Fig 1.
Empirical results from the behavioural online experiment.
Decreasing reaction times as a function of the magnitude of the equal alternatives. X-axis presents mean brightness of equal alternatives (0.3, 0.4, 0.5, 0.6), on a scale of brightness from 0 to 1 in PsychoPy. Y-axis presents mean reaction times, in seconds. Bars show 95% confidence intervals. Participants experienced equal alternative conditions, interleaved with unequal alternative trials in pseudo-randomised order. Participants that performed the whole experiment experienced each equal alternative presentation ten times.
Fig 2.
Empirical results from the slime mould experiment.
Decreasing latencies to reach a food source as a function of the magnitude of the equal alternatives. X-axis presents the concentration in egg yolk of equal food sources (20, 40, 60, 80 g.L−1). Y-axis presents mean latency to reach a food source, in minutes. Bars show 95% confidence intervals. 50 slime moulds were tested for each magnitude for a total of 200 slime moulds.
Fig 3.
Optimal decision boundaries in the space of possible estimates of option values: Linear time costs lead to weakly magnitude-sensitive optimal policies (top row), while geometric discounting of reward leads to strongly magnitude-sensitive optimal policies (bottom row).
In the linear time cost (Bayes Risk) case nonlinear subjective utility changes complex time and value-dependent decision boundaries in estimate space into a simple mostly magnitude-insensitive ‘best-vs-next’ strategy (top row; see [4], Fig 6C). For geometric discounting of rewards over time, optimal decision boundaries are strongly magnitude-sensitive and interpolate between simple ‘best-vs-average’ and ‘best-vs-next’ strategies (see [4], Fig 6). Triangles are low dimensional projections of the 3-dimensional evidence estimate space onto a plane moving along the equal value line, at value v [4]. Dynamic programming parameters were: prior mean and variance
, waiting time tw = 1, temporal costs c = 0, γ = 0.2, and utility function parameters m = 4, s = 0.25 (for the linear time cost) and m = 4, s = 3.5 (for the geometric time cost). Time steps chosen to illustrate boundary collapse.
Fig 4.
Optimal decision boundaries in the space of possible estimates of option values: Optimal policies for linear time cost (Bayes Risk) rapidly transition from approximately linear subjective utility, and hence weakly magnitude-sensitive, decision boundaries in estimate space (Fig 3, top row for s = 0.25; present figure, top row for s = 0.5, to more step-like subjective utility where immediate ‘choose the best’ decision-boundaries are necessarily magnitude-insensitive (bottom row for s = 0.75, and higher values of s).
Triangles are low dimensional projections of the 3-dimensional evidence estimate space onto a plane moving along the equal value line, at value v [4]. Dynamic programming parameters were: prior mean and variance
, and utility function parameters m = 4, s ∈ {0.5, 0.75}. Time steps chosen to illustrate boundary collapse.
Fig 5.
Linear time costs lead to weakly magnitude-sensitive simulated reaction times across a range of nonlinear subjective utility functions for equal value option sets.
Simulation parameters were: prior mean and variance
, observation noise variance
, temporal cost c = 0, waiting time tw = 1, and simulation timestep dt = 5 × 10−3. Lines are the mean reaction time for 104 simulations, 95% confidence intervals are shown as red shading (mostly invisible because smaller than the linewidth). Y-axis made consistent with Fig 6 for comparison. Non-decision time was implicitly zero.
Fig 6.
Geometric discounting of reward leads to strongly magnitude-sensitive simulated reaction times across a range of nonlinear subjective utility functions, with decisions postponed for low equal-value option sets.
Simulation parameters were: prior mean and variance
, observation noise variance
, temporal cost γ = 0.1, and simulation timestep dt = 5×10−3. Lines are the mean reaction time for 104 simulations, 95% confidence intervals are shown as red shading (mostly invisible because smaller than the linewidth). Non-decision time was implicitly zero.
Fig 7.
(A) Stimuli example for human psychophysical experiments: Participants were requested to decide as fast and accurately as possible which of the three stimuli was brighter; they were asked to maintain fixation on the cross at the centre of the screen and minimise distraction for the short duration of the experiment. Unknown to participants, conditions of interest were conditions for which the stimuli had equal mean brightness. (B) Photograph showing a slime mould that chose one food alternative among three equal ones. The slime mould was placed in the centre of a petri dish (60 mm ⌀) filled with agar gel (10 g.L−1) at a distance of 2 mm from each food alternative.