1.) Consider the discrete Bertrand game described in the Oligopoly lecture notes

Question

1.) Consider the discrete Bertrand game described in the Oligopoly lecture notes/video. According to the rules of this game each student selects a number from the set {0,1,2, 3, 4, 5, 6, 7, 8, 9, 10} and is randomly matched with another student. Whoever has the lowest number wins that amount in dollars and whoever has the high number wins zero. In the event of ties, each student receives half their number in dollars. What number would you select if you played this game in our online class? Explain your reasoning. 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.

Explanation / Answer

1.) In the above numbers of set {0,1,2,3,4,5,6,7,8,9,10}

Nash equilibrium will be attained when each candidate selects number 10 because in that case, each will recieve 5 dollars. But since, under Bertrand game each candidate selects the number simultaneously so one cannot observe the outcome of other and vice-versa.

While taking decision on number each student will calculate its return. For example : If first student selects 8 and second student selects 3 then, second student gains 3 dollars while first student wins zero. So, selection should be like that in which student have certainity of some winning. In the above scenario, if first student selects 1 by taking into account that (1) other cannot selects 0 then winning will be zero and (2) if student selects more than 1 then, he will earn 1$ and (3) in the case of tie, each willl earn $0.5. That means in all situations, he will win something.

If student selects number 2 then, it will be the case when other student selects 1 then, first student earns zero. Similarly, for every number from 2 to 10 there is uncertainity of earning any amount.

So, If I would be the player of this game then, I will definately select number 1 as outcome of selecting this number is somewhere certain rather than choosing any other number.

2.) 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

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