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!
