Game theory makes a clear prediction about when people will disclose information, and when they will keep it hidden. The prediction: they will disclose their information when it is better than others expect, and they will refuse to do so only when it is worse than expected. Game theory says that Clinton will choose not to release her speeches, and Trump will choose not to make public his recorded conversations or tax returns, only if they are worse than their voters anticipate.

By Christopher Cotton, Queen’s University

The frontrunners for the Democratic and Republican nominations in the US presidential election seem to have a sharing problem. At least when it comes to sharing information with voters.

Hillary Clinton has declined to release transcripts from speeches she gave on Wall Street. Donald Trump has refused to release his tax returns, and is now refusing to make public recordings of an off the record conversation that he had with the New York Times.

These situations are reminiscent of a classic game theory model involving the strategic disclosure of evidence. In the model, one person chooses whether or not to reveal private evidence about her quality. If she reveals the evidence, then a second person (or group of people) observes her quality and treats her accordingly. If she refuses to reveal the evidence, then the other person forms beliefs about her quality, taking into account the fact that she chose not to reveal her evidence.

When I teach this “game” to my university students, I tell the story of a recent graduate applying for a job and deciding whether to list her GPA on her resume. Or, the story of a firm owner applying for a loan from a bank and deciding whether to share its books with the loan officer.

But, one may also think of a presidential candidate choosing whether or not to release past tax returns, off the record conversations, or speeches, knowing that their content may change voter opinion.

In these situations, game theory makes a clear prediction about when people will disclose information, and when they will keep it hidden. The prediction: they will disclose their information when it is better than others expect, and they will refuse to do so only when it is worse than expected.

Game theory says that Clinton will choose not to release her speeches, and Trump will choose not to make public his recorded conversations or tax returns, only if they are worse than their voters anticipate.

To see why this is, let’s think through Clinton’s choice of whether or not to release her speech transcripts. A similar story could be told Trump’s choice of whether to release his tax returns or recording.

If Clinton does not disclose the transcripts, then voters will make a guess about how good or bad the speeches are, and will vote accordingly. I will refer to the voters’ guess as their expectations about speech content.

If Clinton’s speeches are better than the voters’ expect them to be when she keeps them hidden, then she can improve her chances of winning the election by releasing them. She has an incentive to show the voters that her speeches are better than they think.

On the other hand, if Clinton’s speeches are worse than the voters’ expectations, then she has no incentive to release them. She is better off leaving voters overly optimistic about their content.

The only time that Clinton will not release the speeches is when they are worse than voters think. This means that voters are necessarily too optimistic about the speeches that are not released. If they weren’t, then Clinton would choose to disclose the transcripts. Voters should be reacting more negatively to Clinton’s decision not to release them.

This is true regardless of how pessimistic voters are about the speeches. For example, if Clinton thinks not releasing the transcripts leads voters to have “fairly bad” expectations about the speeches, then she will release the speeches whenever they are better than fairly bad, and she will not release them only if they are worse than that. By not disclosing her speeches, Clinton’s strategy reveals that they are worse than voters expect.

Regardless of how pessimistic voters are about nondisclosure, they are never pessimistic enough.

Similar logic shows that when Trump chooses not to make public his tax return or his conversation with the New York Times, it’s because he thinks that revealing will hurt his campaign. But in the case of Trump, this could mean that the information makes him look more moderate than voters attracted by his extreme campaign rhetoric expect. This is consistent with the speculation that Trump’s conversation with the New York Times shows that he is more flexible on immigration than many of his supporters believe.

Thus far, our discussion has assumed that the candidates, with their teams of campaign consultants, are strategic when choosing whether to release information. But we haven’t required that voters are strategic. In this sense our logic does not represent a complete game theoretic analysis, which typically assumes that all players behave strategically.

Game theory uses strategic reasoning along the lines of “if she anticipates that he anticipates that she anticipates that …” applied over and over again. It solves for a Nash Equilibrium where all players behave strategically while correctly anticipating the behavior of others.

In the above example, Clinton responds strategically to voter behavior, but the voters are not responding strategically enough to Clinton’s behavior. Voters should anticipate Clinton’s strategy and revise their beliefs accordingly, becoming more pessimistic.

When this is the case, Clinton should anticipate that voters are more pessimistic and she should change when she releases the speeches to account for this increased pessimism. But, then voters in turn will anticipate Clinton’s revised disclosure strategy, and will be even more pessimistic when she doesn’t disclose her information. Knowing this, Clinton will again revise her strategy, leading to even more pessimism among voters.

The logic keeps going and going, until we reach the conclusion that when Clinton doesn’t release her transcripts, voters expect that the speeches must have been truly awful. In response, Clinton will release the speeches whenever they are better than the worst possibility. That’s the Nash Equilibrium of the game.

This idea is called the unravelling result in game theory. The term refers to the way beliefs about those who refuse to disclose their information unravel, deteriorating as the “if she anticipates that he anticipates …” logic is applied over and over again.

We know that voters are not always rational, and therefore the extreme prediction of the unravelling result is unlikely to apply in the real world. But the logic does show that as voters (correctly) become more pessimistic about a candidate’s refusal to share information, this puts more pressure on the candidates to disclose information.

If (as game theory suggests) voters think the worst about a candidate who refuses to disclose information, then this puts the greatest pressure on candidates for disclosure. This will lead to greater disclosure of information, and more accurate voter beliefs about the candidates.

According to game theory, voters should see Clinton’s refusal to disclose her speeches or Trump’s refusal to make public his taxes and conversations as a very bad sign, and they should expect the worst. Only when voters are pessimistic enough will the candidates choose to release their information, and this could make us all better off.

This article originally appeared on the author’s personal blog.