Amtrak Trains Fight to Go First
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Only a government regulator would have difficulty with a problem like this. The solution is to allow the party who pays a "goes first premium" to go first. Let them bid for it on the Internet. If passengers want to go first they will willingly pay the additional fare. If not, they will not pay an additional fare, but they may be subject to delays.
Using pricing signals to allocate scarce resources is the only logical way to do so. Otherwise it is just choosing sides.