[Blax0r]

Wilmott, a well-known website for quantitative finance, also has a brainteaser/think tank type message board. There's usually a decent amount of overlap of topics between sbr's think tank and theirs, but there are a lot more Ganchrow-level folks over there.

Anyway, here's the post I saw (http://www.wilmott.com/messageview.c...threadid=81918):

You are watching a (U.S.) college basketball game, and you will receive $500 if you can tell me the winner.
The catch is that you don't need to tell me before the game, but can tell me any time during the game. However, the payoff goes down by $10 for every minute of the game that passes.

Some assumptions:
-College games are forty minutes long, and somehow there are no ties, nor overtime.
-You know absolutely nothing about either team.
-You just get score updates at the end of every minute, you are not actually watching the action. (e.g. after 5 min, TeamA: 8, TeamB: 7)
-If you want, you can wait for the end of the 40th minute to answer (when the game is over and you know the actual winner).

What is your strategy?

- What if the rules changed so that the payoff goes down by $5/min in the first half, but $10/min in the second?

Typical other follow-ups for the interviewee:
- What if the rules changed so that the payoff goes down by only $5/min if either team has a 5 point lead?
- What if the rules changed, so that the payout is now $5,000 and $100/min?
- $50,000 and $1,000/min?
- What if we let you play the $5,000 and $100 game, but ten times, how would it change your strategy?
Etc.

Follow-up for wilmotters:
- What if the rules changed, so that the payout is now $5,000 and $x/min.
What value of x makes this question the most interesting?

I'd like to hear the thoughts of actual sports bettors on making a position on games where you can only see tick-by-tick stat updates rather than the teams' historical data.

[splash]

If you just pick a team at random before the game starts, you have an EV of +$250. If you know absolutely nothing about the teams, it's impossible to guess whether or not it's likely that one team will pull far enough ahead to exceed +$250 EV, especially with the reduced payout. I'd probably just pick a team at random before the game starts.

[FourLengthsClear]

If you just pick a team at random before the game starts, you have an EV of +$250. If you know absolutely nothing about the teams, it's impossible to guess whether or not it's likely that one team will pull far enough ahead to exceed +$250 EV, especially with the reduced payout. I'd probably just pick a team at random before the game starts.

It is impossible to know, but it is entirely possible to estimate the probability and that is the root of this 'problem'.

I will have a go at it later if I have time or nobody else comes up with a viable solution.

[Blax0r]

The only strategy I can think of involves attempting to match the scoring rates to other historical games and hopefully getting a moderately-low variance mean expectation. I'd definitely like to hear your further thoughts (or anyone's ideas really), since in my mind I'm drawing a parallel to a guy who just trades on betfair all day (ie, a practical situation).

[specialronnie29]

consider the problem of with 1 minute left in the game. you know the score. should you bet on a team or wait for the game to end?

Let the score be 'a' for team A and 'b' for team B.

You need to construct an objective function which you then will maximize

one method or way of thinking about it would be to calculate an estimate of each team's 'true' points per minute rate. this estimate is a/39 and b/39. alternatively you can construct an estimate of the difference in pts scored per minute (which is probab ly easier actually)

the difference in score is a-b. suppose this is positive so team A is winning. then of course you want to bet team A if you are going to bet after the 39th minute. if the bet wins you make 110, otherwise 0. the probability of your bet winning is 1 minus the probability of team B overtaking team A. from earlier you have an estimate of (a-b)/39. from the normal distribution we can calculate the probability of this difference in absolute value being greater than the difference (a-b), i.e. the probability of team B covering the difference in the last minute.

this can be generalized to any time period t and then you will choose t to maximize EV

this approach is very simple and likely to be quite wrong as it appears to ignore the variance of the estimates above along the time path. after only 1 period these estimates are likely to be noisy and you face the tradeoff of believing them and getting bet in earlier vs. waiting to get a better estimate.

[Data]

The value of time is a total red herring. It should come into consideration only with MUCH higher stakes. The scenario is the same as betting at +400 line on a 50% prop (bet (not to get) $100 to win 500$). At such odds, one must really hate money to NOT bet before the game starts.

[FourLengthsClear]

The value of time is a total red herring. It should come into consideration only with MUCH higher stakes. The scenario is the same as betting at +400 line on a 50% prop (bet (not to get) $100 to win 500$). At such odds, one must really hate money to NOT bet before the game starts.

You are right in the basis that you have been told that the matchup is between two NCAAB teams.

What if you have no idea about the relative strength of the teams and the matchup could be between the LA Lakers and a team from the Arizona Midget Basketball League? Does your statement hold true?

[Data]

You are right in the basis that you have been told that the matchup is between two NCAAB teams.

What if you have no idea about the relative strength of the teams and the matchup could be between the LA Lakers and a team from the Arizona Midget Basketball League? Does your statement hold true?

It depends on what do you mean by "could be". To make this a different solvable problem you have to state the probability of such a disparity to occur. Speaking from a standpoint that the problem should comply with common sense, the probability of this happening is abysmal and my answer stands.

[donjuan]

It depends on what do you mean by "could be". To make this a different solvable problem you have to state the probability of such a disparity to occur. Speaking from a standpoint that the problem should comply with common sense, the probability of this happening is abysmal and my answer stands.

The problem definitely does not state that they are equal teams nor does it come close to implying such.

[FourLengthsClear]

The problem definitely does not state that they are equal teams nor does it come close to implying such.

.... I fully agree, do you have a point?

[FourLengthsClear]

It depends on what do you mean by "could be". To make this a different solvable problem you have to state the probability of such a disparity to occur. Speaking from a standpoint that the problem should comply with common sense, the probability of this happening is abysmal and my answer stands.

OK then let's put it another way. Can we quantify what degree of expected disparity between the teams would be needed to mean that a +EV of more than 250 is probable?

[Data]

OK then let's put it another way. Can we quantify what degree of expected disparity between the teams would be needed to mean that a +EV of more than 250 is probable?

I do not see the need for quantifying this. The EV of more than 250 is certainly probable but it is highly unlikely. The conditions to satisfy this are like knowing the winner with 100% certainty after 24 minutes or less. Those extremely rare cases can safely be ignored because it is overwhelmingly more likely that the EV will go below 250 and will keep going lower and lower never coming back.

Lets put it this way. To get 250+ EV we need to satisfy
p*P>0.5
where p is the probability that EV will climb above 250 during the game (obviously, p must be p>0.5) and P is the probability of a matchup where p>0.5. Following your example with the Lakers playing the midgets, while p=1, you must have this type of matchup to occur with probability 0.5 or higher to make waiting worthwhile.

[Pokerjoe]

Suppose it's 50% likely that, after 5 minutes, you'll have information (the score) that makes your chance of picking the winner 75%, and 50% likely you're still at 50/50, as you were at the open.

Half the time you, after waiting 5' to make your bet, have an EV of 337.5, and the other half of the time you'd have an EV of 225, for a total EV of 281.25, which is higher than the EV of 250 of just picking a team at the start.

IOW, write the equation, plot the graph. Chance of each given lead, value of each given lead, at each given minute, re decreasing payout.

If you don't have the DB to make such estimates, figure out the equilibrium point (that is, the chances/values needed to equal the initial 250 EV), and guess whether it feels likely to be over or under that point.

But to suggest it isn't possible to gain enough information to make waiting +EV, is wrong.

[TomG]

Graph the payout function over time.

Graph the win probability over time.

Where the lines intersect is the answer.

I think.

[That Foreign Guy]

Assuming no external information (ie you can't see a betfair market or anything about the game once it starts).

The question is basically at what point the value of an extra minute of score data becomes less than x (the penalty).

A minute's value is the % it adds to accuracy of winner estimation * $500. The penalty is $x

so I think we want the first derivative of the accuracy formula to = x

I think the best way to approximate this formula is to look at a database and see what % of teams that were ahead after 1 minute went on to win, what % of teams after 2 minutes etc, and find the line of best fit.

If the payout only goes down when a team is ahead by 5 pts I think you pretty clearly wait till a team goes ahead by 5 before considering a bet. You're getting $250 on random team+5 that way. Looking at number of teams ahead by 5pts after m minutes who went on to win would approximate the accuracy formula.

The increased $ value questions seem like an attempt to measure risk aversion, they don't change the structure of the game in EV terms. Assuming I could find a correct answer to the initial question I would sell some of my action to people smart enough to see the +EV in the larger games.

Assuming no third party investing / hedging is allowed, the larger games payoffs are a larger % of bankroll so I believe this means some sort of reverse Kelly formula applies. By this I mean it seems like there is increased EG in taking the less uncertain paths (waiting for info rather than taking the coinflip early) at the expense of EV. I haven't explained that well though, it's some half remembered log curve utility function stuff.

This would actually be a really fun way to bet on games you know nothing or have no opinion about (you can take a side at any point during the match but the payoff goes down with each minute).

[u21c3f6]

Suppose it's 50% likely that, after 5 minutes, you'll have information (the score) that makes your chance of picking the winner 75%, and 50% likely you're still at 50/50, as you were at the open.

Half the time you, after waiting 5' to make your bet, have an EV of 337.5, and the other half of the time you'd have an EV of 225, for a total EV of 281.25, which is higher than the EV of 250 of just picking a team at the start.

IOW, write the equation, plot the graph. Chance of each given lead, value of each given lead, at each given minute, re decreasing payout.

If you don't have the DB to make such estimates, figure out the equilibrium point (that is, the chances/values needed to equal the initial 250 EV), and guess whether it feels likely to be over or under that point.

But to suggest it isn't possible to gain enough information to make waiting +EV, is wrong.

I agree with the above.

However, unless I had access to this relatively large amount of information boiled down into the exact optimum
minute to wager and/or optimum minute/score combination, I would simply choose a side prior to the game starting.

Joe.

[mr.ed]

Uh...wouldn't exactly call "time" a red herring, as the problem asks what is the optimal time to bet given the scenario presented....so this is all about time and nothing but time. A shoot from the hip "bet before the game" will win you $250 a pop, but this is a problem of maximizing profits.

I could immediately dismiss the notion of betting before the game started, as the penality for waiting just a few minutes was relatively minor compared to the wealth of information it would reveal, namely the better team. I chose to look at the 5 minute mark. Using 3 college basketball seasons (approx 5500 games) the team ahead after 5 minutes wins roughly 2/3 of the time. So, if you just wait 5 minutes to pick a winner your profits will increase to $300 a game. This may not be the optimal time to bet it, but certainly in the ballpark. I would run more numbers but I'm getting sweepy....so I'm content with the fact that I've narrowed it down significantly. I will not do any more work on this as I realize that someone actually offering me this proposition is failry remote......as it involves an individual giving away free money for no apparent reason.

[Data]

Uh...wouldn't exactly call "time" a red herring, as the problem asks what is the optimal time to bet given the scenario presented....so this is all about time and nothing but time. A shoot from the hip "bet before the game" will win you $250 a pop, but this is a problem of maximizing profits.

I could immediately dismiss the notion of betting before the game started, as the penality for waiting just a few minutes was relatively minor compared to the wealth of information it would reveal, namely the better team. I chose to look at the 5 minute mark. Using 3 college basketball seasons (approx 5500 games) the team ahead after 5 minutes wins roughly 2/3 of the time. So, if you just wait 5 minutes to pick a winner your profits will increase to $300 a game. This may not be the optimal time to bet it, but certainly in the ballpark. I would run more numbers but I'm getting sweepy....so I'm content with the fact that I've narrowed it down significantly. I will not do any more work on this as I realize that someone actually offering me this proposition is failry remote......as it involves an individual giving away free money for no apparent reason.

I agree that it could possible to obtain the information that gives more EV for the given prop. The idea of waiting a little time after the game starts seems very promising for this purposes. However, I have two objections.

First, the choosing of the 5 minutes mark screams selection bias. If you do have the numbers then you must as well know the numbers at 1 to 10 minutes marks to give a reasonable assessment.

Second, more importantly, based on the given scenario, the bettor has no information of historical scores broken down by minutes. Collecting this information will take more than 40 minutes resulting in EV of 100.

[mr.ed]

After mulling over the problem for a few minutes, my instincts told me that picking a winner at that 5 minute mark would result in more profits than picking a winner at the opening tip-off. And since I make wagering decisions such as this quite frequently, without any data to support my position, it was nice to run some numbers and conclude that my initial instincts were correct. Is the 5 minute mark better that the 6 minute mark....who knows....my method of collecting this data for the 5 minute mark was rather crude and time-consuming, so I suppose I will never know.Also, if there is an evil man behind the curtain selecting the games, then my conclusions are worthless. Even though the problem didn't state it, my research was based on a randomly selected college basketball game. This must be the criteria, as anything else would make any kind of research and analysis a pointless exercise.

[Data]

After mulling over the problem for a few minutes, my instincts told me that picking a winner at that 5 minute mark would result in more profits than picking a winner at the opening tip-off. And since I make wagering decisions such as this quite frequently, without any data to support my position, it was nice to run some numbers and conclude that my initial instincts were correct. Is the 5 minute mark better that the 6 minute mark....who knows....my method of collecting this data for the 5 minute mark was rather crude and time-consuming, so I suppose I will never know.Also, if there is an evil man behind the curtain selecting the games, then my conclusions are worthless. Even though the problem didn't state it, my research was based on a randomly selected college basketball game. This must be the criteria, as anything else would make any kind of research and analysis a pointless exercise.

Again, the idea is promising. But.

First, at this point it is nothing but a hunch. We do not have the data ready. Without analyzing the historical data, so far, we have no reason to reject the hypothesis that the pregame bet is the best.

Second, so, what exactly mark you are going to bet at based on that hunch? 5 minutes? Why not 4 minutes? Why not 6 minutes? Again, you cannot answer this which means "5 minutes mark" is NOT an acceptable solution.

[ScreaminPain]

Here is a formula for the optimum time to bet a game, such as the one in question:

Take the number of points one team is ahead.
Subtract three.
Add a half-point if the team that is ahead has the ball, and subtract a half-point if the other team has the ball. (Numbers less than zero become zero.)
Square that.
If the result is greater than the number of seconds left in the game, the lead is safe.

If you've got a 10-point lead and the ball with 10 minutes left, is that a safe lead?
Of course not; teams come back from a 10-point deficit all the time. A 10-point lead, plus the ball, gives you a 7.5-point safety margin. It's safe for 56.25 seconds—56, rounded down. With 600 seconds to play, a 10-point lead (with the ball) is 9 percent safe. That doesn't mean a team with a 10-point lead and the ball with 10 minutes to go has only a 9 percent chance of winning. Rather, it means they're 9 percent of the way to having a completely insurmountable advantage.
An 11-point lead with nine minutes to play—we'll let you keep the ball. That's an 8.5-point safety margin with 540 seconds to play; it's 13 percent safe (72.25 divided by 540).
A 12-point lead with eight minutes to play ... that's a 9.5 point margin. It's 19 percent safe (90.25 divided by 480).
A 13-point lead with seven minutes to play ... 26 percent safe.
A 16-point lead with four minutes to play ... 76 percent safe, assuming the team with the lead also has the ball. It's really unusual for a team to come from 16 back with four to play and win, but it does happen. I would guess it happens twice a year somewhere in the world of college basketball

This is NOT my calculations. Found it ?? awhile back.....

[Data]

Here is a formula for the optimum time to bet a game, such as the one in question

Not even close.

[mr.ed]

I acknowledged that picking a team at the 5 minute mark may not be the best strategy. For me to get the optimal result would be too labor-instensive, given my crude methods of data gathering and analysis. I was simply out to prove that picking a team at the 5-minute mark is better than at the opening tip-off. I did so using historical data from 3 seasons. In fact, it is significantly better, which was my hunch and I proved it was correct beyond a shadow of doubt. Again, my results are based on the games being randomly chosen...if there is trickery involved in picking the games (e.g. selecting games where the home team trails by 20 at half time and comes back to win) then all analysis is pointless.

[Data]

I was simply out to prove that picking a team at the 5-minute mark is better than at the opening tip-off.

And you did. But what's the point? How does that help us with the given scenario? What is your final answer?

We have a point A with a reading of 250 and a point B with a reading of 100. Our process goes from A to B. The first null hypothesis is that it is a straight line. The second null hypothesis is that the reading stays at 250 and then drops. We have a hunch that these null hypothesis are wrong and the reading will get above 250 at some point. We need something to reject the null hypothesis. I see nothing thus far.

[saratoga1927]

Give me the $100 at the end of each game.

[mrpooh]

The biggest hurdle for me is the tick-by-tick scoring. In any one minute period a team could have more positions then another thus any one minute time frame's data isn't very useful to look at. If team A scores a 3 pointer at 15:59, team b misses their shot, then team A gets another 3 pointer at 15:01, that one tick shows a 6 point swing. But all that happened was 2 made baskets and one missed. Where as you could have 3 dunk shots with some turnovers and arrive at the same 6 point swing, but you have a greater swing in the game. Plus you never know the fouls in the game, so a team up 7 with 3 to go could have a key player or two almost ready to foul out or already on the bench. With having no knowledge of the game, the only options are either take a team at the start of the game for a coin flip, or take the sure money after the game ends. No data you gather during the game can really help you, because the only data you get is score.

[That Foreign Guy]

The biggest hurdle for me is the tick-by-tick scoring. In any one minute period a team could have more positions then another thus any one minute time frame's data isn't very useful to look at. If team A scores a 3 pointer at 15:59, team b misses their shot, then team A gets another 3 pointer at 15:01, that one tick shows a 6 point swing. But all that happened was 2 made baskets and one missed. Where as you could have 3 dunk shots with some turnovers and arrive at the same 6 point swing, but you have a greater swing in the game. Plus you never know the fouls in the game, so a team up 7 with 3 to go could have a key player or two almost ready to foul out or already on the bench. With having no knowledge of the game, the only options are either take a team at the start of the game for a coin flip, or take the sure money after the game ends. No data you gather during the game can really help you, because the only data you get is score.

LOL wut. You wouldn't take the team that was up 30-12 after 15 minutes?

[mr.ed]

Mr. Pooh may be on to something if you are looking for a perfect record, but the goal is to make as much money as possible, and most of his decisions would be in the final minutes of a game, which is way worse than blindly picking a team at the start.

Congrats for coming up with the right answer, Foreign Guy, in post #15. I could run the numbers and arrive at the best answer using your method but it would take some time that I am not willing to give. I have already proved that picking a team at the 5 minute mark is much better than at the opening tip-off, and that is enough for me.

[mrpooh]

LOL wut. You wouldn't take the team that was up 30-12 after 15 minutes?

Just because a team is up big early on, doesn't mean it wasn't a fluke. How many times in the NCAA tourny do you see a hot shooting 12 seed hit 3's early, only to see a big come back. Sure, the teams has a better chance, but you still have no info about the teams, so it very well could be a 9-18 team that starts out hot vs a 22-6 team that will roll them late

[mr.ed]

Lets say the odds of coming back after being down 30-12 after 15 minutes is 1 in 20. (probably not even that high but for the sake of argument lets go with it.)

Profits in this problem would be 19 X $350 = 6650 / 20 games = $332.50 in profits per game.

If you wait just 2 more minutes before pulling the trigger then you will not equal the profits above, as picking a team after 17 minutes will only get you $330 even if you are right 100% of the time.

It's ok to be wrong once in awhile in this problem...just waiting until a team wraps up the game would be horrible strategy and much worse than flipping a coin at the start.

[mrpooh]

Lets say the odds of coming back after being down 30-12 after 15 minutes is 1 in 20. (probably not even that high but for the sake of argument lets go with it.)

Profits in this problem would be 19 X $350 = 6650 / 20 games = $332.50 in profits per game.

If you wait just 2 more minutes before pulling the trigger then you will not equal the profits above, as picking a team after 17 minutes will only get you $330 even if you are right 100% of the time.

It's ok to be wrong once in awhile in this problem...just waiting until a team wraps up the game would be horrible strategy and much worse than flipping a coin at the start.

The point I'm trying to make is, you might as well guess at the start, because any data you get after the fact, is not reliable. You have a better chance taking out all human judgement at the start, then picking any one point in the game you THINK is the best to pick from

[mrpooh]

Also, I'm assuming this is a one time shot, not something you do with every game. The majority of posts seem to assume it is more then one game. Strategy greatly changes with the number of games

[That Foreign Guy]

Just because a team is up big early on, doesn't mean it wasn't a fluke. How many times in the NCAA tourny do you see a hot shooting 12 seed hit 3's early, only to see a big come back. Sure, the teams has a better chance, but you still have no info about the teams, so it very well could be a 9-18 team that starts out hot vs a 22-6 team that will roll them late

Tell you what, if you don't think an early lead is significant. I will make a bet with you for march madness. I will take the first team to take a 10 point lead in every game, you will take the other team, if no team takes a 10 point lead no bet. I will bet as much as you want for even money on each game or on who has the most wins over the tournament.

If you don't like that I am also willing to blindly bet on the team that is ahead at the end of the 1st quarter.

The point I'm trying to make is, you might as well guess at the start, because any data you get after the fact, is not reliable. You have a better chance taking out all human judgement at the start, then picking any one point in the game you THINK is the best to pick from

You accidentally make a good point here - the value of each minute is not constant - ie optimal strategy is not to create one f(x) for the time value of information, it's to create multiple f(x,p) as the points difference between the two sides at each minute affects the value of that info (10-2 is more significant than 5-5 etc so the next minute has a lower x.

Also, I'm assuming this is a one time shot, not something you do with every game. The majority of posts seem to assume it is more then one game. Strategy greatly changes with the number of games

Unless you're playing for significant sums of money so utility curves start coming in to play just No. No it doesn't.

If you don't see why you have no business betting on sports.

Now, if you have no information about any other historical games etc then maybe correctly solving f(x) is impossible so blind picking at the start is correct but I don't think that was the question. I read it as you have no knowledge of the teams playing in this game or the game progress beyond the score BUT you are free to do whatever CBB research you want.

Related brain teaser for people:

Let's say you're offered this tick by tick betting on CBB but your generous opponent says:

I'll let you choose whatever statistic you want to receive minute by minute updates on and you can choose when to bet at the same terms. Is there anyone who doesn't choose score? If so why?

What about if you're allowed one extra statistic as a freebie?

What if you have to pay $100 for an extra statistic? Is it worthwhile? At what price does your chosen statistic become worthwhile?

[HotStreak]

I say you wait until the end. You are receiving %20 of the prize guaranteed. No matter how much the money is 500 or 50,000. You wait.

There may come a point before the end that it is correct to pick. You have to let it play out. Given the fact that it will cost you nothing, I wait for the lock.

[That Foreign Guy]

I say you wait until the end. You are receiving %20 of the prize guaranteed. No matter how much the money is 500 or 50,000. You wait. There may come a point before the end that it is correct to pick. You have to let it play out. Given the fact that it will cost you nothing, I wait for the lock.

*facepalm*

Not sure if you're trolling here. Hope you are.