Technically it's not competing on price. It's competing on cost, a result of which is often, but not necessarily, a lower price point. Business strategy courses explicitly teach you that simply competing on price (that is, simply lowering prices in hopes that you'll beat the competition), will blow up in your face.
A classic example (presented to my MBA class) was the two adjacent pizza parlors in NYC competing on price and driving the selling price of a pizza down to less than a dollar, whilst a shop a block away and around a corner was able to keep normal pricing.
well, it's not so black and white as only competing on cost or price. it's true that your competitive advantage has to come from lower costs (via operations, marketing, finance, or whatnot), so you do compete with other market participants on cost.
but you also compete on price in the marketplace--not so much as to ignore your costs, as you note, since negative margins generally don't lead to viable businesses, but price competition nonetheless.
your example is the simplest game-theoretic version of price competition, which is more of a teaching model than a practical application.
(dynamic) price discrimination and other marketing tactics are typically employed to ensure positive margins even as you compete on price in the marketplace (e.g., airline seats).
A classic example (presented to my MBA class) was the two adjacent pizza parlors in NYC competing on price and driving the selling price of a pizza down to less than a dollar, whilst a shop a block away and around a corner was able to keep normal pricing.