qre_for_multiple_lambdas.rst

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.

.. tabs::

    .. group-tab:: Table

        ======  =========  =========  =========  =========  ==========
        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
        ======  =========  =========  =========  =========  ==========

        .. code-block:: python

            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

    .. group-tab:: Arrays

        .. code-block:: python

            import sgamesolver
            import numpy as np

            payoff_matrix = np.array([[[10, 1],
                                    [8, 5]],
                                    [[10, 8],
                                    [1, 5]]])
            game = sgamesolver.SGame.one_shot_game(payoff_matrix=payoff_matrix)
            game.action_labels = ['stag', 'hare']

            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 :numref:`qre_for_multiple_lambdas`.

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