Tuesday, November 3, 2009

Nonlinear Utility Functions and Fantasy Football

I realize my blog has been pretty negative lately, since all I've been able to come up with to write about is an endless string of horrendous beats. I was going to write about more of these beats, as I took some gruesome ones yesterday (I 3-bet bravos with AA in the small blind. The board runs out something like T84-J-9, with 3-bets going in on the flop and 2 more on the turn. On the river I ask "Do you have kings? I can beat kings" and just call to see his JJ two outter spike on the turn. At 40/80 it folded to Kobra on the button and he raised. I got tricky and just called with Kd Kc. After I 6-bet the ten-high flop with two diamonds because he was short on chips, the board came running diamonds. He raised me all in for 3 chips on the river and I of course called with the second nuts to see Ad Ac. This type of cooler is supposed to happen something like once every 50,000 hands I think), but decided instead to just give you an overview of them and flesh out a concept I thought a lot about yesterday.

Utility functions are used by economists to determine a person's utility (happiness) based on possible outcomes of a situation. In poker, the correct utility function is one that maps directly (linearly) to dollars. If you win $1 you should be 1 unit of happy. If you win $100 you should be 100 units of happy. If your personal utility function is any different than this you will be at risk for sub-optimal decision making, as you're personal happiness will be offsetting the correct mathematical decision dictated by the current pot odds situation you're facing.

And there-in lay the rub. Most people's utility functions are decidedly nonlinear when it comes to money won or lost. As a simple example, take a look at the illustration on this page, which plots utility as a function of wealth. The basic idea is that each incremental unit of wealth adds less utility (happiness), which his shown by the concave down (negative second derivative) of the graph. In more simplistic terms that are obvious once you think about it, $10,000 will increase the happiness of a homeless person a lot more than it will increase the happiness of a software developer, who in turn will have his happiness increased a lot more than a billionaire would by a gift of $10,000. This also applies to poker in terms of a single session, and is the reason that a lot of recreational players "eat like birds and shit like elephants." The average player gains a lot of utility by booking a small win in a specific game. If he sits down at 20/40 with $300 and wins $300 more, he will often pick up his chips and go home happy as a clam. He won $300 and it makes sense for him to stand up, because he is on a less steep part of the curve above. If he loses back the $300, he will incur a bigger loss of utility than what he stands to gain if he wins a second $300. Therefore, from his somewhat non-rational yet completely valid point of view, the correct strategy is to stand up. An interesting effect of this phenomenon on the other end is the ability to pass one's "threshold of pain" in the losing direction. Many player won't "quit stuck", which is where the aforementioned "shitting like an elephant" comes into play. If they are stuck, even a fairly small amount, they will continue to play in an effort to get even. Eventually they move to an area of the curve in the "loss" section (down and to the left of the part shown on my graph) that is incredibly flat. If a person is stuck $2000, losing $500 doesn't really add a whole lot of pain. It's the very same monetary catastrophe that it was when you lost the first $500, but from experience I can tell you it just doesn't sting quite as much. The correct strategy for this person is actually to keep playing (assuming he is playing in a game in which his EV is neutral), since as he loses he incurs less pain than he gains as he wins. Were this person to break even for the day, he'd be thrilled, whereas the times he turns his $2000 loss into a $4000 he only feels marginally worse.

Deal or No Deal has probably advanced public awareness about utility functions a great deal over the past few years. To anyone not familiar with the game (perhaps I have some readers in Sub-Saharan Africa), it is very simple. You start out with N cases, each of which have some money in them. The amounts of money are known, and typically range from one cent to one million dollars. You pick a case that is "yours" and save it, then open other cases a few at a time to eliminate them. After you open a few the "banker" will offer you a "deal" for a certain amount of money. You can either take the deal or open a few more cases, at which point you'll be offered a new deal. Typically the deal is a fraction of the fair value of the remaining cases expected value (for example if you have 10 cases left with a total of $1.2 million in them, you will never be offered the full $120,000 that represents your expected value). Eventually, when only two cases remain, your final deal will be offered. You can either take it or open your case and win whatever is inside. The implications of a personal utility function here are drastic. I watched a show just last week where a woman had 6 cases left with over $800,000 inside them. The fair value (expectation) of her case at that point was $133,333. She was offered $96,000 and she took it. The funny part is, I would have taken the deal too. I'm in a life situation right now where $96,000, even after taxes, would be life changing money. It would go a long way towards me getting out of this apartment and into a piece of property I could own. My utility function is very nonlinear as it progresses from the $20,000 mark to the $500,000 mark, and therefore I'd have taken the down payment and run.

So where am I going with this? Fantasy football, of course.

The utility function for the performance of your team for fantasy football is yours and yours alone. There are perhaps players out there for whom nothing short of a championship season will afford any sort of happiness whatsoever. Others will glean a good bit of utility simply from making the playoffs. I argue, however, that most fantasy football managers gain the greatest bit of utility simply from not building a team that in the end causes embarrassment. This is my utility function in a nutshell. I don't want to embarrass myself. Therefore, my goal should be to build a team that has the greatest probability of winning at least 6 games (in a 14 game season), as that is my threshold value for "not embarrassing". If you win more games that that, maybe you'll make the playoffs and that'd be fun, and if you do you could win, because we all know that fantasy football playoffs are a complete crap shoot. Here then is my fantasy football "don't draft an abomination" strategy. If you follow these simply steps, your team will win 6 games. I promise. Let us assume a 12-team league with standard rules in which you start 2 running backs, 3 wrs, and a flex each week (a "deep" league true, but fairly standard). The draft has 16 rounds. Here then, are your commandments:

1. Draft a kicker in the 16th round.

2. Draft a TE in the 15th round. Do not fall for Tony Gonzales, Antonio Gates, Jason Witten, or Dallas Clark. There are tons of decent tight ends (Visanthe Shiancoe...Heath Miller...That Havner guy from Green Bay....Vernon Davis...Kellen Winslow....Jeremey Shockey.....you will end up with one of these guys. If you don't, it is because your league-mates are drafting two tight ends, in which case you will win anyway because they are idiots).

3. Draft a Defense in the 14th round.

4. For the three positions above, play the waiver wire relentlessly. Rams playing Detroit this week? Start 'em! Jacksonville's kicker playing Cleveland at home? Time to cut Lawrence Tynes! Basically use only 3 roster spots on these three positions, no matter what.

5. Don't draft a QB before the 9th round unless your league has some weirdo scoring system (like a point per completion which apparently happens in Facebook leagues) or a rule allowing you to flex a QB some number of times (like my CBS Sportsline League does). If it's one QB the whole way, take one in the 9th or 10th round and hope it works out. If you must, draft a second one in 13th round.

6. Draft the last two starting RBs on the board (or at least one of them). This year Julius Jones and Cedric Benson were available in the 7th or even 8th round of many fantasy drafts (later in "less competitive" leagues). Having these guys as your 4th and 5th running backs will basically ensure that no matter what happens, you will field a solid team with a chance to win every single week.

7. On your 16-man roster, end up with at least 5RBs and 6WRs. Keep drafting them. "No Quarterback, eh?" mocks your friend while sipping on his 4th Miller Lite while taking Heath Miller in the 8th round? Pick Ray Rice. "Still no QB?" he says next round? Take Hines Ward.

8. No handcuffs. If you have 5 starting RBs on your team, you don't need handcuffs.

Following this simple 8 point strategy, your team will never suck. No matter what. No amount of injuries can devastate you, and somebody will step up and lead to the promised land of 7 or 8 victories.


dlgilbert4 said...

That is an amazing poast.

Captain R said...

"negative second derivative"? who the f*** talks like that?

You should've written this post 2 months ago. I'm last in my FF league. Who's the commissioner? Yes, that's correct.

Wacky said...

Engineers and mathematicians talk like this.

Utility functions and economics are interesting. Read Freakonomics by Levitt for an interesting applications of economic theory to non economic stuff, very interesting read.

jesse8888 said...

1. Thanks for the toast to my poast.

2. I talk like that, Mr. Commissioner.

3. I've heard that book is awesome, and that the upshot is that McDonald's is structured the same way as a local crack-cocaine distribution network. Confirm/deny?

Burg said...

Unfortunately I am Jesse's Fantasy Football General Manager. My happiness utility function is to "Just Win Baby". I basically disagree with all his tenants except the 1st one. On every pick you take the best available piece of the puzzle. And you can take a flyer on some jamok who is high risk high reward. We do still speak to each other and are 4-4 with a bullet.

Any business major should understand the meaning of a negative 2nd derivative.

Wacky said...

I don't know about awesome, that's a pretty strong word for any book with an economic slant. I read it many years ago, so I don't remember everything, but I do remember not agreeing with all his theses and tenets, nonetheless it was a very interesting read. I highly recommend it for any intelligent person who doesn't mind reading stuff that they don't necessarily agree with. What's the point of that anyways?

Dave said...

Actually, this explains a lot. I agree with "Burg" - I don't care about anything except winning. Of course, Jesse's made the playoffs in my league every year (that run will come to an end this year), I believe, but has never won, so many he's retro-fitting his utility function to his performance.

Freakonomics is definitely worth reading (and the new release: Super-Freakonomics, which I haven't read), but it does have a couple flaws. For my money, Gladwell's books ("Blink", "The Tipping Point", "Outliers") are a bit more enlightening.

jesse8888 said...

See Dad, the basic problem here is that we mixed utility functions. Don't worry, my luck in this league is fantastic to date. We'll make the playoffs at like 9-7 or something similarly absurd then draft a real clunker and take 3rd, the only non-winning position I'm yet to capture.

pdscjr said...

You should be embarrassed for playing in a fantasy football league that has 14 weeks in the regular season...

Also, I don't even draft a kicker.