“So where were we?”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. We are using the same framework Weinstein Gould used – pitch_success, OnBase and wOBAAt the point of the last presentation we determined that: In a 2-1 Count, as opposed to a 0-0 count, there is a significantly different strategy in terms of pitch types thrown Problems with the fixed effects models. I was having massive problems running the fixed effect models within the framework of Weinstien-Gould’s research.With some slight adjustments (with the help of Rob), we were able to come up with some models where we were able to replicate the tests performed in his research.In the following slides, -I will be showing you the conceptual MSNE game for the pitcher vs hitter, -and also the results of the adjusted Weinstein-Gould models First of all I would like to clear up some terminology and give you all a better understanding of the situation I am describing Best way of doing this is by using video.
“I am going to show you a plate appearance”<draw on board>As you watch this I want you to remember that:Remember – 3 strikes is an out, and 4 balls is a walk – a free path to first base. To throw a strike the pitcher has to throw a ball through imaginary strike zone- Width of plate- between knees and lower chestAs soon as the pitcher falls behind in the count, it becomes easier for the hitter to get on base, statistically speaking.This is a particularly good example, as it is an early season game, shows a 0-0 ball game with no runners on base, and early in the game, so this is the first time the pitcher and hitter have met in the game.David Bell, so he was a solid Major League Baseballer who hit career .257 and is 5 years into a 10 year career at this PA. 3 Clubs in 5 years shows that he is only an average player.Tim Hudson, is considered to be an excellent pitcher and at this point was 2 years into his currently active career.So we have a very strong, young pitcher, against a solid Major League journeyman.…<describe the action>2-1 now the hitter is at an advantage statistically – as per this table.Hitter has a statistical advantage in 2-1 count.So…the count has moved to 3-13-1 count is a huge advantage according to our data.In OUR DATA – it’s shown that in a 3-1 count, 3-1 20% increase in getting on base2-2 20% decrease in getting on baseNote that the strategy was all fastballs. Early in the game, 1 out, no runners on base, the strong pitcher will back his fastball against the average hitter.IMPORTANT POINT – Weinstein Gould attempted to prove that the measurables of OBA and wOBA could be determined from the result of 0-0 (or in our case 2-1). This clip shows the ambitiousness of that reasoning.
Show payoff game on board. See Polak Lecture
In our dataset the most populous combination of hitter-pitcher was hitter Brian McCann vs Mike Pelfrey, so I’ve decided to use them in this example.Brian McCann is one of the best hitters in the league and Pelfrey is a promising pitcher.Note that these numbers are examples only, but logically they make sense.If the hitter guesses fastball correctly his chances of success (or utility) are 50%. Which is good in baseball.And we can see that if the players choose incorrectly, then there is an advantage to the player that choose correctly.We can calculate the mixing strategy combination…
Right Column :If Pelfrey is mixing both pitches in a NE then both must yield the same payoff, in which case McCann must be mixing what he is anticipating.[0.5 Fastball, 0.5 Other]Left ColumnIf McCann is anticipating pitches in a NE then both must yield the same payoff, in which case Pelfrey must be mixing what he is pitching.[2/3 Fastball, 1/3 Other]We have proved that in equilibrium the Pitcher must be indifferent between the pitches in his mix, to solve for the batters equilibrium strategy
As discussed in the previous presentation – this fixed effects did not work, and Rob has his concerns about this model in anycase.
As discussed in the previous presentation – this fixed effects did not work, and Rob has his concerns about this model in anycase.
After much angst and trying, we came up with the following model…Not a complex model. Test of hitter specific pitch type strategy in hetereogenous datasetA series of binomial modelsFastball/other, Changeup/otherPerformed a p-value test of whether the hitter_dummies were jointly significant
So, like Jesse Weinstien Gould we were able to prove a hitter specific strategy for the majority of pitches thrown.In our initial tests of difference in # of pitch types between the 0-0 count and the 2-1 count – we already proved a different strategy in 2-1. Our hitter-specific tests verify that for fastball/curve/change- Discuss reasonsRest of the pitch types are indifferently thrown – it suggests that they are slightly more specialist pitches that a pitcher has good control of.Handedness – Assuming most pitchers are RH in sample, they are generally considered to be at an advantage against LH hitters…Therefore as the changeup moves away from the LH hitter, we can see why the hitter specific strategy of more changeups to LH hitters is observed.
So now that we know that pitchers use a hitter-specific strategy overall.So in a homogonised dataset, can we prove that within each pitcher-hitter battle - Is each type of pitch thrown by every pitcher to every hitter has the same effect?Yes.
SHOW PitcherHitterType Dummy on board PitcherID,HitterID, PitchTypeIDEg 599-201-0Next 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.
Weinstein-Gould’s results are replicated!!! This is a pleasing result, however as I mentioned in the last presentation there are issues with Weinstein-Gould’s wOBA measure of success.In terms of the pitch success model, the measurement of MSNE using pitcher-hitter-type dummies, etc is perfectly reasonable.However without considering other variables, it’s very ambitious to attempt to measure the result of the plate appearance with respect to an individual pitch.I will show you why…<show a plate appearance>So, if we think back to the video we saw before… Hudson vs BellAll fastballs – draw strikezone and number pitches- 1 Miss High inside- 2 Miss Low inside-3 Strike Away-4 Ball Away (blown call by umpire)- 5 Strike Low away – hit to short stop for an outAccordingto the Weinstien-Gould Framework/Hypothesis, pitcher, hitter and pitch type dummies are enough to determine the outcome of this plate appearance.Withoutextra information and variables, this is extremely difficult. This is a huge model!Perhaps reasonable for the “next pitch”
Pitchers continually falling behind 2-1 may be indifferent to the hitter, but is it MSNE?
Mixed strategies in baseball Part 2
Pitching Strategies in a Hitters Count: Mixed Strategies in Baseball<br />Greg Powell<br />Masters of Applied Econometrics<br />
So, where were we?<br />Baseball is Big Business – effective strategies are utmost importance<br />MSNE Unobserved in low incentive games<br />Weinstein-Gould attempting to prove MSNE with respect to his 3 measurables<br />PitchSuccess - (Ball/Strike)<br />OnBase– Batter gets on base via hit/walk<br />wOBA – weighted OnBase<br />We are using the frame work Weinstein-Gould used in a 0-0 count into a 2-1 count.<br />Pitching strategy between two counts IS different, statistically<br />MSNE Game – Pitcher vs Hitter – Conceptual Game<br />
2-1 Count – Hitter Advantage = Better Chance of Getting On Base<br />League Average OBA<br />OBA of in 0.333 in 2009 <br />0.325 in 2010 (ESPN 2011)<br />2-1 Count is advantage hitter, statistically – 0.397 OBA<br />After the following pitch:<br />Ball 3-1 OBA = 0.671<br />Strike 2-2 OBA = 0.191<br />
Example Payoff Game<br />MSNE & Baseball<br /><ul><li>Zero Sum Game
Simultaneous moves (mostly!)</li></ul>-Weinstein-Gould Models an M x Nmatrix<br /> M,N = # pitches<br /><ul><li>In Major Leagues, Batters have access to full scouting histories – they will know what pitch types pitchers have in their armoury.
In this example reduced the payoff matrix to a “Fastball” vs “Other” pitch type</li></li></ul><li>Example Payoff Game (cont…)<br />
Calculating the MSNE<br />Pelfrey ROWS payoff FASTBALL against Hitter (q,1-q): q(0.50) + (1-q)(0.60)<br />Pelfrey ROWS payoff OTHER against Hitter (q, 1-q): q(0.65) + (1-q)(0.45)<br />If Pelfrey is mixing in NE then payoffs to McCann FASTBALL/OTHER must be equal<br />q(0.50) + (1-q)(0.60) = q(0.65) + (1-q)(0.45)0.2q + 0.45 = 0.6 – 0.1q q = 0.5 McCann’s NE Fastball Mix<br />McCann Columns payoff FASTBALL against Pitcher (p, 1-p): p(0.5) + (1-p)(0.35)<br />McCann Columns payoff OTHER against Pitcher (p, 1-p): p(0.4) + (1-p)(0.55)<br />If McCann is mixing in NE then payoffs to Pelfrey FASTBALL/OTHER must be equal:<br />p(0.5) + (1-p)(0.35) = p(0.4) + (1-p)(0.55)0.5p + 0.35 – 0.35p = 0.4p + 0.55 – 0.55p0.3p = 0.2p = 2/3 Pelfrey’s NE Fastball Mix<br />
Equilibrium Strategy…<br />Batter McCann Strategy<br />[1/2 Fastball, 1/2 Other]<br />We have proved that in equilibrium the Pitcher must be indifferent between the pitches in his mix, to solve for the batters equilibrium strategy<br />PITCHER Pelfrey Strategy<br />[2/3 Fastball, 1/3 Other]<br />
First Analysis – Hitter Specific Strategies on Heterogeneous Dataset<br />Conventional Baseball Theory suggests that pitchers use hitter specific strategies…<br />In a heterogeneous dataset…<br />…can we prove that pitchers use a hitter-specific strategy?<br />YES!<br />
First Analysis – Hitter Specific Strategies on Heterogeneous Dataset<br />Initial Multinomial Logistic Regression <br />72,445 Rows <br />805 pitcher dummy variable columns<br />965 hitter dummy variable columns<br />Test the hitter dummy variables<br />Null Hypothesis H0: Hitter 1 = Hitter 2 = Hitter n = 0<br />Joint insignificance of Hitter fixed effects = pitcher indifference/MSNE<br />
First Analysis – Hitter Specific Strategies on Heterogeneous Dataset<br />Major Issues with data size/singular co-variance of dummy variables<br />Series of binomial models with only the hitter dummies<br />Fastball = hitter_dummies + e<br />Changeup= hitter_dummies + e<br />Curveball= hitter_dummies + e<br />etc…<br />Test whether hitter_dummies are jointly significant<br />Jointly significant means hitter-specific strategy<br />
First Analysis – Results<br />HITTER SPECIFIC STRATEGY:<br />Fastball <br />Curveball <br />Changeup<br />Handedness Issue<br />RH Hitters – Only fastball <br />LH Hitters – Fastball, Curveball, Changeup<br />Why?<br />Already proved a different strategy in 2-1. <br />Hitter anticipating fastball – less fastballs<br />Curveball difficult to throw for strike – less thrown<br />Changeup perhaps the easiest pitch to throw for a strike without throwing a fastball – more changeups<br />“Pitcher must throw a strike to level count”<br />
Second Analysis – Model (homogenous data)<br />Testing for indifference/MSNE<br />In a homogenised dataset, within every pitcher-hitter battle…<br />…Can we prove that each type of pitch thrown by a given pitcher to a given hitter has the same effect?<br />YES!<br />
Second Analysis – Model (homogenous data)<br />Condensed dataset of 5632 rows of the most populous hitter-pitcher pairs.<br />Similar modeling issues as in the first analysis, ie Singular co-variance , etc<br />Therefore following models run on each of the 3 measures of success:<br />Binomial: Pitch_success = PitcherHitterPitchType + e<br />Linear: wOBA = PitcherHitterPitchType + e<br />NOT ABLE TO BUILD MODEL FOR OBA<br />Joint Test of PitcherHitterPitchType = 0 <br />If insignificantly different from 0 – each pitch thrown with same effect<br />Therefore Indifferent strategy, therefore MSNE employed<br />
Second Analysis - Result<br />Pitch Success Binomial Model<br />FAIL TO REJECT Joint Test of PitcherHitterPitchType = 0<br />Effects Insignificantly different from 0 – each pitch thrown with same effect<br />Pitchers acting indifferently on 2-1 pitch = MSNE!!!<br />wOBA Linear Model<br />Reject Joint Test of PitcherHitterPitchType = 0<br />Effects Significantly different from 0<br />No evidence of MSNE with respect to plate appearance<br />Weinstein-Gould’s results are replicated<br />
Other Issues<br />Pitchers who continually fall behind 2-1 are the poorer pitchers<br />Poorer control<br />Can’t mix optimally, yet they are indifferent to batter<br />It might be indifference, but it might not be mixing optimally<br />Batters who work counts to 2-1 are more selective and better hitters?<br />Weinstein-Gould proposed examining 0-0 to avoid the dynamics of the previous pitch<br />This research shown that the dynamics don’t really matter within his framework – same result<br />
Further Work<br />For publication<br />Examine 1-2 count where the pitchers are ahead in the count<br />Examine Locations<br />Hitters in Majors may actually look for location rather than pitch type…<br />Examine other variables such as:<br /># Outs<br /># Runners on Base<br />Score<br />Etc…<br />
Conclusion<br />In 2-1 count Pitchers mix optimally with respect to Pitch Success measure<br />Plate appearance (wOBA) measures probably cannot be estimated without considering many other variables or building a more complex model<br />PROFESSIONALS USE MSNE (OR AT LEAST ARE INDIFFERENT)!<br />