Chapter 7: Q13E (page 241)

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 the payoffs by a $2 \times 2$ matrix.
(b) What is the value of this game, and what are the optimal strategies for the two players?

Expert verified

The value of the game is $0$ and the optimal strategy of both the player will be equal i.e., $\frac{1}{2}$ .

Step by step solution

(a)It is given that for $R$ to win, the two coins must have same outcome, i.e., either both heads or both tails.

The above condition can be represented as:

The matrix represents the money $R$ got by game.

$H = Head, T = Tail$

$+ 1$ indicates that $R$ got a dollar whereas $- 1$ shows that $C$ got a dollar.

(b)

Let, Probability of $R$ to get head and tail be given as $X_{1}$ and $X_{2}$ .

And, Probability of $C$ to get head and tail be given as $y_{1}$ and $y_{2}$ .

Max: $z$ : Min: $w$

$\begin{matrix} z \leq x_{1} - x_{2} \\ z \leq - x_{1} + x_{2} \\ x_{1} + x_{2} = 1 \\ x_{1}, x_{2} > 0 \end{matrix}$

$\begin{matrix} w \geq y_{1} - y_{2} \\ w \geq - y_{1} + y_{2} \\ y_{1} + y_{2} = 1 \\ y_{1}, y_{2} > 0 \end{matrix}$

The value of this game is $0$ .

Therefore, the optimal strategy of both the player is equal to i.e., $\frac{1}{2}$ .

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