Abstract
The advent of Internet broking pages allows customers to ‘apply’ to a number of different companies at one time, leading to multiple offers made to a customer. The saturated condition of the personal financial products has led to falling ‘take’ rates. Financial institutions are trying to increase the ‘take’ rates of their personal financial products. Applicants for credit will have to provide information for risk assessment, which can be used to assess the probability of a customer accepting an offer. Interactive channels such as the Internet and telephone allow questions that are asked to depend on previous answers. The questions selected need to provide information to assess the probability of acceptance of a particular variant of financial product. In this paper, we investigate a model to predict the best offer to extend next to a customer based on the response for the questions, as well as the question selection itself.
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Acknowledgements
We would like to make special mention of thanks to Fair Isaac Inc. for providing initial funding for this research.
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Appendix
Appendix
Using value iterations as we know that v m(r 1,n 1,r 2,n 2) converges to v(r 1,n 1,r 2,n 2) where v m(r 1,n 1,r 2,n 2) is defined by:
If response is ‘yes’ (y):
where
If the response is a ‘no’ (n):
where
with v 0(r 1,n 1,r 2,n 2)=0 which means trivially v 1(r 1,n 1,r 2,n 2)⩾v 0(r 1,n 1,r 2,n 2). We use this induction to prove v m(r 1,n 1,r 2,n 2)⩾v m−1(r 1,n 1,r 2,n 2).
Since max{a,b}−max{c,d}⩾min{a−c,b−d}, so
So v m+1(r 1,n 1,r 2,n 2) is a monotone increasing sequence that is bounded by
So the iterates of the value iteration will converge to a v(r 1,n 1,r 2,n 2).
We now use the iterates of the value iteration to prove
and
If the response is ‘no’, hence
If this condition holds for m, it will hold for m+1. We use this result to prove some general results in Lemma A.1.
Lemma A.1
-
v m+1(r 1,n 1,r 2,n 2) is
-
i)
non-decreasing in r i ,
-
ii)
non-increasing in n i .
The optimality equations can be rewritten as:
-
i)
If response is ‘yes’ (y):
where
If the response is a ‘no’ (n):
where
The definition for v m+1(r 1,n 1,r 2,n 2) is maintained as profit from an applicant before answering the question and is calculated as follows:
Next, we proceed with proving result (i):
Assume this is true for m−1 with v 0(r 1,n 1,r 2,n 2)=0.
We take the optimality equation from the ‘yes’ category. So,
Trivially, the same proof will show that v m(n,r 1+1,n 1,r 2,n 2)⩾v m(n,r 1,n 1,r 2,n 2) as the equations of v m(n,r 1,n 1,r 2,n 2) and v m(y,r 1,n 1,r 2,n 2) are almost identical except or the q i terms. Hence v m(r 1+1,n 1,r 2,n 2)⩾v m(r 1,n 1,r 2,n 2).
Also
Trivially, the same proof will show that v m(n,r 1,n 1,r 2+1,n 2)⩾v m(n,r 1,n 1,r 2,n 2).
Hence v m(r 1,n 1,r 2+1,n 2)⩾v m(r 1,n 1,r 2,n 2). Since result (i) holds for m, so it will hold for m+1. By induction, this holds then for all m.
We use the same technique to prove result (ii):
Assume that this is true for m−1 with v 0(r 1,n 1,r 2,n 2)=0.
We take the equations from the ‘yes’ response:
Trivially, this proof will show that v m(n,r 1,n 1,r 2,n 2)⩾v m(n,r 1,n 1+1,r 2,n 2).
Hence v m(r 1,n 1,r 2,n 2)⩾v m(r 1,n 1+1,r 2,n 2).
The same proof will also show that v m(n,r 1,n 1,r 2,n 2)⩾v m(n,r 1,n 1,r 2,n 2+1).
Hence, v m(r 1,n 1,r 2,n 2)⩾v m(r 1,n 1,r 2,n 2+1). Since result (i) holds for m, so it will hold for m+1. By induction, this holds then for all m.
Therefore, we have proven the monotonicity of the optimality equations for this case as well.
Theorem A.1
-
v(r 1,n 1,r 2,n 2) is
-
i)
non-decreasing in r i .
-
ii)
non-increasing in n i .
-
i)
Proof
-
Let m → ∞ as in Lemma A.1. □
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Seow, HV. Question selection responding to information on customers from heterogeneous populations to select offers that maximize expected profit. J Oper Res Soc 61, 443–454 (2010). https://doi.org/10.1057/jors.2009.6
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DOI: https://doi.org/10.1057/jors.2009.6