Dr. Strangelove, the classic black comedy produced at the height of the Cold War, mercilessly satirizes the doctrine of Mutually Assured Destruction, which dominated foreign policy in that era.[1] Peter Sellers plays Dr. Strangelove, a former Nazi nuclear scientist called upon to advise the President of the United States after a crazy general loses his mind and sets off a nuclear war. Because of the retaliatory policy mechanisms put in place under the MAD doctrine, the US and Soviet leadership cannot figure out how to stop a nuclear exchange from destroying human civilization. But, you know, in a funny way.
Dr. Strangelove: Mr. President, it is not only possible, it is essential. That is the whole idea of this machine, you know. Deterrence is the art of producing in the mind of the enemy the fear to attack. And so, because of the automated and irrevocable decision-making process which rules out human meddling, the Doomsday machine is terrifying and simple to understand. [smiles gleefully]
...and completely credible and convincing! [2]
Here's where our present-day budget stalemate comes in: it seems like most Americans wish Washington would quit "raising the stakes," and figure something could get done if both parties would stop threatening national ruin. But what if the exact opposite is true? What if Dr. Strangelove is right, and our only way out is to raise the stakes even further? In this post, we'll find the mixed strategy equilibrium of the game of chicken introduced in part 1. Then we'll show why the only way to get to a Grand Bargain is to make the looming crisis so catastrophic politicians are forced to compromise.
President: How is it possible for this thing to be triggered automatically and at the same time impossible to untrigger?
Dr. Strangelove: Mr. President, it is not only possible, it is essential. That is the whole idea of this machine, you know. Deterrence is the art of producing in the mind of the enemy the fear to attack. And so, because of the automated and irrevocable decision-making process which rules out human meddling, the Doomsday machine is terrifying and simple to understand. [smiles gleefully]
...and completely credible and convincing! [2]
Here's where our present-day budget stalemate comes in: it seems like most Americans wish Washington would quit "raising the stakes," and figure something could get done if both parties would stop threatening national ruin. But what if the exact opposite is true? What if Dr. Strangelove is right, and our only way out is to raise the stakes even further? In this post, we'll find the mixed strategy equilibrium of the game of chicken introduced in part 1. Then we'll show why the only way to get to a Grand Bargain is to make the looming crisis so catastrophic politicians are forced to compromise.
Outline of the Game.
We'll use the same game outline explained in part 1 :
In part 1 we saw that there are two pure strategy Nash equilibia: GOP Wins and Obama Wins. The reason both sides can't "just come together," is because the Grand Bargain scenario is not a stable equilibrium. As soon as one side compromises, the other has an incentive not to. No Deal is also not a stable equilibrium, because both players have an incentive to compromise rather than face the consequences of a budget crisis.
Mixed Strategies.
Both sides can still create a credible threat by increasing the *risk* that the partisan forces they've worked into a frenzy will get out of their control, and they won't be able to put the genie back in the bottle before it's too late. For example, in the summer of 2011 Fox News, the media arm of the Republican party, convinced conservatives that the "theory" that we'd default if unless we raised the debt limit was just another liberal conspiracy like evolution, climate change, and babies being born homosexual. This successful farce created the risk that even though the GOP leadership wanted to avoid a government default, at the last minute when there's no room for error, Tea Party Representatives could be forced by their right-wing base to to blow up the deal agreed to by Obama and Boehner.
When Obama and Boehner employ these sort of probabilistic threats, they are using what game theorists call "mixed strategies." A mixed strategy is when a player decides which option to pursue based on a specified probability the player predetermines before the game begins. [3] We'll say the House GOP chooses pr, the probability that they will compromise. Obama chooses po, the probability that the Administration will compromise. This implies a probability of ( 1 - pr ) that the House GOP will not compromise, and a probability of ( 1 - po ) that the Administration will not compromise.
Best Responses.
The interesting thing about mixed strategies is that a player's best strategy is not at all based on their own payoffs, but rather their opponent's. Each player seeks to "keep their opponent guessing" by choosing p so that their opponent is indifferent between their expected payoffs for each available pure strategy. [4] So in our budget chicken game, this means the GOP creates the perception that they have a probability pr of compromising, such that Obama can't increase his expected payoff no matter which strategy he picks. Now let's find these values for pr and po .
To find Obama's "expected payoff" from compromising, we take his payoff if he compromises and so does the GOP (+1), times the probability that the GOP will compromise ( pr ), plus his payoff if he compromises but the GOP doesn't (-2), times the probability that the GOP will not compromise ( 1 - pr ). [5] Then we find Obama's "expected payoff" from not compromising the same way. Finally, we want to know what value of pr the GOP could choose so that Obama couldn't expect to do any better by selecting one strategy over the other, so we set these two expected payoffs equal to each other and solve for pr. So:
Obama's Expected Payoff (Compromise) = Obama's Expected Payoff (Don't Compromise)
This translates to:
1pr - 2 ( 1 - pr ) = 2pr - 10 ( 1 - pr )
Now we simplify this expression and solve for pr :
pr - 2 + 2pr = 2pr - 10 + 10pr
3pr - 2 = 12pr - 10
9pr = 8
pr = 8/9 ~ 88%
So if GOP leaders create the perception that there's an 88% chance that they'll compromise in the end (and thus a 12% that they won't compromise), then Obama won't be able to improve his chances regardless of which strategy he picks.
Now let's calculate Obama's equilibrium choice for po the same way:
GOP's Expected Payoff (Compromise) = GOP's Expected Payoff (Don't Compromise)
1po - 2 ( 1 - po ) = 2po - 10 ( 1 - po )
po - 2 + 2po = 2po - 10 + 10po
3po - 2 = 12po - 10
9po = 8
po = 8/9 ( ~ 88% )
Since the payoffs in this game are symmetrical, the equilibrium values of po and pr are equal, as we would expect.
Expected Payoffs in the Mixed Strategy Equilibrium. [6]
If Obama believes there's a 8/9 chance the GOP will fold, and thus a 1/9 chance they won't, then Obama's expected payoff for choosing to compromise is:
1pr - 2 ( 1 - pr )
= 1 ( 8/9 ) - 2 ( 1/9 )
= 8/9 - 2/9
= 6/9
= 2/3
If they GOP played this mixed strategy but Obama chose not to compromise, then his expected payoff would be:
2pr - 10 ( 1 - pr )
= 2 ( 8/9 ) - 10 ( 1/9 )
= 16/9 - 10/9
= 6/9
= 2/3 ( ~ 0.67 )
So if the GOP plays its equilibrium mixed strategy, then Obama's payoff will be 2/3 no matter what he does. You can verify that the same is true for the GOP if Obama plays the equilibrium mixed strategy, since again, the game is symmetrical.
Raising the Stakes.
Now let's see what happens if we make the consequences of doing nothing even greater. For example, the Administration has been doing everything it can to ensure the pain of the sequester is felt acutely starting now, rather than spreading the cuts over a longer period of time, because they want the up-front damage to create pressure on the GOP. So change the payoffs for not getting a deal done from -10 to be even worse; let's say -100. Everything else is the same:
Now let's find the mixed strategy equilibrium to this new, more dangerous game by the same method as before, but substituting -100 for -10.
GOP leaders choose pr such that:
1pr - 2 ( 1 - pr ) = 2pr - 100 ( 1 - pr )
3pr - 2 = 102pr - 100
99pr = 98
pr = 98/99 ( ~ 99% )
And we already know that Obama would do the same, since the game is symmetrical. So in the first game, each side compromised 67% of the time, but now that the payoffs for "No Deal" are worse, they each compromise 97% of the time. That means in the earlier game, the probability of reaching a grand bargain was ( 88% x 88% ) = 77%, but now that the potential for catastrophe is larger, the probability of the grand bargain is ( 99% x 99% ) = 98%. Politicians are more likely to compromise the more afraid they are of the consequences of failing to reach an agreement.
For this very reason, game theory suggests that increasing the potential damage of the sequester makes both sides better off. Using the same method as before, we find their expected payoff (which in equilibrium is the same for each side, and equal for "compromise" and "don't compromise"):
= 1 ( 98/99 ) - 2 ( 1/99 )
= 96/99 ( ~ 0.97 )
So if the sequester is less damaging and only would only cost them -10, then in equilibrium, each side expects a payoff of 0.67. But if the sequester is more damaging costing each side -100, then their expected payoff is 0.97.
Our counter-intuitive result is that the country is better off the more we increase the potential for economic disaster, because that's the only way they'll be scared into compromising with each other. [7] The most dangerous thing about these sequester cuts is that they are not catastrophic enough to prompt action to avoid them. Our only hope is that the sequester is resolved as part of the debt ceiling debates. Failure to raise the debt ceiling would be such an incredible display of suicidal lunacy that we would surely avoid it.
The real solution would be to figure out how we as a country could stop playing this terrifying game of chicken. But given the way the politician's incentives currently aligned, game theory would tell us that our best chance at compromise is to hope for the worst.
The real solution would be to figure out how we as a country could stop playing this terrifying game of chicken. But given the way the politician's incentives currently aligned, game theory would tell us that our best chance at compromise is to hope for the worst.
Now that the sequester cuts have gone into effect, this specific budget battle has transitioned from a game of chicken to an iterated prisoner's dilemma. Part 3 of this series, we'll see which party is willing to endure the dreadful economic impacts the longest. Who's gonna cave first? Read part 3 >>
