The term "bidding war" doesn't typically apply to initial offers.

The concept of bidding war includes having to pay more than someone else to obtain something. If a market is so hot that the seller has no reason to even wait for competing offers, then the initial offer has to be high enough to make it worth their while. The initial is your bid, and because you predict other buyers will be able to satisfy the seller's demands, you bid defensively by bidding extra.

A single shot isn't a war.

A single shot that decides the outcome is an assassination, and that's generally done to avoid a war that will have worse outcomes, if we want to stretch this metaphor to the breaking point.