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.
I understand that a weakly dominant strategy in a second price sealed-bid auction is to always bid the amount how much the item is worth to you. However I'm having trouble with some example questions that I've been given.
We have $n$ bidders, $n \ge 2$. There is only one object in the auction. Player $i, i = 1, . . . , n,$ evaluates the object by giving it a valuation $v_i$ , where:
$$v_1 > v_2 > v_3 > . . . > v_n > 0$$
Each player i submits a sealed bid $b_i \ , i = 1, . . . , n$. So, we can describe a bidding profile of all players as $(b_1, b_2, b_3, . . . , b_n)$.
Now what I what I want to understand is are both the listed bidding profiles below nash equilibrium?
A) bidding profile $(v_1, 0, 0, . . . , 0)$
B) bidding profile $(v_2, v_1, 0, . . . , 0)$
Surely both are a nash equilibrium because every bidder will bid there valuation for the item thus no one has any incentive to change there bid? E.G a weakly dominant strategy
I just wanted some clarification, any help on the matter would be greatly appreciated.