On the convergence of fictitious play algorithm in repeated games via the geometrical approach

Xiaoguang Yang; Yifen Mu; Zhouming Wu

REVIEW 4 major objections 4 minor 33 references

On the convergence of fictitious play algorithm in repeated games via the geometrical approach

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper proves that continuous-time fictitious play converges to a Nash equilibrium from every starting point in every $3\times3$ game whose indifference point lies outside both players' simplices, using a projection that reduces the…

desk verdict The paper's own Example 4.2 contradicts Theorem 3.1, but the projection method and the class comparison are genuinely useful and deserve referee attention. read the letter →

arxiv 2412.19216 v2 pith:VEN5R5XX submitted 2024-12-26 math.OC

Zhouming Wu , Yifen Mu , Xiaoguang Yang This is my paper · ORCID

classification math.OC MSC 91A2691A05

keywords fictitiousplaycontinuous-timeNashequilibriuminternalindifferentpointprojectionmappingbest-responsedynamicsconvergencegeometricalapproach

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's central claim is that continuous-time fictitious play — the learning rule in which each player best responds to the opponent's empirical play — converges to a Nash equilibrium in every $3\times3$ game whose indifferent point lies outside both players' strategy simplices. This defines a new class of games with the fictitious play property, distinct from the previously known quasi-supermodular games. If the claim is true, it answers a long-standing question about which $3\times3$ games are safe for fictitious play: non-convergence, as in the classic three-by-three counterexample, requires a fully mixed equilibrium in a specific sense. The proof's engine is a projection that collapses each player's strategy triangle to an edge, turning the four-dimensional best-response dynamics into a planar system, plus a redefinition of saddle and sink equilibria for the nonsmooth dynamics. The paper also shows the projection works for degenerate games and some $4\times4$ games, where it can certify non-convergence.

What carries the argument

The central object is the projection mapping $\varphi = \varphi_A \times \varphi_B$, which sends each player's strategy simplex to an edge by drawing the line through a point and the game's indifferent point and recording its intersection with the chosen edge; when the indifferent point is absent, the projection runs parallel to the now-parallel indifference lines. On the resulting unit square, the image of the best-response dynamics is a planar piecewise-linear system (PBRD) with nine labeled cells, and the key identity is the return map $f$ on the entry edge, composed of eight fractional-linear maps that can be written down explicitly from similar triangles. The derivative of $f$ factors into ratios of cell widths and heights and is strictly less than 1, which is the contraction that rules out the endless eight-cell loop that would otherwise allow non-convergence. The paper also introduces geometric redefinitions of saddle and sink Nash equilibria tailored to the nonsmooth vector field, and proves these always exist when equilibria are multiple.

What would settle it

Numerically integrate the continuous-time fictitious play equations (2) for any $3\times3$ game whose indifferent point lies outside both simplices, starting from a point whose projection lands on the eight-cell loop; if the projection's return distance shrinks to zero but the original four-dimensional trajectory's distance to the unique Nash equilibrium does not (for instance, the trajectory keeps switching among all three actions), then Proposition 3.3(2) and Theorem 3.1 are false.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is Theorem 3.1: every continuous-time fictitious play process approaches a Nash equilibrium in every $3\times3$ game without an internal indifferent point (IIP), the unique (possibly negative-probability) strategy profile at which all of a player's pure actions give equal payoff. Since that point lies outside both players' simplices, each player's strategy triangle contains at most two indifference lines, making the new projection mapping well-defined. The projected best-response dynamics live on a unit square divided into nine cells; the paper proves that any trajectory either enters a four- or six-cell path where a known $2\times n$ convergence result forces convergence, or it follows an eight-cell loop controlled by a return map. The map's derivative factors as a product of ratios of cell widths and is strictly less than 1, so the return map is contractive and eventually pushes the trajectory into the convergent regime. For games with multiple equilibria, the paper defines saddle and sink equilibria geometrically, proves a saddle and a sink always coexist, and uses the saddle's separating lines to partition the square so that no trajectory can visit all three actions of both players forever.

Load-bearing premise

The load-bearing premise is that convergence of the projected planar dynamics forces convergence of the original four-dimensional dynamics, even though the projection identifies many different original states with a single point on the plane.

Editorial extensions

If this is right

The paper's Theorem 3.1 adds the entire class of $3\times3$ games without an internal indifferent point to the known class of games with the fictitious play property, and Example 4.1 shows this class is not contained in the quasi-supermodular games previously identified.
Corollary 3.1 turns the long-run pattern of action profiles into a classification of the limit equilibrium: one action profile repeated means a pure Nash equilibrium, two switching profiles mean a saddle, and four switching profiles mean a sink.
Because CFP and BRD follow the same trajectory shapes and differ only in time scaling, the contraction of the return map on the projected square transfers to the original dynamics, giving convergence from every initial condition in the no-IIP class.
The projection method also applies outside the convergent setting: it reproduces a known $4\times4$ non-convergent example and reveals regime shifts in one-parameter families of games, where a continuum of Nash equilibria can emerge and the dynamics may approach the continuum without converging to any point of it.

Reading between the lines

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

Editorial inference: the derivative bound on the return map is quantitative, so the same proof should yield a rate at which trajectories on the eight-cell path approach the contractive regime; a natural extension is an explicit bound on the number of returns needed for $\epsilon$-convergence in games without IIP.
Editorial inference: Proposition 3.3(2) is the step most worth probing, because $\varphi$ identifies whole line segments with single points, so convergence on the projected square does not formally rule out motion along those fibers; simulating the original CFP for several initial conditions with the same projection would test whether the reduction hides any non-convergent behavior.
Editorial inference: the regime-shift examples suggest that the connectivity structure of the Nash equilibrium set can change abruptly along a payoff path while the dynamics remain tame; if pursued further, this connects to the difficulty of homotopy methods for equilibrium computation rather than to fictitious play itself.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague, here's the read. The headline: the main convergence theorem as stated is contradicted by the paper's own Example 4.2, and the proof has load-bearing gaps. Still, this is not a throwaway. The paper introduces a genuinely new geometric reduction and gives a convincing comparison with the quasi-supermodular class, so it is worth a referee's time—but not in its current form.

What is actually new: the class of 3x3 games without an internal indifferent point (IIP), and the projection mapping from the four-dimensional product simplex to the unit square. That mapping is a real idea, and the authors use it to turn best-response dynamics into a planar piecewise-linear system. The Poincaré map contraction for the typical cell path is explicit and checkable. The example showing the class is not contained in the quasi-supermodular games of Berger (2007) is solid. The higher-dimensional extension is speculative but interesting.

Now the soft spots, in order of severity. First, Theorem 3.1 says every CFP approaches equilibrium in every 3x3 game without IIP. The paper's own Section 4.2, Example 4.2 at k = -2/3, is a 3x3 game without IIP, and the text says its trajectory 'surrounds and approaches the set of NEs, but will never converge to any point in the set.' Under the paper's own definition in Section 2.2, convergence means pointwise convergence to a strategy profile. So the theorem is internally inconsistent with the example. The authors hand-wave that degeneracy doesn't affect convergence, but that is exactly the case that breaks it. If the theorem is meant only for nondegenerate games, then the abstract, the theorem statement, and the degenerate section all need revision.

Second, the reduction step, Proposition 3.3, is not proven. The projection is non-injective: a point on the projected edge corresponds to a whole segment in the original simplex. The proof of part (2) is two sentences and invokes Lemma 2.2 after saying the trajectory 'eventually belongs to the 4 cells around NE.' It never rules out motion along the fibers of the projection. That is the bridge from the planar dynamics back to the original dynamics, and it is load-bearing.

Third, the proof of Theorem 3.2 covers one typical case and says 'all the other cases are just similar.' That is not a proof for a theorem stated over all games without IIP. Appendix D enumerates cases but does not carry out the analysis for each.

Fourth, Proposition 3.4 contains an explicitly circular remark: the existence of a sink NE is declared 'a straightforward corollary of our main results Theorem 3.1,' and then Theorem 3.3 uses Proposition 3.4 to prove convergence. The authors claim it is not circular, but it is.

There is also a scope mismatch: the abstract advertises the fictitious play property (discrete-time), but the theorems concern CFP. The discrete-to-continuous gap is left open.

Who this is for: researchers working on fictitious play and game dynamics. The projection idea and the non-overlap example are useful; the convergence claim is not supported as written. I would not cite the main theorem, but I would bring the paper to a reading group to discuss the technique and the failure mode.

Recommendation: send to peer review, with the expectation of major revision or rejection. The authors need to fix the degenerate case, give a real proof of the reduction, and cover all claimed cases. As it stands, the central result is contradicted by their own example.

Referee Report

4 major / 4 minor

Summary. The paper studies continuous-time fictitious play (CFP) in two-player 3x3 games without an internal indifferent point (IIP). It introduces a projection from the four-dimensional strategy space to a planar system, defines saddle and sink Nash equilibria for the resulting non-smooth best-response dynamics, and claims Theorem 3.1: every CFP approaches equilibrium in every 3x3 game without IIP. The paper also claims that the projection method extends to degenerate games and to some higher-dimensional games, and it compares the proposed class with quasi-supermodular games.

Significance. If the main theorem were correct, the paper would identify a substantial new class of games with the fictitious play property and would offer a geometric reduction that could be useful for other game classes. The paper contains concrete examples and simulations, and the geometric projection idea is genuinely interesting. However, the central theorem is contradicted by the paper's own degenerate Example 4.2, and key proof steps are only sketched for a 'typical case' or handled by assertion of similarity. The contribution is therefore not established as it stands.

major comments (4)

[Section 2.2 and Section 4.2, Example 4.2] Section 2.2 defines convergence of CFP as pointwise convergence to a strategy profile (x*, y*). Theorem 3.1 asserts convergence in every 3x3 game without IIP. Example 4.2 with k = -2/3 is presented as a 3x3 game without IIP (payoff matrices (13)), and Section 4.2 states that 'for 3 x 3 games without IIP, degeneracy does not affect the convergency of FP.' Yet the same section and Figure 9 state that the CFP trajectory 'surrounds and approaches the set of NEs, but will never converge to any point in the set.' Approaching a continuum is not pointwise convergence under the paper's own definition. Thus Theorem 3.1 as stated is internally inconsistent with the example. If the theorem is meant only for nondegenerate games, the abstract, Theorem 3.1, and Section 4.2 must be revised; if it is meant as stated, the example must converge pointwise, which the text denies.
[Section 3.3, proof of Theorem 3.2] The proof of Theorem 3.2 is given only for 'the following typical case illustrated by Figure 4', with the statement that 'all the other cases are just similar.' Theorem 3.1 covers all 3x3 games without IIP, and Appendix D only lists possible cases without proving convergence for each one. Since the vector field and the Poincare map depend on the arrangement of best-response regions and indifferent lines, convergence in one configuration does not imply convergence in the others. The same issue appears in Theorem 3.3 ('the proof for case (2) is similar') and in Proposition 3.4, where cases (2) and (3) are deferred to Appendix F. This leaves the main theorem unproven even setting aside the degenerate counterexample.
[Section 3.2, Proposition 3.3(2)] The reduction from PBRD to BRD is not justified. The projection mapping phi is non-injective: each point of the projected edge corresponds to a line segment in the original strategy simplex. The two-sentence proof says only that after PBRD converges, the BRD trajectory 'eventually belongs to the 4 cells around NE' and then invokes Lemma 2.2. It does not rule out persistent non-convergent motion along the fibers of phi within those cells. Because all subsequent contraction arguments are carried out in the projected plane, this step is load-bearing: if the reduction fails, the planar convergence results do not transfer to the original CFP dynamics.
[Section 3.4, Proposition 3.4] Proposition 3.4 asserts the existence of a saddle NE and a sink NE for games with multiple NEs. The proof of the sink part states: 'The existence of sink NE is a straightforward corollary of our main results Theorem 3.1, whose proof only relies on the existence of saddle NE.' But Theorem 3.1 is the statement being proved, and its proof uses Proposition 3.4. Even if the convergence argument uses only the saddle part, the proposition as stated invokes the main theorem in its own proof. The sink existence must be proved independently or removed from the proposition.

minor comments (4)

[Abstract and Section 1] The manuscript uses 'IP' and 'IIP' inconsistently, and contains the typo 'Internal Indiffernt Point'; these should be unified to 'IIP' and 'Indifferent Point'.
[Section 2.1] The definition of the edge of Delta_A writes 'x1 <= 0, x2 <= 0'; the inequalities should be 'xi >= 0' for a probability simplex edge.
[Section 3.2, Proposition 3.3(1)] Proposition 3.3(1) states that phi(x,y) belongs to Z^B_j x Z^B_k if and only if (x,y) belongs to Z^B_j x Z^B_k; the second factor should be Z^A_k, not Z^B_k.
[Throughout] There are numerous grammatical and typographical errors (e.g., 'convergency', 'have being receiving', 'the trajectory approach the continuum') that should be corrected before publication.

Circularity Check

1 steps flagged · score 4.0 of 10

Proposition 3.4's sink-existence claim is derived from the still-unproved Theorem 3.1, an explicit circular dependency; the main convergence proof's saddle-existence branch is otherwise independent.

other [Section 3.4, Proposition 3.4, final paragraph of proof]
"The existence of sink NE is a straightforward corollary of our main results Theorem 3.1, whose proof only relies on the existence of saddle NE. Hence this does not lead to circular argument."

Proposition 3.4 is the lemma used to prove Theorem 3.3 for multiple-NE games, and Theorem 3.1 is the theorem being assembled in this very section. Thus, at the point Proposition 3.4 is proved, Theorem 3.1 has not been established. Deriving the existence of a sink NE as 'a straightforward corollary of our main results Theorem 3.1' makes the proposition's sink-existence assertion depend on the very theorem that Proposition 3.4 is being used to support. The sentence 'whose proof only relies on the existence of saddle NE' does not break the cycle: it only indicates that the saddle half is independently usable, while no independent proof of the sink half is supplied and the proposition's own statement requires it.

full rationale

The paper's central derivation is mostly self-contained: the projection mapping, the planar PBRD analysis, the Poincaré-map contraction in Theorem 3.2, and the combinatorial contradiction for saddle NE in Proposition 3.4 do not fit parameters to the data they claim to predict and do not rely on self-citations. Proposition 3.3(2), which transfers PBRD convergence back to BRD convergence, is under-proved because the projection is non-injective, but that is a proof gap rather than a circular reduction and is not counted as circularity here. The genuine circular step is the final paragraph of Proposition 3.4, where the existence of a sink NE is asserted as a corollary of Theorem 3.1 before Theorem 3.1 is proven. Since Theorem 3.3 uses only the saddle-existence half of Proposition 3.4, the main convergence claim retains independent content despite this circular sub-claim. Separately, and outside the circularity score, Section 4.2's Example 4.2 describes a game without IIP whose projected dynamics 'surrounds and approaches the set of NEs, but will never converge to any point in the set,' which appears to contradict the pointwise convergence asserted by Theorem 3.1; this is an important correctness concern but not itself a circularity, so it is flagged here without increasing the circularity score beyond 4.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The paper introduces no fitted constants; its geometry is expressed through arbitrary positive cell widths p, q, r, P, Q, R, M, m that are valid for all values. The load-bearing assumptions are the nondegeneracy condition (silently assumed), the transfer from projected to original dynamics, and the completeness of the case enumeration. The indifferent point is a new but well-defined mathematical object with computable content.

assumptions (4)

domain assumption The game is nondegenerate and all best response regions have positive measure.
Stated in Section 3.1 and Remark 1. The proof of Proposition 3.2 and the main convergence arguments rely on unique best responses; Theorem 3.1 omits this condition, and the paper's Example 4.2 shows the claim fails for degenerate games.
ad hoc to paper Convergence of the projected best-response dynamics implies convergence of the original dynamics (Proposition 3.3(2)).
The projection is non-injective, so this is a nontrivial reduction. The proof is sketched in two sentences and does not rule out motion along the fibers of the projection.
ad hoc to paper The 'typical case' analyzed in Theorem 3.2, plus 'all the other cases are just similar', covers every geometric configuration of games without IIP.
Appendix D enumerates configurations but does not prove convergence for each; the assertion that the remaining cases are similar is unverified and load-bearing for the theorem.
standard math The indifferent point exists and is unique for almost all games, and for 3x3 games with no indifferent point the indifferent lines are parallel (Lemma 3.1).
Proven in Appendix B via linear algebra; the projection mapping and the class definition depend on this result.

invented entities (1)

Indifferent point independent evidence
purpose: A generalized strategy profile, possibly with negative probabilities, at which every action yields the same payoff; it defines the class of games without an internal indifferent point and anchors the projection mapping.
The point is computed from the payoff matrix via a linear system, so it is independently checkable and its existence is proven for almost all games. It is a mathematical construct rather than a physical entity.

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Pith. "Pith review of On the convergence of fictitious play algorithm in repeated games via the geometrical approach." pith.science (2026). https://pith.science/paper/VEN5R5XX

@misc{pith2026241219216,
  author       = {Pith},
  title        = {Pith review of: On the convergence of fictitious play algorithm in repeated games via the geometrical approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VEN5R5XX}},
  note         = {Machine review of arXiv:2412.19216}
}

abstract

As the earliest and one of the most fundamental learning dynamics for computing NE, fictitious play (FP) has being receiving incessant research attention and finding games where FP would converge (games with FPP) is one central question in related fields. In this paper, we identify a new class of games with FPP, i.e., $3\times3$ games without IIP, based on the geometrical approach by leveraging the location of NE and the partition of best response region. During the process, we devise a new projection mapping to reduce a high-dimensional dynamical system to a planar system. And to overcome the non-smoothness of the systems, we redefine the concepts of saddle and sink NE, which are proven to exist and help prove the convergence of CFP by separating the projected space into two parts. Furthermore, we show that our projection mapping can be extended to higher-dimensional and degenerate games.

Figures

Figures reproduced from arXiv: 2412.19216 by the authors.

**Figure 2.** Three scenarios of φA: When x¯ lies near an edge of ∆A, directly connecting each point with x¯ and the projection mapping φA can be well-defined. This case is shown by the first subgraph. When x¯ lies near a vertex of ∆A as in the second subgraph, we need the extended line connecting each point with x¯ in order to project all the points in ∆A onto the same edge. The third subgraph shows the case that x¯ does not exi… view at source ↗

**Figure 3.** An example of projection mapping φ: the whole strategy simplex ∆A is projected onto the edge e A 1 e A 2 , and ∆B is projected onto the edge e B 2 e B 3 , corresponding to e˜ A 1 e˜ A 2 and e˜ B 2 e˜ B 3 on the square ∆˜ . The vertex e A 3 is projected to a point e˜ A 3 lying in the interior of e˜ A 1 e˜ A 2 . In ∆A and ∆B, the indifferent lines l A jk, lB ij are colored by blue, and they are projected into points ˜… view at source ↗

Figures from the paper (14 more)

**Figure 4.** Figure 4: Similarly, we can define another Poincaré mapping for the path going through 6 cells as [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 4.** Figure 4: An example of the game without IIP with unique mixed NE: the red star profile in [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: Trajectory of CFP and PBRD for Example 3.1: different colors indicate different action profiles. There are [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: The dynamics near saddle NE and sink NE under mapping [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

**Figure 7.** Figure 7: Two cases of saddle NE: in the left subgraph, in Cell I and V the trajectory would move away from the saddle [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

**Figure 8.** Figure 8: Regime shift of CFP dynamics in Example 4.2: when [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

**Figure 9.** Figure 9: CFP in A(− 2 3 ), B : the NE of this games is (x, 1 − x, 0), eB 1 , ∀x ∈ [0, 1], and the trajectory approach the continuum of NE in ∆A, but does not approach any point of it. Example 4.3. We consider another group of games A(k), B also with the parameter k ∈ (−2, +…

**Figure 10.** Figure 10: Regime shift of CFP dynamics in Example 4.3. When [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]

**Figure 11.** Figure 11: The original and projected strategy simplex space of Game (14). In the left graph, the red point represents [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]

**Figure 12.** Figure 12: The Proportion of Matrix without IIP in 100, 000 Samples. Appendix D Enumeration of different games In this section, we will distinguish different games according to the location of indifferent lines and the arrangement of best response regions. As indicated by Propos…

**Figure 13.** Figure 13: Two cases for the location of indifferent lines: in fig. 13a, there is one edge of simplex having two intersection [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]

**Figure 14.** Figure 14: Enumeration of different arrangement of best response regions. Different colored strategy pairs indicate [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]

**Figure 15.** Figure 15: The game with unique and pure NE: the unique NE and its image under [PITH_FULL_IMAGE:figures/full_fig_p021_15.png]

**Figure 16.** Figure 16: Different arrangements of best response regions when the game has a sink NE. [PITH_FULL_IMAGE:figures/full_fig_p022_16.png]

Discussion (0). Continue with ORCID to comment.

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Reviewed August 11, 2026 · model on record in the stance chip above.

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