Chapter 9, Problem 1FCCE

What do the individual entries of a payoff matrix represent?

To determine

The representation of the individual entries of a payoff matrix.

Answer to Problem 1FCCE

Solution:

The individual entries of a payoff matrix represent the payoffs from player C to player R for various moves.

Explanation of Solution

A matrix whose rows and columns are labeled with the strategies of the players is called a payoff matrix.

Let us consider two players R and C, where R denotes row and C denotes column.

Then the payoff matrix is defined as:

$A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1n}\\ a_{21}& a_{22}& \dots & a_{2n}\\ ⋮& ⋮& \dots & ⋮\\ a_{m1}& a_{m2}& \dots & a_{mn}\end{array}\right]$

Where, the player R can make $R_1,R_2,\dots ,R_m$ moves and the player C can make $C_1,C_2,\dots ,C_n$ moves.

Entries of the payoff matrix, $a_{ij}$ represent $R_i$ move the player R and $C_j$ move by the player C.

That is,

$\begin{array}{l}\overline{\begin{array}{cccc}{C}_1& C_2& \dots & C_n\end{array}}\\ A=|\begin{array}{c}{R}_1\\ R_2\\ ⋮\\ R_m\end{array}\left[\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1n}\\ a_{21}& a_{22}& \dots & a_{2n}\\ ⋮& ⋮& \dots & ⋮\\ a_{m1}& a_{m2}& \dots & a_{mn}\end{array}\right]\end{array}$

For example,

Consider the payoff matrix, $A=\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right]$.

If player C selects the first column and player R selects the second row, then C pays $\$4$ to R.