Brainteaser from Wilmott related to sports betting

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.

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.

Originally Posted by 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.

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.

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).

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.

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.

Originally Posted by 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.

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?

Originally Posted by FourLengthsClear

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.

Originally Posted by Data

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.

Originally Posted by donjuan

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?

Originally Posted by Data

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?

Originally Posted by FourLengthsClear

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.

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.

Graph the payout function over time.

Graph the win probability over time.

Where the lines intersect is the answer.

I think.

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).