A new paper on separable vs entangled stochastic choice

[Victor Aguiar]

Dear all, 

My coauthors and I just finished this new short paper 

https://www.researchgate.net/publication/378938384_Entangled_vs_Separable_Choice

Abstract:

We study joint probabilistic choice rules that describe the behavior of two decision-makers, each facing a possibly different menu. These choice rules are separable when they can be factored into autonomous choices from each individual solely correlated through their individual probabilistic choice rules. Despite recent interest in studying such rules, a complete characterization of the restrictions on them remains an open question. A reasonable conjecture is that such restrictions on separable joint choice can be factored into individual choice restrictions. We name these restrictions separable and show that this conjecture is true if and only if the probabilistic choice rule of at least one decision maker uniquely identifies the distribution over deterministic choice rules. Otherwise, entangled choice rules exist that satisfy separable restrictions yet are not separable. The possibility of entangled choice complicates the characterization of separable choice since one needs to augment the separable restrictions with the new emerging ones.

p.s. We use Bell Inequalities used to detect entanglement in quantum physics to be able to characterize our separable model. 

Best

V

[Victor Aguiar]

Dear all, 

We have updated our paper with a full characterization of the separable model. 

Abstract:

We investigate joint probabilistic choice rules describing the behavior of two decision makers, each facing potentially distinct menus. These rules are separable when they can be decomposed into individual choices correlated solely through their respective probabilistic choice rules. Despite its significant interest for the study of peer effects, influence, and taste variation, a complete characterization of these rules has remained elusive (Chambers, Masatlioglu, and Turansick, 2021). We fully characterize separable choices through a finite system of inequalities inspired by Afriat's theorem. Our results address the possibility of entangled choices, where decision makers behave as if they do not communicate, yet their choices are not separable. More generally, we establish that separable joint choice restrictions can be factored into individual choice restrictions if only if at least one decision maker's probabilistic choice rule uniquely identifies the distribution over deterministic choice rules. The no communication condition and the individual restrictions are no longer sufficient in the absence of this uniqueness. Our results offer robust tools for distinguishing between separable decision-making and behaviors beyond mere peer effects such as imitation and cheating. 

https://www.researchgate.net/publication/378938384_Entangled_vs_Separable_Choice

Best

V

[Spiegler, Ran]

Dear colleagues,

How do competitive markets function when consumers aren't all great at inferring latent features from market prices?

Well, here's a new paper, joint with Yair Antler, which tackles this question:

https://www.ranspiegler.sites.tau.ac.il/_files/ugd/4871e3_6e63ed0298994c64a2d3d8896a228b41.pdf

Abstract: We develop a market model in which products generate state-dependent potential hidden charges. Firms differ in their ability to realize this potential. Unlike firms, consumers do not observe the state. They try to infer hidden charges from market prices, using idiosyncratic subjective models. We show that an interior competitive equilibrium is uniquely given by what is formally a Bellman equation. We leverage this representation to characterize equilibrium headline prices, add-on charges and welfare. Market responses to shocks display patterns that are impossible under rational expectations. For example, even though equilibrium prices are fully revealing, they can also vary with consumers' private information.

Thank you,

Rani

Ran Spiegler

School of Economics, Tel Aviv University

&

Department of Economics, University College London

URL: https://www.ranspiegler.sites.tau.ac.il/ 

New book: The Curious Culture of Economic Theory (MIT Press)

X (Twitter) book page: @rani_spiegler

[Victor Aguiar]

Dear all,

A brief update to our 2024 thread: Nail Kashaev, Martin Plávala, and I
have posted a substantially revised version of Entangled vs. Separable
Choice.

What if we could detect when two agents' joint behavior cannot be
explained by social independence, without treating correlation as
coordination or imposing strong assumptions on behavior?

In our benchmark, agents choose as though they were in separate rooms
and unable to communicate at the time of choice. Their actions may
nevertheless be arbitrarily correlated through shared tastes, beliefs,
information, environments, incentives, or randomization devices.
Separability is therefore a revealed-preference notion of social
independence, not statistical independence.

AGENTIC SOCIAL INDEPENDENCE

Now that we live in the singularity, pending confirmation, many
artificial agents will interact in marketplaces. But how can we know
whether they are genuinely acting independently?

We construct a white-hat stress test showing how a designer could
engineer what we call entangled collusion. The resulting joint rule
passes no-signaling and Duggan-Levitt marginal screens while violating
separability. Qwen3-14B is used only for local state inference from
short histories; stationary choices are produced by a deliberately
hand-coded local automaton that implements the protocol. The LLM does
not autonomously discover collusion. The point is that local inference
can support coordinated joint behavior that is invisible in the
marginals but detected by the CHSH test.

EMPIRICAL ILLUSTRATION

Since fútbol is still on our minds after the World Cup, we also apply
the method to UEFA European Championship final-round group matches
from 1980 through 2024. Under an explicit binary-effort-to-outcome
map, the 53-match illustration rejects separability (p = 0.015).

This is evidence against separability under the model, not proof that
teams cheated and not an attribution of intent. Possible explanations
include strategic adaptation to common information, tacit
coordination, omitted state-dependent composition, or failure of the
simplified effort-outcome map.

GENERAL SOLUTION

In the binary-covariate, binary-action case, separability is
equivalent to no-signaling plus Bell-type CHSH inequalities. In
general finite environments, the characterization uses finitely many
virtual replicas and yields a finite system of linear restrictions.
The test requires no utility maximization, equilibrium model,
parametric specification, or conventional rationality assumption.

So yes: decision theory, Bell inequalities, artificial agents, and
fútbol have somehow ended up in the same paper.

Paper:
https://www.researchgate.net/publication/378938384_Entangled_vs_Separable_Choice

A less technical discussion:
https://lnkd.in/eznz_RQA

This paper has been in development since before 2024. It is difficult
and deeply technical, but it speaks to an increasingly important
problem: security and governance in decentralized agentic
marketplaces. What makes me happy about academic research is searching
for the most general and portable abstraction, then writing it as a
letter to the future.

Comments are very welcome.

Best,
Victor