  We give a detailed characterization of optimal trades under budget
constraints in a prediction market with a cost-function-based automated market
maker. We study how the budget constraints of individual traders affect their
ability to impact the market price. As a concrete application of our
characterization, we give sufficient conditions for a property we call budget
additivity: two traders with budgets B and B' and the same beliefs would have a
combined impact equal to a single trader with budget B+B'. That way, even if a
single trader cannot move the market much, a crowd of like-minded traders can
have the same desired effect. When the set of payoff vectors associated with
outcomes, with coordinates corresponding to securities, is affinely
independent, we obtain that a generalization of the heavily-used logarithmic
market scoring rule is budget additive, but the quadratic market scoring rule
is not. Our results may be used both descriptively, to understand if a
particular market maker is affected by budget constraints or not, and
prescriptively, as a recipe to construct markets.
