Inference Rules for Outcome Correspondence

• Inference rules for outcome correspondence are formal mechanisms that connect outcomes from different systems through logical, probabilistic, and causal mappings.
• They employ principles like reflexivity, symmetry, and transitivity to ensure safe reasoning under uncertainty in multi-equilibrium games and conditional event settings.
• These rules support efficient inference by leveraging max-closed structures and algorithmic proofs, enabling polynomial-time verification in specific models.

Inference rules for outcome correspondence arise across logic, conditional event theory, probabilistic inference, causal inference, and game theory, serving as formal mechanisms to infer relationships between the outcomes or (conditional) consequences of structures—deductive systems, games, or statistical models. These rules underpin safe reasoning about how altering a system, its premises, or its structure guarantees certain relationships between the resulting outcomes, even under informational or computational uncertainty.

1. Conceptual Foundations and Formal Definitions

The core notion of outcome correspondence is the formal relation connecting the outcomes of two systems—such as games, probabilistic experiments, or logical sequents—by means of logical, functional, or probabilistic maps. In the context of normal-form games, an outcome correspondence from $\Gamma$ to $\Gamma'$ is a set-valued map $\Phi:A\rightsquigarrow A'$, assigning to each outcome $a$ in $A$ (the outcome space of $\Gamma$) a set $\Phi(a)$ in $A'$ (of $\Gamma'$). This framework supports robust comparisons between systems without unique predictions about the outcome, as is needed in multi-equilibrium games or non-deterministic inference settings (Oesterheld et al., 26 Nov 2025).

In probabilistic logic, outcome correspondence is quantified by conditional random variables—expressing, for example, the logical relationship between a conclusion and its premises within a family of conditional events (Gilio et al., 2018).

Outcome correspondence also underlies causal and associational inference in statistical models, where models encode rules that allow (or preclude) inference about population-level outcome distributions from observed or manipulated data, especially subject to biases such as outcome-dependent sampling (Didelez et al., 2011).

2. Natural Inference Rules for Outcome Correspondence

Inference rules operate on outcome correspondences, propagating the consequences of known or hypothesized primitive relationships. In game theory and binary constraint structures, the basic propagation rules are as follows (Oesterheld et al., 26 Nov 2025):

1. Reflexivity: Every entity corresponds to itself, i.e., $\Gamma \sim_{\mathrm{id}} \Gamma$.
2. Symmetry: Correspondence can often be inverted: if $\Gamma \sim_\Phi \Gamma'$, then $\Gamma' \sim_{\Phi^{-1}} \Gamma$, with $\Phi^{-1}(a') = \{a \mid a' \in \Phi(a)\}$.
3. Transitivity (Composition): Sequential correspondences combine: if $\Gamma \sim_\Phi \Gamma'$ and $\Gamma' \sim_\Psi \Gamma''$, then $\Gamma \sim_{\Psi\circ\Phi} \Gamma''$, $$\left(\Psi\circ\Phi\right)(a) = \bigcup_{b \in \Phi(a)} \Psi(b)$$.
4. Intersection: Overlapping known correspondences allow refinement: if $\Gamma \sim_\Phi \Gamma'$ and $\Gamma \sim_\Xi \Gamma'$, then $\Gamma \sim_{\Phi\cap\Xi} \Gamma'$, $(\Phi\cap\Xi)(a) = \Phi(a) \cap \Xi(a)$.
5. Trivial (All): Any game always trivially corresponds to any other via the “all” map: $\Gamma \sim_{\mathrm{all}} \Gamma'$, where $\mathrm{all}(a) = A'$.

Primitive domain-specific correspondences include isomorphism (games played isomorphically are outcome-correspondent) and strategy elimination (removing dominated strategies induces a natural collapse map) (Oesterheld et al., 26 Nov 2025). In probabilistic logic, conjunction and disjunction, together with logical rules (e.g., De Morgan dualities), serve as the building blocks for determining when outcome inferability is preserved under logical or probabilistic extensions (Gilio et al., 2018).

3. Completeness, Max-Closedness, and Computational Complexity

The standard propagation rules for outcome correspondence are generally incomplete: for arbitrary binary constraint structures, even iterated application of these rules (including finite k-ary extensions) cannot infer every valid correspondence unless $\mathrm{P} = \mathrm{NP}$ [(Oesterheld et al., 26 Nov 2025), Montanari 1974]. This is exemplified by cases where no valid assignments exist for certain pairs, but this emptiness is undetectable by local path-consistency alone.

Completeness is restored in special settings, particularly when all primitive outcome correspondences are max-closed—that is, their value sets respect a closure condition under a total order [Jeavons et al. 1995]. In practice, many natural game-theoretic primitives (dominance elimination, isomorphism of reduced games, etc.) satisfy max-closedness, and in these cases, path-consistency suffices both for soundness and completeness. In such cases, all safe improvement queries can be decided efficiently (polynomial time).

However, in the absence of max-closedness, the problem of deciding if, under arbitrary primitive outcome correspondences, one configuration is a safe improvement over another is co–NP-complete, even when restricted to Pareto preferences (Oesterheld et al., 26 Nov 2025).

The following table summarizes key completeness and complexity properties:

Setting	Completeness of Rules	Complexity
Arbitrary OCs	Incomplete	co–NP-complete
Max-closed OCs	Complete	Polynomial time
Game-theoretic (typical)	Usually Complete	Polynomial time (if max-closed)

4. Logical and Probabilistic Inference Rules

Within the coherence-based theory of conditional events, outcome correspondence is characterized and validated by the behavior of conjunctions and higher-order logical operations. The concepts of p-consistency and p-entailment play a central role (Gilio et al., 2018):

• p-consistency: A family $\mathcal{F} = \{E_i|H_i\}$ is p-consistent if the assessment $(1,\ldots,1)$ for probabilities is coherent.
• p-entailment: $\mathcal{F}$ p-entails $E_{n+1}|H_{n+1}$ iff any coherent extension $(1,\ldots,1, p_{n+1})$ forces $p_{n+1}=1$.

The following theorem is fundamental: for a p-consistent family, p-entailment of $E_{n+1}|H_{n+1}$ holds if and only if the conjunction $C_n = \bigwedge_{i=1}^n (E_i|H_i)$ equals $C_{n+1} = C_n \wedge (E_{n+1}|H_{n+1})$. This captures outcome correspondence as the “invariance” of the joint consequence under extension by the conclusion (Gilio et al., 2018).

A catalog of classic inference rules and their status with respect to p-validity is as follows:

Rule Name	Premises	Conclusion	p-Valid?
And	$\{B|A,\;C|A\}$	$BC|A$	Yes
Cut	$\{C|AB,\;B|A\}$	$C|A$	Yes
Cautious Monotonicity	$\{C|A,\;B|A\}$	$C|AB$	Yes
Or	$\{C|A,\;C|B\}$	$C|(A\vee B)$	Yes
Boole’s example	$\{C|A,\;C|B\}$	$C|AB$	No
Transitivity	$\{C|B,\;B|A\}$	$C|A$	No

When rules are non-p-valid, two methods for repair are used: (A) adjoin a suitable premise (e.g., $A | (A \vee B)$), or (B) impose a logical constraint (e.g., $BC \subseteq A$). This ensures the conjunction with the purported conclusion remains unchanged, thereby restoring outcome correspondence.

5. Graphical and Causal Inference Frameworks

Graph-based methods encode rules for outcome correspondence in statistical and causal inference, particularly under outcome-dependent sampling (Didelez et al., 2011). Directed acyclic graphs (DAGs) or influence diagrams are augmented to include sampling indicators. d-separation or moral graph separation rules then determine when conditional independence holds, enabling the recovery of associational and causal quantities.

Key identifiable rules for outcome correspondence in these models include:

• Collapsibility of Odds Ratios: $Y \perp S | (X,C)$ or $X \perp S | (Y,C)$ implies the odds ratio $\text{OR}_{YX}(C,S=1) = \text{OR}_{YX}(C)$.
• Zero Association Testing Under Biased Sampling: Under the same separations, $Y \perp X | C \iff Y \perp X | (C, S=1)$.
• Causal Identifiability: When covariates $C$ satisfy the causal back-door criterion and sampling dependencies are appropriately blocked, $\text{COR}_{YX}(C) = \text{OR}_{YX}(C, S=1)$.

Inference proceeds by encoding the problem in a DAG, reading separation properties, and validating that the desired property (e.g., test of no effect, collapsibility of a statistical measure) is preserved in the presence of outcome-dependent sampling (Didelez et al., 2011).

6. Algorithmic and Proof-Theoretic Correspondence

Proof-theoretic approaches, as in modal logic, generate outcome correspondence rules algorithmically from axioms using algorithms such as MASSA (Domenico et al., 2022). MASSA constructs analytic geometric rules corresponding to first-order frame conditions, generating cut-free derivations in labelled sequent calculi (G3K). These analytic rules are guaranteed to be admissible—cut elimination holds—whenever the input formula falls into the class of definite analytic-inductive formulas.

The process involves:

• Transformation from modal axiom to negative normal form;
• Identification of the underlying frame correspondent (first-order property);
• Construction of the corresponding geometric rule and cut-free derivation in G3K.

This provides a systematic, algorithmic synthesis of inference rules guaranteeing outcome correspondence between the validity of the modal axiom and the first-order property in the model semantics (Domenico et al., 2022).

7. Illustrative Examples and Applications

Game-theoretic Example: In the “Three Chickens” games, elimination of strictly dominated strategies and isomorphism axioms suffice (via local rules) to prove that one game is a strict safe Pareto improvement of another (Oesterheld et al., 26 Nov 2025).

Probabilistic Logic Example: The conjunction of three conditional events yields a region of coherent assessments bounded by Fréchet–Hoeffding inequalities. Reverse inference (back-propagation) computes all extensions to individual conditional previsions consistent with a given joint conjunction (Gilio et al., 2018).

Causal Inference Example: In a classical case–control DAG, the necessary d-separation holds to permit consistent estimation of causal and associational odds ratios via logistic regression over sampled data, conditional on appropriate covariates (Didelez et al., 2011).

These exemplify the breadth and practical impact of inference rules for outcome correspondence, which arise at the intersection of logic, statistics, game theory, and computational reasoning.

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References:

• "Choosing What Game to Play without Selecting Equilibria: Inferring Safe (Pareto) Improvements in Binary Constraint Structures" (Oesterheld et al., 26 Nov 2025)
• "Generalized Logical Operations among Conditional Events" (Gilio et al., 2018)
• "Graphical Models for Inference Under Outcome-Dependent Sampling" (Didelez et al., 2011)
• "Algorithmic correspondence and analytic rules" (Domenico et al., 2022)