120 may be any additional device or apparatus commonly known and understood in the art without departing from the scope of the present application. For example, the computer device 120 may be the same or similar to the computer system 102. Furthermore, those skilled in the art similarly understand that the device may be any combination of devices and apparatuses.

Of course, those skilled in the art appreciate that the above-listed components of the computer system 102 are merely meant to be exemplary and are not intended to be exhaustive and/or inclusive. Furthermore, the examples of the components listed above are also meant to be exemplary and similarly are not meant to be exhaustive and/or inclusive.

In accordance with various embodiments of the present disclosure, the methods described herein may be implemented using a hardware computer system that executes software programs. Further, in an exemplary, non-limited embodiment, implementations can include distributed processing, component/object distributed processing, and parallel processing. Virtual computer system processing can be constructed to implement one or more of the methods or functionalities as described herein, and a processor described herein may be used to support a virtual processing environment.

As described herein, various embodiments provide optimized methods and systems for allocating limited resources to agents that reveal their stochastic demands on arrival over a finite horizon in a proportionally fair manner that exhausts available resource budgets.

Referring to FIG. 2, a schematic of an exemplary network environment 200 for implementing a method for allocating limited resources to agents that reveal their stochastic demands on arrival over a finite horizon in a proportionally fair manner that exhausts available resource budgets is illustrated. In an exemplary embodiment, the method is executable on any networked computer platform, such as, for example, a personal computer (PC).

The method for allocating limited resources to agents that reveal their stochastic demands on arrival over a finite horizon in a proportionally fair manner that exhausts available resource budgets may be implemented by a Sequential Resource Allocation (SRA) device 202. The SRA device 202 may be the same or similar to the computer system 102 as described with respect to FIG. 1. The SRA device 202 may store one or more applications that can include executable instructions that, when executed by the SRA device 202, cause the SRA device 202 to perform actions, such as to transmit, receive, or otherwise process network messages, for example, and to perform other actions described and illustrated below with reference to the figures. The application(s) may be implemented as modules or components of other applications. Further, the application(s) can be implemented as operating system extensions, modules, plugins, or the like.

Even further, the application(s) may be operative in a cloud-based computing environment. The application(s) may be executed within or as virtual machine(s) or virtual server(s) that may be managed in a cloud-based computing environment. Also, the application(s), and even the SRA device 202 itself, may be located in virtual server(s) running in a cloud-based computing environment rather than being tied to one or more specific physical network computing devices. Also, the application(s) may be running in one or more virtual machines (VMs) executing on the SRA device 202. Additionally, in one or more embodiments of this technology, virtual machine(s) running on the SRA device 202 may be managed or supervised by a hypervisor.

In the network environment 200 of FIG. 2, the SRA device 202 is coupled to a plurality of server devices 204(1)-204(n) that hosts a plurality of databases 206(1)-206(n), and also to a plurality of client devices 208(1)-208(n) via communication network(s) 210. A communication interface of the SRA device 202, such as the network interface 114 of the computer system 102 of FIG. 1, operatively couples and communicates between the SRA device 202, the server devices 204(1)-204(n), and/or the client devices 208(1)-208(n), which are all coupled together by the communication network(s) 210, although other types and/or numbers of communication networks or systems with other types and/or numbers of connections and/or configurations to other devices and/or elements may also be used.

The communication network(s) 210 may be the same or similar to the network 122 as described with respect to FIG. 1, although the SRA device 202, the server devices 204(1)-204(n), and/or the client devices 208(1)-208(n) may be coupled together via other topologies. Additionally, the network environment 200 may include other network devices such as one or more routers and/or switches, for example, which are well known in the art and thus will not be described herein. This technology provides a number of advantages including methods, non-transitory computer readable media, and SRA devices that efficiently implement a method for allocating limited resources to agents that reveal their stochastic demands on arrival over a finite horizon in a proportionally fair manner that exhausts available resource budgets.

By way of example only, the communication network(s) 210 may include local area network(s) (LAN(s)) or wide area network(s) (WAN(s)), and can use TCP/IP over Ethernet and industry-standard protocols, although other types and/or numbers of protocols and/or communication networks may be used. The communication network(s) 210 in this example may employ any suitable interface mechanisms and network communication technologies including, for example, teletraffic in any suitable form (e.g., voice, modem, and the like), Public Switched Telephone Network (PSTNs), Ethernet-based Packet Data Networks (PDNs), combinations thereof, and the like.

The SRA device 202 may be a standalone device or integrated with one or more other devices or apparatuses, such as one or more of the server devices 204(1)-204(n), for example. In one particular example, the SRA device 202 may include or be hosted by one of the server devices 204(1)-204(n), and other arrangements are also possible. Moreover, one or more of the devices of the SRA device 202 may be in a same or a different communication network including one or more public, private, or cloud networks, for example.

The plurality of server devices 204(1)-204(n) may be the same or similar to the computer system 102 or the computer device 120 as described with respect to FIG. 1, including any features or combination of features described with respect thereto. For example, any of the server devices 204(1)-204(n) may include, among other features, one or more processors, a memory, and a communication interface, which are coupled together by a bus or other communication link, although other numbers and/or types of network devices may be used. The server devices 204(1)-204(n) in this example may process requests received from the SRA device 202 via the communication network(s) 210 according to the HTTP-based and/or JavaScript Object Notation (JSON) protocol, for example, although other protocols may also be used.

The server devices 204(1)-204(n) may be hardware or software or may represent a system with multiple servers in a pool, which may include internal or external networks. The server devices 204(1)-204(n) hosts the databases 206(1)-206(n) that are configured to store information that relates to resource availability and information that relates to metrics for resource allocation fairness.

Although the server devices 204(1)-204(n) are illustrated as single devices, one or more actions of each of the server devices 204(1)-204(n) may be distributed across one or more distinct network computing devices that together comprise one or more of the server devices 204(1)-204(n). Moreover, the server devices 204(1)-204(n) are not limited to a particular configuration. Thus, the server devices 204(1)-204(n) may contain a plurality of network computing devices that operate using a master/slave approach, whereby one of the network computing devices of the server devices 204(1)-204(n) operates to manage and/or otherwise coordinate operations of the other network computing devices.

The server devices 204(1)-204(n) may operate as a plurality of network computing devices within a cluster architecture, a peer-to peer architecture, virtual machines, or within a cloud architecture, for example. Thus, the technology disclosed herein is not to be construed as being limited to a single environment and other configurations and architectures are also envisaged.

The plurality of client devices 208(1)-208(n) may also be the same or similar to the computer system 102 or the computer device 120 as described with respect to FIG. 1, including any features or combination of features described with respect thereto. For example, the client devices 208(1)-208(n) in this example may include any type of computing device that can interact with the SRA device 202 via communication network(s) 210. Accordingly, the client devices 208(1)-208(n) may be mobile computing devices, desktop computing devices, laptop computing devices, tablet computing devices, virtual machines (including cloud-based computers), or the like, that host chat, e-mail, or voice-to-text applications, for example. In an exemplary embodiment, at least one client device 208 is a wireless mobile communication device, i.e., a smart phone.

The client devices 208(1)-208(n) may run interface applications, such as standard web browsers or standalone client applications, which may provide an interface to communicate with the SRA device 202 via the communication network(s) 210 in order to communicate user requests and information. The client devices 208(1)-208(n) may further include, among other features, a display device, such as a display screen or touchscreen, and/or an input device, such as a keyboard, for example.

Although the exemplary network environment 200 with the SRA device 202, the server devices 204(1)-204(n), the client devices 208(1)-208(n), and the communication network(s) 210 are described and illustrated herein, other types and/or numbers of systems, devices, components, and/or elements in other topologies may be used. It is to be understood that the systems of the examples described herein are for exemplary purposes, as many variations of the specific hardware and software used to implement the examples are possible, as will be appreciated by those skilled in the relevant art(s).

One or more of the devices depicted in the network environment 200, such as the SRA device 202, the server devices 204(1)-204(n), or the client devices 208(1)-208(n), for example, may be configured to operate as virtual instances on the same physical machine. In other words, one or more of the SRA device 202, the server devices 204(1)-204(n), or the client devices 208(1)-208(n) may operate on the same physical device rather than as separate devices communicating through communication network(s) 210. Additionally, there may be more or fewer SRA devices 202, server devices 204(1)-204(n), or client devices 208(1)-208(n) than illustrated in FIG. 2.

In addition, two or more computing systems or devices may be substituted for any one of the systems or devices in any example. Accordingly, principles and advantages of distributed processing, such as redundancy and replication also may be implemented, as desired, to increase the robustness and performance of the devices and systems of the examples. The examples may also be implemented on computer system(s) that extend across any suitable network using any suitable interface mechanisms and traffic technologies, including by way of example only teletraffic in any suitable form (e.g., voice and modem), wireless traffic networks, cellular traffic networks, Packet Data Networks (PDNs), the Internet, intranets, and combinations thereof.

The SRA device 202 is described and illustrated in FIG. 3 as including a sequential resource allocation module 302, although it may include other rules, policies, modules, databases, or applications, for example. As will be described below, the sequential resource allocation module 302 is configured to implement a method for allocating limited resources to agents that reveal their stochastic demands on arrival over a finite horizon in a proportionally fair manner that exhausts available resource budgets.

An exemplary process 300 for implementing a mechanism for allocating limited resources to agents that reveal their stochastic demands on arrival over a finite horizon in a proportionally fair manner that exhausts available resource budgets by utilizing the network environment of FIG. 2 is illustrated as being executed in FIG. 3. Specifically, a first client device 208(1) and a second client device 208(2) are illustrated as being in communication with SRA device 202. In this regard, the first client device 208(1) and the second client device 208(2) may be “clients” of the SRA device 202 and are described herein as such. Nevertheless, it is to be known and understood that the first client device 208(1) and/or the second client device 208(2) need not necessarily be “clients” of the SRA device 202, or any entity described in association therewith herein. Any additional or alternative relationship may exist between either or both of the first client device 208(1) and the second client device 208(2) and the SRA device 202, or no relationship may exist.

Further, SRA device 202 is illustrated as being able to access a resource availability data repository 206(1) and a resource allocation fairness metrics database 206(2). The sequential resource allocation module 302 may be configured to access these databases for implementing a method for allocating limited resources to agents that reveal their stochastic demands on arrival over a finite horizon in a proportionally fair manner that exhausts available resource budgets.

The first client device 208(1) may be, for example, a smart phone. Of course, the first client device 208(1) may be any additional device described herein. The second client device 208(2) may be, for example, a personal computer (PC). Of course, the second client device 208(2) may also be any additional device described herein.

The process may be executed via the communication network(s) 210, which may comprise plural networks as described above. For example, in an exemplary embodiment, either or both of the first client device 208(1) and the second client device 208(2) may communicate with the SRA device 202 via broadband or cellular communication. Of course, these embodiments are merely exemplary and are not limiting or exhaustive.

Upon being started, the sequential resource allocation module 302 executes a process for allocating limited resources to agents that reveal their stochastic demands on arrival over a finite horizon in a proportionally fair manner that exhausts available resource budgets. An exemplary process for allocating limited resources to agents that reveal their stochastic demands on arrival over a finite horizon in a proportionally fair manner that exhausts available resource budgets is generally indicated at flowchart 400 in FIG. 4.

In process 400 of FIG. 4, at step S402, the sequential resource allocation module 302 receives a request from an entity for a resource for which a predetermined maximum amount of the resource is available for allocation. In an exemplary embodiment, the resource is a limited resource, and there may be an expectation for additional requests for the resource that may be made in a sequential manner within a predetermined time interval, and an irrevocable decision about an allocation of the resource must be made, without knowledge of either the number of additional requests or the amounts of resource to be requested. In an exemplary embodiment, the resource may be any one or more of an economic resource, a power systems resource, a cloud computing resource, or any other suitable type of resource.

At step S404, the sequential resource allocation module 302 estimates a number of future requests and amounts of the source to be requested. Then, at step S406, the sequential resource allocation module 302 calculates a standard deviation of an uncertainty of at least one future request for the resource, and then uses the calculated standard deviation to adjust an estimate of a least one of the amounts. In this aspect, there is a recognition that there may be a tendency to initially reserve too much of the resource for future requests, which then tends to lead to a circumstance in which the allocation of the resource is not optimized, and therefore, the adjusted amount is usually greater than the initially estimated amount.

At step S408, the sequential resource allocation module 302 determines a first amount of the source to be allocated to the requesting entity. In an exemplary embodiment, the determination of the first amount of the resource may be made by applying an algorithm with respect to the predetermined maximum available amount of the resource and a result of the estimates generated in step S404 and as adjusted in step S406. In an exemplary embodiment, the algorithm may be designed to optimize a Nash Social Welfare (NSW) objective that is generalized for sequential settings.

At step S410, when an actual allocation of the resource has been made to the first entity and the predetermined time interval has not elapsed and there is at least one unfulfilled future request, the sequential resource allocation module 302 calculates an updated value of the maximum amount of the resource that is available for allocation based on the actual allocation of the resource that has been made to the first entity, and then repeats the estimating of step S404 and the adjusting of step S406 based on the updated value of the maximum amount of the resource that is available for allocation in order to determine a third amount of the resource to be allocated to a second entity that is different than the first entity.

At step S412, after the predetermined time interval has elapsed and thus, all subsequent future requests have been received, the sequential resource allocation module 302 uses the knowledge of those subsequent requests and the corresponding amounts of the resource that have been requested to determine an optimal hindsight allocation of the resource with respect to the entity from which the original request was received in step S402. Then, at step S414, the sequential resource allocation module 302 uses the optimal hindsight allocation to determine a fairness metric that corresponds to the entity from which that original request was received.

Problem formulation: In an exemplary embodiment, consideration is given to a supplier that has a divisible resource with limited budget size B, and N agents that arrive sequentially over T time steps requesting various numbers of units of the resource, noting that the numbers of units may include fractional numbers. At time tε{1, . . . , T}, agent iε{1, . . . , N} arrives and reveals its demand

X
i
t

∈
ℝ

sampled from distribution

P

X
i
T

⁢

❘
"[LeftBracketingBar]"

X
i
1

,
…

,

X
i

t
-
1

.

It is assumed that each agent has at least one demand over the horizon T. The supplier observes demands

X
t

=

(

X
1
t

,
…

,

X
N
t

)

,

and makes an irrevocable allocation

A
t

=

(

A
1
t

,
 
…
,
 

A
N
t

)

,

where
⁢

A
i
t

∈

ℝ
≥
0

denotes the amount of resources allocated to agent i. It is further assumed that the allocations are immediately fulfilled by the agents and removed from the inventory. Agent i has a utility function u(A_i,X_i), representing its satisfaction with the allocation

A
i

=

(

A
i
1

,
…

,

A
i
T

)

∈

ℝ

≥
0

T

given its latent demands. The utility function is a non-decreasing non-negative concave function. The following utility function is considered, where an agent's utility linearly increases with its total allocation up to its total request:

u
⁡
(

A
i

,

X
i

)

=

∑

t
=
1

T

min
⁢

{

A
i
t

,

X
i
t

}

(

Expression
⁢

1

)

A client with the utility function in Expression 1 only values an allocation in the time window a resource was requested, and not in earlier or later windows, which is suitable in settings where the allocation process is time-sensitive and the supplier is not able to delay the decision-making outside of the current window. If all agent demands X₁, . . . , X_Nare known to the supplier at the time of decision-making, they can be used for determining the allocations. This setting is referred to herein as an offline setting. However, in the online or sequential setting, the latent types of agents are gradually revealed over time, such that the supplier only learns about

X
1
t

,
…

,

X
N
t

at time t.

Notation: (x)⁺=max {x,0}. For vectors X and Y, X≥Y denotes that X_i≥Y_ifor each element i. 0 denotes a zero vector. N(μ, σ²) denotes the normal distribution with mean u and variance σ².

Fairness in Allocation Problems: The supplier aims to allocate the resource efficiently, i.e., with minimal leftover at the end of horizon T, and in a fair manner satisfying welfarist notions of fairness. In an exemplary embodiment, the Nash Social Welfare (NSW) objective may be used for the allocation fairness purpose, due to it satisfying Varian Fairness properties, such as Envy-Freeness, Pareto-Efficiency, and Proportionality. The NSW is defined as

∏

i
=
1

N

⁢

u
⁡
(

A
i

,

X
i

)

w
i

where w_i∈ [Image Omitted] reflects the weight assigned to agent i by the supplier. The NSW is a balanced compromise between Utilitarian Welfare objectives, which maximize the utility of a larger number of agents, and Egalitarian Welfare objectives, which maximize the utility of the worst-off agent. Since

arg

max

A
i

∏

i
=
1

N

u
⁡
(

A
i

,

X
i

)

w
i

=

arg

max

A
i

∑

i
=
1

N

w
i

⁢
log
⁢

u
⁡
(

A
i

,

X
i

)

(

Expression
⁢

2

)

is true, the logarithm of NSW is often used as the objective function in allocation problems. This may be referred to as the log-NSW objective.

In the sequential setting, it is not guaranteed to have allocations that are simultaneously pareto-efficient, envy-free, and proportional. In an exemplary embodiment, one objective in the sequential setting is to find allocations that maximize the expected log-NSW in hindsight, i.e., at the end of the horizon T when all demands X₁, . . . , X_Nare known. Specifically,

arg

max

A
i

∑

i
=
1

N

w
i

⁢

𝔼

X
i

[

log
⁢

u
⁡
(

A
i

,

X
i

)

]

.

(

Expression
⁢

3

)

Fairness Metrics: Let A_i^onlinedenote the allocation vector of agent i given by an online algorithm subject to latent demands, and A_i^hindsightdenote the allocation vector in hindsight after all demands are revealed. The expected difference between the overall allocations for the agents can be used to measure the fairness of the online algorithm. Let

Δ
⁢

A
i

:=

❘
"[LeftBracketingBar]"

∑

t
=
1

T

⁢

A
i

t
,
online

-

∑

t
=
1

T

⁢

A
i

t
,
hindsight

❘
"[RightBracketingBar]"

,

then

Δ
⁢

A
max

=

𝔼
[

max
i

Δ
⁢

A
i

]

,

Δ
⁢

A
mean

=

1
N

⁢

∑

i
=
1

N

𝔼
[

Δ
⁢

A
i

]

.

(

Expression
⁢

4

)

These are respectively concerned with the worst-off agent and the average agent in terms of cumulative allocations in hindsight.

Markov Decision Process Formulation: Determining the optimal allocations under the NSW objective is a sequential decision-making problem as the allocation in one step affects the budget and allocations in future steps. In an exemplary embodiment, the problem may be formulated as a finite-horizon total-reward Markov Decision Process (MDP) modeled as a tuple <{ [Image Omitted] , , P_t, R_t}_{t=1 . . . , T}> denotes the underlying time-dependent state space, is the action space, P_t:××→ describes the state transition dynamics conditioned upon the previous state and action, R_t:×→ is a non-negative reward function, and T is the horizon over which the resource is allocated. The goal of the supplier is to find an allocation policy π_t: [Image Omitted] → mapping the state to an action, in order to maximize the expected sum of rewards

𝔼
[

∑

t
=
1

T

⁢

R
t

(

s
t

,

π
t

(

s
t

)

)

]

.

State Space: The state space [Image Omitted] is time-dependent, and the state size increases with time step t since the state captures the past demand and allocation information. Specifically, the state at step t is defined as s_t=(X^1:t, A^1:t−1, B^t)

- where X^1:t:=(X¹, . . . , X^t), A^1:t=(A¹. . . , A^t), and

B
t

=

{

B

t
-
1

-

∑

i
=
1

N

⁢

A
i

t
-
1

t
≥
1

B

t
=
1

.

(

Expression
⁢

5

)

Action Space: The action space is state-dependent and time-dependent. For any s_t=(X^1:t, A^1:t−1, B^t) ∈ [Image Omitted] , the following expression is true:

𝒜
t

=

{

A
t

∈

ℝ
N

:

∑

i
=
1

N

A
i
t

≤

B
t

}

.

(

Expression
⁢

6

)

The state and action space are both continuous and therefore infinitely large. However, [Image Omitted] is a compact polytope for any s_t∈, and is compact if the requests are bounded.

State Transition Function: Give state s∈ [Image Omitted] and action a∈ the system transitions to the next state s′∈S_t+1with probability P(s,a,s′)=Prob(s_t+1=s′|s_t=s,a_t=a), (Expression 7), where

X
i

t
+
1

~

P

X
i

t
+
1

|

X
i
1

,
…
,

X
i
t

.

Reward Function: The reward at time step t∈{1, . . . , T} is defined as follows:

R
t

(

s
t

,

π
t

(

s
t

)

)

=

∑

i
=
1

N

{

X
i
t

>
0

}

.

w
i

(

U
i
t

-

U
i

t
-
1

)

,

(

Expression
⁢

8

)

where

U
i
t

=

log
⁡
(

∑

τ
=
1

t

min
⁢

{

A
i
τ

,

X
i
τ

}

+
ϵ

)

,

t
∈

{

1
,
…

,
T

}

.

(

Expression
⁢

9

)

U
i
0

=
0

,

and ε is a very small value added for numerical stability. The indicator function

{

X
~

i
t

>
0

}

ensures that only agents having a demand at time t are accounted for. Then, the expected sum of rewards over the entire horizon T is equivalent to the expected log-NSW objective defined in Expression 3 as ε→0.

At time t, with state s_t∈ [Image Omitted] and action (allocation) A^t∈, the optimal Q-values satisfy the following Bellman optimality equation

Q
t

(

s
t

,

A
t

)

=

R
t

(

s
t

,

A
t

)

+

𝔼
[

max

A

t
+
1

∈

𝒜

t
+
1

Q

t
+
1

(

s

t
+
1

,

A

t
+
1

)

]

.

(

Expression
⁢

10

)

The optimal policy corresponding to Expression 10 is denoted by

π
t
★

.

Then, the optimal allocation is the solution to

arg

max

A
t

Q
t

(

s
t

,

A
t

)

.

(

Expression
⁢

11

)

Existence of an Optimal Policy: The state space of the MDP is continuous, the action space is continuous and state-dependent, and the reward is bounded and continuous due to the constant & in Expression 9. Further, it is known that a stationary optimal policy

π
★

=

(

π
1
★

,
…

,

π
T
★

)

exists.

If the action space is restricted to allocations that satisfy

A
i
t

≤

X
i
t

for all i and t, then per Expression 8, the reward at step t becomes

R
t

(

s
t

,

A
t

)

=

∑

i
=
1

N

{

X
i
t

>
0

}

.

w
i

(

log
⁡
(

A
~

i

t
-
1

+

A
i
t

+
ϵ

)

-

log
⁡
(

A
~

i

t
-
1

+
ϵ

)

)

,

(

Expression
⁢

12

)

where

A
~

i
t

:=

∑

τ
=
1

t

⁢

A
i
τ

denotes the cumulative allocated resources to agent i until time t. At time step T,

π
T
★

maximizes Q_T(s_T,A^T)=R_T(s_T,A^T) which does not depend on future allocations. The optimal allocation A^T* can be directly computed from Q_Tsince there is no uncertainty about future demands. Then, the optimal policy for all t can be derived by recursively solving Expression 11 backward in time. This quickly becomes computationally intractable, which motivates an effort to find alternative solutions that trade off sub-optimality for computational efficiency.

Offline Setting-Allocations in Hindsight: If the supplier could postpone the decision-making until the end of the horizon T, it would have perfect knowledge of all demands X¹, . . . , X^T. Let

X
~

i

=

∑

τ
=
1

T

⁢

X
i
τ

denote the total demands of agent i. Then, the supplier solves the following convex program, referred to herein as the Eisenberg-Gale program, in order to maximize the NSW objective in Expression 3 for allocations Ã=(Ã₁, . . . , Ã_N).

max

A
~

≥
0

∑

i
=
1

N

w
i

⁢

log
⁡
(

u
⁡
(

A
~

i

,

X
~

i

)

)

⁢

s
.
t
.

∑

i
=
1

N

A
~

i

≤

B
.

(

Expression
⁢

13

)

With the utility function defined in Expression 1, allocating resources beyond the client's request does not increase the utility. Therefore, solving Expression 13 is equivalent to the following:

max

A
~

∑

i
=
1

N

w
i

⁢

log
⁡
(

A
~

i

)

⁢

s
.
t
.

0

≤

A
~

i

≤

X
~

i

,

i
=
1

,
…

,
N

⁢

∑

i
=
1

N

A
~

i

≤

B
.

(

Expression
⁢

14

)

Then, any distribution of agent i's allocation Ã_iacross the T time steps that satisfies the demand constraint at each step, would be an optimal allocation in hindsight A_i^hindsight, i.e.,

A
i
hindsight

∈

{

(

A
i
1

,
…

,

A
i
T

)

:

0

≤

A
i
t

≤

X
i
t

,

∀
t

,

∑

τ
=
1

T

A
i
t

=

A
~

i

}

.

(

Expression
⁢

15

)

FIG. 5 is an algorithm 500 to be applied for providing an optimal solution to a water-filling problem where an objective is to allocate limited resources to agents with differently weighted importance and the allocation is performed after the requests are observed, in a proportionally fair manner that exhausts available resource budgets, according to an exemplary embodiment.

The optimal solution to Expression 14 takes the form

A
~

i
★

=

min
⁢

{

X
~

i

,
μ

}

,

where μ is a function of budget B and demands {tilde over (X)} such that

∑

i
=
1

N

⁢

A
~

i

=

B
.

Referring to FIG. 5, the solution can be efficiently derived by the water-filling algorithm 500 given as Algorithm 1, and the threshold u can be interpreted as a water level. In line 3 of Algorithm 1, the agents are ordered according to their demands and weights, such that agents with smaller demands and higher weights are prioritized. This ordering determines which agent is used to compute the water level μ. For each selected agent i_j, the condition in line 7 determines whether there is enough budget to fully satisfy agent i_j's demand. If there is enough budget, agent i_jreceives its full request, as shown at line 11, and the supplier moves on to the next agent in order. Otherwise, the available budget is proportionally divided among the remaining agents, as shown at line 8, while ensuring

∑

k
=
j

N

⁢

A
~

i
k

=

B
.

Sequential Setting—Heuristic Algorithm: In an exemplary embodiment, an intuitive online algorithm referred to herein as Sequential Algorithms with Fairness based on Future Estimates (SAFFE) is provided. The SAFFE algorithm uses the hindsight solution corresponding to the offline setting described above. In an exemplary embodiment, by substituting unobserved future requests with their expectations derived from demand distributions estimated from historical data, the sequential decision-making problem may be converted into solving the offline problem in each time step. To this end, at each time step, an estimate of the expected future demands is used to determine the total resources expected to be allocated to each agent by time T. This allows for a reservation of a portion of the available budget for future demands. Specifically, at t=1, . . . , T, the following problem is solved:

max

C
t

∑

i
=
1

N

{

Y
i
t

>
0

}

.

w
i

⁢

log
⁡
(

A
~

i
t

+

C
i
t

)

⁢

s
.
t
.

0

≤

C
i
t

≤

Y
i
r

,

∀
i

,

∑

i
=
1

N

C
i
t

≤

B
t

,

(

Expression
⁢

16

)

where

Y
i
t

=

X
i
t

+

𝔼
[

∑

τ
=

t
+
1

T

X
i
τ

]

,

i
=
1

,
…

,

N
.

(

Expression
⁢

17

)

The indicator function

{

Y
i
t

>
0

}

ensures that consideration is not given to absent agents, i.e., those with no current or expected future demands.

C
i
t

denotes the total allocation for agent i over the period t, . . . . T if it were possible to perfectly estimate future demands and if the arrived as their expectation. In other words,

C
i
t

consists if the allocation in the current time step and the reserved allocations for the future. Then, the current allocation

A
i
t

is a fraction of

C
i
t

derived as

A
i
t

=

C
i
t

⁢

X
i
t

Y
i
t

.

(

Expression
⁢

18

)

FIG. 6 is an algorithm 600 to be applied for providing sequential allocations with fairness based on future estimates in a method for allocating limited resources to agents that reveal their stochastic demands on arrival over a finite horizon in a proportionally fair manner that exhausts available resource budgets, according to an exemplary embodiment. Referring to FIG. 6, similar to the hindsight problem as described above and expressed as Expression 14, it is possible to efficiently solve Expression 16 by using a variant 600 of the water-filling algorithm given in Algorithm 2. At time step t, the water level μ_Tis calculated by accounting for all previous allocations to the agents and their total expected future demands.

SAFFE—Discounted Algorithm: While SAFFE is simple and easy to interpret, it is sub-optimal. In an exemplary embodiment, it may be shown that SAFFE maximizes an upper bound of the optimal Q-values of the MDP as described above. This implies that SAFFE over-estimates the expected reward gain from the future and reserves the budget excessively for future demands. Thus, in an exemplary embodiment, in order to correct this over-reservation, an algorithm referred to herein as SAFFE-Discounted (SAFFE-D) is provided, which penalizes uncertain future requests by their standard deviations. At every step t, the SAFFE-D algorithm computes

Y
i
t

=

X
i
t

+

∑

τ
=

t
+
1

T

(

𝔼
[

X
i
τ

]

-

λ
⁢

std
⁡
(

X
i
τ

)

)

+

(

Expression
⁢

19

)

for some regularization parameter λ≥0 and solves Expression 16. As in SAFFE, the current allocation

A
i
t

is split proportionally from

C
i
t

according to Expression 18. The regularizer λ is a hyper-parameter that can be tuned using historical data to provide the optimal trade-off between consuming too much of the budget at the current step and reserving too much for the future. In this aspect, the uncertainty in future demands decreases as the horizon T is approached, and better performance is expected with a decreasing time-dependent function such as λ(t)=√{square root over (T−t)}λ for some λ>0. Alternatively, the λ(t) can be learned as a function of time.

In an exemplary embodiment, an objective is to design a sequential allocation algorithm that is close to being optimal in hindsight, maximizing the expected NSW objective as provided in Expression 3. In order to determine how sub-optimal SAFFE-D is, an upper bound of the performs gap between SAFFE-D and the hindsight solution is derived in terms of ΔA^maxas shown in Expression 4.

Let A^oracledenote the allocations derived using SAFFE-D in a setting where an oracle has perfect knowledge of the sequence of incoming demands, i.e., when SAFFE-D is used with no stochasticity. This may be referred to has SAFFE-Oracle. The upper bound for ΔA^max, i.e., the distance between A^SAFFE-Dand A^hindsight, may be computed in two steps: by bounding the distance between SAFFE-D and SAFFE-Oracle, and then by bounding the distance between SAFFE-Oracle and hindsight.

SAFFE—Discounted versus SAFFE-Oracle: Under mild assumptions on distribution of requests, the distance between solutions of SAFFE-D and SAFFE-Oracle may be quantified via the use of concentration inequalities for deviation of future requests from their expected values. The following definitions are provided:

Y
_

i
t

=

X
i
t

+

𝔼
[

∑

τ
=

t
+
1

T

X
i
τ

]

+

T
-
t

ξ

⁢

std
⁡
(

X
i
τ

)

(

Expression
⁢

20

)

and

Y
_

i
t

=

X
i
t

+

𝔼
[

∑

τ
=

t
+
1

T

X
i
τ

]

-

T
-
t

ξ

⁢

std
⁡
(

X
i
τ

)

.

(

Expression
⁢

21

)

For simplicity, it is assumed that agent i's requests are independent and identically distributed. Then, for ξ>0, with probability at least 1−ξ, Chebyshev's inequality yields

Y
_

i
t

≤

X
i
t

+

∑

τ
=

t
+
1

T

X
i
τ

≤

Y
_

i
t

.

(

Expression
⁢

22

)

It is further assumed that all agents have equal standard deviations

std
⁢

(

X
i
t

)

,

but expectations may differ. The worst case scenario bound is provided first; in this circumstance, highly unbalanced future requests for different agents are allowed, i.e., there is an agent k such that

Y
_

k
t

≥

Y
_

j
t

for all other agents j≠k.

Theorem 1—Unbalanced Requests Bound: Let

A
i

t
,

SAFFE
-
D

⁢

and
⁢

A
i

t
,
oracle

denote allocations by SAFFE-D for

λ
⁡
(
t
)

=

T
-
t

ξ

and SAFFE-Oracle, respectively. Then, for all agents i,

❘
"[LeftBracketingBar]"

A
i

t
,

SAFFE
-
D

-

A
i

t
,

oracle

❘
"[RightBracketingBar]"

≤

{

2
⁢
N
⁢

(

T
-
t

)

ξ

⁢

std
⁡
(

X
i
t

)

if
⁢

B
t

≤

∑

i
=
1

N

Y
_

i
t

,

4
⁢

(

T
-
t

)

ξ

⁢

std
⁡
(

X
i
t

)

if
⁢

B
t

≥

∑

i
=
1

N

Y
_

i
t

(

Expression
⁢

23

)

with probability at least 1−ξ.

Proof: Intuitively, the discrepancy sales with the number of agents, because if all j≠k submit requests according to their upper bound

Y
_

j
t

,

then water level moves so that their SAFFE-Oracle allocations increase compared to SAFFE-Oracle with

Y
_

j
t

,

and this happens on the account of agent j who now gets less. Finally, using Expression 22 completes the proof, as the discrepancy can be translated between SAFFE-Oracle with

Y
_

i
t

⁢

and
⁢

Y
_

i
t

to that between SAFFE-D and SAFFE-Oracle with

Y
i
t

.

In order to move beyond the worst case scenario of highly unbalanced requests distributions, it is now assumed that

Y
_

i
t

⁢

and
⁢

Y
_

i
t

are same for all agents, i.e., if their requests have same distribution, then the discrepancy between allocation scales better.

Theorem 2—Balanced Requests Bound: If for any two clients i and j the same bounds exist as in Expression 22, i.e.,

Y
_

i
t

=

Y
_

j
t

⁢

and
⁢

Y
_

i
t

=

Y
_

j
t

,

then for all agents i,

❘
"[LeftBracketingBar]"

A
i

t
,

SAFFE
-
D

-

A
i

t
,

oracle

❘
"[RightBracketingBar]"

≤

{

0

if
⁢

B
t

≤

∑

i
=
1

N

Y
_

i
t

4
⁢

(

T
-
t

)

ξ

⁢

std
⁡
(

X
i
t

)

if
⁢

B
t

≥

∑

i
=
1

N

Y
_

i
t

(

Expression
⁢

24

)

with probability at least 1−ξ. The proof is similar to that provided above for Theorem 1.

In an exemplary embodiment, the next step is to show that SAFFE-Oracle achieves optimal allocation in hindsight. Theorem 3: For all i, the following holds:

∑

t
=
1

T

A
i

t
,

oracle

=

A
~

i

,

(

Expression
⁢

25

)

where Ã_iis the solution to the Eisenberg-Gale program per Expression 13. The proof relies on two observations. Firstly, for t=1, the sum of allocation for the current step and the reserved allocations for the future

C
i
1

,

equals the solution Ã_iof the Eisenberg-Gale problem per Expression 13. Secondly, SAFFE-Oracle fully distributes the total reserved allocation for the future by the end of the horizon T.

Theorem 4—Gap between SAFFE-D and Hindsight: In the unbalanced requests setting for the cap between SAFFE-D and hindsight measured by ΔA^maxintroduced in Expression 4, the following holds:

Δ
⁢

A
max

≤

N
⁢

T

3
/
2

ξ

⁢

std
⁡
(

X
i
t

)

.

(

Expression
⁢

26

)

In the balanced requests setting, the following holds:

Δ
⁢

A
max

≤

2
⁢

T

3
/
2

ξ

⁢

std
⁡
(

X
i
t

)

.

(

Expression
⁢

27

)

Proof: Firstly, it is noted that Theorem 3 guarantees that

Δ
⁢

A
max

=

𝔼
[

max
i

❘
"[LeftBracketingBar]"

∑

t
=
1

T

A

t
,

SAFFE
-
D

-

∑

t
=
1

T

A

t
,

oracle

❘
"[RightBracketingBar]"

]

,

as SAFFE-Oracle achieves hindsight solution. Then, Theorems 1 and 2 may be employed together with the triangle inequality in order to obtain an upper bound on the right hand side. In the case of unbalanced requests, this upper bound may be expressed as

Δ
⁢

A
max

≤

N
⁢

T

3
/
2

ξ

⁢

std
⁡
(

X
i
t

)

.

In the case of balanced requests, this upper bound may be expressed as

Δ
⁢

A
max

≤

2
⁢

T

3
/
2

ξ

⁢

std
⁡
(

X
i
t

)

.

Finally, as ΔA^mean≤ΔA^max, Theorem 4 also provides an upper bound on ΔA^mean.

In an exemplary embodiment, the SAFFE-D algorithm may be evaluated under various settings, and its performance may be compared with baseline sequential allocation algorithms in terms of fairness metrics and resource allocation efficiency. IN particular, the metrics may be used to assess how the fairness of SAFFE-D allocations is affected as the budget size, number of agents, and time horizon varies; whether the algorithm favors agents depending on their arrival times or demand sizes; how sensitive the algorithm is to future demand estimation errors; how the discounting in SAFFE-D improves fairness; and how SAFFE-D compares to an allocation algorithm learned using reinforcement learning for the described MDP.

In an exemplary embodiment, consideration is given to the following types of metrics for comparing the fairness and efficiency of sequential allocation algorithms: 1) Log-NSW: Expected log-NSW in hindsight as in Expression 3. 2) Utilization percentage: The fraction of available budget distributed to the agents over the horizon. Due to stochasticity, it is possible to have an overabundance of initial budget such that it exceeds the total demands over the horizon. Therefore, this metric is defined by only considering the required fraction of the available budget B as follows: