Cliffhanger: The fiscal cliff bargaining game

The negotiations over the fiscal cliff in the US are entering its final stages. The current situation is a stand-still, where both sides know the concessions they need to make, but neither is willing to make the first move and loose leverage. The Republicans are unwilling to make concessions on tax increases to high-income earners, while the Democrats are refusing to reign in the welfare system and finally reform it. The situation is close to a classical hawk-dove game (or even better a game of deterrence  in which neither of the two actors are willing to back off.

We've heard arguments from both sides claiming that if no agreement is reached the other side is willing to “push” the US over the fiscal cliff. In particular if either of the sides is unwilling to back off from cutting spending or raising taxes, then both will have to agree to a status quo level of automatic deep spending cuts and big tax hikes. Only then will we experience the full effects of a completely wrong austerity policy. Bottom line is that neither party is willing to let the other party hold the country “hostage”. In this desire they are both very likely to reach a unique Nash equilibrium in which the US is pushed off the fiscal clif (i.e. the status quo policy is achieved). Let's draw a game table: 

Dems/ Reps	Strike deal	No deal
Strike deal	-2,-2	-10,0
No deal	0,-10	-5, -5 *

The unique Nash equilibrium of the game is (no deal, no deal) with a payoff of (-5,-5) to both parties. The reasoning is the following; it is obvious that both parties would receive a higher payoff from cooperation (strike deal) than from defection (no deal), but neither of the parties wants to back down from its proposals, leading to a worse off equilibrium where no deal is reached. If one party backs down and is willing to accept the deal under any circumstance to avoid the fiscal cliff, then the other party may decide to defect on making the deal and force its own terms. This outcome makes the other party worse off since they lose reputation in the face of their own voters. To each party losing credibility of their own voters seems to be a much worse option than resorting to a no deal “fiscal cliff” equilibrium. With neither party willing to back down, the rational result must be a defection strategy from both parties, where the fiscal cliff is applied. 

However, as Axelrod (1984) and many others have proven in many real-life scenarios the cooperative equilibrium is achieved despite the fact that sometimes it’s more rational to choose a defection strategy. This could end up being the case here, but it wasn’t the case in August 2011, during the first big quarrel over the budget deficit. The current conundrum is actually an effect of the inability to make a deal back then, where a status quo policy was set to be initiated in the end of this year, if no solution is reached. 

Let’s draw the dynamic bargaining game (click to enlarge):

Democrats (the President) move first and propose policy q₁ = f(τ₁,y₁), a combination of tax increases and spending cuts.

Tax increases in the case τ₁, are lower than in the status quo, fiscal cliff scenario. These would include only tax increases on the rich, in particular on those with over $250,000 yearly income, raising about $1.6 trillion in the course of the next 8 years. This would still be lower than the status quo case, τ₀, which would raise taxes on almost all Americans (letting the Bush tax cuts expire for all). The y₁ policy implies spending cuts that are lower than the status quo, and represent around $400bn in various cuts and entitlement reforms. If the Republicans accept than this is the equilibrium policy and the deal is reached, with Republicans accepting fully and making no amendments. If they reject then we enter period t = 2.  

In period 2, Republicans make their own proposal and set tax increases to be lower than τ₁, and much lower than the status quo τ₀. On the spending side they propose a platform of higher spending cuts y₂ > y₁, but still not higher than the status quo policy. If the Democrats accept and make no amendments, the equilibrium policy is reached at q₂. If the Democrats reject, the game enters period t = 3. Each policy offered in the next round of negotiations is more preferred to than the previously offered policy q_o ≺ q₁ ≺ q₂ ≺ q₃. The proposed policies can also be modelled as a deviation from each party’s optimal policy, but the intuitive result of the game is the same. 

Period 3 is the last period since there is a time constraint given in the negotiations (31st of December 2012). If until this period of negotiations no deal is reached then the status quo (“fiscal cliff”) is applied. 

In the last period the Democrats again move first and propose lower tax increases than in τ₁, but still higher increases than the previous Republican proposal of τ₂. They propose higher spending cuts than in q₁, but still lower than the Republican proposal of q₂; y₁ < y₃ < y₂. The Republicans now make the final move of the game. Their choice to give in and accept will depend on how close the Democrats were to their most preferred option. If they were rather far from it, the Republicans choose to reject the proposal and lead to the fiscal cliff. A similar scenario occurred last week after the Republican leaders rejected Boehner’s proposals. Instead of being the final period this resulted in yet another round of bargaining. We can enter yet another period in which the Republicans have a chance to make their final proposal, but it wouldn’t hurt the intuitive result of the game. 

Bottom line is, whatever the Democrats move forward as their final proposal before the deadline, the Republicans will have the final choice whether or not to accept it. 

We can introduce time discounting in the game and say that the closer we move to the deadline, the more likely it is for a party to reach a deal, since their voters don’t think of further defection as a sound strategy, but rather as stubbornness. However, if this isn’t the case then the deal will not be reached as long as both parties are too far apart from their optimal combination of tax increases and spending cuts f(τ*, y*). Whether or not the deal will be reached depends on the preferences of each party’s support on how long to keep a strong position and not back off. Recall the first table. The parties are reluctant to back off and cave in to the other party’s position, since they will see this as a potentially worse off outcome than the fiscal cliff (in terms of lost voter support, particularly for the upcoming local elections). The fiscal cliff is a better option since each party can always accuse the other one for not cooperating. 

Commitment device 

One can claim that the status quo, offering a series of automatically induced horror stories, is enough of a commitment device to ensure a cooperative equilibrium, where both parties reach a satisfactory outcome. But the commitment device obviously isn’t strong enough. If we think of this game as a classical cold war nuclear game of deterrence, where both parties have a credible threat they are willing to use, but both of them knows that triggering an action will lead to the least preferable outcome – a nuclear war, the commitment device is much stronger. Because of a strong commitment device, both parties abstain from initiating an action. 

This game, however, is a bit different, since the least preferable outcome is inevitable after January 1st (in the nuclear game, the nuclear war can be prevented by doing nothing, while in this game the fiscal cliff must be prevented by commencing an action). Knowing this both actors should find a strong incentive to cooperate. Imagine if January 1st is the day when the nuclear war starts. Then both actors have really strong incentives to prevent it. In the fiscal cliff scenario, this apparently isn’t the case. In other words, both actors apparently don’t see the fiscal cliff as such a huge threat, even though it is highly enforceable (and credible). 

This is the main issue preventing a cooperative outcome. Apparently the threat isn’t credible enough, or better yet, the threat isn’t assumed to be that devastating. Bottom line is that some members of both parties are rather willing to allow the threat to be achieved than they are willing to make any concessions.

Maybe I can get a good paper out of this?