Problem $1.(20$ points$)$ Two drivers simultaneously approach an intersection from different directions. Each driver can either stop $($S$)$ or continue $($C$).$ The drivers$$ preferences are given in the following payoff matrix:

SC

S $1,11$$,2$

C $2,1$ $0,0$

where the parameter in $(0,1)$ reflects each driver$$s aversion to being the only one who stops.

Find the pure strategy Nash equilibria of this game.

Find the mixed strategy Nash equilibrium of this game.

In the mixed strategy Nash equilibrium what is the likelihood of both drivers continuing $($and thus being in a crash$)?$ How does this probability vary with $?$ Provide some intuition for why this is the case.

How would the equilibria of the game change if $>1?$

Now, suppose that drivers are $($re$)$educated to feel guilty about choosing to continue.

The new payoffs becomes:

SC

S $1,11$$,2$

C $2$$,1$ $$$,$

where $>0$ is a measure of the agent$$s guilt, and in $(0,1).$ Find the new pure and

mixed strategy Nash equilibria. $[$Hint: Your answers may depend on the size of $.]$