A Democratic president and a Republican Congress are at odds over two pieces of legislation. One is a welfare bill and the other is a farm subsidy. Congress loves the subsidy and hates the welfare bill; the president loves the welfare bill and hates the subsidy.

Congress must originate any legislation, after which the president can decide to veto the bill; Congress will be unable to overturn any veto.

Congress is considering whether to propose the subsidy, or the welfare program, or join them both into an omnibus bill. Congress gets a payoff of 4 if the subsidy passes, 3 if the omnibus passes, 2 if nothing passes, and 1 if the welfare bill passes. The president gets a payoff of 4 if the welfare bill passes, 3 if the omnibus passes, 2 if nothing passes, and 1 if the subsidy passes.

a) Draw this interaction as a game in extensive form with Congress proposing legislation first (first stage) and the president either signing or vetoing that legislation second (second stage).

b) Use backward induction to solve for the (pure strategy subgame perfect) Nash equilibrium of the game. Note that a player's strategy should be a plan of action for every one of their decision nodes (more generally, it is a plan of action for every information set). What is the associated equilibrium outcome?

c) Suppose that the president has the power of a line-item veto. In other words, he or she can veto a portion of a bill, while allowing other parts to pass into law. Draw the extensive form of this modified game.

d) Using backward induction determine the (pure strategy subgame perfect) Nash equilibrium and associated equilibrium outcome of this modified game.

e) What bill if any passes in this case? Has the power of the line-item veto been useful for the president in this case?