5) Quantal response equilibria#

Gambit implements the idea of McKPal95 and McKPal98 to compute Nash equilibria via path-following a branch of the logit quantal response equilibrium (LQRE) correspondence using the function logit_solve. As an example, we will consider an asymmetric matching pennies game from Och95 as analyzed in McKPal95.

[1]:

import pygambit as gbt

[2]:

g = gbt.Game.from_arrays(
        [[1.1141, 0], [0, 0.2785]],
        [[0, 1.1141], [1.1141, 0]],
        title="Ochs (1995) asymmetric matching pennies as transformed in McKelvey-Palfrey (1995)"
)
gbt.nash.logit_solve(g).equilibria[0]

[2]:
$\left[[0.5000000234106035, 0.49999997658939654],[0.19998563837426647, 0.8000143616257336]\right]$

logit_solve returns only the limiting (approximate) Nash equilibrium found. Profiles along the QRE correspondence are frequently of interest in their own right. Gambit offers several functions for more detailed examination of branches of the QRE correspondence.

The function logit_solve_branch uses the same procedure as logit_solve, but returns a list of LQRE profiles computed along the branch instead of just the limiting approximate Nash equilibrium.

[3]:

qres = gbt.qre.logit_solve_branch(g)
len(qres)

[3]:

193

[4]:

qres[0].profile

[4]:
$\left[[0.5, 0.5],[0.5, 0.5]\right]$
[5]:

qres[5].profile

[5]:
$\left[[0.5182276540742868, 0.4817723459257562],[0.49821668880066783, 0.5017833111993909]\right]$

logit_solve_branch uses an adaptive step size heuristic to find points on the branch. The parameters first_step and max_accel are used to adjust the initial step size and the maximum rate at which the step size changes adaptively. The step size used is computed as the distance traveled along the path, and, importantly, not the distance as measured by changes in the precision parameter lambda. As a result the lambda values for which profiles are computed cannot be controlled in advance.

In some situations, the LQRE profiles at specified values of lambda are of interest. For this, Gambit provides logit_solve_lambda. This function provides accurate values of strategy profiles at one or more specified values of lambda.

[6]:

qres = gbt.qre.logit_solve_lambda(g, lam=[1, 2, 3])
qres[0].profile

[6]:
$\left[[0.5867840364385154, 0.4132159635614846],[0.4518070316997103, 0.5481929683002897]\right]$
[7]:

qres[1].profile

[7]:
$\left[[0.6175219458400859, 0.3824780541599141],[0.3719816648492249, 0.6280183351507751]\right]$
[8]:

qres[2].profile

[8]:
$\left[[0.6168968501329284, 0.3831031498670716],[0.31401636202001226, 0.6859836379799877]\right]$

LQRE are frequently taken to data by using maximum likelihood estimation to find the LQRE profile that best fits an observed profile of play. This is provided by the function logit_estimate. We replicate the analysis of a block of the data from Och95 for which McKPal95 estimated an LQRE.

[9]:

data = g.mixed_strategy_profile([[128*0.527, 128*(1-0.527)], [128*0.366, 128*(1-0.366)]])
fit = gbt.qre.logit_estimate(data)
type(fit)

[9]:

pygambit.qre.LogitQREMixedStrategyFitResult

The returned LogitQREMixedStrategyFitResult object contains the results of the estimation. The results replicate those reported in McKPal95, including the estimated value of lambda, the QRE profile probabilities, and the log-likelihood.

Because data contains the empirical counts of play, and not just frequencies, the resulting log-likelihood is correct for use in likelihoood-ratio tests. [1]

[10]:

print(fit.lam)
print(fit.profile)
print(fit.log_like)

1.8456097536855862
[[0.615651314427859, 0.3843486855721409], [0.38329094004562914, 0.6167090599543709]]
-174.76453191087447

All of the functions above also support working with the agent LQRE of McKPal98. Agent QRE are computed as the default behavior whenever the game has a extensive (tree) representation.

For logit_solve, logit_solve_branch, and logit_solve_lambda, this can be overriden by passing use_strategic=True; this will compute LQRE using the reduced strategy set of the game instead.

Likewise, logit_estimate will perform estimation using agent LQRE if the data passed are a MixedBehaviorProfile, and will return a LogitQREMixedBehaviorFitResult object.

Footnotes:

The log-likelihoods quoted in McKPal95 are exactly a factor of 10 larger than those obtained by replicating the calculation.

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