gambit-liap: Compute Nash equilibria using function minimization

gambit-liap reads a game on standard input and computes approximate Nash equilibria using a function minimization approach.

This procedure searches for equilibria by generating random starting points and using conjugate gradient descent to minimize the Lyapunov function of the game. This is a nonnegative function which is zero exactly at strategy profiles which are Nash equilibria.

Note that this procedure is not globally convergent. That is, it is not guaranteed to find all, or even any, Nash equilibria.

Changed in version 16.2.0: The Lyapunov function is now normalized to be independent of the scale of the payoffs of the game; therefore multiplying or dividing all payoffs by a common factor will not affect the output of the algorithm.

The criterion for accepting whether a local constrained minimizer of the Lyanunov function is an approximate Nash equilibrium is specified in terms of the maximum regret. This regret is interpreted as a fraction of the difference between the maximum and minimum payoffs in the game.

-d

Express all output using decimal representations with the specified number of digits.

-n

Specify the number of starting points to randomly generate.

-i

New in version 16.1.1.

Specify the maximum number of iterations in function minimization (default is 1000).

-m

New in version 16.2.0.

Specify the maximum regret criterion for acceptance as an approximate Nash equilibrium (default is 1e-4). See Acceptance criteria for Nash equilibria for interpretation and guidance.

-h

Prints a help message listing the available options.

-q

Suppresses printing of the banner at program launch.

-s

Specifies a file containing a list of starting points for the algorithm. The format of the file is comma-separated values, one mixed strategy profile per line, in the same format used for output of equilibria (excluding the initial NE tag).

-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.)

-v

Sets verbose mode. In verbose mode, initial points, as well as points at which the minimization fails at a constrained local minimum that is not a Nash equilibrium, are all output, in addition to any equilibria found.

Computing an equilibrium in mixed strategies of e02.efg, the example in Figure 2 of Selten (International Journal of Game Theory, 1975):

$ gambit-liap e02.nfg
Compute Nash equilibria by minimizing the Lyapunov function
Gambit version 16.2.0, Copyright (C) 1994-2024, The Gambit Project
This is free software, distributed under the GNU GPL

NE, 0.998701, 0.000229, 0.001070, 0.618833, 0.381167