▁they ▁can ▁agree ▁on ▁the ▁relatively ▁lik elihood ▁each ▁will ▁win . ▁And ▁third , ▁there ▁are ▁no ▁direct ▁benefits ▁from ▁fighting . ▁ ▁Thus , ▁using ▁John ▁von ▁Ne umann ▁and ▁O sk ar ▁M orgen st ern ▁expected ▁utility ▁functions , ▁F ear on ▁finds ▁the ▁expected ▁utility ▁for ▁war ▁for ▁states ▁A ▁and ▁B , ▁which ▁are ▁PA ▁- ▁CA ▁and ▁( 1 ▁- ▁PA ) ▁- ▁C B , ▁where ▁PA ▁is ▁A ' s ▁probability ▁of ▁winning ▁a ▁war , ▁CA ▁is ▁A ' s ▁costs ▁for ▁war ▁( pro port ional ▁to ▁how ▁much ▁they ▁value ▁the ▁utility ), ▁and ▁C B ▁is ▁B ' s ▁costs ▁for ▁war ▁( pro port ional ▁to ▁how ▁much ▁they ▁value ▁the ▁utility ). ▁With ▁simpl ification , ▁if ▁X ▁is ▁A ' s ▁share ▁of ▁a ▁peace ful ▁settlement , ▁F ear on ▁finds ▁that ▁peace ▁is ▁better ▁than ▁war ▁when ▁PA ▁- ▁CA ▁< ▁X ▁< ▁PA ▁+ ▁C B . ▁A ▁satisf actory ▁X ▁exist ▁ ▁if ▁PA ▁+ ▁C B ▁> ▁PA ▁- ▁CA , ▁or ▁CA ▁+ ▁C B ▁> ▁ 0 . ▁Because ▁CA ▁and ▁C B ▁are ▁individually ▁greater ▁than ▁ 0 , ▁so ▁is ▁their ▁sum . ▁Therefore , ▁the ▁inequality ▁holds ▁and ▁so ▁some ▁settlement ▁is ▁mut ually ▁prefer able ▁to ▁war . ▁ ▁The ▁question ▁is ▁why ▁two ▁rational ▁states ▁cannot ▁find ▁an ▁X ▁that ▁satisfies ▁both ▁sides , ▁even ▁though ▁one ▁must ▁always ▁exist ▁and ▁war ▁is ▁the ▁worst