We do it both ways. We hold different kinds of re-buy tournaments, and the rules regarding the bounty chip differ depending upon the type of tournament:
1. optional bounty chip
Since the bounty chip is an optional purchase, it is surrendered when a player is all-in and loses. If the player chooses to re-buy, he has the option of buying another bounty chip. If a player is all-in and loses to a non-bounty player, he retains the bounty chip.
2. multiple re-buy tournaments
Each player may re-buy multiple times during the re-buy period (usually the first hour, and only if he was all-in and lost). The re-buy cost is the same as the original buy-in, which includes the purchase of another bounty chip. In most cases, this bounty chip value is small in comparison to the entry fee ($5 bounty for a $40 entry fee, for example).
3. single re-buy tournaments
Each player may only re-buy once, and the re-buy cost does not include the purchase of another bounty chip. In most cases, this bounty chip value is large in comparison to the entry fee ($10 bounty for a $30 entry fee, for example). A player surrenders his bounty chip only when knocked out of the tournament for good. This encourages players to re-buy, to protect their bounty chip investment. Sometimes this format also uses re-buy chips, which can be redeemed for free add-on chips if not used by the end of the re-buy period.
4. super-bounty tournaments
These tournaments combine different portions of the previous formats. A re-buy tournament with a 'super' bounty also has a 'regular' bounty which is relatively small but required ($55 entry fee with a $5 bounty). The super bounty purchase is optional, and is relatively large ($20 for that same $60 entry cost). A player loses the regular bounty when he is all-in and loses, but does not lose the super bounty until completely eliminated from the tournament. A subsequent re-buy of $60 would include another $5 'regular' bounty chip, and he would retain the super bounty chip since he was not completely eliminated. However, if he chose to not re-buy and had lost his all-in to another super bounty player, he would surrender the super bounty chip to that player who knocked him out. If his final elimination is at the hands of a non-super-bounty player, he retains the super bounty which is redeemed for cash.
Not nearly as confusing as all that might seem. Rules are always made clear beforehand, and it allows the action junkies to gamble larger amounts than the tighter players prefer to lose. In one of the tournament league series we play, there is also a house bounty placed on the previous tournament winner, which is won by the player who knocks him out of the tournament for good -- but he retains it if he can and does re-buy (if still during the re-buy period).