nucleation

LIT-1504: Real World Games Look Like Spinning Tops

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Properties4
StatusActive — read in full 2026-10-24 (NOTE-1146) from arXiv v2 (17 June 2020); worth reading for one clean theorem, one general lemma and a hypothesis with evidence. The theorem: where players can signal n bits before the result is settled, every pattern of wins and losses among 2ⁿ strategies, a cycle through all of them included, is realized. The lemma: Nash clusters are weakly ordered by population performance, and beating a population that contains a whole cluster lifts a strategy above it. The hypothesis, that real games are shaped like a spinning top, with cycles thickest at middling strength, rests on sampled empirical games. The abstract says it is proved and the body says it is not. The random-game theorem's proof is defective, and the fixed-memory fictitious play result needs a replacement rule it does not state.
Tagsgame-theory, learning-theory, anthology-candidate
Cited as “paper” byNOTE-1146 — Czarnecki et al. — Real world games look like spinning tops
Cited as “source” byTHEORY-1199 — Real-world games of skill have a spinning-top geometry: their non-transitive part, measured as Nash cluster size, is largest at intermediate strength and narrows toward the strongest and the weakest strategies, THEORY-926 — In a game where each player can transmit n bits before a win or a loss becomes unattainable, every antisymmetric pattern of wins and losses among ⌊2ⁿ⌋ strategies, a cycle through all of them included, is realized by some set of pure strategies, THEORY-930 — In empirical games built from about a thousand sampled search agents in eight OpenSpiel board games, most on 3×3 or 4×4 boards, and in the AlphaStar league, Nash clusters taken from maximum-entropy equilibria are, read against a fitted skew-normal curve with no test, largest at intermediate transitive strength and narrower toward the strongest and the weakest strategies, while Blotto, the disc game, Kuhn poker and noisy Elo games do not show that profile, THEORY-940 — In a finite symmetric zero-sum game, successive Nash clusters are weakly ordered by relative population performance, and a strategy that beats every member of a population containing a whole cluster lies in a strictly earlier cluster

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Czarnecki, Gidel, Tracey, Tuyls, Omidshafiei, Balduzzi & Jaderberg (2020), Advances in Neural Information Processing Systems 33 (NeurIPS 2020) — arXiv:2004.09468

Key takeaways

From the reading, NOTE-1146. Page numbers are arXiv v2’s.

  • Long cycles are generic where play can signal. If each player can transmit n bits before a win or loss becomes unattainable, every antisymmetric win–loss pattern among ⌊2ⁿ⌋ strategies is realized by pure strategies that announce an identity and then throw or take the game (Theorem 1; THEORY-926). Go is at least about 1000-bit communicative, and Tic-Tac-Toe 5.58-bit. The theorem is ordinal: it is about the sign of the outcome.
  • Nash clusters. Repeatedly peel off the support of a Nash of what remains. Earlier clusters are weakly ahead of later ones in relative population performance, and a strategy beating every member of a population containing a whole cluster lies in an earlier cluster (Theorems 2 and 3; THEORY-940). Both hold in every finite symmetric zero-sum game, not only in Games of Skill.
  • The spinning top is a hypothesis. “Naturally, the above does not prove that real world games follow the Games of Skill geometry” (p. 6), against the abstract’s “We prove the existence of this geometry”. The evidence is about 1000 sampled search agents per game in eight small board games, plus AlphaStar’s league. The authors say the league shows “AlphaStar induced game geometry” (p. 18) (THEORY-930).
  • Population size. With a pessimistic oracle, populations below a critical size cycle, and above it they converge to the strongest strategies. In the disc game no size helps, and in an Elo game size 1 suffices.
  • Defects. Proposition 3 needs a replacement rule it does not state. The reader built a three-layer game in which replacing the newest member cycles forever and replacing the oldest converges. Theorem 4’s proof computes the wrong variance (σ²/m⁴ for what is σ²/(2m²) at most) and treats a selected quantity as zero-mean.

Why this is here

Filed 2026-10-24 for the record’s cycles line. Theorem 1 says when cycles are unavoidable: whenever parties can condition on who they face before the outcome is fixed. That bears on CLAIM-451 and TERM-077. Nash clustering is a cardinal layering of a cyclic record, and Theorem 3 is a condition for escaping a cycle (NOTE-1146). It carries anthology-candidate because its subject is training agents for games. It is filed for understanding, not as an instruction for practice.

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