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In his story 'Riders of the Purple Wage' (1967), Philip Jose Farmer
describes a society with something like 'negative money'.  In that story,
wealth is a universal birthright, money is a societal negative, getting
rid of money is everyone's goal and difficult to achieve (a bit analogous
to Losing Chess).  

Would 'Worth than Worthless' pieces necessarily have positive values in
Losing Chess?

'positive values in Losing Chess?'

Clever thought. A bit of analysis seems to tell me that a negative relay Q
would be a good thing to have in Giveaway Chess.

Undoubtedly the best piece to have in Giveaway would be one with huge
mobility and no capture -- in other words, the Ghost!

If you take the game from the Black Ghost page and play giveaway, Black
must surely win by using the Ghost at just the right time to force a
series of captures. Is the -relay Q stronger than that? In fact, in
Giveaway it is weaker because the opponent will simply capture it at first
opportunity!

The interesting thing about the -relay N in the normal game is that you
can get a small positive value from it by trying to get your opponent to
capture it; but in Giveaway there is no such reluctance.

Thus the Ghost is stronger than even -relay Amazon in giveaway chess. In
my opinion, at least.

the idea of negative money has been disussed in great detail, in the n.
california bay area, and now in tulsa, ok on this:
http://www.designhaven.net/blog/carlos/index.php?p=74&c=1 blog.

I invented the Giveaway game with the Ghost and Relay. See /index/msdisplay.php?itemid=MSghostarelay. Both players have one Ghost.

There is multiple purposes of the Ghost. You can use a Ghost at just the right time to force a series of captures, but you can also block opponent's pawns and block your own pieces from being forced to capture.

Ralph Betza's classic brief on negative values. Few today know "nattering nabobs of negativity" (call it NNN) came from Spiro Agnew, who resigned as Vice President just before Richard Nixon resigned as President.

Yet it depends on the position. Betza's first example, Negative Relay Knight with no capturing power that relays its move to opponents actually as described still has blocking power, and we could make up a problem where it figures in a checkmate for owning side. But over-all there is average negative value in such a piece, and barring a weird specific combination in view, you just as well get rid of it but probably cannot. (That would be another study to design, namely show a case when it is worthwhile to capture a NRK.)

NNN are an experimental Chess Different Army and have enhanced Bishop and Rook to go with Negative Relay Knight. Is NNN equal in value to Colorbound Clobberers and to F.I.D.E. and to Nutty Knights and to Pizza Kings?

Under Strangeness in Asymmetrical Relay, Betza suggest further relay complications such as to move as a Rook or capture as a Bishop, and if your own piece is doing so of course it has (further) negative value.

The value of an extra King might also be negative. I once tried the following test (requested by someone on chess.com): play a FIDE army without Queen against a FIDE army where the Queen was replaced by a second King, under rules of absolute royalty (i.e. none of the royals can be exposed to capture). The result was not significantly different from 50%. So on average the extra King is just as much a liability for getting mated as an asset for (defensive)  tactics.

This is not equally distributed over the game, though. In a Pawn ending, a side with two Kings almost always beats a single King very easily. So in the late end-game the extra King is nearly as good as having an extra minor. That means that it should have a significantly negative value in the middle-game to neutralize its effect overall.

I suddenly remembered another variant with a worse-than-worthless piece: 'The Wuss'. This is basically a second royal that is extra vulnarable, because it is only allowed to evade attacks by moving away, and not by capturing the attacker or interposing. The Wuss itself has no capture power at all. It moves a a Queen, though. This piece must also be more of a liability than an asset, although also worth something in a Pawn ending for its ability to block Pawns.