Source
Czarnecki, Gidel, Tracey, Tuyls, Omidshafiei, Balduzzi & Jaderberg, LIT-1504, read in NOTE-1146 from arXiv:2004.09468v2: section 6 and Table 1 (pp. 8–9), supplements D–I (pp. 17–22), Tables 2–3.
What was actually shown
- The data. For each game, about 2000 agents:
- MinMax(d, s) for d = 0–9;
- variants that value the horizon at 0;
- MaxMin(d, s), which seek to lose;
- MCTS with 10, 100 and 1000 simulations;
- 50 seeds each. Each pair played twice, swapping sides, and duplicate rows were removed, leaving about 1000 agents. AlphaStar’s 900-agent league matrix was taken from its authors.
- The profile. Maximum-entropy Nash clustering. For each cluster, its size against its mean RPP against all clusters. A skew-normal curve is fitted and drawn.
- The finding. The eight board games and AlphaStar show sizes rising to a peak and falling toward both ends. Blotto and Kuhn poker grow with strength, the noisy Elo games grow too, the disc game is one cycle, and the Elo game is a chain. In Go 3×3 the top cluster is not trivial.
- What could have come out otherwise. Clusters could have grown with strength, as in Blotto. They did not in these eight games.
What this does not say
- That the shape belongs to the games. The sampler sets the transitive axis through search depth and the spread through seeds, and the authors say it “does not guarantee uniform sampling of strategies” (p. 9). For AlphaStar they say the profile reflects “AlphaStar induced game geometry” (p. 18), shaped by how the league was trained.
- That it is proved. The abstract says so, and the body withdraws it (p. 6). Theorem 1 (THEORY-926) shows long cycles among weak strategies, not narrowing at the top.
- That cluster sizes are robust. Maximum-entropy Nash makes clusters as large as possible. The authors note that minimum-entropy equilibria would give “more peaky tops” (p. 17).
- Anything about large boards. The board games are 3×3 and 4×4 except Tic-Tac-Toe and Connect Four.
Connections
A descriptive picture the record can cite for where cycles are expected in contests of skill: at middling strength, among varied styles. Its evidential weight is that of a sampled, untested profile (NOTE-1146).