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arxiv: 2604.07544 · v1 · submitted 2026-04-08 · 💻 cs.GT

Zero-Sum Fictitious Play Cannot Converge to a Point

Jaehong Moon This is my paper

Pith reviewed 2026-05-10 16:52 UTC · model grok-4.3

classification 💻 cs.GT
keywords fictitious playzero-sum gamesNash equilibriumpointwise convergencegame dynamicslearning in gamesequilibrium set geometrynon-convergence
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The pith

Fictitious play fails to converge to any single Nash equilibrium in certain zero-sum games.

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

The paper establishes that fictitious play, which is known to drive empirical frequencies toward the set of Nash equilibria in zero-sum games, does not always settle on one particular equilibrium point when multiple equilibria are present. The authors construct a class of zero-sum games in which the learning process keeps oscillating among equilibria indefinitely, and this non-convergence holds for any choice of tie-breaking rule. A sympathetic reader would care because the result closes an open question on the exact limiting behavior of this long-studied procedure and shows that setwise convergence is the best one can guarantee in general. The argument rests on two geometric properties of the equilibrium set together with one technical assumption that together force the trajectory to avoid every individual point.

Core claim

We establish that FP does not necessarily stabilize at a single equilibrium. Specifically, we identify a class of zero-sum games where pointwise convergence fails, regardless of the tie-breaking rules employed. We prove that two geometric conditions on the NE set (A1 and A2) and a technical assumption (A3) are sufficient to preclude pointwise convergence. Furthermore, we conjecture that the first two conditions alone may be sufficient to guarantee this non-convergence, suggesting a broader fundamental instability in FP dynamics.

What carries the argument

Two geometric conditions A1 and A2 on the Nash equilibrium set, which (with technical assumption A3) force the empirical-frequency trajectory of fictitious play to avoid converging to any individual equilibrium point.

If this is right

  • In the identified class of games, the sequence of empirical frequencies never approaches any single Nash equilibrium.
  • Convergence of fictitious play remains only set-valued, not pointwise, no matter how ties are resolved.
  • The same non-convergence holds across all tie-breaking rules, so the instability is intrinsic to the geometry of the equilibrium set.
  • The result supports the conjecture that the two geometric conditions alone may already be enough to block pointwise convergence.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the conjecture that A1 and A2 suffice is true, then the same oscillation could appear in any zero-sum game whose equilibrium set has the identified geometric shape, even without A3.
  • Designers of learning algorithms for games might need to add explicit stabilization mechanisms when equilibria form extended sets satisfying those geometries.
  • The same geometric conditions could be checked in other continuous-time or discrete-time dynamics to test whether they produce analogous cycling.
  • Numerical experiments on randomly generated games meeting A1 and A2 would give direct evidence of the oscillation patterns the proof predicts.

Load-bearing premise

The Nash equilibrium set in the constructed zero-sum games satisfies the two geometric conditions A1 and A2 together with the technical assumption A3.

What would settle it

A concrete simulation or analytic proof showing that fictitious play converges to one specific equilibrium point in a zero-sum game whose Nash set meets conditions A1, A2, and A3 would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.07544 by Jaehong Moon.

Figure 1
Figure 1. Figure 1: Max-min graph for the game in (9). The lower envelope [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Max-min graph for the game in (10). The value of the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Sample trajectory of [xˆ(k)]1 for FP applied to the 2×3 game (10), starting from no prior and using uniform random tie-breaking among multiple best responses. The trajectory exhibits persistent oscillations between 1/4 and 3/4. The horizontal axis is logarithmic in k. equilibrium set is non-singleton. In fact, this construction yields an equilibrium set of positive measure. On the other hand, by adding a s… view at source ↗
Figure 5
Figure 5. Figure 5: FP trajectory of Player 1 for the reduced game [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Sample trajectory of Player 1 under FP for the zero-sum [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Sample trajectory of Player 1 under FP for the zero [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
read the original abstract

Fictitious play (FP) is a history-based strategy to choose actions in normal-form games, where players best-respond to the empirical frequency of their opponents' past actions. While it is well-established that FP converges to the set of Nash equilibria (NE) in zero-sum games, the question of whether it converges to a single equilibrium point, especially when multiple equilibria exist, has remained an open challenge. In this paper, we establish that FP does not necessarily stabilize at a single equilibrium. Specifically, we identify a class of zero-sum games where pointwise convergence fails, regardless of the tie-breaking rules employed. We prove that two geometric conditions on the NE set (A1 and A2) and a technical assumption (A3) are sufficient to preclude pointwise convergence. Furthermore, we conjecture that the first two conditions alone may be sufficient to guarantee this non-convergence, suggesting a broader fundamental instability in FP dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that fictitious play (FP) in zero-sum games converges to the Nash equilibrium (NE) set but does not necessarily converge pointwise to a single equilibrium point when multiple equilibria exist. The authors identify a class of zero-sum games whose NE set satisfies two geometric conditions (A1 and A2) together with a technical assumption (A3); they prove that these conditions suffice to ensure that the empirical frequency vector cannot stabilize at any single point, for any tie-breaking rule. They further conjecture that A1 and A2 alone are sufficient and suggest this indicates a broader instability in FP dynamics.

Significance. If the central result holds, it would resolve an open question on pointwise convergence of FP in zero-sum games with non-unique equilibria. Providing explicit geometric conditions under which pointwise convergence fails, together with a proof of sufficiency, would clarify the limitations of history-based learning rules. The conjecture, if established, would strengthen the contribution by removing the technical assumption A3. The work is relevant to algorithmic game theory and dynamics in games.

major comments (2)
  1. [Abstract and §3] Abstract and §3 (Theorem on sufficiency of A1-A2-A3): the proven non-convergence result requires the additional technical assumption A3; the manuscript does not exhibit a concrete game satisfying A1 and A2 for which A3 can be verified independently, nor does it demonstrate that A3 holds in any natural example beyond the constructed class. This makes it unclear whether the demonstrated counter-examples support the broader claim that pointwise convergence 'cannot' occur under the geometric conditions alone.
  2. [§4] §4 (Conjecture statement): the claim that A1 and A2 alone suffice is presented as a conjecture without any partial proof, counter-example search, or reduction showing that A3 is redundant. Because the central title and abstract phrasing suggest a general non-convergence result for zero-sum games with multiple equilibria, the unproven status of the conjecture is load-bearing for the scope of the contribution.
minor comments (2)
  1. The notation for the geometric conditions A1 and A2 is introduced without an accompanying diagram or explicit coordinate-free characterization, which would aid readability when verifying the conditions on the constructed games.
  2. [Introduction] The introduction should include a brief comparison table or paragraph contrasting the new result with prior convergence theorems for FP in zero-sum games (e.g., those establishing setwise convergence).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address each major comment below and describe the revisions we will undertake to improve clarity and scope.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (Theorem on sufficiency of A1-A2-A3): the proven non-convergence result requires the additional technical assumption A3; the manuscript does not exhibit a concrete game satisfying A1 and A2 for which A3 can be verified independently, nor does it demonstrate that A3 holds in any natural example beyond the constructed class. This makes it unclear whether the demonstrated counter-examples support the broader claim that pointwise convergence 'cannot' occur under the geometric conditions alone.

    Authors: We agree that the presentation would benefit from an explicit, verifiable example. The manuscript defines a parametric class of zero-sum games in which A1, A2, and A3 hold simultaneously by construction, and the proof of the theorem confirms that A3 is satisfied throughout this class. To address the referee's concern directly, we will add a low-dimensional concrete instance (for example, a specific 3-by-3 payoff matrix) in which we separately verify A1, A2, and A3, compute the equilibrium set, and illustrate the non-convergence of the fictitious-play trajectory for any tie-breaking rule. We will also revise the abstract to state explicitly that non-convergence is proven under the three assumptions for the identified class, thereby removing any implication that the result holds for A1 and A2 alone. revision: partial

  2. Referee: [§4] §4 (Conjecture statement): the claim that A1 and A2 alone suffice is presented as a conjecture without any partial proof, counter-example search, or reduction showing that A3 is redundant. Because the central title and abstract phrasing suggest a general non-convergence result for zero-sum games with multiple equilibria, the unproven status of the conjecture is load-bearing for the scope of the contribution.

    Authors: We acknowledge that the conjecture is stated without supporting arguments such as a partial proof or systematic search for counter-examples; our primary contribution is the sufficiency theorem under A1-A2-A3. We will revise the abstract and the opening paragraphs of the introduction to delineate clearly the proven statement (non-convergence when A1, A2, and A3 hold) from the open conjecture (whether A1 and A2 alone suffice). The title is intended to reflect the main theorem for the constructed class rather than an unqualified general claim; the revised abstract will ensure this distinction is unambiguous. revision: yes

Circularity Check

0 steps flagged

No circularity: direct proof of sufficiency under explicit assumptions A1-A3 with conjecture left open.

full rationale

The paper's central result is a mathematical proof that geometric conditions A1 and A2 plus technical assumption A3 on the Nash equilibrium set suffice to show that fictitious play fails to converge pointwise in a class of zero-sum games, for any tie-breaking rule. This is presented as a direct derivation from the stated assumptions without any fitted parameters, self-referential definitions, or load-bearing self-citations. The conjecture that A1 and A2 alone suffice is explicitly noted as unproven. No step reduces the claimed non-convergence to a renaming, ansatz, or input by construction; the argument remains self-contained against the geometric properties of the equilibrium set.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of two geometric conditions on the Nash equilibrium set and one additional technical assumption that together define the class of games for which non-convergence is shown.

axioms (2)
  • domain assumption Geometric conditions A1 and A2 on the Nash equilibrium set
    Invoked to characterize the class of zero-sum games where pointwise convergence fails
  • ad hoc to paper Technical assumption A3
    Additional assumption used to complete the proof of non-convergence

pith-pipeline@v0.9.0 · 5451 in / 1309 out tokens · 63850 ms · 2026-05-10T16:52:33.431040+00:00 · methodology

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Reference graph

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