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Unspecified in the NYT: when it comes to housing lotteries, the demand is most significant for lower-income units

There are a couple of odd things about the New York Times article today, Long Lines, and Odds, for New York’s Subsidized Housing Lotteries. First, it states:
The odds of winning the New York Lotto jackpot are, of course, worse (one in 22 million on a $1 play), but the housing lotteries have daunting odds of their own.
Last year, a new building in Greenpoint, Brooklyn drew 58,832 lottery applications for 105 affordable units. Not far behind was the Sugar Hill development in Upper Manhattan, which drew more than 48,000 applicants for 98 apartments.
Given the frequent use of the term "affordable," the article does not make sufficient distinction between subsidized income-linked housing geared to low-income households and that geared to middle-income ones.

Rest assured that the demand is most significant for lower-income units. That's why the middle-income Hunter's Point South Towers in Long Island City need to be advertised.

The searching Forest City?

The article states:
Developers said they had learned to start marketing the apartments early, sometimes years ahead.
“It’s somewhat like finding a needle in a haystack,” said Melissa R. Burch, executive vice president for development at Forest City Ratner Companies, which is preparing for the lottery of 2,250 affordable units at Pacific Park Brooklyn, formerly known as Atlantic Yards.
First, Burch has already left Forest City. It's far more difficult to find certain, better-off households to apply. And, of course, Forest City is not preparing for a lottery for all 2,250 units. The project will take another ten years, so it will be one building at a time.