bimatrix game

class="infobox" | $\backslash mathbf\{A\}\; =\; \backslash begin\{bmatrix\}$ c & d & e \\ f & g & h \\ \end{bmatrix} | $\backslash mathbf\{B\}\; =\; \backslash begin\{bmatrix\}$ p & q & r \\ s & t & u \\ \end{bmatrix}

colspan="2"|$$ \begin{array}{c|c|c|c} & X & Y & Z \\ \hline V & c,p & d,q & e,r \\ W & f,s & g,t & h,u \\ \end{array} A payoff matrix converted from A and B where player 1 has two possible actions V and W and player 2 has actions X, Y and Z

In game theory, a bimatrix game is a simultaneous game for two players in which each player has a finite number of possible actions. The name comes from the fact that the normal form of such a game can be described by two matrices - matrix $A$ describing the payoffs of player 1 and matrix $B$ describing the payoffs of player 2.{{cite web | url=http://www.utdallas.edu/~chandra/documents/6311/bimatrix.pdf | title=Bimatrix games | accessdate=17 December 2015 | author=Chandrasekaran, R}}

Player 1 is often called the "row player" and player 2 the "column player". If player 1 has $m$ possible actions and player 2 has $n$ possible actions, then each of the two matrices has $m$ rows by $n$ columns. When the row player selects the $i$-th action and the column player selects the $j$-th action, the payoff to the row player is $A[i,j]$ and the payoff to the column player is $B[i,j]$.

The players can also play mixed strategies. A mixed strategy for the row player is a non-negative vector $x$ of length $m$ such that: $\backslash sum\_\{i=1\}^m\; x\_i\; =\; 1$. Similarly, a mixed strategy for the column player is a non-negative vector $y$ of length $n$ such that: $\backslash sum\_\{j=1\}^n\; y\_j\; =\; 1$. When the players play mixed strategies with vectors $x$ and $y$, the expected payoff of the row player is: $x^\backslash mathsf\{T\}\; A\; y$ and of the column player: $x^\backslash mathsf\{T\}\; B\; y$.