Last month, Joseph J. Lhota, the chairman of the Metropolitan Transportation Authority, floated an idea: With a fare increase scheduled for next March, why not raise some of the money by eliminating the 7 percent bonus on pay-per-ride MetroCards?

On Monday, as the authority presented four proposals for how to carry out the increase, Mr. Lhota acknowledged that the suggestion had not gone over well.

“I’ve heard the public loud and clear,” he said, adding that he would recommend to the authority’s board that the discount — which gives riders an extra $1.40, for example, with each $20 placed on a card — be kept at least partially intact.

The assessment, rendered only minutes after Mr. Lhota unveiled the authority’s four proposals, signaled the start of a trying season at the authority: the period when all players — the chairman, his board members, and, at a series of hearings next month, the riding public — begin to address the question of how to allocate the burden.