From 2008 to 2013, homicide cases generated 129 reward payments averaging $1,532. There were slightly more rewards related to robbery cases, with payments averaging $916. Rewards on rape cases averaged $1,724.

Over all, payments have declined from 2008 through 2013. Payments to confidential informants, for example, have fallen by 18.9 percent since 2011, according to the data.

Operation Gun Stop has seen its payments fall by 25.7 percent since 2008, when the program claimed 319 guns and 569 arrests. Last year, the program yielded 235 guns and 330 arrests.

John A. Eterno, who co-wrote “The Crime Numbers Game: Management by Manipulation,” suggested that the drop in gun tips was likely an outgrowth of “community alienation” from the Police Department’s reliance on stop-and-frisk tactics.

Eli B. Silverman, Mr. Eterno’s co-author, said that it was “disturbing but not surprising to see these results.”

“Police tactics have consequences,” he added.

A spokesman for the Police Department would not discuss the payments. The department also declined to provide a breakdown of the payments to confidential informants, saying that revealing such details could put the informants at risk.

But in an affidavit filed in State Supreme Court in Manhattan in August, Lt. William McCabe, who supervises the sensitive data unit in the department’s intelligence division, said informants are paid “for many different activities, including spending time with, or performing tasks assigned by, members of the N.Y.P.D.”