Why does integrity matter?

Integrity. We admire it. We demand it in our leaders (and criticise them when they lack it). Organisations (like the police service) write it into their values. We aspire to it ourselves, wanting people to see us as having it. But why does integrity matter? Why all the fuss about it? What’s the basis for it? And do we need to think very carefully about some of life’s bigger questions if integrity is going to make any sort of sense?