▁derivative ▁of ▁this ▁with ▁respect ▁to ▁ ▁and ▁get : ▁▁▁▁ ▁The ▁function ▁is ▁maxim ized ▁when ▁this ▁derivative ▁is ▁equal ▁to ▁zero , ▁which ▁occurs ▁at : ▁▁▁▁ ▁which ▁implies ▁that ▁▁▁▁ ▁but ▁the ▁proportion ▁of ▁winning ▁bet s ▁will ▁eventually ▁conver ge ▁to : ▁▁▁▁ ▁according ▁to ▁the ▁weak ▁law ▁of ▁large ▁numbers . ▁ ▁So ▁in ▁the ▁long ▁run , ▁final ▁wealth ▁is ▁maxim ized ▁by ▁setting ▁ ▁to ▁zero , ▁which ▁means ▁following ▁the ▁Kelly ▁strategy . ▁ ▁This ▁illustr ates ▁that ▁Kelly ▁has ▁both ▁a ▁determin istic ▁and ▁a ▁st ochastic ▁component . ▁If ▁one ▁knows ▁K ▁and ▁N ▁and ▁wishes ▁to ▁pick ▁a ▁constant ▁fraction ▁of ▁wealth ▁to ▁bet ▁each ▁time ▁( other wise ▁one ▁could ▁che at ▁and , ▁for ▁example , ▁bet ▁zero ▁after ▁the ▁K th ▁win ▁knowing ▁that ▁the ▁rest ▁of ▁the ▁bet s ▁will ▁lose ), ▁one ▁will ▁end ▁up ▁with ▁the ▁most ▁money ▁if ▁one ▁bet s : ▁▁▁▁ ▁each ▁time . ▁This ▁is ▁true ▁whether ▁ ▁is ▁small ▁or ▁large . ▁The ▁" long ▁run " ▁part ▁of ▁Kelly ▁is ▁necessary ▁because ▁K ▁is ▁not ▁known ▁in ▁advance , ▁just ▁that ▁as ▁ ▁gets ▁large , ▁ ▁will ▁approach ▁. ▁Some one ▁who ▁bet s ▁more ▁than ▁Kelly ▁can ▁do ▁better ▁if ▁ ▁for ▁a ▁stretch ; ▁someone ▁who ▁bet s ▁less ▁than ▁Kelly ▁can ▁do ▁better ▁if ▁ ▁for ▁a ▁stretch , ▁but ▁in ▁the ▁long ▁run , ▁Kelly ▁always ▁wins . ▁ ▁The ▁he ur istic ▁proof ▁for