Reasoning about influence in politics


Understanding a politician’s influence at first appears to be hopelessly tangled. A politicians’ influence is very tenuously related to the vote they can cast in parliament, and is mediated by a complicated process involving respect for precedent, social consensus, explicit and implicit negotiation, explicit and implicit appeals to popular opinion, and so on. Fortunately, on closer inspection many of these challenges can be ameliorated.

In the following research notes, we introduce an argument that the naive answer is about right: if there are 100 politicians with one vote each, then each politician has about 1% of the total impact of the politicians.

The result is highly useful in making estimates of the influence you might expect to have by becoming a politician, or indeed in any situation when a group of people negotiate over an outcome e.g a company board setting strategy, or a committee of grant makers allocating funding.

Note that the following are only preliminary research notes that were made while doing a case study, and not the results of in-depth analysis, so we’re cautious about the conclusion. Nevertheless, we’re keen to share the ideas and seek feedback.

The argument

Consider a state in which each new government is able to pass a new set of laws, but these laws are only able to deviate very slightly from established precedent. At first, it may appear that this change significantly reduces the influence of each government, because their choices are constrained to lie within a relatively narrow range. But on closer inspection, this state’s respect for precedent has two effects: (1) it constrains the choices of each government to be similar to the choices of their predecessors, and (2) it amplifies the effects of this government’s choices, because those choices will implicitly bind the hands of future governments. Indeed, across a wide range of models these two effects roughly cancel out, so that prima facie widespread respect for precedent has little effect on the influence of each politician.

A similar analysis holds in many other cases. For example, if it is hard for a politician to break with the party line, then (1) this reduces their own freedom, but (2) amplifies the impact of their own maneuvering, by shifting the party line. Similarly, if politicians’ votes are determined by political pressure from other politicians, then this effect cancels out (though when their votes are determined by political pressure from elsewhere, this does decrease their influence—some influence is being shifted from politicians to the group which is doing the pressuring).

The result of this argument is that regardless of how politicians wield influence, the total impact of a single politician tends to be roughly equal to the total impact of all politicians, times their share of influence (which is prima facie distributed uniformly over all players, and may then be adjusted by other considerations which asymmetrically increase or decrease their influence).

So if we would be willing to pay $N to change the behavior of k politicians (perhaps distributed over many years), we should be willing to pay roughly $N / k to change the behavior of each one.

More details

Diminishing or increasing returns?

For a different analytical approach, consider for each i the amount we would be willing to pay to change the behavior of i politicians. As a function of i, we expect this function to be increasing. In some cases this function will have very inhomogenous structure—for example, if we simply needed a majority vote to pass some legislation then we might be willing to pay all $N dollars for the winning vote and nothing anywhere else. But in general we expect the gains to be more diffuse.

As we become more uncertain about the attitudes of other politicians and the mechanisms of the political system, it becomes less and less plausible that our willingness to pay to influence i politicians behaves very strangely near special values of i. So the core question becomes whether we expect returns to be diminishing or increasing.

For political parties increasing returns seem highly plausible, because the parliamentary system is structured to have very abrupt changes near obtaining a majority. But for interests within a party, or particular policy proposals, there aren’t such clean transitions. Indeed, there can’t be such clean winner-takes-all dynamics at every scale: for example, if each party internally used a winner-takes-all system and the party in power had complete authority, then an interest which had less than ½ support could nevertheless influence outcomes by first winning in the majority party, and so the returns to political influence become more diffuse. The issue becomes even more severe if there are winner-takes-all dynamics at more scales.

In general, without some strong reason to expect increasing returns, diminishing returns seem like the default assumption. The first politicians with a certain interest can pursue the most highly-leveraged opportunities to push their agenda, including making trades (implicitly or explicitly), focusing on parts of the political process where they can have the biggest impact, focusing on the most important issues for their interests, and influencing the values of other politicians.


Although this picture suggests that the influence of a single politician is not significantly diminished by the complexities of the political process, it is still the case that the influence of an individual politician is “spread out” by these complexities. For example, a politician exerts much of their influence indirectly in future years (by setting precedent and changing opinion) and are restrained by similar influences exerted in the past. Similarly, a politician may exert much of their influence in domains where they aren’t directly working.

This more diffuse influence may be less useful for certain tasks, especially if we wish to push very specific projects. This doesn’t disrupt the standard arguments that suggest we should face diminishing marginal returns, but it does mean that the influence of a small number of politicians may be less significant than it at first appears—i.e., returns diminish less slowly than we might expect.