# 2nash - Online in the Cloud

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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