SIMPLE PRICING BY PRICE-DIFFERENCE REGULARIZATION
A method and system for determining an optimized price schedule that controls the simplicity of a price schedule while maintaining efficiency in terms of welfare. A welfare function and a simplicity metric representing a simplicity of price schedules are determined. A full objective, corresponding to a comparison of the welfare with respect to the simplicity of each price schedule is determined and an optimized price schedule, representing the schedule having the highest computed full objective, is output.
1 . A method for selecting an optimized price schedule, comprising:
providing a simplicity metric for computing a simplicity term based on weighted differences between prices in a price schedule;
providing a welfare function for computing a welfare term corresponding to the price schedule;
for a plurality of price schedules:
evaluating the simplicity term of each price schedule in accordance with the simplicity metric,
computing a welfare term for each price schedule in accordance with the welfare function, and,
determining a difference between the welfare term and the simplicity term for each price schedule; and
outputting an optimized price schedule selected from the plurality of price schedules, the selected one of the plurality of price schedules being selected in accordance with the determined respective differences.
2 . The method of claim 1 , further comprising:
performing a plurality of iterations, for each iteration:
determining a set of price differences between an initial price schedule and a target schedule which optimize a soft threshold function, the soft threshold function being based on the welfare function for the initial price schedule and simplicity metric for the initial price schedule, subject to a constraint on the set of price differences,
determining a new price schedule by optimizing a hard threshold function that optimizes the welfare function subject to the determined set of price differences, and
computing a value of the new price schedule as a function of the computed welfare term and simplicity term for the new price schedule; and
wherein outputting the optimized price schedule further includes selecting one of the new price schedules, the selected one of the new price schedules being selected based on the values computed over the plurality of the iterations.
3 . The method of claim 2 , wherein the iterations are continued while the computed value of the new price schedule improves or until a stopping criterion is reached.
4 . The method of claim 1 , wherein the price schedule comprises a set of matrices, each of the matrices mapping components of first and second dimensions to a price of an item.
5 . The method of claim 4 , wherein the set of price differences comprises a plurality of vectors of price differences (y j i ), each vector of price differences being computed for a respective one of a set of difference matrices (Y i ) iεA that are computed between the respective matrices of the given price schedule and the target price schedule.
6 . The method of claim 5 , wherein the simplicity metric is a function of the vectors of price differences (y j i ) and corresponding weighting coefficients (α i ) iεA for each of the difference matrices.
7 . The method of claim 6 , wherein the weighting coefficients (α i ) are variables which are a function of demand for the item.
8 . The method of claim 2 , wherein the applying of the soft threshold function further comprises applying a regularization penalty to the simplicity metric.
9 . The method of claim 2 , wherein in the computing the value of the new price schedule as a function of the computed welfare term and simplicity term the differences between the given price schedule and the target schedule are L 0 -regularization terms.
10 . The method of claim 2 , wherein the optimizing of the soft threshold function comprises performing a multi-stage convex relaxation.
11 . The method of claim 5 , wherein the simplicity metric S(p) is of the form:
S ( p )=Σ iεA α i ∥Y i p∥ 0 ;
wherein:
(Y i ) represents the ith difference matrix in the set of the set of difference matrices,
(α i ) represents a weighting coefficient for the respective difference matrix,
(p) represents the price schedule, and
∥Y i p∥ 0 is 0 if the difference vector Y i p is 0 and 1 otherwise.
12 . The method of claim 11 , wherein the computing the value of the new price schedule (J(p)) comprises computing a difference between the welfare function and the simplicity S(p).
13 . The method of claim 3 , wherein the computing of the value of the new price schedule as a function of the computed welfare term and simplicity term for the new price schedule is subject to at least one constraint.
14 . The method of claim 13 , wherein the at least one constraint comprises at least one of:
not allowing a specific price change; and
a maximum number of allowed price changes;
15 . The method of claim 4 , wherein in the simplicity function the weights favor simplicity of the matrices of the price schedule.
16 . The method of claim 1 , wherein the price schedule is a parking price schedule.
17 . The method of claim 4 , wherein the price of the item comprises the price of a parking place.
18 . The method of claim 17 , wherein the dimensions include at least one of the group consisting of arrival time, location, and duration of stay.
19 . The method of claim 1 , wherein the simplicity metric includes a complexity cost associated with a number times a price differs for adjacent hours, a complexity cost associated with a number times a price differs for adjacent half hours, complexity cost associated with a number of times a price differs for adjacent block faces, and a complexity cost associated with a number of times a rice differs for adjacent days.
20 . The method of claim 3 , wherein the welfare function is based on a deviation between the price schedule and the target price schedule.
21 . The method of claim 20 , wherein the welfare function is based on a squared deviation between prices in the price schedule (p) and prices in the target price schedule (y).
22 . A system for selecting an optimized price schedule comprising memory which stores instructions for performing the method of claim 1 and a processor in communication with the memory for executing the instructions.
23 . A computer program product comprising a non-transitory recording medium storing instructions, which when executed on a computer, causes the computer to perform the method of claim 1 .
24 . An optimized price schedule selection system, comprising:
a processor;
a simplicity metric component configured for computing a simplicity term based on weighted differences in a price schedule using a simplicity metric;
a welfare component configured for computing a welfare term corresponding to the price schedule using a welfare function, the welfare function representative of a quantification of a price schedule relative to a selected target; and
memory in communication with the processor, which stores instructions which are executed by the processor for:
evaluating the simplicity term of each price schedule of a plurality of price schedules in accordance with the simplicity metric;
computing a welfare term for each price schedule of the plurality of price schedules in accordance with the welfare function;
computing a full objective of each price schedule as a function of the computed welfare term and simplicity term; and
outputting an optimized price schedule corresponding to a price schedule having a highest computed full objective.
25 . A method for determining an optimized price schedule, comprising:
providing a simplicity metric for computing a simplicity term based on a price schedule;
providing a welfare function for computing a welfare term corresponding to the price schedule;
selectively optimizing at least one parameter associated with the simplicity metric;
maximizing the welfare term with respect to the simplicity metric having at least one selectively optimized parameter associated therewith; and
outputting a new price schedule associated with the maximized welfare term.
26 . The method of claim 23 , wherein selectively optimizing at least one parameter includes applying a multi-stage convex relaxation to the simplicity metric.
27 . The method of claim 23 , wherein the simplicity term comprises a set of L 0 -regularization terms.
28 . The method of claim 23 , wherein the welfare function is of a form corresponding to a deviation of the price schedule from a target price schedule.
29 . The method of claim 23 , wherein the price schedule is a parking price schedule.