Ultrafilter Intelligence Comparators

• Ultrafilter-based intelligence comparators are defined as a framework that aggregates the performance of deterministic Legg–Hutter agents across all deterministic environments using ultrafilters.
• The approach is motivated by social choice theory, yielding robust structural properties such as team-monotonicity and quitter-monotonicity for agent comparisons.
• Incorporating free ultrafilters ties the framework to the Axiom of Choice, resulting in nonconstructive but total preorders that challenge computable implementations.

Ultrafilter-based intelligence comparators constitute an abstract framework for systematically comparing the relative intelligence of deterministic Legg–Hutter agents (DLHAs) by aggregating their performance across all deterministic environments using the mathematical concept of an ultrafilter. This construction, deeply motivated by social choice theory and the Kirman–Sondermann theorem on infinite electorates, provides a family of total, transitive preorders that generalize agent-environment comparisons beyond numeric intelligence assignments. The approach grants access to a robust suite of structural theorems on agent “team” formation and quitting strategies, though the practical realization of free ultrafilter comparators is inherently nonconstructive and depends on the Axiom of Choice (Alexander, 2019).

1. Formal Framework: Environments, Agents, and Values

Let $\mathcal{E}$ denote the class of all deterministic environments and $\mathcal{A}$ the class of all deterministic Legg–Hutter agents (DLHAs). Each environment $e \in \mathcal{E}$ repeatedly maps finite action histories $(a_1, \dots, a_n) \in \mathbb{N}^n$ to $$(r_{n+1}, o_{n+1}) \in \mathbb{R} \times \mathbb{N}$$, where for every agent $A$, the total accumulated reward $\sum_{i=1}^\infty r_i$ converges in $\mathbb{R}$. An agent $A \in \mathcal{A}$ is a function mapping observed histories $(r_1,o_1,\dots,r_n,o_n)$ to actions $\mathcal{A}$0. When $\mathcal{A}$1 interacts with $\mathcal{A}$2, the sequence $\mathcal{A}$3 allows the definition of a value function:

$\mathcal{A}$4

For a fixed environment $\mathcal{A}$5, one writes $\mathcal{A}$6 if $\mathcal{A}$7, setting a local (per-environment) agent preference.

2. Ultrafilter-Based Aggregation: The Election Model

The framework treats environments as “voters” and agents as “candidates”: each environment $\mathcal{A}$8 ranks pairs $\mathcal{A}$9 by the sign of $e \in \mathcal{E}$0. The set of “votes” where $e \in \mathcal{E}$1 does at least as well as $e \in \mathcal{E}$2 is aggregated via an ultrafilter $e \in \mathcal{E}$3.

Definition (Ultrafilter). A set system $e \in \mathcal{E}$4 is an ultrafilter if:

1. $e \in \mathcal{E}$5;
2. If $e \in \mathcal{E}$6 and $e \in \mathcal{E}$7, then $e \in \mathcal{E}$8 (upward closure);
3. For $e \in \mathcal{E}$9, $(a_1, \dots, a_n) \in \mathbb{N}^n$0;
4. For every $(a_1, \dots, a_n) \in \mathbb{N}^n$1, either $(a_1, \dots, a_n) \in \mathbb{N}^n$2 or $(a_1, \dots, a_n) \in \mathbb{N}^n$3 (maximality).

Principal ultrafilters contain a singleton and are thus “dictatorships;” nonprincipal (free) ultrafilters are characterized by the property that no finite set is “large.”

The ultrafilter-based comparator $(a_1, \dots, a_n) \in \mathbb{N}^n$4 is defined by:

$(a_1, \dots, a_n) \in \mathbb{N}^n$5

One writes $(a_1, \dots, a_n) \in \mathbb{N}^n$6 for $(a_1, \dots, a_n) \in \mathbb{N}^n$7, $(a_1, \dots, a_n) \in \mathbb{N}^n$8 for $(a_1, \dots, a_n) \in \mathbb{N}^n$9 in the analogous way; by ultrafilter properties, exactly one of the three holds for any pair (Proposition 3.1). Transitivity is guaranteed (Proposition 3.2) (Alexander, 2019).

3. Structural Properties and Variants

Each ultrafilter $$(r_{n+1}, o_{n+1}) \in \mathbb{R} \times \mathbb{N}$$0 yields a distinct and, if $$(r_{n+1}, o_{n+1}) \in \mathbb{R} \times \mathbb{N}$$1 is free, genuinely nondictatorial comparator. Principal ultrafilters at position $$(r_{n+1}, o_{n+1}) \in \mathbb{R} \times \mathbb{N}$$2 reduce $$(r_{n+1}, o_{n+1}) \in \mathbb{R} \times \mathbb{N}$$3 to comparison in environment $$(r_{n+1}, o_{n+1}) \in \mathbb{R} \times \mathbb{N}$$4 alone. Free ultrafilters aggregate across all environments in a maximally “coherent” way, and the choice of $$(r_{n+1}, o_{n+1}) \in \mathbb{R} \times \mathbb{N}$$5 selects a canonical, but nonconstructive, infinite aggregation—no computable description exists (see Section 5 for an explicit illustration).

Nontriviality (Proposition 3.3):

If $$(r_{n+1}, o_{n+1}) \in \mathbb{R} \times \mathbb{N}$$6 is free, there exist $$(r_{n+1}, o_{n+1}) \in \mathbb{R} \times \mathbb{N}$$7 with $$(r_{n+1}, o_{n+1}) \in \mathbb{R} \times \mathbb{N}$$8 and others $$(r_{n+1}, o_{n+1}) \in \mathbb{R} \times \mathbb{N}$$9 with $A$0; thus the induced order is nontrivial.

4. Structural Theorems: Teams, Splitting, and Quitters

Several structural theorems demonstrate the expressive power of the ultrafilter-based framework:

4.1 Team Formation and Team-Monotonicity

Given agents $A$1, $A$2, define $A$3 as the agent who, after observing $A$4, acts as $A$5 (if $A$6 even) or $A$7 (if $A$8 odd).

Team-monotonicity (Proposition 5):

If $A$9 and $\sum_{i=1}^\infty r_i$0, then $\sum_{i=1}^\infty r_i$1.

4.2 General Splitting: Incomparability

For a general split $\sum_{i=1}^\infty r_i$2 based on any set $\sum_{i=1}^\infty r_i$3 of partial histories, it holds (Proposition 7):

$\sum_{i=1}^\infty r_i$4

Thus, any “team” behaves like one of its members to almost all environments in $\sum_{i=1}^\infty r_i$5.

4.3 Quitters and Truncation

Define $\sum_{i=1}^\infty r_i$6 as the agent behaving like $\sum_{i=1}^\infty r_i$7 until total reward $\sum_{i=1}^\infty r_i$8, then skipping. Two key properties:

• Quitter-monotonicity (Proposition 10): If $\sum_{i=1}^\infty r_i$9 and $\mathbb{R}$0, then $\mathbb{R}$1.
• Trivial-quitter (Proposition 11): With nonnegative rewards, $\mathbb{R}$2; an agent is not outperformed by its own quitter.

4.4 Necessity of the Axiom of Choice

The existence of free ultrafilters is equivalent to the Axiom of Choice. Without AC, nondictatorial (free) ultrafilter comparators cannot be exhibited, and all explicit comparators are principal/dictatorial.

5. Examples of Ultrafilter-Based Comparators

Ultrafilter Type	Construction	Comparative Outcome
Principal	$\mathbb{R}$3	Comparison depends only on environment $\mathbb{R}$4 (dictator)
Free (Nonprincipal)	Not constructible; cofinite sets are “large”	Aggregates across all environments, can disagree with all principal comparators

A canonical toy example considers $\mathbb{R}$5 as “always skip” ($\mathbb{R}$6) and $\mathbb{R}$7 as “always press button 1,” where the reward assigned for $\mathbb{R}$8 depends on the parity of $\mathbb{R}$9. The ultrafilter $A \in \mathcal{A}$0 can be chosen to pick even or odd indices as “large,” yielding $A \in \mathcal{A}$1 or $A \in \mathcal{A}$2 respectively; no principal ultrafilter achieves this.

6. Relation to Numeric and Universal Intelligence Measures

While the ultrafilter approach is abstract and inherently noncomputable, whether analogous structural results—such as team-monotonicity and quitter-monotonicity—hold for concrete numeric universal intelligence measures like $A \in \mathcal{A}$3 (Legg–Hutter), Hernández–Orallo & Dowe, or Hibbard’s measures is unresolved. Specific open questions (Questions 1–5 in (Alexander, 2019)) inquire whether

$A \in \mathcal{A}$4

and whether quitter properties persist under restrictions to skipping-respecting and bounded environments.

7. Interpretative Remarks and Open Directions

The ultrafilter parameter $A \in \mathcal{A}$5 encapsulates choices akin to risk attitudes, discount factors, and environmental prioritizations, unifying these subjective factors into a rigorous, abstract aggregation mechanism. A plausible implication is that intelligence cannot be uniquely captured or “pinned down” by any finite battery of tests, but—conditional on the social-choice axioms of ultrafilter aggregation—a canonical, infinite comparative scheme exists. Extension to randomized agents and stochastic environments presents a natural direction for further work, and the investigation of computable or approximate analogues (e.g., Solomonoff-prior-weighted schemes) remains open. The theory demonstrates that structural properties such as total preordering, team-monotonicity, and quitter-monotonicity, not previously established for concrete intelligence metrics, hold abstractly in the ultrafilter-based setting (Alexander, 2019).