Last time I ran the New York Times Spelling Bee through the machinery of cooperative game theory, treating each day’s word list as a weighted voting game: words are voters, points are votes, Genius is the quota, and a word has power when it’s the one that carries you across the line. The punchline was that the power ranking is boring — a rescaling of the scoreboard — until you replace the fiction that you find words in random order with a model of how people really solve, common words first. Then the ranking shatters, and the day’s pangram, FANCIFUL, suffers the largest collapse on the board. I called the honest, order-aware index the Waggle index (Ed.: two letters off from Wuffle, John. You know Bernie Grofman will notice. Bernie Grofman notices everything.), and I promised you a curse.
Here’s the curse. I said FANCIFUL “falls.” It does, eventually. What it does first is the interesting part.
Turn the dial slowly
The Waggle index has a knob. Call it \(\alpha\), and let me be exact about what it turns, because MOP readers deserve the real definition and not a wave of the hand.
Words get found one at a time. Give each word a findability \(g\) — for now, how common the word is in ordinary English; I’ll say where that number comes from in a moment. The discovery process then draws the next word from those still unfound, in proportion to findability raised to the power \(\alpha\):
\[ \Pr\bigl(\text{next found}=j \mid \text{already found } S\bigr) \;=\; \frac{g_j^{\,\alpha}}{\sum_{k \notin S} g_k^{\,\alpha}}, \qquad j \notin S. \]
That is the whole model. (Formally it is a Plackett–Luce process; equivalently, hand each word an alarm clock that rings after an \(\mathrm{Exponential}(g^{\alpha})\) delay and read the words off in the order they ring.) A word \(i\) is pivotal when it carries the running score across the Genius line \(q\): the words found before it, the set \(B_i\), sum to just under \(q\), and word \(i\)’s own score \(w_i\) finishes the climb. The Waggle index is the probability of exactly that event, over the discovery order above:
\[ W_\alpha(i) \;=\; \Pr\!\left( q – w_i \;\le\; \sum_{j \in B_i} w_j \;<\; q \right). \]
Exactly one word crosses the line in any given solve, so the indices sum to one across the day’s words: \(W_\alpha\) parcels out a single unit of credit for reaching Genius.
Now the dial. At \(\alpha = 0\) every unfound word is equally likely next — \(g\) raised to the zero is one — so the order is a uniform shuffle, and \(W_0\) is precisely the classical Shapley–Shubik index, the thing that merely rescaled the scoreboard last time. At \(\alpha = 1\) you draw in proportion to raw frequency, a faithful sample of the language. And as \(\alpha \to \infty\) the ratio in that fraction runs away toward whichever word is more common, so with probability approaching one you find the single most frequent remaining word next — the order goes rigid, descending by frequency, CALL first every time. Your intuition was right on the nose.
So \(\alpha\) is not skill. It is how heavily your search leans on the corpus — the weight you place on “commonness” when you decide what to reach for next. At zero you ignore the corpus; at one you trust it; past one you obey it. It measures how representative of ordinary usage your guessing order is, and says nothing about how many words you eventually find or how clever you are. There is a real and separate question of what skill does to this dial — a stronger solver might lean harder on common words, or might reach the rare ones just as readily and flatten the whole thing — but that is a measurement problem, and an honest one, and I owe it its own post rather than a hand-wave here.
One loose end I promised to tie: where \(g\) comes from. The frequencies here are usage counts from a corpus of movie and television subtitles — everyday spoken English, not the language of books or of a seasoned solver’s vocabulary.1 It is a learned frequency, drawn from one particular record of how people talk, and swapping the record would swap the dynamics. Which corpus you trust is itself a dial, quietly upstream of \(\alpha\), and we will come back to it.
So the natural thing is to watch a single word’s power as you turn the dial from ignoring the corpus to obeying it. Do that for FANCIFUL and you get this:
The pangram’s power rises as your search grows more representative — about 25% above its blind, corpus-ignoring value — reaches a peak around \(\alpha \approx 0.4\), and only then collapses, sliding to essentially zero as the order goes rigid. FANCIFUL is most powerful not for the corpus-blind guesser and not for the frequency robot, but for a searcher whose order is loosely, humanly representative — leaning on common words without being enslaved to them. Lean harder and you strip the pangram of its leverage. That is the curse, and it is no rounding artifact — the calculation is exact.
Why would leaning harder on common words make the game’s most glamorous word matter less? To see it, I need one idea, and it’s the idea I want you to carry out of this post.
Mean prefix weight
Pick a word. In any given solve, some set of words gets found before it — call that set its prefix — and those earlier words have already put some number of points on the board. The prefix weight is exactly that: the running score at the instant just before you find the word in question. It is not measured in seconds or in keystrokes; it is measured in points already banked.
Prefix weight is the whole game, because a word is pivotal precisely when its prefix weight lands in a narrow band just below the quota. FANCIFUL is worth 15, and Genius sits at 106, so FANCIFUL clinches Genius exactly when the board reads somewhere between 91 and 105 as you find it — high enough that the pangram’s fifteen points finish the job, not so high that Genius already happened without it. Below 91 the pangram lands too early and fifteen isn’t enough. At 106 or above you were already there, and the pangram is a victory lap.
The mean prefix weight is the average of that running score over all the ways a solve might unfold — the expected state of the board at the moment this particular word shows up. It is a location: it tells you where in your accumulation a word tends to arrive. And here is the strange fact that runs the whole post. As you turn \(\alpha\) from blind toward corpus-bound, FANCIFUL’s mean prefix weight moves monotonically — steadily, without reversing — from about 68 down toward 56. It never rises. It drifts away from the pivotal band [91, 105] the entire time. On average, the harder you lean on common words, the earlier the pangram arrives and the less it can decide.
So if you looked only at the average, you’d conclude the pangram’s power falls monotonically, dead boring, and you’d be wrong. The power humps. The average says one thing; the truth says another.
The curse lives in the tails
The resolution is that pivotality does not care about the average prefix weight. It cares about how often the prefix weight lands in a specific window — a fact about the whole distribution, not its center. As your search leans harder on frequency, the heavy common words (FINANCIAL, CLINICAL, CLINIC, FACIAL) stop being coin flips and become near-certainties to be found early. That concentration does two things at once: it drags the average down, and it sharpens the distribution, briefly piling probability right into the [91, 105] window even as the center of mass slides below it. The upper tail fattens before the whole thing deflates. The mean and the pivotality move in opposite directions, and the pangram rides the difference up to a peak before the collapse wins.
This is why the curse is worth more than a chuckle. It is invisible to any summary that tracks where a word typically lands. You have to watch the shape of the distribution, not its middle, and the interesting behavior is exactly the part a mean throws away.
It isn’t just the pangram
FANCIFUL is the marquee case because it is the pangram and it starts on top, but the curse belongs to a whole class of words. Sort the day’s thirty answers by frequency and the humped power curves are precisely the ones in the middle of the distribution — seven of them, a clean band. The ubiquitous words (CALL, FINANCIAL) fall monotonically to zero: always found early, never decisive. The obscure words (CLINICIAN) rise monotonically: always found late, right at the cliff’s edge, where deciding things happens. It is the middle-frequency words that hump — familiar enough to be found before the end, rare enough to sometimes be found right at the line.
And the peak has structure. As the words get rarer, their peaks slide to the right: the more obscure a middling word is, the higher you must turn \(\alpha\) before the common words pile up densely enough to strand it at the edge. The pangram peaks at \(\alpha \approx 0.4\); the rarer middling words peak later, and some of them hump harder than FANCIFUL does — humble CULL nearly doubles its power at the top of its arc. The pangram gets the headline; the phenomenon is a property of the whole middle class.
What the curse is really about
Strip away the bee and here is the shape of the thing. A player’s power is non-monotone in how representative — how corpus-driven — the process around them is. Turning up the force that organizes the order is not uniformly better or worse for any given player’s leverage; there is an interior sweet spot on the dial, and it sits at a different \(\alpha\) for every player. For the pangram it sits early, at a barely representative search. For other words it sits later, or not at all.
If that has a familiar ring, it should. It is a cousin of the swing voter’s curse, where a voter can do better by staying uninformed and letting the informed decide, because more competence in the electorate can wash out an individual’s pivotal role. Different force, same unsettling geometry: turn up the thing that organizes a collective — information there, corpus-fidelity here — and the amount any one member matters need not move in a straight line. I have a longer post owed on that curse, and I’ll pay it.
My corner of the discipline even has a name for the best spot in a game with no stable winner. When no position beats every challenger — the chaos this blog is named for — Grofman and his co-authors, writing under the house pseudonym A. Wuffle, defined the Finagle point: the location from which a candidate needs only the smallest nudge to see off whoever comes along.2 The pangram has one too. Its Finagle point sits at a barely representative search — the intermediate \(\alpha\) where the common words crowd in just so — and the curse is that every turn of the dial toward obeying the corpus is a turn away from it. They named the whole apparatus, with a self-awareness the discipline is not famous for, after Finagle’s Law: no matter what happens, you can come out ahead if you just know how to finagle. For the pangram, read it the other way, and it becomes an epitaph.
There is one more place this dial turns up, and it is not a game. The same knob — in what order do we let the parts of a system enter the accounting? — sits underneath the fashionable machinery that people now use to explain artificial intelligence. It is set to a default there, quietly, almost universally. And as we have just seen, a default is a dangerous thing to set on a dial whose readings do not move monotonically. That is for next week (hopefully). [Glances at calendar.]
With that, I leave you with this.
Notes
1 The findability \(g\) is a word’s usage count in the OpenSubtitles 2018 corpus — roughly the language of film and television dialogue — as compiled in Hermit Dave’s FrequencyWords list (the fifty thousand most common words). Words outside that list are handed a small floor value, so the truly obscure answers are treated as equally rare and, at high \(\alpha\), shuffled among themselves in a tail. A different corpus — printed books, web text, or the actual guessing logs of Bee solvers — would rank the words differently and move every curve in this post. That the choice of corpus is doing quiet work is not a bug in the exercise; it is the exercise.
2 A. Wuffle, Scott L. Feld, Guillermo Owen, and Bernard Grofman, “Finagle’s Law and the Finagle Point: A New Solution Concept for Two-Candidate Competition in Spatial Voting Games Without a Core,” American Journal of Political Science 33, no. 2 (1989): 348–375. A. Wuffle is Grofman’s long-running pseudonym, and the concept is motivated explicitly by the chaos theorem. The finagle radius is the radius of the smallest circle around a position from which some point can defeat any challenger in the space; the finagle point is the position that minimizes it. It is usually impossible to be unbeatable, and nearly always possible to be barely beatable — which is the whole consoling idea.