 nash(1) 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 m[blue]lrslib user manualm[]