I'm trying to understand the basics of game theory and the topic of auctions have come up. What I want to know is second priced bids always a nash equilibrium?

Suppose we have this question.

We consider a second price sealed-bid auction with complete information. We have n bidders, n ≥ 2. There is only one object in the auction. Player i, i = 1, . . . , n, evaluates the object by giving it a valuation vi , where:

v1 > v2 > v3 > . . . > vn > 0

Each player i submits a sealed bid bi , i = 1, . . . , n. So, we can describe a bidding profile of all players as (b1, b2, b3, . . . , bn).

A) Is the bidding profile (v1, v2, v3, . . . , vn), i.e., the one where every player bids her valuation of the item, a Nash equilibrium of the game?

B) Is the bidding profile (v1, 0, 0, . . . , 0) a Nash equilibrium of the game?

C) Is the bidding profile (v2, v1, 0, . . . , 0) a Nash equilibrium of the game?

Surely it's always a nash equilibrium becuase 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 on questions A, B and C as I thought they would all have nash equilibrium regardless.

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