= Log-optimal multiplicative betting
{title2=$g_*=D(p\Vert u/U)-\log U$}
For independent outcomes with positive <probability distribution> $p$, allocations $a_j>0$ summing to one and return multipliers $a_j/u_j$, the <strong law of large numbers> gives asymptotic logarithmic fortune growth $g(a)=\sum_jp_j\log(a_j/u_j)$. <Gibbs inequality> gives $g(a)=g(p)-D(p\Vert a)$, so the unique optimum is $a=p$. Writing $U=\sum_ju_j$ gives $g(p)=D(p\Vert u/U)-\log U$. Thus $U<1$ guarantees positive optimal growth, whereas $U>1$ allows the house to force negative growth by choosing $u_j=Up_j$, but does not guarantee decline for every choice of $u$.
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