Bales Goes Deep: Are Cash Games Really All About Value?

If you had Brian Westbrook on your fantasy team in 2007, you sure do. He was on my team, and when that happened I smashed my computer chair into like eight pieces. That was my third broken computer chair of the season, which I think would be somewhat embarrassing if it weren’t actually totally kick-ass. Long story short: If you haven’t broken items of value over unfavorable results in a fantasy league with a $50 buy-in, I really have no interest in getting to know you.

The Westbrook play was horrible for fantasy purposes, but it was really smart on his part. Unlike most NFL coaches, Westbrook realized that, due to the time remaining in the game, the contest was no longer about point-maximization.

Actually, the goal should never be a blind focus on point-maximization; it should be about maximizing win probability. Now, in many game situations, those two aims are equivalent—point-maximization is often an effective means of maximizing win probability—but it’s important to realize the two are not synonymous.

Daily fantasy owners could learn a lot from Westbrook. Almost everyone I see approaches the game as though it’s one of point-maximization. I believe that’s almost never actually the case.

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DFS and Win Probability

As a daily fantasy sports player, the goal is to win leagues. Sometimes you need to score a lot of points to do that, but making a conscious effort to maximize points will almost certainly not maximize the odds of winning money.

I’ve talked a lot about this concept in the past, specifically in terms of GPP strategy, so I won’t harp on it too much here (you can read more in The Shifting Daily Fantasy Football Landscape). The general idea is that tournaments are primarily about 1) increasing upside and 2) benefiting when others are wrong, i.e. leveraging inefficiencies in player ownership into an edge. Cash games are mostly about narrowing the range of outcomes for your lineup to increase its floor.

I recently had a shift in my cash-game philosophy that embraces a more high-variance approach to head-to-head games. Unlike in 50/50s—leagues in which there’s no additional benefit to finishing higher and higher above the cash line—there is a reward for owners who have outlying scores in head-to-head games (since those pay out in a more linear way).

Anyway, one thing I haven’t really touched on too much is whether or not we should be emphasizing ownership in cash games (to any extent at all). While cash games are mostly games of finding value (however we want to define ‘value’), is it ever the case that it’s smart to not maximize value in cash games?

The Prisoner’s Dilemma

You have probably heard of the prisoner’s dilemma. If not, here is a blurb I just wrote about it, and by that, I mean I copied it from Wikipedia.

Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communicating with the other. The prosecutors lack sufficient evidence to convict the pair on the principal charge. They hope to get both sentenced to a year in prison on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to: betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent. The offer is:

• If A and B each betray the other, each of them serves 2 years in prison
• If A betrays B but B remains silent, A will be set free and B will serve 3 years in prison (and vice versa)
• If A and B both remain silent, both of them will only serve 1 year in prison (on the lesser charge)

It is implied that the prisoners will have no opportunity to reward or punish their partner other than the prison sentences they get, and that their decision will not affect their reputation in the future. Because betraying a partner offers a greater reward than cooperating with him, all purely rational self-interested prisoners would betray the other, and so the only possible outcome for two purely rational prisoners is for them to betray each other. The interesting part of this result is that pursuing individual reward logically leads both of the prisoners to betray, when they would get a better reward if they both kept silent.

This is the quintessential example of game theory, and I believe it has implications for DFS players. Namely, we need to reconsider how we think about what is optimal in cash games.

Before going any further, I want to point out that I’m not talking about fading obvious values in cash games just for the sake of it. In almost every case, I still think you should be playing the highest-value guys.

However, when you have two players rated nearly the same, but you know one is going to be much more popular than the other, I think you could make an argument to give up just a bit of value in favor of the lower-owned player.

For one, you’re putting yourself in position to benefit if everyone else misses on the high-owned option. You could also be hurt a lot by fading him, so it’s a high-variance move and you need to be okay taking on that risk.

If it were just about ownership, I’d still say to try to maximize value in most instances. However, one thing I think people forget about is the rake. If two even players face one another in a head-to-head game, they will both be long-term losers, losing roughly 10 percent of all entry fees to the site. We ain’t playing for free.

Well, the more lineup overlap there is (whether it is in a big 50/50 or a large group of head-to-head matches), the more likely it is that you lose to the rake. If you were theoretically playing someone who you knew had the exact same lineup as you except for one player, would you still face him? Of course not; one player isn’t a large enough edge to overcome the rake. It follows that the smaller the pool of relevant players, the smaller the potential edge, and the more likely it is you get beat by the rake.

Thus, I don’t believe cash games should always be about point-maximization, nor do I believe they should always be about value-maximization. I think you’re generally trying to maximize points and value, but in situations in which you have an under-the-radar player ranked nearly the same as a more popular one, you should side with the latter because it will diversify your lineup and minimize the chances of losing to the rake.

In prisoner’s dilemma terms, you’re remaining silent—something that actually helps your opponent, too, by improving the odds that, if one of you has an edge, you will realize that edge and not lose it to the rake. DFS is approached as though it's zero-sum, but because the sites need to make money, it's really a negative sum game in which players can sometimes cooperate to help one another.