On a simple strategy weakly forcing the strong law of large numbers in the bounded forecasting game

Abstract

Azuma-Hoeffding-Bennett inequality, Capital process, Game-theoretic probability, Large deviation,

Game-theoretic versions of strong law of large numbers for unbounded variables

“…We believe that this again shows effectiveness of game-theoretic proofs as we have shown in our previous works [4,5].…”

“…Then adjusting some constants we can assume hðxÞ $ x 2 for all x without loss of generality. Then and E 0 1 , E 3 , where E 0 1 is given in (4). Therefore Skeptic can force E 3 .…”

“…In Ref. [4] we showed that a single simple strategy based on past averages of Reality's moves forces SLLN for the case of bounded Reality's moves. For the special case of the coin-tossing game path behavior and convergence rate of SLLN can be very explicitly stated [5,9].…”

Implications of contrarian and one-sided strategies for the fair-coin game

“…Since E 2 ⊂ E 1 , forcing E 2 is stronger than forcing E 1 . However the multiplicative strategy in Subsection 2.1 is of interest, because it is a contrarian counterpart of the momentum strategy studied in [2].…”

“…It is only natural to expect that stronger results require more complicated strategies by Skeptic. For example in [2] we have shown that a simple strategy of Skeptic based on the past average of the moves by Reality forces SLLN for the case of bounded Reality's moves. However if Reality's moves are unbounded, strategies for forcing SLLN are much more complicated as discussed in [4].…”