There are two specific problems with government - how big government can be, and who should pay for it. There are two sides - Republican and Democrat.
One solution is to apply Solomon's Rule: divvy it up. One sides controls the size, the other controls who pays for it.
The GOP keeps claiming they want a small government. OK, they get to decide how much taxes to charge. Just let the Democrats decide WHO pays those taxes.
If your side disagrees on how they do their part, you can wheel and deal to arrange a compromise.
If the GOP wants to reduce our tax bite from 2.4 trillion in 2011 to only 2.2 trillion, that's fine. Its a solid 9% tax reduction. Just let the Democrats decide who gets that 9% tax reduction.
If the GOP doesn't like the Democrat's solution, they can negotiate away things. Let's say the Democrats gave a tax split that was 100% taxing the wealthy at 80%. The GOP could agree to raise taxes by $500 billion more dollars if $250 billion of that money came from the poor/middle class. Otherwise, they are stuck entirely with the tax plan the Democrats came up with.
You can do the same with spending.
Have one side decide how much to spend, give the other side power over what to spend it on. The GOP wants a small government, let them limit the size of our spending, but don't be surprised if the Democrats keep all the social programs and reduce the military.
But honestly, in real life this would never work - for two big reason.
The GOP is outright lying about their intentions. They have zero real desire to lower the deficit and zero real desire to reduce the size of the government. They had the power to do that before, under GW Bush, but the government grew. Worse, the new 'T-party' people are too arrogant to compromise and achieve there goals. In effect deficit reduction and smaller government are just buzz words they use to defend attempts to reduce spending on projects they dislike, and use the money on projects they like.
Another problem is tax predictions don't always come true. The GOP loves to predict that lowering taxes raises income, and the Democrats like to say the opposite. Neither have anywhere near a 100% accuracy. I personally would let the person that was most accurate about which way the actual tax changes came out to get to pick which side they do next year.
P.S. The main reason we have these problems is that economics is probably the softest of the soft sciences. That is a psychiatrist is far more likely to be able to predict what a crazy person will do, then an economist can predict what the economy will do. There is an old joke about an economics professor. Each year he uses the same questions - it's just the correct answer that changes.
Honestly, no one truly knows much at all about economics - aside from a few theories that we know don't work very well . And no one - democrat or republican - actively pushes any of the theories that we know don't work.