Joshua Tucker says yes:
…if LeBron waits until every other play has signed, those players will all have made their decisions not thinking they have the maximum chance of winning a championship. Because they value both winning and making money, every one of those players will have signed for more money than they would have needed to sign had Lebron already signed with that team. LeBron, upon joining that team, will therefore be playing with players who were more expensive than they needed to be. This in turn means that whatever team he joins will either (a) have less money to sign LeBron or (b) have less money to sign other players besides LeBron and the free-agents they have already signed. Either way, LeBron gets less of what he wants (defined here as money + likelihood of winning) than if the other free agents had known he was going to be on that team before he signed.
Therefore the converse should also hold: by moving sooner, LeBron should be able to get more of what he wants. By virtue of being the single best free agent available, Lebron instantly adds more to a team’s chance of winning a championship than any other player, and therefore will drive down the cost of acquiring other players to complement him as he seeks out additional championships.
But I don’t think that is right. LeBron needed to find out if Wade and Bosh are willing to take significant pay cuts, to help Miami bring in better players. So far it seems he is learning the answer is “no.”
More formally, you can think of this as threshold and discontinuity issues kicking in. If LeBron signs quickly with Miami, and Wade and Bosh are selfish in pecuniary terms, Miami can’t do much of anything to become a decent contender. That is because the salary cap makes it very difficult to bring in other good players at reasonable cost (the “luxury tax”). No major free agents have stepped forward and shown their willingness to take a big pay cut to play with LeBron.
If that is indeed what has been learned, LeBron now can pit a few other teams against each other — Cleveland, the Lakers, maybe even Phoenix and Houston — and ask how big a financial commitment to winning they are able to make. (Miami of course can be kept in the mix.) It takes a while for those teams to signal their intentions, and that also requires waiting on LeBron’s part, if only to let the bids escalate. That is the way to extract greater sacrifices from other players and also from the owners, a factor which I don’t see Tucker putting at the center of his analysis.
Of course LeBron won’t wait very long. At some point each team has put its best plan on the table and then he will choose (for reasons similar to those outlined by Tucker), which is likely quite soon. Still, it is privately optimal for him to start that process with some waiting and with a minimum of non-committal rhetoric, which is indeed what we are observing.