Theory states that GSP auctions induce truthful bidding. Is it the case that this is true ONLY IF
a) each of the bidders truthfully bids their value ($b_i = v_i$) (each bidder's optimal strategy) AND
b) each bidder assumes that all of the other bidders are also truthfully employing their own optimal strategy ?
My question arises out of the following possible example:
Two bidders bidding for a single item, Vickery sealed-bid second-price auction.
The bidders have equal values for the same item (say $v_1 = v_2 = 10$)
$b_1 = 7$
$b_2 = 9$
$b_2$ will win the auction and pay $7$; with a surplus payoff of $2= (9-7)$. This outcome is equivalent to the example where:
$v_1 = b_1 = 7$
$v_2 = b_2 = 9$
$b_2$ wins in either example but neither bidder is using the optimal strategy in the first example. What logic am I missing?