Explainer: The Median Voter Theorem, Turnout, and the Democratic Primary

[The following is a lightly-edited and annotated transcript of the above video.]

This primary season is a rare time that results from game theory are receiving mainstream media attention. As such, it is worth a moment to talk about how the median voter theorem and turnout are affecting the battle between Bernie Sanders and Joe Biden.

Let’s start with the central argument Biden’s camp is making. It is basically an appeal to the median voter theorem. In practice, the median voter theorem says that, in a two-candidate election, the candidate with the ideological position closest to the median voter is most likely to win.
To diagram this, imagine we put every voter in the United States on a left-right spectrum. Here, to make things simple, let’s suppose there are only five.

We call the middlest-most person the “median voter”.

Now consider the ideological positions of Donald Trump and Joe Biden. Biden is a moderate democrat. He’s left-of-center, but not by too much. Trump, on the other hand, is far to the right.

If everyone votes for the candidate closest to their preferred position, whomever the median voter supports will win the election. This is because everyone on the left or everyone on the right will also vote in the same direction, and that is enough to guarantee more than half the vote.

Here, the median voter supports Biden. As such, Biden is likely to win.

In contrast, Sanders is a very progressive candidate. He has an ideological position far to the left. In fact, it is further to the left from the median voter than Trump is to the right of the median voter. Now the median voter supports Trump, so Trump is more likely to win the election. This is why moderate Democrats say that Biden is the more “electable” candidate.

Sanders supporters have an interesting counterargument. In their view, the median voter theorem isn’t “wrong”, just underspecified. Who the median voter is depends on who is turning out for the election. Their bet is that Sanders will inspire a bunch of young, liberal voters to come to the polls.
And with more liberal voters, the median voter shifts to the left. Now Sanders can win despite his ideology.

There are two counterarguments to the Sanders’ position. First, we haven’t seen young voter turnout rise substantially during the primary season.

Second, by the best we as political scientists can estimate, the turnout necessary to compensate for Sanders’ more extreme ideology would be unprecedented. That doesn’t mean it won’t happen—Sanders’ campaign has been historic for many reasons—just that it isn’t obvious that it will.

If you are interested in learning more about the median voter theorem, click the link for a lecture just on the subject. It’s also a topic I cover in Chapter 4 of Game Theory 101: The Complete Textbook.

Bargaining Theory in Action: Donald Trump and Executive Orders

Over the past seven days, you have either grown to love or hate executive orders. But regardless of your political perspective, Trump’s actions are showcasing one of the major findings of bargaining theory: namely, that proposal power is one of the greatest sources of bargaining power.

What Is Proposal Power?
Put simply, proposal power is the ability to structure the terms of a possible settlement. It is distinct from receivership, which is the ability to say yes or no to any particular offer.

Perhaps surprisingly, whether you can make an offer has an enormous impact on your welfare. Imagine a seller needs at least $30 to sell and a buyer will buy for no more than $50. Clearly, a transaction should take place, as there is a $20 surplus (the buyer’s maximum minus the seller’s minimum).

If the seller can make a take-it-or-leave-it offer, then he can set the price at $50. The buyer, only having the ability to accept or reject, buys the good—it is just enough to convince him to buy.

Now imagine the buyer makes the take-it-or-leave-it offer. She can set the price at $30. The seller, only having the ability to accept or reject, sells the good—it is just enough to make her willing to part with it.

Who proposes the offer makes a major difference. If we endow the seller with the proposal power, she walks away with the entire $20 surplus. Yet if we endow the buyer with the proposal power, the exact opposite happens: the buyer receives the entire $20 surplus. It’s night and day!

You might think this is because of the take-it-or-leave-it bargaining protocol. It is not. If rejecting simply leads to another round of bargaining where the proposer stays the same, the result is identical: the proposer receives the entire surplus.

The lesson here is simple: without the ability to make offers or counteroffers, the deals reached will look very bad for you.

When it comes to executive orders, the POTUS’s proposal power is YUGE.

Proposal Power in Action
In many contexts, proposal power is easy to acquire. At a flea market or car dealership, for instance, nothing stops a buyer or seller from making the initial offer or responding with a counteroffer of their own. They just do it.

Government is different. Some systems have proposal structures baked into the legal system. And this has major consequences for the types of laws that are ultimately implemented.

In the United States, most of the proposal power rests with Congress. That is because only Congress can write laws; the President can merely sign or veto whatever lands on his desk. Thus, if Congress and the President both agree that a law needs fixing, the solution implemented is going to be much closer to what Congress wants than what the President wants.

There is a key exception to that rule, however, and it is what we have been seeing over the last week. Executive orders essentially flip the script—the President institutes the policy, and it is up to Congress to accept it (by doing nothing) or reject it (by creating a bill that undoes the policy, and then overriding the inevitable presidential veto).

It should be obvious that the President choosing his own policies would result in outcomes closer to his ideological preference. I suspect liberals realized this was an issue weeks ago, but only recently realized how bad it would be for them. Bargaining theory goes deeper here and explains this. It isn’t just that the President fares better when choosing the policy than when he signs a Congressional law. It is saying that a strategic, forward-thinking President can set the policy so far to his liking that Congress is just barely willing to go along with it. And if the gulf between Congressional preferences and Presidential preferences is substantial, so too will be the types of policy outcomes associated with executive orders and standard legislation.

This logic sets expectations for what is to come for both fans and foes of Trump. Foes should continue to fear executive orders and should not be particularly worried about laws passed through Congress. Liberals see Paul Ryan as their last remaining hope. And they might find solace there—but only for standard laws. Undoing executive orders requires having a much larger share of Congress on board, so Paul Ryan does not get you very far.

For Trump supporters, it is the opposite: they will find standard laws relatively unexciting but be thrilled with the executive orders.


Bargaining and Supreme Court Nominations

With the death of Justice Antonin Scalia over the weekend, the scramble has begun to make sense of the nomination process. Senate Republicans are (predictably) arguing that the seat should remain unfilled until after the 2017 election, presumably so a Republican president could potentially select the nominee. Senate Democrats and President Obama (predictably) feel differently.

Overall, it seems that people doubt that Obama will resolve the problem. But most of the arguments for why this will happen fail to understand basic bargaining theory. That’s what this post is about. In sum, nominees exist that would make both parties better off than if they fail to fill the vacancy. Any legitimate argument for why the seat will remain unfilled until 2017 must address this inefficiency puzzle.

You can watch the video above for a more through explanation, but the basic argument is as follows. The Supreme Court has some sort of status quo ideological positioning. This factors who the current median justice is, the average median of lower court justices (which matters because lower courts break any 4-4 splits from the Supreme Court), and (most importantly) expected future medians. That is, one could think about the relative likelihoods of each presidential candidate winning an election and the type of nominee that president would select and project it into this “status quo” ideology.

Confirmation of a new justice under Obama would change that ideological position. In particular, Obama’s goal is to shift the court to the left. Republicans want to minimize the shift as much as possible.

Nevertheless, failing to fill the position is costly and inefficient for the court. In other words, ideology aside, leaving the seat unfilled hurts everyone. These costs come from overworking the existing justices, wasting everyone’s time debating these issues incessantly, and generally making the federal government look bad. Due to these costs, each side is better off slightly altering the ideological position of the court in a disadvantageous way to avert the costs.

Visually, you can think of it like this:


(If that isn’t clear, I draw it step-by-step in the video.)

Thus, any nominee to the right of the Republicans’ reservation point and to the left of the Democrats’ reservation value is mutually preferable to leaving the seat unfilled.

This simple theoretical model helps make sense of the debate immediately following Scalia’s death. Senate Majority Leader Mitch McConnell says “The American people should have a voice in the selection of their next Supreme Court justice. Therefore, this vacancy should not be filled until we have a new president.” But what he’s really saying is “Our costs of bargaining breakdown are low, so you’d better nominate someone who is really moderate, otherwise we aren’t going to confirm him.”

And when Senator Elizabeth Warren says…

…what she really means is “Mitch, your costs are high, so you are actually going to confirm whomever we want.”

To be clear, the existence of mutually preferable justices does not guarantee that the parties will resolve their differences. But it does separate good explanations for bargaining breakdown from bad ones. And unfortunately, the media almost exclusively give us bad ones, essentially saying that they will not reach a compromise because compromise is not possible. Yet we know from the above that this intuition is misguided.

So what may cause the bargaining failure? One problem might be that Obama overestimates how costly the Republicans view bargaining breakdown. If Obama believed the Republicans thought it was really costly, he’d be tempted to nominate someone very liberal. But if the Republicans actually had low costs, such a nominee would be unacceptable, and we’d see a rejection. (This is an asymmetric information problem.)

A more subtle issue is that presidents have a better idea of a nominee’s true ideology than senators do. Maya Sen and I explored this issue in a recent paper. Basically, such uncertainty creates a commitment problem, where the Senate sometimes rejects apparently qualified nominees so as to discourage the president from nominating extremists. Unfortunately, this problem gets worse as the Senate and president become more ideologically opposed, and polarization is at an all-time high.

In any case, I think the nomination process highlights the omnipresence of bargaining theory. Knowing the very basics—even just a semester’s worth of topics—helps you identify arguments that do not make coherent sense. And you will be hearing a lot of such arguments in the coming months regarding Scalia’s replacement.