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arxiv: 1907.04272 · v1 · pith:KFEGQVSHnew · submitted 2019-07-09 · 💰 econ.TH

Ordinal Imitative Dynamics

George Loginov This is my paper

Pith reviewed 2026-05-24 23:51 UTC · model grok-4.3

classification 💰 econ.TH
keywords evolutionary dynamicsimitate the better realizationordinal mean dynamicsiteratively dominated strategiesstrict equilibriareplicator dynamicspopulation games
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The pith

An imitate-the-better-realization rule eliminates iteratively strictly dominated pure strategies and stabilizes strict equilibria in population games.

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

This paper introduces an evolutionary dynamics based on the imitate the better realization rule, where agents copy a randomly chosen opponent's strategy only if its realized payoff exceeds their own. The resulting ordinal mean dynamics is polynomial in strategy frequencies. Despite lacking Nash stationarity and payoff monotonicity, the dynamics eliminates pure strategies that are iteratively strictly dominated by other pure strategies and makes strict equilibria locally stable. Comparisons to replicator dynamics reveal equivalence in simple cases but distinct behaviors in more complex games such as Rock-Paper-Scissors.

Core claim

Under the imitate the better realization (IBR) rule, the resulting ordinal mean dynamics eliminates iteratively strictly dominated pure strategies and renders strict equilibria locally stable, despite lacking Nash stationarity and payoff monotonicity.

What carries the argument

The imitate the better realization (IBR) rule, under which agents imitate the strategy of a randomly chosen opponent if the opponent's realized payoff is higher than their own.

If this is right

  • Pure strategies iteratively strictly dominated by pure strategies are eliminated.
  • Strict equilibria are locally stable.
  • The dynamics is topologically equivalent to replicator dynamics in trivial cases.
  • In Rock-Paper-Scissors games the dynamics exhibits the same types of behavior as replicator dynamics but the partitions of the game set do not coincide.
  • In other cases the IBR dynamics exhibits behaviors impossible under the replicator dynamics.

Where Pith is reading between the lines

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

  • The rule might produce different long-run predictions than replicator dynamics in games with cycles or multiple equilibria.
  • Laboratory experiments with human subjects given only pairwise payoff observations could test whether dominated strategies disappear at the predicted rate.

Load-bearing premise

The population is large, matching is random, and agents observe only realized payoffs of a single randomly chosen opponent.

What would settle it

A simulation or experiment showing that an iteratively strictly dominated strategy persists in the population under the IBR rule, or that a strict equilibrium is unstable.

Figures

Figures reproduced from arXiv: 1907.04272 by George Loginov.

Figure 1
Figure 1. Figure 1: Some solution trajectories in games A1 (left) and A2 (right). In the game A1 strategy 3 is dominated by both 1 and 2, strategy 2 is weakly dominated by 1, but once strategy 3 is eliminated, both 1 and 2 coexist. From the perspective of an agent playing strategy 3 the remaining two strategies are equally good. The only case when strategy 1 gains advantage over strategy 2 is when an agent playing strategy 2 … view at source ↗
Figure 2
Figure 2. Figure 2: Some solution trajectories in games A3 (left) and A4 (right). The critical region is z = 1 2 . In game A4 with the reversed order of payoffs the critical region z = 1 2 becomes absorbing, which suggests that strategy 2 survives along any trajectory originating in the interior of the state space. 2.4 Stability of strict equilibria We conclude this section with a stability property for strict equilibria. If … view at source ↗
Figure 3
Figure 3. Figure 3: Some solution trajectories for games A1, B3, and C2. [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Left: the nullclines divide the simplex into 6 regions. Right: solution trajectories from [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The solution trajectory from Figure 5: The solution trajectory [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Some solution trajectories in games Z (left) and W (right). Game W has the self-negating property, so there are closed orbits around the interior rest point, but a part of the interior of the simplex is the basin of attraction of the state x3 = 1. In the time-reversed game x1 = 1 becomes the attractor, while the interior rest point preserves its region with the closed orbits. 6 Conclusion This paper invest… view at source ↗
read the original abstract

This paper introduces an evolutionary dynamics based on imitate the better realization (IBR) rule. Under this rule, agents in a population game imitate the strategy of a randomly chosen opponent whenever the opponent`s realized payoff is higher than their own. Such behavior generates an ordinal mean dynamics which is polynomial in strategy utilization frequencies. We demonstrate that while the dynamics does not possess Nash stationarity or payoff monotonicity, under it pure strategies iteratively strictly dominated by pure strategies are eliminated and strict equilibria are locally stable. We investigate the relationship between the dynamics based on the IBR rule and the replicator dynamics. In trivial cases, the two dynamics are topologically equivalent. In Rock-Paper-Scissors games we conjecture that both dynamics exhibit the same types of behavior, but the partitions of the game set do not coincide. In other cases, the IBR dynamics exhibits behaviors that are impossible under the replicator 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

0 major / 3 minor

Summary. The paper introduces an evolutionary dynamics based on the imitate-the-better-realization (IBR) rule in large-population random-matching games. Agents imitate a randomly sampled opponent's strategy only when the opponent's realized payoff exceeds their own, yielding an ordinal polynomial mean dynamics in strategy frequencies. The central claims are that this dynamics eliminates pure strategies that are iteratively strictly dominated by pure strategies and locally stabilizes strict equilibria, even though it lacks Nash stationarity and payoff monotonicity. The paper further compares the IBR dynamics to the replicator dynamics, establishing topological equivalence in trivial cases, conjecturing matching qualitative behavior (but non-coincident partitions) in Rock-Paper-Scissors, and exhibiting behaviors impossible under replicator dynamics in other games.

Significance. If the derivations hold, the result is significant because it enlarges the class of dynamics known to satisfy iterative dominance elimination and local stability of strict equilibria without requiring payoff monotonicity or Nash stationarity. The polynomial, ordinal character of the IBR flow is a clean, parameter-free construction that directly inherits these properties from the imitation rule. Explicit comparison with the replicator dynamics, including the RPS conjecture and the identification of non-replicator behaviors, supplies concrete, falsifiable distinctions that strengthen the contribution.

minor comments (3)
  1. [Abstract] The abstract states the main claims but does not reference the specific propositions or theorems that establish elimination and stability; adding such pointers would improve readability.
  2. Notation for the mean dynamics (e.g., the precise form of the polynomial vector field) should be introduced with an explicit equation number in the main text to facilitate later references.
  3. The RPS conjecture is presented without a precise statement of the partition of the game set; a short formal conjecture statement would clarify the claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and constructive report, including the accurate summary of the paper's contributions and the recommendation for minor revision. We appreciate the recognition of the significance of the IBR dynamics results on iterative dominance elimination and local stability of strict equilibria without Nash stationarity or payoff monotonicity.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper first defines the IBR rule and the resulting ordinal mean dynamics explicitly in terms of strategy frequencies and realized payoffs. It then asserts that this dynamics eliminates iteratively strictly dominated pure strategies and locally stabilizes strict equilibria. These properties are presented as consequences of the definition rather than inputs to it. No parameters are fitted, no self-citations supply load-bearing uniqueness theorems, and no ansatz or renaming reduces the central claims to the input definition by construction. The comparison to replicator dynamics is an external investigation, not a circular reduction. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; full derivation and modeling assumptions unavailable.

axioms (1)
  • domain assumption Agents imitate the strategy of a randomly chosen opponent when that opponent's realized payoff exceeds their own.
    Core modeling choice that defines the IBR rule and generates the claimed ordinal dynamics.

pith-pipeline@v0.9.0 · 5664 in / 1169 out tokens · 20376 ms · 2026-05-24T23:51:38.169880+00:00 · methodology

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Works this paper leans on

13 extracted references · 13 canonical work pages

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