# A Beautiful Lineup – Nash Equilibrium Lineups Part II

*What follows is more great work from Ari Caroline ( @aricaroline) with his brilliant analysis on lineups. You can catch part 1 here if you missed it.*

**Previously… on Lost**

Hopefully, not all of you were completely lost reading my last post in which we started to introduce Nash Equilibrium Lineups. I do hope to tie up loose ends a bit better than the show did but realistically, we still won’t cover everything in this “episode”.

For those of you for whom this is too abstract or too technical, I’d suggest that you just jump to the conclusions at the bottom of this post. Take note, however, on your way down, of all the colorful interesting charts that you’re missing.

Where we left off:

- There is some measure that we are calling Floor Stretch (made up of speed, ball circulation and some other “intangibles”) that is not fully covered in the box score stats.
- This Floor Stretch measure underlies the value of small players and justifies their inclusion in the lineup despite, on average, having lower raw productivity numbers (ADJP48) than the bigger players.
- These two factors, Floor Stretch and ADJP48, must be balanced against each other by the coach when considering which types of players to include in a lineup.
- Floor Stretch is only important when compared to the other team’s lineup. Since it has to be weighed against decisions by the other team, this brought us into the realm of Game Theory.

Now, if we can figure out a range for what the value of Floor Stretch actually is, position by position, we can use the payoff matrices that we introduced last time to see if there are any Nash Equilibrium around basic lineup types. Given that it is so popular, we’d expect there to be an equilibrium around the classical PG, SG, SF, PF, C lineup. Otherwise, why does everyone continue to stick with it, right?

**The Assumptions**

In a theoretical exercise like this, we usually simplify things by making some important assumptions. Once we’ve established the basic principles, we can try to relax those assumptions and see if the theory still holds. With that…

**Initial Assumptions:**

- Point guards have the highest Floor Stretch value and centers have the lowest.
- We’ll use the centers as our base, giving them a value of 1.00, so the value of the others is as a percent increase on the base. Thus, if an average point guard offers a Floor Stretch value of 1.25, it means that they are 25% more valuable than the centers.
- Floor Stretch increases linearly as the traditional positions get smaller, so PG>SG>SF>PF>C
- This will be the first assumption that we drop, using the “Fiona Rae Floor Stretch Variation”.
- Floor Stretch sums up in the same way that Wins Produced does. So if we have two point guards playing and they each have a Floor Stretch value of 1.25, the cumulative effect is 2.5.
- You could argue that Floor Stretch should have non-linear interaction effects but, for now, don’t.

**The Math: How I’m calculating total team strength and why**

I’m calculating the overall team strength resulting from a particular lineup as the sum of all the raw productivity (ADJP48) multiplied by the difference between the summed Floor Stretch values of the two teams. In mathematical terms:

In short, a team’s raw productivity is either enhanced or penalized based on how their Floor Stretch stacks up against the other team’s Floor Stretch. I chose this calculation as a natural extension to Prof. Berri’s Wins Produced algorithm. We’ve established many times on this blog the strength of Prof. Berri’s algorithm in predicting outcomes and how much can be directly derived from the box score stats. Therefore, instead of trying to add another value element, I’m adding a modifier. If your team has a Floor Stretch advantage of, say, 0.25 as we calculate it, your raw productivity is enhanced by 25%. If, however, you are at a Floor Stretch disadvantage of the same value, your raw productivity is penalized by 25%. Obviously, we have to add some adjustments to account for extreme values. On average, however, the net impact of Floor Stretch on the total ADJP48 across all teams will net to zero.

**A couple more quick definitions and a refresher on the payoff matrices**

In Game Theory, a **dominant strategy** is one that is best for a competitor NO MATTER WHAT the other competitor does. A **dominated strategy**, by contrast, is one that is inferior NO MATTER WHAT the other competitor does.

I don’t want to repeat everything from the first installment but here is a quick refresher on how to read the payoff matrices.

**Now to Floor Stretch!**

Let’s consider two fictional teams – we’ll call them the *Frequentists* and the *Bayesians* – and see how we might estimate a range of values for Floor Stretch. You may think that a made-up, theoretical variable like Floor Stretch could have an infinite range of values. However, our friendly payoff matrices can show that not to be the case. For example, if we assume that an average PG has a Floor Stretch value of 1.34, we’d get the following payoff matrix:

In this example, Floor Stretch is so valuable that the equilibrium falls out on a lineup of just about all PG’s!

Similarly, if we assume Floor Stretch to be too low for an average PG (we use 1.14 in the example below), we get an equilibrium lineup of all Cs!

**Monta Ellis, Dominated!**

The astute observer will notice that I simplified things a bit more in the above matrices, allowing me to use only PGs and Cs. The average ADJP48 that I used (which are real historical numbers!) have lower values for the SGs and SFs than the PGs. Since we’re assuming that PGs have the highest Floor Stretch, that means that **playing an average SG or SF is a dominated strategy! **They have lower ADJP48 scores *and* lower Floor Stretch scores. So, if they’re just average, we’d rather drop the position entirely!

For the sake of simplicity, I also assumed that the Cs and PFs are roughly interchangeable. Using the values that we are working with here, that assumption seems to be “close enough”.

**Goldilocks Floor Stretch**

Now, since every bit of intuitive logic and accumulated experience tells us that lineups made up entirely of centers or entirely of point guards just wouldn’t work well, we know that the Floor Stretch value that we are looking for has to be between the two values that we’ve tried above. In fact, smack dab between them (1.24 for a PG) seems to work quite well with our intuitive sense.

The equilibrium now falls just where we’d expect it to, with a reasonable mix of bigs and smalls.** **

**Ok, but what about Dwayne Wade or Manu Ginobli??**

The assumptions that we’ve made until now center on the **average values** for each position. You’d think, however, that there must be some ADJP48 value such that we could justify a lineup that is a bit more diverse than just centers and point guards. And you’d be right!

However, to arrive at such equilibrium, you’d need some very productive SGs and SFs. I’ve tweaked the numbers below to make them conveniently round but, even so, (given all of our assumptions etc.) a SG has to demonstrate value that is more than 40% greater than the average SG to earn a spot in the lineup!

You can see this looking at the column labeled “Position Upgrade”.

Using these revised ADJP48 scores, we have four, roughly equivalent equilibrium lineups, two of which include SGs, three include SFs, and one which is our traditional PG, SG, SF, PF, C lineup.

**Who makes the cut?**

A natural follow-up question to this last conjecture would be; how many SGs and SFs demonstrate sufficient productivity, by this measure, to justify their position? The answer is that there are a few.

Let’s start with the SGs:

I’m going to leave it to Arturo to correlate the shot selection of these guards with their productivity.

There are also around 10 SFs per year that surpass our more rigorous threshold:

**To recap:**

- We can use our common sense of how the game works to calculate a theoretical range for the value of Floor Stretch
- A “Goldilocks” value for Floor Stretch will yield a Nash Equilibrium for lineups that aren’t all bigs and aren’t all smalls

- As long as an average SG or SF has a lower ADJP48 than a PG and also a lower Floor Stretch value, playing these positions is a dominated strategy and should be discarded.
- There are theoretical ADJP48 thresholds, however, for these positions that would yield Nash Equilibrium lineups that do include these positions
- Only the top tier of SGs and SFs surpass this threshold

**Coming soon to a blog near you…**

We’ll start to relax some of our initial assumptions. What if the hybrid driving ability of SFs actually gives them greater Floor Stretch than the SGs (the aforementioned “Fiona Rae Variation”)? What if Floor Stretch isn’t just additive?

We’ll look more at dominant strategies. How complicated should this be for coaches to implement?

Eventually we’ll get to some really interesting questions. Can we calculate an overall optimal lineup mix (looking at Mixed Strategy Nash Equilibriums)? Can we calculate an actual value for Floor Stretch from the available data? If so, shouldn’t it vary player to player, not just position to position? Can we use this to refine our Wins Produced algorithm? Can we calculate a Wins Produced score for coaches (look for more on this from Arturo)?

Stay tuned…

-Ari