Failures of Contingent Thinking

TL;DR

This paper develops a unified theory of contingent thinking by defining subjective implication and representing a DM's view of uncertainty with an interpretation of uncertainty (IOU) that yields a subjective state-space. It shows how to recover the DM's internal implications from betting data and analyzes how updating can occur through event-based state elimination or through implication-based realizations, with SIDEU capturing decision-making under translation ambiguity. The framework formalizes conditions under which contingent reasoning improves or deteriorates and links misinterpretation of language to violations of state-wise dominance. The approach offers a parsimonious, testable lens for understanding contingent thinking, with implications for designing experiments and mechanisms that limit misinterpretation and enhance reasoning under uncertainty.

Abstract

We present a behavioral definition of an agent's perceived implication that uniquely identifies a subjective state-space representing her view of a decision problem, and which may differ from the modeler's. By examining belief updating within this model, we formalize the recent empirical consensus that reducing uncertainty improves contingent thinking, and propose a novel form of updating corresponding to the agent 'realizing' a flaw in her own thinking. Finally, we clarify the sense in which contingent thinking makes state-bystate dominance more cognitively demanding than obvious dominance.

Figures (5)

• Figure 1: Three IOUs representing the statements from tversky1983extensional. Each statement corresponds to a row; the truth assignment maps each statement to the states in its row with filled in cells. The first panel is the 'objective' state-space and the latter two are logically flawed but consistent with the experimental findings.
• Figure 2: Acts $\color{lam1}f\color{black}\xspace$ and $\color{lam2}g\color{black}\xspace$. For the DM, the act $\color{lam2}g\color{black}\xspace$ is ambiguous, taking multiple values in $w_2$. The statements listed beneath each state are those modeled as true in that state.
• Figure 3: The implication relation from Example \ref{['ex:2']} and two IOUs faithful to it.
• Figure 4: The IOU $(W,t)$ rationalizing subjects behavior; only the primitive statements are represented in each state. The numbers on the bottom row represent the probability of each state according to the experimental design outlined above and given a fixed and independent probability $\gamma$ of $\textsc{p}$, where $\alpha = \frac{\gamma}{108}$ and $\beta = \frac{1-\gamma}{108}$.
• Figure 5: Discretized version of a second-price auction for a bidder with value $v = 4$. The acts $\color{lam2}f_{3}\color{black}\xspace$ and $\color{lam1}f_{4}\color{black}\xspace$ correspond to a bid of three and four, respectively.

Theorems & Definitions (49)

• Example 1
• Example 1: continued
• Example 1: continued
• Definition 1: Subjective Implication
• Definition 2: Nullness and Disjointness
• Example 2
• Definition 3: Interpretation of Uncertainty
• Definition 4: Faithfulness
• Theorem 1
• Example 2: continued