Since then, a number of questions and criticisms have been raised about our work — some offering sensible critiques and alternative perspectives and others simply mistaken. We have responded in print to some of these, and will list some of those responses at the end of this post. Here we will respond briefly to the most important challenges to our research. In brief, we don’t believe that any of these critiques, individually or collectively, undermine our central claims.

1. Majority “win rates” don’t really measure policy influence

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A group’s “majority win rate” is simply the percentage of policies for which the outcome is consistent with the preferences of the majority of that group and inconsistent with the preferences of the majority of the other group. Some critics argue that we were wrong to conclude that the middle-class and the affluent aren’t equally represented because when the majority of one group want a policy that the majority of the other group opposes, the two groups are about equally likely to get their way.

That, at least, was argued by political scientists Omar Bashir and J. Alexander Branham et al. They reanalyzed our raw data (available here) to show that when the majority of middle-income and the majority of high-income Americans want different policies, the middle class “win” by getting what they want 47 percent of the time while the affluent get what they want 53 percent of the time.

But think about this from a policymaker’s point of view. Policymakers have little incentive to lean one way or the other on a policy, if the public (or a particularly influential sub-group of the public) is closely divided. The chart below shows the likelihood that a policy will be adopted depending on the level of support or opposition among affluent Americans (those at the 90th percentile of the income distribution). Differences in degree of support matter most at high levels of support or opposition and least when the affluent are relatively equally divided, with only a slim majority in favor or opposed.

These critics treat a slim majority (say 51 percent) as equivalent to an overwhelming majority (say 91 percent). In the first case, nearly as many people would “win” if the policy change was rejected as if it was adopted; in the second case, refusing to change the policy would be a dramatic deviation from what the public wants.

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If we stop using the majority-plus-one approach of 50 percent support as the dividing line, and replace it with a more politically consequential benchmark of strong support (75 percent or above), the differences between middle-class and high-income respondents is much larger. When only the affluent strongly support a proposed policy change, that policy is adopted 46 percent of the time; when only the middle-class strongly support a policy, that policy is adopted only 24 percent of the time.

The affluent are, not surprisingly, better at blocking policies they dislike than achieving policy change they desire. When a policy is strongly opposed by the affluent (less than 25 percent support) but not strongly opposed by the middle-class, that policy is adopted only 4 percent of the time. But when a policy is strongly opposed by the middle-class but not by the affluent, the policy is adopted 40 percent of the time.

In other words, contrary to the story our critics tell, the well-to-do get the policy outcomes they strongly prefer far more often than do average-income Americans.

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2. “Winning” and influence are two very different things

In our work, we focused on how much influence different groups have over government policy while our critics often focus on how frequently a group sees its preferred policy adopted. We believe that both are important. Even a group with no influence will sometimes, perhaps often, get the policies it wants because it shares the preferences of a powerful group or because the strong status quo in our often gridlocked government means any group is likely to get its way on policies that it opposes.

In our 2014 “Testing Theories” paper we wrote, “ordinary citizens often win the policies they want, even if they are more or less coincidental beneficiaries rather than causes of the victory.” Getting what you from government is important, and average Americans would no doubt be even less enthusiastic about their government than they already are if not for these coincidental policy victories. To that extent, our critics are right to point to the high level of “democracy by coincidence” enjoyed by middle-income and high-income citizens.

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But focusing too fully on the half-full glass of average citizens’ coincidental victories gives a very incomplete picture of our severely biased political system. Actual influence matters, both because there are many important issues on which affluent and less well-off Americans diverge, and because “democracy by coincidence” is a pale imitation of real democracy, leaving the powerless majority dependent on a powerful minority to get the policies they want.

3. The policy preferences of the middle-class and the affluent are correlated but distinct

As our critics note (and as we have noted as well), policies that are popular (or unpopular) with high-income Americans also tend, on average, to be popular (or unpopular) with the middle-class and even with the poor. After correcting for measurement error, for example, middle- and high-income preferences are correlated at a very substantial .78 (Testing Theories, table 2).

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This high correlation reflects the fact that higher and lower income Americans hold similar views in many policy areas. On most aspects of national defense, environmental policy, drug policy, and education, for example the views of the affluent and the poor are nearly identical. Even on redistributive economic policies, there is often more agreement than one might expect. Lower and higher income Americans, for example, are equally supportive of unpaid family leave laws and equally opposed to a national sales tax. (The high correlation across income groups also reflects the high reliability of the aggregated preference measures we used as explained in Affluence & Influence.)

But there are also many important issues on which the affluent and the less well-off want different things. Middle-income Americans, for example, are more likely to support raising the minimum wage or indexing it to inflation; increasing income taxes on high earners or corporations; and cutting payroll taxes on lower income Americans. And they’re more likely to be opposed to tax cuts for upper-income individuals, spending cuts in Medicare, and roll-backs of federal retirement programs. Moreover, whatever divergence we find between middle-class Americans and those at the 90th income percentile are likely to understate the differences between average Americans and the truly rich (those at the 99th or 99.9th income percentiles).

4. The policy preferences of the truly wealthy are even more distinct

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We have lots of data across many decades on the preferences of Americans at the 90th income percentile. Unfortunately, we don’t have such data on the preferences of the truly rich. But we do have some evidence about the truly rich. And that evidence suggests that their views differ from middle-income Americans’ in the same direction as the “merely affluent” 90th percentile — but to a greater degree (Page and Hennessy, Page, Bartels, and Seawright).

For instance, consider a study of Chicago-area millionaires conducted in 2011. Across a range of economic policies, the broader public was far more likely to support government efforts to improve the lives of ordinary citizens than were the millionaires. For example, 68 percent of the public — but only 19 percent of the millionaires — thought the government ought to see to it that everyone who wants to work can find a job. Fully 78 percent of the public — and only 40 percent of the millionaires — thought that the minimum wage should be high enough so that no family with a full-time worker falls below the poverty line. And 58 percent of the millionaires but only 27 percent of the public thought that, to reduce the federal budget deficit, the U.S. should cut spending on domestic programs like Medicare, education and highways.

It would be reasonable to think that these truly rich Americans exert more influence over policy than the merely affluent. If so, then the preference divergences that our critics focus on — between the middle-class and the affluent — are less significant than the larger differences between average Americans and the truly wealthy.

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To assess influence, we must rely on the data we have for the 90th income percentile. As a result, our findings might attribute influence to the merely affluent that really belongs (at least in some substantial measure) to the truly rich. To the extent that this is true, the views of America’s truly wealthy would be the views that matter most. And as the policy preferences of the Chicago-area millionaires suggest, there are huge gaps between what the wealthy prefer and what the majority of the American public wants the government to do on central economic issues.

5. Influence is massively unequal — even when using the “merely affluent” as a proxy for America’s economic elites

Finally, there can be little doubt that influence over government policymaking is massively unequal in America. In our “Testing Theories” paper, we used structural equation models to assess the independent influence of middle-class and affluent citizens as well as interest groups. In “Affluence & Influence” Gilens used very different techniques to assess that influence. However we measured it, the results were the same: the affluent have more power than the rest.

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We can see that unequal influence even if we leave aside complex statistical techniques. Simply tracking policy outcomes by levels of support clearly shows the influence of the well-off and the lack of influence of middle-class Americans.

To see that, we can use the same measure of strong support or opposition to decide what each group strongly prefers: more than 75 percent or less than 25 percent. The table below shows that strong support among the affluent is associated with about a 25 point greater probability of a policy being adopted (holding constant support or opposition among the middle-class) while strong support among the middle-class is actually associated with a small decline in the likelihood that a policy will be adopted (holding constant support or opposition among the affluent).

In other words, strong support among high-income Americans roughly doubles the probability that a policy will be adopted; strong support among the middle class has essentially no effect.

That’s even more true when we look at what each income group strongly opposes. The table below shows that when the affluent strongly oppose a policy, it’s associated with a 25 to 30 percentage point decline in the likelihood that it will be adopted; but when the middle class strongly oppose a policy, that’s associated with a small increase in the likelihood it will be adopted (again, holding constant the support or opposition of the other group).

This same pattern even holds, although less strongly, if we look at 50 percent support instead. When most of the affluent support a policy, it’s about 13 or 14 percent more likely to be adopted than if most of the affluent are opposed (holding constant support among the middle-class). But if most of the middle class support a policy, it’s only 2 or 3 percent more likely to be adopted (holding constant support among the affluent).

To summarize, U.S. government policy depends strongly, although imperfectly, on what the well-to-do want but weakly, if at all, on what middle-income Americans prefer. We believe this massive inequality of influence is a serious indictment of the quality of American democracy.

Should the majority rule?

We would never argue, as one journalist critical of our work suggested, that “democracies should enact the people’s opinions exactly as currently stated.” As Gilens wrote in “Affluence & Influence,” “There are good reasons to want government policy to deviate at times from the preferences of the majority: minority rights are important too and majorities are sometimes shortsighted or misguided in ways that policymakers must try to recognize and resist.”

An ideal democracy, if such a thing can be imagined, would not offer a perfect match between public opinion and government policy, as if people sitting at home were voting directly from their TV remotes or mobile phones. But the gross inequality that our research reveals is strongly undemocratic and incompatible with notions of political equality that most Americans hold dear.

Many Americans voting for outsider candidates believe that government pretty much ignores people like them. We think they’re right.

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Martin Gilens is professor of politics at Princeton University and a fellow at the Center for Advanced Study in the Behavioral Sciences.