THE playoffs in America’s National Football League (NFL), which kicked off on January 8th, are a pretty dull affair for fans of the Buffalo Bills. Like the team’s post-industrial hometown, the once-proud Bills—the only team to ever reach the Super Bowl championship game in four consecutive years—have fallen on hard times. They have not qualified for the playoffs since 1999, the longest drought in the league. This year, the Bills have been in the news more than usual for a team whose 2016 campaign is already over—but for all the wrong reasons. In the season’s final week, they caused a minor stir by benching their quarterback, Tyrod Taylor, with the presumable intention of preventing him from exercising a valuable policy in his contract in the event he suffered an injury. Now it appears that Mr Taylor may have the last laugh: on January 5th, he underwent surgery that may enable him to trigger the clause anyway. This sorry chapter of the Bills’ history with one of their most prized players began last offseason, when they gave Mr Taylor a five-year, $90m contract extension. That deal put him in the middle of the pack at his position in terms of salary—which arguably made him a modest bargain, since his performance statistics were somewhat above average. Unlike in most other professional-sports leagues, not all contracts in the NFL are guaranteed. Instead, teams are free to cut many of their players with few or no financial penalties, as long as they do so in March, well before the season starts. As a result, players on non-guaranteed deals bear the lion′s share of their own risk of injury: whenever they get hurt, which happens quite often in such a brutal sport, their employers can simply cast them off the next year. Because the quarterback is the game’s most important position, most starting signal-callers have enough bargaining leverage to negotiate a deal that gives them two or three seasons’ worth of guaranteed money. However, Mr Taylor only managed to lock down $9.5m, among the lowest totals for a quarterback in the league.

The first conclusion one might draw is that Mr Taylor needs to find himself a new agent. But he did manage to negotiate a pair of beneficial clauses in what looked on the surface like a somewhat unfavourable deal. The first requires the Bills to guarantee another $30.75m if they don’t cut him before March 2017, just one year into his pact. In tandem, a uniquely worded provision stipulates that if he suffers an injury that renders him “unable to perform football services” and the team does cut him before that deadline, they will have to pay him $27.5m to walk away. Combined, the terms meant that the only way Mr Taylor would be left high and dry was if he stayed healthy (preventing his injury payout) and the club still chose to drop him.

Mr Taylor wound up having a fine 2016 campaign, finishing among the top third of the league’s quarterbacks by one measure. Nonetheless, the latter clause still became a point of contention down the stretch, when he was hobbled by a sports hernia. Mr Taylor managed to play through the injury and remain productive. But with one game left in the season, the incentives for the player and his team diverged. The Bills stood nothing to gain from starting Mr Taylor in that contest instead of his weaker backup, since they were already ineligible for the playoffs: it makes virtually no difference whether a team finishes with eight wins and eight losses or seven and nine. (In fact, a defeat would actually be slightly preferable, since it would improve the club’s position to select young players in next year’s amateur draft.) Meanwhile, were Mr Taylor to suffer a devastating injury in the game, Buffalo would owe $27.5m to a player poised to contribute nothing. Rex Ryan, the Bills’ coach, wanted to play Mr Taylor anyway. But the club’s owner, Terry Pegula, would hear none of it: following a heated exchange, he had Mr Ryan fired, and Mr Taylor was duly benched in Week 17.

The Bills believed that this hard-headed choice would enable them to keep all of their options open for the 2017 campaign. But Mr Taylor had a trick of his own to play in response. Four days after his season ended prematurely, the quarterback underwent surgery on his hernia. The expected recovery time is six to ten weeks. Barring a major setback, he should certainly be ready to play by the time pre-season camp starts in July. However, the Bills need to decide whether to retain him by March 11th, which is only nine weeks away. If his rehabilitation period happens to push up towards the latter end of the predicted range, he might not pass a physical examination before the deadline. In turn, that could conceivably trigger his injury clause and net him $27.5m.

There is nothing unusual about incorporating an injury bonus into an NFL contract: players often negotiate them to reduce the risk of signing a non-guaranteed deal. However, they are generally carefully worded, so that they are only activated when an athlete gets injured during one season and remains unable to play at the start of the following one. In this case—as would befit a team that has managed to miss the playoffs (which three-eighths of the league makes every year) 17 times in a row—Buffalo agreed to remarkably vague language in its deal with Mr Taylor. “This contract is very unusual in its lack of specificity,” says Andrew Brandt, a former vice-president of the Green Bay Packers. “The contract leaves itself open to interpretation as to when not being able to play in 2017 means.” Mr Taylor is expected to argue that the clause refers to his condition when Buffalo makes the choice to cut or keep him: if he is unable to play in March, the payout should be his. Conversely, the team is likely to counter that the text refers to the start of next season: as long as Mr Taylor can suit up by the time the first game kicks off in September, he should not be entitled to compensation. Just hours after his surgery, the Bills fired the first salvo in what will probably wind up as a dispute resolved in arbitration, saying that their quarterback “elected” to have the procedure (presumably as opposed to it being medically necessary).

The provision’s unfortunately imprecise language did not oblige the two parties to hurtle towards what now looks likely to wind up as a contentious divorce. The Bills could easily have decided to accept the relatively small risk of a catastrophic injury to Mr Taylor with just one game left in the year, and played him anyway. Alternately, they could have sought to buy out the clause for what would probably have been a modest sum—players and owners in the NFL routinely re-negotiate contracts before they expire.

However, what looks like a poor outcome for both sides would have come as little surprise to students of game theory. The vague wording of the deal created a situation somewhat comparable to the famous prisoner’s dilemma. In this scenario, two inmates are set to stand trial and cannot speak with each other. They will each get a modest sentence if they both refuse to talk, and will serve somewhat more time if they both implicate the other. But if one rats out his partner while the other keeps mum, the former gets off scot-free while the latter spends far longer behind bars.

Similarly, if the Bills and Mr Taylor—who, unlike these proverbial jailbirds, were free to communicate—had sought to coordinate their agendas, they could have preserved their marriage at little cost. But Mr Pegula was determined to eliminate the risk of his worst-case scenario, in which he would cough up $27.5m to Mr Taylor and then have to shop for a new quarterback—just like the prisoner who wants to ensure he does not wind up locked up for years, which would occur if he refuses to point the finger at a collaborator who does not extend him the same courtesy. And once the possibility of exchanging his injury clause for something else of value vanished, Mr Taylor’s optimal strategy became doing whatever he could to maximise the odds of it being activated.

Given this elementary logic, it might seem inevitable that the relationship between the quarterback and his employer would end acrimoniously. However, slightly more advanced chapters of game theory suggest that the Bills may in fact have misplayed their hand. The prisoner’s dilemma is an example of a “single-shot game”: regardless of their decisions, neither inmate expects to have to make the same choice regarding the same conspirator in the future. In contrast, NFL contracts are more like a “repeated game”, in which participants must consider how their actions in one round will affect other players’ decisions later on. On one hand, this was of course the only time that the Bills will need to choose whether or not to bench Tyrod Taylor in order to avoid the risk of activating his injury payout. However, it was far from the only time that they will need to decide between paying up to keep a player happy or maximising their short-term gain. And in repeated games, reputation is everything. As a result, reciprocal altruism is often more attractive than the cutthroat approach that prevails in single-shot situations.

After 17 years of futility, Buffalo was already an unattractive destination for potential free agents. Now, their efforts to court new players may be further hindered by the precedent they set with Mr Taylor of short-changing their employees, and by their messy divorce with Mr Ryan, an experienced coach. Anthony Lynn, Mr Ryan’s talented replacement, also distanced himself from the club’s handling of their quarterback by calling it a “business decision”. He is already interviewing for other jobs. Even if this potential market penalty is hard to measure and probably small, the cost of buying out Mr Taylor’s injury clause with one game to go was likely smaller still.

The final reason to question the Bills’ handling of Mr Taylor is that they probably shouldn’t have been so determined to maintain the ability to cut him for free in the first place. Even though Mr Taylor was playing hurt for much of the year, he is a fine quarterback—decidedly better than three of the 12 starters on this year′s playoff teams, and roughly on par with half of the rest. He is one of the most valuable assets on an otherwise mediocre roster. The $30.75m that Buffalo will be obliged to guarantee if they retain him is more than reasonable for a signal-caller of his calibre: the club would almost surely do worse if it sought to sign a free agent at the position for the same price. Moreover, Buffalo’s abysmal record at attempting to develop young stars does not bode well for their odds of drafting and grooming a suitable replacement. The Bills will probably have to find a new quarterback. But this sorry episode suggests that what might help them most of all is new ownership.