# Gimme the Rock Part 2- The Nash Equilibrium

Posted on 10/30/2010 by

“See, there were these two guys in a lunatic asylum… and one night, one night they decide they don’t like living in an asylum any more. They decide they’re going to escape! So, like, they get up onto the roof, and there, just across this narrow gap, they see the rooftops of the town, stretching away in the moon light… stretching away to freedom. Now, the first guy, he jumps right across with no problem. But his friend, his friend didn’t dare make the leap. Y’see… Y’see, he’s afraid of falling. So then, the first guy has an idea… He says “Hey! I have my flashlight with me! I’ll shine it across the gap between the buildings. You can walk along the beam and join me!” B-but the second guy just shakes his head. He suh-says… He says “Wh-what do you think I am? Crazy? You’d turn it off when I was half way across!”The Joker from The Killing Joke

Note: I screwed up in my numbers when I first posted this and switched Steals for Turnovers (I’m fixing this now).

Yesterday I started my brand new series.  The genesis of this series came from an article on Carmelo Anthony by Jeremy Wagner of Roundball Mining Company:

The motivation behind this piece is that the author, cannot believe the premise that Carmelo Anthony is inefficient. While he admits that he in most cases believes in statistics, in this case the statistics do not match his conclusions drawn from direct observation. Faced with this quandary, he decided that he was going to take a look at the numbers himself and while I admire his efforts I disagreed with his results. He argued that while Melo was not an efficient scorer he could be if he chose to be. I find a fundamental problem with this idea. I think young players can be molded to appropriate behavior by the right system but veterans are who they are for the most part.

There is an ineffable calculus that makes a person an efficient generator of offense. The end goal is simple, the player needs to find the best possible look or find the man that has it and get him the ball without turning it over. The details, is he open, is it a three, will he get an and 1, etc., are not so simple. If the player is successful his team scores. The needs of the many do indeed outweigh the needs of the few or the one. Measuring this is a challenge and by now you know how I feel about challenges.

Challenge Accepted!

Gimme the Rock The Metric

So the goal is to develop a simple measure for the return (points) on investment (possessions) for players on offense (i.e. Player offensive efficiency). My concept is based on the idea that were I an NBA gm paying a player  I would care about getting value (points) from my assets (possessions). I want to do this simply in a way that anyone can understand and using publicly available information. My method will be as follows:

• Look at all the data for the 2008, 2009 & 2010 for every player and get:
• Player Position
• Minutes Played (and eliminate all players with less than 1200 Minutes Played which leaves 383 players)
• Pts48 (points per 48 minutes)
• FGA48 (field goal attempts per 48 minutes)
• Pts per Attempt (pts per attempt = pts48/fga48)
• AST48 (assists per 48 minutes)
• TO48 (turnovers per 48 minutes)
• FTA48. This was added because of two of my fabulous readers (some dude and Man of Steele take a bow)  who pointed out that some free throws while free do end possession. They’re right of course.  So we are going to add the SD-MOS hack-a-shaq correction.
• Offensive possessions used per 48 (FGA48+AST48+TO48 +.44 *FTA48 term (an approximation for possessions used thru Free Throws from Prof. Berri) this is the possesions spent by the player)
• Offense Generated (Pts +Asst *2.68) per 48. The 2.68 is the average points generated per FGA for 2008 thru 2010
• Offense generated per possession used. This is the key measure as it reflects how many points the team generates when the player in question gets the rock.
• Offense Generated at 30 possessions used. Here I’m just projecting every player at an even number of possessions.

In the first piece I Looked at the Six Players from the original Melo Piece:

1. Carmelo Anthony
2. LeBron James
3. Kevin Durant
4. Dwyane Wade
5. Kobe Bryant
6. Kevin Martin

And it looked like this: (Fix #1)

Carmelo is tied with Durant at the rear of the group (Durant is young and improving however).

But this was just the intro.

Now I could look at every player together but I noticed a funny thing:

(Fix #2)

Role & position play a huge role in how you affect the offense so for this series we’ll be looking at players by position and then ranking them. First, we’ll talk about the generals of the offense, the napoleons of the court. Point Guards on deck.

Gimme the Rock Rankings: Point Guards
Basketball is game. Game Theory is the branch of applied mathematics that attempts to mathematically capture and model behavior in strategic situations (that ineffable calculus again) in which the success of the player depends on the choices of others. Sounds very much like what we try to do here. One idea of interest is the Nash Equilibrium. The informal definition of this concept from Wikipedia is as follows:

Informally, a set of strategies is a Nash equilibrium if no player can do better by unilaterally changing his or her strategy. To see what this means, imagine that each player is told the strategies of the others. Suppose then that each player asks himself or herself: “Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, can I benefit by changing my strategy?”

If any player would answer “Yes”, then that set of strategies is not a Nash equilibrium. But if every player prefers not to switch (or is indifferent between switching and not) then the set of strategies is a Nash equilibrium. Thus, each strategy in a Nash equilibrium is a best response to all other strategies in that equilibrium.

To simplify: this is the best possible world for all the players. A point guard should strive to achieve this end state of basketball Nirvana by which the best possible return is achieved for the players on his team. The greater good (team success) is served. My hope is that by looking at Point Guards through the lens of value vs. cost I can show how they succeed at approaching this ideal state. But that’s enough buildup, let’s get to the table:

(Fix #3)

Not surprisingly, Steve Nash is very close to being the best generator of offense among point guards and in the league (I did foreshadow this quite a bit). What is surprising is number one, Jose Calderon. He’s mister efficiency. Among the 79 Point guards who qualified only 16were one std deviation or above from the mean:

Top Tier (>2 std dev): Jose Calderon (Mr. Efficiency), Steve Nash (too many turnovers for Steve), Jason Kidd,Chris Paul

2nd  Tier (>1 std dev):Deron Williams,Chris Duhon, Brevin Knight, Antonio Daniels,Rajon Rondo, Travis Diener,  Anthony Carter,  Steve Blake (#%#% Lakers),  Jason Williams, Jeff McInnis, Carlos Arroyo (Boricua power),Chauncey Billups

A notable non-successful name is Derrick Rose. Rose is below average for a point guard generating 39.7 pts per 30 possessions vs. Calderon’s 50.9 at the top end and vs. say Mo Williams’ 41.5. So Rose is 11.2 points worse than the most efficient point guard in the league and 1.8 points worse than an average one.

With that I end this part. Tomorrow shooters. Comments as always welcome.

(Image courtesy of xkcd.com)

Go here for part 3

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