As most everyone who follows the NBA knows by now, LeBron James has opted out of his contract with the Miami Heat to explore his options. These options include returning to the Heat, or signing with another team. Now, there are lots of constraints on what other teams can pay him, and I will gladly leave this to others to explain in detail. What I want to briefly explore here is whether LeBron is acting in his own best interest by dragging this process out, and whether a game theoretic perspective can help us understand what his best option is.
First, let’s make some simplifying assumptions, as all game theoretic models do. I’m going to assume that basketball players’ decisions regarding the team with which they should sign are motivated by two factors: making money and winning championships. It may be that other factors – such as playing close to home – matter as well, but my approach here is going to focus on money and winning.
Players may face trade-offs between these two goals. Indeed if they don’t, then the decision is simple: in this model, if any player can get paid the most money by the team with which he or she has the best chance of winning a championship, then he or she will simply sign with that team.
Where things get interesting is when the chances of winning a championship go up if a player is paid less money. This could be because by taking less money in a salary-capped NBA the player in question gives his team more money to sign other strong players (e.g., what Phil Jackson wanted Carmelo Anthony to do with the Knicks), or it might be because the best teams don’t have as much money available (e.g., the current champion San Antonio Spurs), or it might be because in the NBA teams are allowed to sign their own players for more money than players from other teams. Either way, players are often faced with a choice between earning more money and being more likely to win.
This leads to a complicated situation when multiple free-agents are trying to decide where to sign. If I’m a free agent, not only do I have to think through what my signing will mean to a team’s ability to sign other players and compete, but I have to anticipate what other players are thinking as well. I have some information – I know how much money every team has to spend – but I don’t know how much everyone else values winning vs. money. Solving this game is complicated, and would normally place a premium on waiting to see what everyone else does first, although even that’s not as simple as it seems because every time another free agent signs, that leaves less money available to sign the next free agent.
So this leads me to my second assumption: LeBron James will add more to any team’s likelihood of winning an NBA championship than any other current NBA free agent. To be clear, this doesn’t mean that any team with LeBron will win the championship, or that other combinations of players might be more valuable than LeBron alone. It simply means that LeBron signing with a team will increase that team’s chance of winning a championship more than any other single free-agent signing with that team. I’d even go so far as to say LeBron would help a team win a championship more than any other player period, but we don’t even need to make that strong an assumption here. Either way, if you don’t buy that assumption, then that’s fine: the argument would then hold for any other player you think is currently the most valuable free agent.
But returning to LeBron, as soon as he signs with any team, the chances that that team will win a championship goes up. Thus for any player who values both winning and making money and is willing to make a trade-off between the two, having LeBron on a team suddenly makes the likelihood of winning go up, and therefore that player should be willing to take less money to play with LeBron.
So let’s think this through: 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.
One other interesting implication of this analysis is that if LeBron intended all along to stay in Miami, he should have either not opted out of his contract or else immediately re-signed a new deal with the Heat. Now obviously the fact that LeBron is taking his time shows that he is not following the logic of the argument I’ve outlined here, but it does make you wonder if the chances of him going somewhere other than Miami might be higher than we think. Of course, Miami can still pay LeBron more than any other team – so this argument may be moot – but the more he time he takes to return to Miami, the more he might hurt his chances of being surrounded by the supporting cast he really wants.