Question 1

Find the value and optimal strategies for the zero sum games corresponding to the matrices \[\text{(a)}\quad \left(\begin{array}{rr} 4 & -3 \\ -9 & 6\end{array}\right)\quad \text{(b)}\quad \left(\begin{array}{rr} 3 & 1 \\ 5 & 7\end{array}\right)\quad \text{(c)}\quad \left(\begin{array}{rr} -3 & -4 \\ -7 & 2\end{array}\right).\]

Question 2

Your friend proposes that you play the following game. Each of you independently chooses heads or tails. If you both call heads your friend will pay you 3 pounds, if you both call tails your friend will pay you 1 pound. Otherwise you pay your friend 2 pounds. Formulate this as a matix game and solve it. Should you agree to play?

Question 3

Use dominance and a graphical method to solve the zero sum game with matrix \[\left(\begin{array}{rr} 0 & 5 \\ 1 & 4\\ 3 &0\\ 2 & 1\end{array}\right).\]

Question 4

Use dominance and a graphical method to solve the zero sum game with matrix \[\left(\begin{array}{rrr} 0 & 8& 5 \\ 8 & 4&6\\ 12 & -4&3\end{array}\right).\]

Question 5

Solve the zero sum game with matrix \[\left(\begin{array}{rrrr} 3 &-2 & 4& 7 \\ -2 & 8&4&0\end{array}\right).\]

Question 6

In this question we will investigate whether bluffing in Poker is ever a good idea, by considering a simplified version of part of the game.

Suppose that player one is dealt a card which is either an Ace or a King (with equal probabilities). Having seen their own card, player one can choose to Fold or Bet. If player one folds they pay player two $1. If player one bets player two either Folds or Bets. If they fold then they pay player one $1. Otherwise player one pays player two $2 if the card dealt was the king, and player two pays player one $2 if the card was an ace.

Represent this game in normal form and solve it.