Mixed Strategy Nash Equilibrium in Baseball: The 2-1 Count

Greg PowellMasters of Applied Econometrics

Research Motivation• How can pitcher’s mix their pitches optimally when hitters have the

advantage in the count?• Interest in game theory, strategy and baseball• Real Time Human Behaviour• What conditions lead to optimal play?• Experimental research overall to reject MSNE. Do professional athletes

use MSNE?• Always wanted to examine the rich, freely available “Gameday” dataset• See what insights can be obtained• 2-1 and one count is an advantage to the hitter. Is there a strategy

where a pitcher can limit damage by the hitter?• Improve my own hitting approach!

Baseball Overview• Major League Baseball

– NY Yankees • $206 million Payroll • $5.5 million median player salary

– Pittsburgh Pirates • $34 Million payroll• $450K median player salary

• Highly paid/motivated players – High stake games– BIG BUSINESS

• 30 Teams, 162 games each (plus playoffs). Lots of heterogeneous and homogenous data.– “Baseball Everyday!”

• Amazing number of baseball literature both published and online (both statistically based and otherwise). http://content.usatoday.com/sportsdata/baseball/mlb/salaries/team

Quick Baseball Terminology!• 2 (Balls) - 1 (strike)– What is a Strike/Ball?– Baseball – Ideal for

statistical research as there are discrete outcomes to each play

– What is a Plate Appearance?

Quick Baseball Strategy

– Pitch Types– Pitcher/Hitter Strategies– 2-1 - Considered a

“Hitter’s Count” or a “Fastball Count”

Methodology – MSNE

• Mixed Strategy Nash Equilibrium– We can see in this

matching pennies game (sigh) that there is no pair of pure strategies that players would switch to if they knew what their opponent would do.

– Players are therefore indifferent to opponent strategy

MSNE B Heads B Tails

A Heads +1,-1 -1,+1

A Tails -1,+1 +1,-1

Methodology – MSNE & Baseball?

• MSNE & Baseball- Zero Sum Game- Simultaneous moves

(mostly!)-Weinstein-Gould Models an

M x N matrixM,N = # pitches

- In Major Leagues, Batters have access to full scouting histories – they will know what pitch types pitchers have in their armoury.

Batter

Pitcher

1 “Think Fastball”

2 “Think Changeup”

3 “Think Curveball”

1 Fastball 11 12 13

2 Changeup 21 22 23

3 Curveball 31 32 33

Literary Review - MSNE• Experimental Research

– In the Lab– Difficult to prove MSNE – why?– Lots of 2*2 Games and matching pennies– Aren’t we sick of matching pennies???!!!!– Maybe we’d detect MSNE in a World Matching Penny Tournament?

• Research in Professional Sports– Soccer (Chiaporri) – Penalty Kicks

• Not Much Data

– Tennis (Walker and Wooders)• Serve/Return of Serve

– Baseball (Weinstein-Gould)• Data collected by humans• MSNE for Pitch outcome. Not for measures of OnBase and weightedOBA• Very ambitious – analyses the first pitch and also the outcome of the plate appearance

Data

• Data from MLB “Gameday” Database • Collected using the Pitch F/x System• 2009-2010 Data – most games included• 72,445 2-1 Count pitches represented by:– 965 Batters– 805 Pitchers

Data – Summary Statistics• Pearson tests show that there IS a statistically significant and

different strategy in a 2-1 count as compared to 0-0. To be expected.

Methodology

• Initial Multinomial Logistic Regression – 72,445 Rows

• 805 pitcher dummy variable columns• 965 hitter dummy variable columns

–

Test the hitter dummy variables• Null Hypothesis H0: Hitter 1 = Hitter 2 = Hitter n = 0• If we fail to reject it shows that the pitchers have an indifferent or

mixed strategy toward hitters• Intuitively this makes sense

Next Steps

• Find a tool that will run my initial multinomial regression analysis

• Test the null hypothesis• Next step is to look at pitcher-hitter

combinations• Weinstein-Gould -