# What is Game Theory?

According to Investopedia, game theory is defined as “a model of optimality taking into consideration not only benefits less costs, but also the interaction between participants.”

There are a bunch of really useful examples of game theory that are quite common, but I like to make up my own illustrations that are less practical and sometimes difficult to understand. That’s what being a writer is all about.

So here’s my example.

Suppose you’re on a ranch in Texas and you’re in a competition to guess the color of cattle in a barn. Are there barns on ranches? Probably, right? Idk. But there’s a barn and there are 10 cattle in it. The cattle are trotting out of the barn one-by-one, and you’re asked to predict the color of each one—black or brown—prior to it leaving the barn.

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So which color should you pick? If you’re trying to maximize the number of correct guesses, you should guess whichever color is the most common for cattle. I don’t know what that is—my Google search for “most common color of steer” didn’t turn up anything useful (one site said red, which seems…not right)—but let’s say it’s brown.

Let’s say 60 percent of cattle are brown, 30 percent are black, and 10 percent are rainbow. So if the game is purely about accuracy, you should select brown every single time. Maybe you’d shift strategies if like five straight black cattle trot out and you figure whoever owns this group of cattle has solely black ones. But if we assume a “random” distribution of cattle based on their actual real-life frequencies, it would make sense to guess brown each time to maximize the long-term hit rate.

That’s pretty easy. Now let’s say you get a specific number of points for each correct guess, and those point totals change based on the color of the cattle. If you were to get 70 points for each correct guess of black and only 30 points for each correct guess of brown, you’d maximize your point total by selecting black each time; you’re effectively getting a 7:3 payoff on a 2:1 bet, which is good.

This is effectively what we do in daily fantasy sports. We estimate the probability of a player performing at X level and Y cost (in terms of either his salary or the opportunity cost of bypassing him). We’re basically looking for quality odds on each player; in the definition of game theory, it’s the “benefits less costs” phrase.

This is typically how we assess ‘value’ in most situations: what’s the potential and what’s the cost? If the former exceeds the latter, there’s a positive expectation. When forced into a zero-sum game of competing minds, however, things change.

If you were to compete against even just a few other people in the insanely stupid cattle-color-guessing game, your “optimal” strategy would change based on what you know about their strategies. The question you should first be asking yourself is,

“What do my opponents know about cattle?”

If you discern that your opponents don’t know **** about cattle, your optimal strategy would almost assuredly be to guess brown each time. Unless your opponents do the same, you’d win more often than anyone else.

But what if your opponents are cattle ranchers and know everything there is to know about cattle, including their color distribution? Now we have a different ballgame because we need to consider not only how to maximize the chances of being right, but also the payoffs if we are right. And if the payoffs make up for taking on reduced accuracy, it might make sense to occasionally steer (great pun) away from brown.

Example. Let’s say you’re competing against four cattle ranchers and you each put down $100 in a riveting winner-take-all game of cattle-color guessing. You pretty much know each of them will select brown each time. Should you do the same, and if not, how often should you move away from brown?

Let’s answer that with a simple thought experiment. If you were to select brown every single time, everyone would tie. If you were to select brown nine times and black just once, however, everyone would tie nine times, and there would be one “meaningful” cattle trot in which you’d more or less have a 30 percent chance to win $500 on a $100 bet. If you play out that scenario 1,000 times, you’d win roughly 300 and lose 700—a gain of $150,000 and a loss of $70,000 for a net profit of $80,000.

So it’s pretty clear it makes sense for you to go against the grain with just a single choice (assuming you can predict your opponents’ actions).

Now what if you got word that one of the participants was planning to utilize the same strategy as you? Would it still be optimal to select black just once? In that case, you’d tie with him 300 times and lose 700 times—a gain of $75,000 (assuming you split the winnings) and a loss of $70,000 for a net profit of $5,000. One small change in the behavior of one opponent would cause your winnings to drop from $80,000 to $5,000, but it would still be the right choice.

Already, it’s clear you need to be concerned about your opponents in a game of this sort—which is basically a very dumbed-down version of DFS. What you really need to think about is 1) the size of the field, 2) how much better you can predict the future than them, 3) how they’ll act, and 4) the payoffs to you if you’re right and/or they’re wrong.

To demonstrate how the size of the field and your opponents’ anticipated actions can alter the optimal strategy, let’s extend the size of the player pool in the cattle-color-guessing game to 101—a field in which you predict 70 people will select all brown and 30 will pick nine brown and one black. It didn’t make sense to select black twice in the prior example, but what about now?

Let’s do the math. In this case, you’d beat everyone nine percent of the time (30 percent times 30 percent), and you’d lose to one of the two groups 91 percent of the time. That would gross you $909,000 and you’d lose $91,000 for a net profit of $818,000. Increasing the size of the player pool dramatically increases the value of being contrarian and using a “sub-optimal” strategy (one that’s sub-optimal in terms of maximizing accuracy, anyway).

### What You’ll Learn in Using Game Theory in Daily Fantasy Football…

At its core, I think game theory’s implementation into daily fantasy sports is pretty simple; we’re still dealing with benefits less costs, but we need to redefine ‘benefits’ and ‘costs’ to incorporate the various unique aspects of a zero-sum game of competing minds.

I’m going to try to help you do that, showing you how to properly use game theory in NFL. After reading this course, you’ll learn:

1) Why game theory matters and how to exploit opponent beliefs

2) Which stats predict performance in daily fantasy football

3) Which stats predict ownership in daily fantasy football

4) How to implement an “antifragile” tournament strategy that benefits from randomness

## Comments

There are so many different course’s by so many different people. They all look good for the NFL. Is there a recommended order of witch ones to start with and what order to go in. Thanks.

Great article! I have been trying to utilize game theory recently and until reading this article, I didn’t realize the correlation between player pool size and contrarian lineup construction. Great explnation.. this is super valuable information for DFS players looking to take the next step

“3) Which stats predict ownership in daily fantasy football”

I’m very curious about this. Just started playing DFS this week. I noticed Carlos Hyde having the highest ownership rate this week, would be interesting in knowing how to predict this and avoid players like that. Possible metric FFP))/ FFP (dateadd(ww,-2,getdate())) larger ratios = Higher ownership?

RotoAcademy Lead Instructor

## RG Overall Ranking

## 2015 DraftKings FFWC Finalist

## 2019 DraftKings FFWC Finalist

Not necessarily. I think it’s just sort of what interests you. The position breakdowns from JMToWin have been a popular starting place, though.

## Blogger of the Month

i don’t totally disagree with what you’re saying..

however, we don’t see ownership levels at 70% in large tournaments (or even 30% very often – referring to your example of a 101-player pool). so i think ownership levels can be blown out of proportion a bit. also, a player is worth rostering no matter what is % ownership may be if you think he will return 3x – 4x his salary on FD (or 4x – 5x on DK). i still believe that 15% ownership is contrarian, since 85% of the player pool will not have rostered that player.

sometimes i think a bayesian perspective is a little more helpful. in retrospect, if it turns out the the optimal scoring lineup for week 2 consisted of players that were all 50% owned, you should have selected them anyway and split the $$ with a lot of people.

in terms of the size of the player pool, most lineups will be unique (based on the number of possible combinations of randomly selecting a team of 9 players). so (as with any player you select), a low % player is only helpful if he scores a lot of points.

## Blogger of the Month

the downside to these ultra-large field tournaments is that many, many more players are being selected which means that the randomness factor to winning increases (and the skill factor decreases).

Thanks Jon, the JMToWin course’s do look very good.

I haven’t made that “next step” yet, so I’m not as deeply into this as some others…but, I think his general point is over the long run, the sub-optimal approach could yield the greatest returns. In the 101-man example you lose 91% of the time, but the other 9% is when you hit it big. There will be plenty of times where the optimal lineup cashes, but even then you need to have someone low-owned to score a bigger payout.

In hindsight, at least for me, this is painfully obvious but I’m usually too nervous to make a sub-optimal lineup. Maybe it’s just that my goals are different, but I’m trying to build my bankroll which means I’m just happy to cash. If I can turn $40 into $60 every day, then I’m ecstatic. Right now, I can’t afford to lose 91% of the time.

I know this is where proper bankroll management comes in, but I’d really like to hear more from the guys that use cash game winnings to pay for their multi-GPP entries.

## 2015 DraftKings FFWC Finalist

To me GPPs are not really about playing “sub-optimal lineups”. For example, Rus Wilson was ~2% owned in the Milly Maker for week 2 and Drew Brees was about 20%. Brees will on average outscore Wilson, so Brees is the “optimal play” in terms of maximizing points, but Wilson is a far better play in GPPS,when taking ownership into consideration is very important. So an optimal lineup for a cash game is likely not the most profitable lineup for a large field GPP.

## Blogger of the Month

@zeechamp, why do you think ownership % for NFL is so important? just factoring in QB, TE, K, D (on Fanduel) – there are 1,048,576 possible player combinations. That’s before factoring in RB and WR.

The equivalent to what you’re saying is that you would like to know how many people play the number 25 in the mega millions before you buy a ticket.

## 2015 DraftKings FFWC Finalist

@zxcvbnm4321 Ownership is the most important thing to consider in GPPs imo.

If you were to enter 2 of the exact same lineups into a GPP, except one of them had a 2% owned QB (with a slightly lower projection) and the other lineup had a 20% owned QB (with a slightly higher projection), the 2% owned QB would be more profitable in the long run.

Maybe the 2% owned QB gets outscored 60% of the time compared to the 20% owned QB, but you gain so much more compared to the rest of the field when the 40% occurs.

## Blogger of the Month

@zeechamp, the problem is that ownership % varies by week. so that 2% owned QB that goes off is now 15% owned the next week. you can’t replay this week over again. do you think ben is going to be 4.1% owned next week?

it’s a faulty premise to think that % ownership stays the same during the season.

again, there are so many possible lineups. consider that there were 21 RB that scored touchdowns and at least 30 WRs that scored touchdowns. the outliers were fitzgerald, allen robinson, travis benjamin, julian edelman, crockett gilmore, deangelo williams, matt jones.

regardless of ownership %, you needed to have as many of these players as possible to win. the lower the ownership for these players would simply mean that the average score for the tournament would be lower (and the score needed to place in the minimum cash). the higher the ownership % of these players simply means that the average score is higher, as is the score for minimum cash and the top overall score.

you can’t tell me with the information from yesterday you have now that you wouldn’t have taken larry fitzgerald even if he were going to be 20% owned.

## 2015 DraftKings FFWC Finalist

@zxcvbnm4321 We don’t care how highly owned a QB is next week..just this week, we don’t have to use the same guy next week. So we missed out on Ben’s big game this week, oh well you just have to focus on the next week.

So say Ben will be 20% this week after a huge game, we might find a guy that’s

## 2015 DraftKings FFWC Finalist

@zxcvbnm4321 hmm it keeps cutting off my text, but take Starks this week if Lacy doesn’t play. He’s min priced and will likely be 30%+ owned. I will have him in every lineup becasue his point/$ projection is so high that on one else will even come close. Basically: don’t fade players just because of ownership, pivot to other guys in similarly good spots that no one is talking about and will be significantly lower owned.

## 2015 DraftKings FFWC Finalist

Hyde is a good example last weekend. He was coming off a huge game on national TV and his price didn’t adjust because he played on Monday so we knew he would be the highest owned player in GPPs. But there were things to worry about with Hyde: the 49ers were traveling across the country to play in a 12:00 game, coming off a short week (after playing Monday), against a team that was on 10 days rest, and the 49ers were 6.5 point dogs (Hyde has not been a pass catching back in the past). So Hyde was by no means a slam dunk and I though there were guys in similar spots that would be far less owned (Tevin Coleman for example)

## Blogger of the Month

@zeechamp: i agree..my general point is that if you are going to win $$ you have to have the best players regardless of ownership. if you think of the distribution of the scores for the tournament yesterday (i.e., normal distribution), everyone in the tail (where you win lots of $$) had many of the highest scoring players. the % ownership of these players merely affects the minimum score needed to cash. if the ownership % for guys like fitz, deangelo williams, etc. were higher that it actually was for this week, it would mean that the minimum score to cash might be around 120 vs the 113.96 that it is now.

People also need to take into account payout. When the payout structure is so top heavy and not relative to the amount of entries… being contrarian isn’t enough! For example i recorded a top 50 out of 114000 in week 1 in the dive and made $75. Relative to the money had in play I didn’t even profit for the week even with finishing inside the top 0.1 . People with smaller bank rolls would be better off playing in 2,000-10,000 man fields with higher buys in so if they hit that top .01 cash it fits better within the payout structure.

This isn’t exactly a pure game theory question, but I’ve wondered if there are people who play the same limited number of players every week regardless of matchups and cost on the idea that they are going to go off a certain number of times a year and you might not be able to time it.

They would all have to go off together on the same week to win a tournament and the odds are extremely long against it, and even then there’s no guarantee you’ll win. Seems to me such a strategy might be better for cash games as a handful of players will go off each week which actually creates more of a floor than a ceiling.

In the very large fields it is even more important to employ game theory. High risk rewards stacks are typically my favorite. If you can find a stack that will be utilized at a very low rate/ IE 2/3%, they can be very profitable long term. The key is finding stacks with very HIGH ceilings that you can match with high ceiling pitching. Obviously I am using baseball here because I play it more often. The same can be done with QB/WR stacks. The more high end would be a QB/WR/RB or QB/WR/TE stack. If you can find a high ceiling stack here, then maybe match with a high ceiling TE you can separate yourself from the field. If you get good at research these are techniques that can help you hit some huge tourneys. The key is you are taking high risk in any GPP. If you are taking the SAME high risk everyone else is it isn’t profitable. Taking high risk, high rewards that only you have is the way to hit it big time. DFS is always a risk but great research + game theory tactics can certainly be profitable long term.

Halter, Bales latest “Fantasy Football for smart people” cover this type of lineup building and philosophy. If you get the book you can get a free academy course and also save big on other products. The book is a great read and pays for itself.

It’s what Vanessa Rousso studied at Duke.

Picking the best players as a strategy only works if you have a crystal ball. Picking the best players before the fact requires you to make projections; whether intuitively or systematically. But after going through that exercise, you need to understand that you may be generally right, or generally wrong. Because of this, it is now important to look at ownership levels. If you have players with similar projections and only want to take one or the other, it only makes sense to pivot to the lower owned. This week, unfortunately for me one of those pivots was odell from Julio. I had both of them, especially with Julio’s toe a slight concern, projected very close to each other. I would never be able to afford both based on my roster construction, so I went with obj. Didn’t work out, but if it did, it would propell me up the standings. One last thing I learned over time is to know what you are playing for. If your playing a large tourney, like the milly to cash out, then don’t play the milly, play 50/50s and such. If you play the milly, it should be for much fewer but larger wins. It amazes me how many friends play in the milly and justify certain picks as safe picks, but don’t feel very highly about their upside. They justify their lower cash rate by the illusion that they have a chance at the million, whereas the truth is that they would be cashing more often with that process in cash games and have absolutely no chance at winning the million.

2012 DSBC Finalist

The best way to guarantee that your lineup is sub-optimal is to make sure your lineups are all 1k or more less than the cap. Even game theory players who regularly play sub-optimal lineups do so with their ownership % predictions in mind, but NOT by trying to make sure all of their 150 lineups are all 1k or not under the cap. This is because they figure their use of low ownership players in their combinations is enough to guarantee sub-optimal conditions for their lineups. Consequently, in extremely large field tournaments with a low number of total games being played, the best strategy is cap relative not expected ownership relative.

2012 DSBC Finalist

Definitely – I have friends who play Goldschmidt regardless of pitching matchup for this reason in their gpps. I try to tell them to only take Goldie when he is facing a stud pitcher (to guarantee getting a stud bat at low ownership), but they are only interested in having him when he homers twice and don’t realize that it doesn’t payoff to do that when he homers twice against a bad lefty, but pays off big when he homers twice in a game started by a pitcher everyone will be afraid of.