000 - 1000)/10000) = -0.0421
2. KC wins SB: 0.36 * log((10000 + 2000 - 1200)/10000) = 0.0277
3. KC loses SB: 0.24 * log((10000 - 1000 + 1800)/10000) = 0.0185

Sum of logs = 0.00403, EG = exp(0.00403) - 1 = 0.00404 = 0.404%

Sorry but you may have to review again because there are more errors:

1. KC out of SB: 0.4 * log((10000 - 1000)/10000) = 0.4 * log(0.9) = -0.0183
2. KC wins SB: 0.36 * log((10000 + 2000 - 1200)/10000) = 0.36 * log(1.08) = 0.0120
3. KC loses SB: 0.24 * log((10000 - 1000 + 1800)/10000) = 0.24 * log(1.08) = 0.0080

Sum of logs = 0.0017

Using natural log? if not multiple by log(10) = 2.303

Nope. Using normal log, both in excel and on a calculator

Try log(10). If you get 1, then you are using log base 10. If you get 2.303, then you are using natural log.

I took your numbers and multiplied by 2.303 and got my numbers, so you must be using log base 10.

Ok I see what you did there. I was confused about what you were doing because I'm used to seeing natural log abbreviated "ln" and base 10 as "log". I agree that the EG is calculated using natural log and that for your original example of KC WE = 60% and fair odds of +150/-150 on the Super bowl then your growth answer and the optimal bet sizes are correct.

Trouble is, you never know when you're going to get fair odds in the future. As we all know, it's pretty rare (especially on a big betting event like the SB). How does your method work if we get more standard odds of +135/-165 (which has a 4.5% hold)?

Advantage players have several ways of finding advantageous odds. Stale lines are probably the easiest way. Promos are also available (although probably not for the limits we want). Exchange betting gets very close to 0EV. Also, if the AP has a belief that the no vig line is a percent or 2 off, then line shopping will likely find a 0EV (or perhaps a +EV) hedge. Similarly, APs who are effective at beating opening lines (usually when lines are least efficient) have a good chance at getting 0 or +EV bets. Wagering between private parties is also possible and can provide a 0 or +EV opportunity.

And as you also noted in your write up, hedges do not need to be 0 or +EV to be effective.

All very fair points, and as you said we're in agreement that even -EV hedges can increase your EG and if you're able to find a 0 or +EV hedge it's really great! You make a terrific point that the "hedgeability" of a bet can actually decrease your risk enough to make it worth betting over the independent Kelly fraction on the original future (or at the very least, betting full Kelly instead of fractional if your original edge is hard to be sure of) and then line shopping for the best hedge value later on.

Also notice that for your optimal future size of 10% ($1000), my formula of f = p - q/b + h gives the correct hedge bet size of 12% for fair odds.

Yes.

Most novice bettors have no idea how much hedging improves one's rate of success.

Not just novices, sir. Everyone.

https://twitter.com/edteach23/status...51256075198467

https://twitter.com/edteach23/status...78314740203523

haha maybe he is a savant, but for the rest of us we should be measuring profit in %, not "chips", and maximizing our bankroll growth instead of betting like it's a pissing contest

Another calc error. I had hedge at fixed odds of 1.5, which is dumb of course.

With the correct hedge odds, KC future bet becomes -EV when their SB win equity is below (1/3)/0.6 = 0.556. But the EG calc for the general case of KC SB win equity = 0.6 is correct, 0.404%.

New chart:
My point in producing the chart is not to imply that one can predict KC SB win equity. Instead, that KC SB win equity is a stochastic (i.e. random variable), and it projects the optimal KC future bet size. If one can estimate a stochastic representing KC SB win equity, then one can better optimize the KC future bet size.

Well you make a great point, sir. I'd like to double check the math and do some more research on the topic, but would you be cool if I wrote about your concept in my next article and credited you with a link to this thread?

Sure, maybe we could do some collaborative work on hedging and EG strategies.

It is not a trivial topic. The math can get somewhat complex. Excel and other tools certainly help.

Sounds like a plan, DM me and we can talk.

You're absolutely right when you say it's not trivial. And while the math is definitely complex (and tools like Excel are great and often needed), one of my goals with my article series is to provide simple formulas when possible for people to use, similar to the independent Kelly formula.

You did a nice job with this SG.

Thanks Iowa! From your posts in the Golf betting thread, I suspect I'm preaching to the choir with you, but hopefully the formulas and visual aids were of value. Are you familiar with the ideas that PokerHero has been talking about ITT, and what are your thoughts on that?