Impact of correlated information on pioneering decisions

Open Access

Impact of correlated information on pioneering decisions

Megan Stickler, William Ott, Zachary P. Kilpatrick, Krešimir Josić, and Bhargav R. Karamched

Phys. Rev. Research 5, 033020 – Published 10 July 2023

Abstract

Normative models are often used to describe how humans and animals make decisions. These models treat deliberation as the accumulation of uncertain evidence that terminates with a commitment to a choice. When extended to social groups, such models often assume that individuals make independent observations. However, individuals typically gather evidence from common sources, and their observations are rarely independent. Here we ask: For a group of ideal observers who do not exchange information, what is the impact of correlated evidence on decision accuracy? We show that even when agents are identical, correlated evidence causes decision accuracy to depend on temporal decision order. The first decider is less accurate than a lone observer, and early deciders are less accurate than late deciders. These phenomena occur despite the fact that the rational observers use the same decision criterion, so they are equally confident in their decisions. We analyze discrete and macroscopic evidence-gathering models to explain why the first decider is less accurate than a lone observer when evidence is correlated. Pooling the decisions of early deciders using a majority rule does not rescue accuracy results in only a modest accuracy gain. Although we analyze an idealized model, we believe that our analysis offers insights that do not depend on exactly how groups integrate evidence and form decisions.

Received 28 March 2023
Accepted 9 June 2023

DOI:https://doi.org/10.1103/PhysRevResearch.5.033020

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Social systems

Decision making models Diffusion & random walks Markovian processes Stochastic analysis Stochastic differential equations

Interdisciplinary PhysicsPhysics of Living SystemsNetworks

Authors & Affiliations

Megan Stickler ^1,*, William Ott ^1,†, Zachary P. Kilpatrick ^2,‡, Krešimir Josić ^1,3,§, and Bhargav R. Karamched ^4,5,6,∥

¹Department of Mathematics, University of Houston, Houston, Texas 77004, USA
²Department of Applied Mathematics, University of Colorado Boulder, Boulder, Colorado 80309, USA
³Department of Biology and Biochemistry, University of Houston, Houston, Texas 77004, USA
⁴Department of Mathematics, Florida State University, Tallahassee, Florida 32306, USA
⁵Institute of Molecular Biophysics, Florida State University, Tallahassee, Florida 32306, USA
⁶Program in Neuroscience, Florida State University, Tallahassee, Florida 32306, USA

^*Corresponding author: mstickler@carthage.edu; present address: Carthage College, Kenosha, Wisconsin 53140, USA.
^†Corresponding author: william.ott.math@gmail.com
^‡Corresponding author: zpkilpat@colorado.edu
^§Corresponding author: kresimir.josic@gmail.com
^∥Corresponding author: bkaramched@fsu.edu

Article Text

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Issue

Vol. 5, Iss. 3 — July - September 2023

Subject Areas

Interdisciplinary Physics

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Images

Figure 1
Agents receiving partially correlated evidence. (a) Agents make a sequence of measurements to decide between alternatives. At each time step agents all make the same observation with probability $c$ and independent observations with probability $1 - c$ . (b) Representative trajectories of the log-likelihood ratios (LLRs) computed at discrete times [Eq. (2)] for $c = 0.25$ and $c = 0.75$ and two agents. Green segments correspond to increments due to common observations, and red segments arise from independent observations. An agent commits to a decision ( $\pm$ ) when the computed LLR crosses $\pm θ$ . (c) Analogous trajectories generated from simulations of the limiting drift-diffusion model [Eq. (3)].
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Figure 2
Impact of the probability of making a common observation, $c$ , on decider accuracy and timing. (a) The probability of a correct decision increases with the order in which the decision is made (solid lines) and varies nonmonotonically with $c$ . The average accuracy computed over all deciders (dashed line) equals the accuracy of a randomly chosen agent and is constant with $c$ . $N = 100$ . (b) The accuracy of the first decider varies nonmonotonically with $c$ , possessing an internal minimum $0 < c_{\min} < 1$ . As $N$ increases, the lowest accuracy decreases. (c) The accuracy of each of $N = 100$ deciders increases with decision order almost monotonically, so the first (last) decider is less (more) accurate than a lone decider for $c \neq 0, 1$ . (d) The time of the decision of $N = 100$ agents as a function of order is approximately invariant to changes in the probability of common evidence, $c$ . We used the discrete LLR model Eq. (2) with $θ = 10$ and binary likelihood functions $f_{\pm}$ as described in Appendix pp1. Specifically we chose $f_{\pm}$ so that the update size is $\pm 0.05$ .
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Figure 3
When evidence is partially correlated, a randomly selected agent is more likely to be the first decider if they are wrong. (a) Joint distribution of the probability that agent 1 decides first ( $F D = 1$ ) and the belief of agent 2 at that decision time, $y_{2} = y_{2} (T_{1})$ , conditioned on agent 1 being right ( $d_{1} = H^{+} = H$ , blue) or wrong ( $d_{1} = H^{+} \neq H$ , red). When $0 < c < 1$ , the accuracy of the first decider is strictly below that of a randomly selected agent (here agent 1, WLOG) because of inequality (6). When $c$ is small, $P^{+} (F D = 1 | d_{1} = H^{+})$ nearly equals $P^{-} (F D = 1 | d_{1} = H^{+})$ (difference indicated by green line), since the joint distributions are approximately reflections of one another, i.e., $P^{+} (F D = 1, y_{2} (T_{1}) | d_{1} = H^{+}) \approx P^{-} (F D = 1, - y_{2} (T_{1}) | d_{1} = H^{+})$ , with equality holding when $c = 0$ . As $c$ increases, the difference $P^{-} (F D = 1 | d_{1} = H^{+}) - P^{+} (F D = 1 | d_{1} = H^{+})$ first grows ( $c = 0.5$ ) and then shrinks ( $c = 0.9$ ), as both terms converge to $1 / 2$ as $c \to 1$ . As discussed in Appendix pp1, each observation changes an agent's belief, $y_{j}$ , by $\pm 1$ ; e.g., when $F D = 1$ and $y_{1} (T_{1}) = \pm 3$ , then $y_{2} (T_{1})$ is also an odd integer. (b) The probability that agent 1 decides first (conditioned on $d_{1} = H^{+}$ and $H = H^{-}$ ) as a function of $c$ peaks around $c = 0.5$ . (c) Colormap of the joint distributions from (a) as functions of $c$ . Here we used the discrete LLR model (2) with $θ = 3$ and binary likelihood functions $f_{\pm}$ as described in Appendix pp1. $f_{\pm}$ are chosen so the update size is $\pm 1$ .
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Figure 4
When beliefs evolve continuously and evidence is correlated, a randomly selected agent is again more likely to decide first if they are wrong. (a) As in the discrete model, the densities $p (F D = 1, y_{2} | d_{1} = H^{+}, H)$ are nearly reflections of one another for small $c$ . By marginalizing over the distribution of beliefs, $y_{2}$ , we can obtain the difference $P^{-} (F D = 1 | d_{1} = H^{+}) - P^{+} (F D = 1 | d_{1} = H^{+})$ . This difference is small when $c$ is small (red bar minus blue bar). As $c$ increases, this difference first increases and then decreases, the latter because each term in the difference converges to $1 / 2$ as $c \to 1$ . (b) The unimodal response of first-decider accuracy as $c$ increases is again due to $P^{-} (F D = 1 | d_{1} = H^{+})$ obtaining a maximum around $c = 0.5$ . We used the macroscopic model (3) with unit drift and variance and threshold $θ = 3$ .
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Figure 5
Pooling choices of early deciders using a majority rule mildly improves accuracy compared to the first decision when evidence is correlated. (a) The group's decision is determined by the majority of the first $N_{pool}$ deciders. For different population sizes, $N$ , the accuracy of the group decision at first decreases as $c$ is increased and can be lower than the accuracy of a single decider in isolation (dashed line). $N_{pool} = 0.2 N$ . (b) Improvement in the accuracy obtained by pooling the first $N_{pool} = 0.2 N$ decisions compared with the accuracy of the first decision drops substantially even for small values of $c$ and is nearly independent of $N$ . (c) The mean time at which the last decider in the pool makes a decision increases with $c$ (solid curves). Dashed curves give the mean time of the first decision. We used the macroscopic LLR (3) with threshold $θ = 10$ .
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