1. ARC finds a way to throw enough money at them they drop their opposition
2. Public outrage at the "Astrodome wreckers" worries the Texans/Rodeo enough that they back down, fearing a collapse in fan/customer support (the Rodeo seems particularly susceptible to this)
3. Behind the scenes arm-twisting by city and county power brokers, esp. on the Rodeo board of directors
Since it seems to be the way things are typically done in Houston, #3 seems the most likely, maybe with a little #1 and a threat of #2 thrown in.