There is one great reason to go to the police: it establishes a paper trail documenting that a crime was committed. It isn't necessary that the police catch the bad guys for that paper trail to be advantageous.
(Examples: police reports make CC disputes and legal declarations much easier and more likely to be given weight as other than self-serving explanations of a deadbeat. It may also trigger insurance policies either for you or for the VPS company.)
Sorry for being pretty brief earlier, as I was on a cell phone. Here's how a police report can help him, in a few ways:
1) In dealing with the company that is billing you, a police report sends a strong, costly signal that "This is not a routine commercial transaction! A crime has been committed!" which will throw an exception in the billing process that will get caught elsewhere in the company. The accounts receivable department is scored on collecting revenue. That is, to a first approximation, the only thing they care about. The legal department, anti-abuse department, etc etc, is not scored on revenue impact.
You're much more likely to have a productive conversation with them that involves waiving the $4k than you will with Accounts Receivable. AR's calculation goes something like "We'd prefer getting $4k over getting $0." The rest of the company thinks "We'd prefer writing off an uncollected invoice -- costs us $0 -- versus spending hundreds of dollars an hour dealing with the legal process for an unbounded amount of time."
2) Assuming that European credit card providers work on the same heuristics as US ones, "I filed a police report" means their response to your chargeback will virtually invariably be "Wait, an actual criminal case? Eff that, sorry merchant, we do not get paid to litigate on your behalf. Best of luck figuring this out after we give the customer their money back." (Conversely they use police reports as a bar for treating claims of fraud seriously, for example, in the case of "family fraud." If you ever say, for example, "My wife bought that but I didn't let her?" the very next question will be "Have you filed a police report against her for stealing from you?" and if you haven't the very next thing they'll say will be "Well, work it out with her then -- the charge stands. Thxmuchbye!")
3) "Theft" is such a wonderfully useful word for dealing with insurance companies. (HNers might say "No theft happened! It was actually..." No, guys, really, theft happened.) It may be covered under your homeowner's insurance -- depends on the particulars of your policy -- and would almost certainly be covered if you had business insurance. It also may be covered under the VPS provider's insurance, in which case they may decide that rather than trying to ring water from a stone (you) that filing a claim is the quicker way to get paid. (I rather doubt, as a businessman with an insurance policy, that they'd jeopardize their future insurability over a $4k claim, because that is tiddliwinks, but risk management is exactly why that policy exists.)
(Examples: police reports make CC disputes and legal declarations much easier and more likely to be given weight as other than self-serving explanations of a deadbeat. It may also trigger insurance policies either for you or for the VPS company.)