[the_fredrik]

Hi there, and welcome to my first real post on this forum!

With the following reasoning I have convinced myself that it
will always be better to be doing arbitrage betting, rather
than using a great betting system with a certain large
long-term edge like 15%. I will try to keep it
resonable teoretical even though Ganchrow will probably find me to practical!

We assume we have a bankroll of $1000
We further assume there is a bet with two outcomes, A and B.
We KNOW that each have a 50% probability of happning.

The highest odds for outcome A we can find is 2.300 (+130)
The highest odds for outcome B we can find is 1.826 (-121)

Bet A seems terrific with a nice 15% edge or an expected return of $1.15 for every dollar bet.
Bet B seems horrible with an expected return of only $0.913 for every dollar bet.
(This can be compared with a "random" bet at -110 that returns $0.9545, so bet B is in some sense twice as bad as a random bet.)

I know there is another optimal strategy in this specific case, but let us take a step back and get a bit practical for now.

Let us choose between long-term strategy 'ARBITRAGE', which is to identify cases like above and do arbitrage betting, and long-term strategy 'PRO-BETTOR', which is to build a quantitative model of the real world that is capable of estimating the probablility of outcomes like A and B above very accurately. (easy, right?


STRATEGY PRO-BETTOR
==================

After thousands of hours of work using our super-brain, we use our model to identify cases with a large 15% edge like the one described above.
We do Kelly optimal betting, betting $115.4 on A.
In the long run this will mean an expected increase of the bankroll by $115.4*0.15 = $17.3 for every bet like this.

STRATEGY ARBITRAGE
================

We simply look for arbitrages manually or with a program, and are regularly identifying 1.77% arbs like the one described above. So we use our whole bankroll to bet $442.4 on bet A and $557.6 on bet B. Thereby locking in a profit of $18 each time we find such a bet.

**************************************** **

Which seems more difficult to you, the first strategy or the second? The amazing this is that with ARBITRAGE you are making More money at Much Less Risk!!
Even though we Know we have a Super-System that is able to identify bets with a 15% edge.

If you add practical matters like:
- your system might not be perfect
- you might want to have less risk than Kelly
- I can keep going

then to me, it quickly becomes clear that in order for the PRO-BETTOR strategy to be superior, you need a much higher edge than 15%! Who can honestly say they are confident that they have a 25% edge when betting?

There are a lot of aspects I have left out, like the number of bets available, the amount of dollars you can bet and so on, but the practical point I am trying to make is this:

Finding 1.77% arbs (rather easy) at a certain pace is roughly equivalent to having a reliable model that produces bets with a 15% edge at the same rate (difficult)

This means that most of us will be better of doing arbitrage - right or wrong??

Fredrik

[slash]

Why on earth did you waste your time writing such bs?

Grab a bear instead and get more out.

[ShamsWoof10]

Well there is NO DOUBT arbitrage is by far better overall then any system out there but the one thing about it is the books make life very unconfotable for you... It's not worth it also because Pinny. is no longer around for US players which makes it less efficient... You've got to have a ton of money to make it worth your time and if you play with enough the books will find a way to break one off in your ass.... They don't seem to care about smaller amounts though as much...

[Ganchrow]

Yes. A Kelly bettor, not constrained by betting limits, would indeed prefer a sure payout of 1.803% to a 50% probability of winning an event paying out at +2.3000. (To be indifferent between the two, the Kelly bettor would require odds of about 2.4611.)

However, a Kelly bettor could do even better by risking half his bankroll on A, and half his bankroll on B. In this manner his bankroll would increase by 15% half the time, and would decrease by 8.68% half the time, implying expected profit of ~3.16% and expected bankroll growth of ~2.48% over time.

I'd also add that in general scalpable bets tend to have rather low limits, which will be a problem for bettors with appreciable stakes. As such, out of necessity as their bankrolls increase, most advantage bettors will have to eventually shift their focus away from pure arbs.

[magnavox]

Why on earth did you waste your time writing such bs?

Grab a bear instead and get more out.

[slash]

Is that you in that picture magna?

Exactly the thing I was talking about. Get out, take a breath of fresh air and have fun.

[magnavox]

Yep, I can assure you, this bear was quite tasty.

[slash]

Yep, I can assure you, this bear was quite tasty.

LOL!!! It took me 2 posts but now I know what you mean

I should probably edit my post to get that spelling error out

[magnavox]

[ShamsWoof10]

I'd also add that in general scalpable bets tend to have rather low limits, which will be a problem for bettors with appreciable stakes. As such, out of necessity as their bankrolls increase, most advantage bettors will have to eventually shift their focus away from pure arbs.

I totally agree that is maybe the biggest factor...

[wack]

Yes. A Kelly bettor, not constrained by betting limits, would indeed prefer a sure payout of 1.803% to a 50% probability of winning an event paying out at +2.3000. (To be indifferent between the two, the Kelly bettor would require odds of about 2.4611.)

However, a Kelly bettor could do even better by risking half his bankroll on A, and half his bankroll on B. In this manner his bankroll would increase by 15% half the time, and would decrease by 8.68% half the time, implying expected profit of ~3.16% and expected bankroll growth of ~2.48%.

I'd also add that in general scalpable bets tend to have rather low limits, which will be a problem for bettors with appreciable stakes. As such, out of necessity as their bankrolls increase, most advantage bettors will have to eventually shift their focus away from pure arbs.

Quick question for the master. Is 50/50 the best bet because the p is 0.5 of each event occurring? Or is it not as simple as that?

[vanman]

arbitrage every time,no risk to money.

[Ganchrow]

Quick question for the master. Is 50/50 the best bet because the p is 0.5 of each event occurring? Or is it not as simple as that?

Actually, it is. It's not at all obvious but the math works out such that in a case of a pure arb within a market, the Kelly optimal weights, assuming no other constraints would be the same as the outcome probabilities.

So for example, given three outcomes with odds of:
o₁=3.9600
o₂=1.6333
o₃=8.0000

and win probabilities of:
p₁=25%
p₂=60%
p₃=15%

we see that a pure arb exists (1/3.9600+1/1.6333+1/8.0000 ≈ 0.98977 < 1) and hence the Kelly optimal weights are (k₁, k₂, k₃) = (25%, 60%, 15%).

Note that if o₁ were twice as high, let's say, at a line of 7.92, the Kelly optimal weights would not change as the arb would still exist.

However, if o₁ were only slightly lower, at let's say 3.800, then no arb would exist and the above equivalence will not hold. The Kelly-optimal weights in this case would be (k₁, k₂, k₃) = (0%, 1.7476%, 3.1068%).

[wack]

Thanks Ganch, that's what I thought

[wack]

Yes. A Kelly bettor, not constrained by betting limits, would indeed prefer a sure payout of 1.803% to a 50% probability of winning an event paying out at +2.3000. (To be indifferent between the two, the Kelly bettor would require odds of about 2.4611.)

However, a Kelly bettor could do even better by risking half his bankroll on A, and half his bankroll on B. In this manner his bankroll would increase by 15% half the time, and would decrease by 8.68% half the time, implying expected profit of ~3.16% and expected bankroll growth of ~2.48% over time.

I'd also add that in general scalpable bets tend to have rather low limits, which will be a problem for bettors with appreciable stakes. As such, out of necessity as their bankrolls increase, most advantage bettors will have to eventually shift their focus away from pure arbs.

Another question - this is an interesting topic. If the odds in the example were +150 and - 121, and the p of both was 0.5, are you saying the optimal strategy is now to just bet the value side of the line in the top paragraph?

[wack]

I'll try to produce this worked example, please point out any errors:

Just backed a team at -357

The other side of the wager is available at +377.

I believe the true price to be -392 (let's assume that is correct).

3 scenarios:

i) I value bet the entire amount. Let's say $1000.

p(win) = 0.79675 in this case. So I have an EV of:

$1000 * 1.28 * 0.79675 = $1019.84 or $19.84 per $1000 invested.

ii) Pure arb. I wager $268.34 on the other side of the line at +377.

No matter what I make a profit of $11.64.

iii) I split my stakes here, on the basis that p(win) = 0.79675

I have $1000 on the team to win (and that's the max I can have). That needs to be 79.675% of my stake.

My total stake is going to be $1255.10. Therefore I wager $255.10 on the other side of the line at +377

If team wins, I win $280 and lose $255.10. Profit of $24.90. This occurrs 79.675% of the time.

If team loses, I lose $1000 and win $961.73. Loss of $38.27. This occurrs 20.325% of the time.

The EV here is $12.06.

Have I made any mistakes here? Any useful comments?

[Wheell]

In terms of sheer EV you seem to be correct, but arbitrage will always produce a lower EV than simply taking the "good" side when risking the same amount of money. Arbitrage elmintates risk and allows you to effectively wager your entire bankroll whereas when taking sides you should only risk a small portion of your bankroll.

[wack]

Ok I worked through another example and found a stranger result

Excuse the complex prices.

3 outcome event.

Probabilities:

i) 0.4608
ii) 0.31696
iii) 0.22222

$1000 on outcome iii) at odds of 5.9339 (+493.39)

Obviously this is a rick. Also outcome ii) is available at odds of 3.3703 (+237.03) (which is also "above the money). Outcome i) is available near-the-money at 2.1368 (+113.68)

Pure value play: (on outcome iii)

$1000 at 5.9339. EV $1318.64, $318.64 profit

"Favoured Hedge" Play:

i) We have our $1000 @ 5.9339. That's all we can get in this case. We assume an unlimited bankroll.

As it is an arbitrage situation we can apply kelly as per the win probabilites as Ganch states above.

Therefore our $1000 @ 5.9339 makes up 22.22% of our stakes on this event. Our total stakes will be $4500 ($1000/0.222222)

We stake $1426.32 on outcome ii) at the best odds (above the money) of 3.3703.

We stake $2073.6 on outcome i) at the best odds (below the money). - 2.1368

Ok we are a few cents out now due to rounding but humour me.

In this situation we expect:

Return of $4430.86 46.08% of the time
Return of $4807.13 31.696% of the time
Return of $5993.90 22.22% of the time

EV $2041.74 + $1523.67 + $1331.98 = $4897.39 for $4500 investment, $+397.39 and worst case scenario we are losing $69.14, the other two outcomes are $307.13 profit and $1493.9 profit.

Straight Arbitrage:

$1000 @ 5.9339 returns $5933.90
Stake $1760.64 on outcome ii) - returns $5933.9
Stake $2777.00 on outcome i) - returns $5933.9

Return of $396.26 profit no matter what.

So the favoured hedge has a tiny E.V. advantage and the straight value bet is at a disadvantage (although admittedly the value bettor would be having a wager on outcome ii) as that is identified as being above-the-money).

My question really is this. If you can close near-to-the money then surely you should do so? No other strategy seems to have any distinct advantage over the arbitrage here.

I hope it doesn't seem too convoluted but its a real-life example which I did arbitrage and was just going through to see if the other methods were superior in any way. I think perhaps the difference between this and the OP example is that the outcome that is being backed to close the arb is near-the-money rather than well under the money? In situations like the above is the consensus that the arbitrage is the superior strategy?

FYI of course it WAS outcome iii) in this instance.

[the_fredrik]

Thanks for the useful replies (and even the not-so-useful ones - I got a good laugh out of the bear!

I thought I would get some good counter-arguments though.

Basically the only one I've got it "sure, you might be right, but generally you can bet smaller with arbs so the dollar gain is worse, especially as your bankroll grows".

I do not understand this argument. I regularly bet $10,000+ on arbitrage-type of situations. Sure, it is easier to find a big arb on smaller markets - but it seems to me that the reason big arbs are rarer on larger markets is that the markets are more efficient - which would make it harder even for the super-system to find a nice EV in these markets.

So to claim that for large markets, there are no arbs but there are large EV bets seems strange to me. Do you see what I am getting at?

I should mention perhaps that I am in Europe so I have access to Pinnacle Sports, Ladbrokes, and other books. The US arb-hunter might have to change profession these days

As for bookmakers cheating me, I only use the A ranked sportsbooks and have been able to avoid this so far. (Though it was close once - see http://www.sportsbookreview.com/forum/players-ta...ean-books.html)

[wack]

Basically the only one I've got it "sure, you might be right, but generally you can bet smaller with arbs so the dollar gain is worse, especially as your bankroll grows".

I do not understand this argument. I regularly bet $10,000+ on arbitrage-type of situations. Sure, it is easier to find a big arb on smaller markets - but it seems to me that the reason big arbs are rarer on larger markets is that the markets are more efficient - which would make it harder even for the super-system to find a nice EV in these markets.

So to claim that for large markets, there are no arbs but there are large EV bets seems strange to me. Do you see what I am getting at?

I see, but you have to look at it this way.

In the first example I gave above:

The dominant strategy was to simply ride the bet. Given that with arbitrage your bankroll is (hopefully) constantly increasing it gets to the stage, even making $10000+ bets, that you can afford to ride the swings and make more profit. The EV was around $8 greater per $1000 invested.

Over time you will find that you cannot make arbitrages for over $10000, for the following reasons:

1) Accounts will be closed. You can open 100 accounts but once they have identified your betting patterns (which if you are betting $5000+ on one side of a line you can be sure they will do) they will restrict or close new ones almost immediately.

2) Markets change and people use automated tools. When mansion started there were commonly arbs between them and pinny for up to $50000. Robots soon closed off those opportunities.

3) More and more people around the world are using arb sites to make money. Some of these people live in countries where $20 a day is a sufficient/decent wage. Tens of thousands of people are getting into it every year, and hence the bookmakers who ARE giving away the value are getting more and more incentive to stop arbitraging. Every time you place a large wager on a soft line you can be sure it is being recorded and your accounts are being flagged.

There are plenty more arguments. One might be to post your Profit and Loss with Pinnacle - are you winning there? How much have you lost if you aren't winning?

[Ganchrow]

Another question - this is an interesting topic. If the odds in the example were +150 and - 121, and the p of both was 0.5, are you saying the optimal strategy is now to just bet the value side of the line in the top paragraph?

No. Optimal strategy for a Kelly bettor unconstrained by betting limits would be to bet 50% of each of the two outcomes.

If the odds in the example were instead +99,900 and -99,800, and each result occurred with 50% frequency, optimal strategy for a Kelly bettor unconstrained by betting limits would still be to bet 50% of bankroll on each of the two outcomes.

I'll try to produce this worked example, please point out any errors:

Just backed a team at -357

The other side of the wager is available at +377.

I believe the true price to be -392 (let's assume that is correct).

3 scenarios:
i) I value bet the entire amount. Let's say $1000.

p(win) = 0.79675 in this case. So I have an EV of:

$1000 * 1.28 * 0.79675 = $1019.84 or $19.84 per $1000 invested.

Looks fine. With proper rounding I'm at $19.9267 or 1.9927%. Assuming total bankroll were $1,000 this would imply expected long-term bankroll growth of -100%.

ii) Pure arb. I wager $268.34 on the other side of the line at +377.

No matter what I make a profit of $11.64.

I have a wager of $268.3664 and a profit of $11.7413 or 1.6048%. Assuming total bankroll were $1,268.3664 this would imply expected long-term bankroll growth of 0.9257%.

iii) I split my stakes here, on the basis that p(win) = 0.79675

I have $1000 on the team to win (and that's the max I can have). That needs to be 79.675% of my stake.

Important note:

You seem to be implying here that your total stake is fairly large and that the only reason you're betting $1,000 is due to house limits. If this is true, then allocating your bets by placing the max on the value bet and then wagering in proportion to the win probabilities of each of the other bets is not Kelly optimal in general.

Using the above example, let's say there's a max bet of $1,000 on the value team, functionally no max on the other team, and that you have a total bankroll of $10,000. The Kelly optimal bet would be the full $1,000 on the fave at -377 with no bet on the hedge.

My total stake is going to be $1255.10. Therefore I wager $255.10 on the other side of the line at +377

If team wins, I win $280 and lose $255.10. Profit of $24.90. This occurrs 79.675% of the time.

If team loses, I lose $1000 and win $961.73. Loss of $38.27. This occurrs 20.325% of the time.

The EV here is $12.06.

I get an EV of roughly $12.1492 or 0.968%. Assuming total bankroll were $1,255.10 this would imply expected long-term bankroll growth of 0.9472%.

[Ganchrow]

Probabilities:

i) 0.4608
ii) 0.31696
iii) 0.22222

$1000 on outcome iii) at odds of 5.9339 (+493.39)
outcome ii) is available at odds of 3.3703 (+237.03)
Outcome i) is available at 2.1368 (+113.68)

Pure value play: (on outcome iii)
$1000 at 5.9339. EV $1318.64, $318.64 profit

Assuming the total bankroll were utilized on this bet this would imply expected long-term bankroll growth of -100%.

"Favoured Hedge" Play:

i) $1000 @ 5.9339. That's all we can get in this case. We assume an unlimited bankroll.

As it is an arbitrage situation we can apply kelly as per the win probabilites as Ganch states above.

Therefore our $1000 @ 5.9339 makes up 22.22% of our stakes on this event. Our total stakes will be $4500 ($1000/0.222222)

We stake $1426.32 on outcome ii) at the best odds (above the money) of 3.3703.

We stake $2073.6 on outcome i) at the best odds (below the money). - 2.1368

Return of $4430.86 46.08% of the time
Return of $4807.13 31.696% of the time
Return of $5993.90 22.22% of the time

EV $2041.74 + $1523.67 + $1331.98 = $4897.39 for $4500 investment, $+397.39 and worst case scenario we are losing $69.14, the other two outcomes are $307.13 profit and $1493.9 profit.

Assuming the total bankroll were utilized on this bet this would imply expected long-term bankroll growth of 7.8181%. However, you state that the player has an unlimited bankroll in this example. Were that so, then this would not represent Kelly optimal strategy. If we assume the player had a $10,000,000 bankroll with a $1,000 max bet on outcome 3, then optimal weights for outcomes 1 & 2 respectively would be $191,060.53, and $369,236.08, implying expected bankroll growth of 0.1142%.

Straight Arbitrage:

$1000 @ 5.9339 returns $5933.90
Stake $1760.64 on outcome ii) - returns $5933.9
Stake $2777.00 on outcome i) - returns $5933.9

Return of $396.26 profit no matter what.

Assuming the total bankroll were utilized on this bet this would imply expected long-term bankroll growth of 7.1558%.

So the favoured hedge has a tiny E.V. advantage and the straight value bet is at a disadvantage (although admittedly the value bettor would be having a wager on outcome ii) as that is identified as being above-the-money).

My question really is this. If you can close near-to-the money then surely you should do so? No other strategy seems to have any distinct advantage over the arbitrage here.

I'm not really sure I understand what you're getting at here nor do I notice anything particularly "strange" about these results. Remember what Kelly does -- it maximizes the expected long-term growth of a bankroll. That's it. It doesn't maximize expected value, it doesn't minimize risk, it maximizes expected growth. And I'm afraid I don't quite understand what you mean by "close near-to-the money".

I think perhaps the difference between this and the OP example is that the outcome that is being backed to close the arb is near-the-money rather than well under the money? In situations like the above is the consensus that the arbitrage is the superior strategy?

For someone who wants to maximize the growth rate of their bankroll? No. It is not. The arbitrage would not be the superior strategy for such a growth-minded individual regardless of what the consensus might be.

[Ganchrow]

I thought I would get some good counter-arguments though.

I did argue my response that for a Kelly bettor, a riskless arb strategy is inferior to a blended risk strategy, and that the Kelly bettor unconstrained by betting limits, when facing a riskless arb) would wager the a proportion of his bankroll on every outcome equal to that outcome's probability.

Basically the only one I've got it "sure, you might be right, but generally you can bet smaller with arbs so the dollar gain is worse, especially as your bankroll grows".

I do not understand this argument. I regularly bet $10,000+ on arbitrage-type of situations. Sure, it is easier to find a big arb on smaller markets - but it seems to me that the reason big arbs are rarer on larger markets is that the markets are more efficient - which would make it harder even for the super-system to find a nice EV in these markets.

So to claim that for large markets, there are no arbs but there are large EV bets seems strange to me. Do you see what I am getting at?

I think what you mean to say is not that you don't "understand", but rather that you think the argument invalid. Nevertheless, I think that most professional sports bettors and most hedge fund traders would tend to agree that riskless arbs are far more difficult to come by on a dollar-for-dollar basis than riskier value plays. Nevertheless, If you're routinely finding $10,000+ arbs, then good for you. I hope it keeps up. Even so, at $10,000 per arb, you'll still eventually hit a capacity constraint. At that point you might decide to reconsider your views on risk...

[the_fredrik]

I see, but you have to look at it this way.

In the first example I gave above:

The dominant strategy was to simply ride the bet. Given that with arbitrage your bankroll is (hopefully) constantly increasing it gets to the stage, even making $10000+ bets, that you can afford to ride the swings and make more profit. The EV was around $8 greater per $1000 invested.

Over time you will find that you cannot make arbitrages for over $10000, for the following reasons:

1) Accounts will be closed. You can open 100 accounts but once they have identified your betting patterns (which if you are betting $5000+ on one side of a line you can be sure they will do) they will restrict or close new ones almost immediately.

2) Markets change and people use automated tools. When mansion started there were commonly arbs between them and pinny for up to $50000. Robots soon closed off those opportunities.

3) More and more people around the world are using arb sites to make money. Some of these people live in countries where $20 a day is a sufficient/decent wage. Tens of thousands of people are getting into it every year, and hence the bookmakers who ARE giving away the value are getting more and more incentive to stop arbitraging. Every time you place a large wager on a soft line you can be sure it is being recorded and your accounts are being flagged.

There are plenty more arguments. One might be to post your Profit and Loss with Pinnacle - are you winning there? How much have you lost if you aren't winning?

1. This has never happened to me on any A rated book. Of course maybe it will. But if you are an advantage bettor and are a long-term winner, you also have this risk. Are you saying that they dislike an arb:er more than an advantage bettor? I do not believe that.

2-3. I have the feeling that the A rated bookies does not dislike arb:ers at all. It is only the smaller bookies that know they might have a soft line that dislikes us.
As for robots, it is definitely a problem as more and more people use them.
Sure, the world is always changing and maybe I will not make any money on this in 2 years. This will be very evident if/when this happens. Maybe the person with the super-system will one day find his system is no longer producing profits. That will be much harder to spot.

The Pinny argument: It is very interesting that you bring this up. I do not have a careful enough bookkeeping in order to answer this question, but I have a definite feeling that I am losing at Pinnacle and winning at other bookies. If this is the case, I have a super-system right there.
Of course, the fact that all US players are gone from Pinnacle might have an effect on our pinny-super system...

[ShamsWoof10]

1.
2-3. I have the feeling that the A rated bookies does not dislike arb:ers at all. It is only the smaller bookies that know they might have a soft line that dislikes us.
As for robots, it is definitely a problem as more and more people use them.
Sure, the world is always changing and maybe I will not make any money on this in 2 years. This will be very evident if/when this happens. Maybe the person with the super-system will one day find his system is no longer producing profits. That will be much harder to spot.

The Pinny argument: It is very interesting that you bring this up. I do not have a careful enough bookkeeping in order to answer this question, but I have a definite feeling that I am losing at Pinnacle and winning at other bookies. If this is the case, I have a super-system right there.
Of course, the fact that all US players are gone from Pinnacle might have an effect on our pinny-super system...

Why would you need a robot account or several if you can get large bets to arb. with? I am assuming either 10K per side is not enough or on smaller markets where the betting limit is much lower then 10K... I believe all books hate arbers because in the long run they all lose...

As far as the system of taking the other side over Pinny..that is exactly what I used to do after I noticed the Pinny. side losing a lot BUT.... the US market being gone has totally changed that.. It still is but not nearly as much... This is the system I used when I do bet but to one book how am I differant then an arber..?

[Ganchrow]

Are you saying that they dislike an arber more than an advantage bettor? I do not believe that.

An advantage player, through his betting, will transmit information to sports books to which they would not already have access.

Arbers, OTOH, are not transmitting any information that is not already available to the book.

[the_fredrik]

I did argue my response that for a Kelly bettor, a riskless arb strategy is inferior to a blended risk strategy, and that the Kelly bettor unconstrained by betting limits, when facing a riskless arb) would wager the a proportion of his bankroll on every outcome equal to that outcome's probability.

Of course. There is just the practical detail of knowing each outcome's probability. If I knew those I'd switch strategy in a second!

I think what you mean to say is not that you don't "understand", but rather that you think the argument invalid. Nevertheless, I think that most professional sports bettors and most hedge fund traders would tend to agree that riskless arbs are far more difficult to come by on a dollar-for-dollar basis than riskier value plays. Nevertheless, If you're routinely finding $10,000+ arbs, then good for you. I hope it keeps up. Even so, at $10,000 per arb, you'll still eventually hit a capacity constraint. At that point you might decide to reconsider your views on risk...

Sure, I am hitting capacity constraints all the time!
But so is the value bettor.

I do believe that if I already had a good system, I would use it to make more money than I do now. But my initial example makes me think it might not be worth the time and effort to try to invent such a system. My time might be better spent grabbing that bear!

I AM working on reconsidering my risk view but again, my initial example does not encourage me..

[Ganchrow]

I did argue my response that for a Kelly bettor, a riskless arb strategy is inferior to a blended risk strategy, and that the Kelly bettor unconstrained by betting limits, when facing a riskless arb) would wager the a proportion of his bankroll on every outcome equal to that outcome's probability.

Of course. There is just the practical detail of knowing each outcome's probability. If I knew those I'd switch strategy in a second!

You're not be completely consistent and are still bringing up a bit of a false comparison.

First, you're claiming that a riskless arb of a market is better than a straight bet on the value side. As evidence, you're comparing the EV from a total riskless arb to the EV from a value bet sized based upon Kelly.

But then, when I point out that simple Kelly is inappropriate in this regard you complain about the difficulty "of knowing each outcome's probability". But you need to know each outcome's probability in order to calculate simple Kelly weightings. If you don't know these probabilities then to what are you comparing your riskless arb?

There's no question about it. If you have absolutely no clue what the underlying probabilities are of a given event you certainly can't do better than a blanket arb of the entire market. But as long as you have some idea (for instance, what you've gleaned from other bookmakers and exchanges), you almost certainly can do better.

[the_fredrik]

Arbers, OTOH, are not transmitting any information that is not already available to the book.

This is only true in theory I believe. (But maybe I should be careful since I don't know what OTOH means)

Are you saying that the bookmaker knows he is exposed to an arb situation but still chooses to keep his odds?
If I was a small bookie and knew my odds together with Pinnacle produced a 2% arb - I would change my odds in a second.
If I was a large bookie with lots of super-smart people and knew my odds together with Pinnacle produced a 2% arb - I might choose to believe I am right and Pinnacle is wrong. In this case I would welcome any arb action on my odds.

So I believe that in the small bookie case I am transmitting information that is new to the bookie.

[Ganchrow]

This is only true in theory I believe. (But maybe I should be careful since I don't know what OTOH means)

Are you saying that the bookmaker knows he is exposed to an arb situation but still chooses to keep his odds?
If I was a small bookie and knew my odds together with Pinnacle produced a 2% arb - I would change my odds in a second.
If I was a large bookie with lots of super-smart people and knew my odds together with Pinnacle produced a 2% arb - I might choose to believe I am right and Pinnacle is wrong. In this case I would welcome any arb action on my odds.

So I believe that in the small bookie case I am transmitting information that is new to the bookie.

OTOH = "On the other hand"

It's information that, if the bookie doesn't already have, could be obtain elsewhere considerably more inexpensively.

Sometime arbs exists for no reason other than a linesmaker's wife called to remind him to bring home milk.

[the_fredrik]

You're not be completely consistent and are still bringing up a bit of a false comparison.

First, you're claiming that a riskless arb of a market is better than a straight bet on the value side. As evidence, you're comparing the EV from a total riskless arb to the EV from a value bet sized based upon Kelly.

But then, when I point out that simple Kelly is inappropriate in this regard you complain about the difficulty "of knowing each outcome's probability". But you need to know each outcome's probability in order to calculate simple Kelly weightings. If you don't know these probabilities then to what are you comparing your riskless arb?

There's no question about it. If you have absolutely no clue what the underlying probabilities are of a given event you certainly can't do better than a blanket arb of the entire market. But as long as you have some idea (for instance, what you've gleaned from other bookmakers and exchanges), you almost certainly can do better.

I am trying to choose between two approaches to betting:

1. pure arbitrage betting

2. creating a model that estimates probabilities and then I should use your advice and bet my entire bankroll divided according to my estimated probabilities instead of only betting the value side, I agree. (maybe a bit adjusted if I am risk-adverse)

I am surprised that 2 is not much much much better than 1 - since it is so much easier!

It is also interesting to me that it is correct to bet a large portion of my bankroll on what we know is a negative-expectancy bet, just to reduce risk.

[Ganchrow]

I am trying to choose between two approaches to betting:

1. pure arbitrage betting

2. creating a model that estimates probabilities and then I should use your advice and bet my entire bankroll divided according to my estimated probabilities instead of only betting the value side, I agree. (maybe a bit adjusted if I am risk-adverse)

Right, but the title of your thread is "Evidence that it is nearly always better to do arbitrage than take a risk". Perhaps i should have been "Evidence that it is nearly always better to do arbitrage than take a risk if you don't know the underlying probabilities"?

Otherwise, if you do know the underlying probabilities (or are able to estimate them sufficiently well by looking at other books' markets) then your theory as promulgated in the title of the thread is inaccurate.

[ShamsWoof10]

A couple questions to GANCH please...

1.. let's say Pinnacle and Bodog have an arb between the two and I use a system that when an arb. exists to bet the bodog side and not the pinnacle side since I assumed bodog wins more often then not... TO BODOG, in relation to information being transmitted, how is this differant from me actually arbitrating..? To bodog it's the samething and it's pinnacle who doesn't get the action from me...

2... Risk free means stress free and I think no worry or stress should definately account for some value.. just ask the guys in the sportsbook withdrawl thread...

Thanks

[Ganchrow]

1.. let's say Pinnacle and Bodog have an arb between the two and I use a system that when an arb. exists to bet the bodog side and not the pinnacle side since I assumed bodog wins more often then not... TO BODOG, in relation to information being transmitted, how is this differant from me actually arbitrating..? To bodog it's the samething and it's pinnacle who doesn't get the action from me...

From Bodog's perspective it's obviously not any different.

2... Risk free means stress free and I think no worry or stress should definately account

Of course. Who says it shouldn't?

[wack]

Assuming the total bankroll were utilized on this bet this would imply expected long-term bankroll growth of 0%.
Assuming the total bankroll were utilized on this bet this would imply expected long-term bankroll growth of 7.8181%. However, you state that the player has an unlimited bankroll in this example. Were that so, then this would not represent Kelly optimal strategy. If we assume the player had a $10,000,000 bankroll with a $1,000 max bet on outcome 3, then optimal weights for outcomes 1 & 2 respectively would be $191,060.53, and $369,236.08, implying expected bankroll growth of 0.1142%.

Assuming the total bankroll were utilized on this bet this would imply expected long-term bankroll growth of 7.1558%.
I'm not really sure I understand what you're getting at here nor do I notice anything particularly "strange" about these results. Remember what Kelly does -- it maximizes the expected long-term growth of a bankroll. That's it. It doesn't maximize expected value, it doesn't minimize risk, it maximizes expected growth. And I'm afraid I don't quite understand what you mean by "close near-to-the money".

For someone who wants to maximize the growth rate of their bankroll? No. It is not. The arbitrage would not be the superior strategy for such a growth-minded individual regardless of what the consensus might be.

By "close near-to-the-money" I mean betting the third outcome of the event (the only outcome for which you are NOT getting a price over and above what it should be) at a price VERY NEAR to what it should be (in the above case 2.1368 when it should be 2.17). Hence closing off the arb rather than leaving it open and just betting the 2 value sides accordingly.

In this situation I managed to close all three sides of the arb "above-the-money". This is obviously an ideal situation - however in two cases I was only just above the money, and in outcome iii) I was well above-the-money. So I should have bet more on outcome iii) to maximize long term bankroll growth?

I don't understand either why a kelly bettor in that situation would bet so big on an outcome which is undervalued when he can only get $1000 on the "value price". Why not just hypothesise that as $1000 is your max bet on outcome iii) that your bankroll is equal to $1000/0.222222 = $4500? Seems counterintuitive that this $1000 value bet would lead you to bet almost $200k on an outcome that is undervalued (albeit slightly).

I'm interested in best way to achieve bankroll growth.

Assuming the total bankroll were utilized on this bet this would imply expected long-term bankroll growth of 0%.

I don't understand why this is?