Whatever get's decided, hopefully it will apply equally for hand rank 1 and hand rank 11.
Although for a shootout the stakes are much lower than for lowball - because in a shootout a "perfect tie" results in the leader choosing casualties first, effectively becoming the "loser" in a way (just not for the sake of actual card effects). In lowball, a "perfect tie" creates a scenario where the phase can't actually resolve!
I could see a few things being considered here if a "perfect tie" was allowed to stand:
1) Redraw lowball hand and try again (with same bonuses, if any).
2) The Winner from last turn carries over to this turn - players get their antes back.
3) The Winner from last turn carries over to this turn - and gets to keep the antes too.
4) All unbooted dudes boot into the Town Square for a grand Shootout (kidding!)
Number 1 may be a little physically card-drawing intensive in some match-ups (think: four Heretic Jokers circulating). Number 2 & 3 fall apart when this happens on the first lowball of the game. Number 4 might make a good game tie-breaker rule...
I am reminded of a rule that has always bothered me. Did anybody watch the Superbowl? I consider it incredibly inelegant to take a game about "skill" and go about making a major decision (such as who gets "initiative") decided by an out-of-game chance element (a coin-flip). This is actually one of the reasons why I like watching Basketball much better than Football.
In Classic, back when Unique dudes were truly Unique (barring Bio-engineering), if both players revealed, say, Jake Smiley in their starting posse, only one player got to keep him. And it wasn't resolved by a stinking coinflip! It was resolved by a Pull, which, being an in-game chance element, I thought was entirely clever. And more importantly, fair. Because you can build for it.
So what if there was a way in DTR, failing the initial Lowball, to determine initiative (for Grifters and other Reacts and maybe even "perfect ties") that utilized in-game chance elements? Where is my Jump Ball!