(Solved): game theory  (1) Rose and Colin play the following game. Rose picks either 1 or 4, and Colin pi ...

(1) Rose and Colin play the following game. Rose picks either 1 or 4, and Colin picks either 2 or 3. If the numbers picked are consecutive, then Colin pays Rose a payoff equal to the product of the numbers, and if they are not consecutive, then Rose pays Colin a payoff equal to the product of the numbers. (Consecutive means that the larger number is one bigger than the smaller: for example, 1 and 2 are consecutive, but 2 and 4 are not.) (a) Write down the payoff matrix for this game. (b) Determine whether there are saddle points. If not, find the optimal mixed strategies each player should use, and find the value of the game.