Systems and methods for dynamic pricing

1. A system comprising:

one or more processors; and

one or more non-transitory computer-readable media storing computing instructions configured to run on the one or more processors and cause the one or more processors to perform functions comprising:

estimating, at a demand estimation system, a demand for one or more items in an electronic basket of items using one or more different demand models;

training, at an elasticity estimation system and a price estimation system separate from the elasticity estimation system, a multi-armed bandit model by executing a first explore stage of a Thompson sampling algorithm operating on the elasticity estimation system, the first explore stage configured to make sub-optimal decisions and gather data by pulling one or more random arms of the multi-armed bandit model;

estimating, at the elasticity estimation system and the price estimation system separate from the elasticity estimation system, an elasticity for the one or more items in the electronic basket of items by executing a second exploit stage of the Thompson sampling algorithm operating on the elasticity estimation system, the

second exploit stage configured to determine a price using the demand and the elasticity by pulling an arm of the one or more random arms with a lowest regret, as determined during the first explore stage;

transmitting the price to a web server; and

facilitating displaying the price on a video screen of a device when the device queries the web server,

wherein estimating the elasticity for the one or more items in the electronic basket of items comprises:

placing a prior distribution over an elasticity parameter for the one or more items in the electronic basket of items;

placing a likelihood model over past revenue for the one or more items in the electronic basket of items; and

estimating the elasticity for the one or more items in the electronic basket of items using the prior distribution over the elasticity parameter and the likelihood model over the past revenue.

2. The system of claim 1 , wherein:

the one or more different demand models comprise multiple demand models;

estimating the demand for the one or more items in the electronic basket of items comprises:

calculating a respective weight for each demand model of the multiple demand models based on a respective accuracy of each demand model of the multiple demand models;

estimating a respective demand for each of the one or more items in the electronic basket of items using each demand model of the multiple demand models; and

combining the respective demand, as estimated by each demand model of the multiple demand models, into a combined estimated demand using the respective weight for each demand model of the multiple demand models.

3. The system of claim 1 , wherein the multi-armed bandit model further comprises an upper confidence bound algorithm.

4. The system of claim 1 , wherein:

the prior distribution over the elasticity parameter for the one or more items in the electronic basket of items comprises a Gaussian prior distribution; and

the likelihood model over the past revenue for the one or more items in the electronic basket of items comprises a Gaussian likelihood model.

5. The system of claim 4 , wherein the Gaussian prior distribution comprises:

Π 0 (γ ★ )=N(μ 0 ,Σ 0 ), wherein:

γ ★ comprises the elasticity parameter;

μ 0 comprises a mean of the prior distribution; and

Σ 0 comprises a covariance matrix of the prior distribution.

6. The system of claim 4 , wherein the Gaussian likelihood model comprises:

l(R t ; Rev t ,γ ★ )=N(R t ;Rev t ,σ 2 ), wherein:

R t comprises a revenue for the electronic basket of items on a day t;

Rev t comprises an average revenue of each item in the electronic basket of items;

γ ★ comprises the elasticity parameter; and

σ comprises a noise variance.

7. The system of claim 1 , wherein:

estimating the elasticity for the one or more items in the electronic basket of items further comprises:

rejecting the elasticity parameter when the elasticity parameter is greater than a predetermined threshold; and

determining the price comprises:

optimizing a concave function using convex constraints.

8. The system of claim 7 , wherein the concave function comprises:

p

t

=

arg

⁢

p

max

⁢

∑

i

∈

ℬ

⁢

p

i

2

⁢

f

i

⁢

,

t

⁢

γ

★

,

i

p

i

⁢

,

t

-

1

-

p

i

⁢

f

i

⁢

,

t

⁢

γ

★

,

i

+

p

i

⁢

f

i

⁢

,

t

,

wherein:

p comprises the price;

i comprises the one or more items;

comprises the electronic basket of items;

f i,t comprises a demand forecast for the one or more items on a day t; and

γ ★ comprises the elasticity parameter.

9. The system of claim 1 , wherein the one or more different demand models comprise:

d

i

,

t

⁡

(

p

i

)

=

f

i

,

t

⁡

(

p

i

p

i

,

t

-

1

)

γ

★

,

i

,

wherein:

d i,t comprises a demand for the one or more items i on a day t;

p i comprises a price of the one or more items;

f i,t comprises a demand forecast for the one or more items on a day t; and

γ ★ comprises the elasticity parameter.

10. A method being implemented via execution of computing instructions configured to run at one or more processors and configured to be stored at non-transitory computer-readable media, the method comprising:

estimating, at a demand estimation system, a demand for one or more items in an electronic basket of items using one or more different demand models;

training, at an elasticity estimation system and a price estimation system separate from the elasticity estimation system, a multi-armed bandit model by executing a first explore stage of a Thompson sampling algorithm operating on the elasticity estimation system, the first explore stage configured to make sub-optimal decisions and gather data by pulling one or more random arms of the multi-armed bandit model;

estimating, at the elasticity estimation system and the price estimation system separate from the elasticity estimation system, an elasticity for the one or more items in the electronic basket of items by executing a second exploit stage of the Thompson sampling algorithm operating on the elasticity estimation system, the

second exploit stage configured to determine a price using the demand and the elasticity by pulling an arm of the one or more random arms with a lowest regret, as determined during the first explore stage

transmitting the price to a web server; and

facilitating displaying the price on a video screen of a device when the device queries the web server,

wherein estimating the elasticity for the one or more items in the electronic basket of items comprises:

placing a prior distribution over an elasticity parameter for the one or more items in the electronic basket of items;

placing a likelihood model over past revenue for the one or more items in the electronic basket of items; and

estimating the elasticity for the one or more items in the electronic basket of items using the prior distribution over the elasticity parameter and the likelihood model over the past revenue.

11. The method of claim 10 , wherein:

the one or more different demand models comprise multiple demand models;

estimating the demand for the one or more items in the electronic basket of items comprises:

calculating a respective weight for each demand model of the multiple demand models based on a respective accuracy of each demand model of the multiple demand models;

estimating a respective demand for each of the one or more items in the electronic basket of items using each demand model of the multiple demand models; and

combining the respective demand, as estimated by each demand model of the multiple demand models, into a combined estimated demand using the respective weight for each demand model of the multiple demand models.

12. The method of claim 10 , wherein the multi-armed bandit model further comprises an upper confidence bound algorithm.

13. The method of claim 10 , wherein:

the prior distribution over the elasticity parameter for the one or more items in the electronic basket of items comprises a Gaussian prior distribution; and

the likelihood model over the past revenue for the one or more items in the electronic basket of items comprises a Gaussian likelihood model.

14. The method of claim 13 , wherein the Gaussian prior distribution comprises:

Π 0 (γ ★ )=N(μ 0 ,Σ 0 ), wherein:

γ ★ comprises the elasticity parameter;

μ 0 comprises a mean of the prior distribution; and

Σ 0 comprises a covariance matrix of the prior distribution.

15. The method of claim 13 , wherein the Gaussian likelihood model comprises:

l(R t ; Rev t ,γ ★ )=N(R t ;Rev t ,σ 2 ), wherein:

R t comprises a revenue for the electronic basket of items on a day t;

Rev t comprises an average revenue of each item in the electronic basket of items;

γ ★ comprises the elasticity parameter; and

σ comprises a noise variance.

16. The method of claim 10 , wherein:

estimating the elasticity for the one or more items in the electronic basket of items further comprises:

rejecting the elasticity parameter when the elasticity parameter is greater than a predetermined threshold; and

determining the price comprises:

optimizing a concave function using convex constraints.

17. The method of claim 16 , wherein the concave function comprises:

p

t

=

arg

⁢

p

max

⁢

∑

i

∈

ℬ

⁢

p

i

,

t

2

⁢

f

i

⁢

,

t

⁢

γ

★

,

i

p

i

⁢

,

t

-

1

-

p

i

⁢

f

i

⁢

,

t

⁢

γ

★

,

i

+

p

i

⁢

f

i

⁢

,

t

,

wherein:

p comprises the price;

i comprises the one or more items;

comprises the electronic basket of items;

f i,t comprises a demand forecast for the one or more items on a day t; and

γ ★ comprises the elasticity parameter.

18. The method of claim 10 , wherein the one or more different demand models comprise:

d

i

,

t

⁡

(

p

i

)

=

f

i

,

t

⁡

(

p

i

p

i

,

t

-

1

)

γ

★

,

i

,

wherein:

d i,t comprises a demand for the one or more items i on a day t;

p i comprises a price of the one or more items;

f i,t comprises a demand forecast for the one or more items on a day t; and

γ ★ comprises the elasticity parameter.

19. The system of claim 1 , wherein the multi-armed bandit model further comprises an ∈ greedy algorithm.

20. The method of claim 10 , wherein the multi-armed bandit model further comprises an ∈ greedy algorithm.