Bargaining Senators (Current Draft)

### If two senators were each about to propose a bill, how might we expect them to bargain about what they will put forward?

Political economist Ecks Wye Zed has already provided a model of what happens when a policymaker can choose what legislation to propose in a one-dimensional environment.  (Imagine that politics can be divided into left-wing and right-wing and that every proposable bill fits somewhere on this one-dimensional scale.)

Current policy (0) will stay in place if no new legislation is voted in, so all senators vote with this in mind:  if they like the proposed legislation q better than 0 they will vote yes; if not, they will vote no.

This is assuming that the senator knows what his fellows will vote for.  Of course this may not be the case but perhaps once we've figured out what senators do with full information and plenty of time to think, we can complicate the model by making them unsure or pressed for time.

The Next Question

Now I wish to ask the question in two dimensions (more would be complicated and probably won't show anything we wouldn't already learn from the two-dimensional case).  This means I must

(1) assume a utility function for the senators

(2) find the Nash solution in each case

(i) find the set of efficient bargains

(ii) map that set to utility-space

(iii) apply Nash's solution

(iv) map the result back to policy-space

(3) draw conclusions about reality based on my opinion of the assumptions v. meatspace

### U=|b¹-q¹| + |b²-q²|

For the sake of simplicity, take the senators' preferences to be oriented like this:

You could get to any other orientation by flipping or rotating the definitions of "right" and "left" and this reduces the amount of cases to consider.

Feasible Set

Based on fellow senators' preferences, the following bills will pass if proposed:

The math behind this.

Efficient Set

Agreeing on a policy pair outside this set would be pareto-inferior to agreeing on a pair inside it.  (That is, one or both senators could gain without the other losing by moving closer to this set.)

The math behind this.

Solution in Utility-Space

Sketched below is the efficient set and where disagreement points (current policy) will map to.

The math behind this.

Solution in Policy-Space

After mapping disagreement points back to policy space, we can see what sorts of current policies will lead to what sorts of bargains between the senators.  (Remember that their preferences are already given and in Nash's theory the utility functions determine the bargaining outcome from axioms - no bargaining evolution takes place as in Rubinstein's model).

Now we have a prediction about what policy will be proposed.  Anyone have an idea of how to test this?

The math behind this.

### U=|b-q|

Feasible Set

Based on fellow senators' preferences, the following bills will pass if proposed:

The math behind this.

Efficient Set

Agreeing on a policy pair outside this set would be pareto-inferior to agreeing on a pair inside it.  (That is, one or both senators could gain without the other losing by moving closer to this set.)

The math behind this.