$1.$ Consider the following game:

W

U

Player $1$

D

$8,3$

$4,2$

$3,7$

Player $2$

X

$0,4$

$1,5$

$0,1$

Y

$4,4$

$5,3$

$2,0$

$9,4$

$3,2$

$9,9$

For each of the following beliefs of player $1$ about player $2\text{'}$s strategies, determine the best

responses of player $1.$ Note that for each belief the first, second, third and fourth numbers are

the probabilities assigned to W$,$ X$,$ Y$,$ and Z$,$ respectively.

a$)$ P$1=(0,0,0,1)$

$2.$ In the game below, denote the probabilities that player $2\text{'}$s belief assigns to the

strategies of player $1,$ U and D by $2($U$)$ and P$2($D$).$

Player $2$

L

M

R

Player $1$

U

$3,2$

$2,5$

$7,3$

D $4,3$

$15,0$

$1,1$

a$)$ State the relationship that holds between pz$($U$)$ and pz$($D$).$

b$)$ Find the expected payoff of L$,$ M and R for Player $2$ given their belief about player $1\text{'}$s

strategies. Then express these expected payoffs only in terms of p$2($U$).$

c$)$ Graph the expected payoffs of L$,$ M and R with respect to p$2($U$).$

d$)$ Determine the best response of player $2$ for different values of P$2($U$).$

e$)$ Which strategies are never a best response for Player $2?$ Does player $1$ have any

strategies that are never a best response? Why or why not?

f$)$

Finally, solve the game through the iterated elimination of "never best response"

strategies $($hint: part e should allow you to eliminate one of the strategies of player $2.$

From that point forward, you will have a $2$ by $2$ matrix, and the iterated elimination of

"strictly dominated" strategies will give you the same answer as the iterated elimination

of "never best response" strategies$).$

$3.$ In the previous game, imagine that neither player can form probabilistic beliefs about

the other player's strategies. Find the maximin strategy of each player. Make sure you

show your reasoning.