True Theory Dominance

• True Theory Dominance is a framework defining global dominance by extending local optimality through order-theoretic and structural criteria across Bayesian, game-theoretic, and dynamical systems.
• It leverages stochastic dominance in Bayesian models and reaction-proof conditions in gradual mechanisms to ensure truth-telling and robust strategy selection.
• The approach applies differential dominance with tractable LMI conditions to analyze stability and performance in nonlinear dynamical systems.

True Theory Dominance refers to structural properties underpinning "dominance" relations in mathematical models—including mechanism design, Bayesian statistics, game theory, and dynamical systems—where the concept extends beyond local optimality to global orderings dictated by model truth, rational strategy chains, or robust dynamical behavior. Across fields, “True Theory Dominance” encapsulates several deep theorems and operational criteria: from stochastic dominance of posteriors conditioned on the true parameter to necessary and sufficient conditions for the dominance of truth-telling in dynamic mechanism implementations, and to the order-theoretic generalizations of admissibility and p-dominance in system theory.

1. Stochastic Dominance in Bayesian Models

In Bayesian inference, True-Theory Dominance denotes the phenomenon where, for any finite parameter space $\Theta$, the random posterior probability assigned to the true parameter $\theta_0$ (when data is generated from $\theta_0$) is stochastically larger—across all increasing functionals—than when the data is drawn from the prior predictive mixture. Formally, for any threshold $t \in [0, 1]$,

$$\mathbb{P}_{\theta_0}\{ \pi(\theta_0|X) \leq t \} \leq \mathbb{P}_\pi\{ \pi(\theta_0|X) \leq t \},$$

and

$$\mathbb{E}_{\theta_0}[f(\pi(\cdot|X))] \geq \mathbb{E}_\pi[f(\pi(\cdot|X))]$$

for any increasing functional $f$ of the posterior (Hart et al., 2022). This property is inherited from the likelihood-ratio ordering, since the density $d\mathbb{P}_{\theta_0} / d\mathbb{P}_\pi$ equals the posterior on $\theta_0$. It provides a universal optimism principle: observing data generated by the true parameter confers, in expectation and stochastically, an advantage over a Bayesian mixture baseline, regardless of the signal structure. Concrete implications span risk-averse decision functions, reputation in asset-pricing, and the interpretation of posterior probabilities as more “optimistic” when the truth is known.

2. Dominance of Truth-Telling in Gradual Mechanisms

In dynamic implementation theory, True-Theory Dominance addresses when truth-telling remains a dominant strategy in mechanisms where agents incrementally reveal private information (gradual mechanisms). For a strategy-proof social choice function (SCF) $f$, truth-telling is dominant in the direct mechanism by construction. However, dynamic game forms can leak incremental information, potentially undermining incentive compatibility.

The dominance of truth-telling is preserved if and only if every "illuminating transformation" (partitioning an agent's information set) is incentive-preserving, or equivalently, if the mechanism is “reaction-proof”: for every pair of immediate-successor information sets of an agent, no other agent can benefit by a deviation after observing which path was chosen (Wang et al., 15 Jan 2025). This equivalence establishes a single local test (reaction-proofness) that guarantees the global dominance of truth-telling. The result applies to gradual implementations of second-price auctions and top-trading cycle algorithms, where typical ascending-price or RDA routines pass these criteria.

Mechanism Property	Characterization	Result
Incentive Compatibility	All illuminating splits are incentive-preserving	Truth-telling dominant
Reaction-Proofness	Local check: No agent can benefit by deviation after any split	Equivalent to IC

3. Dominance Between Chains of Strategies in Quantitative Games

The classical notion of admissibility identifies strategies not dominated by any other. In quantitative games (on finite graphs with real-valued payoffs), not every strategy is either admissible or dominated by an admissible strategy—contrary to classical Boolean settings. To recover a robust rationality notion, dominance is generalized from individual strategies to chains of increasingly strong strategies

$$(\sigma_\alpha)_{\alpha<\lambda},\quad \alpha<\beta \implies \sigma_\alpha \prec \sigma_\beta,$$

indexed by ordinals (Basset et al., 2018).

One chain $(\sigma_\alpha)$ is dominated by another $(\tau_\beta)$, denoted $(\sigma_\alpha) \sqsubseteq (\tau_\beta)$, if for all $\alpha$, there exists $\beta$ such that $\sigma_\alpha \preceq \tau_\beta$. Maximal chains play the rationality role where no further dominance extension exists. In generalized safety/reachability games, chains are uniformly represented by parametrized automata; every dominated finite-memory strategy is dominated either by an admissible finite-memory strategy or by a maximal uniform chain. Decidability of chain membership and comparison is polynomial-time computable.

Illustrative games (e.g., the “help-me?” game) show that no finite-memory strategy is admissible, but the infinite chain of “eventually bailing” strategies—parameterized by a counter—forms a maximal chain, refining the concept of admissibility.

4. Differential Dominance in Nonlinear Dynamical Systems

The theory of differential dissipativity formalizes dominance analysis in nonlinear systems via the concept of p-dominance. For an $n$-dimensional system $\dot{x}=f(x)$, p-dominance (with rate $\lambda$ and storage $V(\delta x)=\delta x^T P \delta x$, $P$ of inertia $(p,0,n{-}p)$) requires

$$\delta x^T( \partial f(x)^T P + P \partial f(x) ) \leq -2\lambda V(\delta x) - \varepsilon \|\delta x\|^2, \quad \varepsilon>0,$$

holding for all $(x,\delta x)$ (Forni et al., 2017). This contracts two cone-fields in the tangent bundle, yielding a dominated splitting of the tangent space into $p$-dimensional non-contracting and $(n{-}p)$-dimensional contracting invariant bundles. Asymptotics are thus constrained to $p$-dimensional flows: $p=0$ gives exponential stability (unique equilibrium), $p=1$ gives multistability, $p=2$ gives limit cycles or heteroclinic connections.

The framework extends to open interconnections, where feedback of two p-dominant systems produces a $(p_1+p_2)$-dominant closed loop under tractable linear matrix inequality (LMI) conditions. The theory includes robustness margins akin to small-gain and passivity theory, and enables rigorous attractor dimensionality classification in high-dimensional nonlinear systems via tractable convex relaxations.

5. Operational and Algorithmic Consequences

True-Theory Dominance underwrites several algorithmic and structural results:

• Bayesian context: The stochastic dominance of posteriors under the true law admits universality, independent of experiment or conditioning, and supports robust optimism results for all threshold-type functionals (Hart et al., 2022).
• Game-theoretic mechanisms: Reaction-proofness and incentive-preserving splits are verifiable by local checks. Applications to auctions and matching markets confirm practical implementability (Wang et al., 15 Jan 2025).
• Quantitative games: Chain recognition and chain comparison among parameterized automata are polynomial-time decidable, offering a constructive route to identifying maximal chains and their dominance relations (Basset et al., 2018).
• Dynamical systems: Differential dominance is computable via LMI relaxations for the Jacobian matrix over invariant sets, and robust under small-gain and passivity uncertainties (Forni et al., 2017).

6. Synthesis and Theoretical Unification

Across these domains, True-Theory Dominance formalizes the elevation of local or one-shot optimality/equilibrium to robust, global, and structural forms:

• In Bayesian inference, dominance of the “true” posterior reflects epistemic optimism about the true model.
• In mechanism design, truth-telling's dominance is preserved only under precise local conditions on information revelation.
• In game theory, maximal chains of strategies transcend classical admissibility, establishing rationality in richer payoff settings.
• In control and dynamical systems, p-dominance generalizes stability to multistable and oscillatory regimes.

A plausible implication is the potential for cross-pollination between these fields: the order-theoretic and information-structural conditions for dominance may have analogs in statistical, economic, and dynamical contexts, offering a unifying language for robust rationality and model superiority in complex systems.