A normative comparison of threshold views through computer simulations

Abstract

The threshold view says that a person forms an outright belief P if and only if her credence for P reaches a certain threshold. Using computer simulations, I compare different versions of the threshold view to understand how they perform under time pressure in decision problems. The results illuminate the strengths and weaknesses of the various cognitive strategies in different decision contexts. A threshold view that performs well across diverse contexts is likely to be a cognitively flexible and context-dependent fusion of several of the existing theories. The results of the simulations also cast doubts on the possibility of a threshold view that is both simple enough to streamline our reasoning while also allowing us to form good action-guiding beliefs.

Appendices

Appendix A: Weibull distribution

The Weibull distribution is a continuous probability distribution defined by the function

$$\begin{aligned} f(x; \lambda , k)={\left\{ \begin{array}{ll} \frac{k}{\lambda }(\frac{x}{\lambda })^{k-1}e^{-(x/\lambda )^k},&{}\quad \text {if }x \le 0.\\ 0,&{}\quad \text {otherwise}, \end{array}\right. } \end{aligned}$$

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where $\lambda \in (0, +\infty )$ is the scale parameter and $k\in (0, +\infty )$ is the shape parameter. The cumulative distribution function of the Weibull distribution is given by

$$\begin{aligned} f(x; \lambda , k)={\left\{ \begin{array}{ll} 1-e^{-(x/\lambda )^k},&{}\quad \text {if }x \le 0.\\ 0,&{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

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Appendix B: Symmetric dirichlet distribution

The Dirichlet distribution is a multivariate generalization of the beta distribution, parameterized by a vector $\varvec{\alpha }$ with real entries. The Dirichlet distribution with parameters $\alpha _1, \alpha _2, \ldots , \alpha _K$ is given by

$$\begin{aligned} f(x_1, x_2, \ldots , x_K; \alpha _1, \alpha _2, \ldots , \alpha _K) = \frac{1}{B(\varvec{\alpha })}\prod _{i = 1}^{K} x_{i}^{\alpha _{i}-1}, \end{aligned}$$

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where $\sum _{i = 1}^{K} x_i = 1$ and $x_1, x_2, \ldots , x_K > 0$. The normalizing constant $B(\varvec{\alpha })$ is the multivariate beta function

$$\begin{aligned} B(\varvec{\alpha }) = \frac{\prod _{i = 1}^{K} \Gamma (\alpha _i)}{\Gamma \left( \sum _{i = 1}^{K}\alpha _i\right) }, \end{aligned}$$

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where $\Gamma$ is the gamma function.

A special case of the Dirichlet distribution is the symmetric Dirichlet Distribution. In a symmetric Dirichlet distribution, the entries of the vector $\varvec{\alpha }$ are all equal. This corresponds to the case where there is no prior information that leads us to prefer one possibility over another. Since all the entries of $\varvec{\alpha }$ are the same, the symmetric Dirichlet distribution can simply be parameterized by a scalar value $\alpha$, often called the concentration parameter. When $\alpha > 1$ the probability is randomly but evenly distributed across the possibilities. When $\alpha < 1$, most of the probability will be concentrated in a few of the possibilities. The symmetric Dirichlet distribution is given by

$$\begin{aligned} f(x_1, x_2, \ldots , x_K; \alpha ) = \frac{\Gamma (\alpha K)}{\Gamma (\alpha )^K}\prod _{i = 1}^{K} x_{i}^{\alpha _{i}-1}. \end{aligned}$$

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Appendix C: Code for the model

The code for the model, written in Python using the agent-based simulation package Mesa, can be found here: https://github.com/alicecwhuang/threshold_code.git.