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## 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]

```

```       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
```