Math for the Sports Minded - Maximizing Recruit Talent w/Avail Cash

rlwjr · 2008-01-08 07:09:33

This is prolly gonna be a long post. Hope to be able to make it clear enough to follow. If you have any further questions after I make an attempt at explaining, please ask me to clarify.

I play an online college football game/simulation. In this game, you compete with other online coaches during recruiting season to land the best recruits you can with the money you have. The farther a recruit is from you, the more they cost. If another coach and yourself both pursue the same recruit, it basically becomes a bidding war, with distance becoming a factor due to cost being related to distance. Each recruit has numerous numerical ratings such as Endurance, Stamina, Speed, Hands, Blocking, etc.

I have built myself a spreadsheet that uses basic fuzzy logic to determine the best recruit for the best cost. In a nutshell, I create a fuzzy rating between 0 and 1 for talent, and a fuzzy rating between 0 and 1 for estimated cost to recruit that individual. Then, I logically AND those two numbers, giving me in essence, a bang for the buck type fuzzy number between 0 and 1. The spreadsheet also calculates a hard estimated dollar figure to recruit, like say, 5230 dollars.

Okay, this is the part where I describe the math problem I need figured out. How would I go about maximizing my bang for the buck rating and it's associated dollar figure, with my available cash? Let me give you a for instance:

I have 35,000 dollars available.
I know I need to recruit 2 Offensive Linemen (OL), 2 Defensive Linemen (DL), 1 Linebacker (LB), and 2 Defensive Backs (DB)
I have a ranked list of the best players for the buck at each position. Some players offer better talent for the dollar than others.
I have an estimated cost to recruit each player.

How do I go about (mathematically) determining the best way to spend my money? I could take the best available at each position, but may still end up spending over my limit. Do you see what I'm after?

I hope that was clear. I'd just like to say in advance, thanks for taking the time to try to help. Even if you can't help directly, I would greatly appreciate any steering you might provide to get me to the right destination!

TheDude · 2008-01-08 08:02:32

There are a number of factors you need to consider. Too many to translate into a reliable mathematical model, in my opinion.

First of all, you need to consider how valuable the recruit is to your team. Let's say you have 2 quarterbacks on your team. If they are both sophomores, or a sophomore and a freshman, you could look for more of a project at quarterback (if you have a statistic for "upside"), or even afford not to recruit a quarterback at all. On the other hand, if your quarterbacks are a junior and a senior you'll need to spend more money to get a quarterback who will be ready to go quickly.

You also need to consider what it could mean to get a big time player on your team. Maybe you already have 3 or 4 running backs, but there's a recruit out there with 99 speed, 99 break tackle, etc. who would single-handedly impact your team. It might be worth your while to over-spend on recruiting such a player, if only to ensure that your opponent doesn't get him.

Speaking of other players, you should know them as well as you know yourself. Study their rosters and their tendencies so you'll know what kind of players they're likely to go after. If they tend to stay close to home to get "value" out of their recruiting budgets, you can afford to underspend on players close to you because you'll have little competition. Then you'll have some extra cash to outspend your opponents for a select handful of highly ranked recruits in their home states.

On the other hand, if your opponents tend to go for the top players regardless of how far away they are they'll spread their recruiting budgets thin. Overspend a little bit on top tier recruits in your own state, while targetting some solid second tier recruits from other states. Even though you won't be able to spend much, you'll have little competition because your opponents will be spending all of their money on just the top tier recruits.

Also, keep in mind what kind of players your're looking for. Do you use a straight ahead running game? Skimp on recruiting receivers and small, quick running backs. Focus on beefy, strong offensive linemen and powerful running backs. If you prefer a wide open passing attack go nuts on quarterbacks and receivers, but settle on lower rated running backs. In particular, look for running backs who have good pass blocking skills but lower than average running skills, these guys will probably not be targetted by other players.

This goes for defense as well. If you prefer to blitz a lot and play man coverage try to get bigger corners who can jam receivers at the line and smaller, faster linebackers who can rush through gaps in the blocking. If you prefer to sit back in zone coverage and let your defensive line rush the passer by themselves get faster corners who can play better in coverage.

Even if it's outside your comfort zone you should try to do the opposite of what everyone else is doing. Most players are probably going to build passing oriented offenses, which will leave most of the good running backs for you. They'll also probably go for the fastest defenders they can get in order to stop the passing game, which will make it easier for you to run all over them. Go for big, strong corners who can jam receivers at the line and blitz with your linebackers and safeties.

As far as how to recruit your players, give yourself a certain amount of money to recruit each position. Then determine the probability that you can get a certain player by spending a certain amount on him. This will help you decide between targetting 3 or 4 offensive linement when you need 2. There's no good mathematical way to do this, just make as good of a guess as you can as to how many players you can get. In some cases you might be better off spending $1200 each on 4 recruits to get 2. Other times you'll be better off spending $1600 each on 3. This is where knowing your opponents' tendencies will help you the most. If you know your opponent will spend whatever it takes to get his top 2 choices at offensive lineman, let him have them and target the next 3 on your list. That way you can get them for cheap and use the leftover cash to snatch his #1 choice at quarterback.

rlwjr · 2008-01-08 08:36:41

I appreciate your help. I don't mean to sound rude at all, honestly, but about 99% of what you suggested is already included in the fuzzy logic generated talent rating. The cost issues have all been addressed in a fuzzy logic generated cost rating. Those two ratings have been logically "AND"ed together to produce the final talent/cost rating. I have a series of sets, ({talent/cost},{estimated total cost}), and a cash limit number. I need to maximize the series sets with the cash limit number. I don't think it's any more difficult than working with 3 numbers. I seem to remember something similar in calculus way back in the day when I last attempted college calc. I may be completely wrong. I just don't know. I'm looking for some help.

TheDude · 2008-01-08 09:58:15

Is your estimated total cost an estimate of what it will take to recruit a given player with 100% accuracy? Meaning, for a given player do you think that if you spend your estimated total cost on him you will get him 100% of the time?

rlwjr · 2008-01-08 10:03:12

Nope.... but that's okay. I intend on holding back an emergency amount of money. Stuff comes up. And figured into that estimated amount is basically the amount I'm willing to spend. If someone outspends me on that amount, I'm out of the bidding war.

And thanks for sticking with me on this!

Last edited by rlwjr (2008-01-08 10:03:38)

TheDude · 2008-01-09 01:34:49

If you want to do this mathematically I think you need to estimate your chance of getting a recruit depending on how much you spend. It doesn't matter if the average recruit has a talent/cost rating of 5 and you find one with a rating of 20, if you only have a 5% chance to get him you're wasting your money. In the best case you would come up with an equation to determine your (estimated) percent chance of landing a recruit after spending x dollars recruiting him. If that's not possible, then at least come up with dollar amounts that you think will give you a 99%, 75%, 50%, and 25% chance of landing him.

Basically, I think you're looking to maximize the expected value of your recruits. For example, let's say that you want 2 offensive linemen, and there are 5 that meet your bare minimum to consider recruiting: A, B, C, D, and E. They have rankings of:
A - 8.5
B - 8.2
C - 8
D - 7
E - 6.5

Then let's say that this table represents how much you have to spend to get them:

99% 75% 50% 25%
A 1000 800 650 450
B 900 725 600 400
C 850 700 500 300
D 700 500 400 250
E 600 450 300 200

Let's also say that your offensive lineman recruiting budget is $1500. This would be enough to guarantee that you can get recruit A and give you either a 75% chance with D, 50% chance with C, or a 25% chance with B. Multiply those probabilities by their respective talent amounts to find your expected total talent for a particular recruiting strategy. Also keep in mind that in this scenario you're only targetting 2 linemen. You'll need to factor in a "punishment" factor if you miss out on one of your targets and finish with just 1 lineman.

You'll end up with an equation that looks something like this:

If you want to maximize your Talent/Cost ratio for a given dollar amount try a couple strategies by hand. You'll make life much easier on yourself if you can find an equation between dollars spent and the probability of landing a recruit. That way you can get one equation to work with. Otherwise, you just have to brute force a couple calculations by hand (or write a computer program to do them for you) of a couple different recruiting strategies, such as targetting 2 or 3 or 4 players, and how much to spend on each of them.

Going back to our example, let's assume you came up with the following equations instead:

A - ChanceToGet = (DollarsSpent / 100)^2
B - ChanceToGet = 10 + (DollarsSpent / 100)^2
C - ChanceToGet = 15 + (DollarsSpent / 100)^2
D - ChanceToGet = 15 + (DollarsSpent / 100)^2.5
E - ChanceToGet = 25 + (DollarsSpent / 100) ^ 2.5

Plug these into the Recruit#_probability in the equation above and you'll get a single equation that encompasses how much total talent you can expect. Divide the whole thing by the amount of money that you spend and you can then maximize your expected talent / dollars spent.

If you do decide to go through all of this work, you'll end up with an equation with many variables. More precisely, it will have a number of variables equal to the number of recruits in your equation plus 1. If I remember my calculus correctly you can use the gradient and an optimization algorithm called gradient descent to find your maximum value. If this is the calculus thing you were talking about, the gradient, let us know and someone here can help you out.

rlwjr · 2008-01-09 03:22:44

Hey man, I REALLY appreciate the detailed response like that!

Yeah, I think the gradient is what I had in mind, but you bring up a couple good ideas at how to approach this. The probability way of looking at it is interesting, but I'm pretty sure I've got the probability locked in pretty good. The hard dollar amount that the spreadsheet is now generating, is an estimated "most I'd have to spend" figure. If I spent over that amount, I'd bail on the recruit anyway. The money you spend recruiting is incremental anyway. For instance, I might open with a Head Coach visit, and an Assistant Coach Visit, totaling around 1,000 dollars. Well, if at that point another coach jumps on the recruit harder than me, I can choose not to spend any more money and pursue a different recruit. That other coach may be closer to the recruit, have more recruiting cash, have more prestige, and I may back out for any of those reasons.

The way I'm envisioning this to work, at that point, I could update my available cash and have an idea of who to pursue next. In that case, I have not spent anywhere my estimated cost, and I can remove that recruit from the spreadsheet. So, I think you are thinking of ALL the factors that go into recruiting, when I am actually not trying to quantify every action. Does that make sense? I still want the system to be flexible so that I can control it. But when I have 15-20 recruits to pursue, I need help keeping track of who is best to pursue for what cash I have available. I'm trying to redshirt my guys around to prevent those big recruiting classes, but that takes time!

So, thanks a ton man. You've helped alot. If you have any ideas where to go from here, that would be great! I think the optimization of the sets is where I need to go, and I have like no calculus training to speak of.

If you'd like more detail on how I am assigning cost estimates or talent ratings, I'd be more than happy to go into that level of detail if it would help. I can even send you the spreadsheet as it stands, or send you a cut and paste of part of it so you can see what I have working so far, and what I'm working with overall. I'm just happy that someone is willing to help me!

Last edited by rlwjr (2008-01-09 03:25:31)

rlwjr · 2008-01-09 05:46:25

I re-read, and re-read, and re-read what ya posted and I thought it might help to post this detail. I see you working with probability, and wanted to show you how I went about doing what is, in my mind, the same thing.

I am by no means the first to try to quantify talent in this game. There are two extremely popular methods that are used, and talent is often discussed with these rating in mind, or used as a reference. I used one of those ratings in my spreadsheet to determine how "popular" a recruit would be, or how sought after he would be. Let's call this rating "A". Now, at Division III, a DL is considered top-notch if his A is above 43. So, in keeping with how fuzzy logic determines truth, I set a minimum (Amin) and a plateau point (Amax), then used the following spreadsheet formula:

IF A < Amin THEN Atruth = 0
IF A > Amax THEN Atruth = 1
OR (A - Amin) / (Amax - Amin) = Atruth

This in effect, gives me a popularity rating. 0.83 is pretty popular and I can expect to have to fight for this guy. 0.33 is not very popular, landing this guy should be easy (cheaper).

Then, using the public available formula for figuring recruiting costs/distance, (I won't bore anyone with this) I figure an absolute maximum I will spend to land this guy, which is, just for your interest, 2 Head coach Visits, 2 Assistant Coach Visits, 4 Campus Visits (these are expensive as hell), and a Scholorship Offer. I take this number and multiply it by his Atruth. I figure this result is the most I should pay for a guy. Then I have already figured the max I would ever spend, which is 1400 miles with those recruiting actions, and a minimum I would spend, which is 1 Head Coach Visit, and a Scholarship Offer at 1 mile from me. I then determine a Cost Truth rating like so, using "C" for cost:

IF C > Cmax THEN Ctruth = 0
OR 1-((C-Cmin) / (Cmax - Cmin)) = Ctruth

This gives me the Cost Rating that I logically AND with the Talent Rating, to produce an overall Talent/Cost Rating. I order them high to low on the spreadsheet, and that is my order or targets, and an estimated max cost is already available for them.

Now, keep in mind, and I should have pointed this out earlier, my own personal evaluation of talent is WAY different then what is popular. I highly value Work Ethic (which helps their stats rise higher over 4 years) and Stamina (which keeps them in the game longer). I value these attributes EXTREMELY high. So, under this system, I am, in an offhand sort of way, also finding recruits that are valuable to me, but may not be popular according to the popular rating system.

Hope all that made sense! I'm not math wiz by any stretch of the imagination, but I am very determined when I set off to learn something. This whole thing started with an interest in wanting to know exactly what "fuzzy logic" was. I just decided to try to apply it to something I enjoy! The ride has been fun and interesting!

Thanks again!

Last edited by rlwjr (2008-01-09 05:52:03)

TheDude · 2008-01-09 07:51:50

Ok, I'm getting a better idea of what you're looking for. I definitely underestimated how much work you already put into this. I have a few more questions now. What are you trying to maximize? Is it the total amount of talent you can recruit with your budget? How do you account for the different positions? Do you value a good fullback as much as you value a good quarterback? How do you penalize yourself for landing fewer recruits than you target for?

Also, I have a question on how your recruiting works. Does each person determine how many trips they're going to take for each player, input it all at once, and see what happens? Or is it more incremental? For example, do you choose on the first day that you want to do a head coach visit, then the next day you decide to bring him in for a campus visit? If so, it sounds like you could really benefit from an online algorithm, which is an algorithm that accounts for unpredicability in the future. The best example would be the stock market. You don't know how stocks are going to perform in the future, but with the right algorithm you can make more money by buying and selling stocks at certain intervals under certain conditions, rather than just buying a stock and holding it for a set length of time before selling it. Using something like this you could determine when it's best to back off a recruit and spend that money elsewhere.

rlwjr · 2008-01-09 08:46:14

YES!! YES!!!! You are exactly on track with the stock market example.

How it works is like this..... the Max players you can have is 50. I have a set number of players that I put at every position. ie 2 QBs, 6 RB's, 5 WR's. So, when recruiting, I know pretty solidly what I need and how many. The only exception is like if I had 5 RB's, and say a REALLY good QB was available close to me, I'd get by with only 5 RB's till next recruiting season, and take the QB to keep him from the schools in my area. So, I definitely will be setting position numbers that I need to fill. Also, you are correct, recruiting is incremental. It happens in cycles. Every three hours is a web update. You have 3 hours to make your recruiting moves, then another update happens. This goes on for 2 days without the recruit being able to sign. During this time you get a status update like, Recruit is Considering us only, Recruit is Considering Us Among Others, Recruit is Undecided. Undecided means you haven't even got on his radar with the money you have spent, Us among others means you and other coaches all have his attention (a bidding war), Us Only means he's decided for the time being that he wants to sign with you unless another school comes in and outspends you enough to pull that status back down to Us Among Others. That make sense? After two days, recruits can sign with you. If your status with them is Considering Us Only, then they will sig as soon as signing becomes available. If you and another coach are still on his radar, then it's either a continued bidding war, or someone gives up and after an undetermined amount of time, the recruit will sign with whoever was ahead. This signing period goes on for about 5 days. Only thing of note here, and I seriously doubt you can predict this or really protect against it, is the cycle that signing first starts, some rich coaches can swoop in, drop an exorbitant amount of cash on a recruit that you are in the green on, pushing you out of the way, them into the green, and the recruit signs with them the first cycle of signings. I've played about 7 total seasons now, and this has not happened to me or anyone I know, but it does happen and is a possibility.

I'm trying to Maximize talent, and Minimize cost. What ever cash I have left over after recruiting, I get to carry on to next recruiting season. So although it's not necessary, building up a bank roll over seasons can be very advantageous. I value all positions pretty equally. With 228 human owned coaches, every last player becomes integral, and player talent is key to this game. It's about 95% recruiting, and 5% gameplanning. Alot like real NCAA football, LOL The penalty for running out of cash is that you get a walkon player that is utterly the bottom of the barrel talentwise. Walkons are a very hurtful thing. It's preached in the strategy discussions that walkons are the absolute worst thing that can happen to you, to be avoided at all costs.

On another note about how incremental the spending is. If I see a middle of the road type player, I'm not going to open my spending on him with alot of cash. I don't want to spend alot on him when I'm not in love with him, only to find someone that is truly in love with him outspend me. I'd rather drop a little, if someone else gets on him, move to a darn near equal talentwise player. Now, if theres a REALLY good player, and I expect alot of interest, I'm very willing to drop my whole max on him in one round in an attempt to discourage other coaches. Like, there's been times when I spent and spent on a guy, never getting on his radar at all. Some other coach dropped alot more than I was willing to drop on him. Hope that makes sense.

Hey man, it really sounds like you're starting to understand what I'm after and might even have a better idea of how to go about it. Sorry if my descriptions haven't been very good. As you can imagine, the spreadsheet is very complicated, and I just didn't really know where to start in describing what I was doing! Once again, I really appreciate the time and effort you've given me with this.

Last edited by rlwjr (2008-01-09 11:45:49)

Math Is Fun Forum

#1 2008-01-08 07:09:33

Math for the Sports Minded - Maximizing Recruit Talent w/Avail Cash

#2 2008-01-08 08:02:32

Re: Math for the Sports Minded - Maximizing Recruit Talent w/Avail Cash

#3 2008-01-08 08:36:41

Re: Math for the Sports Minded - Maximizing Recruit Talent w/Avail Cash

#4 2008-01-08 09:58:15

Re: Math for the Sports Minded - Maximizing Recruit Talent w/Avail Cash

#5 2008-01-08 10:03:12

Re: Math for the Sports Minded - Maximizing Recruit Talent w/Avail Cash

#6 2008-01-09 01:34:49

Re: Math for the Sports Minded - Maximizing Recruit Talent w/Avail Cash

#7 2008-01-09 03:22:44

Re: Math for the Sports Minded - Maximizing Recruit Talent w/Avail Cash

#8 2008-01-09 05:46:25

Re: Math for the Sports Minded - Maximizing Recruit Talent w/Avail Cash

#9 2008-01-09 07:51:50

Re: Math for the Sports Minded - Maximizing Recruit Talent w/Avail Cash

#10 2008-01-09 08:46:14

Re: Math for the Sports Minded - Maximizing Recruit Talent w/Avail Cash

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