Consider a third-price sealed-bid auction auction with three bidder indexed $i=1,2,3$ and private, independent values $v_1 \sim U(0,1)$. Bidders simultaneously and independently make a bid $b_i$
The question then is:
Is it a BNE for each bidder to bid truthfully, i.e. $b_i=v_i$? Explain.
Based on this I formulated this answer:
No. Consider an example that shows that this is not a BNE. Lets assume $v_1 > v_2 > v_3$ and that all players bid their values: $(b_1,b_2,b_3) = (v_1,v_2,v_3)$. Then player 2 does not win the auction, so has payoff zero ($u_2 = 0$). If player 2 increases his bid to $b_2 = v_1 + \varepsilon$, where $\varepsilon > 0$, then player 2 wins the auction and gets payoff $u_2 = v_2 - v_3 > 0$, which is more then when he bids his value $v_2$. Hence, there is a profitable deviation for the player with the middle value, so the proposed equilibrium is not a BNE.
My doubt about this answer is that the bidder with the middle value does not this, since he does not know the values of the other bidders. For rejecting the proposed BNE, do I need to take into account the posterior beliefs of player 2 being the middle bidder (using the CDF of the uniform distribution) or is it enough that in one state of the world (when he is the middle bidder) bidding higher than his value, and higher than player 1's value, is a profitable deviation?