What is equilibrium dependent upon in Generalized Second-Price Auctions? 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?
The surplus is the value of the object obtained minus the price paid. In your first example, the surplus would be $10-7 = 3$. The reason why you don't find examples where BOTH players are not using optimal strategies is because $b=v$ is weakly dominant. Unlike a lot of games where you need to consider best response functions, this one can be solved simply by removing dominated strategies.

Using your first example, there exists no bids that will give a higher payoff than $b_i=10$ regardless of what $b_{-i}$. Suppose we look at player 1, then consider all cases:

$b_2 &lt; b_1 &lt; v_1$: surplus is the same as $b_1=v_1$ 

$b_1 &lt; b_2 &lt; v_1$: surplus is 0, $b_1=v_1$ is profitable deviation

$b_1 &lt; v_1 &lt; b_2$: surplus is the same as $b_1=v_1$ 

$v_1 &lt; b_1$ surplus is negative or 0 (depending on $b_2$), $b_1=v_1$ is profitable deviation 

Also, auctions are usually used for a non dividable object, which means that examples with multiple players with the same value require a tie breaking rule. Whether or not this weak dominance survives depends on this tie breaking rule.