Promotion comparison model
Assumptions:
- Two items have prices p (cheaper) and q (pricier), 0 < p <= q.
- One qualifying pair; no caps, points, coupons, tax, minimum spend, or expiry effects.
- BOGO means cheaper item free: trip=q; average item cost=q/2.
- 50% off each means both items half price: trip=(p+q)/2; average=(p+q)/4.
- Second item half price: cheaper item is discounted: trip=q+p/2; average=q/2+p/4.
- Per-item cost is pair total divided by two items.
- For a one-bottle need, 50% off each permits one item at p/2; BOGO and second-item-half require buying a second item (pair totals above).

Break-even:
BOGO = 50% off each iff q=(p+q)/2, so q=p. If q>p, 50% off each is cheaper.
BOGO = second item half iff q=q+p/2, so p=0; with positive prices BOGO is always cheaper.
