Any bundle in which both Guy and someone else approach Carol yields (c) and has reward value 0. Oh, so that's how it works? No wonder this situation seems out

Any bundle in which both Guy and someone else approach Carol yields (c) and has reward value 0. Oh, so that's how it works? No wonder this situation seems out of balance. Instead, I would suggest that in the p_star case, N is only the number of other guys who (from Guy's perspective) are more interesting (attractive, well-spoken, outgoing) which helps to keep N down for any suitably attractive, confident Guy. That changes Carol's p_none too. Now she's getting the product across some distribution of p_stars from guys around her, each of which is related to the apparent value of the dating pool and individual confidence. I haven't worked the math, but it definitely seems like p_none < 1/2 only when none of the guys are very confident in their worth. I don't really think you needed to use game theory to get that result.