Universal Myopic Mapping Theorem

• Universal Myopic Mapping Theorem is a principle demonstrating that simple, local decision rules yield near-optimal solutions across complex dynamic systems.
• It employs myopic policies based solely on current inputs or short time blocks, leveraging structured topological and probabilistic properties for robust performance.
• Applications include opportunistic spectrum access, universal simulation, and game-theoretic equilibria, with rigorous quantification of convergence rates and error bounds.

The Universal Myopic Mapping Theorem encompasses a class of principles and explicit results showing that complex decision, dynamical, or simulation problems—often involving uncertainty or noise—can be solved near-optimally using myopic, “local” mappings. Such mappings are only reliant on current input or short blocks, operate in a universal or robust manner (i.e., largely independent of underlying system specifics), and are supported by structured topological or probabilistic properties. The theorem’s various manifestations span stochastic control, universal simulation, and game-theoretic equilibrium theory, unifying them under a quantitative framework where optimal or near-optimal behavior emerges from myopic mechanisms which ignore long-term dependencies.

1. Precise Statement and Scope

The Universal Myopic Mapping Theorem asserts that for broad classes of dynamic systems (restless bandits, random variable simulation problems, and strategy correspondences in games), one can construct myopic policies or mappings that—under certain conditions—attain optimality or fastest possible convergence to target outcomes, without dependence on detailed specifications of the underlying system. Key instances include:

• In multichannel opportunistic access with noisy observations, the myopic policy—selecting actions based on immediate belief about the current state—reduces, under reliability constraints, to a round-robin deterministic mapping independent of Markov transition probabilities (0802.1379).
• In universal simulation, myopic “local” mappings can approximate arbitrary target distributions from general classes of seeds with error rates matching those achievable by seed-specific (non-universal) simulators, quantifying convergence precisely in terms of seed entropy or regularity (Yu, 2018).
• In game theory, the existence and robustness of myopic equilibria—where maximum payoff is attained for every action used—are guaranteed if the payoff correspondence satisfies the spanning property, a topological condition ensuring non-trivial coverage, leading to universal existence of solution mappings (Simon et al., 2020).

2. Core Mathematical Constructs

The theorem’s structure derives from the following central constructs:

• Myopic Policy/Mapping: A deterministic or randomized rule, denoted generically as $f$, which bases action solely on current beliefs or inputs (e.g., $a^*(t) = \arg\max_a \omega_a(t)$).
• Semi-Universality: Independence from system parameters (transition probabilities, seed laws) apart from gross qualitative properties (e.g., correlation sign for channels or seed type for simulation).
• Cyclic/Periodic Structure: For restless bandit models, the mapping takes the form of a round-robin circular ordering $\mathcal{C}(1)$, updating action according to ACK/NAK feedbacks.
• Spanning Property (Game Theory): For set-valued correspondences $F$ on compact $W$, the topological spanning property (Čech homology class $[W]$ lies in the image of the inclusion-induced homomorphism) implies existence of myopic equilibria.
• Convergence Rate Quantification: For universal simulation, error rates are given as
  • Continuous case: $$\limsup_{\Delta \to 0} \frac{1}{\Delta} |P_Y - Q_Y|_{\mathrm{TV}} \leq \int |p_X'(x)| dx$$
  • Discrete case: $$E_{\mathrm{KS}}(P_X^n, Q_Y) = \frac{1}{2}(\max_x P_X(x))^n$$
  • Markov case: $E_{\mathrm{KS}} \asymp e^{-n H_\infty}$

3. Principal Results and Proof Basis

The theorem’s key results and supporting arguments are:

• Multichannel Bandit Optimality (0802.1379): Under bounded false alarm probability $\epsilon$, the myopic policy achieves optimality for $N=2$ channels when $\epsilon < \frac{p_{10} p_{01}}{p_{11} p_{00}}$ (positive correlation case), with numerical evidence up to $N=5$ supporting optimality more generally. The policy’s structure leverages a fixed circular order and updates the current channel conditioned on feedback.
• Universal Simulation (Yu, 2018): A sequence of measurable mappings $f_k$ constructed via partitioning the seed space and applying quantile transforms ensures that, for absolutely continuous $p_X$, the simulated output $P_{Y_k}$ converges in TV or KS distance to $Q_Y$. The error bounds match the rate achievable by tailored, non-universal simulators, confirming the myopic mapping can be universal up to sharp constants.
• Myopic Equilibria and Topological Universality (Simon et al., 2020): The spanning property guarantees myopic equilibrium existence even when payoff functions are non-affine or discontinuous. Topological results (Theorems 1–3) show the property is preserved by intersection, composition, and summing, yielding “universal” solution mappings robust to manifold parametrization.

4. Universal Structure and Independence Properties

The theorem is characterized by forms of independence and robustness:

• In restless bandit problems, the mapping’s universality lies in obviating the need for knowledge of Markov transition probabilities once the initial circular order is set, relying only on qualitative information (e.g., positive/negative correlation).
• For universal simulation, the mapping does not require prior knowledge of the seed law and achieves nearly optimal error rates universally over the seed class.
• In game theory, universality results from the spanning property, which guarantees solution existence without additional geometric or convex structural requirements.

The following table summarizes independence properties in major domains:

Domain	Control/Mapping	Independence
Restless Bandit	Round-robin policy	Markov transitions
Universal Simulation	Measurable mapping	Seed distribution law
Myopic Equilibria	Solution correspondence	Payoff multilinearity

5. Methodological Principles and Quantitative Guarantees

Execution of myopic mapping approaches utilizes:

• Partitioning/Quantile Transformation: In simulation, fine partitioning and inverse CDF mapping allow uniform treatment of arbitrary seeds.
• Circular Ordering and Feedback-Driven Updates: In restless bandits, ordered action cycling parameterized by initial belief vector circumvents the need for running belief updates.
• Topological Homology and Set-Valued Correspondence Operations: For games, the use of relative Čech homology underpins robust existence and universality across parameter spaces.

Formal performance guarantees include linear (continuous case) and exponential (discrete/Markov case) error decay rates, with precise dependence on seed regularity or entropy. In bandit control, optimality is rigorously proven for $N=2$ and numerically established for $N\leq5$.

6. Applications and Significance

• Opportunistic Spectrum Access: The semi-universal, myopic policy structure enables low-complexity implementation at the MAC layer, adapting robustly to channel statistics variations (0802.1379).
• Random Variable and Markov Process Simulation: Arbitrary target distributions may be simulated from general seed classes with exactly characterized convergence rates, supporting robust algorithm design in sampling, privacy, and data anonymization (Yu, 2018).
• Game-Theoretic Equilibrium Analysis: Myopic equilibria provide existence and universality results for classes of games with non-convex or non-linear payoff structures, supporting further analyses in repeated/extensive form games and evolutionary dynamics (Simon et al., 2020).

7. Conceptual Unification and Theoretical Implication

The Universal Myopic Mapping Theorem synthesizes principles from stochastic control, ergodic theory, simulation theory, and topological game analysis. It demonstrates that near-optimal solutions can be achieved via myopic, local mappings whose performance is universal (up to sharp error rates or existence quantifiers) insofar as core system properties—such as partition regularity, entropy, correlation sign, or topological spanning—are satisfied. This suggests a broad meta-theorem: universality and near-optimality can be attained by eschewing global knowledge or heavy-tracking updates, provided qualitative system regularities are detectable.

A plausible implication is that for a wide class of noisy, uncertain, or dynamically evolving systems, pursuit of myopic, universal mapping strategies may yield optimal or near-optimal performance bounded entirely in terms of summary parameters (false alarm rate $\epsilon$, seed entropy $H_\infty$, homology projections) rather than full system specification. This principle underpins numerous results and methodologies in modern stochastic processes, simulation, and game theory.