Fig 1.
Probabilistic graphical model (influence diagram) of a single joint action player, that accounts for multiple strategy choices.
At each time step, Player i receives information about partner action (u−i) through their sensory system (yi). Action selection involves the selection of a proper strategy (discrete variable si) and minimizing a cost function (Ji) which depends on both own (ui) and partner’s action (u−i).
Fig 2.
a. Simulation concept. Each player knows the locations of both rabbit (R) and stag (S). They can see their actions and have imperfect information about their partner’s actions (blurred circles). b. cost matrix used in the simulations, in which the cost can be interpreted as a reduced payoff. c. Costs related to the two equilibria are represented in the action space. Cold colors indicate lower costs, and hot colors indicate higher costs. On the left, whichever action the partner selects it is safe for player i to go for the Rabbit solution. On the right, the two players minimize their costs by selecting both the Stag. d. Simulated Stag Hunt game for different combinations of sensory noise () and internal noise or partner predictability (
). Temporal evolution of the probabilities of selecting the SS (blue), the RR (orange). The curves represent game performance at population level, where probabilities (mean ± SE) have been computed over epochs (1 epoch = 40 trials) and averaged over multiple simulated dyads. The horizontal line indicates the chance level (p = 0.25). Bottom: scatter plots of the joint actions (u1, u2) for one representative dyad for each condition.
Fig 3.
a. Experimental apparatus and task—modified from [11]. The partners were connected through a virtual spring. b. Nash equilibria with asymmetric and symmetric via-points locations. c. In different experimental groups, information about partner actions was provided haptically, through the interaction force alone (Haptic group, H); by additionally displaying the interaction force vector on the screen (Visuo-Haptic group, VH), or by also displaying the partner’s cursor (Partner Visible, PV).
Fig 4.
a. Estimated trajectories in the identification procedure (last ten trials), for three representative dyads (one per group). b. Group differences in the estimated model parameters. The box plots display median, 25th and 75th percentiles of the estimated parameters, for H (light blue), VH (pink) and PV (yellow) groups. Asterisks indicate statistically significant differences (*: p < 0.05; **: p < 0.01).
Fig 5.
Experimental results (top) and simulations (bottom) of the 2VP task.
a. Experimentally observed and b). simulated trajectories for selected dyads in the three experimental conditions (H, VH, PV) in the original 2-VP study [11]. c. Experimentally observed and d. simulated minimum distances of player 1 from VP2 (MD12) and player 2 from VP1 (MD21)—mean ± SE over actual and simulated players—in all three conditions.
Fig 6.
Sensory noise modulates the leadership index.
a. Leadership indices as functions of the estimated sensory noise covariance in experimental data (one data point per subject).
was systematically varied, and each
was simulated five times. The lines (mean ± SE) denote the simulation results, for player 1 (blue) and player 2 (red). b. Simulated leadership indices over time (mean ± SE); c. Simulated (left) and experimental (right) leadership indices for VP2 in the three experimental groups, at the end of the training phase. Experimental data from [11].
Fig 7.
Coordination behavior in the symmetric 2-VP task.
Dyads exhibit a variety of coordination behaviors. a. Typical dyad trajectories (from top to bottom: D8, D3, D12) at early and late training of the experiment. Player 1 and Player 2 are depicted in blue and red. These dyads exemplify typical behaviors: players ignoring each other (D8); cycling between opposite strategies (D3); and converging to a coordination (D12). b. Minimum distance of the two players from partner’s via-points. We report population average and standard error over time for the two groups. On the left we report distribution of the average performances at the end on training.
Fig 8.
a. Estimated trajectories. b, c. Estimated model parameters grouped in terms of Learner and Non-Learner and in H and VH groups. Boxplots display median, 25th and 75th percentiles of the estimated parameters values, for Learners (orange) and Non-Learner (green). *: p < 0.05; **: p < 0.01.
Fig 9.
Simulations (bottom) closely resemble Experiments (top), in the temporal evolution of both the minimum distance to partner’s via-point over trials (a) and in the probability of establishing a collaboration (b), in both Learners (pink) and Non-Learners (green).