Pith. sign in

REVIEW 2 major objections 4 minor 51 references

Game connectivity and adaptive dynamics in many-action games

T0 review · 2 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper shows that for a fixed number of players n≥3, the fraction of many-action generic games with a pure Nash equilibrium that are connected tends to the explicit constant 1−ζ_n as the number of actions grows, where ζ_n is small and v

desk verdict This is a serious, mostly sound paper that fills the large-k gap left open by the authors' earlier work, with a genuinely new explicit limiting constant and quite intricate proof machinery; it deserves a real referee. read the letter →

arxiv 2601.05965 v2 pith:JCRPSHTJ submitted 2026-01-09 econ.TH cs.GTmath.CO

Tom Johnston , Michael Savery , Alex Scott , Bassel Tarbush This is my paper · ORCID
classification econ.THcs.GTmath.CO MSC 91A0691A2660J8005C8005C20
keywords gameconnectivityadaptivedynamicsbest-responsegraphspureNashequilibriumrandomgamesGalton-WatsonbranchingprocessesHammingPoissonapproximation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies generic n-player k-action games (no indifferences) and asks: among those that have a pure Nash equilibrium, what fraction are 'connected' — meaning from every action profile that is not an equilibrium, best-response dynamics can reach every equilibrium? In the many-actions regime (k large, n fixed), the answer is not 1: the fraction tends to 1 − ζ_n, where ζ_n is an explicit positive constant. For n≥3 this constant is small (ζ_3 ≈ 0.0132, ζ_4 ≈ 0.00002) and decays rapidly with n, so almost all many-player-many-action games are connected. The paper proves this by analysing random best-response graphs as random subgraphs of directed Hamming graphs, showing the numbers of 'good' and 'bad' sinks are asymptotically independent Poisson variables. As a consequence, a simple adaptive dynamic (best-response with inertia) converges to a pure Nash equilibrium in all but a vanishingly small fraction of generic games that have one.

What carries the argument

The central object is the random subdigraph of the directed Hamming graph, where for each line (a set of action profiles differing in one coordinate) a winner is chosen uniformly and all edges not ending at the winner are deleted; sinks correspond to pure Nash equilibria. The argument identifies 'good' sinks, reachable from many lines, and 'bad' sinks, reachable from few, and shows their counts are asymptotically independent Poisson variables. The probability p that a sink is bad is controlled by the extinction probability of a Galton-Watson branching process with offspring distribution approximating Poi(n−1). Connectivity is established via 'good cycles' in 2-dimensional slices: directed cy

What would settle it

Simulate the random directed Hamming subgraph L(3,k) for large k (e.g., k=10^6), count the sinks reachable from every non-sink and those not reachable from every non-sink, and test whether their joint distribution converges to independent Poisson variables with means 1−p and p, where p=(η_2)^3≈0.0132. Also simulate random generic 3-player k-action games and estimate the connected fraction among those with a pure Nash equilibrium; if it does not approach 1−ζ_3, the theorem is false.

Watch

Extended reading notes

Core claim

For fixed n≥3, as k→∞, the fraction of generic n-player k-action games with a pure Nash equilibrium that are connected tends to 1 − ζ_n, with ζ_n = 1 − (e^{−λ_n}(1 − e^{λ_n−1}))/(1 − e^{−1}), where λ_n is the smallest positive solution to x^{1/n} = e^{(n−1)(x^{1/(n−1)})}. Equivalently, in the random directed Hamming graph L(n,k) in which each line selects a uniform winner, the number of sinks reachable from every non-sink and the number not reachable from every non-sink are asymptotically independent Poisson random variables with means 1−p and p, p=(η_{n−1})^n, giving connected fraction (e^{−p}−e^{−1})/(1−e^{−1}) = 1−ζ_n. Thus a small but non-vanishing constant fraction of many-action games

Load-bearing premise

The proof of Theorem 4 and hence Theorem 1 depends on the claim that, with probability at least 1/100, each 2-dimensional slice of the random Hamming subgraph contains a 'good cycle' of length at least √k with a basin of k^2/(800 log k) vertices (Lemma 11); if this probability were o(1), the strong-connectivity argument would fail and the constant ζ_n would not follow.

Editorial extensions

If this is right

  • For fixed n≥3, as k→∞, the connected fraction tends to 1−ζ_n; for n=3 this is about 0.9868, for n=4 about 0.99998.
  • As n→∞, the connected fraction tends to 1 uniformly over k (Theorem 2), so the many-player and many-action regimes together give connectedness in the double limit.
  • A simple adaptive dynamic — best-response with inertia — converges almost surely to a pure Nash equilibrium in all but a vanishingly small fraction of generic games that have one (Proposition 3).
  • For n=2, the connected fraction tends to 0, so the n≥3 behavior is special.
  • The iterated limits lim_{n→∞}lim_{k→∞} and lim_{k→∞}lim_{n→∞} both give connected fraction 1.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The explicit ζ_n links a purely game-theoretic prevalence question to the extinction probability of a Poisson branching process; this suggests the 'hard' games in the large-action regime are precisely those where a backwards exploration from a Nash equilibrium dies out after seeing few lines.
  • The Poisson-independence structure of good/bad sinks suggests that, for fixed n, the distribution of the number of equilibria reachable from all non-equilibria obeys a simple law; one could numerically test the predicted Poi(1−p) and Poi(p) counts in simulated random games.
  • The contrast between n=2 (fraction → 0) and n≥3 (fraction → 1−ζ_n) indicates a phase transition in best-response connectivity as the number of players crosses 3; this may guide intuition for which games are amenable to decentralized learning.
  • The proof techniques — good cycles in slices and branching-process approximations — might extend to other random structures on Hamming graphs, such as best-response dynamics in games with correlated payoffs or with heterogeneous action sets.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies the typical connectivity of generic n-player k-action games. A game is connected if it has a pure Nash equilibrium and every non-equilibrium action profile has a best-response path to every pure Nash equilibrium. For fixed n≥3 and k→∞, the paper claims (Theorem 1) that the fraction of generic games with a pure Nash equilibrium that are connected tends to 1−ζ_n, where ζ_n is an explicit positive constant decreasing rapidly in n. It also claims (Theorem 2) that as n→∞ this fraction tends to 1 uniformly over all k≥2. The technical engine is a random subdigraph model L(n,k) of the directed Hamming graph, in which each line independently has a uniformly chosen winner. Theorem 4 states that, for n≤k^{1/2−ε}, the number X of sinks reachable from every non-sink and the number Y of sinks not reachable from every non-sink are asymptotically independent Poisson random variables with means 1−p and p, where p=(η_{n−1})^n and η_x is the extinction probability of a Galton–Watson process with offspring distribution Poi(x). The proof uses a branching-process approximation for good/bad sinks, a 'good-cycle' scaffold to find a large strongly connected component, and several union-bound estimates. The paper also derives an adaptive-dynamics consequence (Proposition 3).

Significance. If the results hold, this is a significant contribution to the random-games and adaptive-dynamics literature. The paper resolves the previously open many-action regime, gives an explicit constant ζ_n, and complements the authors' earlier many-player result. The probabilistic technique—combining branching-process extinction with good-cycle ubiquity in 2-dimensional slices—is novel and likely to be useful beyond this setting. The paper is careful in connecting the random-graph model to generic games and in stating the implications for simple adaptive dynamics. The proofs are long and structured, and several estimates (Lemmas 11–14, 27–35) are plausible and checkable. However, two gaps in the written proofs of Theorems 4 and 5 need attention before the paper is fully convincing.

major comments (2)
  1. [§3, Proof of Theorem 4] The proof fixes ε=1/10 and invokes Theorem 7. Theorem 7 requires n≤k^{ε}/log k, i.e. n≤k^{1/10}/log k. The hypothesis of Theorem 4 only gives n≤k^{1/2−δ} for some δ>0, which allows n(k) to grow like k^{1/5}; such functions do not satisfy n≤k^{1/10}/log k. Thus the proof as written does not cover the full stated range of Theorem 4. This is load-bearing for Theorem 4 as a statement about growing n. A simple fix is to split the proof: for fixed n (the case needed for Theorem 1), Theorem 7 applies for large k; for n(k)→∞, p→0 and Lemma 8 (or a direct branching-process bound) implies that bad sinks are o(1) in probability, while the total number of sinks is asymptotically Poi(1), giving the stated limit. Please clarify this case.
  2. [§3, Proof of Theorem 5] In the case k≤δ√(n/log n), the proof claims: 'By Theorem 15, with probability at least 1−2^{−cn}, every non-sink can reach every sink.' This inference is not immediate from the stated Theorem 15, which only gives a dichotomy: every vertex is either reached from at most N_B(1+ε)k log k vertices or from every non-sink. A sink in the first category would violate the desired event. The proof should explicitly rule out the first category for sinks in this regime. The missing observation is that every sink has at least n(k−1)+1 vertices in its basin (all vertices on the n lines through the sink, since the sink wins each such line), and in the regime k≤δ√(n/log n) this is ≫ N_B(1+ε)k log k for large n. This is likely what the assumption n≥2√(n log n) is for, but it is not stated. Please add this argument.
minor comments (4)
  1. [§1.2, Theorem 1] The displayed formula for ζ_n is ambiguous as typeset: 'ζ_n = 1 − e^{−λ_n}(1−e^{λ_n−1})/(1−e^{−1})' should read ζ_n = (1−e^{−λ_n})/(1−e^{−1}), or at least include parentheses around the numerator. The current layout can confuse the reader.
  2. [§5, Lemma 8] Lemma 8 is stated for all n,k≥2, but its derivation in Section 5 relies on Lemma 27, which is stated only for n≤k^{ε}/log k. The coupling argument actually works for n≤k^{ε} when ε<1/2 (the error is then O(k^{2ε−1})), and for n≥k^{ε} there are no ε-bad sinks. The manuscript should state this extension so the reader can verify the unconditional form of Lemma 8.
  3. [§6, Lemma 11 proof] The exploration process starting from a point implicitly assumes the starting point has an outgoing horizontal edge; if the point wins its horizontal line, the process cannot start. This occurs with probability 1/k per starting point, so the constant 1/9 is unaffected asymptotically, but the proof should mention this negligible conditioning issue.
  4. [§3, Proof of Theorem 5] The line 'Suppose also that n is large enough that n≥2√(n log n)' is confusing as written; for large n this is equivalent to n≥C log n for a constant C. The intended use is to ensure n(k−1)+1 ≫ k log k in the small-k case, and this implication should be spelled out.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: central derivation is self-contained; only minor non-load-bearing self-citation.

full rationale

The main derivation of Theorem 4, and hence Theorem 1, is self-contained. Theorem 7 derives the Poisson(1-p)/Poisson(p) counts of good/bad sinks via a method-of-moments proof whose key input is the Galton-Watson coupling in Lemma 27; p=(η_{n-1})^n is an extinction probability computed from Lemma 16, not fitted to the connectivity fraction. The identification of good/bad sinks with sinks reachable/unreachable from every non-sink is proved via Lemmas 12-14 rather than assumed: Lemma 13 shows every non-sink reaches a good cycle, Lemma 14 shows every good sink is reachable from a good cycle, and Lemma 12 puts all good cycles in one strongly connected component. Lemma 11's constant 1/100 is load-bearing but is derived inside the proof using a birthday-problem repeat-time estimate, a bound on the number of cycles, and explicit constants (δ=1/20, λ=34200/929), so it is not an ansatz smuggled in or a conclusion baked into a definition. The only self-citations are to Johnston, Savery, Scott, and Tarbush (2023): the terminology 'connected' and Theorem 15 used in Theorem 5. The paper explicitly states that Theorem 2 is a combination of the two works, and Theorem 15 has its own proof and assumptions that do not include the present target result; under the stated rules this is independent support, not circularity. No fitted parameter is relabeled as a prediction, and no uniqueness or ansatz is imported via self-citation. The score reflects only minor, non-load-bearing self-citation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted free parameters appear in the paper: ζ_n, p, and the various constants are derived from the model rather than tuned to data. The main unproved imported assumptions are standard probability theorems plus one self-cited previous theorem used for the large-n part.

assumptions (5)
  • domain assumption A random generic game is equivalent to independently uniformly choosing one winner on every line of the directed Hamming graph (the best-response winners).
    Section 2; this is the modeling equivalence that turns game connectivity into connectivity of #»L(n,k).
  • standard math Galton-Watson extinction probability identities (Lemma 16, Lemma 19, Corollary 20) and Otter-Dwass tail bound (Lemma 18).
    Used throughout Sections 4-5 to identify p=(η_{n−1})^n and to bound large total populations.
  • standard math Le Cam's theorem (Theorem 25) for total variation between a binomial sum and a Poisson distribution.
    Lemma 27 couples the binomial exploration process to a Poi(n−1) branching process.
  • standard math Serfling's hypergeometric inequality and Chernoff's bound (Lemmas 21-22).
    Used for concentration estimates in Sections 6-9.
  • domain assumption Theorem 15 from Johnston, Savery, Scott, and Tarbush (2023): for k≤δ√(n/log n), every vertex of #»L(n,k) is reachable from at most N_B(1+ε)k log k vertices or from every non-sink.
    Relied on in the proof of Theorem 5/Theorem 2; stated but not reproved in this paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Game connectivity and adaptive dynamics in many-action games." pith.science (2026). https://pith.science/paper/JCRPSHTJ

@misc{pith2026260105965,
  author       = {Pith},
  title        = {Pith review of: Game connectivity and adaptive dynamics in many-action games},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JCRPSHTJ}},
  note         = {Machine review of arXiv:2601.05965}
}
abstract

We study the typical structure of games in terms of their connectivity properties. A game is `connected' if it has a pure Nash equilibrium and there is a best-response path from every action profile which is not a pure Nash equilibrium to every pure Nash equilibrium; a game is generic if it has no indifferences. In previous work we showed that, among all $n$-player $k$-action generic games that admit a pure Nash equilibrium, the fraction that are connected tends to $1$ as $n$ gets sufficiently large relative to $k$. Here, we consider the large-$k$ regime, which behaves differently: we show that the connected fraction tends to $1-\zeta_n$ as $k$ gets large, where $\zeta_n>0$ is an explicit constant. Thus, a constant fraction of many-action games are \emph{not} connected. However, for $n\geq3$, $\zeta_n$ is small and tends to $0$ rapidly with $n$, so as $n$ increases all but a vanishingly small fraction of many-player-many-action games are connected. Since connectedness is conducive to equilibrium convergence, we find a simple adaptive dynamic that is guaranteed to converge to a pure Nash equilibrium in all but a vanishingly small fraction of generic games that have one. We rely on new probabilistic and combinatorial arguments to tackle the large-$k$ regime.

Figures

Figures reproduced from arXiv: 2601.05965 by the authors.

Figure 1
Figure 1. A 3-player 2-action game (left) and its corresponding best-response graph [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

51 extracted references · 6 linked inside Pith

  1. [1]

    Dominance solvability in random games

    Noga Alon, Kirill Rudov, and Leeat Yariv. Dominance solvability in random games. arXiv preprint arXiv:2105.10743, 2021

    arXiv 2021

  2. [2]

    When better is better than best

    Ben Amiet, Andrea Collevecchio, and Kais Hamza. When better is better than best. Operations Research Letters, 49 0 (2): 0 260\,--\,264, 2021 a

    2021

  3. [3]

    Pure N ash equilibria and best-response dynamics in random games

    Ben Amiet, Andrea Collevecchio, Marco Scarsini, and Ziwen Zhong. Pure N ash equilibria and best-response dynamics in random games. Mathematics of Operations Research, 46 0 (4): 0 1552\,--\,1572, 2021 b

    2021

  4. [4]

    A classification of weakly acyclic games

    Krzysztof R Apt and Sunil Simon. A classification of weakly acyclic games. Theory and Decision, 78 0 (4): 0 501\,--\,524, 2015

    2015

  5. [5]

    Simultaneous best-response dynamics in random potential games

    Galit Ashkenazi-Golan, Domenico Mergoni Cecchelli, and Edward Plumb. Simultaneous best-response dynamics in random potential games. arXiv preprint arXiv:2505.10378, 2025

    arXiv 2025

  6. [6]

    Completely uncoupled dynamics and N ash equilibria

    Yakov Babichenko. Completely uncoupled dynamics and N ash equilibria. Games and Economic Behavior, 76 0 (1): 0 1\,--\,14, 2012

    2012

  7. [7]

    Nash equilibria in random games

    Imre B \'a r \'a ny, Santosh Vempala, and Adrian Vetta. Nash equilibria in random games. Random Structures & Algorithms, 31 0 (4): 0 391\,--\,405, 2007

    2007

  8. [8]

    Computing B ayes-- N ash equilibrium strategies in auction games via simultaneous online dual averaging

    Martin Bichler, Max Fichtl, and Matthias Oberlechner. Computing B ayes-- N ash equilibrium strategies in auction games via simultaneous online dual averaging. Operations Research, 2023 a

    2023

Show all 51 references
  1. [9]

    Learning equilibrium in bilateral bargaining games

    Martin Bichler, Nils Kohring, Matthias Oberlechner, and Fabian R Pieroth. Learning equilibrium in bilateral bargaining games. European Journal of Operational Research, 311 0 (2): 0 660\,--\,678, 2023 b

    2023

  2. [10]

    Random Graphs

    B\' e la Bollob\' a s. Random Graphs. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2 edition, 2001

    2001

  3. [11]

    The hidden game problem

    Gon Buzaglo, Noah Golowich, and Elad Hazan. The hidden game problem. arXiv preprint arXiv:2510.03845, 2025

    2025

  4. [12]

    Playing large games with oracles and AI debate

    Xinyi Chen, Angelica Chen, Dean Foster, and Elad Hazan. Playing large games with oracles and AI debate. arXiv preprint arXiv:2312.04792, 2024

    2024 arXiv

  5. [13]

    Basins of attraction in two-player random ordinal potential games

    Andrea Collevecchio, Hlafo Alfie Mimun, Matteo Quattropani, and Marco Scarsini. Basins of attraction in two-player random ordinal potential games. arXiv preprint arXiv:2407.05460, 2024 a

    2024 arXiv

  6. [14]

    Finding pure Nash equilibria in large random games

    Andrea Collevecchio, Tuan-Minh Nguyen, and Ziwen Zhong. Finding pure Nash equilibria in large random games. arXiv preprint arXiv:2406.09732, 2024 b

    2024 arXiv

  7. [15]

    Finding a N ash equilibrium of a random win-lose game in expected polynomial time

    Andrea Collevecchio, Gabor Lugosi, Adrian Vetta, and Rui-Ray Zhang. Finding a N ash equilibrium of a random win-lose game in expected polynomial time. arXiv preprint arXiv:2510.12846, 2025

    2025

  8. [16]

    Emergent alignment via competition

    Natalie Collina, Surbhi Goel, Aaron Roth, Emily Ryu, and Mirah Shi. Emergent alignment via competition. arXiv preprint arXiv:2509.15090, 2025

    2025

  9. [17]

    Connectivity and equilibrium in random games

    Constantinos Daskalakis, Alexandros G Dimakis, and Elchanan Mossel. Connectivity and equilibrium in random games. The Annals of Applied Probability, 21: 0 987\,--\,1016, 2011

    2011

  10. [18]

    Probability of a pure equilibrium point in n -person games

    Melvin Dresher. Probability of a pure equilibrium point in n -person games. Journal of Combinatorial Theory, 8 0 (1): 0 134\,--\,145, 1970

    1970

  11. [19]

    On the structure of weakly acyclic games

    Alex Fabrikant, Aaron D Jaggard, and Michael Schapira. On the structure of weakly acyclic games. Theory of Computing Systems, 53 0 (1): 0 107\,--\,122, 2013

    2013

  12. [20]

    The Theory of Learning in Games

    Drew Fudenberg and David K Levine. The Theory of Learning in Games. MIT Press, 1998

    1998

  13. [21]

    The probability of an equilibrium point

    K Goldberg, AJ Goldman, and M Newman. The probability of an equilibrium point. Journal of Research of the National Bureau of Standards, 72 0 (2): 0 93\,--\,101, 1968

    1968

  14. [22]

    A simple adaptive procedure leading to correlated equilibrium

    Sergiu Hart and Andreu Mas-Colell. A simple adaptive procedure leading to correlated equilibrium. Econometrica, 68 0 (5): 0 1127\,--\,1150, 2000

    2000

  15. [23]

    Uncoupled dynamics do not lead to N ash equilibrium

    Sergiu Hart and Andreu Mas-Colell. Uncoupled dynamics do not lead to N ash equilibrium. American Economic Review, 93 0 (5): 0 1830\,--\,1836, 2003

    2003

  16. [24]

    Stochastic uncoupled dynamics and N ash equilibrium

    Sergiu Hart and Andreu Mas-Colell. Stochastic uncoupled dynamics and N ash equilibrium. Games and Economic Behavior, 57 0 (2): 0 286\,--\,303, 2006

    2006

  17. [25]

    Simple Adaptive Strategies: From Regret-Matching to Uncoupled Dynamics, volume 4

    Sergiu Hart and Andreu Mas-Colell. Simple Adaptive Strategies: From Regret-Matching to Uncoupled Dynamics, volume 4. World Scientific, 2013

    2013

  18. [26]

    Evolutionarily stable strategies of random games, and the vertices of random polygons

    Sergiu Hart, Yossef Rinott, and Benjamin Weiss. Evolutionarily stable strategies of random games, and the vertices of random polygons. The Annals of Applied Probability, 18: 0 259\,--\,287, 2008

    2008

  19. [27]

    Best-response dynamics, playing sequences, and convergence to equilibrium in random games

    Torsten Heinrich, Yoojin Jang, Luca Mungo, Marco Pangallo, Alex Scott, Bassel Tarbush, and Samuel Wiese. Best-response dynamics, playing sequences, and convergence to equilibrium in random games. International Journal of Game Theory, 52: 0 703\,--\,735, 2023

    2023

  20. [28]

    The N ash equilibrium: A perspective

    Charles A Holt and Alvin E Roth. The N ash equilibrium: A perspective. Proceedings of the National Academy of Sciences, 101 0 (12): 0 3999\,--\,4002, 2004

    2004

  21. [29]

    Self-stabilizing uncoupled dynamics

    Aaron D Jaggard, Neil Lutz, Michael Schapira, and Rebecca N Wright. Self-stabilizing uncoupled dynamics. In Algorithmic Game Theory: 7th International Symposium, SAGT 2014, Proceedings 7, pages 74\,--\,85. Springer, 2014

    2014

  22. [30]

    Game connectivity and adaptive dynamics

    Tom Johnston, Michael Savery, Alex Scott, and Bassel Tarbush. Game connectivity and adaptive dynamics. arXiv preprint arXiv:2309.10609, 2023

    2023 arXiv

  23. [31]

    An approximation theorem for the P oisson binomial distribution

    Lucien Le Cam. An approximation theorem for the P oisson binomial distribution. Pacific Journal of Mathematics, 10: 0 1181\,--\,1197, 1960

    1960

  24. [32]

    Probability on trees and networks, volume 42

    Russell Lyons and Yuval Peres. Probability on trees and networks, volume 42. Cambridge University Press, 2017

    2017

  25. [33]

    Regret based dynamics: convergence in weakly acyclic games

    Jason R Marden, G \"u rdal Arslan, and Jeff S Shamma. Regret based dynamics: convergence in weakly acyclic games. In Proceedings of the sixth international joint conference on Autonomous Agents and Multiagent Systems, pages 1\,--\,8, 2007

    2007

  26. [34]

    Payoff-based dynamics for multiplayer weakly acyclic games

    Jason R Marden, H Peyton Young, G \"u rdal Arslan, and Jeff S Shamma. Payoff-based dynamics for multiplayer weakly acyclic games. SIAM Journal on Control and Optimization, 48 0 (1): 0 373\,--\,396, 2009

    2009

  27. [35]

    The component structure of dense random subgraphs of the hypercube

    Colin McDiarmid, Alex Scott, and Paul Withers. The component structure of dense random subgraphs of the hypercube. Random Structures & Algorithms, 59 0 (1): 0 3\,--\,24, 2021

    2021

  28. [36]

    An impossibility theorem in game dynamics

    Jason Milionis, Christos Papadimitriou, Georgios Piliouras, and Kelly Spendlove. An impossibility theorem in game dynamics. Proceedings of the National Academy of Sciences, 120 0 (41): 0 e2305349120, 2023

    2023

  29. [37]

    Best-response dynamics in two-person random games with correlated payoffs

    Hlafo Alfie Mimun, Matteo Quattropani, and Marco Scarsini. Best-response dynamics in two-person random games with correlated payoffs. Games and Economic Behavior, 145: 0 239\,--\,262, 2024

    2024

  30. [38]

    Probability and computing: Randomization and probabilistic techniques in algorithms and data analysis

    Michael Mitzenmacher and Eli Upfal. Probability and computing: Randomization and probabilistic techniques in algorithms and data analysis. Cambridge university press, 2017

    2017

  31. [39]

    Potential games

    Dov Monderer and Lloyd S Shapley. Potential games. Games and Economic Behavior, 14 0 (1): 0 124\,--\,143, 1996

    1996

  32. [40]

    Nash equilibria in random games with right fat-tailed distributions

    Ting Pei and Satoru Takahashi. Nash equilibria in random games with right fat-tailed distributions. International Journal of Game Theory, 52 0 (4): 0 1153\,--\,1177, 2023

    2023

  33. [41]

    Limiting distributions of the number of pure strategy Nash equilibria in n -person games

    Imelda Young Powers. Limiting distributions of the number of pure strategy Nash equilibria in n -person games. International Journal of Game Theory, 19 0 (3): 0 277\,--\,286, 1990

    1990

  34. [42]

    On the number of pure strategy Nash equilibria in random games

    Yosef Rinott and Marco Scarsini. On the number of pure strategy Nash equilibria in random games. Games and Economic Behavior, 33 0 (2): 0 274\,--\,293, 2000

    2000

  35. [43]

    Twenty Lectures on Algorithmic Game Theory

    Tim Roughgarden. Twenty Lectures on Algorithmic Game Theory. Cambridge University Press, 2016

    2016

  36. [44]

    Population Games and Evolutionary Dynamics

    William H Sandholm. Population Games and Evolutionary Dynamics. MIT Press, 2010

    2010

  37. [45]

    Probability inequalities for the sum in sampling without replacement

    Robert J Serfling. Probability inequalities for the sum in sampling without replacement. The Annals of Statistics, pages 39\,--\,48, 1974

    1974

  38. [46]

    A note on the probability of k pure Nash equilibria in matrix games

    William Stanford. A note on the probability of k pure Nash equilibria in matrix games. Games and Economic Behavior, 9 0 (2): 0 238\,--\,246, 1995

    1995

  39. [47]

    Michael Steele

    J. Michael Steele. Le C am's inequality and P oisson approximations. American Mathematical Monthly, 101 0 (1): 0 48\,--\,54, 1994

    1994

  40. [48]

    No-regret learning and mixed N ash equilibria: They do not mix

    Emmanouil-Vasileios Vlatakis-Gkaragkounis, Lampros Flokas, Thanasis Lianeas, Panayotis Mertikopoulos, and Georgios Piliouras. No-regret learning and mixed N ash equilibria: They do not mix. Advances in Neural Information Processing Systems, 33: 0 1380\,--\,1391, 2020

    2020

  41. [49]

    The evolution of conventions

    H Peyton Young. The evolution of conventions. Econometrica, 61 0 (1): 0 57\,--\,84, 1993

    1993

  42. [50]

    Peyton Young

    H. Peyton Young. Strategic Learning and its Limits . Oxford University Press, 2004

    2004

  43. [51]

    Learning by trial and error

    H Peyton Young. Learning by trial and error. Games and Economic Behavior, 65 0 (2): 0 626\,--\,643, 2009

    2009

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.