Question

Consider a second-price sealed-bid auction. Suppose bidders' valuations are v1=10 and v2=10. Select all that apply. (PLEASE EXPLAIN CHOICES COMPLETEY)

a.	One bidder submitting a bid equal to 10 and the other submitting a bid equal to 0 is a Nash equilibrium.
	b.	Bidding a value b1 equal to her own valuation v1 is a weakly dominated strategy for bidder 1.
	c.	Both bidders submitting bids equal to 0 is a Nash equilibrium.
	d.	Both bidders submitting bids equal to 10 is a Nash equilibrium.

Answer 1

Firstly, to understand the solution the definition of Nash equilibrium has to be laid out clearly.

A nash equilibrium is a set of choices of each player such that there is no unilateral profitable deviation possible. That is no one player can be better off by choosing another bundle other than what is currently being chosen, given the other person's choice.

Now, let us consider each option.
Option A:One bidder submitting a bid equal to 10 and the other submitting a bid equal to 0 is a Nash equilibrium.

Let us consider two cases. First when player 1 choses to bid 10, and player 2 choses to bid 0. Second, when player 1 chooses to bid 0, and player 1 chooses to bid 10.

Case 1: When player 1 chooses to bid 10, and player 2 chooses to bid 0.
Player 1 gets the object, and receives a payoff of v1-b2=10-0=10.
Player 2 doesn't get the object, so the payoff is 0.

If player 1 chooses a higher bid, she will have the same payoff, so no profitable deviation there.
If player 1 chooses a lower bid than what she is currently bidding, but higher than player 2, still she will have the same payoff, so no profitable devitation there as well.
If player 1 chooses a lower bid than what she is currently bidding, and equal to player 2's bid, i.e. 0, then the prize is distributed equally, and her payoff is 5, which is less than her ealier payoff of 10, so she is worse off.
Therefore, player 1 does not have a profitable devitation.

If player 2 increases her bid, to 0<b2<10, she still doesn't get the object, so no profitable deviation.
If player 2 increases her bid, to b2=10, then she will get the half the prize and pay half the price, still no profitable devitation.
If player 2 increases her bid, to b2>10, then she will get the prize, but pay v1=10 for it, so essentially payoff is still zero.
Therefore, player 2 also does not have a profitable deviation
Therefore, Option A is correct.

Option B:Bidding a value b1 equal to her own valuation v1 is a weakly dominated strategy for bidder 1.

Let's take player 2's bid as given. So, there will be 3 cases.
1) b2<10.
Given player 2's bid. if player 1 bids b1=10, then he will make positive profit, if he bids b2<b1<10, then also +ve profit. If he bids b1<b2, then he doesn't get the object and makes 0 profit.
Therefore, we have proved that option B is incorrect. We need not investigate further.

Option C: Both bidders submitting bids equal to 0 is a Nash equilibrium.

If both bidders bid equal to 0, then any one can increase her bid and gain positive profit, since she still has to pay 0, and win the object.
Therefore, Option C is incorrect.

Option D: Both bidders submitting bids equal to 10 is a Nash equilibrium.

If for eg, player 1 deviates to a bid higher than 10, then still the payoff is 0, making her indifferent.
If her deviates to a bid lesser than 10, then she loses the object, and still has a payoff of 0.
So, there is no profitable deviation.

Option D is correct.