The governor said his last offer would have raised income taxes only on those earning more than $1 million a year—an estimated 7,700 Minnesotans, or 0.3 percent of all taxpayers, according to the Revenue Department. Republicans rejected the proposal, Dayton said, because they "prefer to protect the richest handful of Minnesotans at the expense of everyone else."

In the end, this is why the Minnesota government shut down:

Today, more than 20,000 state workers are off the job to protect those 7,700 people from a tax increase. That's 7,700 people, remember, who already pay a lower percentage of their incomes in state and local taxes than the average Minnesotan:

In particular, the wealthiest one percent of Minnesota households — those with incomes over $429,000 — paid 9.7 percent of their incomes in total state and local taxes in 2008, compared to the statewide average of 11.5 percent.

The GOP had a couple of "compromise" ideas for a budget agreement. They:

proposed delaying another $700 million in payments owed to schools, which would add to the more than $1 billion the state already owes K-12 schools. Republicans also offered to issue "tobacco bonds" of an unspecified amount to cover any remaining budget gap. Sources said Dayton considered the offer, but he criticized it as unwise borrowing late Thursday.

Another offer:

asked Dayton to accept controversial policy positions the Republicans pushed for this year, including photo ID requirements at the polls and abortion restrictions. An offer sheet provided to the Star Tribune said the policy adoptions were in exchange for "new revenue in a compromise offer."

This in a nutshell is today's Republican party: to protect 7,700 millionaires from slightly higher taxes, they'll shut down state government. But they might be willing to do something on revenue (whether it involved the 7,700 millionaires, we don't know) in exchange for making it more difficult to vote or get an abortion. That last, by the way, is similar to Iowa, where Republicans tried to block the use of Medicaid funds for some abortions; ultimately they compromised on various ways of trying to talk women out of their decision in the guise of "offering information."