Happiness, Expectations, and the First Round

An oft-cited formula that I believe originated in psychology circles is this: 

Happiness = Reality – Expectations.  

At a glance, this formula seems to hold water; it very simply stands to reason that happiness would be a function both of our expectations as well as our realities.

A simple and motivating example:

If in fact Happiness = Reality – Expectations, and your expectations are for your team to win a World Series title, earning a playoff berth (but a first-round exit to go along with that playoff berth) amounts to, well, unhappiness.

An oft-cited formula in economics is the calculation for expected value (EV).  It is given as: 

While perhaps a tad less forthcoming, this too makes some sense once considered briefly.  In short, this formula states that (in finite cases) the expected value of any one thing should be the probability of its occurrence multiplied by the resulting “payoff”, should that occurrence come to pass.  If there are multiple finite outcomes for any type of occurrence (think the six sides on a die, or the 52 cards in a deck), those probabilities and the payouts that correspond to them are simply summed up.

A simple motivating example:

The expected value of getting to roll a single die and collect dollars that reflect the number rolled.  Take note that the expected value in this case is very simply the average roll of any given die.

EV = 1*(1/6) + 2*(1/6) + 3*(1/6) + 4*(1/6) + 5*(1/6) + 6*(1/6)

EV = 3.5

If someone offered to let you roll a die and collect money based on whatever number faces up, that offer is worth $3.50 given this line of reasoning.

A second example:

Is buying a $2 lottery ticket that pays out $75,000 once every 40,000 times worth it? Based on expected value, it is (and this is true for many lotteries) not:

EV = 75,000(1/40,000)

EV = 1.875

$2 in your pocket > the ticket’s expected value of $1.875.

Where is this all going? MLB general managers presumably, like psychologists or economists, wish to maximize their happiness. Part of that happiness originates from the value they can expect from their picks in the June amateur draft.  Of all those picks, players being scooped up in the first round historically represent the greatest chance for a team to earn on their investment in the form of utility on the field.  So here at the intersection of happiness and expectations it seems of interest to evaluate first round picks in so simplistic terms as the value that they are broadly expected to generate for the team betting on them.

In order to do this, first round pick data were aggregated (thanks baseball reference!) from the years 2000-2018 in order to get a better understanding of who has been drafted at the top of heap over the last 19 years.  Supplemental first round picks were also included in this data.

It is of significance that positions lists for these players are very simply the positions that these players were drafted as, not where they ended up.  Mark Teixeira, for instance, is listed as a third basemen in this dataset.  Additionally significant is what I have changed: outfield positions were inconsistently labelled (some obvious CF’s listed simply as OF’s or vice versa); as a result I have changed all outfield titles simply to “OF” to encompass the whole group.  

Pitchers were bisected by handedness leading to a dataset with 8 positional categories: RHP, LHP, 1B, 2B, 3B, SS, C, and OF.  What I wanted to know what simple: which position has been responsible for the most value, on average, in first round picks during these years.  Additionally, what has the average first round draftee been worth?

First, a quick visualization of the whole distribution.

Some thoughts.  This much is clear: pitchers and “elite” up-the-middle position players (SS and, likely, CF’s) dominate in frequency.  In 19 seasons just 19 players classified/drafted as second basemen have been taken in the first round.  While not terribly surprising to see the infield corners underrepresented, the dearth of first round catchers is of note in my opinion.  This data does not classify outfield positions, but I would imagine that this hesitancy to draft players at the corners extends into the outfield and CF picks would make up the lion’s share of players drafted as outfielders.  Nearly twice as many right handed pitchers (308) have been taken in the first round than the next most represented group, outfielders of all specifications.

Evaluating how these picks have done though requires that dataset to be pared down some.  The next graph includes only players drafted from 2000-2010 so that, generally speaking, enough time has passed for these draftees to have made their mark in MLB.  This is a pretty liberal truncation; players like Gerrit Cole (taken in the first round of 2011) will not be included, but I didn’t want to include data for players who have yet to represent reasonably their value in WAR created.

Not too much change from the first, larger, sample.  Of note is the slightly higher rate of first round LHP’s taken over the last eight drafts. Other than LHP’s leapfrogging outfielders in pick frequency, the order of picks remained the same. 

In this final plot, I have simply summed up the total WAR across each position and divided by the number of players who have been drafted there in the first round.  This plot then provides a rough estimate of how valuable, on average, players drafted in the first round at various positions have been.

Here we find, interestingly, that those oft-forgotten first round corner infielders have in fact produced the greatest average WAR among all positions.  Does this mean GM’s would be most happy using all their first round picks on corner infielders? Absolutely not.  Is it safe so expect the most of players drafted in the first round at those positions? Of course not.  This evaluation comes with far more than a grain of salt, for a number of key reasons.  

One point was already touched upon: players simply do not always remain at the position from which they were drafted.  The aforementioned Mark Teixeira, who never played any extended innings as a ML third basemen, contributed 51.8 WAR to a different position (3B) than the one he eventually played (1B).  A second point is the obvious impact of small sample sizes.  Mark Teixeira, and his 51.8 baseballreference WAR alone, likely pushed this group of third basemen to the highest position on this list.  Furthermore, all those smaller-sample positions rate at the top here: C, 1B, 2B, and 3B.  Right handed pitchers, easily the most represented group here, ranks dead last.

The overall average performance of a first round pick regardless of position, our second question, can be determined (as one method) with expected value. Given the percentage of the time a positional player is taken as a first round pick, and the average amount of WAR produced by that position, one can then calculate the expected value of a first round pick.

EVpick = RHP(.373) + LHP(.165) + C(.065) + 1B(.038) + 2B(.028) + 3B(.071) + SS(.105)

EVpick = 4.01(.373) + 5.48(.165) + 6.88(.065) + 9.43(.038) + 7.14(.028) + 9.97(.071) + 5.20(.105)

= 5.51 WAR

This total, very simply, is the average WAR generated (to date) by a players taken in the first round from 2000-2010. Was expected value worth combing through after all? Maybe not, but an interesting lens through which to consider this issue I believe. 

As expected, this average total was more than a full WAR lower when I included players drafted as recently as 2018, given the lack to time of recent draftees to progress through the minor leagues. Evaluating outcomes based on position or even just terms so broadly as this is obviously an exercise of conjecture, but discovering this 5.51 WAR number from historical data is interesting in its own right.