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Frank WakefieldJohn Francis Hylan defeated the reformer John Purroy Mitchel in the 1917 mayoral election for New York City, restoring the power of Tammany at City Hall. Hylan won re-election in 1921 by a wide margin. Hylan was defeated for re-nomination in 1925 by State Senator James J. "Jimmy" Walker.
So if that is 1918 to 1924 in baseball seasons, and Yankee Stadium wasn't built until 1923... that narrows it down a bit.
Steve knows his NYC history.
That guy looking out onto the field may well be Jimmy Walker, who followed Hylan. Walker won election twice, but resigned during the second term and left for Europe, possibly to avoid criminal actions.