Real World Games Look Like Spinning Tops

• The motivating question is whether games people find interesting and engaging share a common structure, with that understanding potentially informing AI solution methods.Source: §1 · p. 1
• The Game of Skill hypothesis represents transitive strength on an upright axis and cyclic, non-transitive dynamics on a radial axis.Source: §1 · p. 2
• The authors report that Theorem 1 establishes long cycles in a rich class of real-world games and that the hypothesised geometry appears in nine such games.Source: §1 · p. 2
• The paper connects population size and diversity with guarantees of transitive improvement, and reports simple learning experiments on empirical games from nine real-world games.Source: §1 · p. 2
• The authors frame the work as defining a game class, establishing spinning-top geometry theoretically and empirically, and giving theoretical arguments about consistent improvement by state-of-the-art algorithms.Source: §1 · p. 3

• Progress through practice corresponds, in game-theoretic terms, to a strong transitive component.Source: §2 · p. 3
• Pure Rock Paper Scissors is not presented as a standalone way for people to measure strategic skill without information about their opponent.Source: §2 · p. 3
• Diverse styles can counter one another despite similar average performance, while their non-transitive effects diminish as skill rises.Source: §2 · p. 3
• Combining progress and diverse styles yields a wide, highly cyclic middle and narrow peaks of highly skilled and very weak strategies.The hypothesised non-transitivity first increases and then decreases along the transitive dimension.Source: §2 · p. 3

• The analysis focuses on finite normal-form games and symmetric, deterministic, zero-sum payoffs, while casting temporally structured games into normal form.The authors note that this casting may hide structure and explicitly draw some insights from game rules rather than payoffs.Source: §3 · p. 3
• For temporally extended games, strategies map states to actions and outcomes follow from the terminal state reached when strategies play.Figure 2 illustrates extensive-form and partial normal-form views, including a symmetrised representation.Source: §3 · p. 4, Figure 2 · p. 4
• Nash clustering indexes the transitive component and uses cluster size to represent non-transitivity; the authors describe it as one useful decomposition, not the only or best one.Source: §3 · p. 4
• In empirical games, trained policies are strategies, and each training step adds an approximate best response to the policy set.Training may select opponents through self-play or distributions of past agents, and the RL algorithm can find weak, exploitable strategies that satisfy its constraint.Source: §3 · pp. 4–5

• For any game at least n-bit communicative, Theorem 1 realizes every antisymmetric win-loss matrix of dimension floor(2^n) using pure strategies.The theorem applies to the win-draw-loss version of the game, which preserves wins, draws, losses, and cycles but can change Nash equilibria.Source: §4 · p. 5
• The result constructs a cycle of length floor(2^n), a lower bound because some games can have longer cycles.Source: §4 · p. 5
• Go is reported as at least 1000-bit communicative with a cycle of length at least 2^1000; 10-minute modern games are reported as at least 36000-bit communicative.The paper also reports Tic-Tac-Toe as 5.58-bit communicative and one-step games such as Rock Paper Scissors as 0-bit communicative.Source: §4 · p. 5
• The paper says the temporal, multistep structure exposes complexity that a normal-form view alone would hide.Source: §4 · p. 6
• The authors distinguish the long-cycle analysis from validating the spinning-top hypothesis, which they address with empirical game-theoretic analysis of real-world games.Source: §4 · pp. 5–6

5 Layered game geometry

• In layered Games of Skill, non-transitive interactions occur within layers, while layer identity represents transitive strength; fixed-memory fictitious play has a convergence guarantee under stated population-size conditions.The required population is at least the size of the lowest occupied layer when at least one strategy is above z; otherwise it must be at least |Lz|.Source: §5 · p. 6
• Figure 3 relates more complex non-transitivity-handling algorithms to larger portions of the spinning top likely represented, assuming near-best agents and the Game of Skill hypothesis.The analysis offers intuition about training-pipeline development, not a ranking of approaches.Source: Figure 3 · p. 7
• Nash clustering repeatedly forms clusters from the support of a Nash equilibrium over remaining strategies, and the resulting clusters are ordered monotonically by Relative Population Performance.The maximum-entropy choice gives a unique clustering, although multiple Nash clusterings may exist.Source: §5 · p. 7
• A population containing a full Nash cluster can guarantee transitive improvement by training against every member, even without weaker policies or prior knowledge of cluster coverage.The resulting strategy belongs to an earlier Nash cluster.Source: §5 · p. 7
• Wider coverage of strategies around current transitive strength supports continued improvement; the required coverage can shrink as agents become stronger, though |Lz| is not readily known without traversing the game tree.The paper presents this as an intuitive mechanism for why finite-memory fictitious self-play can work in practice, not a directly transferable efficient algorithm.Source: §5 · pp. 6–8

6 Empirical validation of Game of Skill hypothesis

• The empirical games approximate underlying games using strategy samples chosen to cover transitive strength, varied random seeds, and losing Alpha-Beta agents, yielding approximately 1,000 agents per game.The payoff table evaluates all matchups after duplicate agents are removed; the sampling procedure does not guarantee uniform strategy coverage.Source: §6 · pp. 8–9
• Real-world game samples show the hypothesised spinning-top geometry, while some classical game-theory games do not; small-board Go retains non-trivial diversity even among its strongest strategies.Blotto cluster sizes keep increasing at higher play levels, unlike the mid-range-peaked pattern described for Go.Source: §6 · p. 9
• The learning experiment replaces the oldest population member with the weakest sampled strategy that beats the whole population on average.The pessimistic oracle is intended to counter bias toward sampling stronger strategies.Source: §6 · p. 9
• When population size exceeds a critical threshold, training converges to the best strategies in almost all experiments; smaller populations cycle, while non-real-world games show contrasting outcomes.The Disc game continues cycling as population grows, whereas monotonic Elo converges with population size one.Source: §6 · p. 9

7 Conclusions

• Games of Skill are presented as including Tic-Tac-Toe, Chess, Go, StarCraft II, and DOTA.The paper says the class is motivated theoretically and empirically.Source: §7 · p. 9
• The paper reports that n-step games have tremendously long cycles and provides mathematical and algorithmic methods to estimate this quantity.Source: §7 · p. 9
• The authors suggest that classical game-theory games outside the Game of Skill class may present challenges not necessarily relevant to developing AI methods for real-world games.They point to population-based learning, League training, PBT, and diversity seeking as directions informed by their findings.Source: §7 · p. 9

B Computing n in n-bit communicative games

• At a decision node, a player can encode log2 |c(s)| bits through an action, plus the minimum future communicativeness across reachable child states; the symmetric measure takes the minimum across players.The recursive calculation yields the game's communicativeness from its starting state, and the stated solver computes the relevant quantities in linear time.Source: B Computing n in n-bit communicative… · p. 14
• Table 2 presents empirical game profiles, with payoff rows sorted by mean win rate and a fitted Skewed Gaussian illustrating the spinning-top shape.Source: Table 2 · p. 15
• For restricted Tic-Tac-Toe policy classes, cycle-length bounds decrease rapidly as MinMax depth increases, and first-move inductive bias changes the geometry analogously.The analysis considers the whole policy space under restrictions rather than relying on strategy sampling, using Theorem 1 as a proxy for non-transitivity.Source: B Computing n in n-bit communicative… · p. 16, Figure 4 · p. 16

C Cycles counting

• For cycle counting, the method focuses on length-three cycles and uses diag(A^3); for longer cycles, diag(A^p) counts closed walks rather than necessarily simple cycles.Source: C Cycles counting · p. 16

D Nash computation

• The Nash solver iterates until a best response confirms the numerical mixture's outcome exceeds -1e-4; maximum-entropy Nash ensures deterministic procedures through uniqueness.The authors note this choice may produce heavier rather than peakier spinning tops.Source: D Nash computation · p. 17

E Games/payoffs definition

• Empirical payoffs are symmetrised and standardized to P′ij := (Pij−Pji)/(2 max |P|), preserving Nash equilibria and transitive strength while keeping values in [-1, 1].The scaling mainly affects random games and Blotto, whose outcome ranges can be wider.Source: E Games/payoffs definition · p. 18

E.1 Real world games

• The experiments use OpenSpiel implementations of Tic Tac Toe, Misere Tic Tac Toe, and Connect Four.Misere Tic Tac Toe is described as a version where a player wins when the opponent makes a line.Source: §E.1 · p. 18
• Misere Tic Tac Toe is configured by wrapping the Tic Tac Toe implementation with the Misere game rule.Source: §E.1 · p. 18

E.2 StarCraft II (AlphaStar)

• The analysis uses the AlphaStar Final league payoff matrix, representing 900 agents across skills, all three races, and unsimplified play.The authors did not run StarCraft experiments; the strategy sample is described as least controlled and shaped by AlphaStar’s distinctive training process.Source: §E.2 · p. 18
• The observed spinning top is an AlphaStar-induced geometry, not necessarily the geometry of StarCraft II itself.The sample is skewed because agents were iteratively trained to beat earlier strategies; exploiters later added strategic diversity.Source: §E.2 · p. 18
• More uniform sampling of weak StarCraft II strategies could yield an upper cone rather than the slightly warped shape observed.AlphaStar’s iterative training conditions sampling on beating previous strategies, and the authors say this may shape the observed form.Source: §E.2 · p. 18

E.3 Rock Paper Scissor (RPS)

• The section describes ordinary RPS as fully cyclic with no pure-strategy Nash equilibrium, while real-life play adds repeated trials, known opponents, biases, and physical tells.The game-theoretic RPS equilibrium is the uniform mixture of strategies.Source: §E.3 · pp. 18–19

E.4 Disc Game

• The Disc Game samples 1000 points uniformly in the unit circle and is described as fully cyclic.Source: §E.4 · p. 19

E.5 Elo game / E.6 Noisy Elo games

• The Elo game samples 1000 player ratings and converts rating differences into monotonic win probabilities; adding Gaussian noise and antisymmetrizing produces noisy Elo games.The noisy Elo examples show Nash supports growing substantially over the transitive dimension.Source: §E.5 · p. 19, §E.6 · p. 19

E.7 Random Game of Skill

• The random Game of Skill assigns pairwise payoffs from random variables W_ij and S_i, using N=1000 samples.The supplied passage specifies the payoff as 2(W_ij − W_ji) + S_i − S_j, with W_ij and S_i drawn from N(0, 1).Source: §E.7 · p. 19

E.8 Blotto

• Blotto becomes more non-transitive with increasing strength; permutation invariance forces uniform play over permutations and grows Nash support.The tested setup uses N=10 units and K=5 fields, producing around 1000 pure strategies.Source: §E.8 · p. 19

E.9 Kuhn Poker

• Kuhn Poker’s sequential, asymmetric policies are symmetrized after average payoffs are computed from 100 simulations per policy pair.The game has 12 information states, and the payoff matrix is asymmetric because of its sequential moves.Source: §E.9 · pp. 19–20

E.10 Parity Game of Skill

• The n-step game at n = 3 exhibits hundreds of cycles and Nash clusters of size 40 among 161 pure strategies, revealing a spinning-top-like profile without sampling.Source: §E.10 · p. 20

F Other games that are not Games of Skill

• Other examples include Kuhn Poker, which appears analogous to Blotto, and pure RPS, whose geometry degenerates to a single point.The section also says Noisy Elo games can produce Nash supports that grow significantly over the transitive dimension.Source: F Other games that are not Games of… · p. 20

G Empirical Game Strategy Sampling

• The empirical strategy set combines MinMax, MaxMin, and MCTS variants, yielding 2000 deterministic pure strategies spanning the transitive axis.MCTS is included because some games are too difficult for depth-9 AlphaBeta to produce strong policies; MinMax(0,s) serves as a random-policy baseline.Source: G Empirical Game Strategy Sampling · pp. 20–21

H Empirical Game Payoff computation

• Payoffs average two matches with swapped first player, then remove duplicate rows, leaving matrices no larger than 2000 × 2000 and typically closer to 1000 × 1000.The averaging symmetrizes games that are not purely symmetric because of turn order.Source: H Empirical Game Payoff computation · pp. 21–22

I Fitting spinning top profile

• The profile-fitting section models the relationship between mean RPP and Nash-cluster size with a skewed-normal distribution and an affine transformation.The authors leave using likelihood under this model as a measure of game-skillness, and its applications, for future research.Source: I Fitting spinning top profile · p. 22

J Counting pure strategies

• Behaviorally distinct pure strategies are counted recursively; Tic-Tac-Toe has approximately 10^567 such strategies, an upper bound on its minimal payoff-describing normal-form size.Behavioral difference need not imply different payoffs, but different payoffs do imply behavioral difference.Source: J Counting pure strategies · p. 22

K Deterministic strategies and neural network based agents

• Although neural agents can be stochastic, they were found to become almost purely deterministic as training progresses, supporting pure-strategy modeling; mixed-strategy extensions remain future work.Source: K Deterministic strategies and neural… · pp. 22–23

Random Games of Skill

• Random games use Qij = Wij + Si − Sj, separating strategy strength Si from pair-specific interactions Wij, and provide a possible model for Games of Skill.The paper says Si captures transitive strength, while Wij encodes interactions specific to πi playing against πj; it also reports that random games exhibit spinning-top geometry.Source: L Random Games of Skill · p. 23
• As σS/σW grows, Nash-cluster counts increase; each cluster’s strategy-strength range is bounded by α.Nash support is highest around zero transitive strength and tends toward zero as strength approaches ±∞.Source: L Random Games of Skill · p. 23
• Under a uniform improvement oracle, E[S̄t+1] > E[S̄t], and population size reduces the variance of W at a quartic rate.The stated expected-strength increase is for the oracle training setup; the paper attributes the variance reduction to population size.Source: L Random Games of Skill · p. 24