NASH(1) 							   lrslib 0.42b 							   NASH(1)

NAME

       nash - find nash equilibria of two person noncooperative games

SYNOPSIS

       setupnash input game1.ine game2.ine

       setupnash2 input game1.ine game2.ine

       nash game1.ine game2.ine

       2nash game1.ine game2.ine

DESCRIPTION

       All Nash equilibria (NE) for a two person noncooperative game are computed using two interleaved reverse search vertex enumeration steps.
       The input for the problem are two m by n matrices A,B of integers or rationals. The first player is the row player, the second is the
       column player. If row i and column j are played, player 1 receives Ai,j and player 2 receives Bi,j. If you have two or more cpus available
       run 2nash instead of nash as the order of the input games is immaterial. It runs in parallel with the games in each order. (If you use
       nash, the program usually runs faster if m is <= n , see below.) The easiest way to use the program nash or 2nash is to first run setupnash
       or ( setupnash2 see below ) on a file containing:

	     m n
	     matrix A
	     matrix B

       eg. the file game is for a game with m=3 n=2:

	     3 2

	     0 6
	     2 5
	     3 3

	     1 0
	     0 2
	     4 3

	     % setupnash game game1 game2

       produces two H-representations, game1 and game2, one for each player. To get the equilibria, run

	     %	nash game1  game2

       or

	     %	2nash game1  game2

       Each row beginning 1 is a strategy for the row player yielding a NE with each row beginning 2 listed immediately above it.The payoff for
       player 2 is the last number on the line beginning 1, and vice versa. Eg: first two lines of output: player 1 uses row probabilities 2/3 2/3
       0 resulting in a payoff of 2/3 to player 2.Player 2 uses column probabilities 1/3 2/3 yielding a payoff of 4 to player 1. If both matrices
       are nonnegative and have no zero columns, you may instead use setupnash2:

	     % setupnash2 game game1 game2

       Now the polyhedra produced are polytopes. The output  of nash in this case is a list of unscaled probability vectors x and y. To normalize,
       divide each vector by v = 1^T x and u=1^T y.u and v are the payoffs to players 1 and 2 respectively. In this case, lower bounds on the
       payoff functions to either or both players may be included. To give a lower bound of r on the payoff for player 1 add the options to file
       game2  (yes that is correct!)To give a lower bound of r on the payoff for player 2 add the options to file game1

	     minimize
	     0 1 1 ... 1    (n entries to begiven)
	     bound   1/r;    ( note: reciprocal of r)

       If you do not wish to use the 2-cpu program 2nash, please read the following. If m is greater than n then nash usually runs faster by
       transposing the players. This is achieved by running:

	    %  nash game2  game1

       If you wish to construct the game1 and game2 files by hand, see the lrslib user manual[1]

SEE ALSO

       For information on H-representation file formats, see the man page for lrslib or the lrslib user manual[2]

NOTES

	1. lrslib user manual
	   http://cgm.cs.mcgill.ca/%7Eavis/C/lrslib/USERGUIDE.html#Nash%20Equilibria

	2. lrslib user manual
	   http://cgm.cs.mcgill.ca/%7Eavis/C/lrslib/USERGUIDE.html#File%20Formats

July 2009							    03/30/2011								   NASH(1)