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Markov Stochastic Choice

Kremena Valkanova

Abstract

We examine the effect of item arrangement on choices using a novel decision-making model based on the Markovian exploration of choice sets. This model is inspired by experimental evidence suggesting that the decision-making process involves sequential search through rapid stochastic pairwise comparisons. Our findings show that decision-makers following a reversible process are unaffected by item rearrangements, and further demonstrate that this property can be inferred from their choice behavior. Additionally, we provide a characterization of the class of Markovian models in which the agent makes all possible pairwise comparisons with positive probability. The intersection of reversible models and those allowing all pairwise comparisons is observationally equivalent to the well-known Luce model. Finally, we characterize the class of Markovian models for which the initial fixation does not impact the final choice and show that choice data reveals the existence and composition of consideration sets.

Markov Stochastic Choice

Abstract

We examine the effect of item arrangement on choices using a novel decision-making model based on the Markovian exploration of choice sets. This model is inspired by experimental evidence suggesting that the decision-making process involves sequential search through rapid stochastic pairwise comparisons. Our findings show that decision-makers following a reversible process are unaffected by item rearrangements, and further demonstrate that this property can be inferred from their choice behavior. Additionally, we provide a characterization of the class of Markovian models in which the agent makes all possible pairwise comparisons with positive probability. The intersection of reversible models and those allowing all pairwise comparisons is observationally equivalent to the well-known Luce model. Finally, we characterize the class of Markovian models for which the initial fixation does not impact the final choice and show that choice data reveals the existence and composition of consideration sets.

Paper Structure

This paper contains 31 sections, 11 theorems, 50 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model
  3. Stochastic choice functions
  4. General choice procedure
  5. Choice procedure without time constraints
  6. Comparability restrictions
  7. Reversible MSC models
  8. Pairwise and fully comparable MSC models
  9. Reversible and fully comparable MSC models
  10. Initial fixation
  11. Related literature
  12. Conclusion
  13. Proofs
  14. Proof of Proposition \ref{['prop:limit']}
  15. Auxiliary results
  16. ...and 16 more sections

Key Result

Proposition 1

A stochastic choice function $\boldsymbol{\rho}(\alpha,{ \boldsymbol{\pi}^{}(M\xspace)},{Q({M\xspace})})$ generated by an ergodic Markov chain converges to its stationary distribution as $\alpha \rightarrow 0$.

Figures (3)

  • Figure 1: Limitations in a decision maker's perception can lead to varying pairwise comparability restrictions depending on the item positioning.
  • Figure 2: A reversible baseline MSC model rationalizing the choice function given in Example \ref{['ex:T1']}.
  • Figure 3: An irreducible baseline MSC model rationalizing the choice function given in Example \ref{['ex:T3']}.

Theorems & Definitions (33)

  • Definition 1
  • Proposition 1
  • proof
  • Definition 2
  • Definition 3
  • Example 1
  • Theorem 1
  • proof
  • Example 2
  • Definition 4
  • ...and 23 more