Accept-Desirability Models: Foundations & Applications

• Accept-Desirability Models are a formal framework that axiomatizes decision-making by encoding acceptance and rejection in convex cone structures.
• They unify classical preference theories, choice correspondences, risk measures, and even quantum updating through coherent choice functions and explicit axiomatizations.
• AD models enable robust individual and collective decision-making, belief revision, and multi-objective optimization via scalable computational methodologies.

Accept-Desirability models provide a rigorous axiomatization for modeling attitudes toward choice, uncertainty, belief revision, and collective decision using primitive behavioral concepts of “accepting,” “desiring,” or “finding acceptable” certain options, sets, or procedures. These models unify and generalize classical preference, choice correspondence, imprecise probability, risk measures, and even quantum updating by encoding accept/reject judgments in abstract convex, conic, or closure-operator–governed structures. Core developments include the linkage between accept-desirability assessments and coherent choice functions, explicit axiom systems characterizing such models, powerful representation theorems, compatibility with AGM-style belief revision (including under conditioning), and nontrivial applications in both individual and social decision theory.

1. Formal Foundations and Axiomatization

Accept-desirability (AD) models are defined on a real vector space $V$ (the “option” or “gamble” space), typically equipped with a background cone $V_\mathrm{acc}$ (accepted options). For any subject, an assessment is a pair $A = \langle A_\mathrm{acc}, A_\mathrm{rej} \rangle$ with $A_\mathrm{acc}, A_\mathrm{rej} \subseteq V$—the sets of accepted and rejected options. Derived sets are:

• Desirable options: $$A_\mathrm{des} = A_\mathrm{acc} \cap (-A_\mathrm{rej})$$
• Indifferent options: $$A_\mathrm{indif} = A_\mathrm{acc} \cap (-A_\mathrm{acc})$$

A model $$M = \langle M_\mathrm{acc}, -M_\mathrm{des} \rangle$$ is an AD model if it satisfies the following axioms (Coussement et al., 22 Dec 2025, Coussement et al., 10 Feb 2025):

• AD1 (Background consistency): $V_\mathrm{acc} \subseteq M_\mathrm{acc}$, $V_\mathrm{rej} \subseteq M_\mathrm{rej}$
• AD2 (Strictness): $0 \notin M_\mathrm{des}$
• AD3 (Deductive closedness): $M_\mathrm{acc}$ and $M_\mathrm{des}$ are convex cones
• AD4 (Sweetened deals): $$M_\mathrm{acc} + M_\mathrm{des} \subseteq M_\mathrm{des}$$

These axioms ensure that accepted and desirable sets are closed under scaling and addition, and prevent contradictory judgments. In the purely desirability-based interpretation, the model is captured as a convex cone $D \subseteq V$ excluding $0$ and including all strictly positive elements (Bock et al., 2019, Zaffalon et al., 2015, Quaeghebeur et al., 2012):

• $0 \notin D$
• all $f > 0$ belong to $D$
• $D$ is a convex cone (positive scaling and additivity)

Further properties—such as Archimedeanity and mixing—yield finer characterizations (see Section 4).

2. Choice Functions and Behavioral Representation

AD models induce choice functions $C_D$ mapping finite option sets $A \subseteq V$ to subsets $C_D(A) \subseteq A$:

$$C_D(A) = \{ f \in A : \forall g \in A,\ (g - f) \notin D \}$$

That is, an option $f$ is chosen if there is no strictly desirable gain in moving from $f$ to another $g$ in $A$. The strict preference relation $f \succ g$ is defined as $f - g \in D$. The representation theorem states that, for coherent $D$, $C_D$ can be written as a union of argmax sets over all strict partial orders extending the structure imposed by $D$ (Bock et al., 2019, Bock et al., 2018). If additional axioms (totality, mixing, or Archimedeanity) are imposed, $C_D$ reduces to selection by a strict total order, lexicographic system, or expected utility, respectively.

The framework generalizes classical choice correspondences, set-valued choice inclusive of imprecise preferences, and connects naturally to economic models such as rational shortlist methods and non-binary choice (0907.5469, Bock et al., 2018).

3. Conditioning, Belief Revision, and Dynamic Aspects

Conditioning in AD models is formalized using linear projections $E^*: V \rightarrow V$ representing “events.” Observing $E$ introduces new indifferences: any two options whose projections $E^*x$ coincide become functionally equivalent post-conditioning. The conditional AD-model is given by (Coussement et al., 22 Dec 2025, Coussement et al., 10 Feb 2025):

$$M|E = \langle M_\mathrm{acc}|E,\, -\left(M_\mathrm{des}|E \cup (V_\mathrm{des} + M_\mathrm{acc}|E)\right) \rangle$$

where $$M_\mathrm{acc}|E = \{x \in V : E^*x \in M_\mathrm{acc}\}$$, $$M_\mathrm{des}|E = \{x \in V : E^*x \in M_\mathrm{des}\}$$.

Belief revision is framed in the style of AGM theory. The revision operator $\mathrm{Revise}(M, E)$ applies conditionalization if possible, else falls back to closure under the event’s kernel. The revision operator satisfies all but two AGM postulates in general; full compliance holds for propositional and precise probabilistic (linear prevision) specializations (Coussement et al., 22 Dec 2025, Coussement et al., 10 Feb 2025).

This dynamic extends to the quantum context by taking $V$ as the space of Hermitian operators and $E^*$ as quantum projectors, yielding updates analogous to quantum measurement (Lüders’ rule) (Coussement et al., 10 Feb 2025).

4. Specialized Structures: Totality, Mixing, and Nonlinearity

The theoretical power of AD models comes from the ability to tailor behavior by imposing further conditions on the desirability cone:

Additional Axiom	Representation Theorem	Behavioral Regime
Totality (D5): For $f \neq 0$, either $f \in D$ or $-f \in D$	$D$ is governed by a single strict total order; $C_D(A) = \argmax_\succ(A)$	Classical deterministic choice
Mixing (D6): $posi(A) \cap D \neq \emptyset$ implies $A \cap D \neq \emptyset$	$D$ is a lexicographic cone; admits lexicographic probability system $(P_1,...,P_k)$	Lexicographic choice
Archimedeanity (D4): For all $f \in D$, $\exists \epsilon > 0 : f-\epsilon \in D$	$D$ is represented by a unique coherent lower prevision; $D = \{f : \underline{P}(f)>0\}$	Coherent expectations (precise or imprecise)

Beyond positive cones, generalized closure operators permit the encoding of nonlinear rationality patterns: any operator $K$ satisfying extensiveness, monotonicity, idempotence, and closure under pointwise dominance generalizes “linear scaling” to more general reward aggregations (Miranda et al., 2022, Bock, 2023). This allows, for instance, nonlinear attitudes as in Allais-type paradoxes, and direct modeling of nonlinear “currencies.”

5. Aggregative and Social Applications

In collective decision and mechanism design, accept-desirability models describe agent approval in terms of acceptable procedures (rules), outcomes, and procedures-for-outcomes (Abramowitz et al., 2022). Agents are characterized by the sets of rules and outcomes they accept, with possible conjunctive/disjunctive or implementation-indifferent (II) rationalities. Acceptance-maximizing mechanisms (e.g., rule selection for amendments, dichotomous choice) systematically maximize the number of agents finding decisions acceptable (acceptance-score $A(d)$). Corresponding optimality and worst-case acceptance rates are characterized for heterogeneous and homogeneous populations.

The same formalism underlies Feasibility/Desirability Games (FD Games), where strategic or evolutionary outcomes are characterized by feasible and desirable transformations; equilibria correspond to sinks or strongly connected components (SCCs) of the accept-desirability graph (0907.5469).

6. Computational and Applied Aspects

Accept-desirability models provide algorithms for conservative inference: given partial assessments (e.g. sets of desirable option-sets), extend conservatively to the least-committal coherent model via closure operators, conic hulls, and limbo resolution (Bock et al., 2018, Quaeghebeur et al., 2012, Bock, 2023). This applies to credal sets, lower previsions, and choice correspondences.

In applied optimization, desirability functions transform individual objectives to $[0,1]$ scales, aggregate via geometric or arithmetic means, and serve as scalarized black-box objectives in multi-objective optimization and hyperparameter tuning (Bartz-Beielstein, 30 Mar 2025). The approach is directly connected to AD models, interpreting the $[0,1]$ “desirability” values as accept-desirability scores for options/outcomes.

7. Unification and Generalization

Accept-desirability models unify probabilistic, decision-theoretic, and social-choice paradigms by representing attitudes to options—gambles, acts, procedures—via primitive accept/reject or desirability structures satisfying closure and coherence principles. The theory subsumes classical expected utility, lower previsions, imprecise probability, and risk measures; it delivers robust belief revision via event conditioning; it admits generalizations to nonlinear, quantum, or multidimensional “currencies”; and it scales to collective and strategic settings through aggregated accept-reject preferences (Bock et al., 2019, Zaffalon et al., 2015, Abramowitz et al., 2022, Quaeghebeur et al., 2012, Bock, 2023). The closure-operator formulation yields a flexible logical foundation for further extensions and clarification of independence, state-independence, and minimality.

A plausible implication is that accept-desirability models can be applied as foundational structures for rational deliberation in artificial intelligence, robust statistics, and social choice, providing a universal language for uncertainty and approval that interfaces naturally with belief revision, learning, and optimization.