reverse parlay

pico · 07-22-08, 03:55 PM

here is an example...in case some of you are not clear what is a reverse parlay, for yesterday's MLB games

4 teams parlay for blue jays ml, cardinals over 9.5, twins ml, and marlins ml

reverse parlay odds: -1750
parlay odds: +1206

negative EV for both bets

Art Vandeleigh · 07-23-08, 10:13 AM

Maybe a situation with unusual correlation would be appropriate.

Let's say 4 teams were fighting it out for the baseball NL wild card spot. On the last say of the season, one team is 1 game ahead of three other teams, and they are the first to play that day.

If you played a reverse parlay (by your definition) of all 4 teams winning, then you would want at least one of the 4 teams to lose to win your reverse parlay.

If the team leading the other 3 teams in the standings lost, then you win your parlay regardless of what happens in the other games.

If they won their game and clinched the wild card, it would become much less likely for all the remaining 3 teams to give the same effort they would give had the team leading the wild card race lost.

I'm sure there are better examples.

Edit: I now saw that you put no correlated bets in parenthesis. Well, maybe you can sneak this one past a D+ book

RickySteve · 07-23-08, 03:38 PM

There's no reason not to do it, but it's a matter of giving players what they want. Ploppies like to bet a little to win a lot, not the opposite.

pico · 07-23-08, 04:16 PM

Originally posted by RickySteve

There's no reason not to do it, but it's a matter of giving players what they want. Ploppies like to bet a little to win a lot, not the opposite.

ploppies? never heard of that term

pico · 07-23-08, 04:27 PM

Originally posted by Art Vandeleigh

Maybe a situation with unusual correlation would be appropriate.

Let's say 4 teams were fighting it out for the baseball NL wild card spot. On the last say of the season, one team is 1 game ahead of three other teams, and they are the first to play that day.

If you played a reverse parlay (by your definition) of all 4 teams winning, then you would want at least one of the 4 teams to lose to win your reverse parlay.

If the team leading the other 3 teams in the standings lost, then you win your parlay regardless of what happens in the other games.

If they won their game and clinched the wild card, it would become much less likely for all the remaining 3 teams to give the same effort they would give had the team leading the wild card race lost.

I'm sure there are better examples.

Edit: I now saw that you put no correlated bets in parenthesis. Well, maybe you can sneak this one past a D+ book

that is a pretty good example. your reverse parlay team 1 winning early game, team 2 lose, team 3 lose, and team 4 lose. or you can wait til early game finish then do a 3-team reverse parlay of team 2 lose, team 3 lose, and team 4 lose.

reverse parlay bets requires a large amont of bankroll to win small amounts, so i would stay away from shit books.

another example i thought about...but might not be that good...is on the opening day, take 4 home opening teams...bet on them with spread/ml. teams tend to play harder on home opener, so reverse parlay on that might be a +EV play.

Ganchrow · 07-25-08, 02:11 PM

I was looking at this a while back and decided I actually preferred the term "negative parlay" because a "reverse" carries with it a very a distinct meaning. Essentially a negative parlay would allow players to "write" their own parlays.

A parlay is nothing more than a series of IF-WIN(-PUSH) bets that are fully capitalized with 100% of the proceeds of the "previous" legs. This corresponds to betting a given unit amount on the "first" bet and rolling over profits into the "next" bet until either a loss is first encountered or the "end" of the parlay is reached (of course due to the communicative property of multiplication the precise parlay ordering is irrelevant, which is why I put "previous", "first", and "next" in quotations). If the "end" of the parlay is reached, then the player's a winner.

Similarly a negative parlay corresponds a series of IF-LOSE(-PUSH) bets and is akin to betting to win a given unit amount on the "first" bet and rolling over all losses into the "next" bet until either a win is first encountered or the "end" of the negative parlay is reached. If the "end" of the neg parlay is reached, then the player loses, otherwise the player wins.

Does this sound at all familiar? Well, it should. A neg. parlay is economically equivalent to a Martingale staking strategy, where the player doubles up after each loss (in the case of the "next" bet at odds equal to the "previous" -- he'll more than double up if the "next" bet is at odds less than the "previous" and vice-versa at greater odds) and keeps playing until he first wins or he reaches a specified max loss amount.

It's easy to derive true neg parlay pricing by appealing to these two basic principles:

If all bets were offered at zero expectation a true odds neg parlay should also have an EV of zero; and
A win after any leg should result always yield a constant total win amount.

So given N bets with the i^th bet offered at decimal odds of d_i, true odds on the neg parlay would be given by:

[nbtable][tr][td]True Neg Parlay Odds = OC( [/td][td][/td][td]OC( d_i ) )[/td][/tr][/nbtable]
where OC(·) refers to the odds complement function, defined (for decimal odds) as:

OC(d) = d (d-1)

The decimal odds complement function maps decimal odds to the complementary odds on the opposing side (i.e., those odds on the opposite outcome that would imply equal magnitude vig, but of opposite sign).

The explanation for all the above may be more intuitive when considering US-style odds. The odds complement function for US would map payout odds of U to payout odds of -U (e.g. from -200 to +200, or from -110 to +110). So if we were looking to find the true neg. parlay odds on a two bet neg. at +200, we'd first calculate true (regular) parlay odds on two bets at -200, which would be +125, and take the complement of that, which would be -125.

We see that if indeed the neg parlay parlay were offered at zero vig with each single bet winning a third of the time then the neg parlay of the two would win 1-( 2 3 )² = 5 9 of the time. 5 9 is of course the implied probability on a line of -125 (hence zero-vig).

So here are some examples of true neg parlay odds assuming all underlying bets at -110:

Code:

Num.	 Parlay	     Neg
Teams	   Odds	    Odds
2	   +264	    -341
3	   +596	    -826
4	 +1,228	  -1,845
5	 +2,436	  -3,984
6	 +4,741	  -8,477
7	 +9,142	 -17,911
8	+17,545	 -37,723
9	+33,585	 -79,328
10	+64,208	-166,699

One important point to note here is that even though a parlay and the corresponding neg parlay of the opposite sides represent mutually exclusive outcomes that span the entire outcome set, they are NOT unbiased indicators of true market price.

This is because while the percent vig on a regular parlay approaches 100% as the number of constituent bets increases, the percent vig on a neg parlay actually decreases as the number of constituent bets increase, approaching 0 as the number of constituent bets approach infinity. The raw dollar vig given a constant to-win quantity, however, still increases without bound. This is precisely analogous to Martingale where as maximum allowable Martingale trials approaches infinity, expected percent loss and expected dollar loss approach zero and infinity, respectively.

Another way to see this is that maximum possible vig on any bet is limited by the odds itself. For example the true odds on a 3-team parlay of -110 singles imply vig of roughly -13.03%. This parlay would win with probability 50%³ = 12.5%

A neg. partay of the opposite teams would win with probability 1-12.5% = 87.5%. 13.03% vig on a bet that wins with 87.5% probability would imply decimal odds of (1-13.03%) 87.5% ≈ 0.99399, which would necessitate the bettor losing his entire stake were his bet to lose and "only" losing 1-99.3399% = 0.6011% of his stake were his bet to "win" -- no one's that square.

So in other words, because maximum % vig is always capped by the bet loss probability, and because loss probability approaches 0 as neg. parlay size approaches infinity, max % vig approaches 0 as as neg parlay size grows without bound.

A comparison of vig assuming single odds of -110 is given here:

Code:

Num.	  Parlay 	Neg Par
Teams	     Vig    	    Vig
2	  8.884%	 3.006%
3	 13.026%	 1.908%
4	 16.979%	 1.168%
5	 20.753%	 0.693%
6	 24.355%	 0.401%
7	 27.793%	 0.227%
8	 31.076%	 0.127%
9	 34.209%	 0.069%
10	 37.199%	 0.038%

So what this ultimately gives us is a market that will be biased in favor of the neg. parlay, potentially offering more value to cost-conscious bettors.

With that said it should be easy to see that an uncorrelated neg. parlay would never be a part of an optimal (unconstrained) Kelly stake for any Kelly multiplier. Insofar as neg. parlays are nothing more than Martingales (although with legs allowed to go off simultaneously), and Martingale requires a bettor to increase his stake after a bankroll loss, a bettor with preferences exhibiting constant relative risk aversion (i.e., a Kelly bettor) would never find this optimal (assuming he were allowed unconstrained exposure to the underlying single bets -- if not then the neg. parlay could indeed serve as a good substitute).

I'll also point out that because Kelly already represents optimal betting strategy (for a Kelly bettor), no additional derivative wagers could possibly improve the unconstrained solution.

So the only real advantages that come to mind with a neg. parlay are:

Utilizing negative correlation -- In the same way that return on regular parlays is enhanced by parlaying positively correlated wagers, so to is return on neg. parlays with negatively correlated wagers.
Circumventing limits on single bets -- this is of greatest value when used in conjunction with a regular parlay, and with longs odds wagers.
Hedging out of prior parlay risk -- The lower vig on neg. parlays makes it possible to completely hedge out of parlay exposure at a reasonable cost. Try doing that with regular parlays.

Anyway, that's what comes to mind. I'm sure there are a few other interesting uses, and if I can think of them I'll add them to the list.

Lastly, I'm going to give my opinion that if properly marketed by the books, recreational players might well be interested in a product such as this. The pitch would be that a player could select maybe 3-5 very large underdogs and would score a nice payday if ANY of his dogs were to win. I think it could certainly work if done right.

Art Vandeleigh · 07-25-08, 03:14 PM

Excellent read as usual Ganch, although I thought your post may be about synthesizing one of these negative patlay beasts.

Before I wrote the example in my previous post in this thread, I considered another situation.

Let's say each of the 4 AL West teams are playing tonight, but none of them are playing against each other.

In addition, each team is assesed a 50/50 chance of winning their game as far as the public is concerned and all the games will go off at the usual vig your book offers (-110, -105, etc.).

However, a very reliable source of yours has given you information that one of the starting pitchers for one of these 4 AL West teams is having some physical problem unknown by the public at large. Your sources estimate is that the pitcher's problem is serious enough such that the team only has a 25% chance of winning their game.

One problem. Your phone call with your source was cut off just as he was about to tell you which team he's on. It's too late now to call him back. You must bet immediately.

Would you be better off playing the opponent of each AL West team individually at -110, or playing a 4 team negative parlay of all 4 teams winning at -110?

I'm quite certain playing them individually is the better choice, since they are uncorrelated, but the vig on a 4 team negative parlay is so low - 1.168% - now I'm not so sure.

Can you starighten me out?

pico · 07-25-08, 04:05 PM

ganch, yes, i know it is martingale strategy, but the reason i proposed it because i was wondering in what special situations would you spot a negative correlated events.

another example i can think of is that during interleague play, AL home teams tend to have a higher chance of winning. so negative parlay all your AL home picks together is a pretty good strategy.

also if today you know 4-5 games are starting rookie pitchers, you know the chances of one of them blow up is very likely, so you do a negative parlay on the overs.

the list goes on and on...i started this thread hoping some people would input some good scenarios where it is a good time to do a negative parlay.

ganch, i changed the title of this thread from reverse parlay to converse parlay because converse seems the most fitting term to describe this type of bet.

tomcowley · 07-25-08, 04:17 PM

Other than the first part about negatively correlated events, your other examples don't give you anything in the parlay. Each ML should already contain the blowup chance, the AL superiority, etc- and if it doesn't, it's better to just bet that ML.

pico · 07-25-08, 04:24 PM

Originally posted by tomcowley

Other than the first part about negatively correlated events, your other examples don't give you anything in the parlay. Each ML should already contain the blowup chance, the AL superiority, etc- and if it doesn't, it's better to just bet that ML.

fair enough, i think those are uncorrelated events

ok, here is a better example, if you know there is a rain storm in the boston-NYC area, but you don't know where the rain cloud is going to be when the game start, you know fielding will be horrible if it rains. you do a converse parlay of the red sox, yankees, and mets games over.

Ganchrow · 07-25-08, 04:28 PM

Originally posted by Art Vandeleigh

Can you starighten me out?

Sure.

The outcomes of the 4 games are indeed negatively correlated, which creates excess value for the neg parlay.

This results in an optimal stake of 7.724% on each of the 4 bets at -110 and 8.329% on the neg parlay.

Ganchrow · 07-25-08, 04:32 PM

Originally posted by picoman

fair enough, i think those are uncorrelated events

You're right. These outcomes (AL home teams winning in Interleague play, et. al.) are uncorrelated.

And that's exactly why the negative parlay would be of no value to a Kelly-type bettor in these situations.

Ganchrow · 07-25-08, 08:26 PM

Originally posted by Ganchrow

The outcomes of the 4 games are indeed negatively correlated, which creates excess value for the neg parlay.

Here's a simple way to determine the correlation between the wager outcomes.

The actual return matrix for a 4-unit total wager at -110 looks like this:

Code:

# Wins	Prob	Return
4	 9.375%	  90.91%
3	31.250%	  43.18%
2	37.500%	  -4.55%
1	18.750%	 -52.27%
0	 3.125%	-100.00%
EV		   7.39%

Tot Var
4-unit		  3.4659
wager

Now we know that for for N uncorrelated bets at the same decimal odds (d) and edge (E), total variance on an N-unit total wager should equal:

σ² = N*(d-1-E)*(1+E)

which in this case would be:

σ² = 4*( 21 11 -1.0739)*(1.0739) ≈ 3.5877

Similarly for N correlated bets at the same decimal odds (d), edge (E), and covariances (σ_xy), total variance on an N-unit total wager should equal:

σ² = N*(d-1-E)*(1+E) + 2*(N-1)*σ_xy

This gives us:

3.4659 = 3.5877 + 12*σ_xy
σ_xy ≈ -1.014%

for a correlation coefficient (between any two bets) of:

ρ = 4* -1.014% 3.4659 ≈ -1.171%

which should be correct barring some unfortunate algebraic or computational error.

Art Vandeleigh · 07-26-08, 05:02 AM

Thanks for the response Ganch.

Let's say I simplified things a bit. Instead of the pitcher being from any team in the AL West, he's on one of the 2 California based AL West teams. Also, the Angels are playing a day game, and the A's a night game, so proceeds (or Matingale) could be played on the second game.

As I understand things, with the betting options the books have available today, there is no way to extract the benefit of negative correlation at the moment, correct? I'm not as familiar as others with all that books have to offer, but an if-lose(-push) betting option seems as futuristic as a negative parlay.

Can you confirm there's no way to do this at the moment even with this simplified example? (or in general, any 2 sequentially occurring events, where you are certain one event has a reduced probabilty of occurring relative to its odds, but don't know which of the 2 events it is)

Ganchrow · 07-26-08, 07:04 AM

Originally posted by Art Vandeleigh

Thanks for the response Ganch.

Let's say I simplified things a bit. Instead of the pitcher being from any team in the AL West, he's on one of the 2 California based AL West teams. Also, the Angels are playing a day game, and the A's a night game, so proceeds (or Matingale) could be played on the second game.

As I understand things, with the betting options the books have available today, there is no way to extract the benefit of negative correlation at the moment, correct? I'm not as familiar as others with all that books have to offer, but an if-lose(-push) betting option seems as futuristic as a negative parlay.

Can you confirm there's no way to do this at the moment even with this simplified example? (or in general, any 2 sequentially occurring events, where you are certain one event has a reduced probabilty of occurring relative to its odds, but don't know which of the 2 events it is)

Well if the information weren't publicly available (which is how I had interpreted your earlier question) then you could simply bet on the first game, wait for the outcome, and then based upon that place a wager on the second game.

Before the first game started, your probability of winning game 1 would be 50%*50% + 50%*75% = 62.5%.

Now if after watching the game you were able to determine whether or not that game had featured the injured pitcher, then your win probability for game 2 would either be 50% (if it did) or 75% if it did not.

OTOH, if you were unable to determine whether or not that game had featured the injured pitcher, then your win probability for game 2 could be determined via Bayesian inference:

Define the following events:

Let IP1 = game 1 AL West Opp has injured pitcher
Let IP2 = game 2 AL West Opp has injured pitcher
Let OW1 = AL West Opp wins game 1
Let OL1 = AL West Opp loses game 1
Let OW2 = AL West Opp wins game 2

We're looking to determine the following two conditional probabilities:

P(OW2 | OW1)
P(OW2 | OL1)

(The event to the right of vertical bar refers to "given" information, so P(OW2 | OW1) should be read, "Probability that AL West Opp wins game 2 given that AL West Opp wins game 1).

We already know the following:

P(OW2|OW1)

P(OW2|OL1)

The above is of couse the analytical solution, which in a simple case such as this could be more easily obtaining by simply noting the outcome matrix:

Code:

Out1	Out2	Prob
W	W	37.5%
W	L	25.0%
L	W	25.0%
L	L	12.5%

From the above we can easily deduce that P(OW2|OW1) = 37.5% / (37.5%+25%) = 60%
P(OW2|OL1) = 25% / (25% + 12.5%) = 66 2 3 %, but this type of methodology can quickly become tedious in more complicated situations.

What's interesting to note is that regardless of the outcome of game 1, a second bet would still be placed on game 2. You'll also note that were an IF-LOSE or negative parlay available (and the bets needed to be placed contemporaneously), an n-Kelly bettor still bet a good portion on the straight bets. Martingale, you'll recall is exceedingly aggressive and it would take much greater magnitude negative correlation than this for such a strategy to dominate. I've yet to work it out, but I do suspect that even with the negative correlation a regular parlay would nevertheless factor in to the optimal Kelly solution.

Although perhaps I've misunderstood your question? (It's almost 8AM after all, and I've yet to get to sleep. Insomnia, believe me you, is quite a bitch.)

Ganchrow · 07-26-08, 07:31 AM

I'll also note that if event A and event B are negatively correlated then events A and ~B as well as events ~A and B would necessarily be positively correlated. (~X refers to NOT event X, in other words the mutually exclusive complement of the event).

As such, were the vig sufficiently low (books paying close to 7:1 on 3-team parlays come to mind), one could still take advantage of negative correlation by betting a number of regular parlays (or IF-WIN bets) on the positively correlated events.

For example, were we given the following contemporaneous bets:

P(A) = 50%
P(B) = 50%
P(A|~B) = 54%
P(B|~A) = 54%

then at true parlay odds, optimal n-Kelly stakes on all 4 parlays would be zero, although at not unheard of parlay odds of +280 (a roughly 4.26% premium over true parlay odds, although still representing 5% vig), optimal full-Kelly stake on each of the A/~B and ~A/B parlays would be 1.44% of BR.

Art Vandeleigh · 07-26-08, 11:02 AM

Originally posted by Ganchrow

What's interesting to note is that regardless of the outcome of game 1, a second bet would still be placed on game 2.

I think this is where my brain froze earlier today.

To sum up (?):

- it is possible (and straightforward) to synthesize a negative parlay for 2 sequential events, where one of the events is expected to have probability of occurence considerably less than the odds indicate (but the specific event is unknown)

-Since Martingale is "exceedingly aggressive and it would take much greater magnitude negative correlation than this for such a strategy to dominate" it would be unlikely to find situations where you can gain very much vs. wagering against the individual events in a 2 event scenario.

Thank you kindly, 8:00 AM insomniac Ganch.

Ganchrow · 07-26-08, 02:29 PM

Originally posted by Art Vandeleigh

it is possible (and straightforward) to synthesize a negative parlay for 2 sequential events, where one of the events is expected to have probability of occurence considerably less than the odds indicate (but the specific event is unknown)

You got it.

I'll just add that this is even true if both events are actually fairly priced as singles (or even -EV due to vig). Negative correlation creates excess value for neg. parlays, after all.

If the bets are contemporaneous then the only way to do this would be with a neg. parlay (or with several regular parlays, assuming the vig isn;t too great).

If the bets are sequential, then this could be synthesized with single bets provided that the correlation were unknown to the market at-large.

This is exactly analogous to finding value when parlaying two positively correlated events, even when each of the singles themselves are already -EV.

Originally posted by Art Vandeleigh

Since Martingale is "exceedingly aggressive and it would take much greater magnitude negative correlation than this for such a strategy to dominate" it would be unlikely to find situations where you can gain very much vs. wagering against the individual events in a 2 event scenario

By "dominate", I meant of "significantly more utility value than the single bets". In the context in which I made this comment we were referring to bets that already had value as singles.

The lower the preexisting single value and the longer the odds on those singles then, all else being, equal the greater the value of the neg. parlay.