## Scenario

Any game in which the identity of the player does not change the relative order of the resulting payoffs facing that player.
In other words, each player ranks the payoffs from every strategy combination in the same order.

## Description

A game is ordinally symmetric if the ordinal ranking of one player's payoffs is equivalent to the ordinal ranking of the transpose of the other player's payoffs.
If the transpose of the other player's matrix is equivalent (not just ordinally equivalent), then the game is

cardinally symmetric (or just symmetric).

## Example

| |
Player 1 |

| |
A |
B |

Player 2 |
A |
1,5 |
0,6 |

B |
3,1 |
4,8 |

Note that player 1's strategies are the matrix:

which transposed gives the following matrix (ranks in parentheses):

1 (3rd) |
3 (2nd) |

0 (4th) |
4 (1st) |

which is ordianlly equivalent to player 2's strategy matrix:

5 (3rd) |
6 (2nd) |

1 (4th) |
8 (1st) |

## General Form

| |
Player 2 |

| |
L |
R |

Player 1 |
U |
a,w |
b,y |

D |
c,x |
d,z |

such that the ranking of a,b,c,d is equivalent to the ranking of w,x,y,z, respectively.

updated: 12 August 2005

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