# Second-price sealed-bid auction uniformly independent with unknown value

a disclaimer before the question: this is a homework problem. I just want some help/push in the right direction, I'm kind of stuck! The problem is as follows:

In a second-price sealed-bid auction for a single item, there are three bidders. The value of their bids are unknown without extensive research, but are uniform and independent values in [0, 10]. Bidders 1 and 2 did their homework, and now know their values as $v_1$ and $v_2$ respectively. They do not know each other's values.

Bidder 3 does not know his value because he did not do his research. He does know that he has the same value as bidder 2, although he does not know what $v_2$ is specifically. He also does not know $v_1$. He knows that both bidders 1 and 2 know their values.

(a) How should bidder 2 bid in this auction? How should bidder 1 bid?

(b) How should bidder 3 behave in this auction?

My attempt:

(a) For any second-price sealed-bid auction, it is always optimal to bid your true value. Thus bidder 2 and bidder 1 will bid $v_2$ and $v_1$, respectively, especially knowing that both of them know their own true values but not each other's.

(b) Bidder 3 only knows that he has the same value as $v_2$ and that both bidder 1 and bidder 2 will bid their true value (since he knows they know their values). The probability of $v_2$ and $v_1$ taking each value from 0 to 10 is $\frac{1}{11}$.

But I'm stuck here because how could he choose an optimal value when he knows nothing about what $v_1$ and $v_2$ are or his own value?

Any guidance is appreciated. Thanks!

(b): Think about it, what is the expected payoff of bidder 3 if he knew his valuation? Suppose $v_2=v_3>v_1$, then the second highest valuation is $v_2$, so he has a payoff of zero when playing his dominant strategy (doesn't matter whether he receives the item or not). Or, if $v_1>v_2=v_3$, then he does not get the item, and also has a payoff of zero.
Now think about what happens if, because 3 does not know his valuation, bids incorrectly. If his bid is $b$, then if $b>v_2$ he gets payoff 0 if $v_1>b>v_2$ or a negative payoff if $b>v_1>v_2$. If $b<v_2$, then he gets payoff 0 in any case. We already established the payoff is zero if $b=v_2$. Hence, no matter what bidder 3 does, the best he can get is a payoff of zero.