Fig. 1
From: Human adaptation to adaptive machines converges to game-theoretic equilibria
Gradient descent in action space (Experiment 1, \(n = 20\)). (A) Each human subject H is instructed to provide manual input h to make a black bar on a computer display as small as possible. The bar’s height represents the value of a prescribed cost \(c_H\). (B) The machine M has its own cost \(c_M\) chosen to yield game-theoretic equilibria that are distinct from each other and from each player’s global optima. The machine knows its cost and observes human actions h. In this experiment, the machine updates its action by gradient descent on its cost \(\frac{1}{2}m^2-hm+h^2\) with adaptation rate \(\alpha\). (C) Median joint actions for each \(\alpha\) overlaid on game-theoretic equilibria and best-response (BR) curves that define the Nash equilibrium (NE) and Stackelberg equilibrium (SE), respectively). (D) Action distributions for each machine adaptation rate displayed by box-and-whiskers plots showing 5th, 25th, 50th, 75th, and 95th percentiles. Statistical significance (\(*\)) determined by comparing initial and final distributions to NE and SE actions using Student’s t-test (\(P < 0.001\) comparing to SE and \(P = 0.2\) comparing to NE at \(\alpha = 0\); \(P < 0.001\) comparing to NE and \(P = 0.5\) comparing to SE at \(\alpha = 1\)). (E) Cost distributions for each machine adaptation rate displayed using box plots with error bars showing 25th, 50th, and 75th percentiles. (F,G) One- and two-dimensional histograms of actions for different adaptation rates (\(\alpha \in \left\{ \text {0,0.003} \right\}\) in (F), \(\alpha \in \left\{ \text {0.3, 1} \right\}\) in (G)) with game-theoretic equilibria overlaid (NE in (F), SE in (G)).
