0.19 Game theory: homework

Determine whether the games are strictly determined. If the games are strictly determined, find the optimal strategies for each player and the value of the game.

1. $\left[\begin{array}{cc}1& 2\\ -2& 3\end{array}\right]$

2. $\left[\begin{array}{cc}6& 3\\ 2& 1\end{array}\right]$

3. $\left[\begin{array}{ccc}-1& -3& 2\\ 0& 3& -1\\ 1& -2& 4\end{array}\right]$

4. $\left[\begin{array}{ccc}2& 0& -4\\ 3& 4& 2\\ 0& -2& -3\end{array}\right]$

5. $\left[\begin{array}{cc}0& 2\\ -1& -1\\ -1& 1\\ 3& 2\end{array}\right]$

6. $\left[\begin{array}{ccc}5& -3& 2\\ 3& -1& 4\end{array}\right]$

Two players play a game which involves holding out one or two fingers simultaneously. If the sum of the fingers is more than 2, Player II pays Player I the sum of the fingers; otherwise, Player I pays Player II the sum of the fingers.

1. Write a payoff matrix for Player I.
2. Find the optimal strategies for each player and the value of the game.

A mayor of a large city is thinking of running for re-election, but does not know who his opponent is going to be. It is now time for him to take a stand for or against abortion. If he comes out against abortion rights and his opponent is for abortion, he will increase his chances of winning by 10%. But if he is against abortion and so is his opponent, he gains only 5%. On the other hand, if he is for abortion and his opponent against, he decreases his chance by 8%, and if he is for abortion and so is his opponent, he decreases his chance by 12%.

1. Write a payoff matrix for the mayor.
2. Find the optimal strategies for the mayor and his opponent.

A man accused of a crime is not sure whether anybody saw him do it. He needs to make a choice of pleading innocent or pleading guilty to a lesser charge. If he pleads innocent and nobody comes forth, he goes free. However, if a witness comes forth, the man will be sentenced to 10 years in prison. On the other hand, if he pleads guilty to a lesser charge and nobody comes forth, he gets a sentence of one year and if a witness comes forth, he gets a sentence of 3 years.

1. Write a payoff matrix for the accused.

2. If you were his attorney, what strategy would you advise?

Determine the optimal strategies for both the row player and the column player, and find the value of the game.

1. $\left[\begin{array}{cc}-1& 1\\ 1& -1\end{array}\right]$

2. $\left[\begin{array}{cc}1& -1\\ -4& 0\end{array}\right]$

3. $\left[\begin{array}{cc}3& -2\\ 2& 4\end{array}\right]$

4. $\left[\begin{array}{cc}-3& 2\\ 1& -4\end{array}\right]$