All oil and gas companies working in inaccessible areas come up against the problem of having to install their own energy infrastructure. In using cutting-edge technologies in electricity generation Gazprom Neft is not only improving energy efficiency but also reducing environmental impacts and increasing its effective utilisation of associated petroleum gas (APG) and renewable energy sources.
Gazprom Neft is involved in field development in remote, undeveloped regions, completely lacking any utilities infrastructure. Under such conditions the question of supplying power to field assets becomes imperative — the key response to which lies in the construction of gas-turbine power plants (GTPPs), thus also addressing the question of effective APG utilisation, used as fuel in independent power generation.
A 84 MW-capacity GTPP was launched at the Vostochno-Messoyakhskoye oil and gas condensate field in the Yamalo-Nenets Autonomous Okrug in September 2016 — the northernmost onshore GTPP in Russia. In order to protect the underlying permafrost the GTPP — which has a total weight of 1,500 tonnes — has been set on supports, with a thermo-stabilisation system set in place two metres above the ground.
In the Balkans, where production volumes are not as high as at Siberian fields, Gazprom Neft's Serbian subsidiary, NIS, is installing small co-generation plants which operate on the principle of converting wellhead gas into heat and power. Thirteen such electricity stations are already in operation.
In order to ensure an uninterrupted power supply to its biggest refinery, NIS, together with Gazprom Energoholding, is building 200 MW combined heat and power (CHP) facility in Pančevo. This is the first combined-cycle facility to be built in Serbia equipped with both steam-power and gas turbine units.
In a gas turbine facility the end-product of fuel combustion turns the turbine, which is located on the same shaft as the generator producing the electricity current. The remaining energy is heating the steam, rotating the steam turbine, which has a second generator attached.
Construction of a combined-cycle CHP plant is a more effective solution since it also has a noticeable environmental benefit in reducing fuel consumption and, consequently, greenhouse gas emissions.
Serbia has become a genuine proving ground for Gazprom Neft in terms of the development of renewable energy technologies. Together with Energowind, NIS is implementing a project for the construction of a windfarm in Plandište (Serbia), involving the installation of 34 wind turbines, which will produce 240 gigawatt hours of electricity per year.
A further area of focus concerns geothermal energy, whereby electricity is generated by steam turbines powered by energy derived from high-temperature underground reservoirs.
NIS already has several successful projects in this area, and in October 2016 signed an agreement with Singapore's Betec on the joint development of active geothermal zones in northern Serbia.
Using alternative energy sources can be effective, even at remote fields. In 2017 Gazprom Neft began pilot testing its 47.5 MW-capacity "Yurta" combined wind and solar electricity plant, in the Yamalo-Nenets Autonomous Okrug, designed to supply electricity to a local transmission station. Hybrid technologies are making possible significant cost reductions in supplying electricity to remote facilities some distance from network infrastructure, by eliminating the need to install transmission lines.


Once you have reached retirement, it's too late to worry about what might have been if you had saved more while you had a job.
The focus on finances changes tack and you need to consider what steps to take to maximise your spending power on a sometimes severely limited budget.
Problems that mount include maintain your home, replacing your car, inflation increasing the cost of living and how to cope with care bills.
Consumer magazine Which? discussed these problems with retired members.
Many agreed one of the biggest financial problems was sourcing a care home for a relative and then working out how to pay the bills.
How much are care fees?
Not everyone has to pay, but the rules are strict.
Anyone with an ongoing healthcare need can ask the National Health Service to pick up the bill and those with savings of less than £23,250 in England, £27,250 in Scotland or £40,000 in Wales are also exempt.
Care fees can range between £32,000 and £72,000 a year.
Inflation is also a fear for many who do not have index-linked pensions as the Bank of England's favoured annual rate of around 2.5% outstrips any rise in savings and investments.
Which? suggests annuities are one solution, but although they offer guaranteed income for life, the rate of return is meagre.
Downsizing is another possibility, but moves can go wrong if you move to an area that has a more expensive cost of living.
Lots of pensioners are worried that tradesmen will rip them off if they need car or property maintenance.
Finding reliable trades is not easy, but try word-of-mouth recommendations from neighbours or web sites that check references like Trusted Traders or Checkatrade.
Some products and services are more expensive as you age – such as travel, car and health insurance.
Which? says it is not uncommon for cover to double in price or for firms to decline the risk leaving a limited market.
All you can do, says Which?, is shop around for the best quote and do not stick to comparison sites as they often do not have links to specialist providers for the elderly.


Company Name: Fischer Auction Co Inc.
It's tragic if you end up losing your vehicle to the loan company for being unable to make the monthly payments on time. Nevertheless, if you are trying to find a used car or truck, looking for cars for cheap could be the best idea. Mainly because banks are usually in a rush to sell these autos and they achieve that by pricing them lower than the marketplace rate. For those who are fortunate you may get a well maintained car with minimal miles on it. But, before you get out your check book and start hunting for cars for cheap in Lincoln commercials, its best to attain fundamental understanding. The following guide is designed to tell you all about obtaining a repossessed car or truck.
First of all you need to comprehend when looking for cars for cheap is that the loan providers cannot quickly take a car from the authorized owner. The whole process of posting notices and also negotiations on terms usually take several weeks. Once the documented owner obtains the notice of repossession, they're already stressed out, infuriated, and irritated. For the bank, it generally is a straightforward business method but for the automobile owner it is a highly stressful situation. They're not only distressed that they're surrendering his or her car, but many of them really feel hate towards the bank. Exactly why do you should be concerned about all of that? For the reason that many of the car owners feel the urge to trash their automobiles just before the legitimate repossession happens. Owners have been known to tear into the leather seats, destroy the glass windows, tamper with the electrical wirings, along with destroy the motor. Regardless of whether that's far from the truth, there's also a good chance the owner failed to do the critical maintenance work due to financial constraints.
Because of this while searching for cars for cheap the cost really should not be the key deciding factor. A lot of affordable cars have really low price tags to take the attention away from the unknown damages. Furthermore, cars for cheap commonly do not have extended warranties, return policies, or the choice to test-drive. For this reason, when considering to shop for cars for cheap the first thing will be to perform a detailed review of the car. It will save you some cash if you possess the required knowledge. If not don't shy away from employing an expert auto mechanic to acquire a thorough report about the car's health.
So now that you've a general understanding as to what to look out for, it is now time for you to search for some cars. There are several diverse locations from where you should purchase cars for cheap. Every one of them comes with their share of benefits and disadvantages. Here are 4 areas to find cars for cheap.
City police departments are a smart place to begin searching for cars for cheap. These are seized autos and therefore are sold very cheap. It's because the police impound yards are usually cramped for space making the police to sell them as fast as they possibly can. Another reason law enforcement can sell these cars for cheap for less money is that they are repossesed automobiles so any revenue that comes in through offering them will be pure profit. The downfall of buying through a law enforcement impound lot is that the vehicles do not have any guarantee. While going to these kinds of auctions you have to have cash or more than enough funds in your bank to write a check to cover the automobile in advance. If you do not find out where you should seek out a repossessed auto impound lot can prove to be a major problem. The most effective and also the simplest way to discover a police auction is usually by calling them directly and inquiring about cars for cheap. Most police auctions often conduct a monthly sale accessible to everyone and also dealers.
Sites for example eBay Motors normally perform auctions and also offer a fantastic spot to locate cars for cheap. The right way to filter out cars for cheap from the ordinary pre-owned cars is to look with regard to it inside the description. There are tons of private dealerships and retailers that purchase repossessed cars coming from financial institutions and then submit it on the net for online auctions. This is an excellent solution if you wish to look through and also review lots of cars for cheap without leaving the home. However, it's wise to go to the dealership and then examine the car personally when you focus on a specific car. If it is a dealer, request for a vehicle examination report and in addition take it out to get a short test-drive.
Most of these auctions tend to be focused towards reselling cars to resellers and vendors instead of private buyers. The particular logic behind that is uncomplicated. Dealerships will always be hunting for good cars for cheap to be able to resale these cars for cheap for any gain. Car or truck resellers also buy several autos at one time to stock up on their inventories. Seek out lender auctions which might be available to the general public bidding. The simplest way to obtain a good price will be to arrive at the auction early on to check out cars for cheap. It's important too not to get swept up in the joy or become involved in bidding wars. Don't forget, you happen to be here to get an excellent bargain and not to appear like a fool that throws cash away.
In case you are not really a fan of going to auctions, then your only real choice is to go to a car dealer. As previously mentioned, car dealers acquire automobiles in mass and typically have got a quality collection of cars for cheap. Although you may wind up paying a little bit more when purchasing through a dealership, these kinds of cars for cheap are generally extensively checked out along with feature guarantees together with free services. One of many negative aspects of shopping for a repossessed automobile from a dealership is that there is hardly a visible cost change when compared with regular used cars. This is mainly because dealerships must deal with the expense of restoration and also transportation to help make these vehicles street worthwhile. This in turn it creates a considerably greater cost.


Did you know that the correct infant car seat placement can save baby's life in case of a car crash?
The best infant car seat placement.
Read our tips on where and how to install a car seat.
It shouldn't have to be stated that every parent wants the maximum safety and comfort for their child in every part of their lives.
With school runs and working lives, driving can be a big part of a modern parent's lifestyle and therefore correct placement of the child seat should be of paramount importance.
Unfortunately this is not always the case as many adults, especially men who usually refuse to read the instructions, often just stick the car seat in anywhere, buckle it as best they can and off they go.
In the unfortunate event of an accident, correct seat placement can mean the difference between life and death.
The following instructions will assist you in placing the car seat in the most efficient way.
Ensure the seat is fitted and secured tightly. There should be no more than an inch movement in any direction and you should be using the correct harnesses and straps that are labeled on the instructions. Don't be afraid to ask for help if the car seat seems to be moving more than necessary, and follow the instructions by the book; don't create some elaborate network of string, tape, buckles and loops.
Place the seat in the center of the vehicle. A recent study has indicated that the rear center of the vehicle is the safest place for the child as this protects it best from side impacts. Parents often put the baby in the front passenger seat or the rear passenger seats but according to studies performed over recent years, based on insurance claims and car accident data, the safest place is in the center of the vehicle in the rear. (University of Pennsylvania School of Medicine).
What about the front seat? You should never place a child car seat in the front seat of the car with an active airbag. In case of a crash the airbag can hit baby's head with such a force that will injure or kill him. In case you do put the baby in the front make sure you swith the airbag of first. If you cannot disable the airbag, consider moving the front passenger seat back at the maximum distance from the dashboard. If you think that having a child in a rear facing position in the back makes it difficult to keep an eye on your baby, consider using a car baby monitor or car mirror. This way there is no need to take your eyes of the road and risk an accident.
Front facing or rear facing seat? You should put off buying a front facing seat until the child is over 2 years of age. Studies have found that the safest position for both the child's safety in the case of an accident, and the child's positional development is rear facing. Just because you want to see your little ones beautiful smile, don't put his or her safety at risk. Most modern seats have angle adjusters installed. The car seat should be at the right angle so your baby's head does not fall forward and close the airways. The correct angle is specified in the manufacturer's manual.
Always read the manufacturer's instructions. Even if you are a macho man and can put an Ikea bookshelf together in 5 minutes, always read the instructions on car seats. Also read your vehicles manual on car seat placement as it will tell you the details based on studies and tests on the cars specs. Each car seat can be different and each vehicle is certainly different so always read all the information before putting a child in the car.
Often people do not have the patience to read the complete instruction manual. Many car seat companies offer complementary videos you can watch on You tube. It is quicker and easier to understand the instructions when you can actually see how it's done. Here are two videos that will help you to install a baby car seat.
Check your car seat manual for recommendations where you can actually place the car seat in your car.
If possible put your baby in his rear facing infant seat in one of the outside seats and the second car seat with a front or rear facing toddler in the middle. As the baby has already extra protection being in his rear facing infant seat, the toddler will get the extra protection from being in the center.
Children placed in the center rear seating position are at a 43% lower risk of getting injured than children placed in one of the rear outside seats.
So only if this doesn't fit, then you will have to use the two outside seats. Whether you place the seat behind the passenger or driver seat, statistics show that there is no safest side.
Fitting 3 car seats may become quite a challenge. So, for your children's safety, I recommend you get the help of a professional. Click here to find the location of a certified child safety seat technician in your area that can teach you how to properly secure your child and solve all your problems with infant car seat placement.


[SOLVED] Possible Conflict With Atahualpa and WP2.9???
Help! I need to change the language to Chinese!
Page linking directly to portfolio site?
[SOLVED] I want to configure my WordPress blog.
[SOLVED] Remove byline or kicker when post is sticky?
[SOLVED] Can't make my blog theme in my language.


text:
Home / Popular Culture & Performing Arts / Animation / Art in Anime
Art in Anime
The Creative Quest as Theme and Metaphor
Art in Anime quantity
SKU: 9780786465613 Categories: Animation, Popular Culture & Performing Arts
Anime, hand-drawn or computer-animated Japanese cartoons, appears in television series, films, video, video games, and commercials, and represents most genres of fiction. This critical study explores anime's relationship with art from a twofold perspective. Drawing from categories as varied as romance, comedy, slice of life drama, science fiction, bildungsroman, and school drama, it examines anime's representation of characters pursuing diverse artistic activities and related aesthetic visions, focusing closely on the concepts of creativity, talent, expressivity and experimentation. Additionally, the analysis engages with anime's own artistry, proposing that those characters' endeavors provide metaphors for the aims and objectives pursued by anime itself as an evolving art form. The cross-cultural resonance of this work makes it relevant not only to anime fans and scholars, but also to those interested in the phenomenon of image-making.
Bibliographic Info: filmography, appendix, bibliography, index
1. Cultural Perspectives 5
2. The Search for a Language 58
3. Mythopoeia 110
4. Performance and Visuality 156
Appendix: Japanese Art Periods Timeline 224
"highly recommended…meticulous"—The Midwest Book Review; "recommended"—Choice.
You're viewing: Art in Anime $35.00

redpajama_set_name: RedPajamaCommonCrawl


text:
The Demise of Ramsey SteamShip Company
Written by Robert Straughton (Ferry and Coastal Correspondent)
Category: Coastal Commentary
I was greatly saddened to learn of the demise of Ramsey SS Co which was wound up earlier this year due to the liabilities of its share of the merchant navy pension fund.
It came to an efficient and relatively painless end, its remaining ships Ben Maye and Ben Varrey being acquired by one of the other few remaining small UK based operations, Absolute Shipping, for whom they continue to trade without change of name.
Ramsey SS Co had been in business for nearly a hundred years originally with small vessels destined to serve Man's several small harbours. Coal and agricultural freights were the staple cargoes for most of their history.
Their Maltese Cross carrying funnels are one of my earliest memories. I would think nothing of walking from my home eight miles along the beach to Whitehaven when invariably there would be a smart Ramsey boat, Ben Ain or Ben Rein, under the coal chute there. On my first visit to the Isle of Man on a Manxman excursion from Workington on the Solway, there was a Ramsey boat offloading in the inner harbour at Douglas. I have never forgotten witnessing her unloading her coal cargo by wicker basket – as recently as the late 1950s! I guess it prevented the coal from being further broken up and dusty.
I also recall the arrival of their colourful motor coasters. That seemed to set them on a more modern trajectory, ahead of their Irish Sea compatriots and rivals, John Monks or Kellys of Belfast.
It remains to be seen whether the trading patterns of the ex Ramsey boats will change radically, but as I write Ben Maye 548,'79 has recently been to Ramsey as well as Hull, Belfast, Rochester, Littlehampton and Runcorn. Ben Varrey 997,'86 is in Antwerp having recently been in the Tees and to Terneuzen, Plymouth, Belfast, Ipswich and Blyth.
Readers may recall the previous vessel sold by Ramsey SS, the little Ben Ellan, which came from the yard of Cook at Wivenhoe as the River Tamar. She was sold in 2010 to an operator in Paramaribo, Suriname, South America, and she has been reported to be moored in the river there though how active she is I have no news. Still, she must have been considered worthy of that long expensive delivery voyage because Paramaribo is a quite a way from Ramsey, and from Wivenhoe for that matter. Brian Climie's stern view of her departing Rotterdam for the tropics clearly shows her Ramsey port of registry under the crudely painted 'Port Vila'.
The UK coastal scene is diminished without the Ramsey SS Co.
More on this and other news in Sea Breezes Magazine - October 2014 Issue


I have found fulfillment in my work. The discoveries and rewards outdistance any creative hurdles. Various mediums stimulate and build into a single piece of art, a few of which are presented here.
Coming from a creative home my mother, Anne Markin, was a sculptress and painter. My father, a doctor, was an accomplished pianist. My middle sister Joy (Connie Holm) is a successful pastel artist. It came naturally to do the same. Happily, I see the creative intelligence in my two sons. Dennis Arsenis, my oldest son, won awards as a designer and hand crafted his own jewelry. William Arsenis, my younger son, is a published writer and a pianist.


Holiday Cottages In UK! One Of Best Option To Choose For A Relaxing Getaway!
United Kingdome, being the most-visited travel destination of the world comprises of thousands of accommodation options. Be it a weekend break, family get-together or simply exploring the beautiful countryside, this country offers accommodation for everyone.
When talking about accommodation, be it hotel, resort or a holiday rental cottage, here at the United Kingdom you will get plenty of options to choose from. Check out some of the most-known countryside's holiday cottages that you can choose for your frequent getaways from the hustling-bustling hectic lifestyles.
A perfect luxury retreat in the Oceanview apartments for individual travelers or romantic roamers seeking for a relaxing city break from the hustle-bustle of their hectic lives. Dolphin View is a beautiful property located in the close proximity to the seaside resort, shopping areas, and dining hotspots.
A beautifully-designed bedroom for two, ideal for cozying up or read some good books before going off into the peaceful sleep.
The open-plan living area comprising of perfect luxury feel, making you feel home-away-from-home.
Yet another beautiful retreat that must be considered while planning for a city break from the hectic life is 15 The Boathouse. Centrally located in Rye, Sussex, 15 The Boathouse is well equipped with all the major amenities of the town at just a very distance away. Ideal for couples, the luxury cottage is a perfect romantic getaway featured with all modern amenities, airy interiors, and cozy lighting.
A beautifully designed and decorated bedroom with a dressing mirror, antique bedside lamps, and other unique artwork items.
The Granary is yet another homely cottage conveniently located in the Abbey Dore village of the Herefordshire's Golden Valley. Spread over 3 acres of rolling countryside, there are a total of three exclusively designed cottages offering luxury accommodation facilities to its guest.
A perfect hideaway spot for families, group of friends or romantic roamers seeking to spend some quality time while making the most of this tranquil Golden Valley region of Herefordshire.
The cottages offer bedrooms with fresh, crisp and cozy ambiance ensuring a restful night sleep and amazing morning with stunning countryside views.
Yet another ground floor holiday apartment ideal for weekend hideaway is the Dashwood, a premier luxury cottage located in the Watchet village of Somerset. Providing its guests a cozy base, the Dashwood holiday apartment is perfect for couples seeking to spend some quality time while admiring the beautiful countryside of Somerset.
A well-designed kitchen with contemporary, high-standard units, indeed giving it a clean and fresh look.
So, travelers by now it must be clear that this weekend where you will be heading and will be staying.


text:
Easkey one step away from Croker
The flags and bunting are flying high in Easkey as their All-Ireland Junior semi-final is finally here on Sunday next when Dessie Sloyan's men take on Red Hughs of Donegal in Ballinamore (throw-in 2pm).
https://www.independent.ie/regionals/sligochampion/sport/gaa/easkey-one-step-away-from-croker-37709595.html
It has been a long wait for the Sea Blues since their Connacht final victory over Clonbur on November 18th but Sloyan admits they are ready and counting down the days to the semi-final with the Killygordon side.
They know that victory on Sunday means a final in Croke Park which is where every player dreams of playing but Sloyan said all focus is on Sunday and the work that is required come throw-in.
He said: "We are all set now we are really looking forward to a big game on Sunday.
"It has been a long wait but it has crept up quick enough. We took a break for a couple of weeks and thankfully the weather has been good too and the pitch is decent so training has been going well.
"The only problem was that there were no teams we could play in challenge matches as they were not training so that was the toughest thing.
"We don't have that big of a big panel so the only part that was tough was not having the match practise so all we could do was train but we are fine and happy enough with how things are looking," Sloyan added.
"I suppose it was nearly 10 weeks since the Connacht final at the end of November which is a long wait but then we knew come Christmas that the game was only 3 weeks away. Everyone is looking forward to it, we have 28 on the panel which is decent enough.
"We have a few missing through injury but it is not too bad we are looking pretty much like a clean bill of health," the manager said.
Back to their dramatic Connacht final win, it was a Ryan McKenna point in the second minute of injury time that sealed the victory for the Sea Blues and paved their way to the semi final. Sloyan admits though it was not easy stuff to watch from the sideline at the end. "Yes it was dramatic I suppose especially on the sideline! But a one-point win is as good as any. It was great to win and I suppose we let them back into it and should have taken our chances but we learn a lot from games like these."
Having a team from the county in an All-Ireland semi-final is good for Sligo football as a whole, he pointed out. "It is great for Sligo football because winning Connacht titles is not that often for us and we will take any as they come. Especially when we beat the Leitrim, Galway and Mayo champions along the way it was huge for us. It shows the depth of talent there in the team and it is important that the lads are doing well and that they push on the in the county too."
The village of Easkey and the surrounding parishes are all getting behind Sloyan's men for Sunday's big match. "In the last week or so it really has been building and in fairness to the local businesses they are getting behind the team and so too is the parish.
"The neighbouring clubs too have been great and really supportive which is important." Their opponents on Sunday in Ballinamore are Red Hughs from Donegal and Sloyan said they know any team in a semi final is going to be tough.
"We know a bit about them and know they are a good, solid team that are very well set up and very difficult to break down. Winning Ulster shows that they are a strong team and it is going to be a tough test but that is what you're expecting when you are in an All-Ireland semi final."
Easkey know they are one step away from playing in Croke Park and what that would mean to the players and management alike. "
Sloyan says though that the main focus is on the semi final. "We are focusing on Sunday. The goal would be playing in Croke Park as a club player. The fact the final is in Croke Park is huge. Very few players get to represent their club in Croke Park and that is the incentive.
"We haven't looked further than Sunday's game, that is the main thing at the moment.
"We had a couple of bad years, things didn't go our way in terms of relegations and all of that but in fairness to the fellas they are young and working hard. It is a very young panel, there are a few older guys too. The main focus is to promote them and get them playing to a better standard, getting the results which they have been doing and as a manager I am very proud of them," Sloyan added.
Throw-in on Sunday in Ballinamore is 2pm.
Sligo Champion
Mayo comfortable winners over Sligo
Austin Garvin This semi-final was a one-sided affair from beginning to end! Mayo was undoubted favourites coming into the game but Sligo's capitulation was totally unexpected. The Yeats County had reached the final of the Philly McGuinness Cup and did reasonably well against Tyrone, but on this occasion they were never competitive.
CT Gaels are Feile champions
The girls from CT Gaels won the John West Feile Peile na nOg Cup for the first time in their short history on Sunday 30th June 2019, at their 3rd attempt after coming up short in 2017 and 2018.
New York GAA side gain valuable experience at Sligo tournament
Last Weekend, the Sligo GAA U16 development squad played host to their New York GAA counterparts over a 3 day visit to the northwest.
The Sound of Music booklet
A booklet of photographs from the recent production of The Sound of Music by the... News
Sligo-Leitrim TD MacSharry says Mercosur deal is bad for farmers
Fianna Fáil TD for Sligo-... News
Latest survey says property prices up 3.5% in last quarter
Property prices in Sligo have risen by €1,000 in the last year, according to the latest... News
229 households were in mortgage arrears at the end of last year
Families in mortgage distress in Sligo have been given fresh hope after the county's upper... News
Lend your help to Swift survey
Since May of this year, BirdWatch Ireland (BWI)... News
Building hit by car for third time
There were traffic diversions in place along the Inner... News
Motorcyclist's mother pleas for driver caution
The mother of a 34-year-old man who was killed after his motorcycle collided with a car outside Lough Rynn Castle has appealed to... News
Gardaí urging public to be wary about bogus Revenue texts
Gardaí in Sligo are urging people not to respond to fraudulent text messages... News
Thought the road was clear
In her interview with gardaí following the accident, Ms Leonard said she did not see the motorcycle 'until the last second' and she... News
Blissfully happy before death
Just days before 34 year old Glenn Carty lost his life in a road accident in August 2017 he had told his mother he was 'blissfully happy'... News


text:
require 'active_support/all'

# utility to generate a random example file for a votation.
puts "hello #{ARGV[0]}, #{ARGV[1]}"

candidates = (1..ARGV[0].to_i).to_a

votes = {}
ARGV[1].to_i.times do |vote_index|
  vote = candidates.shuffle.in_groups_of(candidates.sample).map { |g| g.compact.join(',')}.compact.join(';')
  votes[vote] ||= 0
  votes[vote] += 1
end

file_name = "vote#{ARGV[0]}-#{rand(10000)}.list"
File.open(File.join(File.dirname(__FILE__), 'support', 'examples', file_name), 'w') do |file|
  file.puts ARGV[0]
  votes.each do |key, value|
    file.puts "#{value}=#{key}"
  end
end

puts "file generated: #{file_name}"

redpajama_set_name: RedPajamaGithub


I hereby confirm my child's withdrawal from the above class with effect from the aforementioned date.
Note: Please click the "Confirm Class Withdrawal" button to process the above withdrawal.
You'll be redirected to a confirmation page after the above withdrawal has been successfully processed.


Pitkälampi forest camp is located within 10 minutes drive from Taivalkoski village center. This peaceful resting place lies in true nature by beautiful Pitkälampi lake. Facilities are the main house and traditional sauna cabin by the lake. In the sauna building there also is a cosy cabinet with an open fire place. The main house offers opportunity to be your own chef – prepare yourself a meal on an open fire grill! Not for winter use.
An overnight stay can be arranged in the cabinet for 8 persons.
Pitkälampi area suits nicely also for company meetings or small festive occations.


Support the red, white and blue with this adorable star tin filled to the brim with sixteen assorted patriotic cause ribbons, stars and American flag lollipops.
All lollipops are hand-made, hard-candy in flavors including cotton candy, marshmallow and cherry.
A sweet way to say we love the red, white and blue!


The Toronto Maple Leafs star is pumped for the playoffs, but he is also excited about the release of a new collaboration that saw the young center shoot his own film with teammate Mitch Marner.
The searing spotlight of the playoffs are here and one of the most high-profile series of the first round is a rematch. Last year, the Boston Bruins knocked out the Toronto Maple Leafs in a wild series that featured huge momentum changes and a lot of action. And while the B's got the 'W' in 2018, the Leafs aren't shying away from their rival for the redux.
Playoff success has eluded the Maple Leafs so far in the Brendan Shanahan era and Boston will be a very difficult opponent. If Toronto is going to succeed, the best players must rise to the occasion and that includes star center Auston Matthews.
After hitting a new career-high in goals (37) and points (73), despite missing 14 games due to injury, Matthews is looking forward to his third straight playoff berth.
It has been a pretty exciting season in Toronto already, with the hype starting back in the summer when marquee free agent and hometown boy John Tavares chose the Maple Leafs as his new franchise. Tavares, Matthews, Mitch Marner and Morgan Rielly drove the bus offensively for the squad and as Matthews completes his third NHL season, he contemplated the growth he experienced as a 21-year-old.
To that end, Matthews has been extending his interests off the ice recently, too. Of course there was the fashion shoots he did for magazines in the summer, but there's also a new project he has launched with tech giant Apple, which saw him as the filmmaker of a short starring his good buddy Marner (check it out below), all shot by Matthews by iPhone.
The fact he got to do it with Marner was also pretty apt. Not only is the duo crucial to Toronto's success both now and in the immediate future, but the two youngsters share a close bond off the ice.
And his partner in crime concurs.
When you watch Matthews on the ice, you see a player who thinks the game at a higher level than most. He sees scoring opportunities that others don't and he has the talent to execute those plays. Perhaps that extends to his other interests, too. It certainly seems to be the case with his Apple movie.


text:
\section{Introduction}
Cooperative sensing exploits spatial diversity to improve sensing accuracy in cognitive radio (CR) systems\cite{CoopSenseSurvey}. While most investigations assume a fixed number of fully cooperative secondary users (SUs) with identical sensing capabilities monitoring a single channel, in practice, there are many possible channels for sensing and transmission, and the sensing accuracy varies over the spectrum and among the SUs. In ad-hoc networks, which do not utilize a central controller, and under the hardware constraints, SUs need to choose both the channels to sense and their collaborators\footnote{We consider only cooperation among the SUs. The PU-SU cooperation \cite{MatchingTheory} is out of the scope of this paper.} for spectrum sensing in a distributed manner. Moreover, since simultaneous SU transmission attempts cause mutual interference and/or collisions, resulting in wasted sensing effort and transmission opportunities, an SUs' agreement on sharing detected spectrum holes is also desirable.

Game theory has been utilized recently to model and analyze SU interactions in cooperative sensing \cite{PartitionFormSenseGame,RLiuEvoGame,ZHanCoalSenseGame,MultiChCoalSenseGame,FairPayoff,AuctionBased,Overlapp,HedonicSenseGame} and opportunistic access \cite{HedonicSenseGame,NetwEcon,JWangChSelSurvey}, but, to the best of our knowledge, only the game in \cite{HedonicSenseGame} takes into account transmission (spectrum access) opportunities when making channel sensing and collaboration decisions in a multichannel CR network. However, the game in \cite{HedonicSenseGame} is not suitable for a heterogeneous environment where the contributions of different SUs within a coalition can vary significantly, although such heterogeneity is typical in wireless networks. For example, suppose several SUs sense the same channel and attempt to transmit over it when it is sensed idle using a distributed medium access control (MAC) scheme. If some of these SUs are located in the proximity of the primary user (PU), they are likely to obtain more accurate sensing results, which lead to better transmission chances than for more distant SUs. In such heterogeneous scenarios, the SUs would benefit if the distributed MAC method was replaced by the following agreement on coordinated medium access: SUs with good sensing accuracy would share their sensing results with other SUs in exchange for a higher payoff (e.g., a higher chance or a larger time share of transmission) when a spectrum opportunity is detected. This observation motivates the proposed two-layer game where coalition formation across and within the channels is based on such an agreement on payoff allocation. In contrast, the game of \cite{HedonicSenseGame} forces all SUs sensing the same channel to cooperate altruistically and awards them equally. As a result, selfish SUs might choose to become free-riders \cite{RLiuEvoGame} by taking advantage of other SUs' accurate sensing results without contributing appropriately. This approach undermines fairness among individual SUs and, thus, might discourage SUs from contributing to cooperative sensing, causing the overall network throughput to decline. While a single-layer coalitional game could be devised to meet the proposed fairness and performance objectives, we demonstrate that the two-layer game structure is more attractive in distributed sensing and access due to reduced complexity, delay, and overhead requirements.

Moreover, to facilitate efficient spectrum sensing, we develop constraints on miss-detection (MD) rates of coalition members and provide novel insights into the performance of fusion rules of sensing results for heterogeneous networks under the constant detection rate (CDR) \cite{YCLiangWCNC} constraints.

The \textit{contributions} of this paper are:
\begin{itemize}
	\item Development and analysis of a two-layer coalitional game that includes:
	\begin{enumerate}[(i)]
		\item An efficient, stable, and distributed coalition formation algorithm that assigns SUs to channels within the CR spectrum.
		\item A contribution-based payoff allocation scheme to promote individual incentives for cooperation.
	\end{enumerate}
  \item Improved spectrum sensing approaches:
	\begin{enumerate}[(i)]
		\item A fair distribution method of the integrated network-level primary collision probability constraint among the coalitions and member SUs sharing the same channel.
		\item Demonstration of performance advantages of the AND-rule combining for heterogeneous sensing environments under the CDR constraints.
	\end{enumerate}
\end{itemize}
While we have first proposed the two-layer game in \cite{YLuCISS}, that paper did not contain several key proofs, practical validation, and complexity analysis of the coalitional game and did not address sensing data fusion rules for heterogeneous environments.

The rest of this paper is organized as follows. Table~\ref{Table:Notation} summarizes significant notation. In section~\ref{Sec:Model}, we introduce the system model and formulate the proposed two-layer game, which is then analyzed in Section~\ref{Sec:Game} using coalitional game theory. Simulation results and comparison with \cite{HedonicSenseGame} are presented in Section~\ref{Sec:Sim}, and conclusions are drawn in Section~\ref{Sec:Conclusion}.

\section{System Model and Two-Layer Game Formulation}\label{Sec:Model}
We consider an overlay slotted\footnote{\label{Note:continuousPUtraffic}Our results can be extended to continuous PU traffic by adjusting the primary collision constraints\cite{JWangChSelSurvey}.} CR ad hoc network with multiple SU pairs and multiple channels under i.i.d. Bernoulli PU traffic. An SU can sense and access only one channel at each time slot due to the hardware constraints. The bidirectional interaction and information exchange within the proposed two-layer game framework is illustrated in Fig.~\ref{Fig:TwoLayerGame}: The top-layer game partitions the SUs into $N$ disjoint groups for sensing different channels, based on the expected payoff computed from the bottom layer. At the bottom-layer, a total number of N coalitional games are played in parallel, and the goal is to determine the collaborative structure among the SUs sensing a given channel as well as the payoff allocation. The two-layer game is introduced below and analyzed in Sections~\ref{Sec:Game} and~\ref{Sec:Sim} under several simplified and typically employed assumptions, e.g., negligible control overhead and time-invariant additive white Gaussian noise (AWGN) channels. We discuss the extension of the proposed game to settings where these assumptions are relaxed in Section~\ref{SubSec:Complexity}.
\subsection{The Top-Layer Game}\label{SubSec:TopDef}
\begin{figure}[!t]
    \centering
    \includegraphics[scale=\FigScaleS]{figures/TwoLayerGame}
    \caption{Two-layer coalitional game.}
    \label{Fig:TwoLayerGame}

\end{figure}

The proposed top-layer game is a \emph{hedonic game}, which is characterized by a set of players and a \emph{preference relation}\cite{Hedonic}. The set of players consists of all SUs $\mathcal{M}=\{1,2,\ldots,M\}$ over the set of all channels $\mathcal{N}=\{1,2,\ldots,N\}$, where each SU is associated with some coalition $C^n=(S,n)$\footnote{Note that a top-layer coalition is a two-tuple specifying both the set of SUs $S$ and the channel index $n$ because the same set of SUs $S$ can achieve different throughputs on different channels.}. The SUs in $S=C^n(1)$ sense and transmit over the $n$th channel, i.e., the sensing decision of each SU $m \in S$ is
\begin{equation}\label{Eq:SenseDec}
n^m_*=n\Leftrightarrow m\in C^n(1).
\end{equation}
The utility of the SU $m\in S$ of the top-layer coalition $C^n=(S,n)$ is given by
\begin{equation}\label{Eq:PTU}
x^{mC^n}=a^{mC^n}R^{mn}
\end{equation}
where $a^{mC^n}$ is the allocated individual payoff of SU $m$ provided by the bottom-layer game,  which is a transferable utility (TU) \cite{CoalGameSurvey}, and $R^{mn}$ is the data rate of SU $m$ on channel $n$ given by $R^{mn}=B^{n}\log_2(1+\gamma^{mn})$, a nontransferable utility (NTU) \cite{CoalGameSurvey}, which can vary greatly across different SU pairs due to their spatial separation. Thus, we refer to the utility (\ref{Eq:PTU}) as \emph{partially transferable utility (PTU).}

The utility (\ref{Eq:PTU}) measures the expected data rate of the $m$th SU over channel $n$. Moreover, following the hedonic game formulation\cite{Hedonic}, we define a preference relation that guides an SU when choosing a top-layer coalition. An SU $m$ prefers to move from channel $n$ to $\tilde{n}$ if the following preference relation is satisfied:
\begin{equation}\label{Eq:Preference}
\begin{split}
\tilde{C}^{\tilde{n}}\succ_m C^n\!\Leftrightarrow\! \left\{\begin{matrix}
 x^{m \tilde{C}^{\tilde{n}}}>x^{m C^n} \\
  \smashoperator[r]{\sum_{ i{\in}  \tilde{C}^n(1)}}a^{i{ \tilde{C}^n}}{+}\smashoperator{\sum_{i\in \tilde{C}^{\tilde{n}}(1)}}a^{i \tilde{C}^{\tilde{n}}}{ >}\smashoperator{\sum_{i\in C^n(1)}}a^{iC^n}{+}\smashoperator{\sum_{i\in C^{\tilde{n}}(1)}}a^{iC^{\tilde{n}}}
\end{matrix}\right.\raisetag{2.8\baselineskip}
\end{split}
\end{equation}
where $\tilde{C}^n{=}(C^n(1)\backslash \{m\},n)$ and $\tilde{C}^{\tilde{n}}$ are the top-layer coalitions that form on channels $n$ and $\tilde{n}$, respectively, if SU $m$ makes the move, and $C^n$ and $C^{\tilde{n}}{=}(\tilde{C}^{\tilde{n}}(1)\backslash\{m\},\tilde{n})$ are the existing top-layer coalitions on these channels. Thus, the top-layer coalition $\tilde{C}^{\tilde{n}}$ is preferable to $C^n$ for SU $m$ if (i) its expected data rate (\ref{Eq:PTU}) improves and (ii) the combined (or equivalently, the average) payoffs of all SUs on channels $n$ and $\tilde{n}$ improve. This \emph{social} utility improvement requirement (ii) introduces a loose constraint on selfish and short-sighted individual movements, with the aim to facilitate fast convergence of the partition formation process. In this sense, it is similar to, yet more effective than, the use of ``history sets'' in \cite{HedonicSenseGame} as will be demonstrated in Section~\ref{Sec:Sim}. For example, in Fig.~\ref{Fig:Example}, SU $m=4$ prefers to switch from channel $1$ to channel $2$ if this movement improves not only its individual throughput, but also the combined transmission opportunities (or equivalently, the average payoff) of all SUs currently residing on these channels, i.e., SUs $1$--$4$ and $6$.

Given $(\mathcal{M},\mathcal{N},a^{\mathcal{M}}, \gamma^{\mathcal{M}\times \mathcal{N}})$, the objective of the top-layer game is to form a \emph{Nash-stable} \cite{Hedonic} partition $\mathcal{P}$ (see Fig.~\ref{Fig:TwoLayerGame} and Table~\ref{Table:Notation}), where each player cannot unilaterally improve its utility $x^{m C^n}$ given Nash utilities of other players.

\begin{figure}[!t]
    \centering
    \includegraphics[scale=0.43]{figures/Example}
    \caption{An example of the two-layer coalition structure and SU movements  with $\mathcal{M}=\{1,\ldots,8\}$ and $\mathcal{N}=\{1,2,3\}$. The top-layer partition is ${\mathcal P} = \{ (\{ 1,2,4\} ,1),(\{ 3,6\} ,2),(\{ 5,7,8\} ,3)\}$ and the bottom-layer partitions are ${\rho ^1} = \{ \{ 1,2\} ,\{ 4\} \}$, ${\rho ^2} = \{ \{ 3,6\} \} $ and ${\rho ^3} = \{ \{ 5\} ,\{ 7\} ,\{ 8\} \}$. SU $m=4$ prefers to move from channel $n=1$ to $\tilde{n}=2$ if (\ref{Eq:Preference}) is satisfied with $C^n=(\{1,2,4\},1)$, $C^{\tilde n}=(\{3,6\},2)$,  $\tilde{C}^n=(\{1,2\},1)$ and $\tilde{C}^{\tilde{n}}=(\{3,6,4\},2)$.}
    \label{Fig:Example}

\end{figure}

\subsection{The Bottom-Layer Game}\label{SubSec:BottomDef}
An output partition $\mathcal{P}$ of the top-layer game determines the set of SUs that can sense and transmit over each channel. The SUs sensing the same channel $n$ can further form disjoint bottom-layer coalitions, resulting in a bottom-layer partition $\rho^n$, as illustrated in Fig.~\ref{Fig:Example}. Within each bottom-layer coalition, the SUs exchange and combine their sensing results to improve the overall successful transmission probability of this coalition. The detected spectrum opportunities are then shared among the bottom-layer coalition members in a coordinated manner using a payoff allocation rule, which determines an SU's share of the slot for transmission and provides collision avoidance (cf. Section~\ref{SubSec:BottomSol}). Formally, we define a \emph{coalitional game} $(S,U^n)$ \cite{CoalGameSurvey} for each channel $n$ with
\begin{itemize}
	\item  \emph{a set of players}: the set of SUs $S\subset \mathcal{M}$ sensing channel $n$, i.e., $S=C^n(1)$ obtained from the top-layer game;
  \item  \emph{a value function}: a function $U^n$ that maps any subset of $S$ to a real value.
\end{itemize}

\emph{Within }each bottom-layer coalition, SUs coordinate channel access through negotiation, instead of relying on a distributed MAC scheme. In contrast, MAC is needed \emph{across} multiple bottom-layer coalitions since several such coalitions could simultaneously detect a spectrum opportunity correctly and, therefore, would compete for access. As in \cite{QZhaoTDFS}, we consider the following two MAC options:

\begin{enumerate}
  \item \emph{0/X-model}: The competing SUs do not employ a distributed MAC scheme. Therefore, their transmission collide and all competing SUs fail to transmit successfully.
  \item \emph{1/X-model}: All competing SUs resort to an ideal MAC scheme to obtain equal transmission chances\footnote{We assume negligible control overhead in this random access MAC model.}. After gaining the right to access, the winning SU shares its transmission opportunity with its bottom-layer coalition's members as required by their binding agreement on payoff allocation.
\end{enumerate}
The chosen MAC option affects the players' \emph{noncooperation cost}, which measures the impact of SU collisions on their transmission opportunities. Due to space limitations, we consider only the 0/X- and 1/X-models, which have a high and a low noncooperation cost, respectively. In practice, fair distributed MAC can be realized through random backoff and control message exchange at the cost of some control overhead and/or missed transmission opportunities.

Next, we consider computation of the value function $U^n$ of the coalition $\eta\subseteq S$ on channel $n$, defined as the overall successful transmission probability of $\eta$ on this channel. Given a partition $\rho^n$ on channel $n$, label all other bottom-layer coalitions in $\rho^n \backslash \{ \eta \}$ on channel $n$ as
\begin{equation}\label{Eq:ParitionExcept}
\begin{split}
\rho^n \backslash \{ \eta \}  {=} \{ {\xi _1},{\xi _2}, \ldots ,{\xi _{|\rho^n |- 1}}\}
\end{split}
\end{equation}
and define a binary-valued random vector of length $|\rho^n |{-} 1$
\begin{equation}\label{Eq:X}
\begin{split}
\mathbf{X}_{\rho^n\backslash\{\eta\}}=(X_{\xi_1},X_{\xi_2},\ldots,X_{\xi_{|\rho^n|-1}})\in\{0,1\}^{|\rho^n |- 1}
\end{split}
\end{equation}
where $X_{\xi_i}\!\in\!\{0,1\}$ is an indicator variable for the event that the coalition $\xi_i$ experiences a false alarm (FA), i.e.,
\begin{equation}\label{Eq:Indicator}
\Pr(X_{\xi_i}=x_i)=\Bigg\{\begin{matrix}
P_{\mathrm{FA}}^n({\xi _i}), & \textrm{if }x_i=1 \\
1-P_{\mathrm{FA}}^n({\xi _i}), & \textrm{if }x_i=0.
\end{matrix}
\end{equation}
The coalition values (successful transmission probabilities) for the two MAC models can be expressed as:
\ifdoublecolumn
\begin{align}\label{Eq:CoalValue0X}
U_\mathrm{0/X}^n(\eta;\rho^n)&=\Pr[\text{$\eta$ is the only coalition in $\rho^n$ that}\notag\\
&\text{successfully detects a spectrum opportunity}]\notag\\
&=\beta^n(1-P_\mathrm{FA}^n(\eta))\cdot{\prod\nolimits _{i=1}^{|\rho^n|-1}}P_\mathrm{FA}^n(\xi_i)
\end{align}
\else
\begin{align}\label{Eq:CoalValue0X}
U_\mathrm{0/X}^n(\eta;\rho^n)&=\Pr[\text{$\eta$ is the only coalition in $\rho^n$ that successfully detects a spectrum opportunity}]\notag\\
&=\beta^n(1-P_\mathrm{FA}^n(\eta))\cdot{\prod\nolimits _{i=1}^{|\rho^n|-1}}P_\mathrm{FA}^n(\xi_i)
\end{align}
\fi
and
\ifdoublecolumn
\begin{align}\label{Eq:CoalValue1X}
&U_\mathrm{1/X}^n(\eta ;\rho^n )=\Pr[\text{$\eta$ successfully detects a spectrum}\notag\\
&\text{ opportunity and wins the MAC contention within $\rho^n$}]\notag\\
&=\beta ^n(1-P_\mathrm{FA}^n(\eta ))\cdot\mathbb{E}\bigg[\frac{|\eta|}{|\eta|+J_{\rho^n\backslash\{\eta\}}\left(\mathbf{X}_{\rho^n\backslash\{\eta\}}\right)}\bigg]\\
&=\!\beta ^n(1\!-\!P_\mathrm{FA}^n(\eta ))\cdot\smashoperator[l]{\sum_{\mathbf{x}{\in}\{0,1\}^{|\rho^n|-1}}}\bigg\{\frac{|\eta|\prod \nolimits_{i=1}^{|\rho^n|-1}\Pr(X_{\xi_i}{=}x_i)}{|\eta|+J_{\rho^n\backslash\{\eta\}}(\mathbf{x})}\bigg\}\notag
\end{align}
\else
\begin{align}\label{Eq:CoalValue1X}
U_\mathrm{1/X}^n(\eta ;\rho^n )& =\Pr[\text{$\eta$ successfully detects a spectrum opportunity and wins the MAC contention within $\rho^n$}]\notag\\
&=\beta ^n(1-P_\mathrm{FA}^n(\eta ))\cdot\mathbb{E}\bigg[\frac{|\eta|}{|\eta|+J_{\rho^n\backslash\{\eta\}}\left(\mathbf{X}_{\rho^n\backslash\{\eta\}}\right)}\bigg]\notag\\
&=\!\beta ^n(1\!-\!P_\mathrm{FA}^n(\eta ))\cdot\smashoperator[l]{\sum_{\mathbf{x}{\in}\{0,1\}^{|\rho^n|-1}}}\bigg\{\frac{|\eta|\prod \nolimits_{i=1}^{|\rho^n|-1}\Pr(X_{\xi_i}{=}x_i)}{|\eta|+J_{\rho^n\backslash\{\eta\}}(\mathbf{x})}\bigg\}
\end{align}
\fi
where $\beta^n$ and $P_\mathrm{FA}^n(\eta)$ are defined in Table~\ref{Table:Notation}, and the number of competing SUs for cognitive access is given by
\begin{equation}\label{Eq:J}
\begin{split}
J_{\rho^n\backslash\{\eta\}}(\mathbf{x})={\sum\nolimits_{i = 1}^{|\rho^n | - 1}} ( {1 - x_i)|{\xi _i}|}.
\end{split}
\end{equation}

Since the bottom-layer coalition value $U^n(\eta;\rho^n)$ depends not only on the actions of the coalition members, but also on the actions of other SUs on channel $n$, this coalitional game is in partition form \cite{CoalGameSurvey}. We assume the overall values (\ref{Eq:CoalValue0X}) and (\ref{Eq:CoalValue1X}) of a bottom-layer coalition are dividable and can be transferred among coalition members according to the allocated payoffs $a^{mC^n}$ [cf. (\ref{Eq:PTU})], so that the bottom-layer game has TU. To illustrate, consider the following example. Suppose the overall value of a two-SU bottom-layer coalition $\eta=\{1,2\}$ on channel $n$ is $U^n(\eta)=0.8$, and both SUs agree on a payoff allocation of $a^{1C^n}=0.2$ and $a^{2C^n}=0.6$. Under the 0/X-model, the coalition $\eta$ acquires a transmission opportunity when $\eta$ discovers an idle time slot. On the other hand, under the 1/X-model, the coalition $\eta$ acquires a transmission opportunity when $\eta$ discovers an idle time slot, possibly competes with other coalitions, and finally wins a right to transmit. The acquired transmission opportunities are then shared between the two SUs according to their payoff allocation agreement, e.g., in a probabilistic manner, with the conditional probabilities that SU $1$ and SU $2$ are allowed to transmit for the entire time slot given by $a^{1C^n}{/}(a^{1C^n}{+}a^{2C^n}){=}1/4$ and $a^{2C^n}{/}(a^{1C^n}{+}a^{2C^n}){=}3/4$, respectively. Other schedule-based multi-access methods for sharing the transmission opportunity can be employed. For example, SU $1$ and SU $2$ can transmit in a time-division multiple access (TDMA) manner, for $1/4$ and $3/4$ of the time, respectively\footnote{SU transmissions might be unsuccessful either due to SU collisions (under the 0/X-model) or MD of PU traffic (under both MAC models).}.

\subsection{Cooperative Sensing}\label{subsec:CoopSenseDef}
Next, we present the cooperative sensing scheme at the physical layer for the set of SUs $S$ sensing channel $n$ assuming a bottom-layer partition $\rho^n$ of $S$. The PU transmission is interrupted if one or more bottom-layer coalitions in $\rho^n$ fails to detect its presence, so the integrated MD probability on channel $n$ is given by
\begin{equation}\label{Eq:ChMD}
\begin{split}
P^n_\mathrm{MD}(\rho^n)=1-\prod_{\eta\in\rho^n}\left ( 1-P^n_\mathrm{MD}(\eta) \right ).
\end{split}
\end{equation}
We impose the following constraints:

\noindent \emph{Network-level} constraint \textbf{(C.1)}: $P_\mathrm{MD}^n(\rho^n)=P_\mathrm{MD}^\mathrm{Ch}$ (cf. Table~\ref{Table:Notation})

\noindent \emph{Coalition-level} constraint \textbf{(C.2)}: All equal-sized bottom-layer coalitions should maintain the same MD rate.

\noindent \emph{Node-level} constraint \textbf{(C.3)}: All SUs within a bottom-layer coalition must satisfy the same MD constraint\cite{RLiuEvoGame,YCLiangWCNC}.

The constraints \textbf{(C.1)} and \textbf{(C.2)} are due to the requirement that the set of SUs $S$ sensing channel $n$ should always maintain the integrated MD rate $P_\mathrm{MD}^\mathrm{Ch}$, regardless of how the coalition formation process evolves, which can be satisfied by
\begin{equation}\label{Eq:EtaMD}
P_\mathrm{MD}^n({\eta})=1-(1-P_\mathrm{MD}^\mathrm{Ch})^{|\eta|/{|S|}}
\end{equation}
for all $\eta\in\rho^n$. The constraint \textbf{(C.3)} is motivated by fairness since all SUs should take equal responsibility for PU protection.

Next, we evaluate performance of the \emph{AND-} and the \emph{OR-combining} rules for the proposed constrained spectrum sensing approach. The coalition-level FA probabilities of these fusion rules can be obtained from \cite[eq.~(10--17)]{YCLiangWCNC} assuming AWGN channels\footnote{In practice, these channels may be subject to fading, and the ergodic sensing accuracy probabilities can be obtained by averaging over the fading distribution \cite{CoopSense,MultiChCoalSenseGame,ZHanCoalSenseGame}.}:
\begin{align}
P_\mathrm{FA,AND}^n(\eta ) &=\smashoperator{\prod\limits_{m \in \eta }}P_\mathrm{FA,AND}^n(m)\label{Eq:CoalFAResultAND}\\
&= \smashoperator{\prod\limits_{m \in \eta }}Q\left(\sqrt{2\lambda^{mn}+1}\phi+\sqrt\nu\lambda^{mn}\right)\notag\\
P_\mathrm{FA,OR}^n(\eta ) &=1 - \smashoperator{\prod\limits_{m \in \eta }}(1-P_\mathrm{FA,OR}^n(m))\label{Eq:CoalFAResultOR}\\
&= 1 - \smashoperator{\prod\limits_{m \in \eta }}{\left(1-Q(\sqrt{2\lambda^{mn}+1}\tilde{\phi}+\sqrt\nu\lambda^{mn})\right)}\notag
\end{align}
respectively, with [cf. (\ref{Eq:EtaMD}) and \textbf{(C.3)}]
\begin{align}
\phi  &\triangleq Q^{-1}\big(1-P_\mathrm{MD,AND}^n(m)\big)=Q^{-1}\big( {( 1 - P_\mathrm{MD}^n(\eta ) )}^\frac{1}{|\eta |} \big)\notag\\
&=Q^{-1}\big((1-P_\mathrm{MD}^\mathrm{Ch})^\frac{1}{|S |}\big)\label{Eq:BetaDef}\\
\tilde \phi & \triangleq Q^{-1}\big(1-P_\mathrm{MD,OR}^n(m)\big)= Q^{- 1}\big( 1 - ( {P_{\mathrm{MD}}^n(\eta )} )^\frac{1}{|\eta|}\big)\notag\\
&= Q^{- 1}\Big( 1 - \big( 1-(1-P_\mathrm{MD}^\mathrm{Ch})^{|\eta|/{|S|}} \big)^\frac{1}{|\eta|}\Big)\label{Eq:TildeBetaDef}
\end{align}
where $m\in \eta\in \rho^n$, and $Q(\cdot)$ is the Q-function \cite[eq. (B.20)]{GoldsmithWireless}. First, from (\ref{Eq:BetaDef}) and (\ref{Eq:TildeBetaDef}), the individual MD constraint $P_\mathrm{MD}^n(m)$ depends on the size of the bottom layer coalition to which an SU belongs for the OR-rule, but not for the AND-rule. Thus, when using the AND-rule, SUs do not have to update the MD constraints when computing their values within different bottom-layer coalitions as long as the SU population $|S|$ on channel $n$ does not change. Second, for both rules, as $|S|$ increases, each coalition $\eta$ reduces its MD rate $P_\mathrm{MD}^n(\eta)$ in (\ref{Eq:EtaMD}), leading to increased FA probability $P^n_\mathrm{FA}(\eta)$ in (\ref{Eq:CoalFAResultAND},\ref{Eq:CoalFAResultOR}) and decreased coalition values in (\ref{Eq:CoalValue0X},\ref{Eq:CoalValue1X}). Thus, large values of $|S|$ are penalized, balancing SU competition on all channels.

Finally, our extensive simulation results using (\ref{Eq:CoalFAResultAND}--\ref{Eq:TildeBetaDef}) show that the AND-rule is more suitable when cooperative SUs have heterogeneous sensing capabilities (i.e., different PU-to-SU SNRs) while the OR-rule provides better performance in homogeneous scenarios. Intuitively, if there is at least one SU $m\in\eta$ with favorable PU-to-SU SNR $\lambda^{mn}$, and thus with small FA probability $P^n_\mathrm{FA,AND}(m)$, the resulting $P^n_\mathrm{FA,AND}(\eta)$ is very small since $P^n_\mathrm{FA,AND}(\eta)\leq \min_{m\in \eta}P^n_\mathrm{FA,AND}(m)$ in (\ref{Eq:CoalFAResultAND}). In contrast, for the OR-rule, if only one member SU has very poor sensing capacity, the entire bottom-layer coalition $\eta$ suffers since $P^n_\mathrm{FA,OR}(\eta)\geq \max_{m\in \eta}P^n_\mathrm{FA,OR}(m)$ in (\ref{Eq:CoalFAResultOR}). For example, in Fig.~\ref{Fig:Hetero} we compare the integrated FA probabilities (\ref{Eq:CoalFAResultAND}) and (\ref{Eq:CoalFAResultOR}) for a coalition $\eta=\{1,2\}$ on channel $n$ when the average PU-to-SU SNR of the two member SUs is fixed to $\bar\lambda$. We observe that the OR-rule and the AND-rule achieve their best performance for homogeneous and heterogeneous sensing capacities (given by PU signal strengths at the sensor) of the two SUs, respectively.  The general case is explored below\cite{YLuThesis}.

\begin{figure}[!t]
    \centering
    \includegraphics[scale=\FigScale]{figures/Hetero}
    \caption{Performance comparison of the $P_{\mathrm{FA}}^n(\eta )$ for the AND- and OR-rule under the proposed MD constraints; $\eta=\{1,2\}$; $P_{{\mathrm{MD}}}^n(\eta )=10^{-4}$; $\nu=5$; $\bar{\lambda}=0.5(\lambda^{1n}+\lambda^{2n})=5$ ($7$\,dB).}
    \label{Fig:Hetero}

\end{figure}

\begin{Prop} \label{Prop:Convex} Consider a coalition $\eta$ on channel $n$ and a fixed bottom-layer-coalition-level MD probability constraint $P_{{\mathrm{MD}}}^n(\eta )$\footnote{Due to space limitations, we omit the proof, which shows the desired quasiconcavity and quasiconvexity by establishing the logconcavity of $P_\textnormal{FA,AND}^n(\eta )$ and $1-P_\textnormal{FA,OR}^n(\eta )$, respectively\cite{YLuThesis}.}. The FA probability of $\eta$ for the AND-rule (\ref{Eq:CoalFAResultAND}) [OR-rule (\ref{Eq:CoalFAResultOR})] is a quasiconcave (quasiconvex) function \cite[Section 3.4]{ConvexOpt}, which achieves its global maximum (global minimum) when $\lambda^{mn}=\bar{\lambda}$, $\forall m\in\eta$, subject to an average PU-to-SU SNR constraint
\begin{equation}\label{Eq:SNRConstraint}
\frac{1}{{|\eta |}}\sum\limits_{m \in \eta } {{\lambda ^{mn}}}  = \bar \lambda
\end{equation}
if $\forall m\in\eta$,
\begin{equation}\label{Eq:ConvexCondition1}
P_{{\mathrm{MD}}}^n(\eta )< {(0.5)^{|\eta |}}
\end{equation}
\begin{equation}\label{Eq:ConvexCondition2}
\tilde{\phi}>-\frac{\sqrt{\nu}(2\lambda^{mn}+1)^\frac{3}{2}}{3\lambda^{mn}+2}.
\end{equation}
\end{Prop}
Note that the constraints (\ref{Eq:ConvexCondition1}) and (\ref{Eq:ConvexCondition2}) are mild\cite{YLuThesis}. For example, they can be easily satisfied if: (i) $P^n_\mathrm{MD}(\eta)\geq 10^{-4}$ and $|\eta|\leq 13$ and (ii) ($\nu\geq 56$ for $\lambda^{mn}\geq 0$) or ($\nu\geq 13$ for $\lambda^{mn}\geq 1$) or ($\nu\geq 2$ for $\lambda^{mn}\geq 10$), i.e., when the MD constraint  $P^n_\mathrm{MD}(\eta)$ is not extremely stringent, the coalition size $|\eta|$ is not too large, and the number of collected samples for spectrum sensing $\nu$ is sufficiently large{\footnote{In practice, low PU-to-SU SNR (e.g. $\leq 1$) requires $\nu$ on the order of hundreds to thousands and medium-to-high PU-to-SU SNR (e.g. $\geq 10$) requires less than $10$ samples for good performance (See, e.g., \cite{Tradeoff,CoopSense}).}}.

Based on these observations, in the remainder of this paper we will assume the AND fusion rule to study a typical wireless CR network with a moderate number of cooperating SUs on each channel, which experience heterogeneous PU-to-SU channel conditions.

\section[Two-Layer Coalitional Sensing and Access Game]{Two-Layer Coalitional Sensing and Access Game\footnote{Proofs of Propositions~\ref{Prop:Characteristic} and \ref{Prop:GrandCoalFormation}--\ref{Prop:Converge} can be found in\cite{YLuCISS}.}}\label{Sec:Game}

\subsection{Grand Coalition Formation and Payoff Allocation in the Bottom-layer Game}\label{SubSec:BottomSol}
Consider the bottom-layer game $(S,U^n)$ on channel $n$.

\begin{Prop}\label{Prop:Characteristic}
For the 0/X-model, the game $(S,U_\mathrm{0/X}^n)$:
\begin{enumerate}[(i)]
	\item reduces to a \emph{characteristic-form game}, i.e., the value of a bottom-layer coalition $\eta$ depends solely on the composition of $\eta$ \cite{CoalGameSurvey}: $U_\mathrm{0/X}^n(\eta;\rho^n)=U_\mathrm{0/X}^n(\eta;\tilde\rho^n)\triangleq U_\mathrm{0/X}^n(\eta)$, for any $\eta \subseteq S$ and any bottom-layer partitions $\rho^n$ and $\tilde\rho^n$ of $S$, such that $\eta \in \rho^n$  and  $\eta \in \tilde\rho^n$;

\item is \emph{superadditive} \cite{CoalGameSurvey}, i.e., the SUs benefit from forming larger coalitions. Thus, for any two disjoint bottom-layer coalitions $\eta,\xi \subset S$, $U_\mathrm{0/X}^n(\eta \cup \xi)\geq U_\mathrm{0/X}^n(\eta)+U_\mathrm{0/X}^n(\xi)$;

\item has efficient grand coalition \cite{Efficiency}, i.e., the grand coalition value is at least as large as the combined value of all coalitions in any other partition: $\forall\tilde\rho^n\neq\{S\}$, $U_\mathrm{0/X}(S)\geq \sum_{\eta\in\tilde\rho^n} U_\mathrm{0/X}(\eta)$. This property follows from (ii).
\end{enumerate}
\end{Prop}

\begin{Prop} \label{Prop:EqualEfficient}
Under the 1/X-model, all bottom-layer partitions of $S$ are equally efficient, i.e., for any two partitions $\rho^n$ and $\tilde\rho^n$ of $S$, $\sum\nolimits_{\eta  \in \rho^n } {U_\mathrm{1/X}^n(\eta ;\rho^n)}  = \sum\nolimits_{\eta  \in \tilde\rho^n } {U_\mathrm{1/X}^n(\eta ;\tilde\rho^n )}$.
\end{Prop}
\begin{proof}
Under the 1/X-model, a spectrum opportunity on channel $n$ is wasted if all bottom-layer coalitions make FAs on the PU presence. Thus, for any bottom-layer partition $\rho^n$ of any $S$ on any channel $n$ [cf. (\ref{Eq:CoalFAResultAND})]
\begin{align}
\sum\limits_{\eta  \in \rho^n } {U_\mathrm{1/X}^n(\eta ;\rho^n )}  =& \beta^n(1 - \prod\limits_{\eta  \in \rho^n } {P_{{\mathrm{FA}}}^n(\eta )}\notag\\
=& \beta^n(1 - \prod\limits_{m \in S} {P_{{\mathrm{FA}}}^n(m)}) \label{Eq:EqualEfficientProof}
\end{align}
which is independent of $\rho^n$.
\end{proof}

\begin{Prop} \label{Prop:NegExternality}
Under the 1/X-model, the bottom-layer game $(S,U_\mathrm{1/X}^n)$ exhibits \emph{nonpositive externalities}\cite{Maskin}, i.e., a merger between two coalitions cannot improve the values of other coalitions. Formally, for any disjoint coalitions $\eta ,\xi ,\zeta  \subset S$ and any partition $\rho^n$ of $S$ such that  $\eta ,\xi ,\zeta  \in  \rho^n$, the coalition value
$U_\mathrm{1/X}^n(\eta ;\rho^n )  \geq U_\mathrm{1/X}^n\big(\eta ; \rho^n \backslash \{ \xi\} \backslash \{\zeta \}\cup \{ \xi  \cup \zeta \} \big)$.
\end{Prop}
\begin{proof}
See Appendix~\ref{Sec:NegExternalityProof}.
\end{proof}

\begin{Prop}\label{Prop:GrandCoalFormation}
The grand coalition always forms in the bottom-layer game under the 1/X and 0/X MAC models.
\end{Prop}
The proof of Proposition~\ref{Prop:GrandCoalFormation} is based on Propositions~\ref{Prop:Characteristic}--\ref{Prop:NegExternality} and follows from the superadditivity property \cite{CoalGameSurvey} for the 0/X model and the \emph{weak efficiency} or the nonpositive externality property \cite{Maskin, Efficiency} for the 1/X model \cite{YLuCISS}\footnote{There are several different points of view on the grand coalition formation in the game theory literature. Here we employ the approaches in\cite{CoalGameSurvey,Maskin, Efficiency}.}. Proposition~\ref{Prop:GrandCoalFormation} implies that all SUs sensing channel $n$ cooperate, and MAC is not utilized. However, MAC needs to be assumed to compute the hypothetical payoff that an SU \emph{could have obtained} if it left the grand coalition and/or the \emph{marginal value} it brings into the grand coalition. Such information is crucial when determining the actual payoff allocation to ensure the stability of and the fairness within the grand coalition \cite{GameSurvey}.

Next, we discuss the allocation of the value of the grand coalition $U^n(S;\{S\})$ (hereafter referred to as $U^n(S)$ for brevity) while satisfying every member SU with its allocated payoff. Fair payoff allocation rules for traditional single-layer coalitional games are extensively studied in the literature assuming (hypothetical) breakdown of the grand coalition, e.g., the Shapley value \cite{CoalGameSurvey}, the Owen value \cite{Owen}, the nucleolus\cite{CoalGameSurvey}, the Nash bargaining solution (NBS)\cite{NBS,NBSNeg}, etc. In this paper, we employ NBS due to its computational efficiency\cite{NBS}. The NBS payoff allocation method distributes the overall value of the grand coalition among the individual players according to the \emph{disagreement point}, which characterizes the minimum payoffs the players are willing to accept. Given the disagreement point, NBS splits the overall profit of cooperation equally among the players and maximizes the product of the players' profits\cite{NBS}.

\begin{Prop}\label{Prop:NBS}	For the characteristic-form (cf. Proposition \ref{Prop:Characteristic}) bottom-layer game $(S,U_\mathrm{0/X}^n)$, NBS assigns a payoff, $\forall m\in S$ \cite{NBS,NBSNeg}
\begin{equation}\label{Eq:NBS}
\begin{split}
a^{mC}_\mathrm{NBS,0/X}=\frac{U_\mathrm{0/X}^n(S)-\sum\limits_{i\in S}U_\mathrm{0/X}^n(\{i\})}{|S|}+ U_\mathrm{0/X}^n(\{m\})
\end{split}
\end{equation}
where the disagreement point is $\big\{U_\mathrm{0/X}^n(\{m\}):m\in S \big\}$.
\end{Prop}
\noindent The $m$th component of the disagreement point $U_\mathrm{0/X}^n(\{m\})$ is chosen as the payoff SU $m$ can obtain with noncooperative sensing, i.e., if bargaining fails. From Proposition~\ref{Prop:Characteristic}, $U_\mathrm{0/X}^n(\{m\})\triangleq U_\mathrm{0/X}^n(\{m\};\rho^n)$ for any partition $\rho^n$ that includes $m$ as a singleton, i.e., it does not depend on actions of other SUs. Since SUs with stronger sensing capabilities have higher singleton values $U_\mathrm{0/X}^n(\{m\})$, the NBS (\ref{Eq:NBS}) provides each SU with the allocated payoff commensurate with its sensing contribution, resulting in a fair payoff allocation.

When the 1/X-model is assumed, the selfish individual payoff $U_\mathrm{1/X}^n(\{m\};\rho^n)$ depends not only on that SU's sensing capability, but also on other SUs' reactions when a given SU does not cooperate. In this paper, we employ the \textit{fine} NBS (\textit{f}NBS) \cite{NBSNeg}, which implies all SUs stand alone if bargaining is not successful, i.e., if the grand coalition breaks down.

\begin{Prop} \label{Prop:fNBS}Under the 1/X model, the payoff of each SU $m\in S$ using \textit{f}NBS is \cite{NBSNeg}
\begin{align}\label{Eq:fNBS}
a_{{\mathrm{\textit{f}NBS,1/X}}}^{mC}&=\frac{1 }{|S|}\bigg[U_\mathrm{1/X}^n(\{ S\} ) - \smashoperator{\sum\limits_{i \in S}} U_\mathrm{1/X}^n\left(\{ i\} ;\{ \{ j\} :{j {\in} S}\} \right)\bigg]\notag\\
&\quad\quad\quad+U_\mathrm{1/X}^n (\{ m\} ;\{ \{ j\} :{j {\in} S}\})\notag\\
&= U_\mathrm{1/X}^n(\{ m\} ; \{ \{ j\} :{j {\in} S}\} ).
\end{align}
\end{Prop}
Thus, the allocated payoff probability is simply given by the successful transmission probability that each SU could have obtained individually assuming all other SUs have also formed singletons. However, this does not imply that SUs should deviate from the grand coalition, because each SU is not guaranteed a payoff of $ a_{{\mathrm{\textit{f}NBS,1/X}}}^{mC}$ if it deviates. In fact, if an SU decides to remain isolated, its payoff is at risk due to the nonpositive externalities (Proposition~\ref{Prop:NegExternality}) if other SUs collude. As a result, an SU $m$ might end up with a much worse payoff than its singleton value $U_\mathrm{1/X}^n(\{ m\} {;}{ \{ \{ j\} :{j {\in} S}\}})$. Therefore, every SU has an incentive to join the grand coalition. Moreover, we found that a \textit{coarse} NBS (\textit{c}NBS) \cite{NBSNeg} allocation where all other SUs form a coalition and try to reduce the hypothetical individual payoff of the deviating SU, results in similar throughput and energy consumption as that of the \textit{f}NBS method in Proposition~\ref{Prop:fNBS} \cite{YLuThesis}.

\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\begin{figure}[!t]

\hrule\hrule\vspace{\Extra}
\textbf{Algorithm 1} Bottom-layer payoff allocation
\vspace{\Extra}
\hrule
\begin{algorithmic}[1]
\Require $C{=}(S,n)$, $\{\lambda^{mn}:m{\in}S\}$, $P_\mathrm{MD}^\mathrm{Ch}$, $\nu$, $\beta^n$
\Ensure $\{a^{mC}:m{\in}S\}$
\State $P^n_\mathrm{FA}(m){=}Q\Big(\sqrt{2\lambda^{mn}{+}1}Q^{-1}((1-P_\mathrm{MD}^\mathrm{Ch})^\frac{1}{\left | S \right |}){+}\lambda^{mn}\sqrt{\nu}\Big)$\label{Line:AdaptTh}
\If   {0/X-MAC}{ $\forall m\!\in\! S$:}
\State $U_\mathrm{0/X}^n(S)=\beta^n(1-\prod_{i{\in}S}P_\mathrm{FA}^n(i))$
\State $U_\mathrm{0/X}^n(\{m\}){=}\beta^n(1-P_\mathrm{FA}^n(m))\prod _{i\in S\backslash\{m\}}P_\mathrm{FA}^n(i)$
\State Compute $a^{mC}_\mathrm{NBS,0/X}$ using  (\ref{Eq:NBS})
\EndIf
\If {1/X-MAC}{ $\forall m\!\in\! S$:}
\State {$a_{{\mathrm{\textit{f}NBS,1/X}}}^{mC} = U_\mathrm{1/X}^n(\{ m\} ;\{ \{ j\} :{j {\in} S}\} )$ as in (\ref{Eq:CoalValue1X})}
\EndIf
\end{algorithmic}
\hrule
\caption{Bottom-layer payoff allocation.}
\label{Fig:AlgBottom}
\end{figure}

Moreover, the bargaining is always successful, i.e., $a^{mC^n}\geq U^n(\{ m\})$ for all $m$ and both MAC models (cf. Propositions~\ref{Prop:Characteristic}--\ref{Prop:fNBS} and \cite{NBSNeg}). Once the negotiation is settled, all SUs in $S$ sign an agreement on the sharing of true sensing results as well as the payoff allocation $\{a^{mC^n}:m\in S\}$. Also specified in this agreement are the transmission coordination protocol and the $m$th SU's fair share of a sensed available slot, given by $\{a^{mC^n}/{\sum_{i\in S}}a^{iC^n}:m\in S\}$ (see Example at the end of Section~\ref{SubSec:BottomDef}). All coalition members should thereafter strictly conform to the finalized agreement\footnote{We assume the SUs are selfish, but honest. In practice, an SU may falsify its sensing result or otherwise breach an earlier agreement, which can be detected  by, e.g., a reputation-based approach \cite{Reputation}.}.  The bottom-layer payoff allocation algorithm is summarized in Fig.~\ref{Fig:AlgBottom}.

\subsection{Coalition Formation at the Top Layer}\label{SubSec:TopSol}
In the top-layer game, SUs use (\ref{Eq:Preference}) to determine if switching to another channel is advantageous. The proposed algorithm in Fig.~\ref{Fig:AlgTop} employs a distributed switching scheduling scheme to facilitate fast convergence to an SU network partition. At most one switch is allowed in each time slot. Initially, all SUs actively compete for the right to switch using an out-of-band control channel \cite{ContrChSurvey}. In every time slot, the winning SU $m$ randomly chooses a potential new channel and notifies other SUs in $\mathcal{M}\backslash \{m\}$ about its action by broadcasting (on the control channel) (i) a SWITCH signal if (\ref{Eq:Preference}) holds, (ii) a HOLD signal if (\ref{Eq:Preference}) fails and it still plans to continue searching, or (iii) a SLEEP signal when all switching opportunities have been exhausted unsuccessfully, i.e., the current channel is the most preferable for this SU. Competition for the right to switch continues until all SUs are asleep, indicating convergence of the partition formation process. Note that all SUs (including sleeping SUs) should cognitively monitor the environment changes, e.g., variation of the PU-to-SU SNRs, the PU/SU locations, the number of SUs, the number of channels, etc., and repeat the partition formation algorithm in Fig.~\ref{Fig:AlgTop} when changes occur (See Section~\ref{Sec:Sim}). Assuming that these parameters are fixed, the following Proposition holds:
\begin{Prop}\label{Prop:Converge} The proposed coalition formation algorithm (Fig.~\ref{Fig:AlgTop}) converges in at most $N^M$ switches.
\end{Prop}

\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\begin{figure}[!t]
\hrule\hrule\vspace{\Extra}
\textbf{Algorithm 2} Distributed top-layer partition formation
\vspace{\Extra}
\hrule
\begin{algorithmic}[1]
\Require $\mathcal{M},\mathcal{N},\beta^\mathcal{N},\gamma^{\mathcal{M}\times \mathcal{N}},\lambda^{\mathcal{M}\times \mathcal{N}}$
\Ensure $\mathcal{P}$
\State \textbf{Initialization}: Each SU $m$ randomly senses a channel $n_*^m$
\State $C^n(1)=\{m:n_*^m=n\}$ and $C^n(2)=n$, $\forall n\in \mathcal{N}$
\State $\mathcal{P}=\{C^1,\cdots, C^N\}$ and $\textrm{Action}=\textrm{SWITCH}$
\While{$\mathcal{M}_\textrm{Active}\neq\emptyset$ \textbf{(at each time slot)}:}\label{Line:While}
\If {$\textrm{Action}=\textrm{SWITCH}$}: {$\mathcal{M}_\mathrm{Active}=\mathcal{M}$}
\EndIf
\If {$\textrm{Action}\neq\textrm{HOLD}$}:
\State SUs in $\mathcal{M}_\mathrm{Active}$ contend for the right to switch
\State SU $m\in C^n(1)$ wins and $\mathcal{N}_\mathrm{Candidate}=\mathcal{N}\backslash \{n\}$
\EndIf
\State $m$ randomly chooses channel ${\tilde{n}}\in\mathcal{N}_\mathrm{Candidate}$
\State $\tilde{C}^n=(C^n(1)\backslash \{m\},n)$, $\tilde{C}^{\tilde{n}}=(C^{\tilde{n}}(1){\cup}\{m\},\tilde{n})$
\State $m$ plays the bottom-layer game $(\tilde{C}^{\tilde{n}},U^{\tilde{n}})$ in Fig.~\ref{Fig:AlgBottom}\label{Line:Play}
\If   {(\ref{Eq:Preference}) holds}:\label{Line:CheckPrefer}
\State $a^{iC^n}=a^{i\tilde{C}^n}$, $a^{jC^{\tilde{n}}}=a^{j\tilde{C}^{\tilde{n}}}$,  $\forall i\in C^n(1)$, $j\in C^{\tilde{n}}(1)$
\State  $\mathcal{P}=\mathcal{P} \backslash\{C^n\}\backslash\{C^{\tilde{n}}\}\cup\{{ \tilde{C}^n}\}\cup \{ \tilde{C}^{\tilde{n}}\}$\State $C^n=\tilde{C}^n$, $C^{\tilde{n}}= \tilde{C}^{\tilde{n}}$,  $\textrm{Action}=\textrm{SWITCH}$
\Else :
\State $m$ stays on channel $n$ and $\mathcal{N}_\mathrm{Candidate}=\mathcal{N}_\mathrm{Candidate}\backslash \{\tilde{n}\}$
\If{$\mathcal{N}_\mathrm{Candidate}\neq\emptyset$}: {$\textrm{Action}=\textrm{HOLD}$}
\Else : {$\mathcal{M}_\mathrm{Active}=\mathcal{M}\backslash \{m\}$, $\textrm{Action}=\textrm{SLEEP}$}
\EndIf
\EndIf
\If{a slot is sensed idle by $C^n=(S,n)$} $\forall m\in S$,
\State{fair share of the slot for $m$th SU's transmission is $a^{mC^n}/{\sum_{i\in S}}a^{iC^n}$}
\EndIf
\EndWhile\label{Line:Converge}
\end{algorithmic}
\hrule
\caption{Distributed top-layer partition formation.}
\label{Fig:AlgTop}
\end{figure}

\subsection{Computational complexity and overhead of the two-layer game}\label{SubSec:Complexity}
The computational complexity of the proposed game is dominated by the FA rate computations (line \ref{Line:AdaptTh} in Fig~\ref{Fig:AlgBottom}). For SU $m$ on channel $n$ [cf. (\ref{Eq:CoalFAResultAND}) and (\ref{Eq:BetaDef})], this rate is determined by $\lambda^{mn}$ and SU population size $|C_t^n(1)|$, where $t$ is the time slot index. Each SU $m$ on channel $n$ maintains three values of its FA rate, corresponding to the current and potential SU populations sizes $|C_t^n(1)|\pm i$, $i=0,1$, and this FA rate information is exchanged over the control channel as needed. Thus, the initialization step requires $3M$ FA rate computations. When an SU explores whether to switch to channel $\tilde{n}$ in time slot $t$ using (\ref{Eq:Preference}), it needs to compute its potential FA rate on that channel, corresponding to the updated SU population size $|C_t^{\tilde{n}}(1)|+1$ and its PU-to-SU SNR $\lambda^{m\tilde{n}}$, resulting in one FA computation per time slot. It then requests the FA rate information from SUs residing on that channel. In those slots where an SU $m$ actually switches to channel $\tilde{n}$, new coalitions $\tilde{C}_t^n$ and $\tilde{C}_t^{\tilde{n}}$ form [cf. (\ref{Eq:Preference})], and SU $m$ needs to compute two additional FA rates corresponding to the SU population sizes $|\tilde{C}_t^{\tilde{n}}(1)|\pm 1$ while all other SUs in $\tilde{C}_t^n(1)$ and $\tilde{C}_t^{\tilde{n}}(1)$ have to compute only one additional FA rate each, corresponding to SU population sizes decreased and increased by 2 on their respective channels $n$ and $\tilde{n}$. Since each FA rate computation takes $\mathcal{O}(1)$ time, the total computational complexity of the algorithm until convergence is
\begin{equation}
\mathcal{O}\Big(3M+T_\textrm{Converge}+\!\sum_{t\in\mathcal{T}_\textrm{Switch}}(|\tilde{C}_t^n(1)|+|\tilde{C}_t^{\tilde{n}}(1)|+1)\Big )
\label{Eq:Complexity}
\end{equation}
where $\mathcal{O}(\cdot)$ is the big O notation\cite{CLRS}, $\mathcal{T}_\textrm{Switch}$ is the set of time slots in which an SU switches in the interval $[1,T_\textrm{Converge}]$ and $T_\textrm{Converge}$ denotes the convergence time, i.e., the number of slots needed to execute the while loop from line~\ref{Line:While} to \ref{Line:Converge} in Fig.~\ref{Fig:AlgTop}.

From the above discussion, the communication overhead of the proposed two-layer game mainly results from the action signaling (SWITCH/HOLD/SLEEP) in Fig.~\ref{Fig:AlgTop}, the FA rate information exchange discussed above, and sharing of sensing results. The latter two types of overhead are commonly assumed in the cooperative sensing literature, e.g. \cite{HedonicSenseGame,AuctionBased,ZHanCoalSenseGame,FairPayoff,MultiChCoalSenseGame,Overlapp,RLiuEvoGame}, while the former costs no more than two bits per slot. Moreover, we must emphasize that the exchange of local data rate information $\{R^{mn}:m\in S\}$ is not required.

We note that the two-layer game structure limits the computational complexity, the negotiation delay, and the communication overhead, since an SU that explores switching from channel $n$ to $\tilde{n}$ needs to learn only the FA rates of the SUs currently residing on channel $\tilde{n}$. In contrast, in a one-layer game where SUs can form arbitrary coalitions spanning multiple channels, an SU looking for a switching opportunity would have to negotiate with and obtain data on a much larger set of SUs.

Finally, while we have ignored the overhead of communication and MAC when computing the coalition values in (\ref{Eq:CoalValue0X}) and (\ref{Eq:CoalValue1X}), such overhead is inevitable in practice. Very large overhead might greatly reduce the value of the grand coalition and, thus, its formation on individual channels. Payoff allocation has not received significant attention in this scenario since most results on bargaining games focus on allocating the value of the grand coalition. Moreover, delay and uncertainty may arise in reaching a joint agreement on both the coalition structure and the payoff allocation in the bottom-layer game. Development of novel game-theoretic solutions is required to accommodate such extensions.

\section{Simulation Results}\label{Sec:Sim}
We assume the following simulation setup throughout this section unless otherwise noted: The sensing and slot durations are $5$\,ms and $100$\,ms, respectively. The SU sensing power $P_S$, SU transmission power $P_\mathrm{SU}$, PU transmission power $P_\mathrm{PU}$, and noise power $P_N$ are $10$\,mW, $10$\,mW, $100$\,mW and $0.1$\,mW, respectively. All users are randomly placed in a square region of $100$\,m${\times}100$\,m, and only path loss effects are considered with the pass loss exponent equal to $2$\cite{HedonicSenseGame}. The PU-to-SU SNR is given by $\lambda^{mn}=P_\mathrm{PU}d_{mn}^{-2}/P_N$, where $d_{mn}$ is the distance between PU $n$ and SU $m$. Similarly, the SU-to-SU SNR between two SUs $m$ and $m'$ is given by $\gamma^{mm'}=P_\mathrm{SU}d_{mm'}^{-2}/P_N$, where $d_{mm'}$ is the SU distance. We compare only with \cite{HedonicSenseGame} because, to the best of our knowledge, other related references did not consider joint spectrum sensing and access in a multichannel CR network.

\begin{figure}[!t]
    \centering
    \includegraphics[scale=\FigScale]{figures/DataRate}
		\includegraphics[scale=\FigScale]{figures/EnergyEfficiency}
    \caption{Performance comparison of the proposed two-layer game and the hedonic game in \cite{HedonicSenseGame}.}
    \label{Fig:EnergyEfficiency}

\end{figure}

In Fig.~\ref{Fig:EnergyEfficiency}, we compare the proposed game (i,iii) with the game in \cite{HedonicSenseGame} (ii,iv) for the two MAC models [cf. (\ref{Eq:CoalValue0X},\ref{Eq:CoalValue1X})]. Initially there are $M=10$ SUs and $N=5$ PUs. At time slots $2000$ and $4000$, these parameters change to ($M=10,N=6$) and ($M=14,N=6$), respectively. Each PU uses one channel with $B^n=10$\,MHz exclusively. The channel availability probability is $\beta^n=0.2$. For fair comparison, the AND-rule combining scheme under the constraints described in Section~\ref{subsec:CoopSenseDef} is used for both the proposed two-layer game and the game in \cite{HedonicSenseGame} assuming $\nu=5$ and $P_\mathrm{MD}^\mathrm{Ch}=0.01$. Note that \cite{HedonicSenseGame} aims to maximize the energy efficiency while the objective of the proposed game is to maximize the expected data rate (\ref{Eq:PTU}). Nevertheless, the proposed game achieves better average energy efficiency and throughput in Fig.~\ref{Fig:EnergyEfficiency}(a,d), since
in this game SUs are encouraged to join channels with more detectable PU traffic to obtain higher payoffs, leading to efficient sensing task distribution among the selfish SUs. As a result, more transmission opportunities can be correctly detected in the CR network while maintaining the same energy cost for spectrum sensing.

Moreover, each SU is allocated the access opportunities it deserves and, thus, is provided with sufficient incentives to participate in the proposed game. In particular, as shown in Fig.~\ref{Fig:EnergyEfficiency}(b,c), every SU has higher data rate when playing the proposed game than operating alone (i.e., all SUs are satisfied with their data rates). Despite the fact that the proposed game can have a greater number of dissatisfied SUs in terms of energy efficiency than the game in \cite{HedonicSenseGame} under the 1/X-model [curves iii vs. iv in Fig.~\ref{Fig:EnergyEfficiency}(e)], these SUs experience negligible energy efficiency loss [Fig.~\ref{Fig:EnergyEfficiency}(f)], and the overall energy efficiency is significantly higher for the two-layer game [Fig.~\ref{Fig:EnergyEfficiency}(d)].

\begin{figure}[!t]
    \centering
    \includegraphics[scale=\FigScale]{figures/CoalLevelFA}
    \caption{ Average coalition-level FA rate of the proposed hedonic game.}
    \label{Fig:CoalLevelFA}

\end{figure}

We also note that the changes in the numbers of PUs/SUs at time slots $2000$ and $4000$ do not cause significant disruptions to these games. Finally, while the overall system throughputs and energy consumption in Fig.~\ref{Fig:EnergyEfficiency}(a,d) are very similar under both MAC models, individual SUs report a higher degree of satisfaction under the 0/X-model as shown in Fig.~\ref{Fig:EnergyEfficiency}(b,c,e,f) since this model has a higher noncooperation cost (Section~\ref{SubSec:BottomDef}).

\ifdoublecolumn
\begin{figure*}
\normalsize
\begin{align}
U_\mathrm{1/X}^n(\eta ;\rho^n ) &=\beta ^n(1 - P_\mathrm{FA}^n(\eta ))\cdot\smashoperator[l]{\sum\limits_{\mathbf{\tilde{x}}  \in \{0,1\}^{|\rho^n | - 3}}}\Bigg\{{\sum_{(x_1,x_2)\in\{0,1\}^2}}\frac{|\eta|\Pr(X_{\xi_1}=x_1)\Pr(X_{\xi_2}=x_2)\prod _{i=1}^{|\rho^n|-3}\Pr(X_{\xi_{i+2}}=\tilde{x}_i)}{|\eta|+J_{\{\xi_1,\xi_2\}}((x_1,x_2))+J_{\rho^n\backslash\{\eta\}\backslash\{\xi_1\}\backslash\{\xi_2\}}(\mathbf{\tilde{x}})}\Bigg\}\label{Eq:NegExternalityBefore}\\
{U_\mathrm{1/X}^n(\eta ;\tilde \rho^n )} &=\beta ^n(1 - P_\mathrm{FA}^n(\eta ))\cdot\smashoperator[l]{\sum\limits_{\mathbf{\tilde{x}}  \in \{0,1\}^{|\rho^n | - 3}}}\Bigg\{{\sum_{x_1\in\{0,1\}}}\frac{|\eta|\Pr(X_{\xi_1\cup\xi_2}=x_1)\prod _{i=1}^{|\rho^n|-3}\Pr(X_{\xi_{i+2}}=\tilde{x}_i)}{|\eta|+J_{\{\xi_1\cup\xi_2\}}((x_1))+J_{\tilde\rho^n\backslash\{\eta\}\backslash\{\xi_1\cup\xi_2\} }(\mathbf{\tilde x})}\Bigg\}\label{Eq:NegExternalityAfter}\\
 U_\mathrm{1/X}^n(\eta ;\rho^n )&-U_\mathrm{1/X}^n(\eta ; \tilde\rho^n )=\beta ^n(1 - P_{{\mathrm{FA}}}^n(\eta ))\cdot\smashoperator[l]{\sum\limits_{\mathbf{\tilde{x}}  \in \{0,1\}^{|\rho^n | - 3}}} {\Bigg\{ \prod _{i=1}^{|\rho^n|-3}\Pr(X_{\xi_{i+2}}=\tilde{x}_i)\cdot A(\mathbf{\tilde{x}})  \Bigg\}} \label{Eq:NegExternalityProof}
\end{align}
\hrulefill
\vspace*{-10.2pt}
\label{Fig:Eq:NegExternalityBeforeAfter}
\end{figure*}
\fi

In Fig.~\ref{Fig:CoalLevelFA} we validate the superiority of the AND-rule over the OR-rule for heterogeneous environments (see Section~\ref{subsec:CoopSenseDef}) by comparing the average FA rates of the coalitions formed in the proposed game for the realistic scenario of Fig.~\ref{Fig:EnergyEfficiency}. Not surprisingly, we observe significant sensing accuracy degradation using the OR-rule, which can be explained by diverse PU-to-SU SNRs among the coalition members.

Next, we compare the computational complexity (\ref{Eq:Complexity}) and convergence time in Fig.~\ref{Fig:Complexity} for varying $N$ values, assuming stable SU/PU population and the 0/X MAC model. Note that when an SU determines whether it should switch to another channel, it needs the knowledge of the updated individual FA rates on the current and new channel for all involved SUs in both the proposed game and the game in \cite{HedonicSenseGame}. Thus, the complexity analysis in Section~\ref{SubSec:TopSol} applies to both games. We notice a significant reduction in convergence time and computational complexity when the proposed game is played, due primarily to provision for the social utility improvement in the preference relation (\ref{Eq:Preference})\footnote{Our game is still superior to \cite{HedonicSenseGame} in terms of the data rate, energy efficiency, and fairness when the social improvement requirement is removed.}. Moreover, extensive numerical experiments show that the actual number of switches needed for convergence is much smaller than the crude upper bound in Proposition~\ref{Prop:Converge}.

\begin{figure}[!t]
    \centering
    \includegraphics[scale=\FigScale]{figures/Complexity}
    \caption{Computational complexity vs. $N$; $M=10$ SUs; 0/X-model: (a) convergence time $T_\textrm{Converge}$; (b) number of switches $|\mathcal{T}_\textrm{Switch}|$; (c) number of FA rate computations (\ref{Eq:Complexity}).}
    \label{Fig:Complexity}

\end{figure}

\begin{figure}[!t]
    \centering
    \includegraphics[scale=\FigScale]{figures/MobilityNumOfSwitch}
    \caption{Switch frequency vs. number of channels ($N$) for various traveling speeds ($V$); 0/X-model; $M=10$ SUs.}
    \label{fig:MobilityNumOfSwitch}

\end{figure}

Finally, we study the effect of mobility on the performance of the proposed game. We assume that the nodes initially achieve a Nash-stable SU network partition and then move in randomly chosen independent directions at a constant speed $V$ following the random direction model \cite{MobilityModel}. When a node reaches the simulation region boundary, it moves in a new randomly chosen direction in the next time slot. Fig.~\ref{fig:MobilityNumOfSwitch} illustrates the \textit{switch frequency} given by the number of channel switches per minute in the network \cite{ZHanCoalSenseGame}. We observe that doubling of the mobile speed $V$ results in 10 to 40 additional channel switches per minute. However, for moderate speeds, the switch frequency is insensitive to the numbers of channels in the network. Frequent channel switching increases $|\mathcal{T}_\textrm{Switch}|$ in (\ref{Eq:Complexity}) and, thus, results in much larger  computational complexity. Moreover, the overhead grows with the mobile speed since the need for message exchange increases. While the proposed game adjusts well to sudden changes in the network settings (see Fig.~\ref{Fig:EnergyEfficiency}) and node positions \cite{YLuThesis}, continuous mobility, as in Fig.~\ref{fig:MobilityNumOfSwitch}, prevents convergence and increases complexity and overhead. Thus, alternative methods are required to resolve SU competition and provide high throughput in congested mobile CR networks. For example, in \cite{YLuTVT,YLuAllerton} we have investigated opportunistic sensing strategies that rely on multichannel diversity by adapting the reward of each SU pair to predicted local channel state information prior to sensing.

\section{Conclusion}\label{Sec:Conclusion}
The proposed two-layer game provides a comprehensive coalitional game-theoretical framework for cooperative sensing and access in multichannel multi-SU CR ad-hoc networks. Each SU is provided with the transmission opportunities it deserves and, thus, with sufficient incentives to participate in the proposed game.	Moreover, the contribution-based payoff allocation rule accelerates efficient distributed allocation of the SUs to the channels within the given CR spectrum. We also present a new cooperative sensing approach under MD constraints for guaranteed PU protection in the heterogeneous environment. Finally, we demonstrate that the proposed two-layer game outperforms the approaches in the literature in terms of energy efficiency, throughput, fairness, and is attractive in practical ad-hoc networks due to its relatively low computation load, distributed algorithm implementation, as well as robustness to changes in the number and positions of the SUs.

\ifdoublecolumn
\else
\begin{figure*}
\normalsize
\begin{align}
U_\mathrm{1/X}^n(\eta ;\rho^n ) &=\beta ^n(1 - P_\mathrm{FA}^n(\eta ))\cdot\smashoperator[l]{\sum\limits_{\mathbf{\tilde{x}}  \in \{0,1\}^{|\rho^n | - 3}}}\Bigg\{{\sum_{(x_1,x_2)\in\{0,1\}^2}}\frac{|\eta|\Pr(X_{\xi_1}=x_1)\Pr(X_{\xi_2}=x_2)\prod _{i=1}^{|\rho^n|-3}\Pr(X_{\xi_{i+2}}=\tilde{x}_i)}{|\eta|+J_{\{\xi_1,\xi_2\}}((x_1,x_2))+J_{\rho^n\backslash\{\eta\}\backslash\{\xi_1\}\backslash\{\xi_2\}}(\mathbf{\tilde{x}})}\Bigg\}\label{Eq:NegExternalityBefore}\\
{U_\mathrm{1/X}^n(\eta ;\tilde \rho^n )} &=\beta ^n(1 - P_\mathrm{FA}^n(\eta ))\cdot\smashoperator[l]{\sum\limits_{\mathbf{\tilde{x}}  \in \{0,1\}^{|\rho^n | - 3}}}\Bigg\{{\sum_{x_1\in\{0,1\}}}\frac{|\eta|\Pr(X_{\xi_1\cup\xi_2}=x_1)\prod _{i=1}^{|\rho^n|-3}\Pr(X_{\xi_{i+2}}=\tilde{x}_i)}{|\eta|+J_{\{\xi_1\cup\xi_2\}}((x_1))+J_{\tilde\rho^n\backslash\{\eta\}\backslash\{\xi_1\cup\xi_2\} }(\mathbf{\tilde x})}\Bigg\}\label{Eq:NegExternalityAfter}\\
 U_\mathrm{1/X}^n(\eta ;\rho^n )&-U_\mathrm{1/X}^n(\eta ;\tilde\rho^n)=\beta ^n(1 - P_{{\mathrm{FA}}}^n(\eta ))\cdot\smashoperator[l]{\sum\limits_{\mathbf{\tilde{x}}  \in \{0,1\}^{|\rho^n | - 3}}} {\Bigg\{ \prod _{i=1}^{|\rho^n|-3}\Pr(X_{\xi_{i+2}}=\tilde{x}_i)\cdot A(\mathbf{\tilde{x}})  \Bigg\}} \label{Eq:NegExternalityProof}
\end{align}
\hrulefill
\vspace*{-10.2pt}
\label{Fig:Eq:NegExternalityBeforeAfter}
\end{figure*}
\fi

\appendices
\section{Proof of Proposition~\ref{Prop:NegExternality}}\label{Sec:NegExternalityProof}
Without loss of generality, we assume $\xi=\xi _1$ and $\zeta=\xi _2$ are the bottom-layer coalitions to be merged. The resulting bottom-layer partition is given by $\tilde\rho^n = \{ \xi _1  \cup \xi _2 \}  \cup (\rho^n \backslash \{ \xi _1 \} \backslash \{ \xi _2 \} )$ with $|\tilde\rho^n | = |\rho^n | - 1$. Expanding the bottom-layer coalition value ${U_\mathrm{1/X}^n(\eta ;\rho^n )}$ in (\ref{Eq:CoalValue1X}) over all possible values of $(X_{\xi_1},X_{\xi_2})$ in $\{0,1\}^2$ yields (\ref{Eq:NegExternalityBefore}) at the top of page \pageref{Eq:NegExternalityBefore} [cf. (\ref{Eq:X})]. Similarly, the bottom-layer coalition value ${U_\mathrm{1/X}^n(\eta ;\tilde\rho^n )}$ after the merger is derived in (\ref{Eq:NegExternalityAfter}) by expanding over all possible $X_{\xi_1\cup\xi_2}$ values. Noting that $\rho^n\backslash\{\eta\}\backslash\{\xi_1\}\backslash\{\xi_2\}=\tilde\rho^n\backslash\{\eta\}\backslash\{\xi_1\cup\xi_2\}$ and $P_{{\mathrm{FA}}}^n({\xi _1}\cup{\xi _1})=P_{{\mathrm{FA}}}^n({\xi _1})P_{{\mathrm{FA}}}^n({\xi _2})$ [cf. (\ref{Eq:CoalFAResultAND})], we subtract (\ref{Eq:NegExternalityAfter}) from (\ref{Eq:NegExternalityBefore}) and simplify to obtain (\ref{Eq:NegExternalityProof}) where [cf. (\ref{Eq:ParitionExcept})--(\ref{Eq:J})]
\begin{align}\label{Eq:A}
A(\mathbf{\tilde x})& =\Big(\frac{1}{b_1(\mathbf{\tilde x})}-\frac{1}{b_{12}(\mathbf{\tilde x})} \Big)|\eta|P_{{\mathrm{FA}}}^n({\xi _2})\Big(1 -P_{{\mathrm{FA}}}^n({\xi _1})\Big)\\
{}&+ \Big(\frac{1}{b_2(\mathbf{\tilde x})}- \frac{1}{b_{12}(\mathbf{\tilde x})} \Big)|\eta|P_{{\mathrm{FA}}}^n({\xi _1})\Big(1 - P_{{\mathrm{FA}}}^n({\xi _2})\Big)\notag
\end{align}
with
\begin{equation}
\begin{split}
{b_1}(\mathbf{\tilde x}) &= |\eta | + |{\xi _1}| + \sum\nolimits_{i = 1}^{|\rho^n | - 3} {(1 - {\tilde x}_i)|{\xi _{i {+} 2}}|} \\
{b_2}(\mathbf{\tilde x}) &= |\eta | + |{\xi _2}| + \sum\nolimits_{i = 1}^{|\rho^n | - 3} {(1 - {\tilde x}_i)|{\xi _{i {+} 2}}|} \\
{b_{12}}(\mathbf{\tilde x}) &= |\eta | + |{\xi _1}| + |{\xi _2}| + \sum\nolimits_{i = 1}^{|\rho^n | - 3} {(1- {\tilde x}_i)|{\xi _{i {+} 2}}|}.
\end{split}
\end{equation}
It is easy to show that $A(\mathbf{\tilde x}$ ) in (\ref{Eq:A}) and, thus, the expression in (\ref{Eq:NegExternalityProof}) must be nonnegative.

\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


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