Explaining Human Choice Probabilities with Simple Vector Representations

Abstract

When people pursue rewards in stochastic environments, they often match their choice frequencies to the observed target frequencies, even when this policy is demonstrably sub-optimal. We used a ``hide and seek'' task to evaluate this behavior under conditions where pursuit (seeking) could be toggled to avoidance (hiding), while leaving the probability distribution fixed, or varying complexity by changing the number of possible choices. We developed a model for participant choice built from choice frequency histograms treated as vectors. We posited the existence of a probability antimatching strategy for avoidance (hiding) rounds, and formalized this as a vector reflection of probability matching. We found that only two basis policies: matching/antimatching and maximizing/minimizing were sufficient to account for participant choices across a range of room numbers and opponent probability distributions. This schema requires only that people have the ability to remember the relative frequency of the different outcomes. With this knowledge simple operations can construct the maximizing and minimizing policies as well as matching and antimatching strategies. A mixture of these two policies captures human choice patterns in a stochastic environment.

Figures (15)

• Figure 1: The simplest case: a two-dimensional histogram. On the left there are three histograms showing the proportion of events for each of two bins. The upper histogram is the uniform, denoted as $\vec{u}$: all bins have the same probability. Histograms (such as $\vec{p}$) can be reflected over the uniform to obtain a $\vec{q}$, and vice versa. The angle between $\vec{q}$ and $\vec{u}$ is identical to the angle between $\vec{p}$ and $\vec{u}$. All of the vectors exist in the same plane. Two distributions are opposites if they are an equal angular distance away from the uniform vector and form a 2D-plane with the uniform.
• Figure 2: From the space of histograms to the space of strategies: The vector representing choice proportions or numbers is decomposed into a vector representing the matching and maximizing strategy combination. On the left you see $\vec{b} = (0.85, 0.15)$ and its decomposition of $\vec{b} = \alpha \vec{x} + \beta \vec{m}$ where $\vec{x} = (1,0)$ and $\vec{m} = (0.7, 0.3)$. The coefficients for each of these two strategies gives us a point on the $(\alpha, \beta)$, in this case $(\alpha, \beta) = (\frac{1}{2}, \frac{1}{2})$. Although the graph on the right looks similar to the one on the left the axes represent completely different things. On the left we are in the space of options: one axis for each option. On the right we are in the space of strategy. One axis for each strategy: maximizing ($\alpha$) and matching ($\beta$). The same method works for decomposing hiding choices into a combination of antimatching and minimizing.
• Figure 3: Testing: In higher dimensional spaces the vector representing a person's selection frequencies $\vec{b}$ (black) may not exist on the plane created by a weighted sum of the maximizing $\vec{x}$ and matching strategies $\vec{m}$ (blue region). This error vector $\epsilon$ (red) quantifies the discrepancy, and can be used as a convenient expression of the goodness of the model.
• Figure 4: Three different viewpoints of the probability simplex in 3-dimensional space. Here, the reflection of $\vec{p}$, denoted $\vec{q}$, lies outside of the probability simplex. Performing the Euclidean projection onto the simplex (right most panel) produces the shortest possible vector $\vec{v}$ to the simplex, but does not always shift $\vec{q}$ back along the original reflection trajectory from $\vec{p}$ (as seen in the middle panel).
• Figure 5: Participant view of screen: This is the five-room condition. After selecting the bottom left room in a seek trial the selected room turned grey. The child is then revealed, a notification is presented, and the counters are updated.