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arxiv: 2604.19359 · v1 · submitted 2026-04-21 · 💰 econ.TH · cs.GT

Recognition: unknown

How damaging is zero-sum thinking to an agent's interests when the world is positive-sum?

Mehmet Mars Seven, Shaun Hargreaves Heap

Pith reviewed 2026-05-10 01:12 UTC · model grok-4.3

classification 💰 econ.TH cs.GT
keywords zero-sum thinkingmaximinNash equilibriumpositive-sum gamesPareto dominancecoordination3x3 gamesordinal payoffs
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The pith

Zero-sum maximin thinking is not generally inferior to Nash play in positive-sum games.

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

The authors investigate whether maximin decision rules, a form of zero-sum thinking, reduce agents' payoffs in positive-sum games relative to Nash equilibrium or best-response play. They provide examples where maximin strictly Pareto-dominates Nash equilibria and demonstrate that the set of symmetric 3x3 ordinal games exhibiting this property has the same size as the set where Nash equilibria dominate maximin profiles. This equivalence implies that neither behavior is systematically better. Additional analysis reveals that maximin can also mitigate coordination failures in games with multiple equilibria, yielding higher realized payoffs. These patterns appear frequently enough in the 3x3 game space to suggest zero-sum thinking may persist without strong payoff penalties.

Core claim

The class of symmetric strictly ordinal 3x3 games in which a maximin profile strictly Pareto dominates all Nash equilibria has the same cardinality as the class in which a Nash equilibrium strictly Pareto dominates all maximin profiles. Consequently, maximin behavior is not generally damaging to agents' interests compared to Nash equilibrium in positive-sum environments.

What carries the argument

Cardinality comparison of two classes of games based on strict Pareto dominance between maximin profiles and Nash equilibria in symmetric 3x3 ordinal payoff matrices.

If this is right

  • Maximin outperforms Nash in some positive-sum games, challenging evolutionary displacement arguments.
  • Coordination failures among multiple Nash equilibria allow maximin to deliver superior realized payoffs.
  • Zero-sum thinking linked to maximin will not be readily eliminated by inferior average payoffs.
  • The frequency of maximin advantages in 3x3 games is non-trivial.

Where Pith is reading between the lines

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

  • Broader classes of games or empirical distributions of real-world interactions could reveal whether the balance persists or tilts one way.
  • Models of belief formation might incorporate maximin as a stable heuristic if coordination benefits are present.
  • Experimental tests could measure how often agents choose maximin-like rules in positive-sum settings with multiple equilibria.

Load-bearing premise

The positive-sum condition combined with the restriction to symmetric 3x3 ordinal games adequately represents the strategic situations where zero-sum thinking occurs, without introducing selection bias into the cardinality result.

What would settle it

A full count of all possible symmetric 3x3 strictly ordinal games showing unequal numbers of games in each dominance class, or extension to non-symmetric or larger games where one class dominates in size.

Figures

Figures reproduced from arXiv: 2604.19359 by Mehmet Mars Seven, Shaun Hargreaves Heap.

Figure 1
Figure 1. Figure 1: A symmetric 3 × 3 game where the maximin profile strictly outperforms Nash equilibrium This game has a unique mixed Nash equilibrium s N = (1/2, 1/4, 1/4), a unique maximin strategy sM i = (0, 5/8, 3/8), and a unique minimax strategy s m i = (1/2, 0, 1/2) for each player i ∈ {1, 2}. The maximin strategy guarantees a pay-off of 5.5, whereas the Nash strategy guarantees only 2.5 if player 2 plays z, even tho… view at source ↗
Figure 2
Figure 2. Figure 2: The induced game pay-offs by each rule If zero-sum thinking is associated with maximin, then in a population of Nash equilibrium and maximin players condition (a) holds (6.125 > 5.75) and so does condition (b) (5.75 > 5.625). Thus on a strict comparison of the difference in rule where the only difference in the comparison of the rules are the rules themselves, maximin players do better than Nash equilibriu… view at source ↗
Figure 3
Figure 3. Figure 3: A symmetric 3 × 3 game with ordinal pay-offs in which the maximin profile strictly Pareto dominates the unique pure Nash equilibrium In this game, the Nash equilibrium is given by (y, y) and mutual maximin yields (z, z). Thus, in a population of maximin and Nash equilibrium players, i.e., in the induced 2 × 2 game between y and z strategies, condition (a) holds (6 > 5) and (b) holds (4 > 3). So, in the str… view at source ↗
Figure 4
Figure 4. Figure 4: A game with two pure Nash equilibria in which average Nash-play pay-offs are below the maximin guarantees In particular, suppose each player randomises over the two Nash equilibria strategies with the result that the two Nash equilibria are equally likely and as likely as the two non-Nash outcomes. The average pay-off from Nash is 1 4 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: An unprofitable game Harsanyi (1964) discusses this ‘unprofitable’ game. It has a unique mixed Nash equilibrium, ((4/5, 1/5),(1/2, 1/2)), yielding pay-offs (3/2, 8/5). Player 1’s maximin strategy is (1/4, 3/4), and player 2’s maximin strategy is (2/5, 3/5). These strategies guarantee pay-offs of 3/2 and 8/5, respectively, so each player’s security level is exactly equal to that player’s Nash equilibrium pa… view at source ↗
read the original abstract

We study whether zero-sum decision rules, maximin and minimax, harm agents' interests in positive-sum strategic environments relative to Nash equilibrium behavior or, more generally, than best response behaviour. Contrary to an influential evolutionary view, we give illustrations where maximin serves an agent's interests better than Nash equilibrium behaviour. We also show that these illustration are not atypical or idiosyncratic because, in our main result, the class of such games where a maximin profile strictly Pareto dominates all Nash equilibria has the same cardinality as the class of games in which a Nash equilibrium strictly Pareto dominates all maximin profiles. Thus, neither behavior is generally superior. We further identify additional mechanisms favoring maximin over Nash equilibrium, including coordination failures under multiple equilibria, where maximin can outperform Nash play in realised-pay-off terms. A systematic analysis of strictly ordinal symmetric 3x3 games shows that these effects arise with non-trivial frequency. Our findings, therefore, suggest that the observed rise in zero-sum thinking in many rich countries, when associated with a maximin decision rule, will not be readily displaced through its generation of inferior pay-offs.

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 paper claims that zero-sum decision rules such as maximin are not generally damaging to agents' interests in positive-sum strategic environments relative to Nash equilibrium play. It provides illustrations where maximin profiles strictly Pareto-dominate all Nash equilibria, establishes a main result that the class of such games has the same cardinality as the class where some Nash equilibrium strictly Pareto-dominates all maximin profiles (hence neither is generally superior), identifies additional mechanisms such as coordination failures under multiple equilibria, and reports that a systematic enumeration of strictly ordinal symmetric 3x3 games shows these effects arise with non-trivial frequency.

Significance. If the cardinality result holds under a suitable parameterization of positive-sum games and the 3x3 enumeration is exhaustive within its domain, the work supplies a direct counterexample to evolutionary arguments that Nash behavior must dominate in positive-sum settings and offers a theoretical rationale for the persistence of maximin-style zero-sum thinking. The explicit enumeration in the 3x3 class provides a concrete, falsifiable frequency benchmark that strengthens the claim beyond mere existence proofs.

major comments (2)
  1. [Main result] Main result on cardinality: the claim that equal cardinality between the two classes implies maximin-superior instances are 'not atypical or idiosyncratic' (and thus that neither behavior is generally superior) does not follow without an explicit measure, topology, or density argument on the space of positive-sum games. In the continuum of real-valued positive-sum payoff matrices, both classes can have cardinality of the continuum while differing substantially in Lebesgue measure or Baire category; the manuscript supplies no bijection preserving the positive-sum constraint or any such measure.
  2. [Systematic analysis of 3x3 games] 3x3 ordinal symmetric games analysis: the reported non-trivial frequency of maximin dominance relies on an enumeration whose selection criteria and completeness are not fully specified in a way that rules out post-hoc restrictions altering the cardinality comparison or the relative performance of maximin versus Nash. Without the explicit list of games or the payoff-matrix enumeration protocol, it is impossible to verify absence of selection effects.
minor comments (2)
  1. [Definitions and setup] Clarify the precise definition of 'positive-sum' used throughout (e.g., whether it requires strict positivity of total payoffs or merely non-zero-sum structure) and confirm it is maintained uniformly in both the cardinality argument and the 3x3 enumeration.
  2. [Illustrations] The abstract states that maximin 'serves an agent's interests better than Nash equilibrium behaviour' in some illustrations; the manuscript should explicitly state whether these are strict Pareto dominance or weaker notions and whether they survive best-response dynamics.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which highlight important nuances in interpreting our cardinality result and the transparency of our enumeration. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Main result] Main result on cardinality: the claim that equal cardinality between the two classes implies maximin-superior instances are 'not atypical or idiosyncratic' (and thus that neither behavior is generally superior) does not follow without an explicit measure, topology, or density argument on the space of positive-sum games. In the continuum of real-valued positive-sum payoff matrices, both classes can have cardinality of the continuum while differing substantially in Lebesgue measure or Baire category; the manuscript supplies no bijection preserving the positive-sum constraint or any such measure.

    Authors: We agree that equal cardinality does not establish measure-theoretic or topological typicality; both classes can indeed have cardinality of the continuum. Our use of cardinality is limited to showing that the maximin-superior class is not of strictly smaller cardinality (hence not 'idiosyncratic' in a set-theoretic sense of being negligible by cardinality alone) within the positive-sum games. We will revise the relevant passages to remove any implication of measure-based typicality, explicitly note the absence of a canonical measure or parameterization on the space of positive-sum payoff matrices, and clarify that the result is an existence and cardinality comparison rather than a density claim. This is a partial revision that strengthens the precision of the statement without changing the core theorem. revision: partial

  2. Referee: [Systematic analysis of 3x3 games] 3x3 ordinal symmetric games analysis: the reported non-trivial frequency of maximin dominance relies on an enumeration whose selection criteria and completeness are not fully specified in a way that rules out post-hoc restrictions altering the cardinality comparison or the relative performance of maximin versus Nash. Without the explicit list of games or the payoff-matrix enumeration protocol, it is impossible to verify absence of selection effects.

    Authors: The enumeration covers the complete finite set of strictly ordinal symmetric 3x3 games, where payoffs are distinct ranks from 1 to 9 with no ties and the bimatrix is symmetric. We will add an appendix that states the exact generation protocol (all permutations of ranks assigned symmetrically), reports the total number of such games, and provides either the full list or a categorized breakdown showing the counts for maximin dominance, Nash dominance, and other cases. This will permit direct verification and eliminate any concern about selection effects. revision: yes

Circularity Check

0 steps flagged

No circularity: cardinality result obtained by direct enumeration of finite game space

full rationale

The central claim equates the sizes of two independently defined classes of positive-sum symmetric 3x3 ordinal games (those where maximin Pareto-dominates all Nash equilibria versus those where Nash Pareto-dominates all maximin profiles). This equality is established by exhaustive classification of the finite set of such games under the paper's stated primitives (ordinal payoffs, symmetry, positive-sum condition), without any parameter fitting, self-referential definition of the classes, or load-bearing self-citation. The subsequent inference that neither rule is 'generally superior' is an interpretive step outside the derivation itself and does not create circularity in the mathematical result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, invented entities, or non-standard axioms are described. The analysis implicitly rests on standard finite normal-form game assumptions.

axioms (1)
  • standard math Finite normal-form games with strictly ordinal payoffs
    Required for defining Nash equilibria, maximin profiles, and Pareto dominance comparisons.

pith-pipeline@v0.9.0 · 5494 in / 1259 out tokens · 47382 ms · 2026-05-10T01:12:27.926920+00:00 · methodology

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