One of my colleagues here in the School of Economics and Finance has forwarded me this link, and I have to confess that I haven't actually come across this before! I'm very pleased that I now know of it, as the broken window fallacy seems to fit the economics of stadiums very, very well.
To paraphrase (using a hypothetical example of a $250m stadium), the broken window fallacy occurs when there is destruction (in the stadium's case, physical or economic obsolescence of the facility), and the new stadium is rebuilt. The costs involved in the rebuild are considered by some as beneficial to those who are involved in the rebuild (ie. construction firms, etc). The opportunity cost of the $250m, however, means that amount cannot be spent elsewhere, and from the point of view of society, makes the idea of a rebuilt facility somewhat less appealing. The facility will be considered a bad investment if the opportunity cost is greater than the stadium investment.
What is also important to consider, too, is that the stadium investment is likely to also have impacts on those involved in the rebuild. Construction firms will consider the facility a benefit if they have spare capacity that they can utilise to build the facility. If they don't, however, then the facility will effectively crowd out other work that the firms would otherwise be doing.
As such, is the city that builds a $250m stadium actually better off? One can always glean a simplistic insight into this by examining multipliers for various activities. If a stadium has a multiplier of 1.75, let's say, then all it takes for the facility to make the local economy better off is if no other alternative investment has a multiplier greater than 1.75. If others do, however, the 'gains' are highly unlikely to materialise.