To behave recklessly; to idly squander one's wealth.
Ducks and drakes is the old English name for the pastime of skimming flat stones on the surface of water to make them bounce as many times as possible. There are various names for the game, for example, stone skipping in the USA and stone skimming in the UK; in fact most countries have their own name for it. I doubt that there's a child anywhere in the world who hasn't tried to establish his or her own record. Most people manage seven or eight bounces. In researching this phrase I was surprised to find that the world record, as endorsed by the Guinness Book of Records, stands at 40. That hardly seems possible, but there is video evidence of Kurt Steiner setting that record in 2002.
The pastime surely pre-dates written records. The first known reference to it in print is in The nomenclator, or remembrancer of Adrianus Junius, translated by John Higgins in 1585:
"A kind of sport or play with an oister shell or stone throwne into the water, and making circles yer it sinke, etc. It is called a ducke and a drake, and a halfe-penie cake."
Why that name was chosen isn't clear. Most early citations give the phrase as 'make ducks and drakes' rather than 'play ducks and drakes', so it may be that the circular ripples that are formed evoked images of splashing waterfowl. For example, from the play Dick of Devon, circa 1626:
"The poorest ship-boy Might on the Thames make duckes and drakes with pieces Of eight fetchd out of Spayne."
Around the same time, the use of 'ducks and drakes' to refer to idly throwing something away or squandering resources came into use. That usage was recorded in James Cooke's Tu Quoque, 1614:
"This royal Caesar doth regard no cash; Has thrown away as much in ducks and drakes As would have bought some 50,000 capons."
The adoption of 'play ducks and drakes' meaning to throw away money seems to have come directly from the throwing of stones in the waterside game.
The meaning now seems to have wandered closer toward the 'unreliable and reckless' and away from the original 'idly squandering'. This may be a simple migration of meaning over time, or it may be due to a confusion between 'playing ducks and drakes' and playing fast and loose.