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2.) Continue to consider this discrete Bertrand model, but now assume that each student has a constant cost of 5 that is deducted from all payoffs. So whoever has the low number wins their number, minus 5. Whoever has the high number loses 5 total. In the event of a tie, each student wins an amount equal to their number divided by two, then minus five. Find any Nash equilibria in this game. Explain your reasoning. Hint: It is perfectly fine for both players to have losses in equilibrium! There are more than 1 Nash equilibria.

Solution

Answer- Nash equilibrium is a solution concept of non-cooperative game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy. That outcome will be the nash equilibrium where each student is satisfied and not willing to change. But in the above scenario there is no nash equilibria because :

(i) If each student selects the highest number i.e. 10 then, each student wins [(10/2) - 5] i.e. 0.

(ii) If each student selects the number less than 10 then, outcome of student will be negative.

In the above game, only one student can wins a positive amount. So, there is no Nash equilibria of his game.