I'm trying to understand the basics of game theory and the topic of auctions has arisen. I understand the basic concepts of auctions but I'm struggling with second price sealed-bid auctions when externalities are added. In particular I cannot find a systematic way to determine the strategies which will be adopted by each player, like in the case of the following problem. Do you have some suggestions? Thank you very much.
Consider dividing an indivisible object. Suppose that buyer 1 values a good at v1 while buyer 2 values the good at v2 but incurs a cost of 3 utils if buyer 1 gets the good. Both v1 and v2 are distributed uniformly on [0,10].Suppose the object is allocated via a second price auction (with no reserve price). Does buyer 2 have a weakly dominant strategy?
The formulation with the cost is just a distraction. Since there’s no reserve price, there are only two outcomes; either buyer $1$ gets the good, or buyer $2$ gets it. So a negative value to buyer $2$ when buyer $1$ gets it has the same effect as a positive value to buyer $2$ when buyer $2$ gets it: You can imagine buyer $2$ paying the cost of $3$ up front and winning it back if she wins the good. So the value of the good to buyer $2$ is effectively $v_2+3$. Thus she has the weakly dominant strategy that a buyer has in any second-price sealed-bid auction: to bid the true value of the good to her, $v_2+3$.