The phrase "a set of bets" is correct and usable in written English.
It can be used when referring to multiple wagers or predictions made in a gambling context or in discussions about risk and decision-making.
Example: "He placed a set of bets on the upcoming horse race, hoping to maximize his chances of winning."
Alternatives: "a collection of wagers" or "a group of bets".
For example, if you deviate from the axioms, you will accept a set of bets that is bound to lose money, even though you can see that losing money is inevitable a priori.
He then invokes the Dutch Book theorem to argue that incoherent degrees of belief sanction as fair a set of bets, which cannot be fair, and claims this reveals that incoherence is fundamentally an epistemic, and indeed a logical, defect.
Another problem involving the value of bets arises in establishing the additivity axiom, for even if each in a set of bets is acceptable, it does not follow that they would be jointly acceptable.
The Dutch Book theorem can now be stated: Given a set of betting quotients that fails to satisfy the probability axioms, there is a set of bets with those quotients that guarantees a net loss to one side.
He argued that by themselves the probability assignments do not mean anything at all, and that they can only be interpreted in the context where an agent faces a choice between actions, i.e., a choice among a set of bets.
They then employ the Dutch Book theorem to argue that incoherent degrees of belief cannot in fact be fair betting quotients, and thus that incoherence involves taking a set of bets as fair that cannot be fair, which is supposed to parallel having inconsistent beliefs.
The service is part of a broad set of bets Microsoft is making this fall to help regain ground it has lost to competitors, especially Apple and Google.
The probability function must output some number, and that number will sanction a particular set of bets as desirable.
However, it remains to be shown that avoiding book with such a restricted set of bets suffices to justify adherence to the axioms.
A Dutch Book argument can be constructed for the principle by extending the Dutch Book for finite additivity (Adams 1962; Jeffrey 2004; Williamson 1999) by appealing to an infinite set of bets that each pay $1 if Wi is true for the price pr(Wi).
It is based on the growing affinity between a set of countries all betting on market economies, foreign investment and trade with Asia to achieve development.
