# Fair Division Algorithms for Electricity Distribution

Dinesh Kumar Baghel
Department of Computer Science
Ariel University, Israel
<EMAIL_ADDRESS>
&Vadim E. Levit
Department of Mathematics
Ariel University, Israel
<EMAIL_ADDRESS>
&Erel Segal-Halevi
Department of Computer Science
Ariel University, Israel
<EMAIL_ADDRESS>

###### Abstract

In many developing countries, the total electricity demand is larger than the
limited generation capacity of power stations. Many countries adopt the common
practice of routine load shedding — disconnecting entire regions from the
power supply — to maintain a balance between demand and supply. Load shedding
results in inflicting hardship and discomfort on households, which is even
worse and hence unfair to those whose need for electricity is higher than that
of others during load shedding hours. Recently, Oluwasuji et al. (2020)
presented this problem and suggested several heuristic solutions. In this
work, we study the electricity distribution problem as a problem of fair
division, model it using the related literature on cake-cutting problems, and
discuss some insights on which parts of the time intervals are allocated to
each household. We consider four cases: identical demand, uniform utilities;
identical demand, additive utilities; different demand, uniform utilities;
different demand, additive utilities. We provide the solution for the first
two cases and discuss the novel concept of $q$-times bin packing in relation
to the remaining cases. We also show how the fourth case is related to the
consensus $k$-division problem. One can study objectives and constraints using
utilitarian and egalitarian social welfare metrics, as well as trying to keep
the number of cuts as small as possible. A secondary objective can be to
minimize the maximum utility-difference between agents.

_Keywords_ Load Shedding $\cdot$ Fairness $\cdot$ Cake-Cutting $\cdot$
Fractional Approval Voting $\cdot$ Consensus Division $\cdot$ Bin Packing

##  1 Introduction

Electricity is of utmost importance for the development of social and economic
sectors. Electricity is perishable and should be consumed as soon as produced;
it might be highly expensive to install alternative storage capacity. However,
in this regard, developing countries face many challenges, like
infrastructure, environmental, and sustainability issues Kaygusuz (2012).
Without additional dedicated financial, institutional, and technological
policies, some 15% of the world’s population will still expropriate
electricity, a majority of which lies in developing Sub-Saharan African
countries Kaygusuz (2012). This lack of electricity is a serious impediment to
the development of industries and communities, thereby causing a significant
drag on economic growth. In comparison, the United Kingdom generates on an
average over 30GW of electricity111https://gridwatch.co.uk for a population of
about 68 million people222https://worldpopulationreview.com/countries/united-
kingdom-population, whereas Nigeria, a developing country, generates under
8.5GW333https://www.ceicdata.com/en/indicator/nigeria/electricity-production
of electricity for a population of over 211 million
people444https://worldpopulationreview.com/countries/nigeria-population.
Electricity deficiency aggravates further with the growing population,
development of backward areas, and the ever-increasing use of digital devices.
Because of these factors, emerging countries will continue to face energy
issues in the near future. Nigeria is one such example of an acute power
crisis Gatugel Usman et al. (2015).

It is necessary to have a basic understanding of the electrical distribution
grid abstractly. In a typical power distribution grid, power travels from the
power plant to the transmission substation. It further travels to the
distribution grid, where the power is stepped down so that it can be useful to
homes and businesses. After necessary stepping down, the power leaves this
grid in different directions via a distribution bus. Lines make up each bus on
the network. Each of these lines depicts a group of individual consumers with
varying electrical requirements
555http://www.science.smith.edu/~jcardell/Courses/EGR220/ElecPwr_HSW.htmll.

Due to this significant difference in the electricity demand and generation
capacity of emerging countries, and to maintain the desired frequency of
electric current (which is 50Hz or 60Hz), disconnecting a substantial
proportion of their electrical networks from the supply is the only choice for
many developing countries. This event where a portion of an electricity
network is deliberately disconnected from the supply is termed _load
shedding_. Load shedding, whose sole purpose is to maintain stable balance
within the electricity grid, often results in disconnecting more load than is
required and seriously impacts the comfort of the households in the shedded
region. Load shedding will be present in the near future. Therefore, it is
necessary to improve the electricity availability at the household level as
much as possible when they need it the most on the condition that the
cumulative electricity consumption by the households currently receiving
electricity should not exceed the supply. In addition, this household-level
electricity allocation reduces waste as compared to conventional load shedding
approaches, improves fairness and revenue, and hence increases satisfaction
within the system. Gaur et al. (2017) have described how a variety of factors
can affect the electricity demand during different hours of the day.
Therefore, any electricity allocation solution must consider these
heterogeneous preferences in a complex power system.

In light of the above, we model households as agents and time as a resource,
where each household has its preference for the consumption of electricity at
each time. The problem is similar to the classic problem of _cake-cutting_ ,
where agents have distinctly separate interests over different parts of the
cake. The objective is to come up with a fair allocation of the cake under
these different and conflicting interests of the agents. Let us assume the
time interval to be allocated among agents is $[0,T]$, the supply is
$\mathcal{S}$, and agent $i$’s demand and utility at time $t$ are $d_{i}^{t}$
and $c_{i}^{t}$, respectively. Our objective is to compute an egalitarian
electricity distribution while minimizing the number of cuts.

There are four cases to consider, from simplest to most complex. (1) Identical
Demand, Uniform Utilities, (2) Identical Demand, Additive Utilities, (3)
Different Demand, Uniform Utilities, and (4) Different Demand, Additive
Utilities. In the first two cases, where all agents have the same demand, the
goal is to allocate the time-interval among $n$ agents as per their utility
(or comfort or preference) vector such that, at any point in time, the
cumulative demand of all agents connected to electricity should not exceed the
supply $\mathcal{S}$. In the third case, we have shown it is not optimal to
connect each agent $1/k$ of the time, where $k$ is the optimal number of bins
formed by packing the $n$ agents’ demands. This non-optimality leads us to
devise the new notion of $q$-times bin packing. Finally, in the fourth case,
we prove that egalitarian electricity division, in this case, may require
$n-1$ cuts and is PPA-hard to compute.

##  2 Related Work

Pahwa et al. (2013) have proposed three load shedding strategies at the bus
level: Homogeneous Load Shedding Strategy, Linear Optimization, and Tree
Heuristic. The simple, quick, and inefficient Homogeneous Load Shedding
Strategy cuts a certain percentage of load from all buses in the system. It is
useful for critical situations, but the amount of load shed is a lot higher
than required. Another drawback of this strategy is that buses that may not be
harmed by the original failure undergo load shedding. Linear Optimization uses
linear programming to meet as much demand as feasible in the system. However,
this strategy results in very few buses in the system, shedding completely or
a large percentage of the load. This is an unfair situation of providing
electricity at the cost of a few buses. Moreover, this strategy is
computationally expensive for large systems. They resolved this issue in the
proposed tree heuristic strategy, which forms a tree based on the network
elements. The tree is formed using the initially failed line. It then
disconnects the same percentage from a subset of lines selected in the tree.
However, if the formed tree is very small with the constituent lines of the
tree not carrying enough load, then this strategy may not suffice to keep the
system intact. Moreover, their result shows that this tree heuristic does not
work for all the cases. Although the proposed tree heuristic results in better
load shedding as compared to the homogeneous strategy, it still disconnects
the load at times when households are running indispensable activities.
Therefore, it is more important to take care of the electricity needs at the
household level while developing solutions for electricity distribution.

In accordance with this, Shi and Liu (2015) developed a fair load-shedding
solution where smart agents (buses on the system) communicate with their
neighbors (upstream and downstream) and determine the amount of load they can
shed and their corresponding compensation in real-time. Assuming the
communicating agents are rational, they used linear models of incentives.
Applying such a solution to the electric grid requires advanced information
and communication technologies along with modern sensors and intelligent
protection applications. However, developing countries require less complex
solutions that deal with user-derived heterogeneous preferences. A retrofitted
meter is a traditional meter with embedded units to provide the desired
communication and control functionality. Recent advances in the design of
smart retrofits aimed primarily at developing countries have made it possible
to consider the needs of households within the network (Azasoo and Boateng,
2015; Keelson et al., 2014). Smart retrofitted meters monitor household
demands and plan for electricity ahead of time. Global System for Mobile
Communications (GSM) technology, securely transmitting energy consumption
data, remote connection, disconnection, and displaying user statistics are
some of the features of smart retrofitted meters used to monitor the household
demands and plan for electricity ahead. In the same line, Heggie et al. (2018)
proposed a solution to allocate as much electricity as possible among the
regions in the distribution network. The regulator sets the target proportion
of supply to be delivered to each region. The allocation method will try to
fulfill two objectives: first, minimizing the load shed, and second,
fulfilling the target proportion set for each region as max as possible. At a
more granular level, Azasoo et al. (2019) proposed a mechanism that bridges
the gap in demand and supply by reducing the energy consumption of user-
defined low priority tasks at the household level during peak hours using
smart metering technology. They assumed that agents would report their true
priorities of appliance usage. It is unfair since it may prohibit agents from
performing some essential tasks (assuming all tasks are set to the same
priority and hence, equally important) by reducing energy consumption.

In light of this, Oluwasuji et al. (2018) presented four heuristic algorithms
for fairly disconnecting households from the supply. They focused on
connecting households as evenly as possible in terms of the number of hours.
In evaluating the performance of these alternatives, they used utilitarian,
egalitarian, and envy-freeness social welfare metrics. In continuation with
this, Oluwasuji et al. (2020) suggested a more fair heuristic method. They
formulated the load shedding problem as a multiple knapsack problem (MKP) and
solved it using integer linear programming. A significant problem with their
approach is that they do not consider the number of cuts in the allocation
since this constraint cannot be integrated into their ILP model.

As a result, the fair load shedding (or electricity distribution) problem can
be represented as a fair resource allocation problem where electricity is the
resource to be shared at the household level. As a resource allocation problem
Chao and Hsiung (2016) solved the electricity trading issue in a smart grid.
From the commercial aspect of a smart grid, they proposed a fair electricity
trading method among buyers (smart grid agent demand) and sellers (energy
generated from renewable resources within the smart grid). For the buyer,
electricity is cheaper to trade than to buy from a power utility. In the case
of the seller, it is more profitable to sell other agents within the grid than
to sell to a power utility. In addition, Gerding et al. (2011) solved the
hybrid electric vehicles (EVs) charging problem based on the preferences
reported by the agents (EV owners). Their developed model-free mechanism
solved the problem of coordinating EV charging into the electricity grid to
prevent overloading. They assumed that valuations are non-increasing for each
incremental unit of electricity. Moreover, their mechanism results in certain
units remaining unallocated to ensure truthfulness. However, these unallocated
units are returned to the grid for other use, but in the case when the purpose
is to distribute electricity to households, these unallocated units are
necessarily wasted. Stein et al. (2012) extended this approach for pure EVs to
model-based online mechanism design, which fulfills the EVs preferences using
the pre-commitment notion, i.e., the mechanism is guaranteed to fulfill the
demand for the selected agent. Their model assumes that agents have a value
for a particular amount of resource, which, in contrast to our model, does not
have an additive value for receiving more and values it to $0$ for receiving
less. Miller et al. (2012) proposed a multi-agent-based coordination algorithm
to optimally incorporate the power outputs of renewable generators in the
distribution network. The aforementioned approaches require that agents behave
rationally and genuinely state their values to arrive at a solution, whereas,
in our model, we are generating these values centrally.

Chen et al. (2012) addressed the issue of fairly allocating cooling to
households when the required power to achieve cooling surpasses supply. They
predicted the relationship between power and cooling and proposed the min-max
and proportional fairness approach. Min-max approach determines the common
temperature to set for all the agents. As a result, some agents get the
requested cooling, whereas others are at a greater inconvenience. On the other
hand, the proportional approach allocates a fraction of their requested
temperature and hence the required power to achieve it. However, it is prone
to manipulation as agents can demand a lower temperature than they require.

A different related problem where the amount of available resource is variable
has been studied by Buermann et al. (2020). In their setting amount of
available energy is distributed at every time step, whereas our focus is on
when to allocate energy. In their paper, the valuation function of the agents
is a linear satiable function. It shows that an agent derives some usefulness
even in the case of partial fulfillment of the demand, whereas in our case of
piecewise constant functions, an agent’s utility is significant only if the
requested demand is fulfilled; otherwise, it is zero. Practically we can
explain it as follows: suppose at some time a household is running some
appliance that requires 1kW of electricity to operate; otherwise, it will not
function. During that time, this was the only activity performed by the
household. In this case, during that time, there is no point in fulfilling the
partial demand of the agent as it is of no use to him. It is as good as giving
no electricity.

In another paper, researchers have combined game-theoretic bankruptcy rules
with Nash bargaining to solve the power allocation problem at the province
level Janjua et al. (2021). However, at the agent level, their solution is not
feasible as it allocates a fraction of the demand to agents. It may be the
case that in the absence of full demand, an agent may not be able to finish
some critical work. Ali et al. (2021) have proposed a mechanism that solves
the load shedding in the case when loads can be shifted to some other hour of
the day. However, their solution is not practical in the real world as there
are many activities that can not be shifted to a different time. In the
mechanism design proposed in Akasiadis and Chalkiadakis (2017), the agents
optionally commit to shift a fraction of their load by specifying their
shifting capacity, shifting costs, and the non-peak interval to shift. The
mechanism proposed in this paper shifts the peak load to some non-peak
intervals. It does so by first defining a marginal shifting quantity
corresponding to non-peak intervals and then selecting a group of agents whose
sum is within this marginal quantity. Mechanism incentives agents for their
truthful commitments. However, if necessary, this shifting can cause some
critical work to shift to some other time resulting in a higher impact on
their comfort hence making this approach less relevant. Moreover, it may
happen that coalition strategies used by the mechanism can be unfair as it may
select a few agents again and again, and some agents are never selected.

We focus on cake cutting as a classic resource allocation problem to solve the
fair electricity distribution problem. The problem was first imposed by
Steinhaus Steinhaus (1949). In cake cutting, a cake is a metaphor for the
resource. However, we treat the time interval as a cake, unlike the previous
approaches. A cake-cutting mechanism allocates the divisible resource to
agents with different valuation functions (or preferences) according to some
fairness criteria. A number of cake cutting protocols have been discussed in
(Brams and Taylor, 1996; Robertson and Webb, 1998; Rothe, 2016). In our
problem, time-interval will be allocated among the agents without violating
the supply constraint. The solution to our problem differs from the classic
cake-cutting algorithms in the sense that the sum of the demands of all the
agents connected at time $t$ should respect the supply constraint, and several
agents may share the same piece.

Our goal is to develop fair electricity distribution solutions for households
as the residential sector forms a major part of the grid’s demand. For
example, in Nigeria, the residential sector represents 51.3% of the grid
demand Nwachukwu (2014). This residential electricity consumption is expected
to rise further Amazuilo Ezenugu (2017) with the increase in living standards.
Modeling the solution at the household level will result in better grid
conditions and energy situations. We use the dataset representative of
Nigeria’s household energy consumption as in Oluwasuji et al. (2020).

##  3 Fair electricity distribution model for households and notations

###  3.1 Supply

We denote by $\mathcal{S}\in\mathbb{R}_{>0}$, the total supply of electricity
available for distribution. It is (usually) measured in kilowatts. We assume,
for simplicity, that the supply is constant over time. In general, there can
be alternative renewable energy sources on the network. The power generated
from these resources is typically weather dependent and hence highly variable
in nature, which makes it difficult to know the exact amount of power at the
time of decision-making. Therefore, power generated from these resources is
not considered in the current work. In the future, we may consider a more
general model, in which the supply can change with time.

We model the distribution of electricity as an allocation of a time-interval
$\mathcal{C}=[0,T]$. For example, $T$ can denote a single hour, day, week,
etc.

###  3.2 Demand

We model each household as an agent. We denote the set of agents by $[n]$,
where $[k]=\\{1,\ldots,k\\}$, and $n$ is the number of agents. We define, for
each agent $i$, the hourly demand at time $t$ as $w_{i}^{t}$. Initially, we
assume that the demand is static, that is, $w_{i}^{t}=w_{i}$ for all
$t\in[0,T]$. We hope to generalize the model to time-dependent demand in the
future.

Each agent $i$ should receive a subset $X_{i}$ of the cake $[0,T]$, which
means that the agent is connected to electricity during the time represented
by $X_{i}$. We denote an allocation by $X:=(X_{1},\ldots,X_{n})$. In the
classic cake-cutting problem, the pieces allocated to different agents must be
pairwise-disjoint. This is an important way in which our problem generalizes
the cake-cutting problem: in electricity division, several agents may be
connected at the same time. The requirement is that, at each point in time
$t\in[0,T]$, the total demand of all agents connected at time $t$ is at most
the supply $\mathcal{S}$.

We assume that there are no transmission constraints imposed by the grid to
distribute the supply generated. That is, the only limit on the allocation is
the total supply $\mathcal{S}$.

###  3.3 Utility

Each agent $i$ has a utility function $u_{i}$, which assigns to each time-
interval $Z\subseteq[0,T]$, the utility that the agent gains from being
connected to electricity at $Z$.666 Oluwasuji et al. (2020) use the term
“comfort” instead of utility. They derive the utility of agents by averaging
the demand over the past four weeks, and normalizing it by dividing by the
maximum value. We consider utility and demand to be two different and
independent inputs.

We assume that the agent utilities are additive over time, that is,
$u_{i}(Z\cup Y)=u_{i}(Z)+u_{i}(Y)$ if $Z$ and $Y$ are disjoint time-intervals.
We also assume that $u_{i}$ is non-atomic, that is, $u_{i}([t,t])=0$ for all
$t\in[0,T]$.

###  3.4 Fairness and Efficiency Criteria

We use three measures for the quality of an allocation:

  * •

Its egalitarian welfare $eg$, defined as:
$eg:=\underset{i\in[n]}{\min}\;u_{i}(x)$;

  * •

Its utilitarian social welfare $ut$, defined as: $ut:=\sum_{i=1}^{n}u_{i}(x)$;

  * •

The maximum utility-difference $ef:=\\{\max_{i,j}\\{\lvert
u_{i}(x)-u_{j}(x)\rvert\\}\\}$.

###  3.5 Information considerations

We assume that the demand of each agent is known to the divider. This is a
reasonable assumption, as often the households are equipped with smart meters,
from which we can compute their demands. Additionally, the demand of each
agent is often fixed in advnace acccording to the type of connection between
the agent’s house and the power grid. Therefore, there are no strategic issues
in reporting the demands. In the future, one may consider mechanisms that
incentivize agents to truthfully report the actual amount of electricity that
they need, but in this work we assume that the demand is given.

We are also not considering strategic considerations in reporting the utility
functions. Recently, Tao (2021) has proved the nonexistence of a truthful cake
cutting mechanism even in the simple case of two agents with piecewise-
constant and strictly positive utilities. Truthful cake-cutting protocols are
known only for very special cases (Chen et al., 2013; Bei et al., 2020;
Alijani et al., 2017).

Since our problem is a generalization of cake-cutting, truthful mechanisms are
definitely not possible.

###  3.6 Key assumptions

We make the same assumptions as in Oluwasuji et al. (2020) to solve our
problem. For completeness, we describe these assumptions here:

  1. 1.

The households are equipped with smart retrofitted meters that allow to
connect or disconnect individual households at each time.

  2. 2.

The estimates of demands we receive from households are correct.

  3. 3.

The utilities are unrelated to electricity distribution events: agents may not
be aware of upcoming energy-distribution events, and so do not conduct some
operations in advance.

##  4 Fair Electricity Distribution Algorithms

In this section, we will address the fair electricity distribution problem
from the simplest to the most general case. We discuss the following cases:

  1. 1.

Identical Demands, Uniform Utilities

  2. 2.

Identical Demands, Additive Utilities

  3. 3.

Different Demands, Uniform Utilities

  4. 4.

Different Demands, Additive Utilities

###  4.1 Identical Demands, Uniform Utilities

In this simple version of the problem, all agents have the same demand $d$,
and their utilities are uniform for each time interval. Hence, agents’ only
interest lies in how much time they are connected. Assuming the total demand
is greater than the supply, only $q=\lfloor\mathcal{S}/d\rfloor$ agents can be
connected at any point in time. Each agent is connected a fraction $q/n$ of
the time, and therefore each agent’s utility is $q/n$ of the total utility.

###  4.2 Identical Demands, Additive Utilities

In this case, agents still have identical demands, but may assign different
utilities to different time-intervals of the cake $\mathcal{C}$. Since the
agents’ demands are same, again only $q=\lfloor\mathcal{S}/d\rfloor$ agents
can be connected at any point in time. So, we can create $q$ adjacent copies
of the cake $\mathcal{C}$. Let us call this cake $\mathcal{C^{\prime}}$. Now
divide the cake $\mathcal{C^{\prime}}$ using the Even-Paz cake-cutting
algorithm Even and Paz (1984). The algorithm guarantees that the allocation of
$\mathcal{C^{\prime}}$ is _proportional_ , which means that the utility for an
agent $i$ is
$U_{i}\geq\frac{1}{n}U_{i}(\mathcal{C^{\prime}}\geq\frac{q}{n}U_{i}(\mathcal{C})$.
Thus, we can guarantee the same egalitarian welfare as in the case of
identical demands and uniform utilities.

As an example assume there are $n=4$ agents, each having a demand of $2$.
Assume cake $\mathcal{C}$ is $[0,2]$ and the supply is $\mathcal{S}=4$. The
agents’ utilites are piecewise-constant:

Piece | [0,1] | $[1,2]$
---|---|---
Agent 1: | 0.8 | 0.2
Agent 2: | 0.2 | 0.8
Agent 3: | 0.7 | 0.3
Agent 4: | 0.3 | 0.7

Only $q=\lfloor 4/2\rfloor=2$ agents can be connected at any point of time.
Now, we construct a cake from $2$ adjacent copies of the interval
$\mathcal{C}$. Let’s call this cake as $\mathcal{C^{\prime}}=[0,4]$.

Correspondingly, agents utilities will consist of 2 adjacent copies of their
respective original utilities, i.e.
$u_{1}^{{}^{\prime}}=(0.8,0.2,0.8,0.2),u_{2}^{{}^{\prime}}=(0.2,0.8,0.2,0.8),u_{3}^{{}^{\prime}}=(0.7,0.3,0.7,0.3),u_{4}^{{}^{\prime}}=(0.3,0.7,0.3,0.7)$
Upon applying the Even-Paz cake cutting on $\mathcal{C^{\prime}}$ and then
mapping the allocation to original cake $\mathcal{C}$, agents allocations are
as shown in Figure 1.

(a)

(b)

(c)

Figure 1: Applying Even-Paz to $\mathcal{C^{\prime}}$: (a) First Iteration,
(b) Second Iteration. (c) Mapping $\mathcal{C^{\prime}}$ to $\mathcal{C}$ and
the corresponding allocations.

###  4.3 Different Demands, Uniform Utilities

A slightly more complicated version emerges if agents have different demands
but uniform utilities. For example, we have $n=3$ agents with uniform
utilities. Let’s assume supply is $\mathcal{S}=30$ and the agent’s demands are
$10,20$ and $30$, respectively.

We define a _maximal feasible set_ as a set of agents, the sum of whose
demands do not exceed the supply, and additional agents cannot be added to it
without exceeding the supply. The maximal feasible sets in the above example
are: $\\{10,20\\},\\{30\\}$. Obviously, only maximal feasible sets should be
considered for allocation, otherwise there is unused supply. Now the question
is _what is a fair allocation of time in this case?_

This problem can be represented as a problem of _fractional approval voting_ ,
where each of the agents $i\in[n]$ approves one or more candidates, and the
output is a distribution over all the candidates. Fractional approval voting
and the equivalent terms have been studied in (Aziz et al., 2019; Bogomolnaia
et al., 2005; Duddy, 2015; Brandl et al., 2021). In our case, the candidates
are the maximal feasible sets, and each agent $i$ “votes” for all candidates
containing $i$.

The most basic fairness condition in fractional approval voting is Individual
Fair Share (IFS). IFS says that each agent $i$ should get a utility of $\geq
1/n$ of his/her total utility. In the cake-cutting literature, this condition
is called “proportionality”. In our case, it means that each agent should be
connected at least $1/3$ of the time. There are many allocations that meet
this condition in our example, e.g. allocating $1/2$ of the time to each
maximal feasible set, or allocating $2/3$ to the set $\\{10,20\\}$ and $1/3$
to the set $\\{30\\}$.

But, the IFS condition is too weak: as seen in the case of equal demands, it
is possible to give each agent a utility of $q/n$, which is usually much
larger than $1/n$. There are two ways to strengthen IFS: Egalitarian and Group
Fair Share.

One way is to form groups of agents and treat each group as an individual
agent. Our goal is to give each agent an allocation $\geq r$, where $r$ is the
largest fraction possible given the agent demands. We call this approach the
Egalitarian approach. In the example above, it allocations $1/2$ to each
maximal feasible set.

Another way is the notion of Group Fair Share (Group FS or GFS) which says
that for a group of $q$ agents, the total allocation should be at least $\geq
q/n$ In the example above, it allocations $1/3$ to the $\\{30\\}$ and $2/3$ to
the $\\{10,20\\}$.

To illustrate a potential problem with the GFS criterion, consider the
scenario where the groups are $\\{\epsilon,30-\epsilon\\}$ and $\\{30\\}$. GFS
still results in allocating the groups $2/3$ and $1/3$ respectively, despite
the fact that $30-\epsilon$ and $30$ are almost the same, and hence this
allocation is apparently unfair for the agent with a demand of $30$. Moreover,
for a group like $\\{5,10,15\\}$ and $\\{30\\}$, the GFS allocation will turn
out to be $3/4$ and $1/4$ respectively, which is even more unfair for a large
family having demand $30$.

Minimalistically, we are interested in the “equal treatment of equals"
fairness principle. Therefore, in the egalitarian approach, our goal is to
maximize $r$ so that each agent can be connected at least $r$ of the time.
Finding $r$ is an NP-hard problem (by reduction from Partition). In contrast
to fractional approval voting, the number of candidates is exponential in $n$.
Therefore, it is not possible to check all candidates. It leads us to look for
other techniques like Bin Packing algorithms. In bin packing, we are given a
set of items (agents), and the goal is to pack this set of items into a
minimum number of bins of size $\mathcal{S}$, where the size of each agent is
its demand. Let us assume the number of bins in a bin packing solution for a
problem instance $I$ is $k$. Then each agent in $I$ can be connected $1/k$ of
the time. However, this lower bound is not optimal. For example, if
$\mathcal{S}=2$ and the agent demands are $={1,1,1}$, then bin packing results
in $k=2$, which gives a lower bound of $r=1/2$. But $r=2/3$ is possible, by
connecting $\\{1,2\\}$, $\\{2,3\\}$ and $\\{3,1\\}$ for $1/3$ of the time
each.

This leads us to a new problem, which we call _$q$ -times bin packing_: given
the set of items, the goal is to pack these items into $k$ bins, of size
$\mathcal{S}$ each, such that each item appears in $q$ different bins. We can
connect each bin $1/k$ of the time; thereby, each agent is connected for at
least $q/k$ of the time. Therefore, the following research questions arise:

  1. 1.

Which bin-packing algorithms can be adapted to $q$-times bin-packing?

  2. 2.

Is the $q$-times bin-packing lower bound always tight for some $q\geq 1$?

  3. 3.

Which efficient algorithms can be used to find a group-FS allocation?

###  4.4 Different Demands, Additive Utilities

In the more realistic scenario, agents have different demands with additive
utilities. What fairness guarantees are reasonable in this setting?

Let $r$ be the egalitarian value of the simplified problem in which agents
have _uniform_ utilities. Can we guarantee each agent a utility $\geq r$ with
_additive_ utilities? The answer is positive: the utilities in each minute are
essentially uniform, so we can partition each minute using the egalitarian
solution, and guarantee each agent a fraction of at least $r$ in each minute.

Nevertheless, the problem is that this allocation requires too many cuts.
However, due to practical considerations, we would like to find a solution
involving fewer cuts. The previously posed question can now be rephrased: Can
we guarantee each agent a utility $\geq r$ with _a small number of cuts_?

Let us assume that we have an egalitarian solution $r=q/k$ where $k$ is the
number of bins from $q$-times bin-packing. Insights from the $k$-consensus
division problem can help answer the question. A _$k$ -consensus division_ is
a partition of cake into $k$ pieces such that each agent $1,\ldots,n$ values
those pieces at exactly $1/k$. This partition can be achieved using a bounded
number of cuts ($\\#$cuts $\leq n)(k-1)$)Alon (1987).

Now, we can find a $k$-consensus division among agents $1,\ldots,n-1$ using
$(n-1)(k-1)$ cuts, and let agent $n$ choose the $q$ pieces he prefers. We can
now connect each bin (— maximal feasible set) $1/k$ of the time, and since
each agent appears in $q$ bins, the utility for agent $n$ is $\geq q/k$, and
all other agents will get exactly $q/k$. However, the problem is NP-hard when
$k=2$, even when all valuations are piecewise constant. The following results
hold for $k=2$:

  * •

Computing an approximate $k$-consensus division with the minimal number of
cuts ($n(k-1)$) is PPA-hard (Filos-Ratsikas and Goldberg, 2018; Filos-Ratsikas
et al., 2020).

  * •

Deciding whether there exists a $k$-consensus division with $n(k-1)-1$ cuts is
NP-hard (Filos-Ratsikas et al., 2018)

For $k=3$, we know that consensus division is PPAD-hard (Filos-Ratsikas et
al., 2020). However, no such results are known for $k>3$ since more cuts are
available. We only know that the problem is in PPA-k for any $k$ which is a
power of a prime integer (Filos-Ratsikas et al., 2021). These hardness results
can be summarized as:

#pieces $k$ | Agents | Cuts | Hardness
---|---|---|---
2 | $n$ | $n$ | PPA-hard
3 | $n$ | $2n$ | PPAD-hard
prime power | $n$ | $n(k-1)$ | belongs to PPA-k

So the questions are:

  1. (a)

Can electricity division be done with fewer cuts?

  2. (b)

Can electricity division be done more efficiently?

Formally, we define the _egalitarian electricity division problem_ as follows.

A solution to the electricity division problem is a distribution, $E$, of the
time intervals for each agent. The _egalitarian welfare_ of this distribution
is the agent with the lowest utility: $eg(E)=\min_{i\in[n]}\\{u_{i}(E)\\}$.

INPUT: $n$ agents with different demands and different valuations over a time-
interval; supply $\mathcal{S}$.

OUTPUT: a feasible allocation of connection time among the agents, such that
$eg(E)$ is maximum among all feasible allocations.

######  Theorem 4.1.

Egalitarian electricity division among $n$ agents:

  1. (a)

may require $n-1$ cuts

  2. (b)

is PPA-hard to compute.

###### Proof.

The proof is by reduction from $2$-consensus division.

Given a $2$-consensus division problem with $n-1$ agents with valuations
$v_{1},\ldots,v_{n-1}$, construct an electricity division problem with $n$
agents:

  * •

The supply $\mathcal{S}=n-1$;

  * •

There are $n-1$ agents with demands $1$ and valuations =
$v_{1},\ldots,v_{n-1}$;

  * •

There is one agent with demand $n-1$ and valuation
$(v_{1}+v_{2}+\ldots+v_{n-1})/(n-1)$

In this instance, there are only two maximal feasible sets:
$\\{1,\ldots,n-1\\}$ and $\\{n\\}$. Therefore, the egalitarian division can be
written as $(X_{1},X_{2})$, where $X_{1}$ is the interval in which the set
$\\{1,\ldots,n-1\\}$ is connected, and $X_{2}$ is the interval in which the
set $\\{n\\}$ connected.

In each maximal feasible set, the valuation of all agents is the same.
Therefore, by classic cake-cutting results, it is possible to guarantee each
agent a utility of at least $1/2$. So the optimal egalitarian value is at
least $1/2$.

All agents in the first feasible set value the first piece as at least $1/2$,
i.e.

$\forall i=1,\ldots,n-1:v_{i}(X_{1})\geq 1/2$

and by additivity, values the second piece as at most $1/2$:

$\forall i=1,\ldots,n-1:v_{i}(X_{2})\leq 1/2$

The second piece $X_{2}$ is given to the $n^{th}$ agent whose valuation is the
average of the valuation of all other agents. Its value is at least the
optimal egalitarian value: $v_{n}(X_{2})\geq 1/2$, but $v_{n}(X_{2})$ is the
average of all $v_{i}(X_{2}):\forall i=1,\ldots,n-1$. This implies
$v_{i}(X_{2})=1/2,\forall i=1,\ldots,n-1$. Again by additivity,

$\forall i=1,\ldots,n-1:v_{i}(X_{1})=1/2$

So $(X_{1},X_{2})$ solves the problem of $2$-consensus division. Thus,
consensus division among $n-1$ agents is a special case of egalitarian
electricity division among $n$ agents (when the egalitarian value is $1/2$).

It is known that a 2-consensus division among $n-1$ agents may require $n-1$
cuts Alon (1987), and it is PPA-hard to find an approximate egalitarian
division. ∎

This leads us to the following research questions:

  * (a)

Are $n-1$ cuts (which might be necessary by the reduction from consensus
division) always sufficient for egalitarian electricity division?

  * (b)

How many cuts are needed for a group-FS division?

  * (c)

Can we heuristically find an egalitarian/group-FS division with few cuts on
realistic instances?

##  5 Conclusion and Future Work

In this paper, we studied the fair electricity division problem. It is related
to several classic problems, like proportional cake cutting, bin packing,
fractional approval voting, and consensus division. We have considered the
four different cases for demand and utilities, from the simple one to the more
generic one. We have provided the solution for the rather simplistic case of
identical demand with uniform/additive utilities. We have seen how the novel
concept of $k$-times bin packing relates to the remaining cases of varying
demand with uniform/additive utilities. We have also shown that in the more
general case of different demand and additive utilities, egalitarian
electricity distribution may require $n-1$ cuts and is PPA-hard to compute
4.1. Ideas that we will develop here may apply to other settings of social
choice or fair division in which agents may have different sizes.

From a broader point of view, this paper shows how fair electricity
distribution and related problems are complex and should be studied more. Some
of the results from our work can be applied to the settings where many agents
organized in overlapping groups want to attend or access the resource of
limited capacity Arbiv and Aumann (2022). Our basic model of fair electricity
distribution can be extended to include more supply constraints, such as
supply $\mathcal{S}$ changing over time; a supply network with a capacity
constraint on link capacities; OR it can be extended to include more complex
demands, such as demand $d_{i}$ changing over time, with price acting as an
incentive to agents to consume less electricity during more overloaded hours.
Another interesting direction would be extending this work to a more general
valuation function class.

##  6 Acknowledgment

We want to thank Dr. Aris Filos-Ratsikas, Dr. Enrico Gerding, and Prof. Amos
Yehuda Azaria for invaluable feedback and comments that helped improve this
research proposal.

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# Quantum time in near-horizon zone of black hole

H. Hadi and F. Darabi
Department of Physics, Azarbaijan Shahid Madani University, Tabriz email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We apply quantum description of time, based on Page and Wootters’ mechanism,
for near horizon region of black hole. It is indicated that this method of
quantization of time which is a modified form of PaW approach and has been
proposed in [21] can not be applied in this particular quantum region of black
hole, fully. The time evolving of the system and flow of time, measurement
problem, recording the history of near-horizon zone and also a consistent
definition of the arrow of time are controversial in PaW approach in this
particular region even by considering the previous modifications which exist
in this literature. The clock ambiguity also is investigated and a unique
definition of clock is defined in near horizon region of black hole.

##  1 Introduction

Time indicates itself as a classical parameter in the Schr$\ddot{o}$dinger
equation. This parameter feature of time is as opposed to other physical
quantities such as position or momentum. Physically this interpretation of
time is indicated by a classical clock in the laboratory. It seems that this
is acceptable for all practical purposes , however, we should define a fully
quantum description of time. One can find some of the proposals in [1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11]. There are some controversial issues about these
proposals [12, 13, 14, 15, 16]. One of these proposal is Page and Wootters
mechanism (PaW) which consider time as a quantum degree of freedom [17, 18,
19, 20]. In this paradigm, all variables of the system are defined by quantum
operators and one of them is assumed to be clock system which consider the
dynamical evolution of the rest of system.

However, this proposal criticized by [7] when applied into constrained
systems. In PaW approach prominent problem is in defining the variables as
operators and particularly there is controversial issues to define a certain
variable as clock. In physical constrained systems, the physical observable
variables are those which have vanishing Poisson brackets with constraints. In
other words, the physical variables are invariant under symmetries of theory
(gauge invariant). However in the systems that the Hamiltonian is one of the
constraints, the other constraints which have vanishing Poisson brackets with
Hamiltonian do not evolve and will not be a good candidates for clock. Page an
Wootters tried to overcome this problem by introducing kinematical variables
which do not have vanishing Poisson brackets. Therefore these variables can
evolve. However, this leads to kinematical Hilbert space which does not admit
a probabilistic interpretation.

In [26] by using a different set of physical quantities, it is tried to find a
solution for above problem. This quantities are relational Dirac observables
which are like a ”evolving constants” [5].The mentioned proposal applies the
evolving constant approach to the construction of observables in canonical
quantum gravity with PaW mechanism by a relational time for generally
covariant systems.

The other proposal to overcome the criticism for PaW mechanism is considered
by formalizing measurements through the von Neumann prescription[21]. Which is
extended to generalized observable POVMs. In this approach all quantum
predictions can be obtained by conditioning the global, timeless state
$|\Psi>>$. The correct quantum propagator and also the correct statistic for
measurement which performed at different times are the advantage of this
approach that are absent in PaW mechanism.

In this investigation the review of PaW approach is done for near-horizon zone
of black hole which is a straightforward modification of PaW mechanism for
near-horizon zone which has been done by authors [27]. In addition some of the
criticisms of this mechanism are reconsidered in this particular vicinity of
black hole. The zone of the black hole has unusual features, especially the
vacuum of the near horizon is frozen vacuum [28] which leads to violation of
equivalence principle of general relativity and is constructed by non-local
behavior of vacuum in near horizon region of black hole. To clarify this
feature, we should emphasize that the complementarity concept of black hole
111 The complementarity claims that the space-time created by a black hole
source cannot be probed by static or infalling observer alone but both of them
together can probe the whole space-time seems that is violated by firewall
constructed in horizon which leads to violation of equivalence principle of
GR. In order to get ride of firewall concept for preserving complementarity
and equivalence principle the $ER=EPR$ correspondence [29] has been posed. In
other words the firewall is a proposal to overcome the AMPS’s paradox [30]
however, $ER=EPR$ conjecture overcomes the paradox and also preserves the
equivalence principle and complementarity of black hole.

The AMPS claims that by accepting the complementarity of black hole the three
following assumption leads to contradiction:

  * 1

An evaporating black hole preserves quantum information without destroying it.
This property is called the unitary evolution of the black hole.

  * 2

The infalling observer crossing the event horizon of the black hole does not
recognize any unusual property in horizon.

  * 3

Relativistic effective quantum field theory is consistent for an static
outside observer and he/she can work with the theory.

AMPS argues the late radiation $B$ of an old black hole 222 An old black hole
is the one which radiated half of it radiation away [31] is maximally
entangled with early radiation $R_{B}$. Regarding assumptions 1 and 3 require
the late radiation $B$ to be entangled with a subsystem of early radiation
$R_{B}$. On the other hand, the assumption 2 leads to entanglement between $B$
and the system of interior radiation $A$ of the black hole. Considering these
assumptions all together leads to violation of monogamy of entanglement [32,
33]. It asserts that if two quantum states are maximally entangled then none
of them cannot be entangled with third party. To overcome this paradox AMPS
argues that the entanglement between late radiation $B$ and interior states
$A$ breaks down. This phenomenon leads to release of energy in the horizon of
black hole. This high amount of energy is firewall which also leads to extend
the singularity to horizon and admits that there is no spacetime in the
interior of horizon [30, 34].

One of the solution for AMPS’s paradox is $ER=EPR$ conjecture. This conjecture
overcomes AMPS’s puzzle beside maintaining the equivalence principle in near
horizon of black hole. The $ER=EPR$ proposal claims that the $ER$ bridge has a
correspondence with $EPR$ pair. This means that $ER$ bridge is created by
$EPR$ correlation in the micro-states of two entangled black holes [29]. This
result is based on the works [35, 36]. In other words one can assert that
$EPR$ correlated quantum system has a weakly coupled Einstein gravity
description or $ER$ bridge is nothing but a highly quantum object. Even in a
more provocative idea, some physicist claim that for every singlet state there
exists a quantum bridge of this type. For more discussion of AMPS’s paradox
and some controversial ideas and solutions for it refer to [37]. However, by
considering the $ER=EPR$ conjecture with details for an old black hole one
confronts with violation of equivalence principle in near-horizon zone by
admitting a particular property for this region. This unusual behavior of zone
is a frozen vacuum which an infalling or static observer of a black hole
cannot excite the vacuum there. This feature assigns an unusual behavior for
near horizon that violates the equivalence principle.

In this paper we first review PaW approach which is formulated by measurements
through the von Neumann prescription in section two. In section three we apply
PaW approach for the near horizon vicinity of black holes. In the next
sections we indicate that this approach has some controversial issues in near-
horizon zone. Although some of these problems have been solved in PaW
mechanism in usual quantum mechanics in Hilbert space, we indicate that there
are controversial discussions for PaW approach in near-horizon zone of black
hole, yet. In section four it is investigated that recording history is not
possible by PaW mechanism in horizon zone. The arrow of time is considered for
horizon vicinity in section five. The time ambiguity and measurement problem
for PaW approach in near-horizon vacuum of black hole are considered in
sections six and seven, respectively. At the end we have a conclusion section.

##  2 Review of PaW mechanism through the von Neumann prescription

In this section we review the PaW mechanism from [21] by definition of quantum
operator for time in Hilbert space. This picture of PaW approach claims to
give a consistent quantum description of time, based on PaW conditional
probabilities mechanism. The main structure of this proposal is reviewed
,briefly, in this section but in section seven the complete conditional
probability and measurement problem for PaW approach is given and it is
indicated that this proposal cannot be apply fully for near-horizon zone of
black hole. Now we review the PaW approach which is given by three following
parts.

  * 1)

The Hilbert space $\mathcal{H}_{R}$ of a system can be considered under an
enlarged space as

$\mathbb{H}=\mathcal{H}_{C}\otimes\mathcal{H}_{R},$ (1)

where $\mathcal{H}_{C}$ is the space of ancillary system which is called the
clock system. This system is equipped with canonical coordinates $T$ and
$\Omega$ which represent position and momentum and can be interpreted as time
and energy indicator of evolving system. Canonical indicator of $T$ and
$\Omega$ are defined as follows

$[T,\Omega]=i$ (2)

The index $R$ in Hilbert space $\mathcal{H}_{R}$ is for the ”rest system”
which we will use it when in PaW approach the system is divided in two
subsystems of ”clock” and ”rest”.

  * 2)

One can describe the constraint operator of the system by following relation

$\mathbb{H}=\hslash\Omega\otimes I_{R}+I_{C}\otimes H_{R}$ (3)

where $H_{R}$ is the system Hamiltonian and $I_{R}$ and $I_{C}$ are identity
operator of $\mathcal{H}_{R}$ and $\mathcal{H}_{C}$, respectively.

  * 3)

Next we define a physical vector of the system $|\Psi>>$ which has following
feature

$\mathbb{H}|\Psi>>=0$ (4)

double-Ket $|\Psi>>$ is defined on $\mathcal{H}_{C}\otimes\mathcal{H}_{R}$ .
The state $|\Psi>>$ is physical vector of the model and is constrained by
Hamiltonian $\mathbb{H}$ with null eigenvalue. $|\Psi>>$ is static state and
does not evolve and represents the full history of the system $R$ consistently
with Wheeler-Dewitt equation [38]. One can derive the state $|\psi>_{R}$ of
the system by conditioning on solution $|\Psi>>$ with projection of
generalized eigenvector time operator $T$ which is given by

$|\psi(t)>_{R}=_{T}<t|\Psi>>$ (5)

where $|\psi(t)>$ is time dependent and the time projection of static state
$|\Psi>>$. The time operator has following definition

$\displaystyle T|t>_{C}=t|t>_{C},$
${}_{C}<t^{{}^{\prime}}|t>_{C}=\delta(t-t^{{}^{\prime}}).$ (6)

where $|t>_{C}$ is the eigenvector of time operator $T$ with eigenvalue $t$ in
clock system $C$.

Now the important puzzle is if one can derive the Schrodinger equation from
Wheeler-DeWitt equation (4). According PaW mechanism and by writing equation
(4) in position representation in $\mathcal{H}_{R}$ one can derive the
Schrodinger equation by following steps. First $|t>_{C}$ is procucted from
left over equation (4)

${}_{C}<t|\mathbb{H}|\Psi>>$ (7)

then we have

$i\frac{d}{dt}|\psi(t)>=H_{R}|\psi(t)>$ (8)

where we applied the following relation for $|\Psi>>$,

$|\Psi>>=\int dt|t>_{C}\otimes|\psi(t)>_{R}$ (9)

Although the wheeler-DeWitt equation (4) gives a timeless picture for universe
system with eigenvetors $|\Psi>>$ by null eigenvalues, one can derive from it
the Schrodinger equation which has time interpretation in its formalism.

##  3 Frozen vacuum and quantized time in near-horizon zone of black hole

In this section, we consider the near horizon of black hole and frozen vacuum
of it and then apply the PaW mechanism through the von Neumann prescription
for near horizon vicinity. We review our previous paper [27] which is a
straightforward modification of PaW mechanism of previous section in a
particular region. It is important to note that in PaW approach one can
consider time quantum degree of freedom as an abstract purification space
without any physical relevance. However, considering time quantum degree of
freedom as a dynamical degree of freedom which is connected to a physical
system has more physical significance. This representation of clock to define
time can be described as an operational definition of time which is
appropriate for proper time [22, 23]. In whole of the paper we assign time
quantum degree of freedom as dynamical degree of the freedom which is relevant
to some physical system.

Now the PaW approach is reviewed in near-horizon zone. The quantum state of
the near horizon is the vacuum state as follows

$|\Psi>>=\sum_{n=0}^{\infty}x^{n}|n>_{a}|n>_{b}$ (10)

where $|n>_{a}$ and $|n>_{b}$ are the states of inside and outside of the
horizon, respectively. In whole of the paper, Hilbert space of the interior
and exterior of the horizon are indicated by $A$ and $B$ respectively.
However, the clock system $C$ is related to $B$ and rest system $R$ to $A$.
The Hamiltonian of Hilbert space $\mathcal{H}_{R}$ of the interior of black
hole horizon is written as follows

$H_{R}=-\sum_{n=0}^{\infty}x^{-n}|n>_{aa}<n|,$ (11)

The ancillary system which does the role of clock system is the Hilbert space
$\mathcal{H}_{C}$ of the exterior of black hole’s horizon. The Hamiltonian for
this Hilbert space is

$H_{C}=\sum_{n=0}^{\infty}x^{-n}|n>_{bb}<n|,$ (12)

Notice that $H_{C}$ and $H_{R}$ indicate the local Hamiltonian of exterior
(clock system) and interior (rest system) of the horizon of black hole,
respectively. To obtain the wheeler-DeWitt equation for this mode one can uses
equations (12) and (11) into equations (3),(4) as follows

$\left(\sum_{n=0}^{\infty}x^{-n}|n>_{bb}<n|\otimes
I_{R}-I_{C}\otimes\sum_{n=0}^{\infty}x^{-n}|n>_{aa}<n|\right)|\Psi>>=0$ (13)

where $|\Psi>>$ here is the frozen vacuum state (10).

To derive the proper time evolution of the interior of black hole’s states by
PaW approach the exterior observer can use his/her own subsystem states as
follows

$|\psi(t)>_{b}=e^{-iH_{b}t/\hslash}|\psi(0)>_{b}$ (14)

Then to obtain the proper time evolution of the interior of black hole before
falling into it, the observer can us the following equation

$|\psi(t)>_{a}=_{b}<\psi(t)|\Psi>>=e^{-iH_{a}t/\hslash}|\psi(0)>_{a}$ (15)

to obtain the time evolution of the inside of black hole. For more details and
timeless picture of wheeler-DeWitt equation refer to [27].

##  4 Recording history and flow of time in near-horizon zone

It seems that there is no flow of time in PaW picture, however, the flow of
time is emerged in this picture by dividing the subsystem of rest of the
universe in three sub-parts, ”observer” (is only interpreted as memory), the
”observed” and a sequence of ancillas. These sub-parts can perform
measurements and store their results and then constructing a history [25].

In this section we consider the history recording of near horizon region which
is needed for describing the time flow in PaW approach. In doing so, suppose
an static observer, Bob, hovers near horizon performing measurement on the
observed to record what has happened. We assume the Bob’s memory starts in a
blank state $|r>^{\otimes{N}}_{M}$ at $t=0$. The observed is in state
$|\psi(t)>$ and $|n>_{i}$ for $i=1,2,3,...$ are eigenstates of observable in
subsystem $i$. Now suppose that Bob and the observed evolve under an effective
time-dependent Hamiltonian. Since Bob is in the exterior of horizon $B$ we use
$b$ instead of $i$. Bob measures the observable with an operator in his
subsystem at time $t=1$. In this case the local Hamiltonian operator (12)
would be suitable for measurement. However any operator can be adopted to do
this work and the final result of this section does not change. This operation
changes the state to

$|n>_{b}\otimes|r>^{\otimes{N}}_{M}\rightarrow|n_{saw1}>_{b}|r>^{\otimes{N-1}}_{M}|1>_{b}$
(16)

where $|n_{saw1}>_{b}$ corresponds to state $|1>_{b}$ which is the outcome of
the measurement. A local permutation over observed can lead to, for example,
the following state

$|n_{saw1}>_{b}|r>^{\otimes{N-1}}_{M}|2>_{b}$ (17)

For the next step, Bob measures the state again which result in

$|n_{saw1}>_{b}|n_{saw2}>_{b}|r>^{\otimes{N-2}}_{M}|2>_{b}$ (18)

here the state $|n_{saw2}>_{b}$ corresponds to $|2>_{b}$. Finally a sequence
of events is recorded by Bob such as eigenvalues of
$|1>_{b},|2>_{2},|3>_{3},...$. He does this by a sequence of ancillas
($|n_{saw1}>_{b},|n_{saw2}>_{b},|n_{saw3}>_{b},...$). These states can be the
states of the ”pointers” in near-horizon region where Bob hovers there.

Now suppose Alice an in-falling observer is falling into the black hole. If
she knows the Hawking radiation states $R_{B}$ and without meeting Bob she is
aware of the interior states $A$ by $ER=EPR$ correspondence which asserts that
the interior states $A$ are a map of early Hawking radiation $R_{B}$
($A=R_{B}$ donkey map) [27, 29]. On the other hand, Alice is aware of exterior
states $B$ by her knowledge of black hole which claims that the exterior
states $B$ are purified by interior $A$. However, if Alice meets the Bob she
will confront with contradiction because, according to donkey map ($A=R_{B}$),
we have ($|n>_{R_{B}}\rightarrow|n>_{a}$). Alice without meeting Bob will
confirm that the states $|n>_{a}$ ($A$) purify the subsystem $B$ which are
$|n>_{b}$ but by meeting Bob she admits that what purify $A$ are the states
($|n_{saw1}>_{b}|1>_{b},|n_{saw2}>_{b}|2>_{b},|n_{saw3}>_{b}|3>_{b},...$) .
Anyway this a contradiction and is not possible. Therefore Bob or even Alice
herself are not able to record the history in near-horizon zone.

We considered the subsystem $B$ which play the role of clock system and
divided it into ”pointer” (ancillas), ”observed” and ”observer” (only consist
of memory system), however one can divide the subsystem $A$ ,too, which play
the role of the rest system in PaW approach. Anyway recording the history for
both subsystems are not possible because there is symmetry for subsystems $A$
and $B$. To explain it more, suppose the subsystem $A$ with local Hamiltonian
$H_{R}$ like as previous subsystem $B$ is divided to $|n_{sawi}>_{b}|i>_{b}$
where $i=1,2,3,...$.333Here like as above the subsystem is partitioned in
three sub-parts but for simplicity we consider $A$ by dividing it in two sub-
parts. The final result is independent of more partitioning.. Again we know
from donkey map that the early Hawking radiation $R_{B}$ is a map of interior
space $A$ which can be represented by
$|n>_{R_{B}}\rightarrow|n_{sawi}>_{b}|i>_{b}$. In other words whatever happen
near horizon, the vacuum there is a frozen vacuum (10). For more information
about this behavior refer to [27, 28].

Although the PaW mechanism can be applied in near-horizon zone, recording
history in this region is not possible. Therefore PaW approach cannot overcome
criticism [25] fully and flow of time is a controversial concept here.

##  5 No arrow of time in near-horizon zone

As a whole the PaW mechanism by itself gives rise to time-reversal symmetric
dynamical law like as unitary quantum theory. In other words to recognize the
arrow of time one has to impose a quantity which increase or decrees
continuously under a dynamical law. Entanglement between observer and observed
in the rest universe is one of the proposal for arrow of time in PaW picture
[25]. This definition is not applicable for near-horizon zone when one uses
the PaW approach.

In order to define arrow of time in near-horizon zone we explain it as
follows. Suppose the pointer states in zone of black hole. We will define the
increase of entanglement between ”pointer” and the ”observed” as arrow of
time. One can define the entanglement between the ”observer” and the
”observed” as a criterion of arrow of time, however, we choose pointer instead
of observer. This choice does not make any difference in the physical results
here. Now, assume that the rest system $A$ which is the interior states of
black hole horizon, consist of pointers and observed. The observed and the
pointer in premeasurement process are as follows

$|\psi>_{pab}\approx|n>_{p}\otimes\sum_{n}^{\infty}x^{n}|n>_{a}|n>_{b}$ (19)

where $|n>_{p}$ , $|n>_{a}$ and $|n>_{b}$ are states of pointer, interior and
exterior of the horizon, respectively. The pointer $p$ is in premeasurement
state and is not in any interaction with inside or outside of the horizon,
yet. In this case the pointer can be written by inside states $A$ as follows

$|\psi>_{pab}=\sum_{n}^{\infty}x^{n}|n>_{p}|n>_{a}|n>_{b}$ (20)

equations (19) and (20) are called premeasurement.The pointer also can be
interpreted as memory state for recording to keep track of what happens for
the observed $|n>_{b}$. Then the arrow of time can be described by increasing
the entanglement between pointer and the observed regarding that the
entanglement never decreases on the rest under the evolution generated by
$H_{R}$. However, being of pointer for keeping track of observed in the rest
system is not possible and it leads to contradiction, because the infalling
observer who falls into the black hole can only confront with a frozen vacuum
which is given by (10). In other words the pointer and observed
$|n>_{p}|n>_{a}$, regarding the $ER=EPR$ correspondence, must be a map of
early Hawking radiation states $R_{B}$

$|n>_{R_{B}}\rightarrow|n>_{p}|n>_{a}$ (21)

On the other hand an infalling observer, Alice, by her knowledge of black hole
knows that $R_{B}$ and $B$ are maximally entangled. Alice in her journey of
falling into the black hole, will confront with $B$ and also is aware of what
purifies $B$ is the inside state $A$. If Alice recognizes the early Hawking
radiation $R_{B}$ before premeasurement of (19) and (20) then she will
confront with contradiction because what she expects of inside state $A$ is
only purified state of $B$ and it cannot be $|n>_{p}|n>_{a}$. Therefore in
order to avoid this contradiction, there must be no possibility of pointer $p$
for recording the track of observed. This admits that the criterion of
entanglement between pointer and observed is not applicable in this region.
The frozen vacuum of near-horizon zone is an obstacle to create or excite any
particle or pointer there. Therefore the entanglement is not a suitable
criterion for considering the arrow of time for PaW approach in near horizon
vicinity of black hole.

##  6 Near-horizon zone and the unique clock

The other problem which can be considered in PaW picture is, that one can
partition the total Hilbert space of the universe in infinity many
inequivalent ways into tensor-product structure. Therefore there are several
choice of clock-rest system which is called clock ambiguity [25].

First we consider clock ambiguity by details and then indicate that how it
works in near-horizon zone. The clock ambiguity problem is as follows. One can
choose a suitable orthonormal basis $|k>$ in the overall Hilbert space
$\mathcal{H}$ ,

$|\psi>=\sum_{k}\alpha_{k}|k>.$ (22)

where $|k>$ can be written as $|t>_{C}|\phi_{t}>_{R}$ in a given tensor
product structure $\mathcal{H}=\mathcal{H}_{C}\otimes\mathcal{H}_{R}$. Next,
consider a different state of the universe, such as

$|\tilde{\psi}>=\sum_{k}\beta_{k}|\tilde{k}>.$ (23)

We know that there is a unitary operator $S$ such that
$|\tilde{\psi}>=S|\psi>$. Therefore we have

$|\psi>=\sum_{k}\beta_{k}S^{\dagger}|k>=\sum_{k}\beta_{k}|\tilde{k}>$ (24)

where $|\tilde{k}>=S^{\dagger}|k>$

The clock ambiguity claims that as long as on can choose basis
$|\tilde{k}>=|t>_{\tilde{C}}|\phi_{t}>_{\tilde{R}}$ by a different bipartite
tensor product structure and since there are countless such choices, therefore
it leads to different description of evolution of the rest.

There is no clock ambiguity because of following reasons

  * Case one

: There is no interaction between subsystems $C$ and $R$: In this case there
is no ambiguity in clock definition. The tensor product structure of the
system of clock and rest is $\mathcal{H_{C}}\otimes\mathcal{H_{R}}$ and since
the clock and rest are non-interacting the Hamiltonian is given by
$H=H_{C}\otimes I_{R}+I_{C}\otimes H_{R}$. By applying the PaW approach the
relative state $|\phi_{t}>_{R}$ is evolved by a unitary evolution as follows

$|\phi_{t}>_{R}=exp(-iH_{R}t)|\phi_{0}>_{R}$ (25)

To make it clarify, for any state $|k>\epsilon\mathcal{H}$ and a tensor
product structure by a unitary mapping is written as follows

$|k>=\sum_{a,b}U^{k}_{a,b}|a>_{C}|b>_{R}$ (26)

where $|a>_{C}|b>_{R}$ are some basis states. Two tensor product structures
are equivalent if and only if there elements $U^{k}_{i,j}$ and
$\tilde{U}^{k}_{a,b}$ are related by local unitaries $P$ and $Q$

$U^{k}_{i,j}=\sum_{a,b}P^{a}_{i}Q^{b}_{j}\tilde{U}^{k}_{a,b}$ (27)

These two tensor product structures are equivalent if $S=P\otimes Q$. Then
there is no any ambiguity in defining clock system.

  * Case one

(in near-horizon zone): This case in near-horizon zone is applicable and there
is no any ambiguity at all. Assume the states interior and exterior of the
horizon as $A$ and $B$, respectively. When we restrict our self to $ER=EPR$
paradigm, for black hole the frozen vacuum (10) is the only state in near-
horizon zone. One can define a local unitary operator for subsystems $A$ and
$B$ separately. According to above explanations the ability of defining a
local unitary operator is enough condition to claim that there is no any clock
ambiguity.

  * Case two

: There is interaction between subsystems $C$ and $R$: In this case the new
tensor product structure is not equivalent with old structure. The unitary $S$
has the form

$S=exp\\{-i(S_{C}+S_{R}+S_{CR})\\}$ (28)

where $S_{CR}$ operates as an interaction term between two subsystems of the
original tensor product structure. If the Hamiltonian of the system has
following relation with $S$

$[H,S]=0,~{}~{}or~{}~{}[H,S_{CR}]=0$ (29)

then there is no any concern about ambiguity of the clock because $S$ would
have a trivial local action on state. When $[H,S]\not=0$ then $H$ is the sum
of two non-interacting terms of $C$ and $R$ in the tensor product structure
which defined by $U^{k}_{i,j}$. However in new tensor product structure which
defined by $\tilde{U}^{i}_{a,b}$ the Hamiltonian will have an interacting term
as follows

$H=H_{\tilde{c}}\otimes I_{\tilde{R}}+I_{\tilde{C}}\otimes
H_{\tilde{R}}+h_{\tilde{C}}\otimes h_{\tilde{R}}$ (30)

Therefore, because of interaction terms, which leads to non-equivalent tensor
product structure, there is no ambiguity here too. In other words, although
there is no local unitary operators which provide the condition (27), we have
$|\phi_{t}>_{\tilde{R}}\not=exp(-iH_{\tilde{R}}t)|\phi_{0}>_{\tilde{R}}$.
Therefore there is no clock ambiguity in the case when there is interaction
between clock and rest system. However it is important to note that we have to
define another clock and rest in order to work in this new basis.

  * Case two

(in near-horizon zone): When we apply the second case that there are
interaction terms between subsystems, in the near-horizon zone we have to note
that defining an interaction term by changing the basis and non-unitary tensor
product structures is not possible. Therefor without the possibility of non-
unitary transformation in new tensor product structure in near horizon of
black hole, we would have a unique clock system. To clarify this issue,
suppose $A$ and $B$ are the quantum states of inside and outside of the near
horizon vicinity, respectively. Like as previous sections, $B$ is the clock
system space and $A$ is the system of rest. If there were an interaction
between clock ($B$) and rest ($A$) systems, then the relative state of $B$
would change, for example, from $|n>_{b}$ to $|n>_{\tilde{b}}$. This changes
of basis for $B$ leads to contradiction. Suppose an in-falling observer Alice
who knows the early Hawking radiation states $R_{B}$ want to fall into black
hole. She is aware of $A$ the state of inside by knowledge of her black hole
by the map ($A=R_{B}$) and also knows that $B$ is purified by $A$. If there is
an interaction terms between $A$ and $B$, then Alice will confront with the
states $|n>_{\tilde{b}}$ which are not the purified states of $|n>_{a}$ that
she expected to confront with them.

In order to avoid contraction the only state of the near horizon zone is (10)
and changing of the basis is restricted to only unitary tensor product
structure in this particular region. Therefore there is no possibility of any
interaction between clock and rest systems. In other words all of the tensor
product structure are equivalent in this zone. So there is only one
possibility of constructing clock system and this clock is unique.

##  7 Measurement and time problem in near-horizon of black hole

The PaW mechanism can be applied on a certain system sequentially which is
suitable to describe statistic of measurements. Here we will follow and review
the von Neumann method [24] and [21] but with restricting it to particular
region of horizon . This proposal which overcomes to some criticisms in PaW
picture in usual Hilbert space, is investigated in near-horizon zone and it is
indicated that in this region the method is not applicable.

In doing so, suppose a measurement apparatus which can describe the process in
terms of a memory systems M. Before the measurement the memory system is in a
ready state $|r>_{M}$ and will be in a state $|i>_{M}$ contains the
measurement outcome after. By assuming the measurement is done at time $t_{1}$
one can write the following relation

$|\psi(t_{1})>_{a}\otimes|r>_{M}\rightarrow\sum_{i}\hat{K}_{i}|\psi(t_{1})>_{a}\otimes|i>_{M}$
(31)

Now we put (15) into the equation (31) which is given by

$|\psi(t_{1})>_{a}\otimes|r>_{M}=_{b}<\psi(t_{1})|\Psi>>\otimes|r>_{M}\rightarrow\sum_{i}(\hat{K}_{i})(_{b}<\psi(t_{1})|\Psi>>)\otimes|i>_{M}$
(32)

Now we put equation (10) into equation (32)

$|\psi(t_{1})>_{a}\otimes|r>_{M}\rightarrow\sum_{n=0}^{\infty}\sum_{i}x^{n}(\hat{K}_{i})(_{b}<\psi(t_{1})|n>_{a}|n>_{b})\otimes|i>_{M}$
(33)

where $K_{i}$ are Kraus operators fulfilling the normalization condition
$\sum_{i}K_{i}^{\dagger}K_{i}=I$. For the special case projective
nondegenerate von Neumann measurements, the $K_{i}=|i><i|$ are projectores on
the eigenspaces relative to the eigenvalues $|i>$ of the observable. Therefore
(33) becomes

$|\psi(t_{1})>_{a}\otimes|r>_{M}\rightarrow\sum_{i}\psi_{i}(t_{1})|i>_{a}\otimes|i>_{M}$
(34)

The probability of getting outcome $i$ is as follows

$P(i|t_{1})=||K_{i}|\psi(t_{1})>_{a}||^{2}$ (35)

The state of system $a$ immediately after recorded by memory $M$ is

$|\phi_{i}>_{a}=K_{i}|\psi(t_{1})>_{a}/\sqrt{P(i|t_{1})}$ (36)

Now , we consider the above process in PaW approach in near-horizon zone. In
doing so, we suppose that the interior of horizon of black hole $A$, rest
system $R$ consists of both ”observed” which labeled by $a$ and the ancillary
memory system which labeled by $M$. In order to insert the memory system which
is needed for von Neumann prescription for measurement process, we have to
indicate its role by a Hamiltonian. As we mentioned before the Hamiltonian of
the near-horizon zone without memory system is $H=H_{C}\otimes
I_{R}+I_{C}\otimes H_{R}$. However when we need to insert memory system too
for measurement process the Hamiltonian is given by

$H=H_{C}\otimes I_{R}+I_{C}\otimes H_{R}(t)$ (37)

where $H_{R}(t)=H_{a}(t)+\delta(t-t_{1})h_{aM}$. Therefore the Hamiltonian
(37) is written as follows

$H=H_{C}\otimes I_{R}+I_{C}\otimes H_{a}(t)+I_{C}\otimes
h_{aM}\delta(t-t_{1})$ (38)

Notice that it is assumed $a$ and $M$ except for a strong coupling between
them at time $t_{1}$ and since $M$ is a memory then there is no free dynamic
for it. In other words, $h_{aM}$ is related to a unitary operator which is
responsible of mapping (34).

As we mentioned in the previous sections, recording memory and the existence
of interaction terms in the Hamiltonian are in conflict with near-horizon
properties of black hole. In this prescription of measurement the memory
recording is done by relation (34) and the interaction in the Hamiltonian
indicates itself in (38). However we know that these definitions are not
possible in black hole’s zone.

One of the main criticism to the PaW approach is that the mechanism cannot
provide a correct propagator or the correct two time correlation. This means
that after the first time measurement, the clock remains stuck. One of the
proposal to overcome this criticism is in [21]. We reviewed this proposal in
this section and indicated that it is not applicable for near-horizon zone.
Therefore, the problem of remaining the clock stuck is maintained in black
hole’s zone.

##  8 Conclusion

Here we reconsidered PaW mechanism in near-horizon zone and some of properties
and controversial ideas of it in this particular vicinity. It is indicated
that recording the history of system in near-horizon zone is not possible.
This system is the vacuum state of black hole’s horizon. For an infalling
observer into the black hole or static observer in near-horizon zone recording
history in $ER=EPR$ paradigm and frozen vacuum of near horizon leads to
controversial issues.

The second result is that there is no arrow of time for the vacuum sate of
near horizon. Like as unitary quantum theory there is no arrow of time in PaW
mechanism. If the arrow of time in this approach is considered, it must be
imposed by separate postulate which under dynamical law a particular property
increases. In PaW approach by dividing one of subsystems in two systems
”observed” and ”pointer” which pointer is for recording track of the evolution
of system, and by using entanglement property of these two systems, we
indicated that this entanglement property does not provide a consistent
mechanism for recognizing the arrow of time in near-horizon zone. Although the
entanglement is a good feature for considering the arrow of time in PaW
approach, it is not applicable in this particular area of black hole.

In addition we considered the time ambiguity for the region of near horizon.
It seems that there is no ambiguity in clock definition of system in PaW
mechanism like as there is no any ambiguity that we have for common Hilbert
space. The main result of this part is that, as long as, it is not possible to
have a interaction term between clock and rest systems then there is only one
unique clock by using PaW approach in near horizon region of black hole. In
usual Hilbert space, although there is no ambiguity in clock system and
definition of it, by interaction term we have to define a new clock system. In
near horizon vicinity there is only one unique clock because all of the
changes of the basis are locally unitary which provide the unitary of the
whole system.

For final conclusion we investigated the other problem in PaW approach which
refers to fact that at first measurement, the clock remains stuck. This
feature does not allow a correct propagator or correct two-time correlations.
There is a proposal in this literature to overcome this criticism by using
conditioning through a von Neumann description of the measurement interaction.
We reconsidered this proposal in near-horizon zone. As long as there is no
interaction terms and memory recording in zone then this proposal is not
applicable in this region.

At the end it is important to note that, since near-horizon zone is very
particular region which one expects to have both quantum mechanics and general
relativity there then investigating for quantum gravity in near-horizon is
more helpful that in usual Hilbert space in curved space or weak gravity. Time
problem in quantum mechanics and also in canonical quantization of general
relativity are very controversial issues which, we think, near-horizon zone is
very suitable place for investigating this problem.

##  9 Acknowledgments

This work is based upon research funded by Iran National Science Foundation
(INSF) under project No 99033073.

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