Okay, so I'm hoping to lean on the expertise here for a problem I've been trying to solve (without luck) for the past few days. This isn't a problem I've seen anywhere, just something I came up with that I'm puzzled about.

So, we all know about the famous birthday paradox, where there is a greater than 50% chance of two or more individuals having the same birthday whenever at least 23 individuals are in the same room. We also all know about the famous Fibonacci sequence (i.e., 1, 1, 2, 3, 5, 8, 13 ...).

So, here's the problem. Let's say that there is a party to celebrate Chinese New Year given by a very rich host, and 23 very rich guests are invited by the host. The order of the guests is then randomized and each guest is given a ticket from 1-23 and must enter the host's house in that order. The first guest arrives with $1, the second guest arrives with $1, the third guest arrives with $2, the fourth guest arrives with $3, etc. Everyone agrees that the first guest who arrives that shares a birthday with one of the previous guests or the host must give the money they brought to the host (e.g., if first guest shares their birthday with the host, they must give the host $1, and if the second guest shares their birthday with either the host or the other guest, they must give the host $1, but the third guest will have to give $2 if they share the same birthday as the host or any of the other guests, etc.). We all know that there is better than a 50% chance that the host will get some amount of money by the time all 23 guests come the the house, but the host agrees to pay the 23rd guest the 23rd Fibonacci number worth of dollars (i.e., $28,657) if nobody has a shared birthday after the 23rd guest arrives.

So my question is this: how can we calculate the AVERAGE AMOUNT the host will get (i.e., if this experiment is run infinite times?) It won't be the 23rd Fibonacci number (i.e., $28,657), of course, because many times there will be a shared birthday long before the 23rd guest arrives. Will the host make or lose money, and how much?

TL;DR: Twenty-three (23) "Fibonacci" guests go a host's BD party and agree give the host money corresponding with the Fibonacci number they are given prior to their intrance if there is a shared birthday; otherwise, the host must pay $28,657 to the 23rd guest. Will the host make or lose money, how much, and what's the math to support it?

Hope this makes sense, and I'm looking forward to your responses, as I've banged my head over this one one too many times.