At a certain warehouse, the price of tobacco per pound in dollars, $p$, is related to the supply of tobacco in pounds, $q$, by the formula


Thus the more tobacco farmers bring to the warehouse, the lower the price. However, a price support program ensures that the price never falls below .25 per pound. In other words, if the supply is so high that the price would be below .25 per pound, the price is set at $p = .25$, and a government agency purchases whatever cannot be sold at that price. One day three farmers are the only ones bringing their tobacco to this warehouse. Each has harvested 600,000 pounds and can bring as much of his harvest as he wants. Whatever is not brought must be discarded. Show that there are no Nash equilibria in which $q1 +q2 +q3 < 975000$, exactly one $qi$ equals 600,000, and the other $qi$s are strictly between 0 and 600,000.

I am aware of what a Nash equilibrium is and how to find one. What I am unsure of is how to show a Nash equilibrium doesn't exist under a certain set of conditions.


Without loss of generality, let $q_1'=600000$ and let $q_2'+q_3'=Q<375000$, $q_2',q_3'\in(0,600000)$.

To show that a NE does not exist, you need to demonstrate that at least one player $i$ can earn more by not playing $q_i'$. In other words, you'd show that $q_1',q_2',q_3'$ are not mutual best responses.

Obviously one such candidate is either $2$ or $3$. By increasing his quantity (which he can, because $q_2'$ is necessarily smaller than $600000$), say $q_2'+\epsilon$, Farmer $2$'s payoff is strictly higher by $.25\epsilon$. Therefore, he would not find $q_2'$ a best response to $1$ and $2$'s strategies. So no NE exists under the given conditions.

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