Category Archives: Game Theory

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.