The best way to kill a joke is to explain it. This is a series about explaining jokes. Each installment takes a game people enjoy and runs it through the machinery of cooperative game theory until the fun stops. The wager is that the autopsy is where the biology is — if you want to know how a game really works, it helps to ruin it first. We begin with the New York Times Spelling Bee.

Here is a claim that sounds like a category error: the Spelling Bee is a weighted voting game.

It isn’t a metaphor. Every morning you get seven letters — one of them singled out in the center — and a hidden list of legal words. Every legal word must use that center letter; the other six are optional.¹ Each word is worth points: four-letter words score 1, longer words score their length, and a pangram that uses all seven letters earns a fat bonus. Find enough points and the game promotes you to Genius, at roughly 70% of the day’s maximum. Genius is the whole psychological event of the Bee; Queen Bee, the 100% mark, is for people with more time than sense.

Now read that scoring rule as a legislature. The words are voters. Each word’s point value is its number of votes. Genius is the quota you need to pass. (Ed.: Well, at least it’s a rubric, right Johno?) A word is pivotal exactly when it’s the one that carries you across the Genius line — short before you found it, over the line after. That is the notion of pivotality Shapley and Shubik built their power index on in 1954. So we can ask a slightly deranged question with a fully rigorous answer: on a given day, which words hold the power?

Let me use a real day: Monday, June 29, 2026. Center letter C, letters A C F I L N U, thirty legal words, 152 points on the table, Genius at 106. One pangram, and it’s a good one — FANCIFUL, worth 15.

First, we kill the game the boring way

Compute the Shapley–Shubik index for all thirty words: the share of pivotal moments each word commands, averaged over every order you might find them in. I’ll spare you the combinatorics — there’s a clean dynamic program, and it’s waiting at the end. The result is a small disaster.

The power ranking is the scoreboard. FANCIFUL, the highest-scoring word, holds the most power. The nine-pointers come next, then the sevens, and so on down to the one-pointers, in exact order. The rank correlation between points and power is a perfect 1.00. Cardinally it’s nearly proportional too: the pangram’s power is about what you’d get from eighteen one-point words pooled together, which is about what eighteen votes should buy.

This is a theorem doing its job — Shapley–Shubik is monotone in weight for weighted voting games — but as an insight it’s a face-plant. We deployed one of the crown jewels of cooperative game theory to rediscover the number the game already prints on the screen. The joke, explained, is not funny. If you’ve read enough of this blog you know I think that’s usually the most interesting place to be standing.

The failure is diagnostic. Shapley–Shubik averages over every possible order of discovery, weighted equally. That is the load-bearing assumption, and it is false about you. You do not find words in uniformly random order. You find CALL, then CLINIC, then FACIAL, and forty seconds later you’re grinding for CANNULA and LACUNA and, if the gods are kind, the pangram. The uniform-order assumption is the fiction that makes the index collapse into the scoreboard. Kill the fiction and see what’s left.

The Waggle Index

Let’s model the order you find words in. Give each word a findability — I’ll use its ordinary frequency in English, so common words surface early and obscure ones straggle in late — and compute each word’s share of pivotal moments under that order. Same game, same quota, an honest discovery process.

The scoreboard shatters. The most powerful word on June 29 is no longer the pangram but CLINICIAN, a nine-pointer nobody types until they have exhausted everything easier, which is exactly why it keeps landing right on the Genius threshold, with CALCULI and CANNULA close behind. FANCIFUL, meanwhile, takes the hardest fall on the board, dropping from the single most powerful word — about 10.5% of all pivotal moments — to under 4%. The crown jewel of the puzzle turns out to be a mid-game formality: by the time most solvers spot the pangram they are already past Genius or nearly there, so it brings glory but no leverage. (Though “fall” is too simple a word for what the pangram does. Its power is not even monotone in how well you play — it climbs before it collapses, for a reason strange enough to earn its own Monday. Call it the pangram’s curse.)

And a cluster of respectable, high-scoring words drops to exactly zero. FINANCIAL at nine points, CLINICAL at eight, with CLINIC, FACIAL, CANAL, and CLIFF behind them: ten of the thirty words become what I’ll call behavioral dummies, never pivotal, not because they are small but because they are found before the line every single time. CALL, which the corpus records about 573,000 times, is the purest case — you type it in the first ten seconds and it never once decides anything.

My favorite pair on this board is CLINICAL and CLINICIAN. Nearly the same letters, nearly the same score, eight and nine, separated only by how often anyone says the word. One is common, lands early, and holds zero power; the other is rare, lands late, and holds the most power on the board. Their fates are set entirely by frequency. The points can’t see the difference. Only the order can.

The rank correlation between points and power, a perfect 1.00 under uniform order, falls to 0.37.

The widget above is live — drag the two sliders. At the far left, “blind luck,” the Waggle index collapses back onto Shapley–Shubik exactly. Drag right toward “common words first” and the board reorganizes under your hands.

What just happened here

This is, if you’ll let me connect it to the day job, the same phenomenon I keep finding in less whimsical settings. A metric — the Bee score — was supposed to stand in for something you care about, reaching Genius. Under one assumption about behavior the metric and the target agree perfectly, so perfectly that the fancy analysis is redundant. Under a realistic assumption they come apart, and the metric starts quietly certifying the wrong words as important. The sorting of words into “matters” and “doesn’t matter” was never a property of the words. It was an artifact of an assumption about how they’d be used, and the moment you make the assumption honest, the sorting inverts.

That gap is the whole subject of a book I’m writing with a frequent co-author, and it is faintly hilarious to find it sitting inside a word game about bees.

There’s also an open question underneath this, older than the Bee. Shapley–Shubik and Banzhaf, the two canonical power indices, differ in exactly one respect: the distribution they assume over orders, or over coalitions. Neither is “correct”; each encodes a different theory of how a collective assembles itself. The Bee is a rare case where you needn’t assume the distribution at all. You can measure it — from word frequencies, or from real solver logs — and compute the power index the players themselves generate. There may be a paper in that. There may also be a cease-and-desist from the NYT. I’ll take either.

Why “Waggle”

Every new index needs a name, and this one names itself. When a foraging bee finds a good source she returns to the hive and performs the waggle dance, a figure-eight whose angle and duration tell the other bees where to go and, crucially, in what order the colony should exploit what’s out there. The waggle dance is a discovery-order protocol. An index built entirely on discovery order can’t be called anything else.

It doesn’t hurt that “Waggle” sits one letter from Wuffle — any power-index paper that can tip its hat to A. Wuffle is doing something right — or that it rhymes with the other five-letter Times game I have designs on. Which brings me to the series.

Coming up in Killing the Game

The Waggle index is one face of Shapley’s idea, power as pivotality, but the same primitive wears other masks in other games, and I mean to try them all on. Connections is a partition game, so its natural object is the value of a tile to the group it lands in — cooperative game theory as clustering. Pips is a constraint-satisfaction game, which is really a story about cost-sharing. And Wordle gets its own installment, though there the right Shapley object is not power but attribution: how much did each guessed letter contribute to collapsing the space of possible answers? That last one is the same math the machine-learning world now calls SHAP. They reached it from Shapley too, and it hides a free parameter that decides which feature looks important — but that is a Monday well down the road.

One idea, many avatars, one steadily diminishing quantity of fun. Bring a friend who no longer likes puzzles.

With that, I leave you with this.

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Under the hood, for the formalists

The Waggle index is a random-order value: word \(i\)’s power is its expected marginal contribution to reaching quota, taken over a non-uniform distribution on discovery orders. I model that distribution as a Plackett–Luce / exponential-clock process — word \(j\) carries an independent \(\mathrm{Exp}(f_j)\) time-to-find, with \(f_j = g_j^{\alpha}\) for a findability \(g_j\) (here, frequency), and the order is the sorted vector of clock times. At \(\alpha = 0\) the clocks are exchangeable, every order is equally likely, and the Waggle index is exactly Shapley–Shubik — the sense in which the interactive’s leftmost setting is the classical index.

Computation is exact, not simulated. Conditioning on word \(i\)’s clock time and substituting \(u = 1 – e^{-f_i t}\) collapses the pivot probability to a one-dimensional integral over \(u \in (0,1)\), where at each quadrature node the sub-quota prefix weight is an ordinary subset-sum recursion over the other words. It runs in milliseconds and agrees with the exact Mann–Shapley computation to twelve digits at \(\alpha = 0\). The forward clock is a cousin of the Kalai–Samet weighted Shapley value; the direction of the clock matters, and getting it right is most of the modeling. Code and the full thirty-word table are linked below.

Notes

¹ That center letter is not merely a heavily weighted vote — it holds a veto. A word that omits it does not score less; it fails to exist. Which makes the center letter less like a powerful legislator and more like a permanent member of the UN Security Council, whose lone “no” sinks any resolution no matter how everyone else votes. So the letters carry their own power structure, distinct from the words’ and nowhere near equal — a game we’ll put numbers on in a later Killing the Game. For now, keep your eye on the words.