We introduce three new classes of attention cost functions: posterior separable, uniformly posterior separable, and invariant posterior separable. As with the Shannon cost function, all can be solved using Lagrangian methods. Uniformly posterior-separable cost functions capture many forms of sequential learning and hence play a key role in many applications. Invariant posterior-separable cost functions make learning strategies depend exclusively on payoff uncertainty. We introduce two behavioral axioms, Locally Invariant Posteriors and Only Payoffs Matter, which identify posterior-separable functions as uniformly and invariant posterior separable, respectively. In combination, they pinpoint the Shannon cost function.

The neurotransmitter dopamine is central to the emerging discipline of neuroeconomics; it is hypothesized to encode the difference between expected and realized rewards and thereby to mediate belief formation and choice. We develop the first formal test of this theory of dopaminergic function, based on a recent axiomatization by Caplin and Dean [2008A]. These tests are satisfied by neural activity in the nucleus accumbens, an area rich in dopamine receptors. We find evidence for separate positive and negative reward prediction error signals, suggesting that behavioral asymmetries in response to losses and gains may parallel asymmetries in nucleus accumbens activity.