Collective decision-making under changing social environments among agents adapted to sparse connectivity

TL;DR

An adaptive strategy in sparsely-connected networks based on highly-simplified social information is revealed, providing an important caveat for sociality observed in the laboratory and suggesting a basis for the social dynamics of highly-connected online communities.

Abstract

Humans and other animals often follow the decisions made by others because these are indicative of the quality of possible choices, resulting in `social response rules': observed relationships between the probability that an agent will make a specific choice and the decisions other individuals have made. The form of social responses can be understood by considering the behaviour of rational agents that seek to maximise their expected utility using both social and private information. Previous derivations of social responses assume that agents observe all others within a group, but real interaction networks are often characterised by sparse connectivity. Here I analyse the observable behaviour of rational agents that attend to the decisions made by a subset of others in the group. This reveals an adaptive strategy in sparsely-connected networks based on highly-simplified social information: the difference in the observed number of agents choosing each option. Where agents employ this strategy, collective outcomes and decision-making efficacy are controlled by the social connectivity at the time of the decision, rather than that to which the agents are accustomed, providing an important caveat for sociality observed in the laboratory and suggesting a basis for the social dynamics of highly-connected online communities.

Figures (4)

• Figure 1: Characterising the optimal social information strategy for differing values of social connectivity. Each row shows differing perspectives on the social response by agents habituated to a specific level of social connectivity: $q_{\textrm{habitual}}=0.9$ (top row), $q_{\textrm{habitual}}=0.5$ (middle row) or $q_{\textrm{habitual}}=0.1$ (bottom row). Panels A-C show the probability a focal agent will choose option A, conditioned on the number of previous decisions for A and B observed by that agent. Panels D-F show the same, but further conditioned on the most recently observed decision being for A (thus also ruling out instances where $n_A=0$). Panels G-I show the social response projected onto one dimension: the difference $n_A-n_B$; grey points indicate probabilities arising from specific sequences, while red points show the mean probability for each value of $n_A-n_B$. Panels J-L show the consequence of these decision rules in terms of the distribution of collective outcomes (total number of decisions for A) under habitual conditions ($q_{\textrm{actual}}=q_{\textrm{habitual}}$, where social connectivity remains at the level the agents are adapted to), while panels M-O show the distribution of collective outcomes under conditions of full connectivity ($q_{\textrm{actual}}=1$) if agents continue to apply the behavioural rules optimised for their habitual conditions.
• Figure 2: Characterising the optimal social information strategy in a large ($n = 100$), sparsely-connected network, for agents habituated to differing connectivity ($q_{\textrm{habitual}} = \{0.01, 0.05, 0.1\}$). (A) Agents adopt a decision rule that depends on the observed difference in the number of prior decisions between options ($n_A-n_B$). This rule is essentially independent of the connectivity to which they are habituated. (B) When this rule is applied under the habitual conditions, the observable response to the true difference ($N_A-N_B$) is weaker in conditions of lower connectivity; (C) More sparsely-connected groups are less cohesive under natural conditions (solid lines), as measured by the total number of agents choosing one option. Increasing network connectivity to $q_{\textrm{actual}}=0.2$ and applying the same decision rule results in a substantially more cohesive collective outcome that is independent of the habitual connectivity (dashed line). (D) The degree of consensus ($|N_A-N_B|/(N_A+N_B)$) varies monotonically with the true connectivity of the network and is independent of habitual connectivity. The dashed grey line indicates the expected consensus when agents choose independently.
• Figure 3: Variation of optimal social responses with differing preference alignment $\rho$, private information variance $\epsilon$ and group size $n$. (A) variable $\rho$, with fixed $\epsilon=1$, $n=100$; (B) variable $\epsilon$ with fixed $\rho=0.5$ and $n=100$; (C) variable $n$ with fixed $\rho=0.5$ and $\epsilon =1$. In each case is also plotted the probabilities given by equation \ref{['eqn:alfonso']} (dashed lines), with sociality parameter $S$ given by equation \ref{['eqn:effS']}, demonstrating a close fit between the two formulations across widely varying parameters; (D) the calculated value of $S$ in a group of size $n=100$ as a function of the 'relative social weighting' (RSW), for varying values $\rho$ and $\epsilon$.
• Figure 4: Average payoff obtained by agents in sparse networks of varying sizes as a function of actual connectivity and mean degree. Agents in each network are habituated to a connectivity $q_{\textrm{habitual}} = 5/(n-1)$ implying a mean degree of 5, though the results are not sensitive to this factor as long as habitual connectivity is sparse. (A) Mean payoffs increase as mean degree increases in all networks up to an optimal value, and then fall, with a differing optimal mean degree for different group sizes. The dashed grey lines indicate the average reward for a single agent without any social information (lower line) and the average reward for an agent that always chooses the option with greater utility (upper line); (B) The optimal mean degree generating maximum payoffs varies proportionally to the square root of network size; (C) Mean payoffs increase with connectivity up to an optimal value and then fall, with a differing optimal connectivity for different group sizes; (D) The optimal connectivity varies inversely with the square root of network size.