QUESTION 2 Four bidders bid for a certain object in a third

QUESTION 2 Four bidders bid for a certain object in a third price auction. They respectively value the object at V1-50 V2 70 V3 40 V4-25 Their values are commonly known. Assume that the bids must be integers Ties are resolved at random. What is an equilibrium in (weakly) undominated strategies? B1 69 82 70 B3 40 84 25 B1 70 B2 = 110 B3-50 B4\"35 UES TION 3

Solution

Only the last option is correct because every bid is a best response of the other players bid. Nobody could gain by deviation. For example suppose player 3 bids 111 and wins, now he has to pay 70 ( the new third highest bid), at which he is making a loss. Similarly we can think of all the other bids in the last option and find out that there is no profitable deviation.

Now consider option 1, here clearly player 1 could bid more than 70 and pay 40 and thus get a net payoff of 10 which is greater than the 0, he is getting currently therefore, there is a profitable deviation. Similarly it is true for option 2.

Now consider option 3, clearly player 2 could bid more than 80 and pay 40 and thus get a net payoff of 30 which is greater than the 0, he is getting currently therefore, there is a profitable deviation. Similarly it is true for option 4.