Fig. 3: Game-theoretic model of co-adaptive user–machine systems.
a, Schematic of co-adaptive interfaces in which the user and decoder are modelled as adapting to minimize their own individual cost functions (plots in the inset show data for each agent). b, Visualization of the potential function that describes dynamics within the user–decoder game model (simplified 1D user and decoder; Methods). Two axes represent the scalar values of the user and decoder actions, and the vertical axis represents the value of the potential function. A two-dimensional projection of the potential function is also shown. c, Gradient field of the user and decoder cost functions (with equal penalty terms, λD = λE). The purple (user) and orange (decoder) curves are nullclines (where the agent’s gradient equals 0), which intersect at stationary points (black stars). d, Heat map showing the error decay rate as a function of learning rates in equation (5) with penalty parameters λD, λE = 1/4. e, Error decay rate as a function of decoder learning rate (αD) for two values of encoder learning rate (αE = 3/2 (blue) and αE = 1/2 (red)) corresponding to the horizontal slices in d. f, Stationary values of scalar decoder (orange) and encoder (purple) as a function of decoder penalty parameter (λD) relative to the value of encoder penalty (λE).
