Game Theory $-$ End of Chapter Problem

There are two ice cream vendors, Ben and Jerry, on a crowded strip of beach. There are five permissible locations for ice cream vendors to locate, cleverly named positions $\backslash (1,2,3,4\backslash ),$ and $5.$ Position $1$ is at the far north end of the beach, Position $5$ at the far south end, and Position $3$ in the middle, while Positions $2$ and $4$ are midway between their respective neighbors. Each location may accommodate more than one vendor. There are $1,000$ beachgoers scattered uniformly across the beach $(200$ at each position$)$ who will buy from the closest vendor.

a$.$ Fill in the payoff matrix below, with payouts reflecting the number of customers each vendor attracts. Each vendor gets the $200$ customers in his location plus $200$ customers in each adjoining location. For example: If Ben is at Position $1$ and Jerry is at Position $4,$ Ben will get $400$ beachgoers from locations $1$ and $2,$ and Jerry will get $600$ of the beachgoers from Positions $3,4,$ and $5.$

$\backslash$begin$\{$tabular$\}\{|$l$|$l$|$l$|$l$|$l$|$l$|$l$|\}$

$\backslash$hline $\backslash$multicolumn$\{2\}\{|$c$|\}\{\}$ & $\backslash$multicolumn$\{5\}\{|$c$|\}\{$ Jerry $\}\backslash \backslash$

$\backslash$cline $\{2-7\}$ & Position $1$ & Position $2$ & Position $3$ & Position $4$ & Position $5\backslash \backslash$

$\backslash$hline $\backslash$multirow$\{3\}\{*\}\{$ Ben $\}$ & Position $1$ & & & & & $\backslash \backslash$

$\backslash$hline & Position $2$ & & & & & $\backslash \backslash$

$\backslash$cline $\{2-7\}$ & & & & & $\backslash \backslash$

$\backslash$hline & Position $33$ & & & & & $\backslash \backslash$

$\backslash$hline & & & & & $\backslash \backslash$

$\backslash$hline & & & & & $\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

b$.$ The Nash equilibrium in this game $($where the ordering is $"($Ben$,$ Jerry$)")$ is