251008 - NOT USED - PUMCM for Activity-Based Models

This report demonstrates that the Perturbed Utility Markovian Choice Model (PUMCM) can be effectively applied to Dynamic Discrete Choice Models (DDCM) for activity-based travel demand modeling. The key finding is that PUMCM and traditional logit models solve the same underlying problem through their shared connection to Shannon entropy (Yao and Zhang, Perturbed Utility Markovian Choice Model: Choice Probability Generation Function and Estimation, 2025) , making PUMCM a natural extension of familiar discrete choice methods to sequential decision-making contexts. Main Result: When the perturbation function is Shannon entropy, PUMCM recovers the recursive logit model, enabling both efficient parameter estimation and behavioral simulation.

1. Theoretical Foundation

1.1 The Shannon-Logit Connection

The fundamental insight is that three seemingly different approaches produce identical choice probabilities: A. Random Utility (McFadden, 1974) $U(a)=v_a+{ϵ}_a,{ϵ}_a\sim \text{Gumbel}$ B. Shannon Entropy Maximization (Jaynes, 1957) ${\max }_{\pi }-{\sum }_a\pi (a)\ln (\pi (a))\text{s.t.}{\sum }_a\pi (a)v_a=\bar{v}$ C. Perturbed Utility (Fosgerau & McFadden, 2012) ${\max }_{p\in \Delta }{\sum }_j{p}_j{v}_j-F(p)$ ${\max }_{\pi }{\sum }_a\pi (a)v_a-{\sum }_a\pi (a)\ln (\pi (a))$ All three yield the multinomial logit formula: $\pi (a)=\frac{\exp (v_a)}{{\sum }_{a^{\prime }}\exp (v_{a^{\prime }})}$ Proof (for C → Logit): Taking the first-order condition: $\mathcal{L}(p,\lambda )={\sum }_j{p}_j{v}_j-F(p)-\lambda \left({\sum }_j{p}_j-1\right)$ First-order condition for optimal $p^{\ast }$: $\frac{\partial \mathcal{L}}{\partial p_j}=v_j-\frac{\partial F}{\partial p_j}(p^{\ast })-\lambda =0$ $\frac{\partial }{\partial \pi (a)}\left[{\sum }_a\pi (a)v_a-{\sum }_a\pi (a)\ln (\pi (a))-\lambda \left({\sum }_a\pi (a)-1\right)\right]=0$ $v_a-\ln (\pi (a))-1-\lambda =0$ $\pi (a)=\exp (v_a-\lambda -1)$ Using ${\sum }_a\pi (a)=1$: $\pi (a)=\frac{\exp (v_a)}{{\sum }_{a^{\prime }}\exp (v_{a^{\prime }})}$

1.2 Extension to Markov Decision Processes

PUMCM Framework (Yao & Zhang, 2025): For an MDP with states $S$, actions $A_s$, transitions $P$, utilities $u$, and discount $\gamma$, the agent solves at each state: ${\max }_{\pi (\cdot |s)}{\sum }_{a\in A_s}Q(s,a)\pi (a|s)-F_s(\pi (\cdot |s))$ where $Q(s,a)=u(s,a)+\gamma {\sum }_{s^{\prime }}P(s^{\prime }|s,a)V(s^{\prime })$ is the Q-value function.

Choice Probability Generation Function: $H_s(Q)={\max }_{\pi }\{Q^{\top }\pi -F_s(\pi )\}$ Key Property: The optimal policy is the gradient: ${\pi }^{\ast }(a|s)={\nabla }_a{H}_s(Q(s,\cdot ))$ For Entropy ($F_s={\sum }_a{\pi }_a\ln {\pi }_a$): $H_s(Q)=\ln \left({\sum }_a\exp (Q(s,a))\right)$ ${\pi }^{\ast }(a|s)=\frac{\exp (Q(s,a))}{{\sum }_{a^{\prime }}\exp (Q(s,a^{\prime }))}$ This is the recursive logit model - a direct extension of multinomial logit to sequential choices.

References

Core PUMCM Papers:

• Yao, R., & Zhang, K. (2025). Perturbed utility Markovian choice model. TRISTAN XII.
• Fosgerau, M., & McFadden, D. (2012). A theory of the perturbed consumer. NBER Working Paper.
• Fosgerau, M., et al. (2022). A perturbed utility route choice model. Transportation Research C, 136.

Activity-Based Modeling:

• Västberg, O., et al. (2020). A dynamic discrete choice activity-based travel demand model. Transportation Science, 54(1), 21-41.
• Rust, J. (1987). Optimal replacement of GMC bus engines. Econometrica, 55(5), 999-1033.

Entropy and Discrete Choice:

• Fosgerau, M., et al. (2020). Discrete choice and rational inattention: A general equivalence result. International Economic Review, 61(4), 1569-1589.
• McFadden, D. (1974). Conditional logit analysis of qualitative choice behavior.

Information Theory:

• Shannon, C. E. (1948). A mathematical theory of communication. Bell System Technical Journal.
• Jaynes, E. T. (1957). Information theory and statistical mechanics. Physical Review.