I fail to see how this wouldn't collapse to precommitting and group action immediately. The majority side could always get what it wanted at little to no cost to itself. Republicans for example could put "All of Obamacare, unemployement, Social Security, and any other socialist programs" five times in slight variations, and allocate all their cut votes to it. There are 240 republicans and 195 democrats, which would result (if I understand your system) in a 10% cut to those programs. Or more if you're allowed to count it 5x times. Subsequent iterations of the game would cut it by more and more each year. They would be free to block the other side with their defend votes as usual.
The granularity of nominations would be an issue. You could kill programs with sufficiently precise cuts. For example if you wanted to end foreign wars you could separate out "All gasoline used in the military abroad", "All the food used to feed soldiers abroad", "All the bullets and munitions used abroad", or "All the electricity used by our military abroad". Each one is incredibly cheap (relative to the cost of the entire war) but successfully cutting any one to zero would end the war and achieve your goals. The same method could be used to kill many other programs.
Also, it would be necessary to do cuts before defends. Otherwise, you could have a situation where there are 27 variants of "Cut all of Unemployment Benefits" (with some slight caveat to make them unique), and all members of one side vote to cut the same one while the members of the other side are spread out trying to defend them all.
As it happens, there are more Democrats than Republicans in the Senate. Since in the end both houses must agree on a budget, perhaps the actual thing to do is to give each party the same number of votes, divided in some way between their senators and representatives.
As for whether in fact every year the budget would change in one direction, that assumes that cuts will continue to need to be made. It's not clear how you determine the level of cuts (or increases) to be made under this game. Also, the notion that the party in power has more influence on the budget seems to me to be a benefit from the perspective of democracy, rather than a negative.
I agree that granularity is a potential problem.
The US Congress is trying to resolve the national debt by getting hundreds of people to agree on a solution. This is silly. They should agree on the rules of a game to play that will result in a solution, and then play the game.
Here is an example game. Suppose there are N representatives, all with an equal vote. They need to reduce the budget by $D.
What game-theoretic problems does this game have? Can you think of a better game? Is it politically better to call it a "decision process" than a game?
The main trouble area, to my mind, is order of play. First I said that budget items would be listed by taking turns. The 1..N, N..1 order is supposed to make neither first nor last position preferable. But taking turns introduces complications, of not wanting to reveal your intentions early.
Then I said votes are placed secretly and revealed all at once. This solves problems about game-theoretically trying to conceal information or bluff your opponent. It introduces other problems, such as tragedy-of-the-commons scenarios, where every Republican spends their "defend" votes on some pork in their state instead of on preventing tax cuts, because they assume some other Republican will do that.
Is it better to play "cut" votes first, reveal them, and then play "defend" votes?
Is there a meta-game to use to build such games?