$30$ points$)$ This is a long question, which should give you enough practice in com$-$

puting mixed$-$strategy equilibria. Suppose Ben and Jerry play the following game,

in which Ben is the row player and Jerry is the column player.

l m r

U $111041$

M $232225$

D $531122$

$($a$)$ Are there dominant or strictly dominated strategies in this game?

$1$

$($b$)$ Are there any completely mixed$-$strategy equilibria in the game $($that is$,$ equi$-$

libria in which each player chooses each of his pure strategies with positive

probability$)?$ If yes, find them.

$($c$)$ Are there any mixed$-$pure strategy equilibria in the game $($i$.$e$.,$ equilibria in

which one player plays a pure strategy and the other mixes$)?$ If yes, find them.

Guidance: for each pure strategy of one player, check what the best response is

of the other player. Only if the other player has more than one best response

could he be mixing, so you only have to check pure strategies such that the

other player has more than one best response. In these cases, you have to find

probabilities of mixing for the other player such that the first player would like

to play the pure strategy and not deviate.

$($d$)$ Are there any mixed$-$strategy equilibria in which each player mixes between

two strategies? If yes, find them. Guidance: consider all the combinations of

mixing for each player $($there are $3$

$2$

$3$

$3$

$2=9$ such combinations$).$ For each

combination and each player compute the probabilities of mixing that make the

other player indifferent between the pure strategies he is mixing over. Then

check if any of the players want to deviate to the strategy they are not mixing

over.