gambit-nfg-logit [-d DECIMALS] [-s STEP] [-a ACCEL] [-m MAXLAMBDA] [-l LAMBDA] [-S] [-h] [-q] [-e]
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]. 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.
The default values of these parameters are chosen to work reasonably well on games in which the payoffs of the players are in the range [-10,10]. Since lambda has a cardinal, and not just ordinal, effect on the perturbed payoffs, scaling the payoffs in the game also scales the interpretation of lambda; these parameters should thus be adjusted for use in games with larger payoff ranges.
The default values of the parameters are also chosen with the goal of tracing the entire branch to determine the terminal Nash equilibrium and general shape of the branch relatively quickly. Users interested in using the output of this method for maximum likelihood analysis of data from laboratory experiments should set the initial stepsize smaller, and the maximum acceleration closer to 1, in order to compute more points on the branch.
-d DECIMALSExpress all output using decimal representations with DECIMALS digits.
-s STEPSets 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 ACCELSets 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 MAXLAMBDAStop when reaching the specified value of the lambda parameter. 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 LAMBDAWhile tracing, compute the logit equilibrium points with parameter LAMBDA accurately.
-SBy 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.)
-hPrints a help message listing the available options.
-eBy 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.