Modeling other minds: Bayesian inference explains human choices in group decision-making

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Science Advances  27 Nov 2019:
Vol. 5, no. 11, eaax8783
DOI: 10.1126/sciadv.aax8783
  • Fig. 1 Multiround PGG.

    The figure depicts the sequence of computer screens a subject sees in one round of the PGG. The subject is assigned four other players as partners, and each round requires the subject to make a decision: Keep 1 MU (i.e., free-ride) or contribute 1 MU. The subject knows whether the threshold to generate public goods (reward of 2 MU for each player) is two or four contributions (from the five players). After the subject acts, the total number of contributions and overall outcome of the round (success or failure) are revealed.

  • Fig. 2 Human behavior in the PGG Task.

    (A) Average contribution probability across subjects is significantly higher when the task requires more volunteers (k) to generate the group reward. (B) Average probability of success across all subjects in generating the group reward is significantly higher when k is lower. Error bars indicate within-subject SE (52). (C) Average probability of contribution for each subject for k = 2 versus k = 4. Each point represents a subject. Subjects tend to contribute more often when the task requires more volunteers. (D) Average success rate for each subject was higher for k = 2 versus k = 4. (E) Average probability of contribution across subjects decreases throughout a game, especially for k = 4. Dotted lines are linear functions showing this trend for each k. (F) Average contribution probability across subjects as a function of number of games played. The contribution probability does not change significantly as subjects play more games.

  • Fig. 3 POMDP model of the multiround PGG.

    (A) Model: The subject does not know the average probability of contribution of the group. The POMDP model assumes that the subject maintains a probability distribution (“belief,” denoted by bt) about the group’s average probability of contribution (denoted by θt) and updates this belief after observing the outcome ct (contribution by others) in each round. (B) Action selection: The POMDP model chooses an action (at) that maximizes the expected total reward (∑ri) across all rounds based on the current belief and the consequence of the action (contribution “c” or free-ride “f”) on group behavior in future rounds.

  • Fig. 4 Optimal actions prescribed by the POMDP policy as a function of belief state.

    Plot (A) shows the policy for k = 2 and plot (B) for k = 4. The purple regions represent those belief states (defined by αt and βt) for which free-riding is the optimal action; the yellow regions represent belief states for which the optimal action is contributing. These plots confirm that the optimal policy depends highly on k, the number of required volunteers. For the two plots, the decay rate was 1 and t was 9.

  • Fig. 5 POMDP model’s performance and predictions.

    (A) Average fitting and LOOCV accuracy across all models. The POMDP model has significantly higher accuracy compared to the other models (*P < 0.05 and ***P < 0.001). Error bars indicate within-subject SE (52). (B) POMDP model’s prediction of a subject’s probability of contribution compared to experimental data for the two k values [black circles: same data as in Fig. 2C). (C) Same data as (B) but the POMDP model’s prediction and the experimental data are shown for each k separately (blue for k = 2 and orange for k = 4]. (D) POMDP model’s prediction (blue circles) of a subject’s belief about group success in each round (on average) compared to actual data (black circles, same data as in Fig. 2D). (E) Same data as (D), but the POMDP model’s prediction and actual data are shown for each k separately (blue for k = 2 and orange for k = 4). (F) Same data as (B) and (C) but with the data points binned on the basis of round of the game. (G) Same data as (D) and (E) but with the data points binned based on round of the game.

  • Fig. 6 Distribution of POMDP parameters across subjects.

    (A) Histogram of α1 across all subjects. (B) Histogram of β1 across all subjects. (C) Histogram of the ratio α11 shows a value between 0.5 and 2 for almost all subjects. (D) Histogram of (α1 + β1)/2. For most subjects, this value is between 40 and 120. (E) Histogram of prior belief Beta(α1, β1) translated into expected contribution by the others in the first round. Note that the values, after fitting to the subjects’ behavior, are mostly between 2 and 3. (F) When k = 2, all subjects expected a high probability of group success in the first round (before making any observations about the group). (G) When k = 4, almost all subjects assigned a chance of less than 60% to group success in the first round. (H) Box plot of decay rate γ across subjects shows that this value is almost always above 0.95. The median is 0.97 (orange line) and the mean is 0.93 (green line).

Supplementary Materials

  • Supplementary Materials

    This PDF file includes:

    • Supplementary Text
    • Fig. S1. Distribution and change in belief parameters over multiple rounds.
    • Fig. S2. Data generated by the POMDP model compared to experimental data.

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