Homework $08$: Game Theory Assignment $3$

Question $1.$ For the following games, identify the best response functions of each player, any dominant strategies that exist, which pairs of strategies yield maximum social welfare, and any Nash Equilibria.

Q$1)$a$)$ The Battle of the Sexes $($V$1)$

A couple has agreed to meet this evening, but cannot recall if they will be attending the Boxing Match or a Dance $($and the fact that they forgot is common knowledge$).$ The Person $1$ would prefer to go to the boxing match, and Person $2$ would rather go to the Dance. Both would prefer to go to the same place rather than different ones. The payoff matrix is:

Person $1$

$\backslash$table$[[$Person $2],[,$Boxing Match,Dance$],[$Boxing Match,$4,2,0,0],[$Dance$,0,0,2,4]]$

$B_1($ Boxing $)=$

$B_1($ Dance $)=$

$B_2($ Boxing $)=$

$B_2($ Dance $)=$

Dominant Strategies for Player $1$: $q,$

Dominant Strategies for Player $2$: $q,$

Social Welfare Maximum Strategies: $q,$

Nash Equilibrium: $q,$

Q$1)$b$)$i$)$ Stag Hunt $($V$1)$

Two hunters live in the forest. They will spend the day hunting for either a stag or a hare. If both hunters stalk the stag, they are likely to flush it out and eat like kings. If either hunter chooses to hunt a hare, they will likely catch it and eat well for the night. If just one hunter goes after the stag alone, they are unlikely to catch it$.$

$B_1($ Stag $)=$

$B_1($ Hare $)=$

$B_2($ Stage $)=$

$B_2($ Hare $)=$

Hunter $1$

$\backslash$table$[[$Hunter $2],[,$Stag,Hare$],[$Stag$,2,2,0,1],[$Hare$,1,0,1,1]]$

Dominant Strategies for Hunter $1$: $q,$

Dominant Strategies for Hunter $2$: $q,$

Social Welfare Maximum Strategies: $q,$

Nash Equilibrium:

Math $106$

$1$ of $4$