This is an article I wrote two years ago while I was working at Rotocurve. It is about game theory in MLB DFS. I made some adjustments to it and also wrote a follow-up piece about how different levels of players can use game theory to their advantage. That article really does not make much sense unless you read the original piece, so here is that piece…
The words “game theory” are mentioned so frequently in Daily Fantasy Sports (DFS) that the average new player must think everyone in the industry has a PhD in economics. You do not need a higher level degree in order to understand it. You do not even need to understand it in order to apply it. It does help to know what Game Theory is if you want to apply it correctly though. That is what I am going to do teach you here.
Game Theory is a rather complex idea. It studies the science behind how people make strategic decisions. In plain English, Game Theory dictates what actions each player should take to produce the best possible results for themselves. In DFS terms, this means Game Theory is what we should be using to decide who to roster (actions we take) in order to win the most money (best possible result). If that sounds like something you would be interesting in learning, feel free to read on.
How does DFS fit the Game Theory Model?
Game Theory has been studied in many different ways for many different situations. It is used in Economics, Politics, and even Biology. In order to correctly apply the concepts, we need to know which principles can be applied to DFS. DFS is a game of interdependence, meaning the outcome for each person depends on his/her own actions within a larger group. You make the best lineup in your tournament…you win the most money. Easy concept to understand.
It is also a zero-sum game, meaning the interest of each lineup entered conflicts with that of every other. You do not sum up total entries here, and notice I said the interest of each lineup and not each player. We will get into that when we talk about multi-entry later. Each lineup entered is in conflict with every other lineup that is entered in the same tournament. The idea is that the highest score wins the most money and each lineup is competing against every other lineup for the top spot.
DFS is also a game of simultaneous strategic implementation. This means that everyone submits their lineups at the same time and therefore no one knows what the other person’s strategies are when they make their own decisions. This is especially true of non-late swap sites and opens a whole different can of worms when you introduce late swap into the mix. While late swap sites do offer a tad more complexity here, for purposes of the analysis we will think of it along this line of reasoning and discuss the differences where needed.
Now that we have the ground rules, we can fit it into the framework for discussion.
So How Will This Help Me Make a Lineup?
We know that DFS is a game of interdependence, where each lineup stands alone and either returns us money or not. It is also a zero sum game, so we have to have winners and losers. Therefore, it is in our best interest to make sure our lineups are going to score more than our opponents in order to have a lineup that wins us money.
In tournament play, that usually means we need to have a score in the top 20% of total entries in most Guaranteed Prize Pool tournaments (GPPs). Notice the wording was “score more than our opponents” and not tried to score the most points possible. While you want to score the most points possible, sometimes it makes more sense to not put the highest optimal or projected lineup together and enter it into the tournament. Understanding why is the key to understanding game theory and requires us to talk about risk vs. reward and the concept of logic circles.
Risk vs. Reward
Risk vs. reward is the easier of the two concepts to illustrate and understand, so we will start there. The highest optimal projected player in each spot is the guy who comes with the least amount of risk. He will also be the guy who is likely to be the highest owned player. Since we do not know exactly how things will turn out, we need to weigh playing or fading this guy in terms of our risk of doing so vs. the reward we get for doing so.
If Paul Goldschmidt is facing a bad lefty in Coors Field, he is an obvious play for everyone who does even a little bit of research. The risk of playing a Goldschmidt and having him underperform is much smaller than the risk of playing someone else who may not have as favorable a situation today. The reward part is what most people do not understand about making DFS tournament lineups. The reward part is: how will playing Goldschmidt help you maximize your potential return?
Let’s say he is the most likely high scorer at the position and the odds of him doing so are 20%. That means he will be the highest scorer only 1 out of every 5 times in this situation. That might sound low, but on a full 15 game slate with 30 or more (1B/DHs) players available at his position, that is not far off from the actual odds of a top play on the day. Now, if the ownership on Goldschmidt was 20%, the risk to reward ratio would be 1:1. At 10% ownership he would be a superior play, because the chances of him being the top dog are better than the number of people who would benefit from it. If his ownership was 40%, it would be a horrible play, because while he is the most likely top scorer, the number of people who benefitted from it far outweigh the risk you take of it happening. With 40% of a pool taking that player, it does not even guarantee you a cash spot in the top 20% of a tournament by having him. That is also only for the 20% of the time he does happen to be the top scorer.
Therefore, the risk vs. reward equation is skewed against you if you use him. If he only is the top scorer one out of five times (20%) and only half (20% cash line, 40% ownership) of the people who had him cash, then you really only should expect him to help you cash 1 out of every 10 times you take that bet. Even then, your cash is not guaranteed to be anything more than a minimum cash, because 40% of the people have the same player you do. You are still competing with a large number of just to make it into the top 20% of the tournament.
If you assume a normal distribution, one out of every four cashing rosters will be in each of the 4 quadrants of the top 20%. Meaning you will have one finish in the top 5%, one between 5-10%, one between 10-15%, and a fourth between 15-20%. If the strategy gets you a cash one out of ten times and only one out of four cashes makes the big money, then your chances of making big money by taking the chalk is basically one in forty times doing so with 3-8 other small cashes along the way.
Now let’s assume in example two that another first baseman is 10% likely to be the top scorer, but is only 5% owned on the day. That means that one in ten times that guy will be the top dog and every time that happens you end up in the top 5% with the rest of the people who had him. If you run this scenario forty times like we did to get one top 5% roster with the chalk, you would end up with 4 rosters that all have a chance for a top 5% finish. That means by picking a lower-owned guy with half the probability of being a high scorer, you actually are four times more likely to find yourself in the top 5% of a tournament with a chance for big money. The drawback is you probably only cash half as often, but the potential profit on each of those cashes is much higher.
This is obviously a very linear one-on-one comparison and DFS is not that straightforward. It does help to illustrate the point we are trying to make with how to use Game Theory. In a normal game of DFS, you have 9-12 guys on your roster, so the overlap of how all of them perform collectively will have an effect on your final score. That still does not change the fact that you have to weigh the risk of any one guy being the top scoring option against the reward you get for it. Remember, the goal is to maximize our potential return, not minimum cash a tournament. Minimum cashing tournaments at a 20% clip, as the normal distribution suggests will wipe out a bankroll quickly. Playing it safe with the chalk is rarely the correct option when you use Game Theory to guide your lineup choices.
Of course we did mention that the chalk is still likely to hit 20-30% of the time. You probably still need two or three highly owned plays at a minimum as many top cashing rosters often have. It is rare that all the chalk plays misses on the same day, but our job as players is figure out which guys we can successfully fade and which we cannot. In order to maximize the return of fading a player, the ones we fade should be the ones who present the worst risk vs. reward equation. If we think someone is going to be 30% owned, but only a 10% chance of being one of the top scorers on the day, then that guy is someone we should try to leave off in favor of a different option. This is how we can maximize our potential for a big roster.
What About Price?
You might be saying to yourself that nowhere did we mention price. You would be absolutely right. Price has nothing directly to do with Game Theory and maximizing our potential for a win. Where price plays a role is in our estimations of ownership percentages that guide our risk vs. reward argument. This is also where the concept of the logic circle comes into play. A logic circle is the main way we would use Game Theory to figure out who will and will not be highly owned. Price comes in to play by helping us estimate those choices.
If we know Goldy is in a great spot, we can assume he will be highly owned. How highly owned is a function of what we have to pay for him. If Goldy is $3000 on DraftKings or FanDuel, he will be 80-90% owned. More likely he will be $4500-$5500 and that will keep his ownership in the 20-30% range. His price does not change his projected outcome in any given situation. He is still likely to score the same amount of points in a matchup whether he is $2000 or $5000. Goldy does not care where he is priced and neither should we. This is why you will hear some people say that a guy will be low-owned because he cost X amount of dollars. DFS players making lineups may fade a guy they perceive to be too expensive, but those are guys a savvy student of Game Theory would roster. That player is still likely to score the same amount of points, but he is worth using now if the ownership will be low.
Take a look at these screenshots…this was the day after Kris Bryant’s massive three home-run game. Bryant was 30% owned, but was not necessarily a better player than Arenado, Baez, and Donaldson, all three of whom ended up scoring more DraftKings points for a cheaper price.
DFS players would be better served to stay away from guys the masses consider to be cheap, because the risk/reward of using popular punt options is likely a bad proposition. This is especially true if they have a very low probability of being a top scorer, while the ownership on this highly touted play is high. This is usually the case when you get a guy like Phil Goselin batting cleanup at $2000 in Coors Field. Everyone is saying what a great spot it is and how you should play him, but he is still not likely to be the top scorer. At high ownership, he would not really give you the best chance to move ahead of the pack and make a big score even if he was. The optimal move would be to fade this low probability play in favor of someone with a better chance of being a high scorer at a lower ownership percentage.
The logic circle is a problem in zero-sum games that assumes everyone will act out of the belief that others know that they know that this path is the optimal one. It comes into play when we use our decision making process to try and figure out what the optimal strategy is to help us cash high up in the top of a tournament.
The argument of a logic circle is best illustrated by the differences we see in ownership from the lower buy in levels and the upper levels. A logic circle is best described by the thinking that I know that you know that I know Player X or Team X is in a great spot today and therefore I am going to do this instead because you think I am going to do that. In a tournament with a lot of entries, we assume that everyone knows this.
Therefore, if you are playing in a $3 tournament and know that everyone is going to stack the obvious choice of the Rockies at Coors Field, the logic circle says you should fade the Rockies. By stacking some other team that is the second most likely to be the highest scoring team, we give ourselves a better chance for a high cash finish with a better risk and reward relationship. This is a very viable strategy in lower buy in tournaments, because not everyone uses this deeper Game Theory analysis when building rosters. This is especially true for those who single enter huge field tournaments with 1000s of entries in them.
The Nash Equilibrium, put forth by John Nash (A Beautiful Mind), suggests that each player has no unilateral incentive to deviate from the path that has the highest chance of bringing them the highest score. They mistakenly think that winning a GPP is best accomplished by trying to score the most points. That tends to lead to high ownership on obvious plays. The highest team totals, the team facing the worst pitcher, or whatever team is playing in Coors Field are all good examples of high ownership situations. While these plays are the ones that are the likeliest to score the most points, they are not the likeliest ones to help you win the tournament or position yourself to cash highly in one if the ownership is sky high.
The further up the buy-in chain you move, the more sophisticated the players are. You tend to see lower ownership on the top chalk plays and higher ownership on the second or third level deviations as the buy-ins increase. Most top players understand Game Theory, or at the very least unknowingly deploy it when building lineups. That is why you may see Cargo vs. a right-handed pitcher in Coors Field at 40% in the $3 Moonshot on DraftKings, 30% in the $27 Payoff Pitch, and only 20% owned in the $300 Perfect Game on the same night. This is a great illustration of the difficulties when using Game Theory, but also the advantage you can gain in roster building if you understand it.
How Do I Look at this?
The difficult part of Game theory is projecting what the other players are likely to do. Most people mistakenly assume that everyone is hyper rational. Being hyper rational in DFS MLB means that everyone is thinking that Colorado is going to be chalk and because of this everyone knows they are better off going a different route. If you look at Coors ownership on any given day, you will realize this is not the case. Game Theory does not assume, nor require that everyone is using this hyper rational line of reasoning. It also does not require that everyone has perfect information in order to make the most rational decisions. We will have highly informed hyper rational players and also uniformed chalk players as part of our overall player pool. Realizing you will have each type and in different quantities is how you can get the framework to do your analysis.
In order to perform Game Theory analysis, we need to have a feel for what ownership levels are likely to be on a given slate. This really only comes with experience. There is no exact way to tell what someone’s ownership levels will be, but you can get a pretty good idea of what percentages the chalk plays are likely to be if you know where they have been at that buy-in level in the past.
In order to do a smart analysis, we need to try to think about how every individual DFS player in that prize pool will handle the situation. Then we can aggregate that into a total projected ownership. Despite knowing that chalk is less likely to bring you a big score, people still roll out chalk plays at every buy-in level on every night knowing that those guys will be highly owned. To do the analysis you do not have to assume everyone is thinking about it and trying to maximize their chance of winning. In fact, knowing that a high percentage of people are doing the latter makes ownership rates more predictable for those of us who are thinking about things from a Game Theory prospective. You have to analyze your choices based on the competition and what you perceive they will do.
We know more low stakes players tend to be chalky with lineup construction, so it makes sense to go off the board in lower buy-in tournaments to a second or third option that should be much lower owned compared to the chance of success. With higher stakes players moving to the second and third best options more often, it actually is more beneficial to play more of the chalk at those levels.
Remember that we look at the risk vs. reward argument to guide us in Game Theory. The projections for any player do not change because we play at a higher buy in, but the risk vs. reward equation does if the ownership is different on those levels. A guy may not be a good play if you feel he will be 40% owned in a Moonshot, but he might be the top play if you felt he was going to be 15% owned in the Perfect Game on the same day.
How Does Game Theory Make Me Money?
Here is the entire argument summed up in a few paragraphs…
Most payout structures are top heavy. In order to make more than just double your money, you need to finish in the top 5% assuming an average pay structure where that would be at least three times you buy in with the upside of five to fifty times your buy in if you land in the top 2%. If only 20% of rosters cash in most tournaments, normal distribution would predict that one in five of your rosters returns some money to you.
Now, skilled players probably cash more often than that, but let’s assume normal distribution for the sake of the illustration here. That means if the best you do when you cash is sneak into the back end of the money for double or triple your buy ins, you would be a losing player since you played one unit five times and only returned two or three units back from the total of five you wagered.
Without catching a big score every now and then, you would eventually go broke following this pattern. The most likely scenario we discussed above is that playing chalk will leave you trapped in this vicious cycle. You may wind up cashing more often this way, but it is much tougher to cash highly when so many other people are also on the same plays. You have to outscore 40% of the total pool with your other players to make it to the top and that is very tough to do when so many rosters with so many different combinations of guys exist that also have the same top scoring chalk play you do.
While taking a lower owned play with a lower probability of success will not lead to cashing as often, the position you finish when you do cash is likely to be closer to the top. You only need a roster that has better plays than a smaller percentage of people who also had your lower owned top play to win a tournament now. You may cash less often, but you are more likely to have the big cash when you do. One big cash for 50-100 times your buy in is enough to bankroll your play for a long time. It gives you a chance to not only make a big score, but to become a profitable player over the long haul.
It is tough to convince someone that it is in their best interest to not play the most obvious plays. When you are putting actual money on the line, it just feels right to take the most obvious guys who are most likely to score the most points. Game Theory tells us this is the wrong way to look at things.
Our goal should be to try and put the guys in our lineups that give us the best chance to win it all. That does not mean we need to fade every chalk play, but it does mean that the more chalk we play the more competition we will have for one of those top prizes with the smaller margin for error. The top options are likely to pan out more often, but finding another group of players that can outscore the chalk is more likely to produce a big payout.
Game Theory is the study of strategic decision-making and therefore the teachings dictate what actions each player should take to produce the best possible results for themselves. Broken down in the simplest of terms for DFS, you are more likely to take down a huge pot by playing against the chalk and therefore it is the optimal strategy for GPP tournament plays.