gambit-logit reads a game on standard input and computes the principal branch of the (logit) quantal response correspondence.
The method is based on the procedure described in Turocy [Tur05] for strategic games and Turocy [Tur10] for extensive games. It uses standard path-following methods (as described in Allgower and Georg's "Numerical Continuation Methods") to adaptively trace the principal branch of the correspondence efficiently and securely.
The method used is a predictor-corrector method, which first generates a prediction using the differential equations describing the branch of the correspondence, followed by a corrector step which refines the prediction using Newton's method for finding a zero of a function. Two parameters control the operation of this tracing. The option -s sets the initial step size for the predictor phase of the tracing. This step size is then dynamically adjusted based on the rate of convergence of Newton's method in the corrector step. If the convergence is fast, the step size is adjusted upward (accelerated); if it is slow, the step size is decreased (decelerated). The option -a sets the maximum acceleration (or deceleration). As described in Turocy [Tur05], this acceleration helps to efficiently trace the correspondence when it reaches its asymptotic phase for large values of the precision parameter lambda.
gambit-logit
-d
Express all output using decimal representations with the specified number of digits. The default is -d 6.
-s
Sets the initial step size for the predictor phase of the tracing procedure. The default value is .03. The step size is specified in terms of the arclength along the branch of the correspondence, and not the size of the step measured in terms of lambda. So, for example, if the step size is currently .03, but the position of the strategy profile on the branch is changing rapidly with lambda, then lambda will change by much less then .03 between points reported by the program.
-a
Sets the maximum acceleration of the step size during the tracing procedure. This is interpreted as a multiplier. The default is 1.1, which means the step size is increased or decreased by no more than ten percent of its current value at every step. A value close to one would keep the step size (almost) constant at every step.
-m
Stop when reaching the specified value of the parameter lambda. By default, the tracing stops when lambda reaches 1,000,000, which is usually suitable for computing a good approximation to a Nash equilibrium. For applications, such as to laboratory experiments, where the behavior of the correspondence for small values of lambda is of interest and the asymptotic behavior is not relevant, setting MAXLAMBDA to a much smaller value may be indicated.
-l
While tracing, compute the logit equilibrium points with parameter LAMBDA accurately.
-S
By default, the program uses behavior strategies for extensive games; this switch instructs the program to use reduced strategic game strategies for extensive games. (This has no effect for strategic games, since a strategic game is its own reduced strategic game.)
-h
Prints a help message listing the available options.
-e
By default, all points computed are output by the program. If this switch is specified, only the approximation to the Nash equilibrium at the end of the branch is output.
Computing the principal branch, in mixed strategies, of e02.nfg
<../contrib/games/e02.nfg>, the reduced strategic form of the example in Figure 2 of Selten (International Journal of Game Theory, 1975):
$ gambit-logit e02.nfg
Compute a branch of the logit equilibrium correspondence
Gambit version 16.1.0b1, Copyright (C) 1994-2023, The Gambit Project
This is free software, distributed under the GNU GPL
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...
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