--- title: Protocol-Based Empirical Choice Functions url: https://www.emergentmind.com/topics/protocol-based-empirical-choice-functions type: topic --- # Protocol-Based Empirical Choice Functions Searching arXiv for recent and relevant papers on protocol-based and empirical choice functions. Protocol-Based Empirical Choice Functions designate a family of empirical and decision-theoretic constructions in which the primitive input is not a fully specified utility function on a known state space, but an observed **protocol**: menu choices, budget choices, smart-card route realizations, search-tree pruning rules, or finite families of act–consequence pairs. The common objective is to map such protocols into a choice function, choice correspondence, or empirically disciplined dominance relation, typically under explicit structural restrictions such as utility maximization, coherence, Pareto rationalization, path independence, or revealed preference [2512.05677]. The label is used explicitly in “Empirical Decision Theory” [2512.05677], and several earlier works are naturally interpreted in the same vein: binary demand under varying price–income protocols [1902.11012], coherent extension of partial menu choices [2407.21164], empirical route-choice set construction from smart-card protocols [2503.17370], and safe action-pruning rules in online search [1910.00614]. ## 1. Conceptual scope In the most explicit formulation, an empirical decision problem is a pair \((\mathbb A,\mathscr P_{\mathbf z})\), where \(\mathbb A=\{a_1,\dots,a_n\}\) is a context-consistent finite set of action descriptions and \(\mathscr P_{\mathbf z}\) is a \(\mathbf z\)-protocol: a finite family of act–consequence pairs \(((q_j,p_j))_{j\in\underline N}\) such that each action description \(a_i\) appears exactly \(z_i\) times [2512.05677]. This formulation replaces the classical primitive \(X:S\to C\) by observable records of what happened when an action description was used. The paper’s stated objective is to address optimality “in a radically empirical way” and to derive inferential guarantees through “(I) consistent statistical estimation of choice sets, (II) consistent statistical testing of choice functions with robustness guarantees, and (III) direct inference for empirical choice functions using credal sets” [2512.05677]. The same conceptual move appears, in narrower forms, across several adjacent literatures. In binary demand, the operative protocol is the budget environment \((p,y)\), equivalently the pair of leftovers \((y,y-p)\), and the empirical object is the structural choice probability \(q(p,y)\) or \(q(y,y-p)\) [1902.11012]. In coherent choice-function theory, the primitive data are observed menus together with chosen and rejected options, encoded as an assessment \(\mathcal A\subseteq Q\times\mathcal Q\) [2407.21164]. In public transport route choice, the protocol is a stream of tap-in, transfer, and tap-out events from smart-card data, from which distinct observed paths are aggregated into empirical route sets [2503.17370]. In online policy improvement, a choice function \(\psi:\mathcal{SP}\to 2^A\) specifies which actions are admissible at each search-tree node, so the protocol is the pruning rule itself rather than an ex post dataset [1910.00614]. A unifying feature is that the output is usually **set-valued**. The aim is not always to infer a unique optimal item. Depending on the framework, the output can be a set of non-rejected menu options, a set of undominated bundles, a set of feasible routes attached to an origin–destination context, or a set of search actions retained by a pruning protocol [2407.21164]. This suggests that protocol-based empirical choice functions are best understood as a general family of methods for converting observed or imposed decision protocols into disciplined, usually incomplete, empirical choice correspondences. ## 2. Core formal objects A recurring primitive is the menu-based choice function. In the coherence-based literature, menus are finite nonempty subsets \(A\) of an option space \(V\), and a choice function is a map \(C:Q\to Q\) such that \(C(A)\subseteq A\) for all \(A\in Q\) [2407.21164]. The associated rejection function is \(R_C(A)=A\setminus C(A)\), and the intended semantics are that \(u\in C(A)\) means no option in \(A\) is known to be strictly preferred to \(u\), whereas \(u\in R_C(A)\) means some option in \(A\) is strictly preferred to it [2407.21164]. The 2020 practical inference paper adopts the same formal architecture, with menus \(Q=\{A\subseteq V:|A|<\infty\}\cup\{\emptyset\}\), rejection \(R_C(A)=A\setminus C(A)\), and coherence axioms \(\mathrm{C}_0\)–\(\mathrm{C}_4\) [2005.03098]. In stochastic environments, the basic object is a stochastic choice function \(\mathbb P:X\times\mathfrak X\to[0,1]\), together with the threshold-induced deterministic correspondence \[ C_{\mathbb P,\lambda}(S)=\{x\in S:\mathbb P^\ast(x,S)\ge \lambda\}, \qquad \mathbb P^\ast(x,S)=\frac{\mathbb P(x,S)}{\max_{\omega\in S}\mathbb P(\omega,S)}. \] This “Fishburn family” converts repeated stochastic choice into a one-parameter family of deterministic empirical choice filters [2303.08202]. Rationality is then assessed not directly on \(\mathbb P\), but on the deterministic correspondences \(C_{\mathbb P,\lambda}\) via Chernoff, Condorcet, and No-Cycle conditions [2303.08202]. Binary-demand analysis works with a different but closely related empirical object. An individual chooses \(Q\in\{0,1\}\) under budget constraint \[ N+pQ=y, \] and heterogeneity induces a structural population choice probability \[ q(p,y)=q(y,y-p). \] Rationalizability means the existence of utility functions \(W_0(\cdot,\eta)\), \(W_1(\cdot,\eta)\) and a distribution \(G\) such that \[ q(y,y-p)=\int 1\{W_0(y,\eta)\le W_1(y-p,\eta)\}\,dG(\eta), \] with monotonicity in leftover numeraire at the individual level [1902.11012]. Here the empirical choice function is the full conditional probability surface \((p,y)\mapsto q(p,y)\). A further variant is the latent Pareto-rationalized choice function used in Gaussian-process learning and multi-objective Bayesian optimization. There, observed data are menu-choice pairs \((A_k,C(A_k))\), and the model assumes a latent vector function \({\bf f}({\bf x})=[f_1({\bf x}),\dots,f_{n_e}({\bf x})]^\top\) such that the observed choice set equals the Pareto set of nondominated menu items [2110.08217]. The related Gaussian-process learning paper adopts the same multiple-utility representation, writing a choice function \(C:Q\to Q\) with \(C(A)\subseteq A\) and interpreting the selected subset as the set of strongly Pareto-undominated options [2302.00406]. ## 3. Major protocol types in the literature The literature grouped under this label spans several distinct protocol classes. | Protocol | Empirical object | Representative source | |---|---|---| | Budget protocol \((p,y)\) | Binary choice probability \(q(p,y)\) | [1902.11012] | | Menu-choice assessment \((V,W)\) | Natural extension \(C_{\mathcal A}\) | [2407.21164] | | Smart-card trip protocol | Empirical route set \(C_T^E(o,d)\) | [2503.17370] | | Search-tree pruning protocol | Action choice function \(\psi(p;s)\) | [1910.00614] | | Act–consequence protocol \(\mathscr P_{\mathbf z}\) | Empirical choice sets and tests | [2512.05677] | The budget protocol of binary demand is unusually clean because the protocol variables are exactly the economically relevant leftovers under the two alternatives. The resulting restrictions are “global,” “closed-form,” and invariant to the number or configuration of observed budgets [1902.11012]. In assessment-based choice theory, by contrast, the protocol is finite and menu-specific: one observes chosen subsets and rejected alternatives, encodes each observed rejection as a difference set, and then computes the natural extension consistent with coherence [2407.21164]. Smart-card route choice supplies an observational protocol rather than an elicitation protocol. In the Danish Rejsekort system, passengers must “tap-in at the start of the journey,” “tap-in and tap-out at each transfer,” and “tap-out at the end,” so observed trips can be reconstructed into route alternatives for each stop-to-stop origin–destination pair [2503.17370]. Search-based reinforcement learning occupies another extreme: the protocol is the admissible-action rule at each search-tree node, formalized by a choice function \(\psi\), and the main question is when that protocol guarantees online policy improvement relative to a base policy \(\pi\) [1910.00614]. Taken together, these cases show that “protocol” may denote either an observation design, an elicitation design, an institutional choice architecture, or a computational pruning rule. The common element is that the empirical choice mapping is defined only relative to that protocol. ## 4. Main inferential frameworks One major framework derives **closed-form restrictions** directly from the protocol. In binary choice, rationalizability of \(q(p,y)\) under completely general unobserved heterogeneity is equivalent to monotonicity in \(a_0=y\) and \(a_1=y-p\), and under differentiability to the pair of Slutsky-like inequalities \[ q_p(p,y)\le 0, \qquad -q_p(p,y)+q_y(p,y)\le 0. \] These restrictions are both necessary and sufficient, are global rather than dataset-specific, and yield sharp counterfactual bounds and welfare bounds [1902.11012]. A second framework computes the **natural extension** of partial menu data. For a choice assessment \(\mathcal A\subseteq Q\times\mathcal Q\), consistency means that there exists at least one coherent choice function extending the assessment, and when consistency obtains, the natural extension is \[ C_{\mathcal A}:=C_{O(\mathcal A)}, \] where \(O(\mathcal A)\) is the set of preference orders compatible with the assessment [2407.21164]. The operational criterion is that \(u\in C^{\mathcal G}(A)\) iff there exists a generator set \(G\in\mathcal G\) such that \((A-u)\cap N(G)=\emptyset\), with \(N(G)=\operatorname{posi}(G\cup V_{>0})\) [2407.21164]. The earlier practical paper gives the same conservative logic at the level of coherence axioms and sets of desirable option sets, emphasizing that the method infers exactly those rejections forced by coherence and the observed protocol [2005.03098]. A third framework learns a **latent context-dependent or multi-utility representation** from empirical menus. “Learning Context-Dependent Choice Functions” models a choice function \(c:\mathcal Q\to\mathcal C\) through latent context-dependent utility \(U(\mathbf x,Q)\), with deterministic singleton choice \[ c_U(Q)=\arg\max_{\mathbf x\in Q} U(\mathbf x,Q) \] and deterministic subset choice \[ c_{U,t}(Q)=\{\mathbf x\in Q:U(\mathbf x,Q)\ge t\}. \] Its two main representations are FETA, \[ U(\mathbf x,Q)=U_0(\mathbf x)+\frac{1}{|Q|-1}\sum_{\mathbf y\in Q\setminus\{\mathbf x\}} U_1(\mathbf x,\{\mathbf y\}), \] and FATE, \[ U(\mathbf x,Q)=U'\!\left(\mathbf x,\frac{1}{|Q|}\sum_{\mathbf y\in Q}\phi(\mathbf y)\right), \] which provide permutation invariance and support variable menu size [1901.10860]. The Gaussian-process alternatives replace deterministic networks by GP priors on latent utilities and derive likelihoods for menu/subset observations under Pareto rationalization [2302.00406, 2110.08217]. A fourth framework evaluates stochastic protocols by **thresholded deterministic approximations**. For a stochastic choice function \(\mathbb P\), the Fishburn family \(C_{\mathbb P,\lambda}\) induces a partial ordering \(\trianglerighteq_{\mathrm{rat}}\) over stochastic choice functions, and the parameter-free characterization is \[ \mathbb P\trianglerighteq_{\mathrm{rat}}\mathbb Q \iff \Lambda(\mathbb P)\subseteq \Lambda(\mathbb Q), \] where \(\Lambda(\mathbb P)\) is the set of threshold values at which \(C_{\mathbb P,\lambda}\) fails rationality [2303.08202]. This directly measures how robust revealed-preference coherence is to the thresholding convention used to extract deterministic empirical choice correspondences. ## 5. Structural restrictions, coherence, and procedural composition A large part of the literature studies which structural restrictions on a protocol preserve rationality-like properties. In the coherence approach of De Bock and De Cooman, a coherent choice function satisfies \(\mathrm{C}_0\)–\(\mathrm{C}_4\): nonemptiness, translation invariance, acceptance of uniformly positive options via \(K_C\), positive combination closure, and monotone rejection under menu expansion [2005.03098]. The 2024 natural-extension paper reformulates coherence through sets of preference orders and coherent desirable-option sets, using the bijection \(u\prec_G v \iff v-u\in G\) between preference orders and coherent cones of desirable differences [2407.21164]. Procedural composition introduces another layer. “Lexicographic Composition of Choice Functions” studies \[ \mathcal L_E(C_1,C_2)(Y)=C_1(Y)\cup C_2(Y\setminus E(C_1(Y))), \] where \(E\) is an exclusion rule that maps prior selections into later-stage restrictions [2209.09293]. Its main characterization is that \(\mathcal L_E\) preserves path independence over responsive choice functions if and only if \(E\) is **threshold-linear with cardinal reuse**. In that form, there exist \(t\in\mathbb N\cup\{0,\infty\}\), \(K\subseteq X\), and nested sets \(T^0\subseteq T^1\subseteq\cdots\subseteq T^t\subseteq X\setminus K\) such that for \(|Z|