My mom emailed me yesterday to ask if game theory can help to explain our debt-ceiling crisis. The answer, of course, is yes. I’ve spent a lot of time thinking about financial crises and adjustment policy, and it’s obvious to me that the debt-ceiling stalemate can be understood using canonical works in applied game theory. The most obviously relevant model of stabilizations is Alesina and Drazen 1991, which illustrates why stabilizations are often delayed by examining a dynamic bargaining game where two sides disagree about who should pay the costs of it. To be clear, their model is about a fiscal crisis rather than a debt-ceiling stalemate (which need not be a fiscal crisis unless politicians make it one) but the logic is identical.

(As an aside, this speaks to a broader point I’ve long made to anyone who listens–a small group, admittedly–which is that despite cries from the peanut gallery about how the US financial crisis shows how bankrupt academic economics is, there is absolutely nothing about the US financial crisis [its origins, its effects, how it has played out] that is surprising or confusing given what we know about financial crises. The only way that you can find this surprising is if 1, you don’t know anything about economics, or, equivalently, 2, you don’t think that anything that has happened in any other country’s economic history is relevant for the US.)

Here is what game theory tells us about the debt-ceiling stalemate. Caveats first:

1. This is a loose, and very informal, overview.
2. I really do know a lot about stabilizations. But I know next to nothing about advanced industrial economies with consolidated democratic governments.
3. I know even less about US politics, especially since I’ve been away for a month.
4. Models are by definition wrong. The fact that you can think of an exception proves nothing. Models are like maps: representations of the real world based on assumptions that are known to be incorrect, not reproductions of the real world. They are judged not as right or wrong, they are judged as useful or not. If you can think of a useful extension or simplification that yields an important insight, that’s where you can help.

The players are the voters in each congressperson’s district, the congresspeople, and the President. There is a fraction f of voters in each congressperson’s district who will vote against their representative/senator out of office if s/he votes to raise taxes at all. Independently of that, the probability of winning an election for the congresspeople and the President alike is an increasing function of the state of the economy. Good economy = more likely to be reelected. Congresspeople and the president care only about reelection.

If f = 0 for all districts, then there would be no crisis, because congresspeople would have no incentive not to raise taxes. If f = 1 for all districts, then there would be no crisis, because congresspeople would all have the same incentive not to raise taxes. In reality, f is probably somewhere between .1 and .3 for all likely voters in all districts. Congresspeople with Ds next to their name are more likely on average to draw their support from the fraction of their constituents who do not care exclusively about taxes (that is, 1-f). Congresspeople with Rs next to their name are more likely on average to draw their support from f. f can be located on a unidimensional policy space with unit mass:

Policy Space

0—————-f—————-.5———————————-1

right                   median voter                           left

R voters                                                       D voters

If the debt-ceiling is raised through some mix of increased taxes and spending cuts (the option of just increasing taxes and no cuts appears to be off the table), all congresspeople get the benefit of a better economy, but f vote against the incumbent, which doesn’t matter for D but does for R. If the debt ceiling is raised exclusively through spending cuts, R gets f. D probably pays some cost in this case but I am not aware of any explicitly organized political movement that has made all Ds swear not to cut spending at all (you could add such a group in and it would only reinforce my conclusions). If basic economic logic is right, spending cuts (which would have to be astoundingly large according to the proposals on the table) would lead to worse economic performance over the next electoral cycle (which is the only relevant time period for congresspeople). If the debt-ceiling is not raised, everyone pays the cost of a terrible economy through a default.

Due to the divided government, only proposals with both D and R votes can pass. So the possibilities are

		R Vote
		Cuts	Cuts and Taxes
D Vote	Cuts	Pass	Fail
	Cuts and Taxes	Fail	Pass

So, say that P(reelection | R, cutsonly, passed)–read this as “the probability of being reelected if you are a Republication who votes for only spending cuts and this is passed”–is equal to the value of being consistent (f) plus the benefits of an all-cut economy (e(cuts)).

• P(reelection | R, cutsonly, passed) = f + e(cuts)
• P(reelection | R, cutsonly, fail) = f + e(default)
• P(reelection | R, cuttax, passed) = e(cuttax)
• P(reelection | R, cuttax, fail) = e(default)

Likewise,

• P(reelection | D, cutsonly, passed) = e(cuts)
• P(reelection | D, cutsonly, fail) = e(default)
• P(reelection | D, cuttax, passed) = e(cuttax)
• P(reelection | D, cuttax, fail) = e(default)

Remember, that the congressperson cares only about reelection. The utility of each congressperson is determined solely by the probability of reelection: U(c) = P(reelection|party, vote, outcome). So, the decisions come down to

		R Vote
		Cuts	Cuts and Taxes
D Vote	Cuts	e(cuts) , f + e(cuts)	e(default) , e(default)
	Cuts and Taxes	e(default) , f + e(default)	e(cuttax) , e(cuttax)

It is straightforward to see that so long as f + e(default) > e(cuttax)–meaning that the cost of a default in terms of vote share is worse than the loss of votes for caving to the Ds–any R will vote only for cuts. D’s vote depends on whether e(cuts) <?> e(default). (We know for certain that both e(cuttax) and e(cut) > e(default)).

Much of the debate in Washington right now is about trying to convince congresspeople what those values are. This would be hard enough. But the real problem is that this is a repeated interaction–they play this game every day until August 2. That gives each D and R an additional incentive to bluff by trying to convince the other that these values are different than they actually are. Even if an R believes that caving won’t leave them unelectable (f is very small, and e(default) is much worse than e(cuttax)) they have an incentive to hide that fact (so long as f > 0) to see if they can get the Ds to cave in. Even if a D believes that e(default) is much worse than e(cuts), they will hide this too. The result is a war of attrition, which is what we see.

There’s one more piece to add, which makes things much more complex so I’ve ignored it until now. The probability of reelection given any particular state of the economy is not the same for both R and D. The incumbents (in this case, the Ds) profit more from a good economy than the Rs do, and pay more for weaknesses, but only so long as there’s no default. Most analysts I’ve seen think that the Rs would pay more in voter terms for a default than the Ds would. Rs knows this, so this raises their incentive to allow the economy to fester, but they cannot have a default. So e(default)_R < e(default)_D, e(cuttax)_R < e(cuttax)_D, and e(cut)_R > e(cut)_D). The eventual outcome depends on all of these values, plus f. If my figuring is right, this will lead the Rs to push this till the very end in an attempt to bluff, but then to eventually concede to D.