If, hypothetically, a member of the European parliament was caught on video offering to amend laws in exchange for cash, who would investigate such a thing?

The answer – rather less hypothetical after Britain’s Sunday Times exposed what appears like serious misconduct from four MEPs – depends on who you speak to.

The European Anti-Fraud Office, or OLAF, thinks it should be the one investigating. After all, what’s the point of having a dedicated EU anti-fraud agency if you can’t investigate credible allegations that MEPs are taking bribes?

But when an OLAF team showed up in the parliament to search the MEPs offices last week, they were unceremoniously denied entry by parliamentary authorities. Parliamentarians insisted OLAF was set up to investigate fraud against the EU budget, and it’s not clear that the EU has lost money in this episode.

OLAF disagrees on the limits of its role, but so far hasn’t been anywhere near the MEPs’ offices, which have been sealed off.

Insiders say part of the reason for the stand-off is a power struggle between parliament and the European Commission, the EU’s executive branch, which is ultimately OLAF’s master.

It doesn’t help that the fraud-busters are held in pretty low esteem in Brussels, after years of not proving particularly successful at achieving prosecutions against wrongdoers. Proposals are due out soon from the Commission on how to overhaul the body.

If not OLAF, then, who else could investigate? There’s nobody else at EU level who has the power to do so, so that leaves national authorities, which is exactly who the parliament thinks should be called in.

The argument for national teams is that it’s the courts of those countries that ultimately will render verdict on the MEPs, as parliamentarians are elected according to laws laid down by each country – in this case Austria, Romania, Slovenia and Spain.

Of course, that would mean that even if each parliamentarian had been caught doing exactly the same crime – and it should be pointed out all deny wrongdoing – they might end up with completely different punishments based on whose law is enforced. Hardly a satisfying outcome.

And let’s see whether parliament does indeed grant access to national police or prosecutors. Parliament authorities insist they will, but it will be interesting to see how they respond when uniformed police from one of the member states show up at the parliament’s office. One imagines more than a few triplicate forms might be required here.

A final possibility, pushed by Transparency International, the lobby group, is that it should be the Belgians who take the lead. After all, the parliament is (mostly) based in Belgium, which is where the alleged law-breaking took place.

According to the group:

“[Belgian law] provides a sound legal framework and appropriate sanctions if it is found that criminal proceedings are required. The [fraud law] explicitly makes provision for offences by ‘a person who exercises a public function in a public international organisation’”

That might mean, however, all Eurocrats in Brussels are bound by Belgian law, at least when they are physically based here, which kicks up even more questions. What if all this had happened during a Strasbourg plenary week, on French soil? Or in the constituency office of one of the MEPs, outside of Belgium? And are Commission officials also bound by Belgian law?

Regardless of the outcome of this particular wrangle, it certainly won’t speed up the pace of the investigation – once the investigators are found, that is.

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