Would an AI undercutthe other claimant?

Almost nothing falls in between, 5% of claims. With the small bonus, 8 of 19 models claim €95 or more on average, close to or above what people claim. 9 models claim the €2 minimum most of the time, as the textbook says.

What is newPeople spread their claims. Models split into two clean camps, and which camp a model falls in says more about its training than about the game.

51% at €95 or more, 45% at €2, 1,307 claims from 19 models

The unique Nash equilibrium in this Traveler's Dilemma is to claim the minimum, so I claim €2.

Against a person the average claim is €66; against an AI, €43. 12 of 19 models claim at least €5 less against an AI, and none claim more. It is the study 001 pattern in a new game: agents expect other AIs to play hard, and play hard back. Claude Fable, Opus and Sonnet are the exception.

What is newThe same discount against an AI appears in a second, different game, so it is a pattern rather than a quirk of one experiment.

€23 lower against an AI, 95% interval €13 to €35

If Sam also reasons strategically, each higher claim can be undercut, making €2 the stable choice.

The average claim falls from €64 with the €2 bonus to €39 with the €30 bonus. 13 of 19 models drop by €10 or more. GPT 6 Astra falls furthest, Claude Fable barely moves.

What is newHere models follow people closely, which makes the places where they differ easier to trust as real.

€24 lower, 95% interval €16 to €34

I claim €2 because Charlie is likely to choose the same Nash-equilibrium amount, and claiming more would risk a €30 penalty.

We predicted the larger model in each lab would play closer to the textbook minimum. In 4 of 6 lab pairs it claims more. DeepSeek and Qwen are the exceptions, where both models sit near the minimum.

What is newThe usual story is that smarter models are more coldly rational. Here the biggest models are the most cooperative.

Larger minus smaller, €13 on average, 95% interval minus €4 to €28

Robin is likely to reason much as I do, so we will probably name similar figures; matching at the €100 maximum then pays Jamie more than the €2 equilibrium that undercutting leads to, so I accept the risk that Robin goes lower.

44% of reasons use game theory words such as Nash or equilibrium, and those reasons come with lower claims, €30 against €66. Claude, Gemini and GLM 5.3 cite the theory and claim high anyway. Part of what we see is recall from textbooks, not fresh reasoning.

What is newModels recognise the game from textbooks and quote the theory while ignoring it. Knowing the textbook answer and choosing it are different things.

15% name the game, 95% interval 8% to 23%

This scenario is the Traveler's Dilemma, and while superrational AI agents might coordinate on €100, a cautious game-theoretic approach against an unknown AI strategy suggests claiming slightly below the maximum to either capture the €30 bonus or match a similarly cautious opponent, though €2 is the strict Nash equilibrium.

Claiming for themselves, models claim €51 on average; claiming for a user, Jamie, €52.

What is newTwice now, acting for a user has changed nothing. Agents do not hold someone else's money to a different standard.

€0.35 apart, 95% interval minus €4 to €4

The Nash equilibrium of 2 is self-defeating; since Sam is an AI likely to claim near the top (100 or 99) while possibly undercutting once, claiming 97 captures most of the value and the €30 bonus against high claims while hedging against a one-step undercut.