I don't have any numbers crunched here, so I'm going to be relatively conservative and ask for someone--anyone--to tell me what the San Francisco Giants are doing strategically with their offers to Lincecum.
Let's talk game theory. The basic idea is that when there are two agents with potentially conflicting goals, the best strategy for each depends on their assessment of the best strategy of the other. The Lincecum arbitration case is an excellent example: there are two players, Lincecum and the Giants, in a zero sum game that determines Lincecum's 2010 salary (and beyond). The goal is pick a number that is closer to what an arbitrator would find to be a reasonable salary than your opponent does. Whoever picks the closer number is likely to win. If Lincecum asked for $43 million, the Giants could have offered $2 million and won. Clearly both sides are going to make more reasonable offers than that, but you get the point: the craziness of your offer depends on the perceived craziness of your opponents.
Lincecum's request, $13 million, is not too crazy. Its high, but he has no peers to be compared to. If you base his salary on his performance, and not his peers, $13 million is a bargain. If you look for the closest peer, its Ryan Howard who won his case for $10 million. The Giants offer was just strange: at $8 million, it was well below the previous highest-paid arbitration-set salary. I don't think many people would argue that Lincecum is worth 0.8 Ryan Howards. If Lincecum should be expected to make slightly more than Howard, say $11 million, a conservative figure, Lincecum's offer of $13 million is closer than the Giants $8. So the Giants appear to have low-balled Lincecum; not by a huge amount, but they low-balled him nonetheless. If we agree on a reasonable value of $11, the Giants would be expected to lose, but they could still have a decent chance of winning. In a non-trivial percent of possible universes, their team successfully argues their side and wins the day.
But I'd argue that game theoretic considerations make the Giants offer look really, really bad. First, the outcome of this arbitration hearing won't just affect Lincecum's 2010 salary; it will set the standard for his 2011 and 2012 salary as well. It will be easier for Lincecum to ask for $17 million in 2011 if he is making $13 million in 2010 as opposed to, say, $10 million. So the cost of losing this arbitration hearing is greater than the immediate costs of the $8 million bid. That means that another $2 million spent today saves a potential $4-6 million down the road. This would argue that both players should consider a conservative approach: make sure you win this hearing by bidding close to his true value. Its hard to argue that $8 million is close to his true value.
Second, what the Giants really want to do is sign Lincecum to a long-term deal. Lincecum will have an incentive to do so, given the likelihood for any pitcher to get injured. But by making such a low bid on the arbitration hearing, the Giants have put themselves in an awful negotiating position. Lincecum can expect to win his hearing and thus will be less likely to negotiate. If the Giants had bid, say, $10, there would be a very good chance they would win, which would put more pressure on Lincecum to accept a long-term deal at a price the Giants would prefer. If we were using game theory to build a model of this arbitration process, the potential for injury would be a force that drives down Lincecum's asking price. On the other side, however, it would not substantially affect the Giants estimate of his value. They could be expected to take out injury insurance to protect themselves from the possibility of Lincecum's arm falling off. The net effect is biased in favor of a long-term contract. I think the recent history of long-term deals being struck before arbitration hearings supports this principle.
So if the end goal was to tie Lincecum down for the long term at a good rate, why did the Giants choose to low-ball him? Their most recent offer to Lincecum is being reported to be at $37 for three years: $9.5, $12.5 and $15 respectively. Lincecum is reported to want a deal that starts at $13, not $9.5. Well, of course he does; if he doesn't make a deal, he'll probably make $13 next year. Why start at $9.5? The Giants offer would have had considerably more weight behind it if they had offered $9.5 up front.
In an alternate universe, the Giants offered $10.5 million in arbitration. They then offered $10.5, $13, $15.5 for the next three years. They would have had a very good chance of winning that hearing, and Lincecum would have had very good reason to take the long term deal. In this universe, it would be reasonable to expect him to get tied down for 3 years at under $40 million.
My general point is this: sometimes it is a better strategy to offer to pay more for a good or service. Sometimes a high initial bid will result in a lower long-term cost. The Lincecum arbitration may be just such a case, but the Giants swung and missed completely.