This is the command 2nash that can be run in the OnWorks free hosting provider using one of our multiple free online workstations such as Ubuntu Online, Fedora Online, Windows online emulator or MAC OS online emulator

**PROGRAM:**

**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]

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