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Results · Finite games and operator-algebra type

A finite game that normal hyperfinite strategies approach but never win

Seth Douglas and Nidhal Mghirbi · version 1.0.0 · Zenodo, · 17 pages · doi:10.5281/zenodo.23816141 · PDF

In plain wordsIn an explicit finite game, finite quantum systems come arbitrarily close to a perfect way of winning, yet no normal strategy on hyperfinite systems, the infinite ones built from finite pieces, wins every time; under stated assumptions, quantum field theory strategies also approach but never win exactly.

The result

As the paper states it, in briefAn explicit finite game with 0/1 scoring (561 and 157 questions, 3,696 question pairs) whose perfect correlation finite-dimensional strategies approach, while no perfect strategy has a normal marginal on an injective (hyperfinite) von Neumann algebra of either player. Perfect play lives on the Bellaterra critical-KMS factor of type III1/2, and every perfect strategy contains an expected copy of the group algebra of the free product of three groups of order two. Under the hypotheses of Halvorson’s Proposition 2.31, quantum-field-theory strategies come arbitrarily close but never win exactly.

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Novelty and credit

Novelty
New, with named prior work: earlier value-gap games already exclude perfect normal injective hosts; the distinction here is that the perfect correlation itself is approached. The KMS rigidity is Laca, Raeburn, Ramagge and Whittaker’s; no earlier version of the conjunction was found in a bounded search.

Versionsone, each with its own DOI

Every archived version, newest first. Dates are Zenodo publication dates. All versions: 10.5281/zenodo.23816140, which always resolves to the newest.
VersionDateDOINotes
1.0.010.5281/zenodo.23816141 PDF hosted here.

Paper, code and links

PDF

papers/hyperfinite-v1.0.0.pdf, version 1.0.0, 453,208 bytes, 17 pages.

SHA-256 c754a87071ae9589b8abb88810527521962e8b40205a59dea3dad09f68ffd34c

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Cite

@misc{douglas2026hyperfinite,
  author  = {Douglas, Seth and Mghirbi, Nidhal},
  title   = {A finite game that normal hyperfinite strategies approach but never win},
  year    = {2026},
  version = {1.0.0},
  doi     = {10.5281/zenodo.23816141},
  url     = {https://doi.org/10.5281/zenodo.23816141}
}

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