For example, if there are 3 players then the long-term odds are that each game costs each player 25 cents. If there are 5 players, the average cost goes down to 6.3 cents per game. If there are 7 players, they make on the average 3.1 cents per game. If there are 9 players they make about 9 cents per game.
It isn't clear to me why this should be so.
The issue is in the payout structure you suggest, which is that if you win you get $2, and if you lose, you pay $1. This is not an even-money proposition. If your chances of winning are exactly 1/3, then for every three times you play you will (on average) pay $1 twice and win $2 once, which is break-even. Therefore, you have a positive expectation if your winning chances are any greater than 1/3.
In three-player Flip-Flop, your winning chances are only 1/4, so the three-player game is a bad bet even given this generous payout structure. However, as you add players, your chances of winning tend towards 50% (but never quite reach it). Very quickly, your winning chances will become greater than 1/3, and the game will suddenly have a positive expectation for you, and a negative one for the house.
If the casino wants to guarantee profits, it must adjust its payout structure to an even-money proposition. In other words, losers pay $1, and winners get $1. As you add more players, your winning chances improve, but they're still always slightly less than 50%, so the game will always have a negative expectation for the players.
As a side note, the common parlance in betting is that you pay a certain amount up front (the "bet"), and then if you win you get a certain amount back, while if you lose you get nothing. In this way of speaking, an even-money proposition would be to bet $1 and get $2 back if you win. The bet that you proposed was equivalent to betting $1 and getting $3 back when you win, which is better than even-money.