The Debt Ceiling Game Is Parent-Child With A Bad Nash Equilibrium

The debt ceiling game will only avoid default if it implies economic pain.

Dhaval Joshi,

BCA Research

The debt ceiling standoff can be compared to the parent-child game.

The 'Child" is the Democrats, who can either cut spending very much (be excellent), cut it a bit (be good), or cut it not at all (be terrible). The 'Parents' are the Republicans, who can either refuse to raise the debt limit (punish) or lift it (not punish).

The Republicans' best strategy is to not raise the debt ceiling until the Democrats drastically cut their spending or the economy begins to crash.

As the economy begins to decline, Democrats will be less likely to have to make the necessary spending cuts. However, a recession will most likely hurt President Biden's chances of reelection.

Only if the debt ceiling game ends with economic pain will it avoid default.

The 2001 movie that won four Oscars

A Beautiful Mind

Russell Crowe played John Nash the mathematician, who introduced the concept of 'Nash Equilibrium' and revolutionized the game theory. This is a fantastic movie that you should watch if you haven’t done so already.

Nash made a major breakthrough by defining the steady state in a game by identifying if he could unilaterally change his choices. The Nash equilibrium is this. The Nash equilibrium is reached when all players are aware of each other's rewards (Figure 1).

Our US Political Strategists warned that the Democrats and Republicans were playing a game which posed a greater-than-normal risk of a national debt default.

BCA Research - Shades Of Gridlock: Risk Of US Debt Default

The Democrats must agree to spending cuts in order for the Republicans to raise the debt ceiling. This will prevent the US government defaulting on their debt. The question is: What's the game? What are the payouts? What is Nash equilibrium?

The debt ceiling game is the parent-child game with a bad Nash equilibrium

Parents and game theorists will both recognize that the debt ceiling standoff is a Parent-Child Game, where "the child" has the option to be good or evil, while "the parent" may punish or not. The child in this situation is the Dems, who can either cut spending very much (be good), a little bit (be good), or do not cut any spending at all (be bad). The Reps are the parent who can either refuse to raise the debt limit (punish) or raise it (not punish).

What are the rewards for the Dems and Reps? The payoffs in this case are derived from the ultimate prize of winning 2024's Presidential Election. Each of the six combinations of choices can produce a swing by independent voters. A Reps win is a Dems lose, so each payoff is (X,-X), with the first number being the likely swing for the Reps and the second the mirror image swing from the Dems.

Payoffs consist of two components. The economic impact is the first factor. A US debt default is almost certain to cause a recession. And a recession that occurs within a year of a Presidential Election will usually result in a large swing towards the party not in power, which would be the Reps. This component's payoff would be (5,-5) (Figure 2)

Recently, President Biden acknowledged that this was a gain for the Republicans and a loss for the Democrats.

Biden will take the blame because I'm president and the president is responsible for everything.

The second component is measured by independent voters who are repelled from the party which is non-compromiser and therefore blamed for the recession.

The Reps may lose some votes if they are the non-compromise party. Let's say that the Dems are "quite good", meaning that the cut their spending, but still punish, the Reps. The exact number is not relevant, but the fact that the Reps' loss is smaller than their gain is.

What is the Nash equilibrium for this setup? It is not a good thing.

The same goes for the bad

The Dems have a 'good' strategy, but the Reps' best option is to punish. This is a Nash balance because neither party can benefit from unilaterally changing their choice. As Biden alluded to, for the Reps the gain of a recession (5. -5) is greater than their loss by not compromising (-2. 2.) This gives them a reward of (3. -3) compared with (0. 0).

How to shift into a good Nash equilibrium

Let's now say that the Reps lose more by being non-compromisers than they gain from recession. It could happen one of two different ways. Either the Dems have been'very good,' meaning that they have cut spending significantly, making it difficult to justify "punishment". If the economy begins to crash, then being the non-compromise becomes extremely costly. Let's say that the Reps get paid less for not compromising.

In either case, Nash equilibrium shifts into a good balance in which the Dems reduce spending a great deal in an economy that is not tanking, or reduce spending a little when the economy is in trouble. The Reps' best strategy is to 'don't penalize', which means raising the debt ceiling.

Some Conclusions

I do not intend to take sides in this debt ceiling debate. It is not my intention to take sides in the debt ceiling standoff.

The Reps' best strategy would be to not raise the debt ceiling until the economy begins to crash, as this is the time when being a non-compromise becomes costly (Figures 7 and 8).

It will be difficult for the Dems to cut spending significantly. As the economy begins to decline, the required spending reductions will decrease. However, a recession will likely damage Biden's chances of reelection.

What if, for example, the Dems cut spending in an economy that is not tanking and the Reps raise the debt ceiling? This would mean one of two things:

It is possible that neither the Dems nor the Reps understand the game. Biden's comments suggest they do understand it, if only implicitly.

A second possibility is that Reps are altruistic and put the country ahead of their own political ambitions. Call me cynical but I don't think any political party will give its opponent a pass unless there is a crisis in the country, which is not the case. The Reps will wait until the economy or the stock market starts to crash.

Thirdly, the Reps may be convinced that a downturn is inevitable and so they don't want to risk appearing as the bad guys. Appearing to be on the side of the good guys might be a positive for the Reps but bad for the economy.

The Nash equilibrium of the debt ceiling game is, in summary, a

The same goes for the bad

Equilibrium in which the Dems are "quite good" but the Reps' best strategy is to "punish" until the economy starts to tank. It is only when economic pain is felt that the debt ceiling game will lead to a default. It reinforces our defensive positioning with regard to asset allocation, currency allocation, and sector allocation.