People conspire all the time. Firms collude. Politicians bargain behind closed doors. Militaries deceive their enemies. But other conspiracies are detected, or they do not last long. Why do some remain secret while others are revealed?

A simple game theory model can help illustrate that the probability of a conspiracy being revealed depends on the number of independent choosing individuals whose cooperation is necessary, and the incentives those individuals face. The basic intuition of what makes it hard to keep a conspiracy a secret is not new. But there are interesting implications for institutional design.

Insights from game theory can help us think more clearly about election administration. Those insights can help us see why and how the decentralized structure of American elections dramatically increases the number of independent actors who would have to coordinate to manipulate a national election. That decentralization is one reason to trust U.S. election results.

The decentralization of election administration was a deliberate choice by the constitutional framers. They envisioned that the federal government would retain power to intervene to ensure the survival of the government, but the actual running of elections would remain in the hands of the states (see Federalist 59). While Alexander Hamilton argued for decentralization as necessary to preserve the freedoms of both the states and the federal government, it also has the happy effect of making election fraud much harder to accomplish at a national level.

An Institutional Model of Conspiracy

Let us consider a simple model of a conspiracy. I am using “conspiracy” here in a deliberately broad manner. By “conspiracy,” I simply mean two or more people working together to achieve some goal in secret. Discretion is key to the conspiracy—it will be detected if somebody reveals what they’re trying to accomplish or how. We can model this as a simple probability model:

P_d=1 – (1 – p)^N

Where:

P_d = the probability the conspiracy is disclosed

(1 – p) = the probability the conspiracy is not disclosed

N = the number of people in the conspiracy

Right away, we can see a key relationship: as the number of people involved in the conspiracy, N, rises, the probability of the conspiracy being disclosed increases exponentially. For the sake of numbers, let us assume a 95% probability that the conspirators stay silent. Additionally, we assume each conspirator’s probability of disclosure is independent of each other and identical. (These are simplifying assumptions and not realistic ones, but they help demonstrate the core insight, and that insight isn’t changed by the particulars.)

Having two conspirators results in a 9.75% chance of the conspiracy being exposed:

With 10 people, that chance jumps to 40.1%.

With 50, the probability of exposure is 92.3%.

More generally, we can write the model as:

lim
n → ∞
  [1 − (1 − p)^N] = 1

In other words, as the number of people involved in a conspiracy increases, the probability that the conspiracy is exposed approaches 100%. Even a small number of conspirators greatly increases the probability of the conspiracy being exposed.

Decentralizing the U.S. election system greatly increases the number of independent decision makers whose cooperation is required: each state, county, and polling place has election officials, commissioners, reviewers, etc. That means that by decentralizing election administration across states and local jurisdictions, the Constitution dramatically increases the number of independent actors whose cooperation would be required for coordinated manipulation. Any conspiracy to alter U.S. federal elections would require coordination of thousands, if not hundreds of thousands, of individuals over a vast geographic and political area. In short, the transaction costs of a conspiracy to alter elections rises substantially. The model here predicts that such a conspiracy is theoretically possible, but extraordinarily unlikely.

Now, that is just for a single election. In the case of persistent fraud, then time becomes a factor as well. We can rewrite our model to include time:

P_d(T)=1 – (1 – p)^NT

All the variables are the same, except for T, which is the number of time periods the conspiracy must remain hidden. Again, the problem remains but becomes compounded by the time variable.

What is important to note is that this model does not tell us conspiracies are impossible. Just the opposite: it helps us understand when conspiracies are likely to succeed. To succeed, either the timeframe or the number of conspirators (T or N) need to shrink or the probability of breaking silence (1–p) needs to increase. Conspiracies have succeeded, at least where “success” is defined as “the goal of the conspiracy is accomplished.”

For example, consider Bleeding Kansas. This story provides further evidence that the model is predictive of real life events. Bleeding Kansas did result in electoral fraud, but 1) it was localized and 2) almost immediately detected, which in turn led to the violence. It accomplished short-term goals, but was still detected.

Another example is the Manhattan Project. Some 130,000 people worked on what would become the creation of the atomic bomb. This project was carried out with absolute secrecy. (In fact, many “conspirators” did not know what they were working on until the bombs were dropped.) The N for this project was substantial, and that created risk. So, the federal government and the army worked to reduce the probability that any individual would divulge the secret. Reducing time was inherent to the project: there was a war on and there was pressure to finish the bomb before the Germans. What about (1–p)? We can think of p as a function of economic and social variables: incentives, monitoring, commitment to the cause, etc.

The federal government paid everyone working on the project, from the janitor to the top scientists, extremely well: sometimes four to five times what they could get in the private sector. This was especially true for minorities, who faced substantial discrimination outside the Manhattan Project. Furthermore, World War 2 had a strong patriotic component attached to it. Coupled with strong monitoring, the secrecy of the conspiracy remained.

The Manhattan Project demonstrates just how expensive large conspiracies are to maintain. That project alone consumed about 1.4% of GDP. We can think of that, partially, as the cost of secrecy. And that sort of expenditure is hard to hide. The Germans and Soviets knew we were up to something.

Here are the central insights from this model:

1. As always, institutional design matters. By decentralizing elections, the United States makes it exponentially more difficult to manipulate election results on a large scale, especially at the national level. And, if such manipulation did happen, it would be much easier to detect or uncover
2. Secrecy is a scarce resource just like any other. It is costly to “purchase” through incentives, enforced through monitoring, and so on. The transaction costs of a conspiracy rise rapidly as more people are involved
3. Given 1. and 2., we should ask for strong evidence when we hear allegations of large conspiracies.

A conspiracy is an equilibrium, not a mystery. Like any equilibrium, it survives only if the incentives of its participants support it. When someone alleges a conspiracy, the first question to understand how likely the conspiracy is to be a real one should not be “Could these people coordinate?” but “What incentives kept them coordinated and silent?”

On the flip side, to prevent conspiracies and prevent fraud, we can apply lessons from this model to create institutions that increase the transaction costs of undermining the system: create incentives that make secrecy all the more expensive. Of course, this is easier said than done. If designing incentives were easy, we’d not have these problems in the first place. But what I discuss here does provide a blueprint to help think through these incentives.