Computing QRE for multiple lambdas

When it comes to the QRE homotopy, one might not only be interested in the limiting stationary equilibrium, but also in quantal response equilibria for intermediate values of \(\lambda\). Therefore, sGameSolver can perform path following not only until convergence, but also to specific target values of the homotopy parameter.

Example: Stag hunt

Consider the following version of the stag hunt game.

		player1
		stag	hare
player0	stag	10, 10	1, 8
	hare	8, 1	5, 5

We implement the game and prepare the solver as follows.

TableArrays

state	a_player0	a_player1	u_player0	u_player1	phi_state0
delta			0	0
state0	stag	stag	10	10	0
state0	stag	hare	1	8	0
state0	hare	stag	8	1	0
state0	hare	hare	5	5	0

import sgamesolver

game = sgamesolver.SGame.from_table('path/to/table.xlsx')

homotopy = sgamesolver.homotopy.QRE(game)
homotopy.solver_setup()
homotopy.solver.verbose = 0  # make silent

Let’s define the values of \(\lambda\) that we are interested in and set up an empty container for the corresponding quantal response equilibria. Since the game and thus all quantal response equilibria are symmetric, we only need to keep track of the strategies of player 0.

lambdas = np.arange(0.1, 2.1, 0.1)
strategies = np.zeros(shape=(len(lambdas), 2), dtype=np.float64)

Now we can iterate over all values of \(\lambda\), let sGameSolver compute the corresponding quantal response equilibria and save them in our strategies container.

for idx, lambda_ in enumerate(lambdas):
    homotopy.solver.t_target = lambda_
    homotopy.solve()
    strategies[idx] = homotopy.equilibrium.strategies[0, 0]  # state_0, player_0

Finally, we can use the quantal response equilibria for further analysis or for plotting.

import matplotlib.pyplot as plt
plt.plot(lambdas, strategies[:, 0], label='stag')
plt.plot(lambdas, strategies[:, 1], label='hare')
plt.xlabel(r'$\lambda$')
plt.ylabel('strategy')
plt.legend()
plt.show()

The resulting picture is shown in Fig. 2.

Fig. 2 Quantal response equilibria (all symmetric) in the stag hunt game for different values of the precision parameter \(\lambda\).