# Ari Caroline Introduces Nash Equilibrium Lineups

*The following is some amazing new work from Ari Caroline (our spiffy new template doesn’t make it obvious immediately who the author is). Despite his humility to start the post, don’t be fooled. Ari has put in some amazing work to solving basketball including this. – Editor*

This post is the product of some great dialogue between all of the regular Wages of Wins contributors. I served as the catalyst for the discussion, so I’ve got the writing duties:

## Empirical vs Theoretical

As data scientists, we are pretty easily pigeon-holed as empiricists. We define a problem, gather data

to help diagnose the underlying drivers, analyze, and, to the extent possible, try to let the data tell their own story. Personally, however, I think I’m more of a theorist at heart. I love exploring hypothetical questions not easily found in the data and imagining what basic natural truths would hold sway if the hypothetical became reality.

One of these hypothetical questions emerged from the discussion with Mark Cuban in the comments

following an earlier post. That discussion kind of flew all over the place, but it prompted me to ask

myself about the inherent trade-off between bigs and smalls and what prompts coaches to play the

curious mix that has become the standard NBA lineup.

## Productivity by Position

If we were to look at pure productivity based on the Wins Produced algorithm, using Arturo’s charts

below, you can see that the big players account for a far high % of a team’s productivity:

Arturo has also demonstrated that the average raw productivity at each position is fairly consistent over time:

The one major exception seems to be the jump in productivity of point guards in the 80s. My first instinct was to attribute this all to Magic, but Arturo said that it probably had more to do with the introduction of the three-point shot.

## Why Play Smalls?

One might conclude from the above charts that the optimal strategy should be to play five centers. Of course, we all have an intuitive sense that tells us that wouldn’t be a very good strategy. While teams like the Lakers have had success playing two to three centers simultaneously, playing five centers seems ludicrous. Another team with a couple of fast guards could run circles around the slow bigs and dominate the game. To quote Dr. Berri from a note that he sent during our discussion:

I think teams discovered early on that smaller (i.e. quicker) players could advance the ball quickly down the court. Later on (and this did come later), teams discovered that bigger players had an easier time scoring. Consequently, teams began adding bigger players. But teams could never completely substitute big players for little players because the first function — advancing the ball across the court — has never gone away. Someone has to advance the ball, hence the need for little guys.

One issue in basketball is that scoring has always dominated perceptions. So although teams knew it would help if the little guys would pass the ball to the big guys, the little guys had an incentive to keep the ball and shoot themselves (think Isiah Thomas, Stephon Marbury, Derrick Rose, etc…) I think this issue led teams to start tracking assists (in an effort to give the little guys an incentive to pass).

## Floor Stretch

I also postulated that the smalls could have a somewhat more expansive role. At least in theory, the smalls could amplify the impact of the bigs by spreading out the defense and by pulling them away from the basket. In this sense, anything that spreads the defense (passing, outside shooting, rapid ball movement etc.) would serve to enhance the most basic basketball functions of getting the ball and getting it into the basket as frequently as possible (ignoring defense for the moment). With Dr. Berri’s forgiveness, I’ve decided to lump speed together with all of these other (hypothesized) factors under the general heading of “Floor Stretch”. We’ll use it for an exercise in theoretical sports economics.

Whatever it is that truly makes up “Floor Stretch”, it has to be sufficiently valuable that it offsets the lower raw productivity of the smaller players. Otherwise, we would just have an arms race (legs race?) over who could put out the bigger lineup. It also can’t be too valuable, or else we would revert back to 1940s basketball.

In the Wins Produced algorithm, we take the raw productivity score, ADJP48, and subtract an average base by position. This allows us to say how much more (or less) productive a particular player is than his counterparts at the same position. The implied assumption is that the ** position itself is valuable**. However, that assumption of inherent value in the smaller positions depends heavily on our hypothetical “Floor Stretch” variable. Thus, the assumptions that we make about this hypothetical variable will greatly influence what truly makes an optimal lineup.

## Game Theory and the Nash Equilibrium

The final Wins Produced measure, WP48 (or, more recently, PoP48), has very little Game Theory associated with it*. Whatever the opposing team does, the optimal strategy is to maximize your team’s productivity on the floor. Floor Stretch, however, is really a relative function. Having 5 point guards on the floor only stretches the other team if they don’t also have 5 point guards playing. In this sense, what we really care about is the ratio of Floor Stretch between the two teams competing. Theoretically, the Floor Stretch ratio is what the raw productivity (ADJP48) must be balanced against in order to determine the best mix of players. This, then, gets us into some classical Game Theory.

Thanks to the great mathematician/economist John Forbes Nash (featured in *A Beautiful Mind*), we have a theory for how competitors can reach a steady-state equilibrium in such a contest. If there is a position in which neither player has an incentive to move in either direction, the players will settle at that position until the circumstances change. Let’s use a quick example.

In the above example, various lineups for two teams (A and B) are arrayed against each other in what we call a *payoff matrix*. The projected outcomes for each match-up are scored with positive (red) scores indicating that Team A has an advantage and negative scores (blue) indicating an advantage for Team B. (For now, don’t worry what the numbers themselves mean. We’ll get to that later.) The Nash Equilibrium is the point at which neither team has an incentive to switch to a different lineup as switching will result in an inferior outcome for the team that switches.

In this case, both teams will settle at Lineup 5. As you can see, if you are coaching Team A and both you and Team B puts out Lineup 5, neither team has an incentive to switch. Right now, the two lineups are equivalent, so the net advantage for either team is 0. Looking up and down at the other lineups that you could put out, all of the alternatives put you at a disadvantage relative to your current position. If you try to switch to Lineup 6, your expected outcome is -0.02, meaning that Team B has a slight advantage. Other lineups make the situation even worse, particularly at the extremes. If you put in Lineup 1, your expected outcome drops all the way to -0.14! Team B, though, is in the same predicament. Looking left and right at their alternatives, all are inferior to the lineup that they already have on the floor. The result is that everyone sticks with what they have.

Now, this doesn’t have to occur at a point that is equally advantageous to both sides. Consider the following payoff matrix where the teams are considering different sets of lineups:

In this example, the Nash Equilibrium occurs when Team A plays Lineup A5 and Team B counters with Lineup B6. Team A is at a distinct advantage in this contest (+0.04) but they are still at an equilibrium. All of Team A’s alternative lineups make their situation worse and the same is true with Team B. Clearly, the only way for Team B to get the advantage in this contest is for Team A to screw up and play Lineups A1 or A2 on top or Lineups A8 or A9 at the bottom. If Team B then counters with Lineup B6, they will have the advantage.

## So where are you going with all this?

We’ve established a few things here:

- There is some measure (we are calling it Floor Stretch) that is not fully covered in the box score stats that underlies the value of small players.
- Floor Stretch has to be weighed against the raw productivity from the box stats when considering which types of players to include in a lineup.
- Floor Stretch is only important relative to the other team’s lineup. This brings us into the realm of Game Theory.

If we can make some assumptions about what that Floor Stretch value is, position by position, we can use payoff matrices like the ones above to see if there are any equilibria 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? Well, we’ll see in the next post: *A Beautiful Lineup*.

-Ari

**Ari gained writing duties bringing up such an interesting topic. This comment may put him on deck for a Videocast to discuss this even more!*