Compared to national popular vote, the electoral college favors voters in small states (on average), not large states. It’s because of those extra 2 electoral votes that each state gets!

The other day, in discussing the virtues of the electoral college compared to national popular vote election for president, Jonathan Bernstein wrote that “the big, urban states traditionally did very well in the electoral college. . . . all else equal, a presidential candidate would rather pander to a large state with lots of winner-take-all electoral votes than a small one, which should tend, over time, to balance out the small-state advantage in the Senate.”

I wrote that Bernstein was wrong, that on average, the electoral college benefits voters in small states, in the sense that an individual voter in a randomly-selected small state is more likely to have a decisive vote, compared to an individual voter in a randomly-selected large state. With a national popular vote, of course, all votes are equally likely to be decisive. I presented some graphs and links to some papers supporting my point.

Bernstein replied as follows:

I [Bernstein] don’t think the question of whether a voter in a particular state has a better chance of tipping the election—the question which Gelman’s articles consider—is the same question as whether that state has an electoral college advantage. It’s a piece of it, but not, I don’t think a very important one.

The question of individual voter power looks at it from the point of view of the individual voter. What we want, however, is to look at it from the point of view of the campaigns. What matters there is whether there are differences between states linked to size. As I [Bernstein] said earlier, it used to be the case that big, urban states tended to be closer than the smaller, rural states. . . . if you go after close states, you’ll be going after big states. And so, overall, the big states tend to have an advantage, over and above the advantage that their size alone would give them in a direct vote system.

Let me reply to both points.

First, the probability that a voter in a particular state has a better chance of tipping the election is exactly the same as the probability that a campaign will tip the election by changing one voter’s mind (or by persuading two potential supporters to go out and vote). It is also (to a few decimal places) equal to 1/1000 the probability that a campaign will tip the election by persuading 1000 voters to change their mind, or by persuading 2000 potential supporters to vote, or by persuading 500 voters to change their mind plus persuading 1000 potential supporters to vote.

Let me put it another way. Suppose that in state i, the marginal cost to a campaign of converting one voter (or persuading two supporters to turn out) is x_i. (We have to talk about marginal costs, as there is a diminishing return. There are also possibly other benefits to campaigning in a state, for example to energize local volunteers and campaign contributors, to influence the national media, to help in down-ballot races, etc., but let’s set that aside for now, as these other concerns aren’t really relevant to the Electoral College vs. National Popular Vote debate.) Then under the national popular vote, the decision is simple: to increase your chance of winning the election, you should campaign where 1/x_i is highest. Under the electoral college, though, you should campaign where p_i/x_i is highest, where p_i is the probability that one vote in state i will swing the election. And, as I’ve shown in several articles, under reasonable models p_i is, on average, higher in small states. Thus one effect of the electoral college is to bump up the benefits of campaigning in small states (on average), compared to small states.

It may very well be that, due to economies of scale, p_i is lower in large states, thus making it advantageous for campaigns to devote more effort per voter to large states. Bernstein speculates, “Looking ahead in the long run, it’s probably a lot easier for a party to target big, potentially close states and ignore the little ones, all things being equal.” Or maybe not. For my argument here, it doesn’t really matter. But, in any case, if it’s a good idea to campaign in large states under the electoral college system, it’s even more of a good idea to campaign in large states under the national popular vote.

OK, Bernstein might be right. There might be some nonlinear effect whereby, under the national popular vote, candidates would be expected to campaign everywhere, whereas the electoral college legitimizes state-level campaigning. It’s possible. But that seems like a bank-shot argument to me. Given that, as we’ve found, the electoral college increases the short-term benefits to campaigning in small states compared to large states, it seems unlikely (although, I admit, possible) that this effect would flip in the long term. In any case, I would expect such an effect to be small.

I’m not simply comparing states to each other, I’m comparing states to each other under the electoral college, compared to comparing states to each other under the national popular vote.

Bernstein’s second point is that elections tend to be closer (in percentage terms) in large states than in small states. That’s true. In fact, Jonathan Katz, Joe Bafumi, and I wrote a paper in 2004 on that very topic: Standard voting power indexes don’t work: an empirical analysis. {em British Journal of Political Science} {bf 34}, 657—674., and we found this pattern of elections being very slightly closer in larger jurisdictions to hold in non-presidential elections as well.

But closeness of the election is not the only thing that determines the expected payoff from campaigning in a state. The other factor that comes into play is the number of electoral votes in the state. In fact, if every state had a number of electoral votes proportional to its population, there would be a large-state advantage in the electoral college, just as Bernstein says. But that’s not the way it goes. The small states have electoral votes way out of proportion to their population, and it turns out this makes a difference. We did the math. Take a look again at this graph from our 1998 article:

As you can see, almost all the small-state advantage comes from the states with 3 or 4 electoral votes. And it varies by year (as well as by state). Those extra two electoral votes might not seem like much, but they make a difference. Why do they make a difference? Because elections in large states are only very slightly closer (on average) than elections in small states. Thus it takes only a very small “thumb on the scale” of electoral votes to knock out that large-state advantage.

Back to research

I’m not trying to bang on Bernstein here; I appreciate his open discussion, and all of this makes it clear that we did not fully explain ourselves in our articles as clearly as we could have.

In any case, he made another point that relates very closely to my 2002 article with Jonathan Katz and Francis Tuerlinckx, The mathematics and statistics of voting power. {em Statistical Science} {bf 17}, 420—435. Bernstein writes that the pattern of larger states having slightly closer elections in percentage terms is probably “built into the electoral college system . . . it’s probably true that larger states are less homogeneous and therefore more likely to be competitive.” We discuss this in our 2002 paper and present some mathematical models that have this property. I don’t really know what to do with these models, so I haven’t written anything on them for awhile, but I really like this stuff and would love to pursue them further. Some cool stochastic processes

Bottom line

In summary, if the number of electoral votes in each state were proportional to its population (or its total voter turnout), then Bernstein would be right: the slightly increased closeness (in percentage terms) of elections in large states would result in the electoral college benefiting large states (in the sense of providing an incentive to do more campaigning in those states, compared to what would be done under a national popular vote system). Actually, though, every state gets two free electoral votes. This is enough to give a (small) average increased incentive to campaign in small states, compared to what would be optimal with a national popular vote.

The effect is not large, and it varies by state, but it’s there. If we want to talk about problems with the electoral college, the #1 issue to me is that voters in states such as Massachusetts and Utah don’t get a chance to make a difference. We can also talk about good features of the electoral college. Representation of large vs. small states is way down the list. But if we do want to talk about representation of large vs. small states then, yes, the effect of the two free electoral votes per state outweighs, on average, the effect of larger states having slightly closer elections in percentage terms.

That said, it’s possible that Bernstein is right, that there are some sort of non-monotonic emergent long-term properties of the electoral college system that favor large states, compared to how things would be under a national popular vote. But that seems purely speculative to me. The numbers say otherwise.

Why do I care?

No, this isn’t just a “Someone is wrong on the internet!” story or even a “Someone doesn’t know about my research!” story.

My concern is as follows. The U.S. Senate hugely, hugely overrepresents small states. I haven’t looked at the numbers recently, but a few years ago I checked and found that California had a population equal to the 22 smallest states. Each of those states has two senators. If California were overrepresented in the Senate the way that small states are overrepresented, California would have 44 senators (out of 100). Just imagine the floor debates: “The gentleman from California yields to the gentleman from California,” etc etc etc. Yet we are used to having two senators from Delaware, Wyoming, etc.

OK, fine. It’s in the U.S. constitution. Love it or leave it. It is what it is. But it’s a key aspect of our system and I don’t think it should be minimized. Misunderstandings about the mathematics of voting lead otherwise knowledgeable observers and scholars such as Bernstein to write that the (nonexistent) large-state bias of the electoral college “should tend, over time, to balance out the small-state advantage in the Senate.” That’s just not so. First, the large-state, small-state effect of the electoral college is small. Second, it goes in the wrong direction. It also gives a boost to small states.

7 Responses to Compared to national popular vote, the electoral college favors voters in small states (on average), not large states. It’s because of those extra 2 electoral votes that each state gets!

  1. AmbiValent August 1, 2012 at 3:36 am #

    How does that compare to Penrose’ work? Assuming one state with 1 House seat, one with 16, and one with 49 (and 47 more states not looked at here), I thought Penrose would state each person has an equal chance of influence if EC seats were distributed 1 : 4 : 7, while the EC gives them 1 : 6 : 17?

    • Andrew Gelman August 1, 2012 at 2:55 pm #


      Penrose’s work makes sense under a mathematical model in which voters are voting independently and at random. This model does not fit reality, in particular the closeness (in percentage terms) of elections declines by approximately a factor of N^(-0.1) empirically (where N is the number of voters) rather than N^(-0.5) as would be required for Penrose’s numbers. See my two papers linked above for much discussion of this.

      • AmbiValent August 1, 2012 at 5:50 pm #

        I’m still trying to find out how this translates to the real world. Can I translate the finding that large states aren’t much closer than small ones to the following simplified picture: each state has the same chance to have X% Republicans and leaners, Y% Democrats and leaners, and 10% true independents?

        Because if so, then I’ll have small and large states in which X=Y=45%, and small and large states in which X>=50%, Y<=40% or vice versa.

        In the first case, the 10% independents make the difference. If they vote for Republicans or Democrats independently of each other (since the state is otherwise balanced), I'm back at Penrose – so Republicans and Democrats have a better payoff if they go after the larger balanced states, instead of a number of smaller balanced states with equal total population, even with the 2 EV bonus in the EC.

        In the second case, the independents only matter in the state when there's a landslide – and the state doesn't matter nationally. So in these states, Republicans and Democrats have a better payoff if they have a large number of smaller states, instead of a small number of large states with equal total population, due to the 2 EV bonus in the EC.

        In my opinion, states usually stay in their group, and politicians take the second group as a given once the primaries are over (what would they do? the other guy can win, but they're against him, and they may be closer to a 3rd party candidate, but that one can't win), so politicians fight over the first group, and within in, mainly over the large states.

        If there's an error in my reasoning, please point it out.

        • Andrew Gelman August 2, 2012 at 2:44 am #


          Your model is not terrible, but the mistake in your reasoning is the assumption that the (political) independents vote (statistically) independent of each other. Actually there are statewide and national swings that shift large numbers of voters at once. Penrose’s mathematics works fine if there is independent voting given fixed probabilities but this model does not describe actual vote swings.

          In any case, you can go to our paper and see the data. Indeed, elections tend to be slightly closer in large than in small states, but by a power of about N^(-0.1), not N^(-0.5) as would be required for the methods of Penrose, Banzhaf, etc., to work.

          This is a subtle point which is why my colleagues and I wrote two papers on the topic!

  2. Morriss August 3, 2012 at 7:44 am #

    ” My concern is as follows. The U.S. Senate hugely, hugely overrepresents small states…. OK, fine. It’s in the U.S. constitution. Love it or leave it.. “

    Your newly-discovered Big/Small State issue was in fact a huge, divisive issue at the founding United States Constitutional Convention. The solution was famously known as “The Great Compromise”:

    A Congress with two houses.
    One house based on proportional representation.
    Another house based on equal representation.

    The U.S. Senate with equal state representation is not some casual happenstance… and it is deliberately counter-balanced by the House of Representatives with proportional state representation.

    The U.S. Electoral College system is not some mindless accident either. The American citizenry directly elect state representatives– whose only purpose is to then elect a Federal chief executive officer (President). The Federal Government was designed as a voluntary association of sovereign “states” … a critical point lost on most Americans today.

    Do you object to Representative-Democracy in general — or only to its specific form in the American “Electoral College” ?

    Perhaps you prefer the enlightened Parliamentary method of selecting a chief executive (Prime Minister) used in Great Britain, Canada, Germany, etc ?? In those parliamentary systems… the chief executive is selected by the legislature — not by the people directly. Our Electoral College is actually closer to a popular-vote system than most modern industrial nations… if that’s your preference.

    Difficult to discern what your overall point is here.

    • Andrew Gelman August 3, 2012 at 8:23 am #


      My overall point is that residents of the U.S. are represented roughly proportionally in the U.S. House and haphazardly in the Electoral College (depending on whether these people are living in swing states). Meanwhile, residents of small states are vastly overrepresented in the U.S. Senate.

      My particular point is that the Electoral College causes overemphasis on swing states (as is well known) along with a very small average overrepresentation of residents of smaller states. Thus it is not correct to say that the Electoral College balances out the small-state bias of the Senate.

      In my post above, I am comparing the Electoral College to a national popular vote system. Nowhere am I considering a system where the chief executive is selected by the legislature, so I don’t know where you were getting that from. Nor did I say that large/small state comparisons are “newly discovered.” I don’t know where you got that from either!

  3. Andrea October 22, 2012 at 2:02 pm #

    Regardless, Republicans in California and New York haven’t had their votes for president counted for in decades. It’s like never getting asked to the dance, but always having to make the homecoming queen look pretty.
    It sucks.