ONE of the biggest social experiments ever undertaken, the Department of Housing and Urban Development’s experimental housing allowance program began with great fanfare in 1973 and is ending quietly in 1979-1980. It has involved more than 25,000 families in 12 metropolitan areas, at a cost that is expected to add up to $160 million. As is true of most ambitious programs, it owes its origin to the convergence of several different lines of thought on how government should cope with a problem. The problem in this case was how to improve housing conditions for low-income people. Two key ideas prompted the experiment.  One was that the best way to help families who needed better housing was to give them money that they could use on their own, instead of building subsidized housing for them. The other was that the best way to learn how a new approach would work in practice was to conduct a large-scale social experiment, following a systematic design and using control groups to check the validity of the results.