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What preferences can - and cannot - predict in multi-agent online learning

Omar Abbadi, Rida Laraki, Panayotis Mertikopoulos

arXiv:2608.13810v1cs.GTcs.LG

TL;DR

The paper asks whether a game’s preference graph can determine the stable long-run outcomes of learning dynamics. It compares preference-based stability with dynamic stability, showing that preferences characterize stability for subgames but do not suffice beyond them.

  • Problem

    The paper asks whether a game’s preference graph can determine which regions of strategy space are stable long-run outcomes of learning dynamics.

    Source: §1 · p. 2

  • Method

    It compares preference-graph closure under better replies with dynamic stability and uses an energy function to establish attraction and stability for club sets.

    Source: §1 · p. 3, §5.1 · p. 10

  • Results

    Preference closure is necessary for dynamically stable sets and characterizes stability for subgames, but may fail to control span stability beyond subgames.

    Source: §1 · p. 3, §5.2 · p. 11

  • Takeaways & Limitations

    Preference structure constrains stable outcomes of regularized learning but is insufficient to determine them, and radness is not necessary in general.

    Source: §7 · p. 14

    A straightforward payoff-condition characterization of asymptotically stable sets at the pure-strategy level remains open, and radness is not necessary in general.

    Source: §7 · p. 14

Abstract

from arXiv · show

We examine the interplay between ordinal, preference-based solution concepts in games and the long-run behavior of game dynamics, asking in particular to what extent the combinatorial data of a game -- its preference graph -- determine the outcomes of no-regret learning dynamics -- such as follow-the-regularized-leader (FTRL). In one direction, we show that the skeleton of every dynamically stable set (i.e. the set of pure profiles it contains) must also be preferentially stable, that is, it must be closed under profitable deviations. We then ask the converse question: when do preferences determine the long-run behavior of the players' learning dynamics? We begin by showing that preferences characterize asymptotic stability in the case of subgames -- i.e. subsets of pure profiles obtained by restricting players' action sets. Beyond this case however, the equivalence between dynamic and preferential stability collapses: concretely, we construct a three-player game with a preferentially stable set whose span is dynamically unstable, showing in this way that preferences do not suffice as a criterion of dynamic stability. We then bridge this gap via the notion of resilience under aggregate deviations, an easy-to-check payoff-based condition that guarantees asymptotic stability of arbitrary spans of pure strategies.

1. Introduction

The paper asks whether ordinal preferences determine stable long-run outcomes under FTRL, a no-regret learning dynamic. It finds that the preference graph constrains stability, characterizes it for subgames, but does not suffice beyond them, where payoff-based resilience can guarantee stability.

1. Introduction

  • Different cardinal payoff realizations of the same preference ordering can produce different long-run outcomes, a concern for preference learning [17] [30] and AI-safety reward specification [2].The paper frames this as a reason numerical rewards may matter when implementing an ordinal specification.Source: §1 · pp. 2–3
  • Every asymptotically stable set’s pure-profile skeleton must be closed under better replies; attractors contain the mixed region spanned by connected pure profiles they contain.Thus, if the preference graph is strongly connected, FTRL has no proper attractor.Source: §1 · p. 3
  • A subgame is asymptotically stable if and only if it is closed under better replies; this also sharply characterizes attractors in weakly acyclic games.The result complements prior work on replicator dynamics and stochastic FTRL variants [10] [14], and extends a characterization previously known only for potential games.Source: §1 · p. 3
  • A preferentially closed set can have an unstable span even from full-support initialization, refuting a stronger FTRL conjecture by Biggar & Papadimitriou [8] while leaving their original conjecture open.The counterexample contains a mixed local source.Source: §1 · pp. 3–4
  • Resilience under aggregate deviations (rad) is a payoff-based condition guaranteeing asymptotic stability for arbitrary games and sets of pure strategies.Radness generalizes pure Nash equilibrium setwise and refines better-reply closedness; unlike a condition for sink equilibria under two-player replicator dynamics, it applies to arbitrary player counts and sets.Source: §1 · pp. 3–4
  • A separate result shows that spans of club cycles attract under replicator dynamics; this and radness establish stability for some sink equilibria, including Jordan’s Matching Pennies six-cycle, but neither result implies the other.A complete understanding of learning-game attractors remains elusive.Source: §1 · p. 4

2. Preliminaries

The preliminaries define a finite normal-form game through its players, actions, and payoffs, then extend play to mixed strategies and expected payoffs. The resulting payoff field records each action’s payoff against opponents’ mixed strategies.

Game and strategies

  • A finite normal-form game consists of players with finite action sets and payoff functions over pure action profiles.A pure profile α specifies one action for each player; α−i denotes the opponents’ actions.Source: §2 · p. 4
  • Each player’s mixed strategies are probability distributions over their actions, and their product forms the game’s mixed-profile strategy space.The probability assigned to action αi by mixed strategy xi is denoted xiα.Source: §2 · p. 4

Payoff field and equilibria

  • Payoffs extend to mixed profiles by expectation, with payoff-field component viα(x) equal to player i’s payoff from action αi against opponents’ mixed strategies.The expected payoff under x is ui(x) = ⟨vi(x), xi⟩.Source: §2 · p. 4

3. Preferences, learning and stability

The section defines preference-based solution concepts using unilateral-improvement graphs, then introduces FTRL and stability notions for describing learning dynamics.

3.1. Preference graphs and ordinal stability.

  • The preference graph represents weakly profitable unilateral deviations as directed arcs; reciprocal arcs are ties, which can be resolved by arbitrarily small game perturbations.Source: §3.1 · p. 5
  • Clubs are closed under weakly profitable deviations, s-clubs under strict improvements, and a strongly connected club set is a sink equilibrium and minimal club set.A singleton is an s-club exactly when its profile is a pure Nash equilibrium, and a club exactly when it is a strict Nash equilibrium.Source: §3.1 · p. 6
  • Spans associate pure-profile sets with mixed-strategy regions and subgames with product faces; skeleton recovers H from span(H), but not arbitrary mixed sets.For example, the span of the skeleton of a set of fully mixed strategies is empty.Source: §3.1 · p. 6

3.2. Regularized learning and dynamical stability.

  • FTRL is a broad no-regret family in which players balance accumulated payoff scores against regularization; in continuous time, regret is uniformly bounded by the regularizer’s range.Source: §3.2 · pp. 6–7
  • Regularization shapes strategy dynamics: steep regularizers preserve the initial support face and yield a globally well-posed, face-invariant flow under mild regularity assumptions.Non-steep regularizers may select boundary points, which are typically reached in finite time.Source: §3.2 · pp. 7–8
  • For FTRL, asymptotic stability combines Lyapunov stability and attraction; strategy-flow attractors are invariant asymptotically stable sets tested across X, unlike FTRL’s choice-map-restricted starts.Source: §3.2 · p. 8

4. Implications of dynamic stability

The section links ordinal properties of the preference graph to the dynamical stability of strategy regions, showing that preferences constrain stable outcomes and that connectivity can force attractors to include entire mixed spans.

Implications of dynamic stability

  • Every set attracting under strategy dynamics has a club skeleton.For the strategy flow, face invariance lets the argument use the direction of better replies along a one-dimensional face.Source: §4 · p. 9
  • Every set stable under FTRL has a skeleton that is s-club.The proof uses dual scores: a strict better reply would let one player’s score difference grow until their mixed action shifts outside the stable neighborhood.Source: §4 · p. 9
  • If a strategy-dynamics attractor contains a strongly connected set H, it contains span(H); a strongly connected preference graph therefore admits no proper attractor.The argument uses chain transitivity, generalizing the second theorem of [10], and induction over faces using the fact that no asymptotically stable set lies fully in the interior [22].Source: §4 · p. 9

5. Implications of preferential stability

Preferential stability characterizes dynamic stability for subgames, but not for arbitrary spans: a preferentially stable set can have an unstable span. Weak acyclicity further yields a sharp characterization of minimal attractors, while resilience under aggregate deviations is outside the supplied passages.

5.1. When preferences are enough.

  • Every club subgame has a span that is asymptotically stable under FTRL.A Fenchel-gap energy decreases at a rate proportional to the probability mass outside the span, giving an inward drift even for non-steep regularizers; the approach improves on [13] [14].Source: §5.1 · p. 10
  • For subgames, club status is equivalent to FTRL asymptotic stability when there are no ties, and to being an attractor under strategy dynamics.The FTRL equivalence assumes the game has no ties; the strategy-dynamics equivalence does not state that assumption.Source: §5.1 · p. 10
  • In weakly acyclic games without ties, minimal strategy-dynamics attractors are exactly strict Nash equilibria.Johnston et al. [38] show weak acyclicity becomes overwhelmingly likely at an exponential rate among games with a pure Nash equilibrium when players greatly outnumber each player’s maximum actions, so the characterization applies to typical games in that regime.Source: §5.1 · p. 11

5.2. When preferences are not enough.

  • A 2 × 2 × 2 game has a unique proper club set whose span is unstable under FTRL, demonstrating that preferential stability does not suffice for arbitrary spans.The instability occurs when a player can deviate outside the span before others move toward its included faces; the minimal attractor containing the set is not itself a span of pure profiles.Source: §5.2 · p. 11
  • The counterexample is a dynamically unstable sink equilibrium without a pure local source, refuting a possible strengthening of Biggar and Papadimitriou’s conjecture.Unlike their cited replicator-dynamics analysis, this example escapes from full-support initializations under every instance of FTRL.Source: §5.2 · p. 11

6. Recovering dynamic stability

Preference information alone cannot determine dynamic stability, but payoff-dependent resilience conditions recover stability guarantees for spans of pure profiles. The section establishes separate guarantees for strategy dynamics and replicator dynamics.

6. Recovering dynamic stability

  • Radness uses payoff flux—the aggregate of players’ instantaneous gains from unilateral deviations toward an outside profile—so it strengthens ordinal closure with payoff magnitudes.The flux corresponds to signed preference-graph edge weights; strict radness requires the defining inequalities to be strict.Source: §6 · pp. 11–12
  • Rad sets are s-club, while s-rad sets are club, linking payoff-dependent resilience to closure under profitable deviations.This is the implication established by Proposition 3.Source: §6 · p. 12
  • Every s-rad set H has a span that is an attractor of the strategy dynamics (SD).The proof uses an energy based on the Fenchel coupling that vanishes exactly on span(H) and dissipates near it.Source: §6 · p. 13
  • If H is both rad and a club, its span is an attractor of the replicator dynamics (RD); without ties, every rad set has this stability.The proof uses total outside mass as a local energy that decreases near the span.Source: §6 · pp. 13–14
  • Four games differing in one payoff entry share a preference graph yet exhibit increasingly complex dynamics and convergence to different faces.This illustrates why preferences alone may not determine long-run behavior.Source: §6 · p. 11, Figure 5 · p. 14
  • The replicator-dynamics proof makes the local energy decrease by using clubness to find, for each relevant outside profile, a neighboring profile in H with negative flux.The resulting negative contribution dominates the second-order outside-mass term near span(H).Source: §6 · p. 14
  • Radness gives a general explanation for stability under regularized learning beyond the ad hoc proofs for the 6-cycle in Jordan’s Matching Pennies and Shapley’s game hexagon.Those sets are club and rad but not s-rad; their replicator stability had previously been established with game-specific arguments.Source: §6 · p. 14

7. Concluding remarks … B.1. Sink equilibria.

Preference structure constrains the stable outcomes of regularized learning but does not determine them; whether asymptotically stable sets have a simple pure-strategy payoff characterization remains open. The appendices place this conclusion alongside prior work and define sink equilibria as terminal components of the preference graph.

7. Concluding remarks

  • Preference structure constrains, but does not determine, stable outcomes of regularized learning; radness is sufficient but not necessary for asymptotic stability, leaving a simple payoff-based characterization open.The open question is whether asymptotically stable sets can be characterized by payoff conditions checked at the level of pure strategies.Source: §7 · p. 14
  • Four games with the same preference graph and no proper attractors have markedly different limiting behaviors, and a boundary-containment conjecture for replicator or FTRL limit sets remains open.The authors note that establishing the conjecture likely requires new ideas.Source: §7 · p. 15

Appendix A. Further related work

  • Attractors in replicator dynamics trace to evolutionary game theory, where Ritzberger and Weibull showed that subgames closed under better replies induce attracting faces.This is the replicator special case of Theorem 3.Source: Appendix A · p. 15
  • Replicator dynamics are the continuous-time limit of exponential weights, and FTRL provides a broader unified framework for studying stable long-run outcomes of no-regret learning.Exponential weights is one of the few quintessential no-regret algorithms [15]; the cited connections are for replicator and for regularized dynamics.Source: Appendix A · p. 15
  • In zero-sum games the sink-equilibrium span is the replicator’s unique global attractor [11], while harmonic games are recurrent under FTRL and potential games yield convergence to Nash equilibria.Outside these structured classes, long-run learning behavior in finite games remains poorly understood.Source: Appendix A · p. 15
  • Preference-graph research established that replicator minimal attractors exist and contain sink equilibria [10], while the Fenchel gap extends club-subgame stability results to steep and non-steep regularizers.Papadimitriou and Piliouras conjectured a correspondence between minimal attractors and sink equilibria; the Fenchel-gap result is for continuous-time FTRL and improves on the Bregman-divergence approach [14].Source: Appendix A · pp. 15–16

Appendix B. Preference graphs

  • The preference graph has pure profiles as vertices and an arc between profiles that are comparable for some player.The passage defines an arc when profiles agree on all actions except that player’s action, though its displayed notation is truncated.Source: Appendix B · p. 16

B.1. Sink equilibria.

  • A sink equilibrium is a nonempty sink strongly connected component of the preference graph, equivalently a minimal club set closed under better replies and strongly connected.Once a better-reply path enters one, it can continue within the set but cannot leave; sink equilibria are called so in [12].Source: §B.1 · p. 16

B.2. Examples. … Appendix D. Regularizers and choice maps

The examples show that preference-graph structure and payoff-based deviation conditions capture different features of games, while the appendices define the dynamical concepts and regularizer assumptions used in the analysis.

B.2. Examples.

  • The Prisoner’s Dilemma has a unique sink and strict Nash equilibrium, Matching Pennies has a strongly connected preference graph, and coordination has two strict equilibria.In the Prisoner’s Dilemma all preference-graph paths flow to (D, D); Matching Pennies’ graph is a directed 4-cycle; the coordination game’s two sink vertices are (A, A) and (B, B).Source: §B.2 · pp. 16–17
  • A 2 × 2 × 2 game has an s-rad sink, whereas Jordan’s Matching Pennies is rad but not s-rad.For the first game, Φ(β, α) ≤ −1 outside the sink for every β in it; in Jordan’s game, Φ takes values 0 or −2 outside the sink, and adaptive play can cycle rather than converge to Nash equilibrium.Source: §B.2 · p. 18
  • Shapley’s game is rad but not s-rad, while an asymptotically stable club subgame can fail to be rad.For Shapley’s sink, Φ(β, α) ≤ 0 outside the sink, with equality in some cases; a club subgame has a deviation with Φ = 2 > 0, showing radness is not necessary for asymptotic stability. Shapley’s game was an early example of learning failing to converge to Nash equilibrium under fictitious play.Source: §B.2 · pp. 19–20

Appendix C. Attractors and chain transitivity

  • The appendix treats flows as continuous maps satisfying the flow identities and states that the paper’s flows arise from globally well-posed ODEs with Lipschitz vector fields.A flow obeys ϕ0 = IdX and ϕt+s = ϕt ◦ ϕs.Source: Appendix C · p. 20

C.1. Attractors.

  • Asymptotic stability means stability plus attraction, and compact invariant asymptotically stable sets are attractors.An attractor has a neighborhood whose trajectories approach it; the appendix also gives a lemma on restricting an attractor to an intersecting compact invariant set.Source: §C.1 · p. 20

C.2. Chain transitivity.

  • A compact invariant set is internally chain transitive exactly when the restricted flow has no proper attractor.Internal chain transitivity requires pseudo-orbits between every pair of points in the set.Source: §C.2 · p. 21

Appendix D. Regularizers and choice maps

  • The appendix assumes decomposable regularizers that are continuous and smooth in the interior, with a condition yielding strong convexity on each strategy simplex.The strong convexity is with respect to the Euclidean norm.Source: Appendix D · p. 21

D.1. Choice maps. … Appendix E. Strategy dynamics

The appendix establishes key properties of regularized choice maps and Fenchel couplings, illustrates them with standard regularizers, and develops well-posedness results for continuous-time FTRL. It also specifies conditions under which the induced dynamics can be described globally in strategy space.

D.1. Choice maps.

  • Strong convexity makes each regularized choice map single-valued and Lipschitz continuous, with image equal to the domain of the regularizer’s subdifferential.The choice map is also invariant to adding a constant to every score coordinate.Source: §D.1 · p. 21
  • For steep regularizers, the choice map’s image is the relative interior, and its restriction to scores modulo constant shifts is injective.Equivalently, identical choices imply that the score vectors differ by a constant shift.Source: §D.1 · p. 22

D.2. Bregman divergence and Fenchel coupling.

  • The Fenchel coupling is a primal-dual distance: it vanishes exactly when the strategy equals the choice map, and agrees with the Bregman divergence under the stated support condition.When the regularizer is differentiable at the choice, the extended Bregman divergence reduces to the usual Bregman divergence.Source: §D.2 · p. 22

D.3. Examples of regularizers.

  • The entropy regularizer is steep, yields the softmax choice map, and induces Kullback–Leibler divergence on the specified support domain.It satisfies the strong-convexity condition with K_i = 1 and s_i(p) = p.Source: §D.3 · p. 23
  • The quadratic regularizer satisfies the strong-convexity condition with K_i = 1 and s_i(p) = 1, but is non-steep.Source: §D.3 · p. 23
  • A power-family regularizer is steep for λ ∈ (0, 1) and non-steep for λ ∈ (1, 2]; it includes the Euclidean and square-root cases.The entropy regularizer is recovered in the limit λ → 1, up to affine terms on the simplex.Source: §D.3 · pp. 23–24

Appendix E. Strategy dynamics

  • For every initial score vector, continuous-time FTRL has a unique global score-space solution because the composed payoff-choice map is Lipschitz continuous.Source: Appendix E · p. 24
  • With a steep regularizer, the facewise strategy-space ODE applies globally along each trajectory; defining a global strategy-space vector field requires stronger regularity of s_i.Source: Appendix E · p. 24, §E.1 · p. 25

E.1. Construction of the strategy flow.

Under Assumption E.1, the strategy dynamics define a unique global flow that leaves every face invariant. The flow is also a patchwork of facewise FTRL dynamics, and its time reversal is the strategy flow of the negated game.

E.1. Construction of the strategy flow.

  • Assumption E.1 makes each h_i steep by requiring s_i to be globally Lipschitz on [0, 1] with s_i(0) = 0.Source: §E.1 · p. 25
  • Under Assumption E.1, each S_i is bounded away from zero on X_i, so the choice map π_i is Lipschitz continuous.For each player, some action probability is at least 1/|A_i|; continuity and strict positivity of s_i on (0, 1] yield the positive lower bound.Source: §E.1 · p. 25
  • Under Assumption E.1, the ODE has a unique global solution and induces a continuous strategy flow that leaves each face of X invariant.The vector field is globally Lipschitz; zero coordinates remain zero, and the same invariance holds in reverse time.Source: §E.1 · pp. 25–26
  • The time-reversed strategy flow coincides with the strategy flow of the game with payoffs negated.The reversed flow solves the ODE with vector field −F, which is induced by replacing the payoff field v with −v.Source: §E.1 · p. 26
  • The strategy flow is a globally Lipschitz continuous patchwork of facewise FTRL dynamics.On a face F, the strategy-flow trajectory from x0 in F coincides with the corresponding facewise FTRL orbit.Source: §E.1 · p. 26

Appendix F. Omitted proofs from Section 4 · F.1. Stability and attraction.

The omitted proofs show that FTRL eliminates dominated mixed strategies and that asymptotically stable sets must include pure profiles. They connect stability to profitable deviations, while showing how score initialization can expose a strict better reply even without face-invariance.

Appendix F. Omitted proofs from Section 4

  • Under FTRL, every dominated mixed strategy becomes extinct along every orbit.Source: Appendix F · p. 27
  • No subset of a face’s relative interior is asymptotically stable under FTRL; with steep regularizers, every asymptotically stable set under (SD) intersects the pure-profile vertices.Source: Appendix F · p. 27

F.1. Stability and attraction.

  • Face-invariance and elimination of dominated strategies make edge trajectories follow the preference-graph orientation toward a profitable unilateral deviation.The argument also applies to any flow satisfying both properties.Source: §F.1 · p. 27
  • Every asymptotically stable set under (SD) contains pure profiles closed under better replies.The proof combines intersection with the vertex set and closure under better replies.Source: §F.1 · pp. 27–28
  • Without face-invariance, large initial score gaps keep opponents near their actions while the deviator’s score shifts toward a strict better reply, forcing exit from a stable neighborhood.The proof bounds opponents’ distance over a prescribed time window and makes the better reply dominate other actions by its end.Source: §F.1 · pp. 28–29
  • The FTRL stability proof rules out a strict better reply from a pure profile in a stable set when the reply lies outside that set.It assumes such a deviation and constructs an orbit that exits the stable neighborhood.Source: §F.1 · p. 28

F.2. Connectedness. · Appendix G. Omitted proofs from Section 5 · G.1. Energy functions and asymptotic stability.

A strongly connected pure-profile set has an internally chain transitive mixed span, and any attractor containing it must contain that span; if the whole preference graph is strongly connected, the full strategy space has no proper attractor. Under FTRL, a local energy function is sufficient for asymptotic stability.

F.2. Connectedness.

  • If H is strongly connected, its mixed span is internally chain transitive under the strategy flow induced by (SD).The proof establishes that the span is invariant and admits no proper attractor.Source: §F.2 · p. 30
  • Any attractor containing a strongly connected pure-profile set H must contain span(H).This follows because the restricted flow on span(H) is internally chain transitive and therefore has no proper attractor.Source: §F.2 · pp. 30–31
  • When the whole preference graph is strongly connected, the full strategy space X admits no proper attractor.Source: §F.2 · p. 31

G.1. Energy functions and asymptotic stability.

  • A local energy function is Lipschitz and C1, vanishes exactly as Q(y) approaches S, and strictly dissipates between positive energy levels below a threshold.Source: §G.1 · p. 31
  • If a nonempty closed set S admits a local energy function, then S is asymptotically stable under FTRL.Source: §G.1 · p. 31
  • For sufficiently small initial energy, trajectories stay near S, decrease across positive energy bands, and approach S as energy tends to zero.The energy remains below its initial bound and ultimately converges to zero, which forces strategy-space distance to S to converge to zero.Source: §G.1 · pp. 31–32

G.2. When preferences are enough.

For subgames, better-reply closure of the pure profiles exactly characterizes when their spans attract strategy-flow dynamics; a Fenchel-gap local energy function supports the forward implication.

G.2. When preferences are enough.

  • A subgame B is closed under better replies if and only if span(B) is an attractor for the strategy flow; the forward direction uses a Fenchel-gap local energy function.For B closed under better replies, the Fenchel gap is a local energy function for span(B).Source: §G.2 · pp. 32, 35, 37
  • Strict Nash equilibria are attractors, and in weakly acyclic games without ties, every minimal attractor is a strict Nash equilibrium.The converse characterization uses a better-reply path from a pure profile in the minimal attractor to a pure Nash equilibrium.Source: §G.2 · p. 37
  • A finite game is an ordinal potential game if and only if its preference graph is acyclic.An ordinal potential strictly increases along directed edges; conversely, a topological ordering of an acyclic preference graph defines such a potential.Source: §G.2 · p. 37

G.3. When preference are not enough.

The counterexample shows that preferential stability does not guarantee FTRL stability: the unique club set’s span is dynamically unstable. Near a restricted-face equilibrium, player 3’s profitable deviation helps drive the instability.

G.3. When preference are not enough.

  • On the top face FT, x*=(1/2, 1/2, 1) is a restricted-game Nash equilibrium, yet player 3 strictly profits by deviating to B.Source: §G.3 · p. 38
  • There is a fixed neighborhood U of S such that every neighborhood V of S contains an admissible initial condition whose FTRL trajectory leaves U; hence S is not stable under FTRL.Source: §G.3 · p. 42
  • The dual attractor of BLB contains S but is not itself a span of pure profiles.Source: §G.3 · p. 42
  • The example also fits Biggar & Papadimitriou’s local-source instability result: x* is a local source and a quasi-strict Nash equilibrium of −u on X.Source: §G.3 · p. 42
  • The escape is witnessed by full-support initializations converging to x*, and the proof applies to every FTRL dynamics satisfying the regularizer assumptions, unlike ’s (RD)-only result.Source: §G.3 · p. 43

Appendix H. Omitted proofs from Section 6

The omitted proofs develop local energy constructions and facewise stability arguments for spans of pure profiles, including a simpler entropic construction for sets closed under better replies. They use these results to establish an attractor result for replicator dynamics by proving stability on each face and patching the resulting neighborhoods.

Lemma H.1

  • H is rad (respectively s-rad) exactly when every excluded profile has nonpositive (respectively strictly negative) maximal Φ over each face of span(H).The lemma extends the pure-profile condition to mixed profiles on faces.Source: Appendix H · p. 43

Lemma H.2

  • For s-rad H and steep h, the constructed function F̄_H is a local energy function for span(H).The proof verifies regularity, positive semi-definiteness, and dissipativity.Source: Appendix H · pp. 43, 44, 46

Proof of Theorem 4

  • The proof patches facewise basins to establish that span(H) is an attractor for the strategy flow.The argument uses steepness and strict radness on the relevant face restrictions.Source: Appendix H · pp. 46–47

Entropic setting

  • In the entropic setting, the proof uses the simpler energy W̄_H∘Q and states it is local energy when H is rad and closed under better replies.The supplied passage does not specify the energy function’s target set.Source: Appendix H · p. 47

Appendix H. Omitted proofs from Section 6

  • E = W̄_H ◦ Q is a local energy function for S = span(H), yielding asymptotic stability.The proof obtains strict negativity of the energy derivative near S but outside it, then applies the local-energy criterion.Source: Appendix H · p. 51
  • On each face F meeting S, H_F is rad and closed under better replies, so span(H_F) is asymptotically stable for the face-restricted dynamics.The proof applies the local-energy result to the subgame on F and then transfers stability to the strategy-flow restriction.Source: Appendix H · p. 52
  • In the entropic case, patching the facewise neighborhoods proves that S = span(H) is a compact, invariant attractor under replicator dynamics.Since faces are invariant, trajectories starting in the patched neighborhood approach the corresponding facewise span, which is contained in S.Source: Appendix H · p. 52

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