  A gambler walks into a hypothetical fair casino with a very real dollar bill,
but by the time he leaves he's exchanged the dollar for a random amount of
money. What is lost in the process? It may be that the gambler walks out at the
end of the day, after a roller-coaster ride of winning and losing, with his
dollar still intact, or maybe even with two dollars. But what the gambler loses
the moment he places his first bet is position. He exchanges one distribution
of money for a distribution of lesser quality, from which he cannot return. Our
first discussion in this work connects known results of economic inequality and
majorization to the probability theory of gambling and Martingales. We provide
a simple proof that fair gambles cannot increase the Lorenz curve, and we also
constructively demonstrate that any sequence of non-increasing Lorenz curves
corresponds to at least one Martingale.
  We next consider the efficiency of gambles. If all fair gambles are available
then one can move down the lattice of distributions defined by the Lorenz
ordering. However, the step from one distribution to the next is not unique. Is
there a sense of efficiency with which one can move down the Lorenz stream? One
approach would be to minimize the average total volume of money placed on the
table. In this case, it turns out that implementing part of the strategy using
private randomness can help reduce the need for the casino's randomness,
resulting in less money on the table that the casino cannot get its hands on.
