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[Number Six]

That's the whole thing with trading odds, isn't it?
What they don't tell you is that you still need the insight to find a trade that is misvalued for your particular needs.

It's not a new concept at all and while it's sound enough, the execution might be something else.

Now he's loses very close to half his life's savings.

You have dumbed your example down to such a level I'm surprised you even think it's still valid.
The old timer is just another ignorant gambler playing a game blind, it's not surprising he's about to lose his entire life savings.
In this case proportional betting isn't going to save him, but neither is anything else for that matter.
Quitting, maybe, before being entirely broke would be the best course of action.

[Dragoner]

Where did I write to the order of wins and losses?

You mentioned losing 15 in a row, that suggests the order of wins and losses.

But doubling a stake is a lot harder than losing half of it.

Staying with the same 5% proportional scenario, doubling your BR takes 15 wins. That doesn't seem too hard comparing to the 15 losses it takes to lose half of it. It is a bit harder to double it, that is why you need 0.513 win rate and not just 0.5. But that still isn't "a lot" harder.

I don't like your math.  (I could tell right off that you're not a professional mathematician, and certainly never a gambler.) I really don't like the way in which you're trying to suck gamblers in with it.

Calculations can be either correct or wrong. You are welcome to correct any of my calculations. Saying you don't like it is just the same thing when you picked at me writing decimal odds (2 for even money bet), while that is widely used in sports betting. These arguments are just invalid.

[Bayes]

I just wanted to point out that the chart in my previous post is misleading, because with the win rate at 51% (which is what it is in the simulation), at 5% staking you will eventually lose your entire bank. In my defence, I'd forgotten that when I previously coded this staking plan, I used an edge of 5% not 2% (which meant that the win rate was > the required 0.513).


So the results shown in the above plot were a fluke. If I'd shown results for more spins the final result would have been a loss of bank.


Thanks to Dragoner who sent me a pm explaining the math (hey Archie, how come you didn't pick up on that? you're a "real" mathematician).

[Dragoner]

The more trials we have, the closer win rate gets to the probability of a single trial. Like a coin toss with 50% probability.
If we have 10 trials, the probability of winning 40%-60% (4 or 5 or 6 tosses) is about 66%.
If we have 100 trials, the probability of winning 40%-60% (40 to 60 tosses) is already more than 96%!!!
That is how win rate gets closer and closer to the probability of a single trial in the long run.

So if we have a 55% probability for a single trial, then if we repeat the same trial, we get closer and closer to 55% win rate. This is why actually the long run is our friend if we have +EV betting opportunity. (This is a very big if though... It is not easy to come by +EV bets)


With 5% proportional betting you have a threshold at roughly 0.513.

If we have the probability of a single trial above this threshold, we expect to make money.
If we have a win rate above this threshold, we actually made money.


If you have a win rate above this threshold it is impossible to arrange wins an losses in an order where you lose money. Because the order doesn't matter. You can arbitrarily arrange huge losing streak(s) in a result, if your win rate is above the threshold you can't possibly have a losing result.


If someone is interested in how we get the threshold:
The end result (final bankroll) is BR*0.95^nl*1.05^nw (where BR=initial bankroll, nl=number of losses, nw=number of wins)
To break even we need the multiplier at 1 (that makes our final bankroll=initial bankroll):
0.95^nl*1.05^nw=1
0.95^nl=(1/1.05)^nw    ---> because 1/1.05^nw=(1/1.05)^nw
nl=log_0.95((1/1.05)^nw)
nl=nw(log_0.95(1/1.05))

Win rate is nw/(nw+nl).
Replacing nl, that leaves us with a win rate of
nw/(nw+nw(log_0.95(1/1.05))) = 1/(1+(log_0.95(1/1.05))) which is a little less than 0.513

[Bayes]

edit: just realized I made a mistake in the calculations. I'll be back!

[Dragoner]

There is a typo in your starting point, it is supposed to be a multiplication instead of addition. Then the rest is correct. And looks simpler than what I did.


You can draw the function f(x) = 1 / (1 - log(1 + x) / log(1 - x)) with:http://rechneronline.de/function-graphs/
You can set it up to only show values between 0 and 1, and you can see the relation of the bet proportion and the corresponding win rate threshold.
It is just the threshold though, not the optimal bet proportion.

[Bayes]

There is a typo in your starting point, it is supposed to be a multiplication instead of addition.

Yep. I thought there was another problem, but it was ok. I'm repeating the calculation here for anyone who's interested.


We start off with this equation which has to satisfied in order to break even for a particular value of x (the proportional staking percentage):


(1 - x)^L(1 + x)^W = 1

taking logs gives


Llog(1 - x) + Wlog(1 + x) = 0


=> L = -Wlog(1 + x) / log(1 - x)


Now, let the win rate be R, then R = W / (W + L)


and substituting for L we have


R = W / (W - Wlog(1 + x) / log(1 - x))


factoring and cancelling:


R = 1 / (1 - log(1 + x) / log(1 - x))


Thanks for the link to the plotting software, but it was just as quick to write a little program which stepped through values of x. Here are the results for x ranging from 1% to 10% in steps of 0.5% (note that these results are only valid for bets at 1:1 payout).



% of bank  Win rate threshold
   
   1.00               0.502500
   1.50               0.503750
   2.00               0.505000
   2.50               0.506251
   3.00               0.507501
   3.50               0.508752
   4.00               0.510003
   4.50               0.511254
   5.00               0.512505 <-- Dragoner's result from his above post
   5.50               0.513757
   6.00               0.515009
   6.50               0.516261
   7.00               0.517514
   7.50               0.518768
   8.00               0.520021
   8.50               0.521276
   9.00               0.522530
   9.50               0.523786
  10.00              0.525042


By the way, the Kelly criterion says that you should bet a % equal to your edge in order to maximise growth rate. So in my simulation where the edge was 2%, 2% is the optimum percentage to stake each bet.

[Bayes]

Who said I was trying.  Your previous graph, along with many of Dragon's assumptions, involve so many novice misconceptions that its programing shouldn't have been a consideration.  I fairly stated and attested to a couple of those problems, but you and Dragron cast that away as trolling.

Ok, please point out the misconceptions. And be specific. The problem is that your writing is so ambiguous and incoherent that it's very hard to know what you're talking about most of the time. Please try to be clearer.


And yes, in my judgement, you were trolling. A couple of examples:

Saying that real mathematicians properly write the odds is not "picking" on specific calculations.  Which did I "pick on", your simplistic calculations?

Oh really?  Where did you get that math postdoctorate again?  Remind us.

Attacking Dragoner for not writing odds in the "proper" way is just nit-picking. Anyone who actually does any betting knows that odds in decimal format are commonly used these days, and the second example is just a personal attack because the validity of an argument doesn't depend on anyone's credentials.

Another example: you wrote

Some old guy with $200,000 "to his name" loses fifteen bets in-a-row some day.

Then when Dragoner replied saying that the order of bets is irrelevant, you answered

Where did I write to the order of wins and losses? 

As it stands, this makes no sense, but "fifteen bets in-a-row" clearly implies that order is relevant, so you at least implied it, but then denied it.

There are other examples but I can't be bothered to list them all.

[Turner]

@Bayes
You are being a bit generous with the phrase "most of the time"

[Bayes]

@Bayes
You are being a bit generous with the phrase "most of the time"

Turner, yep... 


Look at this; deliberately unintelligible:

But *** hand over a ******** of ********** only to **** it ******* by a ***** of ***** **********?

I've said it before and I'll say it again. Cryptic speech is a form of passive aggression. The intention is to create a feeling of insecurity and doubt in the listener/reader.

[Number Six]

The internet is a very poor place to seek out actual expertise. 

That is according to your opinion. Besides, the value of information is subjective to the reader, depending on your level of intelligence ie if you can understand it, and whether it is actually relevant to your cause. Of course, some information is not accurate; that is not unbelievable, there are literary, conceptual and factual mistakes in all places. In a way I also happen to see your point (in the context of casino games) - everyone can claim to be an expert without actually proving anything.

Perhaps, you mean "dumbed up", ie, put into a form in which pseudo rocket scientists can "make sense" of it?

Your example still does not really address why proportional betting would not be mathematically logical when you have an advantage. Moreover losing 12 or 15 bets in a row seems more applicable to a -EV casino game as oppose to sports betting, which this thread is meant to be about. It also doesn't represent a sound investment of time or money. It would be wise to test for variance of the bet and also its volatility, with that in mind the individual would know what losing streaks to expect and how much cash was needed to get through them. As a sidenote, that is why long simulations of betting systems are important, and testing "real" play samples aren't. A 5% stake, then, might prove to be too high, though 5% as a maximum is generally recommended for sports. Up to that point, entering a game without that information appears ignorant, and even with an advantage it would still be possible to lose. It's possible to contrive a statistical system for sports betting that offers a long term advantage, with the right information you can create your own probabilistic odds and then proceed to find trades which are misvalued. Back testing the selection method against previous results would enable you to fine tune your formula until it generates favourable results. According to the EV, proportional betting would absolutely be the best staking plan.

[Dragoner]

By the way, the Kelly criterion says that you should bet a % equal to your edge in order to maximise growth rate. So in my simulation where the edge was 2%, 2% is the optimum percentage to stake each bet.

For this we can draw a function of average growth rate for 0.51 win rate: f(x)=(1-x)^0.49*(1+x)^0.51
[attachimg=1]
It shows why 2% is a good idea for this win rate. It also shows, that if we bet over 4% we should expect a decay, because the average growth rate goes below 1.

[Bayes]

Archie,


I don't have the time or inclination to deal with your endless sophistries, and what would be the point anyway? you would just twist things to make it look as though you "really" meant something else much more reasonable. So knock yourself out, but I think you'll find not many members are interested.


BTW, I didn't approve your first post. And for the record, I haven't deleted any of your posts until that one. I did accidentally delete ONE of Turner's a few days ago, for which I apologised.


You can slug it out with Dragoner, that is if he can be bothered to respond.

[Bayes]

@ Dragoner, thanks for posting the graph. 

[Bayes]

I don't know what you're talking about. To my knowledge, none of my posts have been deleted.


You may have been making a point, but it was hardly clear what that point was. There was nothing wrong with the simulation, given the assumptions, and if I hadn't done it Dragoner wouldn't have pointed out my mistake.


I don't know why you think the model and Dragoner's calculations are "simplistic". What do you think needs to be taken into account that was left out?