, ▁denoted ▁as ▁, ▁we ▁differenti ate ▁the ▁above ▁expression ▁and ▁set ▁this ▁equal ▁to ▁zero . ▁This ▁gives : ▁▁▁▁ ▁R ear r anging ▁this ▁equation ▁to ▁solve ▁for ▁the ▁value ▁of ▁ ▁gives ▁the ▁Kelly ▁criter ion : ▁▁▁▁ ▁For ▁a ▁rig orous ▁and ▁general ▁proof , ▁see ▁Kelly ' s ▁original ▁paper ▁or ▁some ▁of ▁the ▁other ▁references ▁listed ▁below . ▁Some ▁corre ctions ▁have ▁been ▁published . ▁ ▁We ▁give ▁the ▁following ▁non - rig orous ▁argument ▁for ▁the ▁case ▁with ▁ ▁( a ▁ 5 0 : 5 0 ▁" even ▁money " ▁bet ) ▁to ▁show ▁the ▁general ▁idea ▁and ▁provide ▁some ▁ins ights . ▁ ▁When ▁, ▁a ▁Kelly ▁bet tor ▁bet s ▁ ▁times ▁their ▁initial ▁wealth ▁, ▁as ▁shown ▁above . ▁If ▁they ▁win , ▁they ▁have ▁ ▁after ▁one ▁bet . ▁If ▁they ▁lose , ▁they ▁have ▁. ▁Suppose ▁they ▁make ▁ ▁bet s ▁like ▁this , ▁and ▁win ▁ ▁times ▁out ▁of ▁this ▁series ▁of ▁ ▁bet s . ▁The ▁resulting ▁wealth ▁will ▁be : ▁▁▁▁ ▁Note ▁that ▁the ▁ordering ▁of ▁the ▁wins ▁and ▁losses ▁does ▁not ▁affect ▁the ▁resulting ▁wealth . ▁ ▁Suppose ▁another ▁bet tor ▁bet s ▁a ▁different ▁amount , ▁ ▁for ▁some ▁value ▁of ▁ ▁( where ▁ ▁may ▁be ▁positive ▁or ▁negative ). ▁They ▁will ▁have ▁ ▁after ▁a ▁win ▁and ▁ ▁after ▁a ▁loss . ▁After ▁the ▁same ▁series ▁of ▁wins ▁and ▁losses ▁as ▁the ▁Kelly ▁bet tor , ▁they ▁will ▁have : ▁▁▁▁ ▁Take ▁the