Prof. Rick Hasen (Election Law Blog) has an intriguing theory on this:
I assume that Judge Walker, like most lower federal court judges, does not like to be reversed. The opinion he issued today is no doubt the most famous (perhaps also the most important) that the Judge will issue in his life....
If Judge Walker denies a stay, then the request for a stay will be filed with the motions panel of the Ninth Circuit. As I explained, this month’s motions panel is tilted liberal, so it is a good (but by no means certain) bet that if Judge Walker denies a stay, the Ninth Circuit will deny a stay too. I have little doubt that if the Ninth Circuit denies a stay, that a stay request will be filed with Justice Kennedy, who will then refer the matter to the whole court.
Getting an emergency stay request before Justice Kennedy during the Court’s summer recess is going to put a lot of pressure on the Justice to decide this matter quickly, and without the opportunity for the reflection that we’ve heard Justice Kennedy engages in when considering more difficult cases. The pressure of time could lead him to grant the stay, and to be put off by the plaintiffs for having brought the case in the first place.
In contrast, if Judge Walker grants a stay pending appeal (meaning no change in the marriage rules pending appeal), the the appeal proceeds on a more leisurely pace.... Eventually, [there will be] a cert. petition filed before the Supreme Court. This gives Justice Kennedy more time for reflection. He could be more amenable to upholding the decision under this more leisurely scenario.... [G]ranting the stay will lower the temperature of this case, and that might be the best way for the judge to get an affirmance down the line.
I’m not sure whether this is right, but I thought it was worth passing along. Of course, the assumption behind it is that the swing voters on the Court (and Justice Kennedy is the likeliest one) find the legal questions to be difficult. Naturally, if they think the case has an easy answer one way or the other, it’s unlikely that the timing of their first decision about the lower court opinion will matter. But if they think the case is hard, then it does seem possible that (1) they might take a different view when considering the case at the normal relatively leisurely pace rather than on a quick decision about the stay, and (2) the normal human tendency to stick with the first decision you make — even when the decision arises in a different procedural posture, and is deliberately designed to be tentative and not binding — might come into play.