Game Theory, Draft Strategy, Equilibrium and the 4for4 Advantage

Game Theory, Draft Strategy, Equilibrium and the 4for4 Advantage

By Kevin Zatloukal (4for4 Contributor), last updated Aug 26, 2014

Kevin Zatloukal's picture

Kevin is a Ph.D. computer scientist. His doctoral work at MIT was focused on quantum algorithms. During the fantasy offseason, he teaches computer science at the University of Washington, his alma mater.

Follow Kevin on Twitter: @kczat.

Drafting is one of the most exciting parts of fantasy football and also one of the most challenging. Unlike the weekly task of choosing which players to start, where each manager can make an optimal decision in isolation, the fantasy draft pits managers directly against one another.

Weekly projections from 4for4 make it easy to know which players to start and even the best streaming options. When it comes to drafting, though, player projections unfortunately aren't enough. Simply choosing the available player with the highest projection is unlikely to get you the best possible team. To make the best decisions, you need to consider not just who is available in the current round but also who will be available later. And that depends on the decisions made by your opponents.

You could assume players will be taken at their average draft position (ADP) from recent drafts. But since you are not planning to simply take the player with the highest ADP for each pick, why should you assume your opponents will?

Perhaps your opponents all plan to take running backs early (say, RBx3). In that case, your best response might be to wait on RBs (the "Zero RB" strategy). But what if your opponents all plan to take receivers early? In that case, your best response might actually be RBx3. And what about quarterback and tight end and defense? How should you respond if your opponents take the elite options early? How should you respond if they don't?

These sorts of questions demonstrate the challenges of the fantasy draft. Your best choice in early rounds depends on who will be available in later rounds, which depends on the choices made by your opponents. If you knew your opponents' strategies, then you could determine your best response. Likewise, if they knew your strategy (and each others), then they could determine their best response. Unfortunately, none of us knows our opponents' strategies.

How can we untangle this knot? Fortunately for us, some of the cleverest people who lived in the 20th century, like the mathematicians John von Neumann and John Nash, wrestled with these questions and came up with solutions. (Sadly, von Neumann passed away long ago, but Nash is still with us. I strongly suggest you avoid him in MFL10s.) Below, we will discuss some of their insights.


Game Theory in Science

John von Neumann studied the mathematics of games in the early 1900s, founding the area now known as "game theory". His key insight was that many games have what is called an "equilibrium". This is a set of strategies for the players with a special property: each player's strategy is the best response to the strategies of the other players. In other words, if each player believes the others are playing these particular strategies, then they have no incentive to change to any other strategy — it would give them no better results.

What does this tell us about strategies used by real players? Suppose that the game is being played repeated or that players are practicing many times before the real game (e.g., mock drafting). And suppose that, each time they play, the players choose new strategies in response to the strategies they saw from their opponents previously. In many cases, it is possible to prove mathematically that the players will eventually choose the equilibrium strategy. And in any case, we can be certain that if at any point the players choose the equilibrium strategy, then they will stick with that strategy from then on because no player can do better by switching to another strategy.

Game theory is not used exclusively (or even primarily) for studying games, however. One of the first applications, made by von Neumann himself, was to the study of economics. When an economic situation can be modeled as a game, then using the same reasoning as above, economists usually will predict that the real players in the economy will play the equilibrium strategies. And in most situations, they have found these predictions to be correct.

Game theory is used in other sciences as well, such as political science and psychology. However, in my opinion, its most interesting applications are in biology.

In biology, scientists usually do not think of games as being played by animals but by their genes. Each gene encodes a strategy into the DNA of the animal. As in any game, the success of that strategy depends on what strategies are played by all the other animals (i.e., on the genes of all the other animals). The genes that are successful in this generation will have more offspring, and those genes will then be more prevalent in the next generation. In other words, we think of each generation as a new game, where the strategies played in the current game will depend on which strategies were most successful in the previous one, just like we said above.

One of the most famous examples of game theory in biology is understanding why nearly all species have children that are male 50% of the time and female 50% of the time. To understand this, suppose that some species of animal has three types of genes: one gene gives 90% odds of having male children, one gives 50% male, and the last gives 10% male.

If the previous generation all had the first of these genes, then the current generation would be 90% male and 10% female. In that case, many males will not be able to reproduce because there are so few females. (To keep things simple, let's assume that each animal mates only a couple times in its life. So suppose that these are either insects with very short life spans or fantasy football writers.) This means that the gene that gives 90% female children will be very successful, the gene that gives 50% of each will be mildly successful, and the gene that gives 90% male children will be unsuccessful. We can see that this strategy is not an equilibrium: the players would be better off switching to the 90% female strategy.

On the other hand, if the previous generation all had genes for 50% male / 50% female children, then the current generation would be 50% male and 50% female. In that case, all the strategies would be equally successful. That means this strategy is an equilibrium, and there is no evolutionary pressure to change strategies. For that reason, biologists are not surprised that nearly all species in nature use this strategy.

My favorite example of game theory in biology is lizard mating strategies. This species of lizard has genes for three different mating strategies. The first gene produces "ultadominant" males that defend a large territory from other males and mate with all the females in their territory. The second gene produces "dominant" males that defend a small territory with only a single female mate. The third gene produces "sneakers", males that disguise themselves as females and sneak into the territory of the ultradominant males to mate with females.

The ultradominant strategy is successful against the dominant strategy but unsuccessful against sneakers. On the other hand, the dominant strategy is successful against sneakers because they can easily recognize the one female they live with. (The lizards only need to recognize their mate, not remember their birthday.)

Mathematically, this game is equivalent to one we are all familiar with: rock/scissors/paper. Each strategy beats one strategy and loses to another. Unlike the case we just saw, this is a game with no equilibrium. Whatever the opponent's strategy, you are better off switching to the one that beats theirs. (And if you are already using that strategy, then your opponent would be better off switching to the one that beats yours.)

So what do biologists predict will happen in this situation? They predict that the strategies will cycle: if the ultradominants are most prevalent in one generation, then the dominants will be most prevalent next, followed by the sneakers and then by the ultradominants again. When biologists watched the lizards in nature, they found that this is indeed what occurs.

In summary, the concept of an equilibrium in game theory has been surprisingly successful at understanding both the games you and I play and those that arise in economics, biology, and other sciences. When an equilibrium exists, we often find that real players are using that strategy, and when one does not exist, we find other behavior like the cycling of lizard mating strategies.

Now, let's see what this can tell us about fantasy drafting.

Non-Equilibrium Draft Strategies

We'll begin by reviewing a couple of popular fantasy football draft strategies.

First, consider the strategy of drafting a quarterback late (to be concrete, say, in the 8th round or later). Could this be an equilibrium strategy? Well, suppose you knew that all of your opponents were waiting until the 8th round to draft a quarterback. Would you wait until the 8th round as well? No! The best response for you would be to take P. Manning or A. Rodgers in the 7th round. Hence, we can conclude that an equilibrium strategy cannot always draft a quarterback (that) late.

Next, consider the strategy of drafting only running backs for the first two or three rounds (a strategy that has been popular in past years). Could this be an equilibrium strategy?

In order to answer that, we need to know the best response when all your opponents are using RBx2 or RBx3. And one way to find the best response is just to try them all and see which is best. For each possible choice of positions to draft (over the first 8 rounds), we can simulate a draft against 11 opponents that always take running backs up until sometime in the 3rd round. (After that, the opponents will use a basic value-based drafting strategy.) At the end of each draft, we get a roster of players, which we can evaluate using 4for4 projections.

What we find is that the optimal strategy is to avoid running backs all together until the later rounds. (The sole exception is that, if we have one of the first 4 or 5 picks, then we should take a running back with that pick. But afterward, we should avoid running backs until late.) In other words, the best response to opponents with RB-heavy strategies is the "Zero RB" strategy.

This tells us that RBx2/RBx3 cannot be an equilibrium strategy. Although I didn't perform this calculation, we would expect that the best response when our opponents are all starting WRx2 or WRx3 is to avoid WRs until late (the "Zero WR" strategy). Hence, that also cannot be an equilibrium strategy.

The strategies just considered — late-round QB, RBx3, and WRx3 — have many proponents. Yet, none of these is an equilibrium. Does that mean we are stuck in a rock/scissors/paper game, where each year the popular strategy is the best response to last year's popular strategy? (Last year it was RBx3; this year it is Zero RB; next year it is who knows what?) Below, we will discuss the evidence that this is not the case, evidence that there is in fact an equilibrium draft strategy.


The Equilibrium Draft Strategy

In our discussion of draft strategies above, we skipped the class of strategies that are probably the most widely used: value-based drafting (VBD).

As a reminder, here is how value-based drafting works. We start with projections for the fantasy points to be scored by each player, which we can get from 4for4. Then, rather than simply ordering players by their projected fantasy points, we do something slightly more clever.

For each position, we choose a "baseline", usually the number of points to be scored by some player at that position who is drafted in the late-middle rounds. For each player, we compute their projected fantasy points minus the baseline for their position, which is called their "value". Now, we simply order players by their value and, with each of our picks, take the available player with the highest value. This is how the 4for4 Top-200 tool, with the baselines calculated based on your roster configuration.

The logic of value-based drafting is straightforward: the worth of each player is not how many points they will give us over the season but rather how many more points they will give us than the alternatives available late in the draft. For example, the value of each quarterback is how many more points they offer than a late-round quarterback.

In some sense, the option of taking a late round quarterback is built into VBD. Unlike a strict late-round QB policy, VBD adapts to the situation. If your opponents over-value elite quarterbacks, then VBD will end up taking a QB late. However, if an elite QB drops enough, then VBD will advise you to take him.

I said above that VBD is probably the most widely used strategy. That claim is based mostly on anecdotal evidence. However, ADP data also has some interesting statistical properties that are consistent with VBD. In particular, historical ADP data shows that players are taken in such a manner that average points vs ADP graph for each position has approximately the same slope. This is exactly what would occur if all drafters were using VBD. More on this later on.

To determine whether an equilibrium exists amongst VBD strategies, I performed simulations to find the outcome with drafters using a wide range of different baselines at each position. (Each choice of a set of baselines constitutes a strategy, in game theory terms.) I tried this out for both 10- and 12-team leagues, using both standard and 0.5 PPR scoring, and with different roster configurations. In each case, there was one equilibrium VBD strategy. That is, there was one choice of baselines such that no drafter can do better by switching to any other baselines.

As it turns out, though, this strategy is not just an equilibrium amongst VBD strategies. No drafter can improve by changing to any fixed-order strategy (like RBx3). No drafter can improve by always taking a QB late, nor by avoiding running backs until the late rounds (Zero RB). In other words, the equilibrium VBD strategy is an equilibrium over all of the draft strategies we have considered, which gives us good reason to think that this is truly an equilibrium over all draft strategies.

What are the baselines for the equilibrium VBD strategy? They are essentially the worst starter baselines, give or take a few at each position. This is similar to the baselines used in the 4for4 Top-200 tool. [1]

When all drafters use this strategy, they all get reasonable, if unspectacular, results. But since it is an equilibrium, no individual drafter can improve their results by changing to another strategy,

As an aside, the simulations also show what situations achieve the largest advantage for you over the other drafters: instead of playing the equilibrium strategy, your opponents try to load up on the same position (e.g., WRx4). You follow their lead with your first pick but then, for the next few rounds, pick up the values at all the other positions. This can lead to over 10 more projected points per game. On the other hand, if your opponents play the equilibrium strategy, then your best response is to do the same, which will give you similar results to the other drafters.


An Example of the Equilibrium Strategy

To see the strategy in more detail, let's look at the equilibrum for a common league setup: 10 teams, standard scoring, and a roster with 1 QB, 3 RBs, 2 WRs, and 1 TE. (We'll ignore DST and K.) This should also be very similar to a league with rosters allowing 2 RBs, 2 WRs, and 1 flex, when standard scoring is used.

The simulations mentioned above tried hundreds of different strategies, but here, we will focus in on some familiar ones. The following table shows the average starter points for five different strategies. The row indicates which strategy you are using and the column indicates which strategy (all) the opponents are using. In each column, the bolded row shows your best response against the opponents' strategy.

  Equilibrium Late QB Mid QB RBx3 Zero RB
Equilibrium 1374.6 1384.8 1382.7 1457.5 1477.0
Late QB 1367.5 1378.4 1365.4 1428.5 1475.7
Mid QB 1367.7 1390.1 1373.8 1428.0 1475.7
RBx3 1361.0 1378.1 1361.4 1373.9 1476.9
Zero RB 1347.2 1351.1 1349.9 1423.3 1376.3

The first thing to notice is that, in the first column, when the opponents are playing the equilibrium strategy, your best response is to also play the equilibrium strategy. This demonstrates that the equilibrium strategy is what it purports to be, an equilibrium for the game.

When drafting from the third slot, for example, here is a typical lineup chosen by the equilibrium strategy:

QB M. Ryan    
RB A. Peterson G. Bernard J. Bell
WR J. Jones V. Jackson  
TE D. Pitta    

In this slot, the equilibrium strategy ends up with a late-round QB, like this, about half the time (more on this in a moment) and a late-round TE, like this, about half the time. We can see that the strategy balances value at all the positions, taking both quality RBs and WRs early and taking elite QBs and TEs early when they are available.

Looking back at the matrix of results above, the second and third columns show the results when the opponents are playing the late-round QB and mid-round QB strategy, respectively. This version of the the late-round QB strategy prefers to take a QB in the late-middle rounds but will not allow P. Manning to slip beyond the 3rd round. The mid-round QB strategy is the same but doesn't allow Manning to slip beyond the second round. The equilibrium strategy, on the other hand, will take Manning at the end of the first round. (Don't read too much into this, however, since we are only using starter points to evaluate strategies. If we put some weight on backup RBs and WRs as well, then Manning would likely drop to the end of the second round.)

When the opponents are playing the late-round QB strategy (the second column), we can see we are better off playing a mid-round QB strategy, taking Manning one round earlier. In fact, we are even better off taking Manning in the late first round (the equilibrium strategy), but since Manning will still be available in the second round, we would expect that it is better to wait until the second round to take him. The table shows that this is indeed the case.

This example shows that the equilibrium strategy is not the best response against all possible strategies. It is the best response in 4 of the 5 cases shown here, but remember that the definition of an equilibrium is simply that the strategy is the best response when the opponents play that strategy as well. It need not be the best response in all cases, though the table here shows that it generally works quite well.

When drafting from the third slot, as before, both the mid-round QB and equilibrium strategies give the same results when opponents play the late-round QB strategy. Here is a typical lineup for both strategies:

QB P. Manning    
RB A. Peterson R. Jennings L. Miller
WR A. Brown K. Allen  
TE K. Rudolph    

Because the equilibrium strategy will take A. Peterson with the first pick and its second pick isn't until the late second round, both strategies end up taking Manning in the second round. In the 9th or 10th slot, however, the equilibrium strategy would take Manning a round earlier instead of an elite WR or TE. As the table above shows, that is a bad trade-off.

In the second column of the table above, we saw that the mid-round QB strategy is the best response to the late-round QB strategy. In the third column, we see that, when the opponents play the mid-round QB strategy (Manning in the second round), our best response is to play the equilibrium strategy that takes him even earlier.

When we are drafting from the third slot, the mid-round QB strategy will have already taken Manning about half the time. However, the equilibrium strategy can still outperform the mid-round QB strategy even at this slot by taking the next elite QB (A. Rodgers) slightly earlier then the mid-round QB strategy. Here is a typical lineup for the equilibrium strategy when the opponents play mid-round QB:

QB A. Rodgers    
RB A. Peterson B. Sankey L. Miller
WR A. Jeffery V. Jackson  
TE V. Davis    

Given these results, we might wonder whether we can beat the equilibrium strategy by taking Manning even earlier than the late first round, say, in the middle of the first round or maybe with the first pick! However, since the equilibrium strategy is an equilibrium, we know that this would not do better. Why not? If the opponents all take quarterbacks earlier than the equilibrium strategy, then the equilibrium strategy will end up with a late round QB. The simulation results show that the extra value picked up at the other positions will more than make up for the loss at QB. In other words, the equilibrium strategy takes QBs at just the right point: if opponents take QBs any earlier, we win by taking a late round QB, and if the opponents take QBs any later, we win by taking an elite QB.

Lastly, let's focus on the last two rows and columns, which contain the RBx3 and Zero RB strategies. Here is what the table above showed for these strategies:

  RBx3 Zero RB
RBx3 1373.9 1476.9
Zero RB 1423.3 1376.3

We can see that, when the opponents all play RBx3, we are better off playing Zero RB: take all the elite options at the other positions while the opponents load up on RBs. On the other hand, if the opponents all play Zero RB, then we are better off loading up on RBs. In short, we are always better off playing the strategy not played by your opponents, an example of what game theorists call an "anti-correlation game".

That said, even though Zero RB seems like the ideal response to RBx3 opponents, looking back at the fourth column of the larger table above, we see that the equilibrium strategy actually does even better. To see why, let's look at the picks made by the Zero RB strategy. Drafting from the third slot, here is a typical lineup when opponents play RBx3:

QB P. Manning    
RB L. Miller B. Tate M. Jones-Drew
WR C. Johnson J. Jones  
TE J. Graham    

As expected, the Zero RB loads up on elite talent at QB, WR, and TE in exchange for later round options at RB. Now, let's see a typical lineup for the equilibrium strategy when opponents play RBx3:

QB P. Manning    
RB A. Peterson B. Tate M. Jones-Drew
WR C. Johnson B. Marshall  
TE J. Cameron    

The equilibrium strategy starts by taking Peterson in the first round. As a result, it doesn't get Graham, instead getting Cameron. Based on the projections, this is a good tradeoff. In other words, by eschewing RBs, even with the first 4 picks of the draft, the Zero RB strategy is throwing away value. (After the first 4 picks, the two strategies should behave almost identically against RBx3 opponents.)

As we can see from these examples, the equilibrium strategy takes a balanced approach, valuing all positions equally. If opponents eschew some positions in favor of others, then the equilibrium strategy responds by taking the value at the other positions, in most cases giving optimal results. And if our opponents also play this strategy, then they leave nothing on the floor for us to pick up, so we can't do any better than to just follow suit.

What is this magic strategy? As mentioned above, it is VBD with some set of baselines. For this particular league setup, the baselines chosen were 300, 130, 135, and 100 points for QB, RB, WR, and TE, respectively, which is essentially the worst-starter method. (As mentioned above, when we take into account the value of backups, we would likely want to lower the baselines for RB and WR.)


Do Real Drafters Use the Equilibrium Strategy?

If we were scientists, now that we believe an equilibrium strategy exists, we would likely predict that real drafters would be using this strategy. Is that what we see?

Even though we don't know the projections used by the average drafter, it does seem clear that sharp ADP data is broadly consistent with VBD using something like a worst starter approach. Furthermore, as mentioned above, historical ADP has the statistical properties we would expected from VBD. Finally, the fact that ADP data for standard and PPR leagues are fairly similar is not only surprising but also what we would expect from a VBD approach.

Another way to test the idea that average draft positions come from applying VBD ranking to some (unknown) projections is to reverse engineer projections from ADP data and see if the results are sensible. In particular, if we assume that the projections for each rank at WR are in line with the average points for that rank over the last 3 seasons [2] and we assume 3-year average points for worst starters at the other positions, then we can compute what the projections must have been for each QB, RB, and TE in order to produce the (known) average draft positions using VBD.

I performed this experiment using the ADP data shown on 4for4. The results were positive. Though the projections produced in this manner were probably nowhere near as accurate as those available from 4for4, they look like something that which a reasonable person might believe. In particular, they accurately reflect some of the peculiarities of these particular players: the top 3 QBs are a big step above other QBs, the top 3 TEs (and particularly, Jimmy Graham) are a big step above other TEs, and so on. So it seems fair to say that ADP data could have been produced by applying VBD to a reasonable set of projections.

All together, then, there is plenty of evidence to believe that the average drafter is using something close to the equilibrium strategy.

It is worth pointing out, however, that ADP is an average over a large number of drafters. Individual drafters may use all sorts of different strategies, which may be far from the equilibrium strategy. It is only the average ("wisdom of the crowd") draft positions that are consistent with the equilibrium strategy. Also, ADP data at any specific fantasy draft site is hugely influenced by the default rankings of that site. In other words, not all ADP is created equal.


Bottom Line: The 4for4 Advantage

In this article, we have seen evidence that there is an equilibrium draft strategy, one where no drafter can get better results by switching to any other strategy. We have also seen evidence that real drafters are using the equilibrium strategy or one similar to it.

If we were scientists, this would be a very pleasing result: game theory leads us to make a prediction about real fantasy drafters which is backed up by the data. (*cough* Nobel *cough*) However, as fantasy drafters ourselves, this result seems slightly depressing: if real drafters are playing the equilibrium strategy, then there is no way for us to gain an advantage, at least not through our draft strategy.

Any analysis of something as complicated as draft strategy will have shortcomings, and the analysis above is no different. However, there is one shortcoming I want to address here.

The simulations described above assumed that all parties had equal information about draft prospects. In particular, all drafters were using 4for4 projections. In reality, not all drafters use 4for4 projections, which have been demonstrated to be substantially more accurate than even the average expert's projections.

Thus, a 4for4 subscriber goes into a draft with an informational advantage. The current projections show many instances where players are likely to be available two or more rounds after when they should be taken. Such an opportunity can easily amount to 1-2 fantasy points per game. Getting a handful of such players would add up to an enormous advantage, nearly as much as what we would get if we could talk our opponents into loading up on WRs while we get all the value in the other positions.

The moral of the story is this: don't expect to beat your opponents with a clever draft strategy; expect to beat them with a basic VBD strategy and the information advantage you get from 4for4 projections.



[1] The simulations introduce randomness into each manager's baselines in order to smooth out any peculiarities in the data. This has the result of washing out differences of a few positions in the baselines, so these results can only narrow down the equilibrium to within that range.

[2] I used wide receivers since per rank points were more consistent for WRs than other positions.

Filed Under: Preseason, 2014

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