# Nash equilibrium in second price sealed-bid auction

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.

## 1 Answer

In scenario $$A$$, only the highest bidder bids their true value and everyone else bids $$0$$. None of the other players can increase their payoff by changing their strategy alone, as no one values the item above $$v_1$$. This makes it a weakly dominant strategy.

In scenario $$B$$, the bidder valuing the item at $$v_1$$ bids $$v_2$$ and the bidder valuing the item at $$v_2$$ bids $$v_1$$, with $$v_1 > v_2$$. In this scenario, Player 2 is actually bidding $$v_1 - v_2$$ above their true value, making it beneficial for this player alone to switch to bidding lower than $$v_2$$ to obtain $$0$$ value instead of his current negative value. Thus this is not a weakly dominant strategy.