Question $2$

Two animals, A and B$,$ are fighting over some prey. Each can behave like a

dove or like a hawk. The best come for each animal is that in which it acts

like a hawk while the other acts like a dove; the worst outcome is that in which

both animals act like hawk. Each animal prefers to be hawkish if its opponent

is dovish and dovish if its opponent is hawkish.

Note that for each of A and B$,$ his$/$her available action is to take $$Dove$$ or

$$Hawk$.$ Therefore, for A$,$ his strategy set is specified by SA $\{$dove$,$ hawk$\}.$

Likewise, for B$,$ her strategy set is specified by SB $\{$dove$,$ hawk$\}.$ Then, a

strategic$-$form game hN ; SA$,$ SB ; $($uA$,$ uB $)$i with N $=\{$A$,$ B$\}$ is specified by: for

the payoff function uA : SA $$ SB $$ R of the player A$,$ we have:

uA $($hawk$,$ dove$)=6$

uA $($dove$,$ dove$)=5$

uA $($dove$,$ hawk$)=2$

uA $($hawk$,$ hawk$)=0$

while for the payoff function uB : SA $$ SB $$ R of the player B$,$ we have:

uB $($dove$,$ hawk$)=12$

uB $($dove$,$ dove$)=10$

uB $($hawk$,$ dove$)=4$

uB $($hawk$,$ hawk$)=0.$

Given this game, answer the following questions:

$(1)$ Describe the game matrix of this game.

$(2)$ Consider the mixed strategy extension of this game.

$2$

$(2-$a$)$ Then, dfine the expected payoff functions of players A and B respec$-$

tively.

$(2-$b$)$ Derive the best response correspondences of players A and B respec$-$

tively.

$(2-$c$)$ Compute what pure$-$strategy Nash equilibria of this game are.

$(2-$d$)$ Compute the mixed$-$strategy Nash equilibrium in this game