On March 1st, the dreaded "sequester" went into effect: draconian spending cuts triggered when Congress failed to reach a budget compromise. [1] But wait -- weren't we told that the sky would fall if these cuts took place? To many Americans, it seems like this game of chicken was a more like a case of chicken little. But make no mistake: if the sequester continues much longer, we risk throwing the economy back into recession. [2] Unlike other man made budget disasters like the debt ceiling crisis, the sequester will cut away economic prosperity month by month, and the average citizen might not feel the effects of this slow bleed until it's too late. [3] Now that we've passed the March 1st deadline and the cuts are already in effect, it's a question of which party is willing to endure the blame the longest. The game is now less like teenagers driving towards a daring collision, and more like a couple of 4th graders competing to see who can bite their tongue the hardest: whether you win or lose, you're still pathetic.
I'll model the negotiations over the sequester as an repeated prisoner's dilemma game. [4] The conclusion is that since the pain of the sequester happens so slowly and there's no clear-cut deadline, neither political party has an immediate incentive to compromise, regardless of how long the sequester lasts. The sequester is far more dangerous than even the debt ceiling crisis, because while the consequences of US default would be worse than the consequences of the sequester cuts, the sequester negotiations are structured such that the equilibrium outcome is for the sequester to stay in effect indefinitely.
Game Outline.
Here's the set up. Think back to before the initial March 1st deadline. Obama and the GOP can each chose either "compromise" or "don't compromise." For Obama, compromising means accepting some entitlement cuts and folding on his demand for new revenue. For the Republicans, compromising means accepting tax increases. If both sides compromise we get a bipartisan Grand Bargain, which is best for the country overall. If only the GOP compromises, then Obama wins. If only Obama compromises, then the GOP wins. But if neither compromises by March 1st, then the sequester cuts go into effect, which cause a slow drag on the economy that could eventually pull us into another recession.
The difference between this and other budget games like the debt ceiling is that the pain is linear, not binary: if they don't reach a deal today they can always get around to it next month. If the cuts were only in effect for one month, less damage would be done than if they were in effect for several months. So for instance, let's suppose (without loss of generality) that it would take about a month to hammer out a deal and push the bill through Congress. Then if Washington missed the March 1st deadline, they guarantee at least one month of economic pain. But they could still get a deal done before April and only suffer the first month's damage. So every month that we don't reach a deal, we play the sequester game again, causing more economic pain for which both sides will be blamed.
The game illustrated below represents a single round (which we're imagining lasts one month.) Obama's payoffs are blue and the GOP's are red. The payoffs for each player are ranked based on what I'll argue are each side's relative preferences, but the precise magnitudes are just for the sake of concreteness.
Dominant Strategies and Nash Equilibrium for a Single Round.
First off, let's find the best strategies for each player if the game only lasted one round. If Obama knew that the GOP was definitely going to compromise that round, Obama would be better off not compromising, so he could get a payoff of +2, instead of +1 if he compromised as well. But if Obama knew that the GOP was definitely not going to compromise that round, then he would still be better off not compromising ( payoff = -1/2 ) than he would be compromising ( payoff = -2.) Since the game is symmetrical, the same is true for the GOP: they'd be better off choosing "don't compromise" regardless of which strategy Obama chooses. Therefore, "don't compromise" is the dominant strategy for both parties. A player has a "dominant strategy" when this strategy would make them better off no matter what the other player does. [5]
If both players chose their dominant strategy (which is "Don't Compromise"), then the Nash Equilibrium for a single round of this game would be "No Deal." [6]
You may recognize this game as a classic prisoner's dilemma: everyone would be better off with the Grand Bargain, but both sides' dominant strategy is to not compromise. [7] So instead, we end up with one more month of the sequester. This is structurally different than the chicken game we illustrated in previous posts because in a game of chicken, if one party knew the other would never fold, they would prefer to give in than endure the sudden crash. But in the game above, we've divided the pain into manageable monthly chucks, so in each round, each side would rather endure one more month with the sequester cuts than let the other party win the whole enchilada.
Congratulations, You've Won Another Round.
Things get more complicated because the game doesn't end until somebody compromises.
Let's say t represents the number of months the sequester stays in effect. The game goes on until the mid-term elections in November 2014, when each side will have to endure the voters' animosity. So the sequester game is played from March 2013 to November 2014, which is 20 months, so the game ends after t months where t ≤ 20 .
Let's say t represents the number of months the sequester stays in effect. The game goes on until the mid-term elections in November 2014, when each side will have to endure the voters' animosity. So the sequester game is played from March 2013 to November 2014, which is 20 months, so the game ends after t months where t ≤ 20 .
Each side is penalized -1/2 "payoff point" for each month ( t ) that the sequester is in effect. So if we get a Grand Bargain after the sequester has been in effect for t months, then each side gets a payoff of 1 - (1/2) t. If Obama folds in the end and the GOP wins after the sequester has been in effect for t months, then Obama gets - 2 - (1/2) t and the GOP gets 2 - (1/2) t. Similarly, if Obama wins after t months of the sequester, then Obama gets 2 - (1/2) t and the GOP gets -2 - (1/2) t. If neither side is willing to compromise and the sequester stays in effect for the full 20 months before the mid-term elections (so t = 20), then both sides get a payoff of (-1/2) t = (-1/2)(20) = -10. You will recall that in part 1 this was the worst possible outcome in that single-round game.
Humor me with a thought experiment. If Obama knew that the GOP was going to fold after 7 months of the sequester being in effect, then he would be willing to hold out 8 months, because then his payoff would be 2 - ( 1/2 ) ( 7 ) = -3/2, which is bad, but still better than if he gave up immediately for a payoff of -2. But then why would the GOP bother holding out that long if they knew they were just going to fold in the end for a payoff of -2 - (1/2)( 7 ) = -11/2? If they were planning on eventually caving anyway, then why not compromise immediately and avoid the pain of the sequester altogether for a less bad payoff of -2? The problem, again, is that the game is symmetrical so the same logic applies to both sides. If the GOP knew that Obama also thought this way and was going to fold in the first round, then the Republicans would be better off refusing to compromise.
And yet we're left with a circular riddle, because if both sides chose their dominant strategy for each individual round, which is "Don't Compromise," then no deal would ever be agreed to, and both sides would end up way worse off than if they had just folded in the first round.
And yet we're left with a circular riddle, because if both sides chose their dominant strategy for each individual round, which is "Don't Compromise," then no deal would ever be agreed to, and both sides would end up way worse off than if they had just folded in the first round.
Is There Any Way Out?
Game theory tells us that in certain repeated prisoner's dilemma games, mutual compromise can be a stable equilibrium -- but only under certain conditions. [8] For example, Obama could say, I'll compromise initially and trust the GOP to compromise too, but if they break that trust in the first round, then I won't ever compromise again. The idea is then that the GOP would see that it is in it's long term interest to perpetually compromise with Obama rather than perpetually failing to reach a deal and letting the sequester go on indefinitely.
But this strategy of perpetual compromise only works if neither side knows when the game is going to end, because otherwise each side would renege on their commitments since they know that after the last round their won't be any consequences for breaking their tacit agreement to work together. [9] In the game illustrated above, the game is known to end as soon as either one of them compromises. For example, if Obama were to compromise in the first round, then the game would be over. Therefore, if the GOP backstabbed him, there would be no second round in which he could retaliate. Knowing this, Obama wouldn't compromise in the first place, and neither would the Republicans.
So the answer is no, there is no way out -- at least not the way we've modeled the game here. Each party will play their dominant strategy each round of this prisoner's dilemma, which is to not compromise, and the sequester cuts will continue for the full 20 months between the beginning of the sequester and the 2014 mid-terms, yielding the worst possible payoff for each side.
Conclusion.
The sequester cuts were supposed to be so painful neither side would go through with them, forcing compromise. But without a clear deadline, the sequester game does the exact opposite. Our only hope is that these automatic cuts will be dealt with as part of negotiations over the debt ceiling, which does have a clear deadline, and will cause immediate devastation upfront if no deal is reached.
To be frank, you don't need to take a math class to see that Washington isn't compromising anytime soon. It isn't particularly bold of me to "predict" that partisanship will prevail. The horrifying and plainly observable truth is that our democracy has become structurally incapable of solving problems through peaceful negotiation between opposing parties. Our country is divided now more than any time since the Civil War. This division has produced and is reproduced by a constant war between two partisan factions -- a real, not metaphorical, economic war where each side threatens to destroy the country if their demands aren't met. We shouldn't be cynical -- we should be terrified.
What we need is an outside intervention to fundamentally restructure the negotiations. Our only hope for escaping this death trap is if the American people stand up and demand that our leaders start playing a different game altogether.
Why is our country so divided? In my next post, I'll look at the structural reason America's heart is split in two.
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