Q13E Matching pennies. In this simple... [FREE SOLUTION]

91影视

Algorithms

Sanjoy Dasgupta, Christos H. Papadimitriou, Umesh Vazirani

Computer Science

1st Edition 路 333 pages

ISBN: 9780073523408

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

Short Answer

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

Represent the payoffs by a matrix.

(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:

	$H$	$T$
$H$	$+ 1$	$- 1$
$T$	$- 1$	$+ 1$

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.

Calculate the value of this game and the optimal strategies for the two players

(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: $w$

$\begin{matrix} z 鈮� x_{1} 鈭� x_{2} 鈥夆赌夆赌夆赌夆赌夆赌夆赌� \\ z 鈮� 鈭� x_{1} + x_{2} 鈥夆赌夆赌夆赌� \\ x_{1} + x_{2} = 1 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆€� \\ x_{1}, x_{2} > 0 鈥夆赌嬧赌嬧赌夆赌夆赌� \end{matrix}$