Real Dominance: Concepts and Applications

• Real dominance is a multifaceted concept that defines asymmetric relations based on trajectories, genotype contrasts, coordinate inequalities, and payoff comparisons.
• It spans applications from modeling pedestrian navigation and genetic fitness to structuring order in graph drawing and approximate search.
• The concept further informs nonlinear system analysis and game theory by certifying low-dimensional asymptotic behavior and operationalizing emerging strategic hierarchies.

Real dominance is a polysemous technical term whose meaning depends on disciplinary context. In current research it denotes, among other things, a psychological trait manifested in pedestrian motion, a context-dependent property of the genotype–phenotype–fitness map, a partial order on points in real coordinate spaces, a certificate that nonlinear asymptotic behavior is effectively $p$-dimensional, and a state-by-state ordering of strategies in decision theory. Despite these differences, the term consistently marks an asymmetric relation that is operationalized from trajectories, genotype contrasts, coordinate inequalities, differential inequalities, or payoff comparisons rather than left at the level of informal description (Randhavane et al., 2018, Li et al., 2023, Lau et al., 23 Jun 2026, Forni et al., 2017, Gnedin, 2011, Rachum et al., 2024).

1. Socially enacted dominance in pedestrian and robot navigation

In socially aware navigation, pedestrian dominance is treated as a psychological trait that manifests in everyday social interactions through movement and navigation behaviors. Dominant pedestrians tend to be assertive and expect others to accommodate them; submissive pedestrians tend to yield, avoid, and circumvent conflicts. The motion cues identified in this formulation are path directness, speed, proxemics, and reactivity to neighbors: dominant pedestrians take straighter, more direct paths toward goals and pass through groups rather than detouring, walk faster on average, keep larger interpersonal distances, and are less influenced by many neighbors while planning less far ahead (Randhavane et al., 2018).

The operational model is the Pedestrian Dominance Model (PDM). It uses five low-level parameters from a 2D Reciprocal Velocity Obstacles motion model: Neighbor Distance, Maximum Neighbors, Planning Horizon, Effective Radius, and Preferred Speed. These parameters are estimated per pedestrian via Bayesian learning from observed trajectories. The perception study underlying the model used 390 MTurk workers, four simulated scenarios—Pass-Through, Corridor, Standing Group, and Narrow Exit—and four adjective ratings on 5-point Likert scales: Submissive, Withdrawn, Dominant, and Confident. The scalar dominance label is constructed as

$$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$

After normalization around default parameters, dominance is predicted by multiple linear regression:

$$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$

The reported interpretation of the coefficients is that faster walking and larger personal space correlate with higher perceived dominance, less planning ahead correlates with higher dominance, considering fewer neighbors correlates with higher dominance, and Neighbor Distance has a small positive and negligible effect. The model is fit by ordinary least squares on 48 videos with leave-one-out cross-validation, with average absolute error approximately $0.15$, corresponding to approximately $85\%$ accuracy on a $0$–$1$ dominance scale.

At runtime, the workflow is to track pedestrians, estimate motion-model parameters from partial trajectories, normalize the parameters, and apply $d = \mathbf{w}^\top x$. Reported average time per frame for motion-model parameter computation ranges from roughly $0.022$ to $0.061$ seconds depending on scenario density and frames, while dominance computation itself is reported in microseconds per pedestrian per frame. The navigation layer then uses complementary dominance: for pedestrians with dominance values $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$0, the robot aims at

$$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$1

and selects robot motion parameters by solving

$$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$2

The same paper extends the idea to autonomous vehicles by modifying AutonoVi proximity costs via

$$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$3

so that high-$$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$4 pedestrians induce vehicle yielding, whereas low-$$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$5 pedestrians induce continued motion subject to safety constraints.

The formulation is explicitly limited. Dominance cues beyond trajectories—gaze, posture, gestures, facial expressions, and audio—are not modeled; high-density crowds may reduce accuracy; social hierarchies are context-dependent; inter-rater reliability statistics were not reported; and the approach assumes reliable pedestrian tracking together with Reciprocal Velocity Obstacles as an adequate local model. A notable behavioral outcome in the Narrow Exit scenario is that very dominant agents exited approximately $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$6 faster than the most submissive agent.

2. Context-dependent dominance in genetics

In genetics, dominance is not treated as a fixed property of an allele but as an emergent, context-dependent feature of the genotype–phenotype–fitness map that can change across genetic backgrounds, environments, sexes, and lineages. For a biallelic locus with genotypes $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$7, $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$8, and $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$9, and trait value or fitness $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$0, the standard dominance coefficient is

$$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$1

with $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$2 indicating additivity, $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$3 recessivity, $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$4 complete dominance, $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$5 underdominance, and $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$6 overdominance. The paper distinguishes dominance of phenotype from dominance of fitness, noting that nonlinear genotype–phenotype and phenotype–fitness mappings imply that the two need not coincide (Li et al., 2023).

The central claim is that observed dominance varies with genetic background because alleles at other loci modify the heterozygote’s value relative to the homozygotes. In the two-locus Cockerham/Kempthorne expansion,

$$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$7

effective dominance at locus $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$8 depends on the state at locus $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$9. The background-specific effective dominance is

$0.15$0

This makes explicit that additive-by-additive, dominance-by-additive, additive-by-dominance, and dominance-by-dominance epistasis can all shift the observed value of $0.15$1.

The paper also places variable dominance within variance decomposition:

$0.15$2

where $0.15$3 includes $0.15$4, $0.15$5, and $0.15$6. Background-dependent dominance makes $0.15$7 context-specific and redistributes non-additive effects into epistatic variance. The review further argues that apparent dominance can arise from nonlinear mappings such as pathway saturation, threshold traits, network buffering, and canalization.

A minimal gene-centric account of variable dominance is given by Wade’s three-locus model. With focal locus $0.15$8 and modifiers $0.15$9 and $85\%$0, and with allele $85\%$1 scaling the effect of $85\%$2 by $85\%$3 while allele $85\%$4 scales the effect of $85\%$5 by $85\%$6, the expected phenotypes are

$85\%$7

$85\%$8

Overdominance occurs for $85\%$9 and $0$0, whereas underdominance occurs for $0$1 and $0$2. The paper uses this to argue that the allele-frequency dynamics of modifiers can generate the full spectrum from recessivity to overdominance.

The empirical review spans cis-regulatory position effects, dominance modifiers of resistance, sex-specific, mating-type-specific, and lineage-specific dominance, and variable dominance in hybrid incompatibilities. The Drosophila white/apricot system is presented as a classic example in which eye-color dominance at white changes depending on phase with the wild-type apr regulator. Other cases include compensatory modification of diazinon resistance in Lucilia cuprina, background- and strain-specific resistance dominance in Daphnia magna, sex-dependent dominance at VGLL3 in Atlantic salmon and in a rainbow trout migration supergene, mating-type modification in yeast evolve-and-resequence crosses, and dominance modulation of male sterility QTL in Drosophila pseudoobscura/persimilis. A recurring misconception addressed by the review is therefore that dominance can be read off from a focal locus alone.

Methodologically, the paper recommends background-specific estimation,

$0$3

together with regression or ANOVA models that include interaction terms, reciprocal crosses that toggle sex or mating type, phased cis/trans constructs, lineage swaps, and QTL-by-QTL mapping. This suggests that “real” dominance in genetics is not a constant attribute but a measured property of a specified background.

3. Real-coordinate dominance in search and graph drawing

In interval-predicate approximate nearest neighbor search, real dominance denotes a partial order on pairs of real numbers that compares coordinates with possibly opposite inequality directions on different axes. A common order is that $0$4 dominates $0$5 iff $0$6. The Unified Dominance Graph adopts an equivalent orientation tailored to interval predicates, using transformed coordinates in which a data point $0$7 is valid for query $0$8 iff

$0$9

Containment, overlap, enclosed-by, and other closed two-bound conjunctive predicates are mapped into this normalized dominance predicate by choosing which interval endpoints become $1$0 and $1$1, and whether one coordinate is negated. For example, overlap is represented by

$1$2

and object-contains-query is represented by

$1$3

Canonicalization uses the nearest transformed data boundaries,

$1$4

with the lemma that raw and canonical queries select the same valid set. UDG then stores embedding vectors together with transformed coordinates, labels edges by rectangles in the canonical dominance grid, and adds validity-preserving patch edges to maintain navigability under restrictive filters. Reported worst-case bounds are $1$5 index size and $1$6 indexing time, while average-case bounds under independence assumptions are $1$7 index size and $1$8 indexing time. On SIFT1M, DEEP1M, DBpedia-OpenAI, S&P 500 stock ranges, and Nasdaq event-time, the reported metrics are Recall@10 and QPS, with selectivities from $1$9 to $d = \mathbf{w}^\top x$0; the paper reports stable high-recall/high-QPS behavior and Recall@10 $d = \mathbf{w}^\top x$1 up to 10M vectors for the primary predicates (Lau et al., 23 Jun 2026).

A different geometric use appears in dominance drawings of directed acyclic graphs. There, a dominance drawing in $d = \mathbf{w}^\top x$2 dimensions is a mapping $d = \mathbf{w}^\top x$3 such that

$d = \mathbf{w}^\top x$4

Incomparability is therefore represented by coordinate crossings: neither point dominates the other componentwise. The key structural parameter is the width

$d = \mathbf{w}^\top x$5

which equals the minimum number of channels in a channel decomposition. The algorithmic result is a $d = \mathbf{w}^\top x$6-dimensional dominance drawing for any $d = \mathbf{w}^\top x$7 with

$d = \mathbf{w}^\top x$8

with coordinate assignment in $d = \mathbf{w}^\top x$9 time after $0.022$0 preprocessing to build a compressed transitive closure. This yields the bound

$0.022$1

The paper further introduces transitive modules and the dimensional neck $0.022$2, with

$0.022$3

and a module-based construction with total time

$0.022$4

Both lines of work treat dominance literally as an order on real coordinates, but one uses it for valid-subgraph restriction in approximate search, while the other uses it to encode reachability exactly in $0.022$5 (Ortali et al., 2019).

4. Dominance as low-dimensional asymptotic structure in nonlinear systems

In differential dissipativity theory, real dominance means that the asymptotic behavior of a nonlinear system is effectively $0.022$6-dimensional in the sense of a dominated splitting of the linearized flow. For

$0.022$7

the prolonged dynamics on the tangent bundle are

$0.022$8

A constant symmetric tensor $0.022$9 with fixed inertia defines the differential quadratic storage

$0.061$0

and the cone fields

$0.061$1

For an LTI system $0.061$2, $0.061$3-dominance with rate $0.061$4 is certified by the existence of $0.061$5 with inertia $0.061$6 such that

$0.061$7

strictly if $0.061$8. For nonlinear systems, differential $0.061$9-dominance requires

$$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$00

This is the core certificate from which cone contraction and dominated splitting follow (Forni et al., 2017).

The strict case yields a reduction principle on $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$01-limit sets. On a compact invariant set $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$02, the tangent bundle splits as

$$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$03

with $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$04 and $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$05, invariant under the linearized flow. The dominant directions lie in $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$06, while the $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$07 directions are transient. The principal asymptotic consequences are listed explicitly: $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$08 recovers contraction and convergence of every bounded solution to a unique fixed point; $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$09 implies that every bounded solution converges to a fixed point, not necessarily unique; and $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$10 implies convergence to a simple attractor, with a generalized Poincaré–Bendixson result stating that if a compact forward invariant set contains no fixed points, then the $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$11-limit set of any point in it is a closed orbit.

The framework extends to open systems by differential $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$12-dissipativity. For $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$13, $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$14, with supply rate

$$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$15

one asks for

$$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$16

The paper emphasizes that the interconnection theory closely mimics classical dissipativity theory: differential passivity and differential small gain preserve $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$17-dominance under appropriate feedback conditions. For example, the small-gain supply is

$$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$18

Certification is algorithmic. The recommended sequence is to use spectral analysis of $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$19 to select $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$20 and $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$21, construct a finite convex relaxation of the Jacobian family, and solve the relaxed LMIs for a constant $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$22 while verifying its inertia afterward. The paper presents canonical examples: strict $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$23-dominance for a damped mechanical system with convex spring, strict $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$24-dominance for Duffing-type springs, preservation of $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$25-dominance under DC motor actuation, and strict $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$26-dominance for a PI-controlled system whose bounded trajectories converge to a periodic orbit. The explicit limitation is that the constant flat metric $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$27 is restrictive, the differential inequality is uniform and global, and the certificates imply topological conclusions on $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$28-limit sets rather than directly constructing attractors.

5. Dominance margins and robustness away from equilibria

The theory of dominance margins generalizes classical gain, phase, and disk margins from $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$29 stability to $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$30-dominant behaviors such as multistability and oscillations. For a minimal SISO LTI transfer function

$$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$31

the analysis is performed on the shifted transfer function

$$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$32

An interval $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$33 is a $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$34-gain margin at rate $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$35 for a strictly $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$36-dominant plant when, for all $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$37, the Nyquist plot of $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$38 yields exactly $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$39 clockwise encirclements of $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$40. Likewise, an interval $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$41 is a $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$42-phase margin when the rotated Nyquist plot $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$43 yields the same encirclement count for every $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$44 in that interval (Padoan et al., 2019).

For Lur’e systems,

$$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$45

the relevant robustness object is the $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$46-disk margin associated with a differential sector $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$47 satisfying

$$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$48

The paper defines $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$49 as the geometric disk, complementary disk, or half-plane determined by the sector endpoints. If the Nyquist plot of $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$50 does not intersect this region and produces the appropriate encirclement count, then the closed loop is strictly $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$51-dominant with rate $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$52. The corresponding frequency-domain condition is a circle criterion tailored to $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$53-dominance. An algebraic restatement is

$$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$54

with the appropriate reversal when $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$55.

The shifted rate $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$56 plays a central role because it creates spectral separation between dominant and transient modes. The paper introduces

$$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$57

as a convenient measure correlating with the width of dominance margins. Increasing $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$58 can enlarge admissible gain and phase intervals by moving the shifted poles away from the imaginary axis, although the “optimal $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$59” depends on the plant.

The illustrative example is a mass–spring–damper system,

$$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$60

with $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$61, $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$62, $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$63, and

$$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$64

With saturated proportional feedback $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$65 and $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$66, the nonlinearity lies in the differential sector $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$67, and for $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$68 the shifted Nyquist plot yields strict $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$69-dominance, corresponding to global bistability. With integral action,

$$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$70

the relevant transfer function is $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$71, and for $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$72 with $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$73 the paper certifies strict $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$74-dominance, hence oscillatory behavior; it states that $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$75 is the 2-disk margin for the chosen $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$76. The same section stresses the limits of the framework: $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$77-dominance does not classify attractors beyond $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$78, strictness is needed to avoid marginal cases, and the paper does not provide LMI-based tests for Lur’e systems, relying instead on geometric Nyquist and circle-criterion arguments.

6. Strategic and emergent dominance in games and multi-agent learning

In decision theory, real dominance is a property of strategies rather than states or trajectories. In the Monty Hall problem, with state space $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$79, strategies $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$80 and $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$81, and utility $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$82, weak dominance is defined by

$$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$83

The formal model has door set $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$84 with $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$85, car location $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$86, an admissible host rule $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$87, and strategy $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$88 where $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$89 is the initial choice and $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$90 maps the offered door $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$91 to $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$92. The payoff identities are

$$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$93

The crucial theorem is the relabel-and-switch lemma: for any strategy $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$94 that is not always-switching, there exists some $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$95 with $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$96, and the always-switching strategy $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$97 with $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$98 weakly dominates $$d = \frac{\big(V_{\mathrm{dom}} + V_{\mathrm{conf}} + 6 - V_{\mathrm{sub}} + 6 - V_{\mathrm{with}}\big) - 4}{16}.$$99 state-by-state. This does not use any prior distribution over car placement or host behavior beyond admissibility. Under a prior $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$00 over $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$01, the always-switching strategy $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$02 wins with probability $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$03, so the Bayesian-optimal rule is to pick the least likely initial door and always switch. In the minimax formulation, the value is

$$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$04

which becomes $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$05 in the classical three-door case (Gnedin, 2011).

A separate but related use appears in multi-agent reinforcement learning, where dominance is operationalized as an emergent social convention. In the Chicken Coop environment, two agents repeatedly play a partially observable stochastic game with two stable pure-strategy Nash equilibria, and aggressiveness is defined by

$$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$06

A dominance relation is declared by thresholding asymmetry:

$$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$07

while rapport is

$$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$08

Group-level dominance is then represented as a tournament

$$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$09

with exactly one directed edge between each unordered pair. The learning setup uses PPO with learning rate $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$10, $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$11, $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$12, 512 episodes per generation, and separate neural networks for each agent. The Chicken payoffs satisfy $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$13 and are set to

$$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$14

Under these conditions, the paper reports that populations of RL agents can invent, learn, enforce, and transmit a dominance hierarchy without explicit programming or intrinsic rewards (Rachum et al., 2024).

The paper introduces hierarchy-comparison metrics. Dominance Hierarchy Distance is

$$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$15

with RDHD restricting to a subset $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$16, and Dominance Hierarchy Transmission Fidelity defined by

$$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$17

Across $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$18 populations with $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$19, the paper reports 248 distinct tournament hierarchies out of $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$20 possible, with only 17 intransitive. When $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$21 experienced agents are transplanted into a new $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$22 population, the median DHTF among naive agents is approximately $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$23. An observation ablation using Opponent Perception Accuracy values in $$d = \mathbf{D} \cdot \begin{pmatrix} \frac{1}{13.5}(\mathrm{Neighbor~Dist} - 15) \ \frac{1}{19.5}(\mathrm{Max~Neighbors} - 10) \ \frac{1}{11.5}(\mathrm{Planning~Horizon} - 24) \ \frac{1}{0.85}(\mathrm{Radius} - 0.8) \ \frac{1}{0.5}(\mathrm{Pref~Speed} - 1.4) \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 0.01 & -0.07 & -0.41 & 0.05 & 0.14 \end{pmatrix}.$$24 shows a positive correlation between OPA and mean rapport, which the paper interprets as evidence that identification of opponents causally enables hierarchy formation.

The conceptual claim made there is that these hierarchies are “real” because they are behaviorally enacted, enforced, stable over time, and transmissible. At the same time, the paper adds an ethical caution: dominance can function as a coordination convention in mixed-motive settings, but if deployed in systems that interact with humans it can introduce inequities or encode unfair social stratification. This contrasts with the Monty Hall setting, where dominance is purely decision-theoretic and normatively eliminative: it serves to discard strategies, not to organize agents into a hierarchy.