Q$.4($b$)$ Two Lassi vendors, A and B operate in Pilani. $($Lassi is a yogurt$-$based drink popular in North India$).$ The population of Pilani can be placed in two categories, the folks inside BITS, Pilani campus, and the locals. In a particular month, the vendors expect to sell $10,000$ glasses of the drink, out of which the campus folks are expected to drink $6000$ glasses while the locals would consume the remaining $4000.$ The vendors can set their per$-$glass prices at Rs$.40$ or Rs$.50.$ It has been noted that the campus folks do not care for the price difference and choose the vendors randomly, while the locals only buy from the cheaper vendor. $($If vendor A sets his price at Rs$.40$ and B at $50,$ the campus folks' business will get equally divided between the two vendors at $3000$ glasses each while all the locals would buy from Vendor A alone. The vice versa is also true.$)$ Assume that the cost of preparing Lassi is zero. We shall structure the interaction u$$ing the principles of game theory. $($Please note that this is a one$-$shot and simultaneous game where the firms would decide their prices simultaneously, and they cannot be changed mid$-$way.$)$

$($i$)$ The pricing options the vendors can exercise are the "strategies" employed in the game. As you can see, four strategic combinations are possible for the two vendors. Calculate the vendors' overall profits $($pay$-$offs$)$ for the four strategic options and fill in the pay$-$off matrix below. The overall profit would include profits from both types of customers, campus folks, and locals.

$[4]$

$\backslash$table$[[,$B$\text{'}$s Strategy$],[\backslash$table$[[$A$\text{'}$s$],[$Strategy$]],,$Rs$.40,$Rs$.50],[$Rs$.40,,],[$Rs$.50,,]]$

$($ii$)$ Solve the game theory problem whose matrix is given below. Show your workings.

$[4]$

$\backslash$table$[[,$B$\text{'}$s Strategy$],[,,$Strategy$-$I,$\backslash$table$[[$Strategy$-],[$II$]]],[,\backslash$table$[[$Strategy$-],[$I$]],(200,200),(280,150)$