NAME
Game::Theory::TwoPersonMatrix - Analyze a 2 person matrix game
VERSION
version 0.1503
SYNOPSIS
use Game::Theory::TwoPersonMatrix;
$g = Game::Theory::TwoPersonMatrix->new(
1 => { 1 => 0.2, 2 => 0.3, 3 => 0.5 },
2 => { 1 => 0.1, 2 => 0.7, 3 => 0.2 },
payoff => [ [-5, 4, 6],
[ 3,-2, 2],
[ 2,-3, 1] ]
};
$g->row_reduce();
$g->col_reduce();
$p = $g->saddlepoint();
$o = $g->oddments();
$e = $g->expected_payoff();
$c = $g->counter_strategy($player);
$g = Game::Theory::TwoPersonMatrix->new(
1 => { 1 => 0.1, 2 => 0.2, 3 => 0.7 },
2 => { 1 => 0.1, 2 => 0.2, 3 => 0.3, 4 => 0.4 },
payoff1 => [ [5,3,8,2],[6,5,7,1],[7,4,6,0] ],
payoff2 => [ [2,0,1,3],[3,4,4,1],[5,6,8,2] ],
);
$t = $g->mm_tally();
$m = $g->pareto_optimal();
$n = $g->nash();
$e = $g->expected_payoff();
DESCRIPTION
A Game::Theory::TwoPersonMatrix
analyzes a two person matrix game of player names, strategies and utilities ("payoffs").
The players must have the same number of strategies, and each strategy must have the same size utility vectors as all the others.
Players 1 and 2 are the "row" and "column" players, respectively. This is due to the tabular format of a matrix game:
Player 2
--------
Strategy 0.5 0.5
Player | 0.5 1 -1 < Payoff
1 | 0.5 -1 1 <
A non-zero sum game is represented by two payoff profiles, as above in the SYNOPSIS.
METHODS
new()
$g = Game::Theory::TwoPersonMatrix->new(
1 => { 1 => '0.5', 2 => '0.5' },
2 => { 1 => '0.5', 2 => '0.5' },
payoff => [ [1,0],
[0,1] ]
);
$g = Game::Theory::TwoPersonMatrix->new(
payoff1 => [ [2,3],[2,1] ],
payoff2 => [ [3,5],[2,3] ],
);
Create a new Game::Theory::TwoPersonMatrix
object.
Player strategies are given by a hash reference of numbered keys - one for each strategy. Payoffs are given by array references of lists of outcomes. For zero-sum games this is a single payoff array. For non-zero-sum games this is given as two arrays - one for each player.
expected_payoff()
$e = $g->expected_payoff();
Return the expected payoff value of a game.
s_expected_payoff()
$g = Game::Theory::TwoPersonMatrix->new(
1 => { 1 => '(1 - p)', 2 => 'p' },
2 => { 1 => 1, 2 => 0 },
payoff => [ ['a','b'], ['c','d'] ]
);
$s = $g->s_expected_payoff();
Return the expected payoff expression for a non-numeric, zero-sum game.
Using real payoff values, we solve the resulting expression for p in the eg/ examples.
counter_strategy()
$c = $g->counter_strategy($player);
Return the counter-strategies for a given player of a zero-sum game.
saddlepoint()
$p = $g->saddlepoint;
If the 2x2 zero-sum game is strictly determined, the saddlepoint is returned. Otherwise undef
is returned.
oddments()
$o = $g->oddments();
Return each player's "oddments" for a 2x2 zero-sum game.
row_reduce()
$g->row_reduce();
Reduce a zero-sum game by identifying and eliminating strictly dominated rows and their associated player strategies.
col_reduce()
$g->col_reduce();
Reduce a zero-sum game by identifying and eliminating strictly dominated columns and their associated opponent strategies.
mm_tally()
$t = $g->mm_tally();
For zero-sum games, return the maximum of row minimums and the minimum of column maximums. For non-zero-sum games, return the maximum of row and column minimums.
pareto_optimal()
$m = $g->pareto_optimal();
Return the Pareto optimal outcomes for a non-zero-sum game.
nash()
$n = $g->nash();
Identify the Nash equilibria in a non-zero-sum game.
Given payoff pair (a,b)
, a is maximum for its column and b is maximum for its row.
SEE ALSO
The eg/ and t/ scripts in this distribution.
"A Gentle Introduction to Game Theory"
http://www.amazon.com/Gentle-Introduction-Theory-Mathematical-World/dp/0821813390
http://books.google.com/books?id=8doVBAAAQBAJ
AUTHOR
Gene Boggs <gene@cpan.org>
COPYRIGHT AND LICENSE
This software is copyright (c) 2014 by Gene Boggs.
This is free software; you can redistribute it and/or modify it under the same terms as the Perl 5 programming language system itself.