Maybe it's different for you, but where I live (in a largeish city in the US), most stores don't have direct competitors within easy walking distance. You've assumed a walk across the street, but out here it's probably a mile and a half drive and 2 unpleasant parking lots. That changes a lot of your scenarios -- while you'd be silly not to walk next door to save a few bucks, the time investment in many US cities is often too high to be worth going to cheaper shops. Looked at in another way, if I'm looking for something specific and I discover that I can get it from a store that is in close range, the value proposition of immediacy and reduced frustration for me can lead to a sale for that retailer even if their price isn't great.
You're also considering the value proposition symmetrically, when you should be considering it asymmetrically. That is, you're thinking "the store that sells this for $15 wouldn't use this app, because then the store selling it for $10 would steal all their business." That's a crappy value proposition for the $15 store, but a great one for the $10 store. Some stores might not use it, but those with better prices or better selections would basically consider it free advertizing.
Also, I'm pretty sure you're wrong about retailers not being able to compete if their prices are higher, and about this sort of app destroying those businesses. We already have this scenario with grocery stores, where you can compare them just by reading the paper (or the online ads), and yet certain more costly stores still survive. Maybe it's location/convenience, maybe it's unique selection, maybe it's customer service or return policies, or maybe it's that cute girl ringing up your groceries.
I think, ultimately, you're conceiving of this app as "how do I get the best price on X within a Y block radius?", whereas I'm conceiving of it as "can I get X from anywhere at all within a Y block radius?" and only secondarily "can I save money on X?" Phrased my way, there's an obvious value proposition for retailers.
You're also considering the value proposition symmetrically, when you should be considering it asymmetrically. That is, you're thinking "the store that sells this for $15 wouldn't use this app, because then the store selling it for $10 would steal all their business." That's a crappy value proposition for the $15 store, but a great one for the $10 store. Some stores might not use it, but those with better prices or better selections would basically consider it free advertizing.
Also, I'm pretty sure you're wrong about retailers not being able to compete if their prices are higher, and about this sort of app destroying those businesses. We already have this scenario with grocery stores, where you can compare them just by reading the paper (or the online ads), and yet certain more costly stores still survive. Maybe it's location/convenience, maybe it's unique selection, maybe it's customer service or return policies, or maybe it's that cute girl ringing up your groceries.
I think, ultimately, you're conceiving of this app as "how do I get the best price on X within a Y block radius?", whereas I'm conceiving of it as "can I get X from anywhere at all within a Y block radius?" and only secondarily "can I save money on X?" Phrased my way, there's an obvious value proposition for retailers.