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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.

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Could you clarify your description of the Matching Pennies game? I take it R and C each predict an outcome. Is one coin tossed? Or do each toss a coin? What do you mean if "both outcomes": both predictions, or the actual result(s) of the tossed coin(s)? As it is currently written, I'm afraid your description renders your question rather un-anwerable. –  amWhy Nov 2 '12 at 18:38
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1. This does not appear to be a question about linear programming. 2. What criterion do you use for optimality? There are many (though probably they all give the same solution for this game). –  Austin Mohr Nov 2 '12 at 18:41
    
Your question is similar to question 5 here: austinmohr.com/Math_170_(Spring_2012)_files/test1sol.pdf –  Austin Mohr Nov 2 '12 at 18:45
    
Sorry but "What is the value of the game": it passes time (in a rather unimaginative way), and "What are the optimal strategies for the two players": to be friends of course. –  dardisco Aug 24 '13 at 6:51
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1 Answer 1

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 $y_T = y_H = {1 \over 2}$

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What are the optimal strategies?? –  Rheem Nov 2 '12 at 23:16
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