Discussion on question by Sarah V.P: How to reason about bet on horses?

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08:10

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Does anyone have any rules/strategy or formulas for approaching betting games? My friend told me that I should try to find the 'equlibrium'? I found myself having problems reasoning through these kind of questions. For example, I have the following question. Here are the odds for a horse race: ...

lulu

This is too vague. Professional gamblers often rely on information which leads them to believe that the posted odds are not good predictors of what will happen, and they bet according to that. If, on the other hand, you are treating the posted odds as accurate predictors, then each bet has expectation $0$ or lower (accounting for the spread taken by the house) .

Rodrigo de Azevedo

Possibly related

@lulu Which odds? The bookies are not in the prediction business. Rather, they are in the market-making business. I do agree that the question is too vague. However, there is hope that it can be refined and made less vague.

lulu

@RodrigodeAzevedo I don't understand, I didn't say anything about bookies. Professional gamblers (in horse racing and such) rely on information. "Horse $X$ ran badly in her last two races so the odds are long, but those were after rain and $X$ is much faster when it is dry", that sort of thing. They bet against the market because they know stuff the market doesn't know, or at least stuff that the market hasn't priced in appropriately. Perfectly sensible strategy, if you are an expert with good access to information.

Rodrigo de Azevedo

@lulu Yes, you dismissed the idea that odds are estimates of probabilities. However, in my humble opinion, the idea that (fixed) odds are "noisy probabilities" (rather than "prices") is so dangerous that it needs to be dismissed in a very emphatic manner. Since this question is phrased in terms of fixed odds, I suspect it may be a question about arbitrage betting, which relies on exploiting short-lived imbalances rather than acquiring better information on the horses.

lulu

@RodrigodeAzevedo I don't understand. Of course odds imply probabilities. These are the "market based" probabilities. It is perfectly possible for a rational gambler to disagree with the market. Why not? Indeed, there are professionals who make a living off of this. Not sure what point you are trying to make.

@RodrigodeAzevedo Since you can't generally short a bet at a horse track, and the house takes a healthy spread, arbitrage is hard to find. In principle, I could imagine some of the derivative bets (combination plays and such) might conflict with the bets on individual races, but combination bets are typically set by the house precisely to cancel arbitrage (and the OP didn't provide any combo bet pricing anyway).

@RodrigodeAzevedo I think you are reading too much into a poorly thought through question. I think the OP just wanted to know if there was a mathematical reason to choose between a long shot bet and a more nearly even one, which, absent information about utility functions and such, there is not.

Rodrigo de Azevedo

08:10

@lulu Which odds? Which market? Fixed odds are prices. The bookies just want to manage their risk by creating incentives for bettors to take risk off the bookies' books. They don't care if their fixed odds are good at playing the role of "subjective probabilities". However, if I properly recall, bets on horses are often parimutuel, which is an entirely different thing. The fact that a question on betting on horses is phrased in terms of fixed odds suggests to me that it's purely academic — a cute toy problem to motivate those being introduced to, say, linear programming.

lulu

@RodrigodeAzevedo Again, I never said a word about bookies nor their intents. Sure, bookies (at least good ones) are just market makers. So what? When the pool is fixed (if you want to speak of pari mutuel betting) that establishes a probability. The bookie has no view on that, but again, so what? It is still an implied probability. With which someone might agree, disagree, or have no opinion about. And professional sports gamblers are a real thing. People are good at it (some people, anyway). I am not discussing abstract matters.

@RodrigodeAzevedo If you want to focus on real world issues, the presence of a large house spread arguably does make the long odds bets look better. If a gambler needs a certain amount of cash, it could well be better to go for it in a small number of bets. Otherwise, they bleed cash steadily to the house. In pari mutuel systems, the spread is a lock, independent of the implied odds. Other systems are different.

Rodrigo de Azevedo

@lulu Since the OP wrote "here are the odds for a horse race", I assume that the odds are posted by a bookie. Hence, the bookie's incentives matter. If Barcelona plays against Liverpool, bettors in Spain and bettors in the UK may bet very differently. Bookies then adjust their (fixed) odds to manage their own risk. Arbitrageurs then come in and force the odds to converge. Then, some other bettors have the opinion that the implied probability is off and their betting forces the odds to converge to an implied probability that may be a decent "subjective probability".

lulu

@RodrigodeAzevedo Again, I have no guess what point you are making. You bring up pari mutuel betting, in which the bookie is just a guy who walks around taking dollars from people. That person couldn't care less what bets people are making. Perhaps (just guessing here) you are trying to claim that horse racing is a perfected market in which the pool has gathered all the information so it is somehow impossible for a gambler to know things that aren't generally known (or properly appreciated). If that is your point, you are simply wrong. It's hard work, but people do it.

Rodrigo de Azevedo

@lulu My point is that you are implying that betting markets are somewhat efficient, though not perfectly so. They are efficient when prices can converge. However, it's 2022 and some prices cannot converge due to, say, legal and regulatory risks. If you don't care about such risks, you can profit from others' fears. I agree that pretty much all of what you wrote is an accurate picture of the reality of, say, (early) 2012.

lulu

@RodrigodeAzevedo I never once claimed the markets were efficient. In fact, by asserting that it was possible for a highly informed professional to make superior judgments on a race, I am strongly suggesting that they are not efficient. The presence of externalities which distort the market only strengthens my point. An expert on those externalities could exploit them.

Rodrigo de Azevedo

08:10

@lulu Yes. Indeed, we do agree on something. The reason I claimed that thinking of fixed odds as implied probabilities is dangerous is that such odds converging to decent implied probabilities requires a level of efficiency that may not exist. The betting ecosystem functions well when each kind of player can actually play its role. Introducing legal and regulatory risks can disturb the betting ecosystem to the point where it ceases to function well. Better not assume that the ecosystem is healthy.

lulu

@RodrigodeAzevedo You are overcomplicating things. If I am offered a bet at a fixed probability, I am free to disagree with that probability. The fact that the implied probability was influenced by a whole host of factors need not deter me. Sure, I should probably understand why the bet is "off" according to my view, if for no other reason than to reassure myself that I am not missing something. But that's it.

Rodrigo de Azevedo

@lulu I would phrase it as follows — I am interested in pure arbitrage (i.e., linear programming), whereas you are interested in statistical arbitrage (i.e., probability). PS: after writing this comment, I checked your list of badges and was not surprised to find a gold badge on probability ;-)

lulu

@RodrigodeAzevedo But again, pure arbitrage isn't relevant here, since we don't have that sort of information. We'd need something like pricing in other markets (in the very distant past, you could arb bets between two cities, for example) or on derivative bets. Something like that. Here? Well, one might remark that the implied probabilities sum to something greater than $1$ , but I attribute that to sloppy writing rather than an exploitable opportunity.

RyRy the Fly Guy

The entire point of this post was to ask whether there was a general approach to handling gambling problems, and everyone here is instead nitpicking the OPs example about horse racing, which actually has nothing wrong with it. The house offers a payout on possible outcomes, and one has determined by some means what the probabilities of those outcomes are, which may or not be accurately reflected by the payout. It's not rocket science.

Rodrigo de Azevedo

@RyRytheFlyGuy "The house" is not a well-defined thing. Unless the question is a toy problem for preparing for interviews at, say, Jane Street, the economics of betting markets actually is the one interesting part. The mathematics of it is kind of trivial.

RyRy the Fly Guy

08:10

The mathematics of it is what the OP is asking about, not some treatise on betting markets.

Rodrigo de Azevedo

@RyRytheFlyGuy For someone whose answer has not yet received a single upvote, it seems to me that you might be a bit too opinionated. If the OP does not know anything about betting markets, then the OP won't ask good questions on betting.

RyRy the Fly Guy

If the math of my response is incorrect, then please illuminate me.

Rodrigo de Azevedo

@RyRytheFlyGuy I haven't read your answer. I haven't downvoted it either. Investing some serious time in reading answers from fellow users whose work I am not acquainted with is a luxury I cannot afford anymore, unfortunately.

RyRy the Fly Guy

Well, i have no problem with someone being opinionated if they are correct, and the net sum of votes is not always a reflection of that. Most people have TLDR attitude, and it seems one of my downvotes came from the OP herself lol, the very person who is not clear what the correct answer is.

Rodrigo de Azevedo

@RyRytheFlyGuy I suspect the OP has not even found the right question yet, let alone the right answer. Which is one reason I haven't read your answer (yet). I suspect that the OP will come to regret the question, delete it, and your answer will vanish before I actually find the time to read it.

RyRy the Fly Guy

08:10

I think you are right about that. i apologize for being rude. There is a way to take issue and be respectful, and that hasn't been me in this conversation.

Sarah V.P

@RodrigodeAzevedo Look that was the question given to my friend in an interview. I don't think there is a right answer. I am just interested in the approach on how to reason about this! You don't want to invest serious time in reading the answers and yet you are investing serious time arguing.

@RyRytheFlyGuy I did not down vote you, but I did not accept it either. There isn't a correct answer. I am skeptical about the first part because for a 7:2 odds, if pink wins, I get 7 + 2 = 9 dollars or 9/2 odds. Just think about the inverse, for example, if you have 2:7, I would win 2 + 7 also but now I can lose 7 instead of 2. Wining only 2 and not get your 7 back isn't winning exactly.

RyRy the Fly Guy

@Sarah you do not appear to understand what odds means. $7:2$ odds implies that for $\$2$ risked, you can win $\$7$ more. So if you bet $\$2$ and win, then you get your $\$2$ dollars back plus $\$7$ more. $2:7$ odds implies that for $\$7$ risked, you can win $\$2$ more. So if you bet $\$7$ and win, then you get your $\$2$ dollars back plus $\$7$ more.

Sarah V.P

@RyRytheFlyGuy Yes that what I mean

RyRy the Fly Guy

ok. I think part of the issue here is that you have posted multiple questions about several topics in one post. You want a general approach to betting problems... then you ask about trading stocks... then you ask about a very specific example in horse betting. I would split these up and ask specific questions about each.

Sarah V.P

@RodrigodeAzevedo Why don't you lecture me then if it is trivial? You spent your whole life good will hunting, but the only good that will come is when you start helping!

RyRy the Fly Guy

08:10

Imagine the horse race is performed $1000$ times and every time you bet on the purple horse. In approx. $1000p$ races, purple wins and you make $1000p\$\frac{9}{2}=1000p\$4.5=\$4500p$; in approx. $1000q$ races, blue wins and you lose $1000q\$1=\$1000q$; and in approx. $1000(1−p−q)$ races, black wins and you lose $1000(1−p−q)\$1=\$1000(1−p−q)$

So your reward over $1000$ races is approx $\$4500p−\$1000q−\$1000(1−p−q).$ That is your expected reward over $1000$ races. You have to take the reward from all bets you expect to win and subtract from it the cost of all bets you expect to lose

You will have to explain why you believe $E[purple] = \frac{9}{2}p$

Sarah V.P

Let us continue this discussion in chat.@RyRytheFlyGuy

Rodrigo de Azevedo

@SarahV.P If you invest some time in refining your question, I will vote for reopening. I did not vote for its closure.

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$Mathematics$ Discussion on question by Sarah V.P: …