I am kind of lost on this problem and would like it if I can get help on this.
Matching Pennies. In this simple two player game, the players (call them R and C) each choose an outcome, heads or tails. If both outcomes are equal, C gives a dollar to R; if the outcomes are different, R gives a dollar to C.
A. represent payoffs by a 2 x 2 matrix.
B. What is the value of the game, and what are the optimal strategies for the two players?
Explain why your answer is optimal.
EDIT (per amWhy)
To better understand what is being asked here, I found the following description of the game in a discrete math text (many varieties of this problem exist):
"Consider the game of matching coins. Two players A and B each toss a coin. If the coins match i.e. both are heads or both are tails, A gets rewarded; otherwise B gets rewarded."
Translating per this problem: Two players R and C each toss a coin. If both coins match, C gives a dollar to R; otherwise, R gives a dollar to C. So in this case, if both coins match, R gains a dollar, C loses a dollar; if coins don't match, C gains a dollar, R loses a dollar.
Best Answer
The Nash equilibrium strategy for a 2-person zero-sum game can be computed through linear programming. In your case, the payoff-matrix is $$ \begin{array}{r}H \\ T \end{array} \left[ \begin{array}{rr} -1 & 1 \\ 1 & -1 \end{array} \right] $$
or $$ e_{ij} = \left\{ \begin{array}{rr} 1 & i = j \\ -1 & \mbox{otherwise} \end{array} \right.$$
The LP is $$\mbox{minimize} \ v$$ subject to $$ - y_H + y_T \le v \\ y_T - y_H \le v \\ y_T + y_H = 1 \\ y_T, y_H \ge 0 $$ which is clearly optimal at $y_T = y_H = {1 \over 2}$