1. Mixed Strategy Nash Equilibrium in Baseball: The 2-1 Count Greg Powell Masters of Applied Econometrics
2. 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!
3. 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
4. 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?
5. Quick Baseball Strategy Pitch Types Pitcher/Hitter Strategies 2-1 - Considered a “Hitter’s Count” or a “Fastball Count”
6. 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
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9. 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
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11. 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.
12. 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
13. 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 -
Editor's Notes
My research is on Mixed Strategy Nash Equilibrium within the context of baseball and more specifically the contest between pitcher and hitter in a 2 balls 1 strike count. This is an important count strategically, as the hitter has the advantage. The study will focus on 3 measures of success for the pitcher – immediate pitch success, and overall success of the plate appearance.My research will attempt to demonstrate that baseball pitchers can mix their pitches for optimal outcomes when faced with a hitters count.The pitcher-batter confrontation is an ideal theatre for Mixed strategies to be discussed - pitchers need to mix their strategies to balance the need to throw his best pitches, vs preventing the hitter from anticipating what comes next. I was also interested in studying this topic, as it has broader implications for real time human behaviourWhat conditions lead to optimal play/decision making? Experimental research poor in proving MSNE – studies in professional sport have restored this balance somewhat.- Improve my own approach!-- First I’d like to go over a bit of baseball terminology for those who aren’t familiar with the topic!
As you can see from the financials – Baseball is BIG BUSINESS!!!Any advantage a player, a coach or team management can gain by instructing their players what to do in certain situations, or identifying strategies to beat other teams or draft players, means huge financial savings/gains, and also more wins on the scoreboard.
Show strike zone on the white board.Show ball on black.Show foul territory (foul strikes)How many ways to get to 2-1. F B BB F BB B FS B BB S BB B SThere might be serial correlation issues here with regard to the outcome of the 2-1 pitch. Weinstein Gould avoided this issue, by selecting to observe the 0-0 pitch!Plate Appearance – is the completion of the batters turn to bat (out or gets on base)
While baseball is mostly a simple game of bat vs ball, at the top level is a game of strategy.The different pitch types looked at in this study are:4-seam Fastball – generally easiest pitch to hitSplit-finger FastballCut FastballCurveballChange-upSliderSinkerKnuckleballScrewballChoice of pitch between manager, catcher and pitcher – using signalsGoto white board2-1 Count – the hitter has the advantage and is looking for a “good pitch to hit” – usually a fast ballPitcher knows this, but also does not want to fall behind 3-1 (and risk a walk)Weaker hitter – aggressive - pound the zone. Weaker hitter knows thisStrong hitter – walk ok, throw edges, both players should be patient!Going to jump to methodology now…
This table is the classic “Matching Pennies” Game - There is no PURE strategyPennies match, Player A keeps coinsPennies don’t match, Player B keeps coinsIf each player knew what the other would do, there is no pair of pure strategies that either player would want to stick to.The strategy is therefore INDIFFERENT
So transferring that knowledge of basic MSNE to baseball this is the kind of game we are going to play.In our game however, even if the batter “Think’s” correctly there is no guarantee of success. By the same token, even if the pitcher throws the pitch the batter is not expecting, there is no guarantee of success on his part either.
There is extensive literature on MSNE in experimentsMukherjee and Sopher2*2 games- Ochs3 2*2 Games with varying Nash EquilibriaPlayers would generally respond above the nashequlibria even when reaching a steady state.- Shachat and Swarthout – similar to ochs- 2*2 Games with complete historiesMcCabe proved MSNE amongst players But I guess the issue is: Aren’t we sick of matching matching pennies?- Do players have enough history to compete properly? Subjects have certainly not devoted their lives to playing trivial 2*2 games or matching pennies. Given that most of the subjects are from educational environments (Shachat uses undergraduate students),Game repetititions of 200 – are results being affected by subject tedium- Soccer – proof of MSNE among penalty kicks – 459 kicks – is this enough data?Simultaneous games Kicker Left/Kicker Right Goald right/Goalie left- Proves MSNE – Kickers act as if goalies are identical – indifferent to individual goalie strategy- Tennis – Modelled as 4 2*2 Games – lots of data, clear MSNE – compared to card matching (O’niell) with no MSNE. Evidence of serial correlationSimultaneous games Server Left/Server Right Receiver “thinks right”/Receiver “thinks left”- Baseball – very ambitious – proves MSNE on 0-0 pitch, but not for plate appearance. Not sure if this is possible!!! A plate appearance can be very long!- PitchOnBase-wOBA – contribution to runs- Professional Athletes have devoted their lives to perfecting their craft. And they have coaching staff assisting them with scouting information on opposition teams and players. And there are high incentives to perform both financially and in terms of continuing careers.- The critical difference between experimental and
The Pitch F/x system is a computerised system similar to “Hawkeye” that is installed in every Major League Stadium. It records the x,y co-ordinates and other information.It’s a consistent and high tech system. Weinstien-Goold’s data was collected by hand.
Gameday data feeds into the MLB application
Go through statistics – significantly different strategy employed in 2-1 count.Talk about less 4 seam fastballs.
Following Weinstein-Gould’s research
Next step to find the tools, and then move onto the next part of Weinstein-Gould.If pitchers using MSNE – coefficients fore Beta2 for PitcherHitterPitchType should be jointly insignificant.Beta2 coefficients for a particular pitcher should be jointly insignificant if that pitcher is using MSNE.
