State–Action Rational Inattention

• State–action rational inattention is a bounded-rationality framework that models decision-making as stochastic channels constrained by mutual information.
• It employs Bayesian predictive approaches and Gibbs–Boltzmann policies to optimally trade off utility gains against information costs.
• The model applies to a range of domains, from discrete choice and control problems to dynamic principal-agent and retirement policy applications.

State–action rational inattention is a family of bounded-rationality models in which the decision object is a stochastic channel from states to actions, and the channel is constrained or penalized by the mutual information between the two. In the finite-alphabet formulation, the primitives are a state space $S=\{s\}$, an action space $A=\{a\}$, a prior $p(s)$, and a utility function $u:S\times A\to\mathbb R$; the agent selects $\pi(a\mid s)$, with induced joint law $P(s,a)=p(s)\pi(a\mid s)$ and action marginal $q(a)=\sum_s p(s)\pi(a\mid s)$. Recent work places this familiar construction inside a Bayesian predictive framework and shows that the classical rational-inattention family is recovered as a special case (Polson et al., 25 Dec 2025).

1. Primitive objects and state–action representation

The canonical state–action model specifies a payoff-relevant state $s\in S$, an action $a\in A$, and a randomized policy $\pi(a\mid s)$. The policy is simultaneously an action rule and an information structure: once the state is realized, the agent does not necessarily transmit it perfectly into action, but instead chooses according to a state-contingent distribution. The object of analysis is therefore not only which action is optimal, but how much state dependence the channel $A=\{a\}$0 should exhibit.

A central feature of this representation is that explicit signals can be bypassed. In the Bayesian predictive formulation, the same channel can be interpreted as predictive reporting: upon observing $A=\{a\}$1, the agent draws a report $A=\{a\}$2, and the report is identified with the predictive distribution

$A=\{a\}$3

under the joint law $A=\{a\}$4. This equivalence ties state–action rational inattention to proper scoring, posterior refinement, and rate–distortion theory rather than treating it as an isolated discrete-choice device (Polson et al., 25 Dec 2025).

The same state–action architecture also appears in formulations where the “state” is not an exogenous environment variable but an internal support representation. In the single-cycle rate–regret formulation, the source variable is a full-support state $A=\{a\}$5, the action set is a finite $A=\{a\}$6, and the consequence geometry $A=\{a\}$7 determines the expected utility

$A=\{a\}$8

This shifts attention from representing the world faithfully to preserving only those distinctions that are action-relevant under the current payoff structure (Walsh, 28 May 2026).

2. Information constraints, utility trade-offs, and regret formulations

In the classical finite state–action formulation, one may either constrain mutual information or price it. The capacity-constrained problem is

$A=\{a\}$9

and the Lagrangian form is

$p(s)$0

The two formulations are equivalent via convex duality. Here

$p(s)$1

The parameter $p(s)$2 is the information-price, i.e. the shadow price of the capacity constraint (Polson et al., 25 Dec 2025).

A related but distinct formulation replaces raw utility loss by regret distortion. Let the full-information optimal policy be

$p(s)$3

with ties broken by a fixed convention. The regret distortion is then

$p(s)$4

For a stochastic action channel $p(s)$5, the expected regret is

$p(s)$6

The rate–regret problem minimizes information subject to a regret budget,

$p(s)$7

or, equivalently,

$p(s)$8

This formulation distinguishes action adequacy from reconstruction fidelity, information-bottleneck prediction, and standard rational inattention (Walsh, 28 May 2026).

In controlled Markov processes, the same principle appears as a steady-state information constraint on stationary randomized policies. With state space $p(s)$9, action space $u:S\times A\to\mathbb R$0, transition kernel $u:S\times A\to\mathbb R$1, one-stage cost $u:S\times A\to\mathbb R$2, invariant distribution $u:S\times A\to\mathbb R$3, and policy $u:S\times A\to\mathbb R$4, the average-cost problem becomes

$u:S\times A\to\mathbb R$5

where

$u:S\times A\to\mathbb R$6

This yields a convex program over the joint measure $u:S\times A\to\mathbb R$7 (Shafieepoorfard et al., 2015).

3. Gibbs–Boltzmann policies and soft-max structure

A standard variational argument yields the fixed-point system for the information-penalized optimum: $u:S\times A\to\mathbb R$8 Equivalently,

$u:S\times A\to\mathbb R$9

Under mild regularity—finite utilities and finite $\pi(a\mid s)$0—for each $\pi(a\mid s)$1 there is a unique maximizer $\pi(a\mid s)$2 of $\pi(a\mid s)$3, and it satisfies this Gibbs–Boltzmann form (Polson et al., 25 Dec 2025).

If $\pi(a\mid s)$4 is taken uniform, or interpreted as a prior over reports, the rule reduces to the familiar softmax or multinomial logit form,

$\pi(a\mid s)$5

With $\pi(a\mid s)$6, often called the temperature or information-capacity parameter,

$\pi(a\mid s)$7

As $\pi(a\mid s)$8 $\pi(a\mid s)$9, choices become deterministic; as $P(s,a)=p(s)\pi(a\mid s)$0 $P(s,a)=p(s)\pi(a\mid s)$1, the channel approaches uninformative uniform randomization (Polson et al., 25 Dec 2025).

The regret-based state–action model has the same rate–distortion Gibbs form, but with regret in place of utility: $P(s,a)=p(s)\pi(a\mid s)$2 Here the limiting behavior is reversed in sign because $P(s,a)=p(s)\pi(a\mid s)$3 prices regret rather than rewarding utility. As $P(s,a)=p(s)\pi(a\mid s)$4, the channel collapses toward the zero-rate limit, namely a deterministic default action $P(s,a)=p(s)\pi(a\mid s)$5. As $P(s,a)=p(s)\pi(a\mid s)$6, suboptimal actions become exponentially unlikely and the channel converges to the full-information policy $P(s,a)=p(s)\pi(a\mid s)$7 (Walsh, 28 May 2026).

Dynamic state–action models preserve the same structure. In the retirement model with hidden pension-policy state, the first-order conditions produce a logit-form action rule in which the exponent contains current-period utility plus discounted continuation value, divided by the attention-cost parameter $P(s,a)=p(s)\pi(a\mid s)$8. The solution is a fixed point in the unconditional action distribution $P(s,a)=p(s)\pi(a\mid s)$9 and the conditional policy $q(a)=\sum_s p(s)\pi(a\mid s)$0 (Hentall-MacCuish, 2019).

4. Predictive foundation, Shannon uniqueness, and information geometry

A common misconception is that Shannon mutual information enters state–action rational inattention only as an imposed processing cost. In the Bayesian predictive approach, the agent is instead evaluated by a strictly proper local scoring rule after reporting a predictive distribution. By Bernardo’s theorem, together with Dawid’s formulation, a scoring rule that is local and strictly proper must be the log-score,

$q(a)=\sum_s p(s)\pi(a\mid s)$1

Under this rule, the expected gain from moving from the prior $q(a)=\sum_s p(s)\pi(a\mid s)$2 to the posterior $q(a)=\sum_s p(s)\pi(a\mid s)$3 is exactly

$q(a)=\sum_s p(s)\pi(a\mid s)$4

Information costs therefore need not be assumed: they arise as expected predictive utility (Polson et al., 25 Dec 2025).

The uniqueness claim is stronger than a generic appeal to entropy. Bernardo’s characterization of proper local scoring rules, combined with Shannon’s branching or additivity axiom and amalgamation invariance, implies that the logarithmic score is the unique refinement-coherent way to reward probabilistic reports, and mutual information is the unique measure of expected gain from predictive refinement. This gives Shannon entropy an endogenous status within the model rather than a purely axiomatic or reduced-form one (Polson et al., 25 Dec 2025).

Within the same framework, several familiar models appear as geometric specializations. Discrete choice yields multinomial logit and Luce’s IIA under entropic regularization. In Gaussian learning, the information-penalized rule reproduces James–Stein shrinkage as optimal capacity allocation. In Gaussian linear-quadratic control with

$q(a)=\sum_s p(s)\pi(a\mid s)$5

the optimal channel is linear–Gaussian,

$q(a)=\sum_s p(s)\pi(a\mid s)$6

with

$q(a)=\sum_s p(s)\pi(a\mid s)$7

and mutual information

$q(a)=\sum_s p(s)\pi(a\mid s)$8

The family $q(a)=\sum_s p(s)\pi(a\mid s)$9 is described as a one-parameter Gibbs manifold; mutual information is the canonical Bregman divergence induced by Shannon entropy; the Fisher metric governs local identification; and varying $s\in S$0 traces the efficient frontier of $s\in S$1 (Polson et al., 25 Dec 2025).

5. Action-sufficient compression and two-dimensional generalizations

In the rate–regret formulation, the exact state–action object is the partition of support states induced by the optimal action map. Two support states $s\in S$2 are policy-equivalent under $s\in S$3 when

$s\in S$4

The coarsest exactly action-sufficient compression is the quotient map

$s\in S$5

Any compression that fails to respect $s\in S$6 must sometimes change the optimal action. The paper’s interpretive claim is that robust single-cycle arbitration does not require preserving all support, but it does require preserving the distinctions that consequence geometry makes action-relevant (Walsh, 28 May 2026).

This perspective also clarifies a recurrent misunderstanding about what is being compressed. The objective is not reconstruction of $s\in S$7 per se, and not merely content-only or scalar-confidence-only arbitration. The latter fail whenever their induced partitions cross action boundaries. Approximate sufficiency is instead defined by bounded expected policy regret, so the action channel is judged by consequence-sensitive regret rather than fidelity to the original support state (Walsh, 28 May 2026).

A different extension adds a second dimension of uncertainty: states of the world $s\in S$8 and hedonic characteristics $s\in S$9. The decision maker chooses a joint distribution $a\in A$0 on $a\in A$1 subject to Bayes-plausibility and marginal-feasibility, with total information cost

$a\in A$2

where $a\in A$3. The resulting conditional choice probability is the weighted multinomial logit

$a\in A$4

At $a\in A$5, this is exactly the Matějka–McKay state–action RI formula,

$a\in A$6

and the marginal $a\in A$7 can have zeros and is not unique. At $a\in A$8, one recovers pure entropic choice,

$a\in A$9

For $\pi(a\mid s)$0, the model admits a unique interior solution and no corner solutions (Engh, 8 Aug 2025).

The two-dimensional model also imposes over-identifying restrictions under product entry. If a new characteristic changes $\pi(a\mid s)$1 to $\pi(a\mid s)$2, then for any two characteristics $\pi(a\mid s)$3 and any state $\pi(a\mid s)$4,

$\pi(a\mid s)$5

is independent of $\pi(a\mid s)$6. If $\pi(a\mid s)$7 is known, $\pi(a\mid s)$8 can be backed out from the data (Engh, 8 Aug 2025).

6. Dynamic control, principal design, and applied domains

In average-cost control of Markov processes, state–action rational inattention becomes a constrained optimal-control problem over stationary randomized policies. With transition kernel $\pi(a\mid s)$9 and stage cost $A=\{a\}$00, the steady-state problem can be written as a convex program in the joint measure $A=\{a\}$01, with affine flow-balance constraints and the information constraint $A=\{a\}$02. The Lagrangian introduces a multiplier $A=\{a\}$03 for the information constraint and a function $A=\{a\}$04 for stationarity, and the resulting Bellman-error decomposition yields an information-constrained Bellman equation. In linear-quadratic-Gaussian settings, the theory leads to an information-constrained discrete algebraic Riccati equation and a Gaussian controller that injects enough noise into the action to satisfy the information cap (Shafieepoorfard et al., 2015).

In retirement applications, the state–action model is used to study costly attention to pension-policy uncertainty. The visible state is $A=\{a\}$05, the hidden state $A=\{a\}$06 is the true State Pension Age, and the household carries a belief $A=\{a\}$07 over $A=\{a\}$08. The attention technology is represented directly by an action channel $A=\{a\}$09, with mutual information

$A=\{a\}$10

and flow cost $A=\{a\}$11. Because $A=\{a\}$12 updates only through the chosen noisy policy, beliefs are history dependent, misbeliefs persist endogenously, and retirement bunching at eligibility emerges when uncertainty is suddenly resolved. In UK data, endogenous misbeliefs explain $A=\{a\}$13 of the excessive drop in employment at the State Pension Age, costly attention makes the SPA up to $A=\{a\}$14 less effective at increasing old-age employment, and information letters improve welfare and increase employment (Hentall-MacCuish, 2019).

Principal-design models use the same state–action logic but alter the attention cost or the informational architecture. In the tractable quadratic framework of Lipnowski, Mathevet, and Wei, the agent’s material payoff is $A=\{a\}$15, attention cost is

$A=\{a\}$16

and the first-order condition implies $A=\{a\}$17. Optimal information policies must satisfy an order-IC inequality between posteriors; full disclosure is not always optimal; and in the three-state example optimal disclosure changes from downplaying the state to exaggerating the state as attention costs increase (Lipnowski et al., 2020).

In dynamic principal–agent problems with multiple agents and multiple channels, state–action rational inattention is implemented with explicit encoders $A=\{a\}$18 and a decoder $A=\{a\}$19. The principal’s reward subtracts channel-specific mutual-information costs,

$A=\{a\}$20

The deep reinforcement-learning framework RIRL uses Gaussian encoders, a recurrent network, GAN-style discriminators for on-the-fly mutual-information estimation, and policy-gradient updates. In the single-step setting, RIRL yields wages consistent with theoretical predictions, while non-zero attention costs lead to simpler but less profitable wage structures and increased agent welfare. In sequential multi-agent environments, inattention to agents’ outputs closes wage gaps based on ability differences, whereas inattention to agents’ efforts induces a social dilemma dynamic in which agents work harder but essentially for free (Mu et al., 2022).

Taken together, these strands show that the state–action rational inattention model is not a single application-specific construction but a general information-constrained decision template. Across predictive reporting, action-sufficient compression, discrete choice, retirement, contract design, and stochastic control, the recurring structure is a channel from state to action, an information measure—usually mutual information—and an optimal randomization rule of Gibbs form. A plausible implication is that many behaviors labeled as soft choice, regularization, screening, sparsity, or boundedly rational policy simplification can be represented within one common state–action geometry, provided the source variable, distortion criterion, and dynamic constraints are specified explicitly.