 It is easy to show how it is possible to make book against someone with betting quotients that violate the probability axioms. Let \(Q(H)\) be the agent’s betting quotient for \(H\). Assuming that the agent’s betting quotients violate the axioms, a bookie can guarantee himself a profit by placing bets with the agent as described below. For simplicity the stake is set here at $1, but the following recipes for constructing a book against such a person are easily adapted for other stakes.