What does "voting with their bellies" mean?
The real challenge is the public's appetite. There's a burger restaurant boom in major cities. People are voting with their bellies and it's not normally mushroom burgers they're after!

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This is a variation on the idiom 'voting with your feet'.

If people aren't happy somewhere, they 'vote ( i.e. express their disapproval) with their feet', or in other words, they leave. Or if a place is good, the public will vote ( i.e. express their support) with their feet, and go there.

So if people vote with their bellies, they are showing their approval of these restaurants by  going there in large numbers and eating.

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It means people choose their desire food by their stomach.