Managing agent agreement??
Don't you need to sign a managing agent agreement before anything is done, and if that agreement is terminated before an agreed period (usually 12 months I think?), then you have to pay out the agreement??
So, if you signed an agreement with multiple agencies, you would have to pay out the agreed period of rental commissions to the losing agencies (unless you were able to get the agencies to agree to the multiple agency agreement, but there would still be fees to pay for their time, advertising, effort etc).
It would also be a headache to keep track of the keys, who was doing what (as mentioned before), and you could burn your bridges with a few managing agents should you wish to change at a later date.