Download your free LightScribe program here. It is a basic, no frills program but I tried it and it works great.
There is a video along with a poll by John Ziegler that reinforces the notion that the lowest common denominators in society elected THE ONE, Barack 'Barry' Hussein Obama, to be the 44th president of the United Sates.
As a famous man (I can't remember exactly who...) once said, "Father, forgive them; for they know not what they do."
With 96.85% of precincts counted, Minnesota's incumbent Republican Senator Norm Coleman is leading wacko Al Franken by the narrow margin of 12,513 votes.
In Alaska, recently convicted felon, incumbent Republican Senator Ted Stevens, is leading his Democratic challenger Mark Begich by 4,526 votes after 72.1% of precincts have been counted. Senator Stevens will likely leave the senate at some point in the not to distant future should he be reelected. He would be replaced by someone chosen by Governor Sarah Palin. However, said appointment will only be for 90 days when a special election must be held.
Update: In Minnesota, with 98.55% of the vote counted, the Minnesota Secretary of State is reporting that Coleman is now leading by only 6,352 votes.
It's only 7:30 P.M. P.D.T. and the presidential election is over. Barack Hussein Obama is the big winner. John McCain should have agreed to be Hillary's running mate. He would have had an office in the White House, even if he wasn't sleeping there.
If this election proves anything it is that racism is alive and well in America, if you are black that is. Fox News is estimating that 95% of black people voted for Obama. 95%!!! Can you imagine what the blacks would say if 95% of whites voted for McCain?
I watched Obama's acceptance speech given before a delirious crowd of well over 100,000 supporters at Grant Park in Chicago and, as usual, the speech was filled with little more than soaring, yet empty, rhetoric. The only thing missing from the setting was a gold crown. The crowd included Oprah Winfrey, who was hanging on Obama's every word, and Jesse Jackson, shown weeping uncontrollably throughout most of the speech.
The hard part for Obama is just beginning. He will now have to govern. He won't be able to vote 'present' any more. Decisions will have to be made. His first decision will likely be who gets to sleep in the Lincoln bedroom first, terrorist William Ayers or the anti-American Reverend Wright. I'm betting on Wright.
Labor unions, the same labor unions that have ruined the textile, automotive and airline industries, must be ecstatic now that the Democrats have control of both houses of congress and the presidency. Union leaders will be gearing up to press their highest legislative priority, the deceptively named Employee Free Choice Act (EFCA), which seeks to replace secret ballot elections with publicly signed union cards. Should it become law, prices will undoubtedly be going up at Walmart. Non-union Walmart has long been a target of labor organizers. In Canada, Walmart merely closed one store where employees managed to organize into a union. Unfortunately, Walmart will not be able to hold off union organizers armed with the Employee Free Choice Act in the United States.
I had to laugh while watching the election coverage on CNBC after Barry's (Barack's) speech. A black man, whom I was not familiar with, spoke up and disparaged trickle down economics. Nope. There would be no more trickle down economics under President Barack Hussein Obama. No sireee. I guess he figures 'trickle up' will work better.
During the next four years I pledge to help him work on the 'trickle up' economic theory. I am going to hire less staff to do landscaping, painting, cleaning, and other general work at home and at work. I am going to spend less, much less as it were, on discretionary items. I pledge that there will be no purchases of new cars; no new refrigerators; no new washing machines or dryers; no new furniture; no new computers; no new printers; and, the most difficult item to personally give up, there will be no new CD's or DVD's. I urge others to join me - let's give Obama's trickle up economic theory a chance.


text:
AB InBev CEO defends mega-beer merger to Congress
Congress wants to ensure there are more choices than just Bud and Miller.
Anheuser-Busch InBev and Molson Coors sought to assuage concerns that a "Mega Beer" merger in the global market would not hurt the vibrancy of smaller craft beers, battling against accusations from industry insiders that another big acquisition would be bad for consumers.
Two of the beer industry's leading chief executives were challenged at a hearing held by the U.S. Senate's subcommittee on Antitrust, Competition Policy and Consumer Rights on Tuesday. The main point of contention: how AB InBev's BUD -0.17% roughly $104 billion acquisition of SABMiller would affect the distribution of thousands of beer brands and styles across the U.S.
AB InBev CEO Carlos Brito sought to assure Congress that his company's deal wouldn't hurt competition. He pointed to a side deal that AB InBev preemptively inked with Molson Coors TAP -0.17% , agreeing to sell the MillerCoors joint venture in the U.S. in a deal valued at $12 billion. With that deal, Molson Coors will get full control of roughly 26% of the U.S. beer market – which it partly controlled through a joint venture with SABMiller. AB InBev, meanwhile, would maintain a market leadership position that would be unchanged at 45%.
"The purpose of this transaction is to enhance our ability to serve new markets, particularly in Africa, Asia and Central and South America," Brito said during his testimony. "What this combination is not about is changing our competitive position in the U.S. beer market."
Beer's 'golden age'
"America is in a golden age for beer," said Craig Purser, president and CEO of National Beer Wholesalers Association, in a statement to the committee. "The true winner is the American consumer. But what makes this choice possible? A robust and competitive system of distribution."
As Fortune reported last week, the nation now has more breweries than ever before: 4,144 in 2015, exceeding a historic high set in 1873. Craft beer volume represented just 1% of the overall beer industry in 1994 but stands at over 11% today.
Craft brewers have been the growth engine for the broader beer industry, a strong performance that has led to more mergers and acquisitions. Senators from several states lauded the vibrancy of the craft beer movement in their states, while also seeming cautious about how a big merger in the beer aisle could hinder growth.
Purser and other beer insiders testified that they were concerned that the combination of AB InBev-SABMiller, a company analysts often call "Mega Beer," would pressure the "three tier" system utilized by brewers and other alcoholic beverage producers. Essentially, brewers brew the beer, wholesalers distribute it, and retailers sell it. Those parties are mainly supposed to remain separate.
"Nobody wants to sit at a bar and find that their only choice is between a Bud and a Miller," said U.S. Senator Chris Coons of Delaware, summing up the concerns the committee had about distribution.
Distribution disruption
Purser asserted AB InBev has sought to disrupt that system by buying up more distributors and pressuring independent brewers via incentives to stock Budweiser and AB InBev beers first, and give less or no space to craft competitors. Some experts who testified before Congress requested conditions be considered for post-merger conduct if the U.S. government were to sign off on the deal. Most of the requests were related to distribution and capacity.
Though worries about hop purchasing and pricing floated throughout the hearing, distribution was by far the biggest concern. Major distributors often have exclusive deals with either AB InBev or MillerCoors, though they also distribute other beers like Corona and Sam Adams.
AB InBev has been accused of seeking to curb the distribution of craft beers. AB InBev also owns 21 wholesalers, though Brito said the company works with over 500 wholesalers in total and only controls roughly 8% of InBev's distribution tier in the U.S. After some pointed questions, Brito said AB InBev would vow to only buy up "around 10%" of the wholesale channel.
Bob Pease, CEO of the Brewers Association, said that the more than 2,800 beer manufacturers he represents are required by law to go through distributors to get their beers on retail shelves today. He contends AB InBev has scooped up distributors to cut off rivals' growth.
Connecticut U.S. Senator Richard Blumenthal grilled Brito, securing a verbal agreement by the executive that the merger wouldn't result in the termination or re-negotiation of existing distribution deals today. Molson Coors CEO Mark Hunter said his company only owns one distributor—and had no plans to buy more.
Hunter aimed to assure Congress that the deal would maintain a strong No. 2 player in the market. He said the only change would be that MillerCoors beers, which include Miller Lite, Blue Moon, and Coors, would have one shareholder rather than two. And because 70% of profits at Molson Coors would be derived from the U.S. market, it can be expected to be a major priority at Molson.
"I am very clear about the independence of Molson Coors," Hunter said. "Our transaction doesn't injure competition, but enhances it."
By John Kell
Source: Fortune
Danone unveils plan to cut methane emissions
The company expects to eliminate 1.2 billion tons carbon dioxide equivalent of methane emissions by the end of the decade. The company says that it already reduced its methane emissions by around 14% between 2018 and 2020.
US researchers land US$2.4M to turn food waste into "affordable" bioplastics
The "first-of-its-kind" pilot project will develop and demonstrate an affordable modular bioprocessing system to produce biodegradable bioplastics from food waste diverted from landfills. The three-year grant will test the scalability and feasibility of the conversion on a national and global scale.
Arkeon partners with ICL to transform CO2 into proteins
Arkeon is allying with specialty mineral giant ICL to support the scaling of its fermentation bioprocess that converts CO2 into the 20 proteinogenic essential amino acids needed in human nutrition. The process, hailed as carbon negative, is based on the use of archaea, a group of microorganisms that naturally feeds off the greenhouse gas.

redpajama_set_name: RedPajamaCommonCrawl


text:
\section*{Acknowledgments}
We thank A.L. Stella and S. Ruffo for useful remarks.
FB also acknowledges collaboration with C. Tsallis and
L.G. Moyano
with whom initial interest for this problem has been
shared,
and F. Leyvraz, A. Rapisarda and H. Touchette for discussions.

redpajama_set_name: RedPajamaArXiv


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Mindanao's rocky mountain high
By Bernard L Supetran Philippine Daily Inquirer June 24,2018
Overview Nature and Culture Park
Up to the late 1990s, the upland province of Bukidnon lived up to the title of John Denver's popular song, with emphasis on the word "rocky."
Commuting then on an ordinary bus, the shortcut between the cities of Cagayan de Oro in the north and Davao in the south was like running through the gauntlet, what with the infinity of unpaved and dusty roads.
When the national highway was completely paved in the early 2000s, traveling the Davao-CDO corridor become a pleasurable experience—with the panoramic long and winding roads.
Most of the roads are undergoing expansion into four lanes to make it compliant with the standards by the public works department due to increased vehicular traffic in the once frontier area.
Still, even by public transport, travel is a breeze with the comfy aircon buses which are, believe it or not, far more advanced than the "rolling coffins" plying the main thoroughfares of Metro Manila.
A landlocked plateau, Bukidnon has no airport, and for this odyssey, we fly to Davao and take the northward journey.
Pit Stop 1: Quezon
After more than two hours of travel via the picturesque BuDa or Bukidnon-Davao Road, your first toilet break is the Overview Nature and Culture Park in Quezon town, a landscaped area that has a soothing vista of rolling hills and meandering roads.
At 915 meters above sea level, the viewdeck is often windswept and kissed by fog because of its altitude.
For a quick bite, pull over at the roadside Coffee Highway, a fast casual dining outlet known for its sugarcane juice and special coffee blend.
A few kilometers away is the Kiokong Eco-Tourism Area, an extreme adventure where one gets to camp at the vertical bivouac, a two-meter wide, 400-ft cliff ledge.
The reserve is dotted with limestone cliffs that make it a haven for rock climbers.
For a less buwis-buhay activity, you can trek down to what locals dub as "Bluewater," a bluish subterranean icy natural cave pool where you can dip for a rejuvenating bath.
Pit Stop 2:
Philippine Carabao Center
An unique spot in Maramag town is the Philippine Carabao Center which claim to fame is the dairy product outlet at the Central Mindanao University where guests can partake of their favorite products or bring home as pasalubong.
Products include fresh milk, choco milk, pastillas (milk candies), white cheese, flavored milk, and milk bars, which are produced from carabao milk.
For those interested in farm tourism, this one should not be missed.
Pit Stop 3: Valencia
This once bucolic ranch town is now the province's boomtown, commercial hub, and "bedroom" with its array of lodging facilities and small shopping malls.
Those who want a blend of country charm and modern comfort can hit the sack at Double M Hotel, arguably the most stylish in the city and the home of Bukidnon Roadhouse Cafe, Totsy's Cakes and Pastries, and Tirso's Deli and Roasters, all of which have been part of Valencia's culinary history.
A must see is the placid Lake Apo, a 24-hectare rift framed by mountains ideal for rafting and fishing with its floating cottages.
Once hailed as the cleanest lake in Region 10, this upland body of water can be reached via a single motorcycle from the city terminal.
Pit Stop 4: Lantapan
This interior town is off the beaten path, but the trip to Café sa Bukid, a cozy farm-themed resto that serves delectable locally-harvested food and drinks, is definitely worth the time.
Further inland is the village of the Talaandig, an indigenous group that has managed to preserve its tribal culture despite the strong influx of modernization.
Found in the settlements around Mt. Kitanglad Range Natural Park, guests can enter their ancestral domain, but only after a coordinated visit where they undergo the traditional welcome rituals, which includes among others, a sacrifice of a live chicken.
The community immersion is a unique experience that will give you a deep appreciation of their spirituality and way of life.
Pit Stop 5: Malaybalay
This provincial capital city is home to the iconic Monastery of Transfiguration, a hilltop pyramid-shaped Benedictine church designed by National Artist Leandro Locsin.
It offers a tranquil place where one can enjoy serenity in prayer and the beauty of nature.
The monastery is open to the public for masses, and the occasional breakfast with the monks where guests can savor the organic Monks Blend products.
Also within the city is the American-era picture-pretty capitol building, and the Kaamulan Park right behind, a 500-hectare greenery that abounds in pine trees, and is a sanctuary for diverse variety of flora.
An upcoming novelty attraction is the hobbit-inspired lodge by wild bird photographer Benjamin Maputi Jr., who established the award-winning Mt. Kitanglad Agri-Ecological Techno-Demo Farm and the Maputi Bird Preserve Camp, which are also in Malaybalay.
Pit Stop 6: Panika Landmark
If you're hunting for a colorful Instagrammable subject, this must be it. Standing proudly along the Sayre Highway in Impasugong town is the oversized replica of the panica, the native women's headgear worn during special events.
Don't miss the seven larger-than-life monuments representing the original tribes in the province—Bukidnon, Higaonon, Talaandig, Manobo, Matigsalug, Tigwahanon, and Umayamnon.
Impasugong is also regarded as the tribal capital, as well as home of cowboys with its sprawling cattle ranches located in the town's hinterlands.
Pit Stop 7: Manolo Fortich
Named after the province's pioneering governor, this resort town was placed on the tourist map by the Dahilayan Adventure Park, which has Asia's longest dual zipline—a two 840-m zipline with a 100-m drop.
It also has a 320-m and 150-m dual zipline for beginners, along with the ropes course, wall climbing, and the first extreme pendulum at Dropzone featuring a 120-ft freefall into a man-made lake.
Another magnet of the town is Kampo Juan, an adventure and camp whose centerpiece is the unicycle or bicycle on a 600-ft long cable at 100 feet above the ground, a 520-ft long zipline, a 110-meter hanging bridge, and a heritage house and eco lodge.
A very recognizable spot within M Fortich is Camp Philips, the sprawling pineapple plantation of Del Monte Philippines and the biggest of its kind in Asia.
Also within the town is Mangima Canyon Springs and Country Park, a 6-ha municipal resort surrounded by hills and mountains, designed for events meetings and recreation, with swimming pools, horseback riding field, and an off-road track for motocross.
After going through this mountain high route, the ideal exit point is Laguindingan Airport, which is about 45 minutes away from Cagayan de Oro City.
John Denver would have been pleasantly surprised to know something close to his beloved Colorado exists half a world away.


text:
Quick links... Browns Indians Cavaliers Ohio State 2021 NFL Draft
SportsBasketballCavaliers
Cavs' owner Dan Gilbert gives statement after LeBron announces his move to the Lakers
Posted: 6:29 AM, Jul 02, 2018
By: Kaylyn Hlavaty
Ronald Martinez

OAKLAND, CA - JUNE 19: Cleveland Cavaliers owner Dan Gilbert, holds the Larry O'Brien Championship Trophy after the Cavaliers defeated the Golden State Warriors 93-89 in Game 7 of the 2016 NBA Finals at ORACLE Arena on June 19, 2016 in Oakland, California. NOTE TO USER: User expressly acknowledges and agrees that, by downloading and or using this photograph, User is consenting to the terms and conditions of the Getty Images License Agreement. (Photo by Ronald Martinez/Getty Images)

After the heartbreaking news that LeBron James decided to take his talents to the west coast, Cavs owner Dan Gilbert released a statement early Monday morning thanking the King for ending the city's 52-year drought by delivering a championship many thought they would never see.
Gilbert said the championship united Clevelanders—a moment on June 19, 2016, that fans will never forget.
Read the following statement below:
"We will always remember the evening of June 19, 2016 as the Cleveland Cavaliers, led by LeBron James, ended the 52-year drought delivering the long elusive championship that many thought they would never see... A championship that united generations of Clevelanders, both living and passed.
Virtually anyone with roots in Northeast Ohio paused and felt the memories of the past and the utter joy that the burden of the so-called 'curse' was finally a thing of the past. Cleveland, Ohio was the home of a championship team for the first time since 1964. Words do not express the meaning and the feeling this accomplishment brought to the people of Northeast Ohio.
None of this would have happened if LeBron James did not agree to come back home and lead the Cavaliers to the promised land. The entire Cavaliers franchise thanks LeBron for that precious moment and for all of the excitement he delivered as he led our team to four straight NBA Finals appearances.
LeBron is a family man, first. We wish his kids, his wife Savannah, his mother Gloria, and LeBron himself nothing but the best in the years and decades ahead. LeBron's connection to Akron, Cleveland and all of Northeast Ohio will most certainly endure as his commitment to the region and his support of many important causes has been impactful to so many kids and families.
LeBron, you came home and delivered the ultimate goal. Nothing but appreciation and gratitude for everything you put into every moment you spent in a Cavaliers uniform. We look forward to the retirement of the famous #23 Cavs jersey one day down the line..."
James agreed to sign a four-year, $154 million contract with the Los Angeles Lakers. This is the third time in eight years James has changed teams.


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Another engineer scam, but he's actually in Lagos, Nigeria. I accepted his request because he Zoosk; Not rated yet Something didn't seem right; claims to be a widower and going to visit sick father in Nigeria. The information they provide sounds like the story above. They wanted money to help with other things. They also lurk in chat rooms and social networking sites as well as Christian and other religious-based dating sites. I corresponded with her for two or three Nigeria scam us army Not rated yet I started chatting to a guy. Mining machine went wrong ands then he asks for money and obviously David R Irvin Not rated yet Fake military, and my friend sent him 900 dollars and he said he was stationed in Tennessee but she sent money to him in Africa. He's originally from Paris France. Love Anyone hear of John Williams.
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Q: SQL syntax error on JOIN statement $stmt = $dbo->getConnection()->prepare("SELECT count(character_name)
            FROM playercharacter
            JOIN character
            ON (playercharacter.character_id = character.character_id)
            WHERE user_id = :user_id");

giving me

Fatal error: Uncaught exception 'PDOException' with message
  'SQLSTATE[42000]: Syntax error or access violation: 1064 You have an
  error in your SQL syntax; check the manual that corresponds to your
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  (playercharacter.character_id = character.character_id)' at line 3' in
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  PDOStatement->execute(Array) #1 {main} thrown in
  /home/content/50/10835750/html/dogdays/game/private.php on line 43

A: character is a Reserved Keyword and happens to be the name of your table. In order to avoid syntax error, the table name should be escaped using backticks. Ex,
SELECT  count(character_name)
FROM    playercharacter JOIN `character`
            ON (playercharacter.character_id = `character`.character_id)
WHERE   user_id = :user_id

*

*MySQL Reserved Keywords List
If you have the privilege to alter the table, change the table name that is not on the Reserved Keyword List to prevent the same error from getting back again on the future.

redpajama_set_name: RedPajamaStackExchange


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Gurgaon: 13-Year-Old Dalit Girl Raped, Killed; Landlord's Relative Held
The police said that landlady's brother Praveen Verma has been arrested on the complaint of the girl's father who made a PCR call when the accused allegedly tried to force the family to cremate the body.
The father has alleged in the FIR that landlady's brother Praveen Verma, along with others, killed his daughter.
2021-08-31T15:34:05+05:30 Gurgaon: 13-Year-Old Dalit Girl Raped, Killed; Landlord's Relative Held
Published: 31 Aug 2021, Updated: 31 Aug 2021 4:39 pm
In yet another shocking incident, a 13-year-old Dalit girl from north Delhi's Narela was allegedly raped and killed at neighbouring Gurgaon by a relative of her landlord.
In an FIR filed with the police, the girl's father has said, "On July 17, my landlord's wife said that her sister-in-law has given birth to a child and she was taking my daughter along with her to her brother's home in Gurgaon. She also said that my daughter would stay there and could play with the daughter of her brother."
However, around 3 pm on August 23, the father was informed by his landlord that his daughter has died, the FIR sates. Around 7 pm, they took the body in a private ambulance to the girl's Narela residence for cremation, it added.
When the father got suspicious, he made a PCR call and Narela police station staff reached the spot, and took the body to the Babu Jagjivan Ram hospital in Jahangirpuri, the FIR stated.
The Gurgaon Police said it has registered a case under sections 302 (murder) and 120B (punishment of criminal conspiracy) of the Indian Penal Code and SC/ST Act.
They said further sections were added in the FIR after the postmortem report confirmed sexual assault, and Verma was arrested.
Earlier this month, a nine-year-old Dalit girl was alleged rape and killed in Delhi Cantt area.
The girl died under suspicious circumstances even as her parents alleged that she was raped and forcibly cremated by a crematorium's priest in southwest Delhi's Old Nangal village.
With PTI inputs
FIR Against 6 After Video Of Them Thrashing Man Inside Graveyard Goes Viral In MP
ISL: FC Goa Sign Dylan Fox From NorthEast United
Outlook Web Bureau Delhi Child Rape Rape Case Dalits National


Current cancer treatment methods and drugs are a good example of a war scenario. Bombard an area or type of cells and destroy whatever you can in the "hope" of destroying the malicious source. In contrast advances in genetic engineering allow us to deal with the route and source of the issue.
Going to war for humanitarian reasons are two contradictory words. similar to the famous American "friendly fire". War/fight is the animalistic reaction when we "think" every other method has failed. Struggle over power.
Your question: Should we in EU and USA stop fighting other people's wars and focus on rebuilding our economies?
The wars consequence for any country on either end is devastating. If the Iraq war cost US in excess of 1$ trillion, I am sure the money could have been put to better use in order to topple Saddam and his regime.
I have often argued with friends that in these cases, if you drain a country of its brains, then the country will practically come to cease. What's a country with no work force or a country that no one wants to live in! Imagine what EU & US could do with 1$ trillion and the best pool of scientists and other academics and entrepreneurs. Of course it is not that feasible in practice, but hopefully will serve as a better starting point than war.
How many average Libyans are there?


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Q: Integrate: $\int\limits_0^{2\pi } {\exp \left( {ar\cos ({\psi _a} - \theta ) + b{r^2}\cos ({\psi _b} - \theta )....} \right)} d\theta $? I am trying to find the following integral in the close form:
$\int\limits_0^{2\pi } {\exp \left( {ar\cos ({\psi _a} - \theta ) + b{r^2}\cos ({\psi _b} - \theta ) + c{r^3}\cos ({\psi _c} - \theta )} \right)} d\theta $
Where, $a,b,c,r$ is non-negative real number.
I was trying to use the form of zero-order modified Bessel identity that is,
Assuming $a > 0$,
$$ I_0(x) = \frac{1}{\pi}\int_0^\pi e^{x\cos\theta}d\theta $$
Where $I_0(x)$ is a modified Bessel function of the first kind.
My approach:

*

*Converting ${ar\cos ({\psi _a} - \theta ) + b{r^2}\cos ({\psi _b} - \theta ) + c{r^3}\cos ({\psi _c} - \theta )}=Kcos({\psi _K}-\theta)$. Then use the identity mentioned above. But I am not sure if it is possible to convert ${ar\cos ({\psi _a} - \theta ) + b{r^2}\cos ({\psi _b} - \theta ) + c{r^3}\cos ({\psi _c} - \theta )}=Kcos({\psi _K}-\theta)$?

*Is there any other approach that I might take? Any suggestion would help.

A: I shall try to make it completely general.
Changing notations, the agument of the exponential function is
$$S=\sum_{n=1}^p a_n r^n \cos (\psi_n-\theta) $$ Expanding the cosines, we have
$$S=\Big[\sum_{n=1}^p a_n r^n \cos (\psi_n)\Big]\cos (\theta)+\Big[\sum_{n=1}^p a_n r^n \sin (\psi_n)\Big]\sin (\theta)$$ that is to say
$$S=A_c \,\cos (\theta)+A_s \,\sin (\theta)$$ and then the problem is
$$\int_0^{2\pi} \exp\Big[A_c \,\cos (\theta)+A_s \,\sin (\theta)\Big]\,d\theta=2 \pi  I_0\left(\sqrt{A_c^2+A_s^2}\right)$$


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Getting through the New South Wales' Drought
This year's high temperatures have made 2018 NSW's driest year since 1965. Our drier than expected winter has left many farmers with failing crops, limited water supplies and declining livestock feed. Many locals say this is New South Wales' worst drought since 1902.
In areas such as Coonabarabran, Coolah and Mendooran, water restrictions are already in place. Some towns have a level 6 water restriction, limiting them to two full loads of laundry a week and three-minute showers. Businesses are banned from washing commercial vehicles and can only use reclaimed water for public gardens and bowling greens.
Wildlife in NSW is greatly affected as well. In Sydney, flocks of parrots set up home in the capital's parklands as they struggle to find new sources of food. Being well-watered and fertilised, Sydney's parklands provide parrots with relatively abundant amounts of fruits and seeds to feed on.
Battling a Crippling Drought
The drought has left many farmers and business owners devastated.
Orchadier Lynette Rideout tells 9 News that she can no longer sustain her drought-hardy Christmas trees. She has no choice but to leave them to die. Even if temperatures and the weather become fairer, it will take years for her trees to grow and her business to recover. Local charity Dilly Drought Drive donates some water to Orchadier's orchard but the quantity is not enough and she is keeping her reserves in case bushfires break out.
In NSW's central-west region, farmers are importing livestock feed as far away as South Australia. Mixed sheep and cattle farmer Robert Taylor is struggling to keep his family afloat and his livestock alive. He is currently shearing sheep, but with the amount of money needed to feed his livestock, his family has no income.
The NSW government is continuously reassessing the aid it is providing to better help NSW's struggling farmers. Aside from a pledged $1billion, the government has put in provisions to allow farmers to invest in infrastructure to store feed, and to secure low-interest loans. The government is also allotting $6.3 million to help farmers suffering from mental health issues.
Cultivating Positivity
Amid the dire circumstances, locals are finding means to stay positive and help each other.
Farmer Cassandra McLaren started a Facebook page named One Day Closer to Rain to give locals a venue to talk about what they are going through and give each other support. Today, the group has over 14,000 members with many people wanting to offer their support. Through the page, farmers in need of hay or other means of income have been given aid.
Schools have been very creative in helping students cope. Extracurricular activities such as plays, woodwork and wall-painting have been provided to give students a project to focus on and help them stay positive. In Trundle Central School, Principal John Southon has upgraded the school showers and opened them to the community to let his students and family wash. He also brought in a therapy dog for the children's wellbeing.
There are also numerous individuals and private groups collaborating to help in NSW. Organisations such as Buy a Bale of Hay, Drought Angels, Rural Aid Australia and Lions Need for Feed are all doing their part to help NSW.
Supporting Local Communities
Rapid Plas has been supporting NSW's local communities for over 25 years. Our team comes to work with the aim of helping regional communities solve water management needs. For questions on our water tanks and livestock feeding solutions, contact us today.
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Planting the Seeds of Urban Farming
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PM says minimum wage hike is a matter of negotiation
Foto: DELFI / Karolina Pansevič
With Lithuania's Tripartite Council set to discuss an increase in the minimum monthly wage on Tuesday, Prime Minister Saulius Skvernelis would not say by how much it could be raised in early 2019.
"The Tripartite Council will now sit down for talks and we'll listen to what expectations the trade unions and employers have," Skvernelis told reporters on Monday.
"The government, as an active member of the Tripartite Council, will stick to the principle (...) that the minimum monthly wage should be between 42 and 48 percent of the average wage ", he said.
After the Seimas approves the proposed tax reform, which provides, among other things, for merging employee and employer social insurance contributions, the formula for the ratio between the minimum and average wage be adjusted as well, Skvernelis said.
The Tripartite Council, which brings together the government and a part of employers and trade unions, is to return on Tuesday to discussions on a minimum wage increase. This time, the minimum wage is expected to be set based on a formula, but some employers want it to be improved.
Several hundred people demanded better health funding in Vilnius
Lithuanity Code (#LietuvybėsKodas). Projection – outpost of high technologies
The council decided last year that the minimum wage should account for 45-50 percent of the average wage.
The minimum wage was last year raised by 20 euros to 400 euros.
Source: BNS


CALLIHAN, MOSIS A. (?–?). Mosis A. Callihan (Morris Callahan) was a partner of Allen Vince as one of Stephen F. Austin's Old Three Hundred colonists. The pair received title to a league of land now in Harris County on August 3, 1824. Callihan probably died before September 30, 1825, for about that date Humphrey Jackson wrote Austin that Nicholas Callahan, the brother of Morris Callahan, deceased, was getting a power of attorney from his father to settle the estate.
Handbook of Texas Online, "CALLIHAN, MOSIS A.," accessed April 22, 2019, http://www.tshaonline.org/handbook/online/articles/fca20.


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\section{Introduction}
\label{Introduction}
Perturbative methods find large utilisation for studying a number of problems in physics, spanning from classical to quantum mechanics~\cite{broer01,bransden01}.
Furthermore, as we know from the theory of chaos~\cite{smith01}, even small perturbations can have drastic effects on the dynamics of some systems, as those particularly sensible to initial conditions. The logistic map, for instance, is very useful for studying these phenomena. Also, the theory of chaos has the merit to have widely popularised fascinating results, as the so called 'butterfly effect', being nowadays mentioned both in books and movies (e.g.~\cite{movie01}), although sometimes without proper scientific care.

Perturbations have also been used in evolutionary game theory for studying the emergence of cooperation~\cite{Capraro2018,castellano_rmp09}, as recently reported in~\cite{amaral_javarone_2020}.
Notably, the latter shows that small and random perturbations on the payoff matrix, of social dilemmas, can strongly influence the strategy equilibrium of a population. However, while their effect on the main diagonal has been clarified, i.e. they support cooperative behaviours, the effect given by off-diagonal perturbations is still unclear.
Thus, here we aim to clarify this specific aspect of the perturbative method presented in~\cite{amaral_javarone_2020}, and to obtain a complete description of the dynamics of evolutionary games, affected by small and random perturbations on their payoff matrix.
To this end, our analyses consider a variety of conditions, as three different update rules, and various games (e.g. Stag-Hunt, Snow-Drift, and Harmony Game).
For the sake of clarity, and for making the manuscript self contained, the main model is fully described in the following section, with all relevant information.
Before moving further, let us give a brief introduction on the framework of Evolutionary Game Theory (hereinafter EGT) that we use for our investigation.

EGT is a suitable framework to study the emergence of cooperation among selfish individuals~\cite{Smith82, Nowak2006, Perc2017}. While cooperation still represents a lively challenge~\cite{Pennisi2005}, EGT has given many important insights on how such a phenomenon can spontaneously emerge at many scales in biological and social systems. Among the most studied mechanisms are kin selection~\cite{Hamilton1964}, direct and indirect reciprocity \cite{trivers_qrb71, axelrod_s81}, network reciprocity~\cite{Nowak1992a, wardil_epl09, wardil_pre10}, group selection~\cite{wilson_ds_an77} and heterogeneity~\cite{perc_bs10, Amaral2016, Amaral2015}.
Specifically, heterogeneity (sometimes dubbed as diversity) was found to be a fundamental element for supporting cooperative behaviours in evolutionary systems~\cite{santos_jtb12, wang_z_pre14b}. A group composed of individuals with different skills can easily become more effective in several challenges. At the same time, group diversity can shield the possible flaws of individual members while sustaining the whole group in a synergetic manner. This rationale has been applied to economies, enterprise communities, vaccination models, and even in biological evolution~\cite{Stewart2016, santos_jtb12, Qin2017a, zhang_xd_cpb12, RajibArefin2019}. Besides, that represents a fundamental aspect in genetic algorithms as well~\cite{Holland1992}, whose core mechanism is based on an heterogeneous population of candidate solutions that evolves towards a (sub-)optimal solution.
In EGT, heterogeneity can be related to different aspects of a game, e.g. incentives, interaction topology, learning rates and dynamics. Previous investigations, as~\cite{ Szolnoki2019, Takesue2019, Amaral2018, Szolnoki2018, Zhou2018, Qin2017a, Mann2016, perc_pre08}, showed that heterogeneity can support cooperative behaviours in many competitive scenarios. At the same time, other studies as~\cite{gracia-lazaro_pnas12}, reported small experimental differences between results achieved on heterogeneous and homogeneous networks, when humans play the Prisoner's Dilemma. As another example, findings reported in~\cite{Javarone2017} shows a relationship between the group size of a community and its heterogeneity. This rich spectrum of results highlights the need to further study this subject and to fully understand its range of effects.

As observed in~\cite{amaral_javarone_2020}, perturbations on the payoff matrix of dilemma games actually are responsible for a form of population heterogeneity, as they provide players with a diversified perception of risks and rewards.
For this reason our investigation, while focused on clarifying the role of perturbations in social dilemmas, can provide findings of potential interest for the debate on the relationships between heterogeneity and cooperation in evolutionary games.
Also, the relevance of studying the effects of an heterogeneous risk perception is particularly motivated by a simple fact of everyday life, e.g. individuals facing the same situation will have different risk perceptions~\cite{ Perc2017}.
Financial trading, emergency medicine~\cite{lawton01}, and even poker games~\cite{javarone_poker_a}, are just a few examples we can mention to appreciate the relevance of an heterogeneity in risk and reward perception in a system. For instance, rational, but risky decisions can be essential to achieve favourable outcomes, and individuals actually show a variety of behaviours in the above listed activities.
The generality of these considerations makes the payoff values suitable for being considered as stochastic variables, instead as fixed values (as per the classic EGT approach).

One way to study the influence of such random variations is to represent each different environmental condition as a new factor in the equations of a model. Following this method, many authors have done important advances in the understanding of how a number of conditions can drive the system dynamics, such as resource heterogeneity~\cite{Vicens2018}, different behaviours~\cite{Fang2019, Liu2019}, seasonal variations~\cite{Szolnoki2019}, diverse learning rates~\cite{Deng2018}, different death rates~\cite{Junior2019}, interaction topologies~\cite{Szabo2007}, and so on. However, another way to understand these phenomena is to study the behaviour of a population whose evolution can be affected by a payoff matrix constantly perturbed by stochastic noise with zero mean value~\cite{Su2019, Zhang2013, perc_njp11}, irregardless of its origin.
In other words, as the environmental perturbations are very diverse and frequent, we can suppose that the sum of infinitely many small perturbations acts as a stochastic perturbation around an average value. Indeed, the central limit theorem indicates that the sum of all these uncorrelated perturbations would probably behave as a Gaussian noise. In this scenario, we are not interested in every single source of perturbation, but rather, in their collective effect.

We deem relevant to mention previous seminal works as~\cite{Perc2006, Perc2006a, Perc2006b, tanimoto_pre07b, hofbauer_jet07}, that (to the best of our knowledge) first studied such disorder in EGT. More recently, the effect of payoff noise on phase transitions has been the core of many researches that shed light on how different types of perturbations can lead to the emergence of cooperation~\cite{amaral_javarone_2020, Szolnoki2019, Su2019, Stollmeier2018, Alam2018, Hilbe2018, Javarone2016b, Tanimoto2016, Yakushkina2015, wang_z_pre14b, Zhang2013, anh_tpb12}. So, here, starting from the model proposed in~\cite{amaral_javarone_2020}, which introduces payoff perturbation in the context of the imitation update rule for all payoff entries, we perform a full analysis under different settings, and then we study the microscopical mechanism that leads to the observed macroscopic results.

After introducing the proposed model in Section~\ref{sec:Model}, we show the results of numerical simulations (i.e., Section~\ref{sec:Results}) and, eventually, in Section~\ref{sec:Conclusions}, we discuss main findings and their possible implications.

\section{Model}
\label{sec:Model}
For the sake of simplicity, we consider two-strategy games, where players can either cooperate (C) or defect (D). Mutual cooperation yields a payoff $R$ (reward) and mutual defection yields $P$ (punishment). A defector receives a payoff equal to $T$ (temptation) when interacting with a cooperator, that in turn receives a payoff equal to $S$ (which usually stands for Sucker's payoff). Then, we introduce the effects of small payoff perturbations, say $\varepsilon$, randomly occurring on the off-diagonal elements of the payoff interaction matrix.
As result, the payoff matrix of any two-player interaction can be written as follows
\[
 \bordermatrix{~ & C & D \cr
                  C & R & S+ \varepsilon _S \cr
                  D & T+ \varepsilon _T & P \cr},\]

\noindent where $T\in[0,2]$ and $S\in[-1,1]$. Without loss of generality, we set $R=1$ and $P=0$. A similar approach was also introduced in~\cite{Perc2006a}.
It is worth clarifying that $\varepsilon_T$ and $\varepsilon_S$ are independent random variables with zero average value ($\bar{\varepsilon}=0$), drawn from a uniform distribution unless stated otherwise. At each new player interaction, the fluctuation is randomly drawn again, meaning that the noise is not cumulative, nor fixed in time or space. This is similar to the annealed disorder case of condensed matter physics~\cite{Tanimoto2016, Fiore2018}.

Here, the control parameter for the proposed model is the perturbation strength $D$, where for a uniform distribution we set $-D<\varepsilon<D$. It is useful to remark that this payoff matrix parametrization spans four different classes of games in the $\{T,S\}$ parameter space: prisoner's dilemma (PD), snow-drift (SD), stag-hunt (SH), and harmony games (HG)~\cite{Szabo2007, perc_bs10} ---see Figure~\ref{diagram}. We emphasize that even if the average value of the perturbation is zero, it can be able to (locally) change the game class to a more friendly or more competitive environment from time to time. However, in the long run, there should be, on average, no unilateral contribution to either.

\begin{figure}
  \includegraphics[width=6cm]{1-eps-converted-to.pdf}
  \caption{$T\times S$ parameter space with $R=1,P=0$, spanning four classes of games. The payoff fluctuation acts over $T$ and $S$ simultaneously and uncorrelated. Note that local fluctuations can lead players to play different classes of games depending on the fluctuation strength.}
  \label{diagram}
\end{figure}

A general core aspect of evolutionary games is given by the strategy update dynamics. We can understand it as a two-step process where: $(1)$ players interact with their neighbours accumulating a payoff, then $(2)$ they may change strategy according to an update rule. Such rules can be defined by taking into account different aspects of any given system. Here we consider the usual Imitative rule \cite{Szabo2007}, the Ising (or Glauber Dynamics) rule \cite{Amaral2017}, and the Dynamic Win-Stay-Lose-Shift (WSLS) rule \cite{Amaral2016b}. This is done so we can study the robustness of the effects created by payoff perturbation since previous works have extensively shown how the update rules can lead to different behaviours~\cite{Takesue2019, Fang2019, Szolnoki2018, Danku2018, Amaral2018, Dercole2019}.

We first present the most usual update rule, the imitative dynamic. Accordingly, one player, say $i$, updates its strategy by comparing its payoff with that of one randomly chosen neighbour, say $j$. Player $i$ adopts the strategy of player $j$ with probability
\begin{equation}\label{imitateeq}
p(\Delta u_{ij})=\frac{1}{ 1+e^{-(u_{j}-u_{i})/k} },
\end{equation}
\noindent where $k$ is the irrationality level in the decision process~\cite{Szabo2007} (similar to the temperature in statistical physics), while $u_{i}$ and $u_{j}$ represent the player $i$'s and $j$'s payoff. We use $k=0.1$ for all simulations unless stated otherwise. The imitation rule is a non-innovative dynamic~\cite{Szabo2007,Nowak2006}, because a player can only change its strategy selecting among those available in the population. This is one of the most explored rules in EGT, and it is associated with the replicator dynamics observed in biological evolution~\cite{Smith82}. Because of the extensive research on this setup, we will focus our analysis mainly on this model.

We present next the so-called Ising rule (also known as Glauber Dynamics \cite{Glauber1963}), where players change strategy with probability
\begin{equation}\label{isingeq}
p(\Delta u_{i})=\frac{1}{ 1+e^{-(u_{i}^*-u_{i})/k} }
\end{equation}
\noindent with $u_{i}$ player $i$'s payoff, and $u_{i}^*$ its potential payoff at a next iteration, if it changed strategy while everything else (i.e. the strategy set of its neighborhood) remained the same. Recently, the Ising rule has been studied in~\cite{Amaral2017, Amaral2018, szabo_jtb12b}, showing that it leads to very different dynamics when compared to imitation models.
In mathematical terms, this setting is equivalent to a Monte-Carlo process used to describe the dynamics of spins in a Heisenberg model~\cite{binder_prb80}. In the context of EGT, this update rule is regarded as a player asking himself what would be the benefits of changing its strategy to a different one. This is closely related to rational analysis of a situation (see also~\cite{Javarone2016d} on this topic), instead of the reproduction of the 'fittest' behaviour.

Lastly, we also implement the Win-Stay-Lose-Shift (WSLS) update rule with dynamic aspiration~\cite{Amaral2016b}. The WSLS strategy relies on cognitive capabilities, instead of replicating process~\cite{bonawitz_cg14, perc_pone11, pacheco_prl06, chen_jsm15}. In this case, players change strategy depending on the degree of satisfaction with their current payoff, in comparison to the average payoff of their neighbourhood.
The probability of a chosen player to change its strategy to the opposite one is given by:
\begin{equation}\label{wslseq}
p(\Delta u_{i})=\frac{1}{ 1+e^{-( \bar{u}-u_{i})/k} },
\end{equation}
\noindent where $\bar{u}$ is the average payoff of the player $i$'s neighborhood.

To implement such dynamics we use an asynchronous Monte-Carlo protocol, so that a random player, say the $i$-th, is selected and its cumulative payoff, as well as those of its first and second degree neighbours, are calculated. Then the $i$-th player can change its strategy according to the defined update rule. One Monte Carlo step (MCS) consists of this process being repeated until each player has had the opportunity to change its strategy (that is, $N$ times). So, we wait for the system to reach a dynamic equilibrium (around $10^4$ MCS's), and then we average the values over the final $1000$ steps. This is repeated for $50-100$ different samples with the same parameters. A square lattice, with Von Neumann neighbourhood and periodic boundary conditions with $N=10^{4}$ individuals, is the background for our simulations. A detailed discussion on Monte Carlo methods in evolutionary dynamics is provided in~\cite{landau_00, Szabo2007}.

\section{Results}
\label{sec:Results}
In this section, we show the results of numerical simulations of the proposed model. For the sake of clarity, irrelevant error bars are hidden. Figure~\ref{timeaverage} a) presents the average cooperation fraction, $\rho(t)$, of a population as time passes, considering the imitative rule. Also, it shows the evolution of cooperation for the parameters $T_c=1.04$ and $S=0$. We remind that for the imitative model with no payoff perturbation, $T_c=1.04$ corresponds to the phase transition point where cooperation becomes extinct. This was chosen to highlight the positive effect of payoff perturbation, since its strongest effect happens near the phase transition boundary. Remarkably, adding a perturbation as low as $D=0.05$ can prevent said extinction, and for $D=0.2$, cooperation can increase even up to $40\%$. We note that this increase in cooperation also happens for a wide range of $T$ values. Qualitatively similar results were obtained for the Ising and WSLS update rule regarding the temporal evolution of the population.

\begin{figure}
  \includegraphics[width=8.5cm]{2a-eps-converted-to.pdf}
  \includegraphics[width=8.5cm]{2b-eps-converted-to.pdf}
  \caption{Average evolution of cooperation, $\rho(t)$, for for $100$ samples using the imitative update rule. We set $T_c=1.04$ in a), the critical extinction point for cooperation. Even when the noise is around $5\%$ of the $T$ value, cooperation can re-emerge. Figure b) presents similar analysis for $T=0.96$. Notice that the $x$ axis is logarithmic. }
  \label{timeaverage}
\end{figure}
The analysis of the temporal evolution indicates that the noise affects almost immediately the cooperative behaviour. Using the definitions of Tanimoto~\cite{Tanimoto2016}, we can observe that the ``Endurance'' phase (where cooperation initially falls) is drastically affected. Notably, even for $MCS<20$, the noise enhances cooperation preventing its extinction. The ``Expanding'' phase is therefore reached for very short times, allowing cooperation to flourish. Also note that Figure~\ref{timeaverage} b) presents the average temporal behavior for $T=0.96$. We observe that the perturbation can be detrimental to cooperation if $T<1$, that is, for more fraternal games. Note however that this detrimental effect is weaker than the positive effect when $T>1$.

Next, we focus on the final equilibrium fraction of cooperators, $\rho$, for the whole range of $T$. For simplicity, we present results for the weak prisoner's dilemma configuration (i.e. $S=0$). We ran simulations for different $S$ values, and the general trends are maintained. Figure~\ref{Tvar_uniform} presents the outcomes achieved by using the uniform noise distribution in all considered update rules (i.e. Imitative, WSLS and Ising). Its inset shows $(\rho-\rho_0)$, where $\rho_0$ is the cooperation value for the case $D=0$. This is especially useful to compare the effects of the payoff perturbation with the unperturbed case, filtering off the effects of varying $T$.
It is interesting to note that there is a general trend in all models, i.e. the noise increases cooperation in the PD region ($T>1$), and such enhancement grows with the noise amplitude, $D$. We stress here how the three update rules have very different dynamics, and that even so, the general effect of the payoff noise was maintained in all three cases.
The inset allows understanding the level of enhancement caused solely by the noise. In particular, the Imitative and WSLS rules present a clear peak, that grows with $D$ while maintains its position (relative to $T$) regardless of the noise level.
It is also very interesting to note that this peak happens in $T_c=1.04$ for the imitative model, and after that, the positive effect begins to decline. This again reinforces that the benefit of payoff perturbation is most strong near the phase transition. At the same time, for $T<1$, the noise can dampen cooperation in different manners for each case. Using the imitative updating rule, the related drop is very shallow ($~15\%$) and almost independent of $D$. That is similar to results achieved by using the WSLS rule. On the other hand, the Ising rule presents a strong drop in $(\rho-\rho_0)$, which is very dependent on $D$.

\begin{figure}
  \includegraphics[width=8.5cm]{3a-eps-converted-to.pdf}
  \includegraphics[width=8.5cm]{3b-eps-converted-to.pdf}
  \includegraphics[width=8.5cm]{3c-eps-converted-to.pdf}
  \caption{Average final cooperation level ($\rho$) as a function of $T$ for the imitative (a), WSLS (b) and  Ising(c) models using the uniform noise distribution. The inset shows $\rho-\rho_0$, where $\rho_0$ is the cooperation value for the case $D=0$.}
  \label{Tvar_uniform}
\end{figure}

To observe how general the effect can be, we also analysed a similar setting but with a Gaussian distribution, instead of a uniform one. In this case, we set our control parameter as $D=\sigma$ (i.e. the standard deviation of the distribution). Let us stress that we cannot directly compare the control parameter $D$ for both cases since, in the uniform distribution, $D$ is the range of the distribution, whereas in the Gaussian case, $D$ is the standard deviation. Even so, $D$ relates to the perturbation strength in both scenarios. The results are very similar for the three update rules, with minor quantitative differences. The general difference being that for Gaussian distributions, cooperation is more enhanced than for uniform ones given the same value of $D$. This can be credited to the fact that in a Gaussian distribution with $\sigma=D$, although most perturbations will be equal or smaller than $D$, there will be rare stronger perturbations occasionally. This, in turn, translates to a more noisy system that can strongly enhance cooperation.

In figure~\ref{imitcompare} we present a comparison between the uniform and Gaussian distributions, using the imitative update rule. For the sake of clarity, we present only $D=0.2$ and $D=0.5$, but the effect was similar for all range of $D\in [0,1]$. As expected (assuming the same control parameter $D$ for both cases), the Gaussian distribution has a more pronounced effect. This suggests that the cooperation enhancement stems from rare and influential fluctuations, that are more common in the Gaussian distribution. This is in accordance with a similar hypothesis presented in \cite{Perc2006, Perc2006a, Perc2006b}.

\begin{figure}
  \includegraphics[width=8.5cm]{4-eps-converted-to.pdf}
  \caption{Average final cooperation level as a function of $T$ for the imitative model, comparing the uniform (continuous lines) and Gaussian (dashed lines) distribution. Here $D$ stands for the strength of the perturbation. In the uniform distribution $D$ is the range of the perturbation while in the normal distribution it is the standard deviation.}
  \label{imitcompare}
\end{figure}

Considering the imitative rule, we varied the noise level, $D$, for different $T$ values. Results are presented in Figure~\ref{vardelta}. The perturbation has a positive and continuous effect on the cooperation in regions where $T>1$, and a (small) detrimental effect if $T<1$. Specifically, the strongest cooperation enhancement happens exactly at the phase transition ($T_c=1.04$). We see that payoff perturbation enhances cooperation for egotistic games ($T>1$) while it can dampen said cooperation for fraternal games ($T<1$).

\begin{figure}
   \includegraphics[width=8.5cm]{5-eps-converted-to.pdf}
  \caption{Average final cooperation level as a function of perturbation strength $D$ for different values of $T$. Black circles represent the critical point $T_c=1.04$, were the perturbation effect is stronger.}
  \label{vardelta}
\end{figure}

To understand whether the cooperation boost is caused by any kind of random fluctuation, we compare the effects of payoff perturbation with the noise generated by high irrationality values, $k$, ~\cite{Javarone2016d, vukov_pre06}. As demonstrated by Szabó et al \cite{szabo_pre05}, irrationality can have beneficial effects on the maintenance of cooperation for some parameter regions. Nevertheless, this is not a linear effect, and there is an ``optimal'' irrationality level, after which the system starts to behave randomly, destroying cooperation. Figure~\ref{rho_k_D} illustrates the final average cooperation level as a function of irrationality $k$ for different payoff perturbation strengths. We see that the payoff perturbation boosts cooperation regardless of the irrationality level. Even more, we see that the cooperation enhancement granted by intermediate levels of irrationality is additive with the beneficial effect of the payoff perturbation (more strongly present in the region $0.15 < k < 0.6$). We used $T=1.04$ for the figure, but this effect remained similar for different $T$ values and different update rules in our simulations.

\begin{figure}
   \includegraphics[width=8.5cm]{6-eps-converted-to.pdf}
   \caption{Average final cooperation level as a function of irrationality, $k$, in the phase transition $T=1.04$ for the imitative model. As expected, there is an optimal irrationality level for $D=0$. Even so, payoff fluctuations always increases cooperation for the whole range of explored $k$ values.}
  \label{rho_k_D}
\end{figure}

We proceed to analyse which microscopic mechanism is responsible for the cooperation enhancement due to payoff perturbation. It is worth noting that lattice snapshots were not useful in this regard in all three update rule settings. The population has the same general spatial distribution with and without perturbation, i.e., in both cases, cooperators form the usual clusters surrounded by a sea of defectors. The main difference being only that the total fraction of cooperators is greater as we include perturbations. While this does not present the cause of the cooperation enhancement, we see that it is not directly related to a strong spatial distribution effect.

We stress that while the perturbations have zero average value, they can locally change the game class being played at each round. Based on this fact, we analyse how players (locally) fluctuate to a more fraternal or egotistic game. To do so, we define for each player, at each Monte-Carlo step, the variable $\phi=\varepsilon_S-\varepsilon_T$. If $\phi>0$ ($\phi<0$) the game being played at a given time will be more fraternal (egotistic). Note that for every player, $\bar{\phi}=0$ for long times. Next, we obtain at each time step the population fraction of cooperators and defectors that had $\phi>0$ in the previous round (named fraternal cooperators, $C_F$, and fraternal defectors, $D_F$). We do the same for cooperators and defectors with $\phi<0$ (named egotistic cooperators, $C_E$, and egotistic defectors, $D_E$). By doing so, we can understand if players strategies are correlated with $\phi$ even if $\varepsilon$ is randomly drawn at every interaction.

Figure \ref{flux} presents the average evolution of the four sub-populations for the imitative model in the region where the perturbation has its strongest effect, i.e. $T_c=1.04$. The main effect of payoff perturbation is to separate the sub-populations of players who play, on average, more fraternal or egotistic games. Specifically, fraternal cooperators ($C_F$) grows more than any other sub-population, being followed by the egotistic defectors ($D_E$). It is important to note however that even if the perturbations are symmetric, $C_F>D_E$.

\begin{figure}
   \includegraphics[width=7cm]{7-eps-converted-to.pdf}
  \caption{Average fraction of each sub-population over 100 Monte-Carlos runs. Here we used $D=0.3$ and $T=1.04$, where cooperation would be extinct if not for the payoff perturbation. Fraternal cooperators ($C_F$) quickly dominate the population being followed by Egotistic Defectors ($D_E$), Fraternal defectors ($D_F$) and lastly  Egotistic Cooperators ($C_E$).}
  \label{flux}
\end{figure}

To further explore this, we obtained the fraction of all sub-populations for different $T$ values. The results are shown in Figure~\ref{fluxtransi}. In it, we can see that this effect happens for all the relevant $T$ range. On average, sites that have fluctuations leading to $\phi>0$ will tend to become cooperators with a higher frequency than sites that had $\phi<0$ will become defectors. Note that the symbols in figure~\ref{fluxtransi} present the (normalised) fraction of cooperators that are fraternal, i.e. $C_F/\rho$ (and the normalised fraction of defectors that are egotistic, i.e. $D_E/(1-\rho)$). This is  especially important, since as $T$ increases, the total fraction of cooperators decreases, but even so, $C_F/\rho>D_E/(1-\rho)$. In other words, even if total cooperation decreases, the perturbation still induces a flux of cooperators to sites where more fraternal games were played. This, in turn, sustains cooperation for greater ranges of $T$.

\begin{figure}
    \includegraphics[width=7cm]{8-eps-converted-to.pdf}
  \caption{ Average final fraction of each sub-population (lines) as we vary $T$. Here we used $D=0.3$. Symbols presents the average normalised fraction of cooperators that are fraternal, i.e. $C_F/\rho$, and the average fraction of defectors that are egotistic, i.e. $D_E/(1-\rho)$. Sites that play more fraternal games on average will tend to become cooperators at a higher frequency then egotistic sites will become defectors.}
  \label{fluxtransi}
\end{figure}

This analysis was also performed for different perturbation strength values ($D$) to observe how it can affect the sub-populations. The results are shown in Figure~\ref{fluxDtransi}. As expected, the increase in both sub-populations of cooperators is monotonous with $D$. The inset shows the difference $(C_F-C_E)$ as well as $(D_E-D_F)$. In it, we can see that even if a stronger perturbation can lead to a higher value of $(D_E-D_F)$, this value is always lower than $(C_F-C_E)$. In other words, agents who play more fraternal sites on average will have a greater probability of becoming cooperators. On the other hand, the opposite is not true for players that have played, on average, more egotistic games, as their average fraction will be smaller than that of the cooperators. This is the main mechanism behind cooperation enhancement. Simulations for the WSLS and Ising update rules remained similar with small quantitative differences, showing how robust this phenomenon is.

\begin{figure}
    \includegraphics[width=7cm]{9-eps-converted-to.pdf}
  \caption{ Average final fraction of each sub-population (lines) as we vary $D$. Here we used $T=1.04$. The inset shows the difference of the ,average, fraternal and egotistic sub-populations, $(C_F-C_E)$ as well as $(D_E-D_F)$.}
  \label{fluxDtransi}
\end{figure}

Let us summarise the main mechanism that can increase cooperation in the proposed model; while players have symmetrical perturbations, defectors locally benefit from a more egotistic game, but cannot maintain said benefit as their clusters do not obtain benefits from the perturbation ($P$ is not perturbed). At the same time, cooperators can form spatial structures that are more robust against negative perturbations, heaping the long term benefits of positive perturbations.  In other words, cooperators are able to form positive feedback loops with the local positive payoff fluctuations, while defectors cannot. This effect is very similar to what was observed in~\cite{Amaral2016, szolnoki_epl14b}, where a very similar positive feedback loop helps the promotion of cooperation for the so called multi-games. Indeed, this seems to be a general property of evolutionary games, i.e., cooperators can benefit from a variety of fluctuations using positive feedback loops, whereas defectors cannot.

To conclude, we present the final average cooperation level for the whole parameter space $T \times S$ considering all update rules, i.e. Imitative (Figure~\ref{phaseimit}), Ising (Figure~\ref{phaseising}) and WSLS (Figure~\ref{phasewsls}). For all figures, we report the case for a high perturbation ($D=0.5$) in a) and the difference between the perturbed and unperturbed case, i.e. $(\rho-\rho_0)$,  in b) for the whole parameter space. We note that the parameter space for the unperturbed case can be found in the literature in~\cite{Amaral2016b} for the Win-Stay-Lose-Shift, in~\cite{Amaral2017} for the Ising model, and in~\cite{Szabo2016a} for the imitative model.

\begin{figure}
  \includegraphics[width=5cm]{10a-eps-converted-to.pdf}
  \includegraphics[width=5cm]{10b-eps-converted-to.pdf}
  \caption{ $T \times S$ parameter space for the imitative model. Final cooperation fraction is represented in colours. a) high perturbation ($D=0.5$) and b) difference $(\rho-\rho_0)$. }
  \label{phaseimit}
\end{figure}

\begin{figure}
  \includegraphics[width=5cm]{11a-eps-converted-to.pdf}
    \includegraphics[width=5cm]{11b-eps-converted-to.pdf}
  \caption{$T \times S$ parameter space for the Ising model. Final cooperation fraction is represented in colours. a) high perturbation ($D=0.5$) and b) difference $(\rho-\rho_0)$.}
  \label{phaseising}
\end{figure}

\begin{figure}
  \includegraphics[width=5cm]{12a-eps-converted-to.pdf}
    \includegraphics[width=5cm]{12b-eps-converted-to.pdf}
  \caption{$T \times S$ parameter space for the WSLS model. Final cooperation fraction is represented in colours. a) high perturbation ($D=0.5$) and b) difference $(\rho-\rho_0)$.}
  \label{phasewsls}
\end{figure}

This analysis indicates that the perturbation changes the population dynamics mainly near phase transition regions. By using all three updating rules, we can observe the general effect that perturbation increases cooperation in the direction of more selfish games (i.e. high $T$, low $S$), while it does the opposite for altruistic games. We also note that the detrimental effect on cooperation is usually lower than the benefit for regions with high $T$ and a low $S$.
Eventually, we deem especially interesting that by using the Ising rule (see Figure~\ref{phaseising}), the perturbation increases the anti-coordination area linearly with the perturbation strength (top right region of the $T\times S$ parameter space).

\section{Conclusions}
\label{sec:Conclusions}
In this work, we studied payoff perturbations considering simple social dilemmas, with the main goal to clarify and quantify their effects when only the off-diagonal is interested, following the results reported in~\cite{amaral_javarone_2020}.
Beyond analysing perturbative methods in evolutionary games, the proposed model has the potential to actively contribute to a vivid debate, i.e. that on the relationship between heterogeneity and cooperation.
We emphasise that the form of heterogeneity we consider is related to that of risk, and reward, perception, i.e. an aspect covering a fundamental role in many social systems, and that might also be relevant, using a different interpretation, in other contexts.

Then, while~\cite{amaral_javarone_2020} focused on a general range of effects resulting from perturbed payoffs, here we concentrate on the strategy equilibrium reached by a population in presence of perturbations acting only on the rewards of betraying and the risks involved in getting betrayed (i.e. the temptation and the sucker's payoff). Hence, cooperators playing with cooperators, and defectors playing with defectors, are not affected by perturbations. In addition, here we analyse a wider range of game parameters, i.e. near and far phase transition points, in order to identify those values that allow perturbations to actively affect the dynamics of the game.

The evolution of strategies has been studied by Monte Carlo simulations, arranging players on a square lattice with periodic boundary conditions. To verify if the observed effects were robust in different settings, we considered three different update rules, Imitation, Win-Stay-Lose-Shift and Ising. We also analised the effects of Gaussian and uniform noise distributions.

Results show that perturbations can have more pronounced effects on the population dynamics mainly near the phase transitions. Notably, the average evolution of a population showed that the perturbations act very quickly, affecting mainly the initial evolution. Also, this effect is proportional to the perturbation amplitude when $T>1$. At the same time, the dampening in cooperation observed for $T<1$ seems to be very small in comparison.
In addition, the overall effect of the perturbation seems to be independent of the specific choice of the update rule, highlighting its robustness.
Then, the analysis related to the combined effect of heterogeneity and irrationality showed that these two perturbation sources are quite independent each other.

In order to understand the microscopic mechanism responsible for the cooperation enhancement near phase transitions, we studied the sub-population of cooperators and defectors that, locally, played more fraternal or egotistic games at each time-step. This is in line with previous works, which proposed that the benefit of diversity would be related to the random changes in the game class (Prisoner's Dilemma changing to Stag-hunt or Snow-Drift)~\cite {Perc2006, Perc2006a, Perc2006b}. This approach allowed to unveil what was responsible for the general effect observed in all cases. On average, we found that the payoff perturbation offers no unilateral contribution for more egotistic of fraternal games. Even so, it can locally promote cooperation for players that had a more positive perturbation. This effect can lead to the formation of stronger cooperative clusters, while defectors cannot benefit from said phenomena in the long run as they do not benefit from mutual support (see also~\cite{Javarone2016b}). On the other hand, in~\cite{tanimoto_pre07b} authors presented arguments on the irrelevance of game class changes for some contexts. While this seems to be an open topic, our initial results indicate that the random fluctuations have deep effects on the phase transitions of the model, being able to improve cooperation thanks to the unusual states found near phase transition points.

Summarising, our main finding is that off-diagonal perturbations cannot be considered as a trivial form of noise, as they seem to selectively enhance cooperation when defection is pervading the population, while they are able to support defection as the population begins to cooperate.
Therefore, in our view, the proposed model offers interesting insights on a relevant aspect of heterogeneity, not limited only to the studying of social phenomena. Notwithstanding, since several details might deserve further attention, we report those we consider potentially more interesting. For instance, the proposed model could be analysed considering populations arranged on complex networks~\cite{Wu2017, Battiston2017}, social behaviours like conformism~\cite{galam_ijmpc08, Galam2004, Javarone2016c} could be combined with random perturbations and, as suggested in~\cite{Stollmeier2018}, also some clinical investigations could find useful to represent specific biological phenomena by the proposed method.

Finally, we see as especially interesting the connection of the payoff perturbation with recent works connecting evolutionary game theory with the Hamiltonian description of physical systems ~\cite{Nakamura2019, Szabo2016a}. These effects could be further studied in the light temporal Griffiths phases~\cite{Vazquez2011, Fiore2018}. Such an exotic state is known to appear near phase transitions in epidemiological and magnetic systems with directed percolation universality class, which is the same phase transition as the imitative model~\cite{Szabo2007}.

\begin{acknowledgments}
This research was supported by the Brazilian Research Agency CNPq (proc. 428653/2018-9).
\end{acknowledgments}


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