Fig. 2

Conjectural variation in policy space (Experiment 2, \(n = 20\)). Experimental setup and costs are the same as Fig. 1A,B except that the machine uses a different adaptation algorithm: in this experiment M iteratively implements policies \(m = L_M h\), \(m = L_M + \delta\) to measure and best-respond to conjectures of the human’s policy and updates the policy slope \(L_M\). (A) Median actions, conjectures, and policies for each conjectural variation iteration k overlaid on game-theoretic equilibria corresponding to best-responses (BR) at initial and limiting iterations (BR\(\phantom{0}_0\) and BR\(\phantom{0}_\infty\), respectively) predicted from Stackelberg equilibrium (SE) and consistent conjectural variations equilibrium (CCVE) of the game, respectively. (B) Action distributions for each iteration displayed by box-and-whiskers plots as in Fig. 1D. (C) Policy slope distributions for each iteration displayed with the same conventions as B; note that the sign of the top y-axis is reversed for consistency with other plots. Statistical significance (\(*\)) determined by comparing action distribution at iteration \(k=9\) to SE and CCVE using Hotelling’s \(T^2\) test (\(P < 0.001\) for SE and \(P = 0.06\) for CCVE). (D) Cost distributions for each iteration displayed using box-and-whiskers plots as in Fig. 1E. (E) Error between measured and theoretically-predicted machine conjectures about human policies at each iteration displayed as box-and-whiskers plots as in B,C. (F,G) One- and two-dimensional histograms of actions for different iterations (\(k=0\) in F, \(k=9\) in G) with policies and game-theoretic equilibria overlaid (SE and BR\(\phantom{0}_0\) in F, CCVE and BR\(\phantom{0}_\infty\) in G).