How large of an impact would an "alternative voting system" have on US politics?

The overwhelming majority of elections in the US use the plurality voting system (also called first-past-the-post). Each voter casts a vote for a single candidate, and the candidate with the most votes wins.

Despite its simplicity and ease of use, plurality voting is not ideal. One major problem is the spoiler effect, which can cause undesirable candidates to win because third parties split the vote. For those interested in the specifics, here is a table of many voting systems and their properties.

It has also been theorized that plurality voting favors a two-party system of government.

There are a few countries which have implemented alternative voting systems, which attempt to eliminate the problems with plurality voting. Australia and India use instant-runoff voting, which eliminates the spoiler effect. Some organizations have used more complicated systems, such the Schulze method (which meets the Condorcet criterion).

Let’s imagine that elections in the US were changed from plurality voting to one of the alternative systems. What effects would that have on the political system?