According to the Archives Mailbag of the Sporting News, the phrase is borrowed from the card game bridge. The third and final game in bridge is called the "rubber game." A set of three games is also called a "rubber."

In baseball, however, a "rubber game" doesn't necessarily refer to the third game of a tied series, but the last and deciding game of any series. The seventh game of the World Series, for example, is a classic rubber game.
Source(s):
http://www.sportingnews.com/archives/sports2000/players/155407.html


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