Equivalent Logit Choice Probabilities from Optimization and Randomization

• Authors: Martí Mestieri Ferrer and Jordan J. Norris
• BSE Working Paper: 1547 | January 2026
• Keywords: discrete choice, entropy, logit, Stirling's Approximation
• JEL codes: C25, D61, C60

We uncover an equivalence between two opposite behavioral microfoundations for the standard multinomial logit choice model: 1) agents optimize their choice accordingly to the random utility model under Gumbel idiosyncratic shocks; 2) agents randomize over options subject to a minimum utility requirement. Both generate identical multinomial logit choice probabilities, yet have different welfare implications: wel-fare is strictly lower under randomization, since only optimizing agents select into options with favorable realizations of idiosyncratic shocks.