Problem $1.(8$ points$)$ Reporting a crime with unknown number of witnesses. Consider a variant of problem $2$ in Problem Set $6.$ There are two players, each of whom does not know if she is the only witness to a crime, or if there is another witness. Let denote the probability each witness assigns to being the sole witness. There are $3$ possible states: only player $1$ witnesses the crime, only player $2$ witnesses the crimes, and both players witness the crime. $[$To be clear. We know that both players witnessed the crime. But they don$$t$.]$ The players must simultaneously decide whether to call the police $($C$)$ or not $($N$).$ Each player who reports incurs a cost c $>0.$ Both players assign a value v $>$ c to the police being informed. Only one player need come forward for this to occur. Hence, reporting agents receive a pay$-$off of v $$ c$,$ and non$-$reporting agents receive a payoff of v $($if the other agent reports$)$ and $0$ otherwise. $1.$ Find a condition on under which the game has a pure strategy Nash equilibrium in which both players call the police. $2.$ When the condition is violated, find the symmetric mixed strategy Nash equilibrium of the game.