$1.$ Bud and Wise are the only two producers of aniseed beer, a New Age product designed to displace root beer. Bud and Wise are trying to figure out how much of this new beer to produce. They know:

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i$).$ If they both produce $10,000$ litres a day, they will make the maximum attainable joint economic profit of $$200,000$ a day, or $$100,000$ a day each.

ii$).$ If either firm produces $20,000$ litres a day while the other produces $10,000$ litres a day, the one that produces $20,000$ litres will make an economic profit of $$150,000$ and the other will incur an economic loss of $$50,000.$

iii$).$ If both produce $20,000$ litres a day, each firm will make zero economic profit.

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a$).$ Construct a payoff matrix for the game that Bud and Wise must play.

b$).$ Find the Nash equilibrium for the game.