Is It a Coin Flip or Is It Justice? It Could Be Both.

Centuries of legal research have shown that judges often make decisions based on bias, chance, or other nonlegal considerations. For example, one famous study from 2011 found that judges ruled more leniently in the morning and after lunch and were stricter immediately before a food break.

“This is just one example within hundreds of papers that say judges are arbitrary,” says Alvaro Sandroni, a professor of managerial economics and decision sciences. “Or sometimes the Supreme Court comes to mind because they tend to be predictable in making decisions based on their political preferences and not by the law. So, they are either lawless or arbitrary.”

Often, these conclusions are framed as evidence that the justice system is broken or corrupt. But Sandroni worked with Leo Katz at the University of Pennsylvania to determine if game theory could provide a less-cynical explanation.

Their game-theory model produced a surprising takeaway: because of the types of cases that reach trial, it often makes sense for judges to rule at random.

“The argument that we make is that if the system works the way it should work, then in some cases, essentially all decisions will look like they are random or politically motivated,” Sandroni says. “So to distinguish between what a good judge does and what a bad judge does, it’s not as easy as to simply look at their decisions.”

Rules of the game

Game theory is an approach to understanding real-world systems, such as economics or diplomacy, by mathematically modeling the strategies of its participants and how they interact.

The approach has often been used to study legal proceedings, usually by assessing the competing interests of opposing sides in a case. In Sandroni and Katz’s model, called “The Judging Game,” they introduce judges as a third party with motivations of their own: to minimize errors and effort.

“Our game is mostly between the judges and the litigants,” Sandroni says. “It’s not so obvious that they are in a strategic situation. So there was an attempt to demonstrate that.”

The researchers’ model builds on a commonly observed phenomenon in legal scholarship called the selection effect. Because many legal conflicts have a clear winner based on facts or precedent, attorneys for both sides will often push to settle these easy cases out of court.

“People know that if you send a trivial case to court where everybody recognizes who is right and wrong, the judge will rule correctly, assuming that the judge is honest,” Sandroni says. “And so people settle the case, because why incur all these costs to get an outcome that was clear from the beginning?”

That means that the cases that do reach the courtroom are often evenly balanced with no obvious ruling, which leads to a provocative implication.

“If all the cases that go to court were hard and had no resolution, there wouldn’t be any incentive for the judge even to look at it,” Sandroni says. “Judges would just say, ‘OK, I know that no matter how much I struggle with it, in the end I would be doing the functional equivalent of flipping a coin or just ruling based on my bias.’”

Optimizing attention

The researchers’ game takes this observation about the selection effect one step further. If only hard cases reach court, a rational judge doesn’t need to pay attention and can rule arbitrarily. But if litigants know that’s happening, they will bring more easy cases to trial, hoping to win the “coin flip.” And because judges want to avoid making errors, they will start paying attention again.

“It’s not stable for judges to pay 100 percent attention; if they did, then they would be wasting their time,” Sandroni says. “It’s also not stable for judges to pay zero attention, because then they’re going to start getting a lot of obvious cases and ruling incorrectly on them.”

Eventually, this push and pull reaches an equilibrium where a mix of hard and easy cases goes to court, the researchers observed. And while this theoretical scenario seems extreme compared with the real world—where you wouldn’t expect to find a judge dozing off through an entire trial—more-realistic behaviors could produce a similar outcome, Sandroni says.

“It doesn’t have to be a scenario where some judges will pay attention and some judges don’t. It might be that all judges pay attention to some cases, or some fraction of the cases, or to some degree of intensity. But it cannot be their full attention, because if it were, then only hard cases [would] go to court, and they’d be wasting their time.”

Arbitrary but functional

The ambiguous balance of hard cases also opens up room for judicial bias, whether political or legal. Even if the judge appears to fairly weigh all the merits of a case and writes a lengthy written opinion to support their ruling, their decision in a difficult case may have been initiated by the smallest of subconscious nudges.

“The idea is that if you are unbiased, your ruling is arbitrary, and if you’re biased—even if this bias is minimal—then you rule as if that’s the only thing that matters,” Sandroni says. “Because if the case is very balanced, then any tiny bias will push the scale.”

Though that conclusion questions the ideal of impartial judges, Sandroni says that this phenomenon is a positive sign for the legal system, just as it would be for a well-functioning company that resolves most of its problems before they reach the CEO’s desk.

“The more arbitrariness, the better the system is,” Sandroni says. “That sounds counterintuitive, but we’re not saying that judges should rule by flipping a coin. We’re saying that if the judges get into a situation where there is nothing more to be done apart from flipping a coin, that’s a sign that the system is working.”