Illusion of Control in a Brownian Game

Both single-player Parrondo games (SPPG) and multi-player Parrondo games (MPPG) display the Parrondo Effect (PE) wherein two or more individually fair (or Llosing) games yield a net winning outcome if alternated periodically or randomly. (There is a more formal, less restrictive definition of the PE.) We illustrate that, when subject to an elementary optimization rule, the PG displays degraded rather than enhanced returns. Optimization provides only the illusion of control, when low-entropy strategies (i.e. which use more information) under-perform random strategies (with maximal entropy). This illusion is unfortuntately widespread in many human attempts to manage or predict complex systems. For the PG, the illusion is especially striking in that the optimization rule reverses an already paradoxical-seeming positive gain - the Parrondo effect proper - and turns it negative. While this phenomenon has been previously demonstrated using somewhat artificial conditions in the MPPG (L. Dinios and J.M.R. Parrondo. Europhysics Letters 63, 319 (2003); J.M.R. Parrondo et al. Advances in Condensed Matter and Statistical Mechanics, eds. E. Korutcheva and R. Cuerno, Nova Science Publishers, 2003), we demonstrate it in the natural setting of a history-dependent SPPG.