Now that the tears of merriment have dried on your cheeks and you've got your breath back, you can actually turn to a real-life example of what happens when you do indeed have two competition authorities. And they fall out with each other.
Australia's got the ACCC: if you've got a merger for 'clearance' (no competition issues), you go to them. And it's got the Australian Competition Tribunal: if you've got a merger for 'authorisation' (competition issues, but overall net benefit), they'll handle it. And then along comes the "mega-merger of the nation's two biggest gambling companies", as the Sydney Morning Herald called it here (possibly $): Tabcorp and Tatts Group, both ASX-listed.
You'll find the full history of what happened next if you google, but save yourself the bother and go to Melbourne Law School Professor Julie Clarke and her very useful Australian Competition Law blog, especially this post about the Tabcorp/Tatts deal. It helpfully includes links to media coverage, too.
The gist: Tabcorp/Tatts tried the ACCC first. No deal, the ACCC saw actual or potential competition problems (here's the Statement of Issues). So Tabcorp/Tatts bailed out and went to the Tribunal.
"Competition problems? What competition problems?" - the Tribunal didn't see them. Au contraire: "The creation of the Merged Entity will lead to greater competition particularly in online wagering" (my emphasis, from para  of the judgement). On net benefits, it saw at  assorted ups and no downs: "The benefits to the public which the Tribunal has found to exist, and which it has taken into account, are substantial. There are no material detriments weighed in the balance which are of significance or likely to arise that outweigh the benefits". Slam dunk - authorisation approved.
I can only imagine the gasket-blowing within the ACCC. But once the little red dots in front of their eyes had cleared, they took the Tribunal to the Full Federal Court for judicial review, on three grounds. CrownBet, a Tabcorp/Tatts competitor, added a fourth: it was a courageous (in the Sir Humphrey Appleby sense) claim that the Tribunal had lost its mind. Predictably, that claim crashed and burned, some bits burning more fiercely than others (at ,"We have experienced some difficulty in distilling CrownBet’s submission about this into a readily comprehensible form").
Not so the ACCC's. Two of its claims got knocked back, but one made it, as you can read in paras  to  of the judgement: the Tribunal missed a detriment (reduced competition between Tabcorp and Tatts in online betting), and you can't be sure of the overall net benefit if you've missed a detriment. So it's back to the Tribunal to fix, and in the meantime the merger is in limbo.
So it's been an interesting exercise. As a general principle, I like the idea of more checks and balances when it comes to the exercise of regulatory (or any other) powers: as one example our High Court 'inputs methodology merits review' of how the Commerce Commission goes about the business of price control was well worth doing. But you wonder when you see two competition authorities slugging it out. That's beginning to look like overkill in a country already liberally equipped with multiple layers of governments and regulators - not that we in New Zealand can point fingers too vigorously, with our 78 territorial authorities for a total population the size of Melbourne.
By the by, we may hear a little more of this case, as it had something useful to say about how regulators should go about the weighing up of benefits and detriments, which is one of the bigger issues in the current NZME/Fairfax merger appeal.
You'll recall that the Commerce Commission took the view that the merger would involve a loss of 'plurality', less variety in the range of opinions on offer in the mainstream media. And while it didn't know what the value of that detriment was, it looked to be significantly larger than the claimed merger benefits.
Can that rough and ready, 'in the round' assessment fly? Or are you prohibited from saying one thing is bigger than another, if you have no real clue how big the first thing is?
Here I'm straying well off the economists' reservation into the lawyers' farmland, but I reckon the Federal Court's judgement is helpful to the 'in the round' approach. Here's what it said (shorn of unnecessary bits):
 Having examined the benefits and detriments resulting from, or likely to result from, the proposed acquisition, the Tribunal is then to determine whether the overall benefit is ‘such’ that the acquisition should be permitted. This requires a balancing exercise to determine the public benefit. The Tribunal has referred to this as a balance-sheet approach ... and this is an informative metaphor. It may suggest, however, that the detriments are to be deducted from the benefits leaving only a net benefit. This is informative but may be likely to be a little unrealistic. Many of the benefits and detriments will be incommensurable and possibly unmeasurable as well. To take an example from this case: how does one weigh the improved efficiency of the wagering market against the perils of problem gambling? It seems to us that the benefits and detriments may more usefully be assayed by means of a process of ‘instinctive synthesis’ sometimes referred to in the law surrounding the formulation of criminal sentences where a similar problem is encountered ...This may be referred to as weighing, but to refer to balancing, or a balance-sheet approach, may suggest that the essential qualitative assessment has a greater degree of precision than the statutory subject-matter permits ...
 ... much administrative decision making involves the weighing of imponderables or incommensurables. It would be unworkable to require the Tribunal explicitly to give a weight to each benefit and we would strain to avoid such a construction were it necessary. It is not, however, necessary so to strain. Section 95AZH(1) does not require what the ACCC suggests. The assessment of benefits and detriments must be complete and the Tribunal must, no doubt, weigh them. This is not necessarily, however, an arithmetical or accounting process. As we have said above, it may involve an instinctive synthesis of otherwise incommensurable factors."An instinctive synthesis of otherwise incommensurable factors": I like it, and so I imagine will the Commerce Commission's lawyers. Though I have to point out that we beat the Aussie Federal Court to the concept: back in 2011 many of us were already "internalising a really complicated situation in my head".