Debt Ceiling Deadlock 2013: The Senators Who Really Hold the Key

We are approaching the third week of the U.S. government shutdown. In just three days, the federal government would default on its debt. In the event of a default, it is inevitable that its rippling effect would reach every nook and corner of today's connected world. As a result, financial markets everywhere, not just in the U.S., would feel this bone-chilling effect.

Negotiations within Congress as well as between Congress and Senate (and the White House) have already broken down. At this point, the fate of this crisis clearly lies in Senate. So, the question is: Which senators hold the key to breaking this deadlock? In other words, what group of senators would be able to influence at least 60 senators to vote "yes" on a deal brokered by them?

This is game theory on a grand, public display. The "players" here are the members of the two major parties in Parliament. Their actions are purely strategic. One anticipated outcome of this "game" is that a bipartisan group of senators would broker a deal in Senate. It would then be sent to Congress at the last minute to avoid the much dreaded default. The Republican-controlled Congress would have two options---either pass that deal or bear the blame for the ensuing default. Considering several recent polls that show that the public is mostly blaming the Republican Party for the government shutdown, it is likely that Congress would pass a Senate-brokered deal.

Going back to my previous question, what group of senators would be influential to brokers such a deal? One part of my PhD Dissertation actually deals with questions like these in strategic/game-theoretic settings. The general approach there was to first design a mathematical model of influence among the "players of the game" (that is, the senators in this case), then learn the parameters of the model (such as how much Senator X influences Senator Y) from the historical voting data, and finally, compute the stable outcomes (in mathematical term, pure-strategy Nash equilibria) of the game to answer such questions. These stable outcomes can be thought as potential voting outcomes in Senate.

Here, I will use that approach to evaluate the influence of various possible groups of senators on their peers. For example, as of today, intense negotiations have been going on between the Senate Majority Leader Harry Reid (D, NV) and the Senate Minority Leader Mitch McConnell (R, KY). Previously, a bipartisan plan, led by Senators Susan Collins (R, ME) and Joe Manchin (D, WV), was proposed without much success. One interesting question is: What would happen if Senators Reid and McConnell now join that bipartisan plan? Will it pass with at least 60 "yes" votes? Another interesting question is: Is there a small number of senators who could help Senators Reid and McConnell pass their deal on the Senate floor (by joining their coalition)?

According to our model, if all the democratic senators, including Senator Reid (D, NV) are predisposed to voting "yes" and if in addition, Senator McConnell (R, KY) also votes "yes," then the bill will pass for sure with at least 60 votes (i.e., there is no chance of a filibuster). However, if we don't assume that all the democratic senators are on board, then it requires a "large" body of eight influencing senators to pass the bill without any filibusters. One such influencing body of senators, according to our model is, Reid (D, NV), McConnell (R, KY), Inouye (D, HI), Johnson (R, WI), Sanders (I, VT), Conrad (D, ND), Portman (R, OH), and Collins (R, ME). This is a bipartisan group of four republican senators and four democratic (or democratic-minded independent) senators.

As statistician George Box's famously said, "All models are wrong, but some are useful," there is no way validating a model of influence like ours. However, we definitely see a lot of anecdotal evidence that our model is capturing various strategic aspects of the deals in Senate. First, our model predicts that we would need a bipartisan group of influencers that consists of roughly the same number of democratic and republican senators for making a deal in Senate. Second, in Senates prior to the 112th Senate, our model predicts that certain groups of only six senators would have been sufficient to reach a complete consensus (i.e., if those six senators voted "yes" then everybody else would also vote "yes"). An influencing group of only six senators is no longer sufficient for the 112th Senate. It requires a body of eight influencing senators just to gather 60 "yes" votes to avoid filibusters (as opposed to a complete consensus of 100 "yes" votes). Furthermore, our model also predicts that reaching a complete consensus in the 112th Senate would require an influence body of 10 senators (as opposed to six senators in prior Senates)! This certainly testifies to the general perception of polarization in Congress these days.

Fiscal Cliff: Using Game Theory to Predict the Outcome

The dreaded "fiscal cliff" is only hours away. Suddenly, Jay Leno's explanation of its danger may not seem to be humorous to many: “It's 4am for our economy, and Lindsay Lohan is behind the wheel.” We know that the President, the senators, and the congressmen are all working round to clock to strike a deal. News outlets are bringing in updates every few minutes. Apparently, not only the traders in Wall Street, but also the whole of Europe and Asia are anxiously waiting to see what the outcome would be. What will happen? Will the country be able to avoid the cliff? I will leave a direct yes/no answer to this question to our local psychic on Middle Country Road. I will rather give you an indirect answer that is based on a game-theoretic prediction.

My PhD advisor Luis E. Ortiz and I developed a game-theoretic model of influence among networked individuals in order to predict outcomes of “strategic” interactions among these individuals. An example of a strategic interaction in the context of the U.S. Congress is how the senators come to the bargaining table in order to reach a deal. We know that some of them are very "influential" in such dealings. Our goal was to design a mathematical model of influence among individuals that can be used to predict the outcomes of various strategic situations.

In our model, individuals (for example, senators) have a binary choice of action +1 and -1 (for example, +1 may denote voting "yes" and -1 "no" on a legislation issue). An individual influences another, which is quantified by what we call an influence factor. An influence factor might be a positive number (for example, from a Democratic to another Democratic senator), a negative number (for example,  from a Democratic to a Republican senator), or zero (no influence). Each individual also has a threshold value, which signifies how much one can tolerate others' "total influence" on him before adopting (or "playing") the action +1. So, how is this total influence obtained? Consider the following example.

Here, Senator DeMint is shown to be influenced by four other senators, some influence factors being positive, some negative. Senator DeMint's threshold is +1 (shown by the red line on the barometer). Now, suppose that Senators Paul, Johnson, Sanders, and Schumer are playing the following actions: -1, +1, -1, +1, respectively (where +1 denotes voting "yes" and -1 "no"). To obtain the total influence on Senator DeMint, we first calculate two sums. First, we sum up the influence factors from those senators who are playing +1 and then we sum up the influence factors from the rest. Finally, the total influence is obtained by subtracting the second from the first sum, which is +4 in this case (shown in green on the barometer). The "best response" of Senator DeMint to these four senators' actions would be to play +1, because the total influence (+4) has exceeded his threshold (+1). If the total influence were below his threshold then his best response would have been to play -1. 

In this example, we are shown only a small part of the whole picture (for example, we are not shown which senators influence Senator Schumer). When we consider the whole picture, we can use one of the most central solution concepts of game theory in our context. This solution concept is called Nash equilibrium, due to John Nash, a beautiful mind. A Nash equilibrium (a "pure-strategy" Nash equilibrium to be technically correct) is defined by a joint-action (that is, one action for each individual) where each individual plays his best response to what the other individuals play. A Nash equilibrium underscores stability, because no individual gains anything by unilaterally deviating to another action (that is, everybody is playing his best-response, simultaneously). Therefore, when we say "stable outcomes," we mean Nash equilibria of the "game." In the U.S. Congress case, an example of a stable outcome is roll-call vote on a legislation issue. Therefore, Nash equilibria of our model/game can be thought of instances of roll-call votes.

Question: Where do we get these numbers (threshold, influence factors)? These numbers were learned from the senators' previous voting records using machine learning techniques. The goal is to have the Nash equilibria of the game capture as much of the previous records as possible. In addition, we also want the Nash equilibria to predict future happenings beyond just the past records, which is technically known as "generalization."

Question: How do we use this model for prediction? Consider, for an example, the debt-ceiling bargaining of 2011. A bipartisan group of six senators, known as the gang-of-six, was formed in earlier to avoid a crisis by reaching a deal. In our model, we translate a question like "how influential was this group?" to a question like this: "What are the Nash equilibria that we have when the senators in this group votes 'yes,' and in how many of these Nash equilibria do we have at least 60 'yes' votes (that is, the bill would pass)?"

Game theory has long been celebrated as a mathematical theory of conflicts that reliably encodes strategic aspects of interactions. In addition, our model of influence is rooted in the widely used threshold models by Mark Granovetter in the context of mathematical sociology. Using our model, we can ask question like these: What would happen if the gang-of-eight senators, together with the Senate Majority Leader Harry Reid and the Senate Minority Leader Mitch McConnell, are able to reach a deal? Would they be able to influence at least 50 other senators to join them (and thereby, pass the bill that they advocate)? I will first give you a little more background on our model (which may be skipped) and then give the answers to these questions in the context of the fiscal cliff situation.

A Few More Words on Our Model (may be skipped)

Many complex real-world networks exhibit strategic aspects of behavior. An example is the network of influence among the U.S. senators, where collective voting outcomes may be treated as end results of complex interactions among the senators. The question of “how” such a collective outcome emerges in a “crowd” or a “gathering” has a long and rich history in sociology (see, for example, David Miller’s book on Collective Behavior and Collective Action). While explaining how a collective action takes place is a scientific pursuit of utmost importance, our focus is rather on an engineering approach to “predicting” stable outcomes in a networked population setting. Our approach is not to go through the fine-grained details of a process, but to adopt the solution concept of pure-strategy Nash equilibria (PSNE) to characterize stable outcomes.

There are several reasons why we deliberately focus on the outcome and not on the process. First, the process of forward recursion (see Mark Granovetter’s “Threshold Models of Collective Behavior”), very often used by economists and computer scientists in studying dynamics of interaction, implicitly assumes non-negative influence. However, in real-world scenarios such as the U.S. Congress, negative influences may be perceived to be common. Second, seldom do we have enough information to model the process of a complex interaction. For example, the Budget Control Act of 2011 was passed by 74-26 votes in the Senate on August 2, 2011, ending a much debated debt-ceiling crisis. Despite intense media coverage, it would be difficult, if not impossible, to give an accurate account of how this agreement on debt-ceiling was reached. Even if there were an exact account of every conversation and every negotiation that had taken place, it would be extremely challenging to translate such a subjective account into a mathematically defined process, let alone learning the parameters and computing stable outcomes of such a complex model.

Our framework of influence can be briefly outlined as follows. We first model a networked scenario as a parametric graphical game, which we call an “influence game.” In practice, the structure and the parameters of influence games are learned from behavioral data. We then perform inferences, such as interventions, by computing or counting PSNE of influence games. In an earlier paper (AAAI-11: “A Game-Theoretic Approach to Influence in Networks”), we laid out the theoretical foundation of influence games, including the complexity of and algorithms for computing PSNE of influence games. We proposed a new approach to identifying the most influential nodes in a network and demonstrated this new approach by computing the most influential senators in the 110th U.S. Congress.

Before performing the following analysis, we learned the parameters of an influence game among the U.S. senators of the 112th Congress (data from May 9, 2011 to August 23, 2012) using a machine learning technique developed by my advisor along with my academic brother Jean Honorio.

The Gang-of-Six Senators

First, let us revisit the debt-ceiling crisis of 2011. During that time a bipartisan group of six senators, called the gang-of-six senators, was formed to address the debt-ceiling problem. It consisted of Senators Chambliss (Rep, GA), Coburn (Rep, OK), Crapo (Rep, ID), Conrad (Dem, ND), Durbin (Dem, IL), Warner (Dem, VA). This group received much media spotlight while trying to reach a deal on debt-ceiling between the Democrats and the Republicans. Despite their best efforts, they were unable to avoid the so-called debt-ceiling crisis toward the end of July 2011. 

Using our model to infer the set of stable outcomes consistent with the gang-of-six senators voting “yes,” we find that there are 11,106 stable outcomes in this set. Among these, 27% stable outcomes consist of less than 50 “yes” votes in each outcome and 90% stable outcomes less than 60 “yes” votes. So, according to our model, the gang-of-six senators are not collectively “powerful enough” to lead to a majority outcome or to prevent a filibuster scenario.

The Gang-of-Eight Senators

The gang-of-eight senators was formed recently to avoid the fiscal cliff situation. In addition to the gang-of-six senators, this group has two new members: Senators Michael Bennet (Dem, CO) and Mike Johanns (Rep, NE). 


Our analysis shows that there are 4,279 stable outcomes where the gang-of-eight senators reach a consensus. Among these, only 16% outcomes have less than 50 “yes” votes. So, this group is apparently more influential than the gang-of-six in gathering up a majority. More importantly, if the Senate Majority Leader and the Senate Minority Leader (Senators Harry Reid and Mitch McConnell, respectively) were able to reach a compromise and join the gang-of-eight senators in a deal, then there are 68 stable outcomes, among which only 3% have less than 50 “yes” votes. There is, however, a possibility of a filibuster, since this group of 10 senators lead to less than 60 “yes” votes in 67% of the outcomes.

Final WordsAccording to our model of influence, if the Senate Majority Leader and the Senate Minority Leader, along with the gang-of-eight senators, can reach a consensus, then in 97% of the stable outcomes, the majority of the senators will vote “yes” on a fiscal cliff bill. Below is a table that summarizes how likely each senator is to join the consensus of these 10 senators (that is, the percentage of stable outcomes where a senator votes “yes” when influenced by these 10 senators voting “yes”).

Senator	% of stable outcomes where senator votes "yes"	Comment
Akaka D HI	72.06%
Alexander R TN	38.24%
Ayotte R NH	27.94%
Barrasso R WY	73.53%
Baucus D MT	75.00%	YES >= 75%
Begich D AK	63.24%
Bennet D CO	100.00%	YES >= 75%
Bingaman D NM	80.88%	YES >= 75%
Blumenthal D CT	75.00%	YES >= 75%
Blunt R MO	88.24%	YES >= 75%
Boozman R AR	100.00%	YES >= 75%
Boxer D CA	67.65%
Brown D OH	70.59%
Brown R MA	29.41%
Burr R NC	100.00%	YES >= 75%
Cantwell D WA	33.82%
Cardin D MD	72.06%
Carper D DE	55.88%
Casey D PA	39.71%
Chambliss R GA	100.00%	YES >= 75%
Coats R IN	67.65%
Coburn R OK	100.00%	YES >= 75%
Cochran R MS	92.65%	YES >= 75%
Collins R ME	39.71%
Conrad D ND	100.00%	YES >= 75%
Coons D DE	55.88%
Corker R TN	26.47%
Cornyn R TX	97.06%	YES >= 75%
Crapo R ID	100.00%	YES >= 75%
DeMint R SC	11.76%	NO >= 75%
Durbin D IL	100.00%	YES >= 75%
Enzi R WY	73.53%
Feinstein D CA	67.65%
Franken D MN	88.24%	YES >= 75%
Gillibrand D NY	51.47%
Graham R SC	25.00%	NO >= 75%
Grassley R IA	98.53%	YES >= 75%
Hagan D NC	79.41%	YES >= 75%
Harkin D IA	58.82%
Hatch R UT	27.94%
Heller R NV	25.00%	NO >= 75%
Hoeven R ND	94.12%	YES >= 75%
Hutchison R TX	98.53%	YES >= 75%
Inhofe R OK	88.24%	YES >= 75%
Inouye D HI	27.94%
Isakson R GA	100.00%	YES >= 75%
Johanns R NE	100.00%	YES >= 75%
Johnson D SD	64.71%
Johnson R WI	95.59%	YES >= 75%
Kerry D MA	73.53%
Kirk R IL	70.59%
Klobuchar D MN	48.53%
Kohl D WI	70.59%
Kyl R AZ	23.53%	NO >= 75%
Landrieu D LA	77.94%	YES >= 75%
Lautenberg D NJ	19.12%	NO >= 75%
Leahy D VT	44.12%
Lee R UT	11.76%	NO >= 75%
Levin D MI	32.35%
Lieberman ID CT	38.24%
Lugar R IN	54.41%
Manchin D WV	45.59%
McCain R AZ	23.53%	NO >= 75%
McCaskill D MO	60.29%
McConnell R KY	100.00%	YES >= 75%
Menendez D NJ	45.59%
Merkley D OR	67.65%
Mikulski D MD	88.24%	YES >= 75%
Moran R KS	97.06%	YES >= 75%
Murkowski R AK	35.29%
Murray D WA	33.82%
Nelson D FL	85.29%	YES >= 75%
Nelson D NE	26.47%
Paul R KY	35.29%
Portman R OH	38.24%
Pryor D AR	35.29%
Reed D RI	10.29%	NO >= 75%
Reid D NV	100.00%	YES >= 75%
Risch R ID	100.00%	YES >= 75%
Roberts R KS	97.06%	YES >= 75%
Rockefeller D WV	92.65%	YES >= 75%
Rubio R FL	38.24%
Sanders I VT	35.29%
Schumer D NY	100.00%	YES >= 75%
Sessions R AL	77.94%	YES >= 75%
Shaheen D NH	42.65%
Shelby R AL	77.94%	YES >= 75%
Snowe R ME	25.00%	NO >= 75%
Stabenow D MI	32.35%
Tester D MT	75.00%	YES >= 75%
Thune R SD	100.00%	YES >= 75%
Toomey R PA	33.82%
Udall D CO	73.53%
Udall D NM	82.35%	YES >= 75%
Vitter R LA	88.24%	YES >= 75%
Warner D VA	100.00%	YES >= 75%
Webb D VA	91.18%	YES >= 75%
Whitehouse D RI	10.29%	NO >= 75%
Wicker R MS	98.53%	YES >= 75%
Wyden D OR	66.18%