“Strength & Numbers”: Is a Weak Argument Better Than A Strong One?

Thanks to Kevin Collins, I saw this forthcoming article (described succinctly here) by Omair Akhtar, David Paunesku, and Zakary L. Tormala.  In a nutshell, the article, entitled “The Ironic Effect of Argument Strength on Supportive Advocacy,” reports four studies that suggest “that under some conditions…in particular, presenting weak rather than strong arguments might stimulate greater advocacy and action.”

This caught my attention because I think a lot about information and, in particular lately, “advice” in politics.  One of the central questions (in my mind at least) in political interactions is when communication can be credible/persuasive.  I was additionally attracted, given my contrarian nature, to the article because of statements such as this:

[These findings] suggest, counterintuitively, that it might sometimes behoove advocacy groups to expose their supporters to weak arguments from others—especially if those supporters are initially uncertain about their attitudes or about their ability to make the case for them. (p. 11)

Is this actually counterintuitive? I would argue, unsurprisingly since I’m writing this post, “no.”  Why not?

I have two simple models that indicate two different intuitions for this finding, both in a “collective action” tradition.  In addition to sharing a mathematical instantiation, also common to the motivations behind both of these models is the fact that the article’s findings/results are largely confined to individuals who already supported the position being advocated.  For example, weak “pro-Obama” arguments were more motivating than strong “pro-Obama” arguments among individuals who supported Obama prior to exposure to the arguments. (The effect of argument strength was insignificant and actually in the opposite direction among those who did not support Obama prior to exposure.)

My focus on collective action in this post is justified because the 4 studies reported in the paper each examined advocacy for a collective choice (either an election of a candidate and adoption of a public policy). Thus, in all studies, advocacy can potentially have an instrumental purpose: secure the individual’s desired collective choice.  Accordingly, suppose that an individual i is predisposed to support/vote for President Obama.  To keep it simple, suppose that advocacy is costly—if advocacy is less likely to affect the outcome of the election, then individual i will be less likely to perceive advocacy as being “worth it.”

Collective Action: Complementary Strength and Numbers. The question is how individual i should react to hearing a pro-Obama argument from a “random voter.”  If the argument is weak, should individual i—who, remember, already supports Obama—view advocacy as more or less likely to affect the outcome?

Well, suppose for simplicity that the election outcome is perceived to be a function like this:

$f(q,n) \equiv \Pr[\rm{Obama\; Wins}] = \frac{ q * n }{1+ q * n}$,

where q>0 is the quality of the best argument that can be made for Obama (a content-based persuasive effect), and n>0 is the number of advocates for Obama (a “sample size”-based persuasive effect).  Then, being a bit sloppy and using derivatives (approximating n being large), the marginal value of advocacy is

$\frac{ q }{1+ q * n} (1-f(q,n))$

and, more importantly, the marginal effect of quality on the marginal value of advocacy (the “cross-partial”) is

$\frac{2 n^2 q^2}{(n q+1)^3}-\frac{3 n q}{(n q+1)^2}+\frac{1}{n q+1}$

The key point is that, for reasonable range of parameters (specifically in this case, if n and q are both larger than 1), increasing the perceived quality of the best argument that can be made for Obama reduces that the marginal instrumental value of advocacy for an Obama supporter.  Note that the perceived quality of the best argument that can be made for Obama is a (weakly) increasing function of the observed quality of any pro-Obama that one is presented with.  In other words, observing a higher quality pro-Obama argument should lower an Obama support’s motivation to engage in advocacy.

Collective Action: Increasing Persuasive Strength. For the second model, let’s pull “numbers of advocates” out and, instead, let’s modify the election outcome model as follows:

$f(q) \equiv \Pr[\rm{Obama\; Wins}] = \frac{ q}{1+ q}$,

where q>0 is the quality of the best argument that is made for Obama.  Now, add a little bit of heterogeneity.  Suppose that a(i) is the quality of the best argument that individual i “has” in favor of Obama.  This, at least initially, is private information to individual i, and suppose it is distributed according to a cumulative distribution function G.  Suppose for simplicity that the argument to which individual i is exposed is the best he or she has yet seen (this isn’t necessary, but allows us to get to the point faster), and denote this by Q.  Furthermore, suppose that individual i will find it worthwhile to advocate (i.e., spread/share his or her own pro-Obama arguments) if a(i)>Q. (This is similar to assuming that advocacy is costless, but this is not important for the conclusion.) Then what is the probability that individual i will find it strictly worthwhile to advocate after observing an argument of quality Q?  Well, it is simply

$1-G(Q)$

Since G is a  cumulative distribution function, it is a (weakly) increasing function of Q.  Thus, 1-G(Q) is a (weakly) decreasing function of Q.  Again, observing a higher quality pro-Obama argument should lower an Obama support’s motivation to engage in advocacy.

What’s the point?  Well, first, I think that information is a very interesting and important topic in politics—that’s “why” I wrote this. But, more specifically, it is ambiguous how to interpret the subsequent lobbying/advocacy behaviors of individuals with respect to varying qualities of information/arguments offered by others when individuals expect that the efficacy of their lobbying/advocacy efforts is itself a function of the quality of the argument.  In these examples, in other words, individuals might not be learning just about (say) Obama, but also about how effective their own advocacy efforts will be.  If this is the case, I humbly submit that the findings are not at all counterintuitive.

With that, I leave you with this.