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How to Make Figures for Formal Articles

Posted on August 10, 2020 | Leave a comment

My last post on how I write formal articles stressed the importance of figures. I thought it might be helpful to expand on that idea. In short, figures are a critical part of the communication process. Good figures separate the well-executed papers from the just good ones. They will reinforce your key points, allow you to write about your results in a more engaging manner, and help you learn more about your model in the process.

Put differently, you need figures in your paper. And the better they are, the better it is for you. These are my thoughts on them.

Why Should I Have Figures?
There are two connected reasons to have figures in your papers: clarity and readability. The majority of formal papers attempt to convince the reader that something that is not obviously true is, in fact, true. We jump through all sorts of hoops to accomplish this. Unfortunately, those hoops are often dense or dull (inclusive or). A picture is worth a thousand words, and replacing a thousand words of abstract mathematics is a win.

Regardless of the mathematics, you want to invite your readers to actually read your manuscript. If someone is skimming through articles, a 10,000 word wall of text is not going to stop them. It also will not stop someone who has read the first five pages and getting bored from closing the window.

But imagine instead one of these readers encounters a figure that shows them something really strange. Someone who was not planning to read your paper might want to learn how the figure came to be. And it might also convince your slipping reader that the ride is worth continuing.

What Should Be Made into a Figure?
My rule of thumb when making figures is to pretend that I am doing an invited talk for the paper I am writing. What do I want to put in the slides to illustrate all my main points? I do not want a slide that says “Proposition 3: The probability of war is nonmonotonic in its costs.” That is boring. I should instead have a plot that illustrates the point. After thinking about all of those slides I would want to make, I will then put almost all of them in the paper itself.

More concretely, I almost always want an equilibrium plot that varies a parameter of interest and shows how the equilibrium changes. These come in two forms. The first is one dimensional. If the narrative you are telling only focuses on one variable, then put that variable on a number line. Add some cutpoints on that number line, and describe the equilibrium behavior associated with it.

Here is an example from my Doubling Down paper:

We are varying the quantity of a CDF evaluated at a particular value. Because it is a CDF, we have bounds on 0 and 1. There are two cutpoints also labeled along the axis that separate the three equilibrium parameter spaces. The top part gives a little context of what is happening in the equilibrium and matches that description to the corresponding proposition in the text. Keep in mind that these are not hard rules about what should be in the figure—what you ultimately put into it depends on what you think is important to communicate with the reader.

The second type figure is two dimensional. These are more complicated to build but are necessary if the narrative is on how large some quantity is related to another. The first step is to put all of the cutpoints in terms of one parameter. This becomes your vertical axis. Your horizontal axis is the other variable of interest. The equilibrium regions of the corresponding figure are now shapes that you can fill in with the critical information.

Here is an example from my Bad-Faith Cooperation paper:

The two key parameters we vary in this figure are the cost and effectiveness of resistance. The labels tell the reader what type of signaling strategy the types adopt for each parameter space and how much resistance the actor chooses within the equilibrium space.

To actually draw two dimensional figures correctly, you need to use realized values of the parameters. It will often take some finagling to get the ideal spacing for each region and to make sure that the each parameter space you want to cover exists for those realized values. As a result, there are usually a lot of moving parts in the final figure you draw. I try to compensate for that by removing any extras that do not absolutely need to be in the figure. This is why the two dimensional figure does not have axis labels for the cutpoints, unlike the one dimensional figure.

Outside of equilibrium plots, you should consider drawing figures for any comparative statics of interest. Any nonmonotonicities or discontinuities always make the cut. Here is an example of that from my Credible Commitment in Covert Affairs paper:

The horizontal axis varies how much trouble an executive gets in if a covert action is exposed. Even if I have lost a reader during the modeling section, there is a good chance that this figure will pull them back in. The executive would seem to be hurt by the potential to get into trouble. The black line on the figure would appear to confirm that. But why does it suddenly go up (red) and then level off (gray)? That is puzzling. I may just have pulled the reader back into the text to find out the answer.

For the most part, the only comparative statics do not receive corresponding figures are basic monotonic relationships, especially if the plot would be just a straight line. Even then, if something strictly increases before becoming flat, I may still want to include that.

How Do I Make Better Looking Figures?
Now that we have established why you should have figures and what is worth figuring, you may want to spend some time making those figure look more attractive. I have two tips here.

First, I strongly recommend TikZ. It integrates seamlessly with TeX, so you can insert math expressions at the proper cut points. It offers precision in what you draw that you cannot obtain from computer-assisted hand drawing. You can even use functions to draw your lines, meaning that you are visualizing exactly what the value is and not some approximation. And it is rendering all of this as a PDF, so your figures will scale nicely with the rest of your document.

The other major recommendation I have is to use color. Oftentimes, adding color will reduce the complexity of the figure, and it also helps make the figure pop out of the document. For equilibrium plots, this means giving each region its own color. Narrating the figure becomes easier, now that you can describe regions by color and not by location.

For estimated quantities, this means giving each value drawn its own color. Actor A’s utility could be black, while Actor B’s utility could be red. If there is only one line to be drawn, then sticking with black and white is fine. Sometimes, though, you will still want to use multiple colors for the same line. This can help distinguish between two regions of the figure as you are narrating it in the text. The utility plot above is an example of that.

The one word of caution I have here is that you are a researcher and probably not a graphic designer. When you start adding a ton of different colors, it is easy to inadvertently create clashing palette. The simple solution to this is to rely on preexisting sets of colors.

These are easy to find online, and universities conveniently give all of the information you need. A simple Google search consisting of the university name + identity (e.g., University of Pittsburgh identity) usually gives you what you need. The key piece of information is the RGB values for the brand colors. You can then define those colors in the preamble of your TeX file (e.g., \definecolor{cardinal}{RGB}{140,21,21}), and use the newly named color in the TikZ commands. I think that Stanford and University of North Carolina are particularly good for these purposes.

One thing to be mindful of is whether the colors are distinguishable in grayscale. Publishers now commonly allow for colored images appear in the online version of articles, but whatever goes to the printer needs to be black and white. Moreover, a peer reviewer may use an office printer to print your color PDF before reading it. Verifying that every color is distinct in grayscale will reduce that reviewer’s frustration.

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How I Write Formal Articles

Posted on May 25, 2020 | 1 comment

Suppose you want to write a formal theory paper. Below is the template I use to do this. I do not always follow these rules. But whenever I break them, I usually justify to myself first why it is a good idea to sidestep the norm.

The Introduction
My introductions usually have a set formula:

  1. Begin with an anecdote that motivates the main point of the paper.
  2. Generalize that main point.
  3. Pivot to how existing work does not address that main point.
  4. Describe the model setup.
  5. Give the results and basic intuition.
  6. Explain empirical content, if there is any. For quantitative work, this means describing the type of regression you are doing and one or two key substantive effects. For qualitative work, this means describing the case, how the central issues of the model were in effect, and how the outcome fits expectations.
  7. A paragraph or two of related work. Note that this may not be necessary depending on the extent of the comparison in (3) and whether there is a motivation section below.

Of these, I think (5) is the biggest problem I see as a peer reviewer. There are way, way too many papers that will say things along the lines of “increases in income decrease the probability of terrorist attacks” full stop. The intuition explaining the connection will not appear for the first time until page 18 or so. This fundamentally misses the point of doing formal theory. We are not interested in the what. We are interested in the why. Formal theory helps elucidate mechanisms. If you are not elucidating the mechanisms in the introduction, you are not writing an effective paper.

To that point, I find it helpful as a reader (and a reviewer) when this part begins with “The model produces x main results. First, …” Then each subsequent x – 1 paragraphs then explains the other results. This gives a good benchmark to the reader of what to expect in the paper and think about what a model look like that would be good for addressing those issues.

Motivation, Sometimes
I have the most variance in what comes next. Sometimes, it is straight to the model. Other times, I give a deeper explanation for why I am building the model that I am.

An underappreciated aspect of formal theory is that it just an exercise in mapping assumptions to conclusions. As the saying goes “garbage in, garbage out.” If the assumptions you put into a model make little sense, then there is no reason to pay attention to whatever the model outputs. Thus, if readers may view the assumptions your model makes as controversial, this is the time to defend them.

Sometimes, this is unnecessary. For example, if the model takes an existing approach and adds uncertainty, then you probably only need a couple of citations in (3) from the introduction to take care of it. Otherwise, I think through the main critical assumptions from the model. I then begin the second section by listing them. The following paragraphs take each assumption and motivate them. Basically, this is an exercise in going through the existing literature to demonstrate that your assumptions have merit. Key places to draw from are:

  1. existing models that use the assumption in a different context (e.g., models of war have uncertainty over resolve, but the standard models of terrorism do not)
  2. quantitative literatures that establish stylized facts that the theoretical literature has not yet developed
  3. qualitative studies that devote the entire work to motivating the same point you want to make

Of these, (3) is the most useful and the type I try to emphasize.

There are two important notes to this section. First, it is not a literature review. You are not just rehashing what the literature says about a particular subject. You are motivating assumptions. Everything you write should be geared toward that.

Second, this is a good way to come up with research ideas in the first place. As a general exercise, whenever I read through the literature, I think about what assumptions are out there and whether they appear in the more specific areas I work in. When there is a mismatch, it is worth spending some time to think about whether those alternative assumptions fundamentally alter existing ideas.

The Model
My modeling sections usually follow a basic formula:

  1. Introduce the players, moves, and payoffs in that order. For most models worth exploring, drawing a game tree is often more cumbersome than it is helpful to the reader. Bulletpoint lists are often more useful for illustrating this.
  2. Describe any conditions on parameter spaces. For example, corner solutions often complicate the math without providing any extra insight. If that is the case, describe what you are assuming, give the explicit mathematical expression (perhaps in a footnote), and explain why the reader should not care about this.
  3. Give any baseline results that are necessary to understand what is to come. For example, if you are working on an incomplete information game, explain the results of the complete information game first. Sometimes, these will be so straightforward that you can do this in a couple of paragraphs without the need to have formal propositions. Do this if you can. Other times, the baseline results are themselves of theoretical interest. In this case, use the formula below.
  4. Give a proposition. Propositions are usually if-then statements. The “if” part should be an intuitive meaning and parameter space. For example, “Suppose costs are sufficiently high (i.e., c > mk – d).” The “then” part is the strategy or outcome that is worth exploring.
  5. Explain the intuition of the proposition. Do not get bogged down in the calculations. But at the same time, do not be afraid to explain the derivation of cutpoints. Some cutpoints appear to be incredibly complicated but are in fact straightforward comparisons. This can give the reader greater insight as to where the relationships are coming from.
  6. Repeat (4) and (5) until equilibrium is exhausted.
  7. Recap using an equilibrium plot. Almost every paper benefits from one of these.
  8. Give the interesting comparative statics, either as propositions or remarks. Provide the intuition just as you would with the equilibrium. Plot the comparative static.

The plot part is the thing I see as the easiest way to improve papers. A good rule of thumb is to pretend every paper you are writing is going to be used as a job talk paper. Then think about what slides you would want to present to illustrate the key points. For example, if you had a slide that said “the probability of war increases in the cost of fighting,” you would not want to leave it as just that. You would want the next slide to show a plot with cost on the x-axis and the probability of war on the y-axis. After going through this mental exercise, every visualization of the results should go in the paper.

Empirical Evaluation
This section may or may not exist. Some models require so much space that doing any sort of empirical evaluation is not impossible given the 10,000 word limit you have to aim for to fit most outlets. Otherwise, there are two ways to go here.

Option 1 is to do some sort of qualitative examination. Hein Goemans and I have written about this in Security Studies. If you want to go down this route, you should read that.

The main trap I see when papers take qualitative approach is matching outcome to outcome. For example, the model might predict that poor people commit terrorism, and then the case study talks about how poor people commit terrorism in a certain country.

This misses the point of doing formal theory. As I described above, models map assumptions to conclusions. Case studies should do the same. In other words, I take the three or so assumptions that are key to the model’s mechanism. I then motivate why those assumptions held in the particular case. Only then is the outcome variable worth mentioning. But the key here is to establish that the incentives that the model describes was key to the actors’ reasoning. (Or at least those incentives plausibly drove it. There are many cases where finding a smoking gun would be a ridiculous expectation. If that is the case, then you should make an argument about why it is ridiculous.)

Option 2 is a quantitative examination of a comparative static. Most of this follows the basic quantitative paper template, so there is not much more to say here. The only thing worth adding is that you need a subsection that pivots the comparative static to a hypothesis that you can test. (Comparative statics are true statements. Hypotheses are things that may or may not be true of data.)

Conclusion
I think conclusions are overrated, so I have a simple formula for this:

  1. Recap the main findings.
  2. Describe takeaways for policymakers.
  3. Consider what extensions to the model might be interesting for future theoretical research.
  4. Explain how empirical scholars might wish to address the findings.

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