Question: Who did The Ballplayer in Insignificance play for?

Context paragraphs:
Joe DiMaggio: Joseph Paul DiMaggio (November 25, 1914 – March 8, 1999), nicknamed "Joltin' Joe" and "The Yankee Clipper", was an American Major League Baseball center fielder who played his entire 13-year career for the New York Yankees.  He is perhaps best known for his 56-game hitting streak (May 15 – July 16, 1941), a record that still stands.

James Walker (actor): James Walker (born James S. Chalton, 1940–2017) was a British actor active in films and on television between 1980 and 2006.  His career highlights included a season between 1992 and 1994 with the Royal Shakespeare Company in Stratford-upon-Avon – including a foreign tour with "The Winter's Tale" – and taking the role of Albert Einstein in Terry Johnson's play "Insignificance", in Harrogate in 1999.

Anempathetic sound: Anempathetic sound in a film is the opposite of empathetic sound: it consists of music or sound effects that exhibit an indifference to the current tone, emotion, or plot-point of the film.  This type of sound can thereby enhance a sense of tragic apathy and insignificance, as when a radio continues to play a happy tune when a character dies, or in Hitchcock's "Psycho" the continued sound of the shower running after Marion Crane has been killed, as if nothing has happened.

Insignificance (film): Insignificance is a 1985 British comedy-drama film directed by Nicolas Roeg, produced by Jeremy Thomas and Alexander Stuart, and adapted by Terry Johnson from his play of the same name.  The film is set in 1954, with most of the action taking place in a hotel room in New York City.  The action revolves around the interplay of four characters who represent iconic figures of the era - Marilyn Monroe, Joseph McCarthy, Joe DiMaggio, and Albert Einstein - called The Actress, The Senator, The Ballplayer, and The Professor, respectively.

Ed Sprinkle: Edward Alexander Sprinkle (September 3, 1923 – July 28, 2014) was an American football player.  He was known to many as "The Meanest Man in Pro Football" and was nicknamed "The Claw."  He played for 12 seasons with the Chicago Bears of the National Football League and is credited with calling attention to the NFL's defensive players.  Bears coach George Halas referred to Sprinkle as "the greatest pass-rusher I've ever seen" and "a rough, tough ballplayer, but not a dirty one."  At first he played on both defense and offense.  He caught 32 passes for 451 yards and seven touchdowns during his career.  His ability to rush opposing quarterbacks, however, soon made him a defensive specialist Earning 4 Pro Bowls.  While accused of "dirty play" and unsportsman-like conduct during his career, Sprinkle claimed that his aggressive play was similar to that found all over the NFL throughout the 1950s.  According to Sprinkle, "We were meaner in the 1950s because there were fewer positions and we fought harder for them.  It was a different era."

Supporting paragraphs: Insignificance (film) Joe DiMaggio
Answer: the New York Yankees