writing

The Probability Telephone Game

Nov 2026

Adults don’t play enough children’s games anymore; and - even more so - they add far less statistics to them than I’d like1.

In the telephone game2, you whisper a secret message from child to child. As a game it has little going for it3. There is no winning; you’re merely bemused by the frailty of language, as sensible, nay literary, phrases like “Life must be lived as play” become “He bites snails”4.

2 of 51 phrases · 165 logged players
“She says hi”La Farge, USA → North Wales, USA → Mokena, USA · survived 21 players
“Clouds travel around the world”Talkeetna, USA · survived 3 players
Click a gap to reveal the phrase that lasted longest unchanged within it. From the Global Gossip Game 2013: the English-only stretch to Talkeetna, Alaska. Starts after the last change of language on the route to Talkeetna, Alaska.

The Probability Telephone Game

The game gets more interesting when you shrink the message. Instead of a whole phrase, pass a single probability word and its number, turning one into the other each round. And rather than children, we’ll use machines and university-educated adults.

For the machines, first ask a model to put a probability (90%) into words (“highly likely”); then spin up a fresh copy and ask what number those words meant.

number 1 → word 1 → number 2 → word 2 → … for k rounds

▼65
95% → «almost choice-free» → 5% → → «unlikely» → 30%
Probability0%25%50%75%100%01word2word3word4word5word6wordRounds (number → word → number)“almost certain” → 95% (42 games). Click to follow one.“almost certain” 95“almost certainly” → 90% (13 games). Click to follow one.“almost certainly” 90“more likely than not” → 51% (38 games). Click to follow one.“more likely than not” 51“it's a toss-up” → 50% (12 games). Click to follow one.“it's a toss-up” 50“unlikely” → 20% (3 games). Click to follow one.“unlikely” 20“highly unlikely” → 10% (22 games). Click to follow one.“highly unlikely” 10“highly unlikely” → 5% (27 games). Click to follow one.“highly unlikely” 5
Average drift after 6 rounds: 11.8 points; 11% of games flipped sides of 50%

You might imagine this is a bit artificial; why would they have no context for each of these ‘rounds’? In fact this happens all of the time in current LLM setups, which pass messages between subagents; or sub-sub-agents; or sub-sub-sub-agents (you get my point). More recently this has expanded into swarms or teams of agents working together on problems. Each one writes, summarises and paraphrases previous work, and in doing so changes the meaning.

The games above show how much. Gemini 3.5 Flash flips about one game in nine from likely to unlikely, or back, within six rounds. Current frontier models (Fable 5.2; GPT 6.1) are getting closer to the ideal - straight, granular lines, where a model reads back exactly what it meant - and rarely flip from a yes to a no. Even then, they still contain attractors, values the model converges on: for Fable 5.2, everything from 80% to 98% ends up as “almost certain” (95%).

You might also assume this is the stochastic randomness of LLMs - you’d be wrong. Games do not degrade at random; they drift towards the terms each model has an innate preference for5. The words do more of the pulling than the numbers: each model reaches for a narrow stock of favourite phrases, and those phrases become the attractors.

What if we add humans to the mix?

Despite the amusing mental image, I couldn’t sit 100 people in a row and have them whisper numbers to each other. Instead, we can use some of our existing data - in my probability survey people told me what number they’d give to a series of phrases. Giving us a mapping from phrase to number; and we can also read that in the reverse direction to get the mapping for number to phrase.

So a human game runs like this. Start at 90%. Person A is a random respondent who read some word as about 90 - say “highly likely” - so that is what they write. Person B is another random respondent; whatever number they gave “highly likely”, say 80, is what they read. Person C then writes a word they would read as 80, and so on for six rounds 6.

Human
▲93
2% → «remote chance» → 6% → → «highly likely» → 95%
Probability0%25%50%75%100%01word2word3word4word5word6wordRounds (number → word → number)“highly likely” → 90% (8 games). Click to follow one.“highly likely” 90“highly likely” → 85% (6 games). Click to follow one.“highly likely” 85“very good chance” → 80% (8 games). Click to follow one.“very good chance” 80“realistic possibility” → 75% (7 games). Click to follow one.751“better than even” → 60% (7 games). Click to follow one.“better than even” 60“about even” → 50% (7 games). Click to follow one.“about even” 50“almost no chance” → 5% (9 games). Click to follow one.“almost no chance” 5“almost no chance” → 3% (5 games). Click to follow one.“almost no chance” 3“almost no chance” → 2% (6 games). Click to follow one.“almost no chance” 2“impossible” → 0% (5 games). Click to follow one.“impossible” 0
1 “realistic possibility” 75%
Average drift after 6 rounds: 25.8 points; 23% of games flipped sides of 50%

Well… humans are pretty bad at this7. We are far less consistent readers of these words, and we drift far quicker. After a single round we have already shifted ~14pp, and by the sixth round nearly a quarter of our games (23%) have flipped the decision, crossing 50% from positive to negative or back. The best models don’t flip at all.

Human → Gemini 3.5 Flash → Human
▼88
98% → «certain» → 100% → → «unlikely but possible» → 10%*
Probability0%25%50%75%100%01word2word3word4word5word6wordRounds (number → word → number)“almost certain” → 95% (23 games). Click to follow one.“almost certain” 95“very likely” → 90% (18 games). Click to follow one.“very likely” 90“very likely” → 85% (11 games). Click to follow one.“very likely” 85“more likely than not” → 51% (5 games). Click to follow one.“more likely than not” 51“toss-up” → 50% (15 games). Click to follow one.“toss-up” 50“unlikely” → 20% (7 games). Click to follow one.“unlikely” 20“highly unlikely” → 13% (5 games). Click to follow one.“highly unlikely” 13“highly unlikely” → 10% (14 games). Click to follow one.“highly unlikely” 10“highly unlikely” → 7% (6 games). Click to follow one.“highly unlikely” 7“highly unlikely” → 5% (9 games). Click to follow one.“highly unlikely” 5
Average drift after 6 rounds: 20.2 points; 12% of games flipped sides of 50%

Put a model in the loop and things improve. Alternate people with even the rather poor Gemini 3.5 Flash - a person writes, Gemini reads it and writes its own phrase, the next person reads that - and the average drift falls from 26 to 20 points, while the flip rate halves to 12%.

Occasionally a human game pulls off a heroic rescue8: it wanders so far that a yes becomes a no, then finds its way home. The best one starts at 2%. Someone calls that “unlikely”, the next person reads it as 30; that becomes “probable”, read as 85, then “almost certainly”, 92. Then 92 is written as “almost certainly not” - which can only happen because two of our 99 respondents read that phrase as 90 or more, presumably missing the “not” - and the game falls back to 10, 3 and 5.

Human
▲3
2% → «unlikely» → 30% → «probable» → 85% → «almost certainly» → 92% → «almost certainly not» → 10% → «almost no chance» → 3% → «highly unlikely» → 5%
Probability0%25%50%75%100%01word2word3word4word5word6wordRounds (number → word → number)“highly likely” → 90% (8 games).“highly likely” 90“highly likely” → 85% (6 games).“highly likely” 85“very good chance” → 80% (8 games).“very good chance” 80“realistic possibility” → 75% (7 games).751“better than even” → 60% (7 games).“better than even” 60“about even” → 50% (7 games).“about even” 50“almost no chance” → 5% (9 games).“almost no chance” 5“almost no chance” → 3% (5 games).“almost no chance” 3“almost no chance” → 2% (6 games).“almost no chance” 2“impossible” → 0% (5 games).“impossible” 0
1 “realistic possibility” 75%
Average drift after 6 rounds: 25.8 points; 23% of games flipped sides of 50%

It is a rescue very much by two wrongs making a right: an odd reading of “unlikely” as 30 sends the game the wrong way, and an odd “almost certainly not” brings it back. Models can’t pull this off; they are too consistent to be ‘lucky’. Of all the model games that flipped across 50%, not one crossed back.

Section titled Footnotes

Footnotes

  1. I am, of course, joking; however there is a long history of taking ‘simple’ games and expanding them out far beyond their need - for example the Optimal Strategy in “Guess Who?”: Beyond Binary Search; or the use of Rock-Paper-Scissors to decide whether Christie’s or Sotheby’s would sell a $20m art collection. Christie’s won with scissors, on the advice of a precocious pair of 11-year-old twins: rock is too obvious, so your opponent will play paper - and naive players do favour rock. Sotheby’s took no advice, played paper, and lost the sale. ↩

  2. In the UK, when I grew up it had a slightly sordid name - “Chinese whispers” - why it is called that no-one really knows; but it isn’t a good look. Before that it was known as “Russian Scandal”, so we - the Brits - really can’t help but indoctrinate patronising superiority in our youth. France equally does not come off well - téléphone arabe (“Arabic telephone”). Truly us and the French exhibit the narcissism of small differences — or just “two cocks on one dunghill”. ↩

  3. So why has it lasted? Because it says a lot about co-operation and community. Each time a message passes between people there is a small loss, as each person pattern-matches it onto their own priors and assumptions. The same happens between LLMs and the question is how badly the errors pile up - and do we risk turning a clear cut no into a yes. ↩

  4. From the 2012 Global Gossip Game, played by 237 people across libraries in 12 countries. ↩

  5. People do the same. Frederic Bartlett’s Remembering (Cambridge University Press, 1932) passed the folk tale ‘The War of the Ghosts’ from person to person. With each retelling it got shorter and more conventional: canoes became boats, and seal-hunting became fishing. Odd details were rationalised, so “something black came out of his mouth” became the soul leaving the body. Eventually it settled into a stable story, one familiar to its audience. ↩

  6. Nobody in a real game should see more than the phrase handed to them; as such ‘simulating’ them as independent entities should be perfectly valid. As a check, I built the model games the same way from one-round data, and they match the genuine serial games I played with the models. ↩

  7. Another win for our AI Overlords ↩

  8. Dropout’s Game Changer: Season 8, Episode 6, “Noise Some Makes” has the best example of this where: “The Evolution of Man Diagram but Each Step Leans Further and Further Back into Limbo” -> “It’s naked limbo, but there’s monkeys in line… but there’s a man doing naked limbo.” -> [acting out naked limbo w/ monkeys] -> “man playing limbo with a group of monkeys, possibly as metaphor for evolution” ↩

  9. Du, Lu & Qu (2026), Possible or Definite?. One round of summarisation over 1.2k clinical documents loses >1/2 the original uncertainty.

  10. Perez et al. (2024), When LLMs Play the Telephone Game. LLM-only with forced evolution (models are told to rephrase; take inspiration; or continue). They see biases to settle into attractor states.

  11. Belem et al. (2026), From “May” to “Is”: Certainty Distortion in Language Model Rewriting. Models repeatedly paraphrase their own scientific/medical output. 3/4 change their certainty, with an overall positive trend.

  12. Tang, Shen & Kejriwal (2026), An Evaluation of Estimative Uncertainty in Large Language Models. A study on GPT-3.5/GPT-4 — they found far more significant divergences; and numbers->words was poor. Largely this has been solved by capability.

  13. Belem et al. (2024), Perceptions of Linguistic Uncertainty by Language Models and Humans. Ten models turn ‘uncertainty phrases’ into numbers; scored against a human population. The most interesting result is that the numbers are swayed by whether the hedged statement is in fact true.

  14. Sileo & Moens (*SEM 2023), Probing neural language models for understanding of words of estimative probability, one of the earliest probes of these words, and of whether models can combine them (“A is likely, B is impossible”: so how likely are both?). Again largely held back by the older class of models.