TY - GEN

T1 - A lattice of gambles

AU - Cuff, Paul

AU - Cover, Thomas

AU - Kumar, Gowtham

AU - Lei Zhao, Zhao

PY - 2011/10/26

Y1 - 2011/10/26

N2 - A gambler walks into a hypothetical fair casino with a very real dollar bill, but by the time he leaves he's exchanged that 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 intactor 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 value, and he can't get back to the original distribution. 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 any fair gamble is available then one can move down the lattice of distributions with respect to the Lorenz ordering. 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 can get its hands on.

AB - A gambler walks into a hypothetical fair casino with a very real dollar bill, but by the time he leaves he's exchanged that 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 intactor 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 value, and he can't get back to the original distribution. 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 any fair gamble is available then one can move down the lattice of distributions with respect to the Lorenz ordering. 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 can get its hands on.

UR - http://www.scopus.com/inward/record.url?scp=80054801918&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80054801918&partnerID=8YFLogxK

U2 - 10.1109/ISIT.2011.6033851

DO - 10.1109/ISIT.2011.6033851

M3 - Conference contribution

AN - SCOPUS:80054801918

SN - 9781457705953

T3 - IEEE International Symposium on Information Theory - Proceedings

SP - 1762

EP - 1766

BT - 2011 IEEE International Symposium on Information Theory Proceedings, ISIT 2011

T2 - 2011 IEEE International Symposium on Information Theory Proceedings, ISIT 2011

Y2 - 31 July 2011 through 5 August 2011

ER -