A decision-maker facing uncertainty can acquire costly information before choosing. Rational inattention models this: the agent selects a signal structure (how to partition the state space into distinguishable categories) that maximizes expected utility minus the mutual information cost of the signal. The output is a stochastic mapping from states to actions, where the randomness reflects deliberate imprecision — the agent doesn't pay for distinctions that don't affect the optimal action.
The paper reveals this as optimal transport. The prior distribution over states must be transported to a posterior distribution over state-action pairs, and the cost of transport is the entropic divergence. The rational inattention problem IS a regularized optimal transport problem, where the entropy regularization corresponds to the information cost.
The nesting matters: the agent first chooses how much to learn (the transport plan), then chooses an action for each realized signal (the inner optimization). This two-stage structure makes it a nested transport problem — the outer problem selects the coupling between states and signals, the inner problem selects the action for each signal.
The transport formulation extends previous results from finite choice sets to arbitrary choice sets. When the action space is a continuum (how much to invest, where to allocate resources), the finite-choice-set theory breaks down. The optimal transport framework handles continuous actions by the same mechanism as discrete ones — the regularized transport plan determines the optimal level of attention everywhere simultaneously.
Attention as optimal transport. The currency is bits; the goods are states; the market clearing condition is Bayes' rule.