Let G=(V,E) be a simple graph and for every vertex v∈V let L(v) be a set (list) of available colors. The graph G is called L-colorable if there is a proper coloring φ of the vertices with φ(v)∈L(v) for all v∈V. A function f:V→N is called a choice function of G if G is L-colorable for every list assignment L with |L(v)|=f(v) for all v∈V. Let size(f)=∑v∈Vf(v) and define the sum choice number χsc(G) as the minimum of size(f) over all choice functions f of G.The chromatic sum Σ(G) of G is the minimum sum of colors taken over all possible proper colorings of G.In this paper we consider the functions g(n)=max{χsc(G)∕Σ(G):Σ(G)=n} and h(n)=max{χsc(G)∕Σ(G):|V(G)|=n}. We conjecture g(n)=Θ(logn), h(n)=Θ(logn), and g(n)≤h(n) for all n. We present lower and upper bounds for g(n) and h(n) and partial proofs of the conjectures.