Let q1,q2,…,qn be the optimal values such that the expected amount of dollars you pay back is minimized. The probability that team i wins is pi, and if they win we have to pay back qiBi dollars. Thus our total expected payback will be S=∑i=1nqipiBi under the constraint that q1+….+qn=1.
We will find a relationship between qi and qj for each i,j. Consider what happens when we only vary qi and qj and fix the rest. All terms of Sstays constant except qipiBi+qjpjBj. SInce S is minimum so is this term. Let qi+qj=k. Then xpiBi+k−xpjBjmust be minimum at x=qi. Remember that the first derivative is 0 at minima. Thus differentiating at x=qi, we have −x2piBi+(k−x)2pjBj=0⟹k−xx=pjBjpiBi⟹qjqi=pj,bjpi,bi for all i,j
In other words, qi is proportional to pibi. Let ai=pibi.Then it is a matter of simple algebra to show that