The Game in Game Theory


Two children have a slice of ice cream cake to share between them. The longer they debate over what a “fair share” would constitute for each of them, the more the cake melts. It is, in fact, entirely possible that the cake will have melted by the time they reach a consensus.

Keep that image in mind while I talk about the most exciting part of preseason football: contract holdouts!

Rookie QB Brady Quinn made headlines by holding out for weeks after signing. Larry Johnson (KC Chefs) held out until last week before signing a 5-year, $43.2M extension. Asante Samuel (NE Patsies) just announced his return yesterday, though he’s still ducking efforts to wear the “franchise” tag. And Michael Strahan (NY Football Giants) is, as of this writing (Tuesday, August 28th, 9:00ish AM), still undecided. The man might very well hang up his cleats (sources say).

So what goes on during a contract holdout? What are the costs and benefits of avoiding training camp? Is the risk worth the reward? To find out, we turn to one of the classics of game theory, Thinking Strategically, by Avinash Dixit and Barry Nalebuff.


For decades, the nation of Israel had a policy of never negotiating with terrorists. The government declared this to discourage hostage-taking: if capturing a civilian, or a politician, will never pay off, why bother? Israel did this to build a reputation of not being someone to screw with. The downside to having this sort of reputation, of course, is that it turns potential hostages into instant victims. There’s always a cost to talking tough.

No GM wants to cave if one of his players pouts and refuses to show up for training camp. To do so would immediately send a signal that this sort of behavior works and would open the floodgates next summer. It’s in every owner’s interest to stay firm.

Asante Samuel knows his game theoryAt the same time, however, the players aren’t chess pawns. There’s a world of difference between an Asante Samuel and a Randall Gay (2 tackles in 3 games last year). Coach Belichick may play it off like Samuel’s return is no big deal, but the defense he spent the entire offseason crafting in his Fortress of Solitude probably hinged more on a guy who caught 10 picks last season than a guy who sat for 12 games.

Samuel knew he was valuable to the Patriots. The question was: how valuable? That’s the kind of question that a holdout is meant to uncover – by watching the management’s change in behavior as the clock ticks and the options narrow.


Every good game theory textbook will cite Dr. Strangelove; Dixit and Nalebuff’s Thinking Strategically cites it two or three times. Stanley Kubrick’s darkly comic masterpiece highlights the key problems of negotiation. The USSR creates a machine that will blanket the Earth in radiation if there’s a nuclear detonation on Russian soil. It cannot be switched off; it cannot be reasoned with. This is an ironclad commitment strategy. The Russians clearly don’t want to extinguish all life on the planet, but by locking themselves into that strategy they make themselves credible bargainers. If you want to see what the downside of commitment is, watch the movie; I don’t intend to spoil it.

A player who doesn’t sign his contract by the time regular season play starts becomes a free agent. However, if a player does not sign with a team by the Tuesday after the 10th week of the season, he must sit out the rest of the year. Few if any contract holdouts come to this point and are typically resolved before regular season play.

(Edit: Tom D. adds a few details that correct my notion of free agency; see below. I feel the thrust of the argument still stands)

How can a player prove that his threat to not play is credible? An uncertain player might continue to show up to training camp and work out, all the while kvetching to his agent, in an attempt to keep all the bases covered. Such a move would instantly be seen as weak.

Larry Johnson has Avinash Dixit on his cellBut a player who boycotts training camp not only hurts the team who wanted to sign him – he hurts himself as well. He doesn’t run plays, so he won’t function as smoothly during gametime. He’s not working out as intensely, so his conditioning won’t be as good. He’s probably generating some bad will among his teammates, who are running two-a-days in the August sun while he’s customizing his uniform in Madden 2008. And if the team is fining him for not showing up at camp (as the Chiefs did Larry Johnson to the tune of $14K a day), it’s costing him money on top of everything else.

Skipping camp is a player’s way of telling the team, “I’m serious. I’m not coming back until I’m sure my demands have been satisfied.” In Larry Johnson’s case, it may have paid off: he is, as of August 21, the highest paid running back in the NFL.

Reasoning Backward

Let’s return to the melting ice cream cake from earlier. Every second these two children spend negotiating over the proper share, they lose out on more cake. If they don’t come to an agreement in about ten minutes, the end result will be no cake for anyone. This puts the child holding the cake knife in the strongest position: he could wait until 9 minutes have elapsed and offer the other kid the chance to lick the knife. It’s an insulting offer, but if the only alternative is no one gets anything, the poor kid might take it.

As I said above, free agency reigns until the Tuesday after Week 10. On the Monday after Week 10, the GM could offer a free agent the league minimum (about $275K a season for a QB). The alternative is $0, so the free agent should take it. The only reason he wouldn’t, in fact, is if he wants to preserve his reputation (see above) as a hard negotiator, or to demonstrate credibility by committing (see above) to a losing strategy. But let’s ignore those for now.

Knowing that the inevitable choice on the last day is “either $275K or nothing,” we can figure out what the inevitable choice would be on the next-to-last day. It’s clearly worth it to the team to have another QB on their roster – it takes pressure off the starter, it increases the depth of the coach’s strategy, etc. That value can be quantified by some amount of money: let’s say $10K. If the free agent QB is a savvy bargainer, he’ll want as much of that money as possible. So on the next-to-last day, the free agent’s “walking price” is $285K – $275K for the last day of trading, plus $10K for being on the team for a day. A team that wants to acquire him two days before the acquisition deadline will have to pony up $285K for the privilege.

And since we know what the inevitable choice will be on the next-to-last day, we also know what the inevitable choice will be on the next-to-next-to-last day. And so forth.

Michael Strahan reads Thinking Strategically in air-conditioned comfortThis is why contract holdouts never last until November and usually don’t even last into the regular season. Both the player and the team have a pretty good idea of the player’s worth. It’s not perfect: hence the negotiation. But both bargainers can estimate what the player would be worth as a free agent. They can also estimate how much it would be worth to have him on playing with them for a week. And they can reason backward from there to determine the point at which it’d be worthwhile to sign him.

Michael Strahan has reportedly been considering retirement. Some might say this is a threat to keep the Giants at the table, but Strahan might have reasoned backwards from Week 10 and not liked what he saw. The Giants have an adequate replacement for him – third year defensive end Justin Tuck. The difference between Tuck and Strahan might be a fairly narrow margin, so the marginal value of another week of Strahan playing might be pretty low. The New York sports media is waiting to see what Strahan or the Giants will decide – but they’ve reasoned backwards from the future, so I think they already know.

Final Thoughts

So is holding out on your contract a good strategy? The answer is: it depends. You have to have a good sense of your value – not just as a player, but as a part of the team that you’re negotiating with. You have to have a strong reputation as a bargainer. And you have to be willing to suffer a little in order to show commitment to your strategy.

There are times when contract holdouts will work – and there are times when they don’t. This may be a little premature, but I think Brady Quinn’s holdout didn’t work. I’m going to predict (with about 65% confidence) that the Notre Dame vet will have cause to regret his holdout down the line.

Brady Quinn never learned to readA rookie QB is like a junk bond – an incredibly speculative investment that could pay off tremendously (Tom Brady) or crash and burn (Ryan Leaf). In order to prove your worth, you need to demonstrate your skills on the field. Brady Quinn missed 11 days of practice. And all the perfect preseason games in the world count for nothing if you’re not the starter – which Brady Quinn isn’t. The fewer games Quinn plays, the less everyone sees of him. The less people see of him, the less he gets to play.

Quinn’s signed with the Browns for 5 years, so he’s in it for the long haul. And he will probably get enough chances to play down the line that he’ll prove himself a worthwhile investment. But it’s like starting to invest at age 28 instead of age 24 – those extra 4 years of savings would have compounded into another $100K by the time you retired. Every game that Quinn doesn’t start is a lost investment opportunity.

Having a perfectly accurate sense of your own financial value is not only possible, as Doctor Strangelove put it – it “is essential!”