How Casting Spells Should Really Work

Warning: extreme MTG nerdery ahead.

Starting with the Magic Origins update bulletin, I’ve observed the rules manager/team trying to work out the process of casting a spell, trying to allow what should be legal (using Bestow when you can’t cast creatures), disallow what shouldn’t be legal (casting Squee out of Ixalan’s Binding), and not twist themselves into a pretzel of incoherent nonsense in the process… which is what happened with the latest update to 601.3e. Well, actually it started in the previous update with Mystic Forge rulings, but they doubled down on that mistake here and made it really bad.

Assuming (as always) no other relevant cards/effects, and speaking normatively throughout, if you have a Mystic Forge out and a Deathmist Raptor on top, you shouldn’t be able to cast it. You can’t normally cast the top card of your library. It’s not an artifact or colorless nonland card, so Mystic Forge shouldn’t let you cast it. Done. Full stop. Nothing lets you cast that object, so you can’t take the game action of starting to cast it. This shouldn’t even be a question.***

With the new update, Cascade on a 3 mana spell, which reads “… until you exile a nonland card whose converted mana cost is less than this spell’s converted mana cost” still (correctly) doesn’t recognize Beck//Call as a card with CMC<3 because it’s CMC 2+6=8…. while at the same time, Kari Zev’s Expertise, which says “You may cast a card with converted mana cost 2 or less from your hand…” somehow DOESN’T even recognize that it’s an 8 CMC card. And Brazen Borrower’s Petty Theft can be cast from the graveyard using Wrenn and Six’s emblem. What in the actual fuck. Eli- C’mon man.

In the explanation, Eli said “if you’re allowed to cast a spell with a certain mana cost or color…” and I don’t know if that’s just a typo or an actual misunderstanding, but Kari Zev’s Expertise says CARD, not SPELL, and it has to SAY card and MEAN card for anything to make any sense, and the failure to properly separate thoughts about cards and thoughts about spells seems to be at the root of all of the issues.

The process of casting a spell can be reduced to

Casting an object
as a spell with certain characteristics/targets/etc
at a particular time

The tricky part is that the properties of the spell are not known at the start of casting, and the allowable times (and other things) depend on those properties. Furthermore, since the gamestate changes between 1 and 2 when the object goes on the stack, what’s allowed and prohibited can change in the middle of casting (e.g. Squee and Ixalan’s Binding). None of this is a problem, or even particularly complicated, but the algorithm to process it all has to be correct or it can spit out some fantastic levels of nonsense.

Let’s start with part 1, casting an object. Their original attempt was that you can literally cast anything- a card from your hand, from your opponent’s library, whatever, and it would be dealt with later if it was illegal. This was.. not a good idea, although it had the seed of one. Since we don’t know what the spell is going to look like at this point, ignoring spell prohibitions here is correct, but something has to control what cards can be cast. As it turns out, the correct answer to that is the same as the answer to why you can’t just put your opponent’s entire hand into his graveyard whenever you feel like it- you can only take legal game actions, and that isn’t one. The path they went down instead, trying to determine what objects can be put on the stack based on what their spells might end up looking like, is effectively a category error. The beginning of casting is about CARDS, not SPELLS.

There are *no other prohibitions* on what objects can be cast. The set of legal objects should be defined *constructively*- cards in your hand by rule and whatever other objects effects allow you to cast (cards in the graveyard with Flashback, cards CMC<=2 in hand while resolving Kari Zev’s Expertise, copies from Isochron Scepter, etc). Combined with the rules on timing and priority, which needs a small (obvious) rewrite to 117.1 to comply, we get the proper result

“A player with priority can cast a castable card” with the set of castable cards defined constructively as above. There’s no way to cast an object at a random time because the only times casting something is an allowed game action are when you have priority or when something is telling you to during its resolution. There’s no way to cast a random object because it’s never a legal game action. This *does* allow you to start casting a sorcery during your opponent’s turn, or even start casting a land from hand, but this is actually fine and no different than starting to cast a spell you can’t choose targets for, which has been CompRules-legal the whole time. The rest of 117 needs to be updated to “normally”, etc. to fit with this paradigm.

There are no cards that prohibit casting *cards*, so there are no prohibitions to worry about here (there are a couple that prohibit playing lands based on characteristics, but playing lands uses a special action and they don’t change characteristics in the middle of that action, so it’s not a problem). All affirmative prohibitions (e.g. “Players can’t play creature spells”) affect *spells*, not *cards*, so there’s nothing else to worry about here.

As far as dealing with the spell aspect, once the object is on the stack, there’s no reason at all to intervene before the legality check in 601.2e as long as the previous steps can be followed (immediate rewind if you can’t choose legal targets, etc). No other objects move, so no more information can be leaked, and 601.3 is basically completely wrong/useless as it currently exists. The only necessary checks in 601.2e are

Land spells are illegal to cast (regardless of other types)
If the spell was cast while the player had priority, the casting is illegal unless it is an instant, has flash, or was cast during the player’s main phase when the stack was empty. (“as though it had flash” gets by this, obviously)
If an effect prohibits the spell’s casting, considering the spell’s properties at this time, the casting is illegal.

That’s it… almost. Squee is still escaping from Ixalan’s Binding. A Grafdigger’s Cage variant that functioned while in the graveyard could (conceivably) be successfully cast from the graveyard. The fix for that is trivial though. The set of spell prohibitions is locked in before the object is put on the stack, so prohibitions that depend somehow on the to-be-cast object itself still apply, and then that set of prohibitions is what’s checked in 601.2e. Squee is stuck. So is my cage.

And that’s it for real. To summarize:

The player picks a castable CARD (object), either a card in hand or an object an effect allows to be cast, completely ignoring any properties the resulting spell may or may not have, any spell-casting prohibitions, and the possibility/impossibility of completing announcement successfully.
The set of spell-casting prohibitions is locked in.
The object is put on the stack
Announcement proceeds up to 601.2e (if possible, rewind if not)
Legality vs. the prohibitions in step 2 is checked using the spell’s current properties
If legal, proceed to 601.2f. If illegal, rewind.

***If you’re wondering how to verify a morph was properly cast with Mystic Forge, sure, it’s ugly, but verifying ANY morph was ever properly cast in any way is ugly, but we survived multiple formats with morph and manifest legal and played at the same time, and this is nothing in comparison.

The Bill James support measure is not objective

As discussed in The Independent Chip Model of Politics and HoF Voting and Bill James and the Trump polarization problem, the system doesn’t work. There was a twitter exchange where Bill seemed hard-pressed to say anything more profound than the trivial and self-referential “the algorithm prints the output of the algorithm”. If Bill were right, and the algorithm actually measured any objective value, then it shouldn’t much matter what sets of polls were run as long as there was enough mixing in the poll groups to have a link to everybody (polls of only R candidates and polls of only D candidates don’t say anything about how Rs would do against Ds, etc).

If the ICM distribution in the other post held, it wouldn’t matter if the population were sampled with one poll including everybody, sets of 4-person polls, sets of 3-person polls, sets of 2-person polls where one matchup was polled 100x more than the rest, or anything else. They would all generate the same support scores. This is the only distribution with that property. We know the ICM distribution doesn’t reflect reality, but how much does that make the support scores sampling-method-dependent? Well…..

As a toy model of the election, start with 4 candidates A/B/C/D who follow the ICM distribution with starting stacks/support in a 4:3:2:1 ratio. From this, we can calculate the probabilities of each of the 4! order preferences. For example, ABCD order has a probability of (4/(4+3+2+1)) * (3/(3+2+1)) * (2/(2+1)) = 13.33% and DCBA has a probability of (1/(4+3+2+1)) * (2/(4+3+2)) * (3/(4+3)) = 0.95%. Since we know everybody’s order preferences, and we make the friendly assumption that we can always sample the population completely and that the preferences never change, we can generate the result of any poll and calculate support scores. In this example, no matter how we poll, the 4:3:2:1 ratio holds and the support scores (normalized to add up to 10,000) are A: 4,000 B: 3,000 C: 2,000 D: 1,000.

Now let’s throw a wrench in this by adding candidate T who gets 40% in every poll he’s involved in and is everybody’s first or last choice. For simplicity, and to make a point later, we’ll treat this as a population with 48 order preferences, 24 with T in 1st followed by the ABCD ICM distribution above for 2nd-5th and 24 with the ABCD ICM distribution for 1st-4th followed by T in 5th (P(TABCD) = .4 * 13.33%, P(ABCDT) = .6* 13.33%, etc). Now we’ll poll this in different ways. Because we can generate any poll we want, we can poll every possible combination once and see what the support scores are. The only variable is how many candidates are included in each poll, and that gives the following support scores:

# in polls	5 (1)	4 (5)	3 (10)	2 (10)
T	4000	3340	2513	1429
A	2400	2628	2891	3189
B	1800	2001	2243	2525
C	1200	1351	1551	1813
D	600	681	802	1044
A/ABCD	40	39.4	38.6	37.2
B/ABCD	30	30.0	30.0	29.5
C/ABCD	20	20.3	20.7	21.2
D/ABCD	10	10.2	10.7	12.2

Even with T thrown in, the relative behavior of the ICM-compliant ABCD group stays mostly reasonable regardless of which poll size is used. T, however, ranges from a commanding first place to a distant 4th place depending on the poll size. Even without trying to define “support” in a meaningful, non-self-referential way, it’s obvious that claiming that any one of the 4 aggregated support numbers *is* T’s support (and that the other 3 are not) is ludicrous. The aggregation clearly isn’t measuring anything when it can massively flip 3 ordinal rankings based only on changing poll sizes.

Integrating different factions (that are strongly ICM-noncompliant with each other) into one list doesn’t work at all- the algorithm can spit out a random number, but it’s hugely dependent on procedural choices that shouldn’t make much difference if the methodology actually worked, so any particular output for any particular choice clearly doesn’t mean anything, and there’s almost no point in even calculating or reporting it.

The Independent Chip Model of Politics and HoF Voting

I’d talked about the Bill James presidential polls before, and he’s running a similar set of polls for HoF candidates that have a similar kind of issue. For whatever reason, this time around I realized that his assumptions are the same as the Independent Chip Model (ICM) for poker tournament equity based on stack sizes. If you aren’t familiar with that, then assume we have 4 players, A with 40 chips, B with 30, C with 20, and D with 10. Everything else being equal, A should win 40% of the time. The ICM goes further than that, and for predicting the probability of second place, uses calculations of the form

Assuming A wins, what are the odds B gets second: Remove A’s chips, and then B has 30/(30+20+10)=50% of the remaining chips, so B is 50% to get second *assuming A wins*.

If you don’t see the analogy yet, the ICM takes as input the stack sizes, which are identical to the probability of finishing first, and uses the first-place percentages to calculate the results of every poll subset. Bill James runs polls and uses the (first-place) percentages to calculate every head to head subset. The ICM assumption that to calculate the result between B/C/D, you just ignore A’s chips, is equivalent to the Bill James assumption that A’s support, if A is not an option, will break evenly among B/C/D based on their poll percentage.

That assumption doesn’t hold in politics, for reasons discussed before, and it doesn’t hold for HoF voting because different people prefer different player types even beyond the roid/no roid dichotomy. As it stands, in the linked poll, Beltre would almost certainly be the leader in 4th-place rankings with ~70% 4th place votes and an average finishing position near or even above 3.0. He’d likely get stomped in every head-to-head matchup, lose the overall rating, etc, but by using only first-place%, he looks like the clear second-preferred candidate, which is obviously very, very wrong.

It could have gotten even worse if Bonds didn’t dominate the roid vote. Let’s say we had a different poll, Beltre 30%, Generic Roidmonster 1 (23.33%), Generic Roidmonster 2 (23.33%), Generic Roidmonster 3 (23.33%) where (if people ranked 1-4) the Roidmonsters were ranked randomly 1-3 or 2-4 depending on whether or not the voter was a never-roider or not. In this case, each Roidmonster would have an average finishing position of 2.3 (Beltre 3.1) and would win the head-to-head with Beltre 70-30… yet Beltre wins the poll only counting first-place votes.

It’s clear that the ICM/James assumptions are violated, and violated to where they’re nowhere close to reality, in polls like this. In the same poll without Bonds, the Bonds votes would go overwhelmingly to Clemens and A-Rod, even though ICM/James assume a majority would go to Beltre. Aggregating sets of votes is going to keep a lot of the same problems because the vote share of any two people in a poll is (well, can be) strongly dependent upon who else is in the poll. The ICM/James model are built on the assumption of independence there, but it’s clearly not close to true in HoF voting or in politics.

MPL/Rivals qualification via Arena is rigged super-hard against outsiders compared to qualification via tabletop

WotC announced its long awaited roadmap for competitive play today, and it’s a mix of good and bad IMO. The tabletop part of the announcement looks pretty good. We don’t have exact details, and there could be (and probably will be) some issues with balancing records needed to qualify for future invites, etc, but the path is clear. Do well at a PTQ or GP, put up good (but not lottery-ticket-difficulty) results at the regional PTs, and you get to keep playing. Do better than that and you qualify for the big PT finals. The regional PTs are like a Rivals league for the PT, and it’s a solution that allows more players into that ecosystem (meaning deserving ones are more likely to enter it and rise to the top) while also keeping travel costs more under control. GPs matter again, and officially partnering with major tournament series (SCG, etc) is a no-brainer and better than the informal discretionary awards that WotC sometimes gave out to their high finishers in the past.

Tabletop

When it comes to qualifying for the next season’s Rivals via tabletop, it’s an advantage to start in the Rivals series as opposed to starting from scratch because Rivals players get automatic invites to all the regional PTs even if they bomb out in the season’s first event. A random who bombs in the first event has to requalify from scratch, but given the unlikelihood of being a top-12 player anyway with a disaster finish in one of the season’s three events, it’s not that huge a handicap. The much bigger handicaps are a random having to be qualified for the *first* event in a season to have the best chance of accumulating the requisite point total, as qualifying later than that basically only serves the purpose of trying to do well enough to auto-qualify for the first event of the next season. Rivals members automatically get to play in the last PT of a season to try to auto-qualify for the first event of the next season. I don’t see a trivial fix for that if the seasonal nature of MPL/Rivals is taken as a given, but these are handicaps that clearly can and will be overcome regularly. Plenty of new players will make Rivals via tabletop.

Arena

The Arena qualification system has several serious flaws. The first is that the tournaments are much smaller, so a random has to hit a lottery-ticket finish in one of two tournaments with a gigantic field (top-16 out of several thousand) to even qualify for an invitational. Compared to the tabletop system, where over 1000 people enter the ecosystem across bunches of tournaments, it’s absurdly more difficult. Second, even if you qualify and do well- or even win- the corresponding invitational, there’s *absolutely no mechanism* for auto-qualifying to the next invitational other than hoping WotC likes you and gives you a discretionary invite, and we already have precedent for the previous PT champion not getting invited. This is in direct contrast to the tabletop system and doesn’t make any sense to me.

Because of this, starting in Rivals is a GIGANTIC advantage when it comes to qualifying for next season’s Rivals via Arena. Rivals players *automatically* qualify for either 2 or 3 invitationals (depending on unreleased details), but even a random who qualifies and does well in the first event has to requalify from scratch to play another one or rely on WotC’s generosity. Somebody might do that, but there are *twelve* spots up for grabs via Arena, and there’s no way in hell there are going to be 12 multi-time qualifiers across three 2x-16-spot-thousands-of-entrants qualifying events. Rivals players effectively get to play one or two extra point-earning events, which is completely unfair and an advantage that doesn’t exist in tabletop.

That’s not even touching on the discretionary invite system. I’m not going to bother talking about the 6 discretionary invites to the Rivals series, because I don’t have anything new to add to that conversation, but having extra discretionary invites to the invitationals on top of that is exactly the same problem as in the previous paragraph. A discretionary invite to a regional PT isn’t that big a deal. Every random with a shot at top-12 is already playing in it. A discretionary invite to an invitational is literally an extra point-earning event that some other people in contention don’t get to play in.. which is the kind of thing WotC just got roasted for, giving LSV a discretionary invite and an extra point-earning event in the race for Worlds. This level of thumb on the scale is a *really* bad look for the supposedly merit/performance-based slots.

In conclusion, there needs to be a merit-based way for randoms who qualify for an invitational via Arena to stay qualified for the next invitational, and it has to be realistic, not “win” or “make top 4” or something crazy. Discretionary invites to invitationals either need to go away completely or to award no points to the Rivals race (or towards Worlds, etc).

Be very, very skeptical of Judge Academy

For the non-Magic: the Gathering audience, some tournament judges filed a lawsuit alleging that they should have been classified as employees. IANAL, and a lot of the issues are beyond the scope of what a non-expert can discuss at all with any confidence, but one aspect- that for-profit companies cannot accept volunteer labor- is one, that by a plain English reading (again, IANAL) they clearly violated thousands of times in the past, including with me personally. Of course, I was “volunteering” with the implicit understanding that I was going to get “gifted” promotional cards and game product worth well into the hundreds of dollars afterwards, and I did.

That’s shady enough domestically with US citizens, but when US citizens went abroad to “work as a volunteer”, and WotC brought foreigners to the US to “work as volunteers”, they were almost certainly running afoul of various labor laws, and on at least one occasion, a judge made the mistake of saying “work” at the border and despite further explanation still wasn’t allowed in the country. So when there’s talk about WotC playing fast and loose with legal obligations around labor.. it’s because they clearly did. And still might, I don’t know.

To avoid these headaches, WotC (purportedly) ditched its judge program and there is a new organization to replace it, Judge Academy. JA held an AMA on Reddit and while it isn’t worth reading the whole thing, they were incredibly evasive on a large number of questions, said they couldn’t disclose financials (for no actual legal reason), and their responses included some gems like (in response to why they aren’t a non-profit), “We also felt it was import not to compete with organizations like the Red Cross for the charitable support being given by these companies.” Really. That happened. So the whole thing looks.. uh.. shady AF.

Looking into the incentives of all those involved and the methods of leverage that can be exercised makes it look even worse than just a shady money grab. Going down the line of what each party wants:

WotC: be free from the legal headache, still effectively control the judge program, spend as little money as possible

Major tournament organizers: still have a competent potential staff pool without spending any new money to train it, keep judges from organizing to ask for more money

High-level judges: still have paying jobs, improve the pay-to-mentoring/training ratio, not have to spend infinite time on the logistics of certification

Now, JA provides certification testing and foils for $100. Without the foils, effectively nobody- and certainly not enough people to bother running a business with staff- would pay $100 to certify. Major tournament organizers would have to pick up the slack and do it at no cost to the trainees. So JA’s business is *entirely* dependent on WotC providing foils that can be resold for over $100 on the secondary market, and they have *zero* recourse if WotC decides they don’t want to do that anymore, either by stopping the foil supply altogether or intentionally sending them garbage to distribute. If WotC does, JA disappears instantly, and both sides are well aware of that. There’s also no chance (for various reasons outside the scope of this post) that JA actually has a contract stipulating a minimum resale value of foils.

So JA *cannot cause trouble for WotC* or WotC just kills it. JA *cannot disobey WotC* or WotC just kills it. Despite being legally separate entities, JA is 100% WotC’s butt muppet. JA has less wiggle room than if they were actually all WotC employees because at least then they’d have some workplace protections, which is kind of ironic given the whole context.

Big TOs will be happy with this for several reasons. The first is that they don’t have to put more of their own resources into maintaining a qualified staff pool around their regions. The second is that because JA is essentially existentially forbidden from causing any trouble, it’s not going to agitate for better working conditions/compensation, and any energy directed at lobbying JA, or any misunderstanding that JA might ever do that is less energy directed at anything that could affect TO bottom line.

There’s not going to be any elected representation for obvious reasons. JA cant cause trouble, so they’re going to vet staff carefully and only work with people who “get it”. And by “get it”, I mean understand that JA is not an organization for judges, it’s an organization that exists to be WotC’s butt muppet, keep big TOs staffed and happy, and get Tim and some high-level judges paid.

There’s no financial transparency- and some combination of incompetence/misrepresentations/blatant lies whenever financials are discussed- because the whole arrangement is shady AF on every level. There’s no way they’re going to go from “shady and opaque” to providing a line by line accounting of their revenue and expenditures that shows everybody exactly how messed up the whole situation is. If JA were an organization for judges, they’d be happy to prove it with financials- and if they were legally registered as one of several types of organizations for judges, they would HAVE TO prove it with financials- but again, they’re not an organization for judges and they’re simply choosing not to be transparent.

When an organization is de facto funded by somebody else (by foils laundered to cash through subscriptions), is incentivized to act in somebody else’s interests, has denied to enter into any obligation to act in your interests, has refused to allow you any ability to determine that it is acting in your interests, and has dodged/obfuscated/misrepresented/outright lied to you repeatedly when these issues are raised, you have to be an absolute fool to trust that it really is going to act in your interests and not somebody else’s.

Mythic Championship III Day 1-Blatant viewer manipulation and group breakdowns

First off, the level of view-count fraud was absolutely out of control today. The bullshit today (ht: darrenoc on reddit) isn’t particularly different than the bullshit they pulled with the Mythic Invitational, but the actual viewership today was anemic to begin with. From the time I first checked, around the start of round 2, until the end of round 8, the number of people in chat (chat being sub-only is ~irrelevant to this) was between 9,000 and 11,500. Since 70-75% of viewers in most large channels are logged in to chat, that’s a real viewership of 12k-16k. Going slightly above that isn’t impossible, but not by too much.

The nominal viewership I saw got as high as 65k, which means that literally 75-80%, or very close, of the reported viewer count was completely fake. Once WotC stopped paying for new fake views, and the numbers started decaying as the day wound down, total views dropped from the 60-thousands to the 20-thousands while the actual people logged into chat- representative of real viewers- stayed in the same 9k-11.5k range. It’s utterly and blatantly fraudulent. There’s a long section about WotC’s viewer fraud in this Kotaku article (open it and ctrl-f magic), and if it’s correct, WotC is spending *hundreds of thousands of dollars per event* for the sole purpose of creating transparently fraudulent viewer numbers.

That’s utterly disgusting.

On to the actual day 1 results.. I’m sure there will be several metagame breakdowns posted elsewhere, so I’m not bothering with that, especially since I had to go derive and input round 7 and 8 results by hand because the official page had this……………..

lolround7

and round 8 results still aren’t up, but I was mainly curious how the different kinds of players did.

I classified the players into 4 groups: MPL members, pros/ex-pros, challengers, and invited personalities from the extra 16 invites (lists at the bottom of the post). The only questionable classification was former PT champion Simon Görtzen, who does commentary now instead of playing. I put him with the pros/ex-pros based on his pro history and that he wasn’t one of the extra invites. These are the performances of each group vs. each other group.

left vs. top	MPL	Pro/ex-pro	Challenger	Personality
MPL	42-42	19-22	27-18	18-12
Pro/ex-pro	22-19	11-11	7-6	6-3
Challenger	18-27	6-7	8-8	5-5
Personality	12-18	3-6	5-5	9-9

Combining the group performances and looking at day 2 conversion rates (not counting the 4 MPL players with byes into day 2) gives

	vs. out of group	out of group win%	day 2	day 2 advance
MPL	64-52	55.2%	6/28	21.4%
Pro/ex-pro	35-28	55.6%	5/13	38.5%
Challenger	29-39	42.6%	1/13	7.7%
Personality	20-29	40.8%	0/10	0%

Looks like the pros crushed it, taking it to the MPL 22-19 while the MPL went 45-30 against the challengers and personalities. There’s a marked difference between those who are/have been at the top of the game and those who’ve never come close.

————————————————————————————————————

Player lists (Bold = day 2)

MPL:

Alexander Hayne
Andrea Mengucci
Andrew Cuneo
Autumn Burchett
Ben Stark
Carlos Romao
Christian Hauck
Eric Froehlich
Grzegorz Kowalski
Janne Mikkonen
Javier Dominguez
Jean Emmanuel Depraz
Jessica Estephan
John Rolf
Lee Shi Tian
Lucas Esper Berthoud
Luis Salvatto
Marcio Carvalho
Martin Juza
Matthew Nass
Mike Sigrist
Paulo Vitor Damo da Rosa
Piotr Glogowski
Reid Duke
Seth Manfield
Shahar Shenhar
Shota Yasooka
William Jensen

Pros/ex-pros:

Allen Wu
Andrew Elenbogen
Ben Hull
Corey Burkhart
Greg Orange
Kai Budde
Kentaro Yamamoto
Luis Scott Vargas
Noah Walker
Ondrej Strasky
Raphaël Lévy
Simon Görtzen
Wyatt Darby

Challengers:

Alexey Shashov
André Santos
CJ Steele
Eric Oresick
Evan Gascoyne
Marcin Tokajuk
Matias Leveratto
Montserrat Ayensa
Nicholas Carlson
Patrick Fernandes
Takashi Iwasaki
Yuki Matsuda
Yuma Koizumi

Personalities:

Amy Demicco
Ashley Espinoza
Audrey Zoschak
Emma Handy
Giana Kaplan
Jason Chan
Jeffrey Brusi
Nhi Pham
Teresa Pho
Vanessa Hinostroza

Bill James and the Trump polarization problem

Bill has been running a series of polls with various candidates for president matched against each other, in an attempt to create a ranking system like this current one at time of writing. There’s no question that the polls can be used to create the rankings, but for the rankings to be meaningful, preferences need to have certain properties, and they don’t appear to.

Starting with Bill’s college football example, if a team is expected to be remotely competitive against Clemson (national champion), they’re also expected to crush UTEP (horrific), and if a team’s game is expected to be remotely competitive against UTEP, they’re expected to get obliterated by Clemson. There’s no concept of a team that’s 30% against both, or 60-40 vs UTEP and 40-60 vs Clemson (while Clemson is 99%+ against UTEP). The football season works as a reasonable approximation because the teams can be given a rating, on one axis, and every pairwise comparison of teams is expected to play out “close” to the rating difference.

If teams with pathological properties in the previous paragraph existed, it would be *impossible* to give them any ratings that wouldn’t produce a bunch of wildly inaccurate predictions, and at that point, it’s not clear what the numbers would represent (since they can’t be used for pairwise or group predictions without a bunch of grievous errors), or what the point of the exercise would be. Any time the pairwise matchups depend on multiple axes- something beyond the assigned rating from Bill’s system- it’s possible for the exercise to go completely haywire.

In college football, the secondary axes, the matchup-specific details, aren’t nonexistent, but they’re much smaller than the primary axis overall quality difference, so the rating method basically works. If a major secondary axis were added where every team were randomly assigned one of rock-paper-scissors at the beginning of the season, and the “winner” started ahead 14-0, the overall ratings wouldn’t be particularly different (treating the 14 points as legitimately scored for the rating calculations), but the pairwise rating-based game predictions would be utterly haywire because they’d have no idea when one of the teams was going to start with two free touchdowns, and *every* possible set of ratings is going to go totally bonkers with game predictions under that setup. That example is totally contrived of course, but the general point is that it’s *impossible* to represent a system with multiple significant axes with one rating number and have it reliably mean anything prediction-wise.

Politics has the obvious multiple axes of party affiliation and candidate preference within the party. and while party affiliation is not absolute, a significant number of people are going to order their preferences as either (almost any D > almost any R) or (almost any R > almost any D), which means that you run into the “team that’s 30% against both UTEP and Clemson” problem. There were 6 polls with Trump against 3 Democrats and he polled 26%-29% against fields headlined by everybody from Warren or Biden down to Booker or Abrams. My guess is he wouldn’t go far above 29%, if at all, even in a field headlined by Inslee. Warren and Biden smash Booker and Abrams head to head, but Trump polls the same against all of them (the only variability is when a second R is included).

There’s no single rating to give Trump that doesn’t go completely bonkers with his predictions against most of the range of D candidates. The system just doesn’t work at all with Trump. Trump is basically his own axis, even stronger than R/D alone, because he’s so polarizing. 26-29% rank him above all Ds, and 70%+ of the poll respondents rank him near last place. The system looks reasonable for Ds relative to other Ds because none of the leaders are super-polarizing relative to the others right now, but that’s not a thing that ever has to be true or stay true.

Because Trump can’t be rated properly by this method, and rating other Rs alongside Ds is also going to be super sketchy for similar reasons, either always including Trump and ignoring his number for ranking the Ds against each other or only including Ds to begin with both seem like improvements, although the latter comes with guaranteed-R voters voting on Ds, which isn’t necessarily ideal.

The quantitative effect of voting machine vulnerabilities in the US

TL;DR Democrats can’t win the presidency in 2020 without flipping deep red (R+15 or more in 2016) states or at least one, and probably two states that Rs won in 2016 that have multiple severe election security vulnerabilities.

Election security has been a hot-button topic lately, but I have yet to see any articles about how much these vulnerabilities allow the 2020 election to be manipulated. As an introduction to election security issues, I highly recommend watching why electronic voting is a bad idea (short and entertaining, trust me), and if you want a more academic take, this recent paper discusses the issues with ballot-marking devices (BMD). This blog is in complete agreement with the paper that the only legitimate use for ballot-marking devices is for those who are physically incapable of hand-marking a paper ballot by themselves, but still doesn’t consider them a necessity for that purpose (states can have voter-assistance protocols and only use hand marked paper ballots). BMDs and other voting machines are technologies that have absolutely no reason to exist for the general population, but thanks to ignorance and good old-fashioned corruption, we’ve given corporate handouts of hundreds of millions of dollars in return for machines that are worse than worthless and compromise the very ability of a fair and verifiable election to exist in many jurisdictions.

This post only covers threats that result in the final vote count not reflecting the votes that were cast. Compromising the list of eligible voters, engaging in a variety of forms of voter suppression, and packed courts simply refusing to accept the results even after a recount are also dangers, but they’re beyond the scope of this post. This post would not be possible without the resources at Verified Voting, and unless sourced otherwise, information about voting methods in use are from there. Electoral college maps are from 270towin.com. Links to recount statutes were mostly found on Ballotpedia.

There are three major types of voting equipment in use.

The hand-marked paper ballot, generally read by an optical scanning machine. While the video above is correct that the scanning machine is vulnerable to attacks, the defense to these attacks is the ability to hand-count ballots, and having a candidate-funded recount always available by law is the ultimate backstop against scanner attacks. The robustness of various recount schemes will be discussed in the state-by-state section later.

Ballot-marking devices are extremely expensive pencils that fill out a ballot that a scanner then reads. While useful for the small number of physically impaired voters, as the paper above notes, in practice it’s difficult to make sure that they’re working properly on election day. Quoting from the paper, “half of voters don’t look at their ballot printout at all, and those who do look for an average of 4 seconds”. They’re brutally vulnerable for down-ballot races, and even for the top race (e.g. President), attacks to change the overall margin by 1-2% are close to undetectable under optimistic assumptions and possibly even 5% or more under real-world conditions. The defense to BMD attacks is simply to not use them, or at worst for only the physically impaired to use them. Because they create a ballot that then has to be scanned, scanner attacks from the previous section are also still in play. These are bad- really bad- but at least they aren’t…

Direct recording electronic systems (DRE) record votes directly on the machine itself. This is obviously a complete security disaster. Some machines also create a paper ballot, which would make them similar to the BMD group- if they work properly. The ES&S ExpressVote XL and Dominion ImageCast Evolution have a ridiculous security flaw that allows users to irrevocably decline to review their ballot, AND THE MACHINE ONLY PRINTS THE BALLOT AFTER THAT- whatever ballot it feels like printing because the voter can’t detect it anymore. The Dominion ImageCast X can’t do that, but it can print on the ballot after it has been “verified” for the last time. Because the ImageCast X can only fill in races where the voter didn’t record a vote, that flaw is much more limited in scope, especially for top-of-the-ballot races, but it, and all other DREs with a paper trail, are at *least* as bad as BMDs above. The defense to DREs is to dump them all in the bottom of the sea.

This post is going to focus on these weakness, but if you really feel like being depressed, there are a lot of security weaknesses that we’re not addressing here. It’s really difficult to overstate the attack surface all of these electronics allow, and none of them would be more than mere annoyances if hand-marked paper ballots with a hand recount always available were adopted everywhere. But alas..

This is a map of the 2016 election (deep red (e.g. OK, LA) = Trump crushed, pink (e.g. FL, PA) = Trump won small, etc).

2016base

Deep blue and deep red states aren’t going to be examined- if any are flipped legitimately, the election is almost certainly over, and hacking them is unnecessary and too obvious. Looking at the competitive states, some are pretty boring from a security perspective.

These states all have hand-marked paper ballots and hand recounts always available (if the candidate is willing to pay for it, of course).

Oregon (D+11).

New Mexico (D+8)

Colorado (D+4)

Maine (D+3) / Maine District 2 (R+10)

Minnesota (D+1.5)

Michigan (R+0.2)

Nebraska District 2 (R+2)

These states use all or predominantly hand-marked paper ballots, but have issues with their recount protocol (ranging from likely inconsequential to extremely vulnerable)

New Hampshire (D+0.3) All paper ballots, hand recount available if the margin of victory is within 20%. This is a stupid rule, but cheating blatantly enough in 2020 to produce R or D +20 in a competitive national election would be even more stupid.

Connecticut (D+13) All paper ballots, hand recount mandatory in very close races *or* if the election moderators suspect shenanigans. Any R win outside the margin of error would qualify as shenanigans unless the national election is a bloodbath in Trump’s favor. This is a horrible system that just isn’t likely to be exploited here. Discretion on whether or not a recount is ever performed shouldn’t exist and certainly shouldn’t belong to one party.

Iowa (R+9) All paper ballots, recount always available, but the election officials have the discretion to recount ballots with machines again instead of by hand.. which removes the ability to correct machine attacks. As in Connecticut, this discretion shouldn’t exist.

Virginia (D+5) All paper ballots, recount only available in very close elections. Furthermore, recounts for optically scanned ballots are *rescanned by machine only*. The machines are supposedly tested before the recount, but that obviously doesn’t defend against certain attacks. This statute is completely insane in two ways. If one side simply cheats *a lot*, there’s no recount available at all, and never doing a hand recount, when a major purpose of a recount is to fix machine screwups and defend against machine cheats, is inexcusable.

Arizona (R+3) Mostly paper ballots by mail (~80% of votes) and a mix of paper and BMDs at precincts. The only recount available is if the vote tally is within 0.1%. There’s a mandated pre-count check on some machines that compares machine and hand counts for 1-2% of ballots before counting all the ballots. That’s trivially defeated by telling the machines to be honest for the first X ballots, and the centralized locations of vote-by-mail counting makes it possible to change *a lot* of votes by compromising a very small number of machines in one place, and the lack of a recount allows it to work. This is effectively the exact system that the video was warning against re: blindly trusting scanning machines.

These states use significant numbers of BMDs or BMD-equivalent DREs, but have hand recounts always available.

Once the electronics are introduced for the act of marking a ballot, they become an attack surface along with the scanners used to count those ballots. In this group, the scanner vulnerabilities are mitigated by the hand recount availability, but the BMD vulnerabilities discussed above remain. And for those who trust in machines and state election officials and all, these machines were already a disaster without being maliciously attacked.

Nevada (D+2) Almost entirely Dominion ImageCast X (one that can still print on ballots after voter verification) with no plans that I can find to dump this before 2020. Amusingly, a quirk in Nevada law dating to the 1970s requires an option for “None of the above candidates” in every race, allowing voters to affirmatively mark a ballot for “Nobody” instead of simply leaving the contest blank…. which mitigates the ImageCast X design flaw of being able to print votes in contests the voter left blank, making it effectively a BMD.

Ohio (R+8) has a mix of paper and BMD/DREs now, but it’s pushing towards paper ballots and simple BMDs. No DREs are certified so far for 2020, and hopefully that will continue to be true.

These states use significant numbers of BMDs or BMD-equivalent DREs and have no hand recount always available, making them doubly vulnerable

North Carolina (R+3) Mix of paper ballots, BMDs, and DREs with a paper trail in 2016. The main current DRE (iVotronic) is getting decertified for 2020. Counties appear to be individually responsible for selecting new systems that either use hand-marked paper ballots or mark a paper ballot, and that leaves the possibility of a significant number of ExpressVote XLs and ImageCasts appearing on the scene, which would warrant an even worse grouping. Even if it’s “just” a lot of new BMDs, recounts are only available for races within 1%, leaving the scanners vulnerable as well.

These states will have no ability to conduct a verifiable close election in a statewide race, either because they use enough machines with no paper trail or use enough ExpressVote XLs and/or ImageCast Evolutions to render the paper trail meaningless.

Wisconsin (R+0.7%) Mix of paper ballots and a wide variety of BMDs, and DREs with a paper trail, including 10.7% of municipalities using ImageCast Evolutions. Furthermore, Wisconsin only allows recounts in very close races, so all three avenues are vulnerable- the scanners, the BMDs, and the ImageCast Evolutions.

Delaware (D+11) Wasting tens of millions of dollars to replace everything with ExpressVote XLs for 2020. This is completely insane. Recounts are only available in close races (as useless as recounting fabricated ballots would be).

New Jersey (D+14) mishmash of different DREs with no paper trail and no coherent plan to not be quite vulnerable in 2020. A couple of counties might move to something less awful, but not enough to matter. Candidates can pay for a recount… except there aren’t any ballots to count again.

Texas (R+9) Has tons of DREs with no paper trail and no plan to change that for 2020.

Florida (R+1) Mostly paper, but enough DREs with no paper trail to flip any legitimately close election. If this is somehow remedied by 2020, the recount law is terribly deficient- only races within 0.25% get hand recounts and races within 0.25%-0.50% get a machine recount, which would put Florida in the Arizona group.

Georgia (R+5) currently uses all DREs with no paper trail. There is talk of replacing these machines by 2020, but given the entanglements between Georgia politicians and ES&S, it would almost certainly be with ES&S ExpressVote XLs, which means replacing no paper trail with a potentially fake paper trail.

And then there’s Pennsylvania..

Pennsylvania is currently a dumpster fire like Georgia or Texas, overrun by DREs with no paper trail. The governor is making a hard push to replace these by 2020, which may or may not work. And when it does “work”, places like Philadelphia can just buy ExpressVote XLs and not really make progress on the security problem. It’s not clear exactly how bad election vulnerability in PA will be, but it’s horrible now and it seems unlikely that what needs to happen to secure it- almost all current DREs removed and almost no ExpressVote XLs and ImageCasts installed- will actually happen, or probably even come close to happening.

What does this all mean?

Under the hypothetical 2020 scenarios where Democrats do a bit better across the board (if Rs do, voting security doesn’t matter since they’re winning in a landslide), let’s look at what happens with a little malfeasance that benefits Republicans. Under the following rules:

Democrats hold every state they won in 2016 (Rs don’t try to rig NJ or DE because it’s too obvious and leave NV alone since it’s a bit risky to flip, say, 5% off BMD deficiencies on the top-ballot race)

Republicans or Republican-aligned interests take the low-hanging fruit and rig the horribly vulnerable elections in states they control (TX/GA/AZ/FL/IA) as well as holding all the deep red states.

We get this map, marking “decided” states in deep color, states/districts with R>=+5 in 2016 in pink, and everything else a tossup

2016simplerig2

Democrats need 38 more (269-269 is a R win), which is not a simple ask. Ohio (R+8) was significantly red and will be BMD-infested at best. Michigan,Nebraska District 2, and Maine District 2 are fair, but WI and likely PA are both dumpster fires in terms of security, and NC is a real mess as well.

It’s *literally impossible*, unless D’s flip one or more deep red states somehow, to win in 2020 without taking a state that has serious-to-extreme election security flaws, and most likely more than one. Breaking this down a little further, if D’s lose Michigan (a fair state), they’re also almost guaranteed to lose OH that they lost by 8% more in 2016, meaning they have to sweep WI/PA/NC, all of which have major security issues. I wouldn’t want to be the Ds there even if those elections were fairly counted. Assuming the Ds win MI, which they likely do in a legitimate win, gives this map

winmich

which clarifies things considerably. Ds have 2 paths, winning PA, NE 2 (R+2), and ME 2 (R+10), which is a really tough ask, or winning any 2 of OH, WI, PA, NC. That’s a 2016 R+8 state and 3 states that Rs won in 2016 with disaster-level election security problems. If you take election fairness seriously at all, this has to be terrifying.

There is a path out for Ds here, but it requires strong leadership and decisive action on something that’s still a fringe issue to most people in and out of government, and it goes against entrenched corporate interests, so I know damn well it’s never going to happen, but the proper course of action security-wise is for WI and NC to immediately decertify all electronics, including scanners, and for PA to decertify all electronics except for scanners. PA can allow scanners because it has a solid recount statute, but WI and NC don’t (unless their Republican-controlled legislatures decide to pass one), so the only safe method for them for now is good old hand-counting.

And don’t forget the various forms of attacks on the voter rolls that are already happening that weren’t discussed here……..

P.S. Comments are moderated. Corrections or additions are welcome. Legitimate questions and interesting content are welcome. JAQing off and random partisan hackery won’t be approved.

Trust the barrels

Inspired by the curious case of Harrison Bader

baderbarrels

whose average exit velocity is horrific, hard hit% is average, and barrel/contact% is great (not shown, but a little better than the xwOBA marker), I decided to look at which one of these metrics was more predictive. Barrels are significantly more descriptive of current-season wOBAcon (wOBA on batted balls/contact), and average exit velocity is sketchy because the returns on harder-hit balls are strongly nonlinear. The game rewards hitting the crap out of the ball, and one rocket and one trash ball come out a lot better than two average balls.

Using consecutive seasons with at least 150 batted balls (there’s some survivor bias based on quality of contact, but it’s pretty much even across all three measures), which gave 763 season pairs, barrel/contact% led the way with r=0.58 to next season’s wOBAcon, followed by hard-hit% at r=0.53 and average exit velocity at r=0.49. That’s not a huge win, but it is a win, but since these are three ways of measuring a similar thing (quality of contact), they’re likely to be highly correlated, and we can do a little more work to figure out where the information lies.

evvehardhit

I split the sample into tenths based on average exit velocity rank, and Hard-hit% and average exit velocity track an almost perfect line at the group (76-77 player) level. Barrels deviate from linearity pretty measurably with the outliers on either end, so I interpolated and extrapolated on the edges to get an “expected” barrel% based on the average exit velocity, and then I looked at how players who overperformed and underperformed their expected barrel% by more than 1 SD (of the barrel% residual) did with next season’s wOBAcon.

Avg EV decile	>2.65% more barrels than expected	average-ish barrels	>2.65% fewer barrels than expected	whole group
0	0.362	0.334	none	0.338
1	0.416	0.356	0.334	0.360
2	0.390	0.377	0.357	0.376
3	0.405	0.386	0.375	0.388
4	0.389	0.383	0.380	0.384
5	0.403	0.389	0.374	0.389
6	0.443	0.396	0.367	0.402
7	0.434	0.396	0.373	0.401
8	0.430	0.410	0.373	0.405
9	0.494	0.428	0.419	0.441

That’s.. a gigantic effect. Knowing barrel/contact% provides a HUGE amount of information on top of average exit velocity going forward to the next season. I also looked at year-to-year changes in non-contact wOBA (K/BB/HBP) for these groups just to make sure and it’s pretty close to noise, no real trend and nothing close to this size.

It’s also possible to look at this in the opposite direction- find the expected average exit velocity based on the barrel%, then look at players who hit the ball more than 1 SD (of the average EV residual) harder or softer than they “should” have and see how much that tells us.

Barrel% decile	>1.65 mph faster than expected	average-ish EV	>1.65 mph slower than expected	whole group
0	0.358	0.339	0.342	0.344
1	0.362	0.359	0.316	0.354
2	0.366	0.364	0.361	0.364
3	0.389	0.377	0.378	0.379
4	0.397	0.381	0.376	0.384
5	0.388	0.395	0.418	0.397
6	0.429	0.400	0.382	0.403
7	0.394	0.398	0.401	0.398
8	0.432	0.414	0.409	0.417
9	0.449	0.451	0.446	0.450

There’s still some information there, but while the average difference between the good and bad EV groups here is 12 points of next season’s wOBAcon, the average difference for good and bad barrel groups was 50 points. Knowing barrels on top of average EV tells you a lot. Knowing average EV on top of barrels tells you a little.

Back to Bader himself, a month of elite barreling doesn’t mean he’s going to keep smashing balls like Stanton or anything silly, and trying to project him based on contact quality so far is way beyond the scope of this post, but if you have to be high on one and low on the other, lots of barrels and a bad average EV is definitely the way to go, both for YTD and expected future production.

Uncertainty in baseball stats (and why DRC+ SD is a category error)

What does it mean to talk about the uncertainty in, say, a pitcher’s ERA or a hitter’s OBP? You know exactly how many ER were allowed, exactly how many innings were pitched, exactly how many times the batter reached base, and exactly how many PAs he had. Outside of MLB deciding to retroactively flip a hit/error decision, there is no uncertainty in the value of the stat. It’s an exact measurement. Likewise, there’s no uncertainty in Trout’s 2013 wOBA or wRC+. They reflect things that happened, calculated in deterministic fashion from exact inputs. Reporting a measurement uncertainty for any of these wouldn’t make any sense.

The Statcast metrics are a little different- EV, LA, sprint speed, hit distance, etc. all have a small amount of random error in each measurement, but since those errors are small and opportunities are numerous, the impact of random error is small to start with and totally meaningless quickly when aggregating measurements. There’s no point in reporting random measurement uncertainty in a public-facing way because it may as well be 0 (checking for systematic bias is another story, but that’s done with the intent of being fixed/corrected for, not of being reported as metric uncertainty).

Point 1:

So we can’t be talking about the uncertainties in measuring/calculating these kinds of metrics- they’re irrelevant-to-nonexistent. When we’re talking about the uncertainty in somebody’s ERA or OBP or wRC+, we’re talking about the uncertainty of the player’s skill at the metric in question, not the uncertainty of the player’s observed value. That alone makes it silly to report such metrics as “observed value +/- something”, like ERA 0.37 +/- 3.95, because it’s implicitly treating the observed value as some kind of meaningful central-ish point in the player’s talent distribution. There’s no reason for that to be true *because these aren’t talent metrics*. They’re simply a measure of something over a sample, and many such metrics frequently give values where a better true talent is astronomically unlikely to be correct (a wRC+ over 300) or even impossible (an ERA below 0) and many less extreme but equally silly examples as well.

Point 2:

Expressing something non-stupidly in the A +/- B format (or listing percentiles if it’s significantly non-normal, whatever) requires a knowledge of the player’s talent distribution after the observed performance, and that can’t be derived solely from the player’s data. If something happens 25% of the time, talent could cluster near 15% and the player is doing it more often, talent could cluster near 35% and the player is doing it less often, or talent could cluster near 25% and the player is average. There’s no way to tell the difference from just the player’s stat line and therefore no way to know what number to report as the mean, much less the uncertainty. Reporting a 25% mean might be correct (the latter case) or as dumb as reporting a mean wRC+ of 300 (if talent clusters quite tightly around 15%).

Once you build a prior talent distribution (based on what other players have done and any other material information), then it’s straightforward to use the observed performance at the metric in question and create a posterior distribution for the talent, and from that extract the mean and SD. When only the mean is of interest, it’s common to regress by adding some number of average observations, more for a tighter talent distribution and fewer for a looser talent distribution, and this approximates the full Bayesian treatment. If the quantity in the previous paragraph were HR/FB% (league average a little under 15%), then 25% for a pitcher would be regressed down a lot more than for a batter over the same number of PAs because pitcher HR/FB% allowed talent is much more tightly distributed than hitter HR/FB% talent, and the uncertainty reported would be a lot lower for the pitcher because of that tighter talent distribution. None of that is accessible by just looking at a 25% stat line.

Actual talent metrics/projections, like Steamer and ZiPS, do exactly this (well, more complicated versions of this) using talent distributions and continually updating with new information, so when they spit out mean and SD, or mean and percentiles, they’re using a process where those numbers are meaningful, getting them as the result of using a reasonable prior for talent and therefore a reasonable posterior after observing some games. Their means are always going to be “in the middle” of a reasonable talent posterior, not nonsense like wRC+ 300.

Which brings us to DRC+.. I’ve noted previously that the DRC+ SDs don’t make any sense, but I didn’t really have any idea how they were coming up with those numbers until this recent article, and a reference to this old article on bagging. My last two posts pointed out that DRC+ weights way too aggressively in small samples to be a talent metric and that DRC+ has to be heavily regressed to make projections, so when we see things in that article like Yelich getting assigned a DRC+ over 300 for a 4PA 1HR 2BB game, that just confirms what we already knew- DRC+ is happy to assign means far, far outside any reasonable distribution of talent and therefore can’t be based on a Bayesian framework using reasonable talent priors.

So DRC+ is already violating point 1 above, using the A +/- B format when A takes ridiculous values because DRC+ isn’t a talent metric. Given that it’s not even using reasonable priors to get *means*, it’s certainly not shocking that it’s not using them to get SDs either, but what it’s actually doing is bonkers in a way that turns out kind of interesting. The bagging method they use to get SDs is (roughly) treating the seasonal PA results as the exact true talent distribution of events, drawing from them over and over (with replacement) to get a fake seasonal line, doing that a bunch of times and taking the SD of the fake seasonal lines as the SD of the metric.

That’s obviously just a category error. As I explained in point 2, the posterior talent uncertainty depends on the talent distribution and can’t be calculated solely from the stat line, but such obstacles don’t seem to worry Jonathan Judge. When talking about Yelich’s 353 +/- 6 DRC+, he said “The early-season uncertainties for DRC+ are high. At first there aren’t enough events to be uncertain about, but once we get above 10 plate appearances or so the system starts to work as expected, shooting up to over 70 points of probable error. Within a week, though, the SD around the DRC+ estimate has worked its way down to the high 30s for a full-time player.” That’s just backwards about everything. I don’t know (or care) why their algorithm fails under 10 PAs, but writing “not having enough events to be uncertain about” shows an amazing misunderstanding of everything.

The accurate statement- assuming you’re going in DRC+ style using only YTD knowledge of a player- is “there aren’t enough events to be CERTAIN about of much of anything”, and an accurate DRC+ value for Yelich- if DRC+ functioned properly as a talent metric- would be around 104 +/- 13 after that nice first game. 104 because a 4PA 1HR 2BB game preferentially selects- but not absurdly so- for above average hitters, and a SD of 13 because that’s about the SD of position player projections this year. SDs of 70 don’t make any sense at all and are the artifact of an extremely high SD in observed wOBA (or wRC+) over 10-ish PAs, and remember that their bagging algorithm is using such small samples to create the values. It’s clear WHY they’re getting values that high, but they just don’t make any sense because they’re approaching the SD from the stat line only and ignoring the talent distribution that should keep them tight. When you’re reporting a SD 5 times higher than what you’d get just picking a player talent at random, you might have problems.

The Bayesian Central Limit Theorem

I promised there was something kind of interesting, and I didn’t mean bagging on DRC+ for the umpteenth time, although catching an outright category error is kind of cool. For full-time players after a full season, the DRC+ SDs are actually in the ballpark of correct, even though the process they use to create them obviously has no logical justification (and fails beyond miserably for partial seasons, as shown above). What’s going on is an example of the Bayesian Central Limit Theorem, which states that for any priors that aren’t intentionally super-obnoxious, repeatedly observing i.i.d variables will cause the posterior to converge to a normal distribution. At the same time, the regular Central Limit Theorem means that the distribution of outcomes that their bagging algorithm generates should also approach a normal distribution.

Without the DRC+ processing baggage, these would be converging to the same normal distribution, as I’ll show with binomials in a minute, but of course DRC+ gonna DRC+ and turn virtually identical stat lines into significantly different numbers

NAME	YEAR	PA	1B	2B	3B	HR	TB	BB	IBB	SO	HBP	AVG	OBP	SLG	OPS	ISO	oppOPS	DRC+	DRC+ SD
Pablo Sandoval	2014	638	119	26	3	16	244	39	6	85	4	0.279	0.324	0.415	0.739	0.136	0.691	113	7
Jacoby Ellsbury	2014	635	108	27	5	16	241	49	5	93	3	0.271	0.328	0.419	0.747	0.148	0.696	110	11

Ellsbury is a little more TTO-based and gets an 11 SD to Sandoval’s 7. Seems legit. Regardless of these blips, high single digits is about right for a DRC+ (wRC+) SD after observing a full season.

Getting rid of the DRC+ layer to show what’s going on, assume talent is uniform on [.250-.400] (SD of 0.043) and we’re dealing with 1000 Bernoulli observations. Let’s say we observe 325 successes (.325), then when we plot the Bayesian posterior talent distribution and the binomial for 1000 p=.325 events (the distribution that bagging produces)

325posterior

They overlap so closely you can’t even see the other line. Going closer to the edge, we get, for 275 and 260 observed successes,

At 275, we get a posterior SD of .13 vs the binomial .14, and at 260, we start to break the thing, capping how far to the left the posterior can go, and *still* get a posterior SD of .11 vs .14. What’s going on here is that the weight for a posterior value is the prior-weighted probability that that value (say, .320) produces an observation of .325 in N attempts, while the binomial bagging weight at that point is the probability that .325 produces an observation of .320 in N attempts. These aren’t the same, but under a lot of circumstances, they’re pretty damn close, and as N grows, and the numbers that take the place of .320 and .325 in the meat of the distributions get closer and closer together, the posterior converges to the same normal that describes the binomial bagging. Bayesian CLT meets normal CLT.

When the binomial bagging variance starts dropping well below the prior population variance, this convergence starts to happen enough to where the numbers can loosely be called “close” for most observed success rates, and that transition point happens to come out around a full season of somewhat regressed observation of baseball talent. In the example above, the prior population SD was 0.043 and the binomial variance was 0.014, so it converged excellently until we ran too close to the edge of the prior. It’s never always going to work, because a low end talent can get unlucky, or a high end talent can get lucky, and observed performance can be more extreme than the talent distribution (super-easy in small samples, still happens in seasonal ones) but for everybody in the middle, it works out great.

Let’s make the priors more obnoxious and see how well this works- this is with a triangle distribution, max weight at .250 straight line down to a 0 weight at .400.

290 successes, .0143 vs .0140

325 successes, .0144 vs .0148

350 successes, .0143 vs .0151

325 successes, .0144 vs .0148

290 successes, .0143 vs .0140

The left-weighted prior shifts the means, but the standard deviations are obviously about the same again here. Let’s up the difficulty even more, starting with a N(.325,.020) prior (0.020 standard deviation), which is pretty close to the actual mean/SD wOBA talent distribution among position players (that distribution is left-weighted like the triangle too, but we already know that doesn’t matter much for the SD)

360 successes, .012 vs .015

325 successes, .012 vs .015

290 successes, .012 vs .013

Even now that the bagging distributions are *completely* wrong and we’re using observations almost 2 SD out, the standard deviations are still .014-.015 bagging and .012 for the posterior. Observing 3 SD out isn’t significantly worse. The prior population SD was 0.020, and the binomial bagging variance was 0.014, so it was low enough that we were close to converging when the observation was in the bulk of the distribution but still nowhere close when we were far outside, although the SDs of the two were still in the ballpark everywhere.

Using only 500 observations on the N(.325,.020) prior isn’t close to enough to pretend there’s convergence even when observing in the bulk.

325500pa

The posterior has narrowed to a SD of .014 (around 9 points of wRC+ if we assume this is wOBA and treat wOBA like a Bernoulli, which is handwavy close enough here), which is why I said above that high-single-digits was “right”, but the binomial variance is still at .021, 50% too high. The regression in DRC+ tightens up the tails compared to “binomial wOBA”, and it happens to come out to around a reasonable SD after a full season.

Just to be clear, the bagging numbers are always wrong and logically unjustified here, but they’re a hackjob that happens to be “close” a lot of the time when working with the equivalent of full-season DRC+ numbers (or more). Before that point, when the binomial bagging variance is higher than the completely naive population variance (the mechanism for DRC+ reporting SDs in the 70s, 30s, or whatever for partial seasons), the bagging procedure isn’t close at all. This is just another example of DRC+ doing nonsense that looks like baseball analysis to produce a number that looks like a baseball stat, sometimes, if you don’t look too closely.