James Comey, the former deputy attorney general, told a riveting story at today's Senate Judiciary hearing of a March 2004 hospital-room encounter at which, in Comey's words, then White House counsel Alberto Gonzales tried to "take advantage" of a critically ill John Ashcroft to get him to certify a national security program about whose legality the Justice Department had profound doubts.
According to Comey, Gonzales and then White House chief of staff Andrew Card visited an exceedingly sickly Ashcroft to get him to override the Justice Department's concerns about the program which Comey wouldn't identify but is widely known to be the National Security Agency's controversial domestic eavesdropping project which the Bush Administration calls the Terrorism Surveillance Program.
The story of Ashcroft beating back Gonzales's attempt to get him to sign off on the program had been reported before. But this was the first time one of the main characters at the meeting had publicly talked about it, providing a wealth of details.
Comey, who was acting attorney general during Ashcroft's illness, described rushing to George Washington Hospital after he learned that Gonzales and Card were headed there, and beating them to Ashcroft's room where Ashcroft's anxious wife awaited.
He recalled putting in a call to Federal Bureau of Investigation director Robert Mueller who immediately headed for the hospital himself but first ordered his agents to not allow Comey to be removed from the hospital room by anyone.
Comey told how when Gonzales and Card arrived, they walked over to Ashcroft's bed and after a brief greeting, announced why they were there which was followed by what sounds like a scene from a Hollywood film. A very weak Ashcroft, who had appeared disoriented to Comey just minutes earlier, lifted his head from the pillow and said very cogently why he refused the certify the program, a position based on discussions he had with Comey and other Justice officials before he got sick, then said they should be talking to Comey anyway since he was the acting AG. Then Ashcroft lowered his head back to the bed.
Comey remembered how the two men left the room without saying a word to him and how later he got an angry call from Card.
In short, based on what Comey described, Gonzales may be one of the few people in Washington who could successfully make his predecessor, John Ashcroft, look like a civil libertarian and liberal hero.
The sense one gets is that there were some very serious Republican officials in Washington who were very much worried about what they saw as sinister behavior by some other very serious Republican officials. Remarkable stuff.
SEN. SCHUMER: Thank you. Okay, we're going to get right into the questioning, because Mr. Comey does not have an opening statement. As I said in my opening remarks, many have been concerned that Alberto Gonzales has made the Justice Department a mere extension of the White House, where independence takes a back seat to service to the White House, where the rule of law takes a back seat to the political needs of the president's party.
Before we get to the other issues, I want to go back to an incident from the time that Mr. Gonzales served as White House counsel. There have been media reports describing a dramatic visit by Alberto Gonzales and Chief of Staff Andrew Card to the hospital bed of John Ashcroft in March 2004, after you as acting attorney general decided not to authorized a classified program. First, can you confirm that a nighttime hospital visit took place?
MR. COMEY: Yes, I can.
SEN. SCHUMER: Okay, can you remember the date and the day?
MR. COMEY: Yes, sir, very well, it was Wednesday, March the 10th, 2004.
SEN. SCHUMER: And how do you remember that date so well?
MR. COMEY: This was a very memorable period in my life, probably the most difficult time in my entire professional life. And that night was probably the most difficult night of my professional life, so it's not something I'd forget.
SEN. SCHUMER: Okay, were you present when Alberto Gonzales visited Attorney General Ashcroft's bedside.
SEN. SCHUMER: And am I correct that the conduct of Mr. Gonzales and Mr. Card on that evening troubled you greatly.
SEN. SCHUMER: Okay, let me go back and take it from the top. You rushed to the hospital that evening. Why?
SEN. SCHUMER: Please. I'll give you all the time you need.
MR. COMEY: Okay, sir, I've actually thought quite a bit over the last three years about how I would answer that question if it was ever asked, because I assumed that at some point I would have to testify about it. The one thing I'm not going to do and be very, very careful about is, and because this involved a classified program, I'm not going to get anywhere near classified information. I'm also -- am very leery of and will not reveal the content of advice I gave as a lawyer or deliberations I engaged in. I think it's very important for the Department of Justice that someone who held my position not do that.
SEN. SCHUMER: In terms of privilege.
MR. COMEY: Subject to that, I -- and I'm uncomfortable talking about this, but I'll answer the question.
MR. COMEY: I -- to understand what happened that night, I've kind of got to back up about a week.
MR. COMEY: In the early part of 2004, the Department of Justice was engaged -- the Office of Legal Counsel, under my supervision, in a reevaluation both factually and legally of a particular classified program. And it was a program that was renewed on a regular basis and required signature by the attorney general certifying to its legality. And the -- and I remember the precise date; the program had to be renewed by March the 11th, which was a Thursday, of 2004. And we were engaged in a very intensive reevaluation of the matter.
SEN. SCHUMER: You thought -- just assuming -- you thought something was wrong with how it was being operated or administered or overseen?
MR. COMEY: We had -- yes, we had concerns as to our ability to certify its legality, and -- which was our obligation for the program to be renewed.
The attorney general was taken that very afternoon to George Washington Hospital, where he went into intensive care and remained there for over a week. And I became the acting attorney general. And over the next week, particularly the following week, on Tuesday, we communicated to the relevant parties at the White House and elsewhere our decision that, as acting attorney general, I would not certify the program as to its legality, and explained our reasoning in detail, which I will not go into here, nor am I confirming it's any particular program. That was Tuesday that we communicated that.
SEN. SCHUMER: What's his name?
MR. COMEY: -- David Ayres -- that he had gotten a call from Mrs. Ashcroft, from the hospital. She had banned all visitors and all phone calls. I hadn't seen him or talked to him, because he was very ill. And Mrs. Ashcroft reported that a call had come through and that as a result of that call, Mr. Card and Mr. Gonzales were on their way to the hospital to see Mr. Ashcroft.
SEN. SCHUMER: Do you have any idea who that call was from?
MR. COMEY: I have some recollection that the call was from the president himself, but I don't know that for sure. It came from the White House. And it came through, and the call was taken in the hospital.
So I hung up the phone, immediately called my chief of staff, told him to get as many of my people as possible to the hospital immediately. I hung up, called Director Mueller and -- with whom I'd been discussing this particular matter and who had been a great help to me over that week, and told him what was happening. He said, "I'll meet you at the hospital right now."
Told my security detail that I needed to get to George Washington Hospital immediately. They turned on the emergency equipment and drove very quickly to the hospital.
SEN. SCHUMER: What was your concern? You were in obviously a huge hurry.
MR. COMEY: I was concerned that given how ill I knew the attorney general was, that there might be an effort to ask him to overrule me when he was in no condition to do that.
MR. COMEY: I was worried about him, frankly. And so I raced to the hospital room, entered, and Mrs. Ashcroft was standing by the hospital bed. Mr. Ashcroft was lying down in the bed. The room was darkened. And I immediately began speaking to him, trying to orient him as to time and place, and try to see if he could focus on what was happening. And it wasn't clear to me that he could.
SEN. SCHUMER: And at that point, it was you, Mrs. Ashcroft and the attorney general and maybe medical personnel in the room. No other Justice Department or government officials.
MR. COMEY: It was just the three of us at that point. I tried to see if I could help him get oriented. As I said, it wasn't clear that I had succeeded. I went out in the hallway, spoke to Director Mueller by phone. He was on his way. He handed the phone to the head of the security detail, and Director Mueller instructed the FBI agents present not to allow me to be removed from the room under any circumstances. And I went back in the room. I was shortly joined by the head of the Office of Legal Counsel, Assistant Attorney General Jack Goldsmith, and a senior staffer of mine who had worked on this matter, an associate deputy attorney general.
So the three of us Justice Department people went in the room. I sat down.
SEN. SCHUMER: Can you just give us the names of the two other people?
MR. COMEY: Jack Goldsmith, who was the assistant attorney general, and Patrick Philbin, who was associate deputy attorney general.
I sat down in a(n) armchair by the head of the attorney general's bed. The two other Justice Department people stood behind me. Mrs. Ashcroft stood by the bed holding her husband's arm, and we waited. And it was only a matter of minutes that the door opened and in walked Mr. Gonzales carrying an envelope and Mr. Card. They came over and stood by the bed, greeted the attorney general very briefly. And then Mr. Gonzales began to discuss why they were there, to seek his approval for a matter, and explained what the matter was, which I will not do.
And Attorney General Ashcroft then stunned me. He lifted his head off the pillow and in very strong terms expressed his view of the matter, rich in both substance and fact, which stunned me, drawn from the hourlong meeting we'd had a week earlier, and in very strong terms expressed himself, and then laid his head back down on the pillow. He seemed spent, and said to them, "But that doesn't matter, because I'm not the attorney general."
SEN. SCHUMER: But he expressed his reluctance or -- he would not sign the statement that they -- give the authorization that they had asked. Is that right?
MR. COMEY: Yes. And as he laid back down, he said, "But that doesn't matter, because I'm not the attorney general. There is the attorney general," and he pointed to me -- I was just to his left. The two men did not acknowledge me; they turned and walked from the room. And within just a few moments after that, Director Mueller arrived. I told him quickly what had happened. He had a brief -- a memorable, brief exchange with the attorney general, and then we went outside in the hallway.
Now, just a few more points on that meeting.
First, am I correct that it was Mr. Gonzales who did just about all of the talking; Mr. Card said very little?
SEN. SCHUMER: Okay. And they made it clear that there was in this envelope an authorization that they hoped Mr. Ashcroft, Attorney General Ashcroft would sign?
MR. COMEY: Yeah, in substance. I don't know exactly the words, but it was clear that's what the envelope was.
SEN. SCHUMER: And the attorney general was -- what was his condition? I mean, he was -- he had -- as I understand it, he had pancreatitis, he was very, very ill, in critical condition, in fact?
MR. COMEY: He was very ill. I don't know how the doctors graded his condition. This was -- this would have been his sixth day in intensive care, and as I said, I was shocked when I walked into the room and very concerned as I tried to get him to focus.
SEN. SCHUMER: Right. Okay, let's continue. What happened after Mr. Gonzales and Card left? Did you have any contact with them in the next little while?
MR. COMEY: While I was talking to Director Mueller, an agent came up to us and said that I had an urgent call in the Command Center, which was right next door. They had Attorney General Ashcroft in a hallway by himself, and there was an empty room next door that was the Command Center. And he said it was Mr. Card wanting to speak to me. I took the call; Mr. Card was very upset and demanded that I come to the White House immediately. I responded that, "After the conduct I had just witnessed, I would not meet with him without a witness present." He replied, "What conduct? We were just there to wish him well." And I said again, "I -- after what I just witnessed, I will not meet with you without a witness, and I intend that witness to be the solicitor general of the United States."
SEN. SCHUMER: That would be Mr. Olson.
MR. COMEY: Yes, sir. Ted Olson.
"And until I can connect with Mr. Olson, I'm not going to meet with you." He asked whether I was refusing to come to the White House. I said, "No, sir, I'm not. I'll be there. I need to go back to the Department of Justice first," and then I reached out through Command Center for Mr. Olson, who was at a dinner party. And Mr. Olson and the other leadership of the Department of Justice immediately went to the department, where we sat down together in a conference room.
MR. COMEY: And talked about what we were going to do. And about 11:00 that night -- this evening had started at about 8:00 when I was on my way home -- at 11:00 that night, Mr. Olson and I went to the White House together.
SEN. SCHUMER: Just before you get there, you told Mr. Card that you were very troubled by the conduct from the White House room and that's why you wanted Mr. Olson to accompany you -- without giving any of the details, which we totally respect, in terms of substance.
Just tell me why. What did you tell him that so upset -- that so upset you? If you didn't tell him, just tell us.
MR. COMEY: I was very upset. I was angry. I thought I had just witnessed an effort to take advantage of a very sick man, who did not have the powers of the attorney general because they had been transferred to me. I thought he had conducted himself -- and I said to the attorney general -- in a way that demonstrated a strength I had never seen before, but still I thought it was improper. And it was for that reason I thought there ought to be somebody with me if I'm going to meet with Mr. Card.
Asked about Comey's testimony at the White House briefing this afternoon, White House Press Secretary Ton Snow merely said: "His testimony can stand on its own.'' And so it does.
Just another day in the office for these corrupt thugs.
Gonzo should go before there's no one left at DoJ. Fall on your sword, man!
I've been waiting for the moderate conservatives to stand up and take their party back from the NeoCon fanatics in the Bush administration and members of their fanclub on the Swamp like Brucie and Little Johnny Squarepants, I hope this is the beginning of the end for the NeoCons.
Just when you think TeamBush can't possibly go any lower.
They're probably going to give Gonzo a medal if history is an indicator.
It's going to be an enojoyable experiance for me watching the Republican nominee for President in 08 run his campaign in the backdrop of the cesspool of corruption that is the Bush administration.
I think I can hear the 08 GOP nominee's theme song playing right now.
"Should I stay (with W.) or should I go, now"
Geez... I mean... just... Christ.
That's just disgusting. It makes me sick. These people need help. Are they totally without conscience?
Whether you are a member of the looney left or wingnut right, or just a concerned citizen, we should all be fighting for the return of habeus corpus and the elimination of illegal surveillance of American citizens. This AG is not only incompetent and partisan, he does not understand the law. He is an embarrassment to all lawyers.
Shouldn't this be a news story on the front page of the Tribune's web site? It could replace the "Seinfeld Returning to Prime Time" that has been there all day. The New York Times and Washington Post are treating it as a news story, and rightly so.
Thank you for this entry. I was wondering if the MSM was even going to report it, especially now that the rest of the week will be all "Falwell" all the time.
What a travesty, possibly even impeachable.
I don't really care. What I want to know is which one of them was the better singer.
Attorney General John Ashcroft sings a rousing rendition of a song he wrote called "Let The Eagles Soar."
Hard to imagine how he could have lost an election to a dead guy.
You just don't get that kinda talent on karaoke night at the local pub!!!
Wow, these guys even get me feeling sorry for John Ashcroft.
"SEN. SCHUMER: You thought -- just assuming -- you thought something was wrong with how it was being operated or administered or overseen?
MR. COMEY: We had -- yes, we had concerns as to our ability to certify its legality, and -- which was our obligation for the program to be renewed."
Even the Justice Department came to realize that the program was of questionable legality. That's what set off this whole pathetic scene.
Why the front page "Did Gonzo try to take advantage of Ashcroft"? Isn't it obvious from your story that he did?
And I agree with Jeff above. This should be front page, as it is on the NY Times site (lead story).
Jeff hit it on the nail. The Trib has become some sort of sorry trash tabloid paper in many ways. Politics are so many times void from the front pages of it.
but hey, Seinfeld will save our world.
His only out is "the best of his ability" is no where good enough.
He should be impeached for subverting the Constitution.
When will Congress get some guts?
This sounds more like an episode of the Sopranos than the United States Justice Department and Executive Branch.
I suppose in their minds, the ends justify the means. And any means can be justified in terms of National Security.
What will it take for all of us to rise up and demand accountability?
Well .... there they go again. Even when you thought they can't get any lower, we find out something even more hateful and disgusting! If we're getting this much garbage while the Cheneybush regime is still IN power, what will we find starting in 2009? Maybe we should impound all documents of the Cheneybush regime right now, before they destroy any more evidence. John Ashcroft as a defender of freedom? Absurd - - except in the Bush League. When Ashcroft looks good, the alternative must be truly repugnant. Now we see it is! Let's skip impeachment of Cheneybush & their lapdog, Alberto, and lock them all up in Guantanamo for oh, say .... forever! With a daily water-boarding.
What kind of slime bag does it take to make Ashcroft seem like a victim and a defender of freedom?
Hey where are JD, John D, Bruce, Jerry White and the gang to defend the Administration on this one?
Have we finally found an act so completely indefensible that they won't even try to spin it?
This is from an administration who's President promised to restore dignity to the White House? It is disgusting what these people are doing to our country.
What else would you expect from the guy who was the torture advocate for the White House? Congress should start impeachment procedings for Gonzales right now.
Tony, my apologies for accidentally thinking you were affiliated with Bruce, John D and that bunch. My mistake for typing before thinking.
Gonzales is an absolutely horrible man. What happened to "The buck stops here" ??
....soooo did he fire all those attorney's when he got home?
I 'm confused on what this has to do with your other manufactured outrage of the attorney firings.
Is this all you can find on him, since the left wing "fishing" exhibition hasn't found any illegal activity from him?
Are you going to parade in people to make him look bad instead?
Have we heard from Card or AG on his side of the story?
Wake me up when he lies under oath and/or obstructs justice.


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Spinoza Prize winner: 'I embrace the fuzzy nature of ecology'
Professor of Global Flyway Ecology awarded Spinoza Prize
Theunis Piersma, professor of Global Flyway Ecology at the University of Groningen, has been awarded a Spinoza Prize. The Netherlands Organisation for Scientific Research (NWO) announced this today.
Theunis Piersma. Photo: NWO/ Ivar Pel
Piersma is free to spend the prize, € 2.5 million, on any part of his research he wants: 'There's much more to migratory birds than just a genome. They have an incredible ability to adapt to their demanding way of life and changes to their environment. We want to understand the interaction between that flexibility and the environments in which they grow up.'
Theunis Piersma is not only professor in Groningen, he is also a senior Wadden researcher at the NIOZ, the Royal Netherlands Institute for Sea Research.
Migratory birds do very complicated things. They travel the world in fascinating ways. And they make choices. As the seasons pass, they not only select their stopping places but also the times when they move from one place to another, which is often thousands of kilometres away. Making a mistake could cost them their lives.
Knot. Photo: Jan van de Kam
Flexibility problem
The wide range of species in nature, biodiversity, is under great pressure. In a world dominated by humans, those species able to adapt to quick changes are the ones with the best chance of survival. So what is behind the ability to adapt? Migratory birds embody that part of the biodiversity that links our country with a large part of the rest of the world. They are a good model for investigating the flexibility problem.
In a short film on the University of Groningen website, Theunis Piersma says that for the last 50 years the emphasis in biological research has been too much on the role of genes, and that the role played by the environment – even in ecology – has been underestimated. This mainly hard scientific approach has its limitations because: 'Ecology is much more fuzzy-edged. I really enjoy embracing the fuzzy nature of it and then discovering the wider picture', says Piersma.
Spoonbills. Photo: Jan van de Kam
Followed for life
'We don't know much yet about how the lifetimes of birds and migratory birds take shape. There are definitely hereditary aspects to migration, but throughout their lives birds also learn a lot from each other, and they make their own observations. For example, they react to the circumstances they encounter during their journeys. Technological developments, for example ever-smaller transmitters and new satellite and internet technology, now make it possible to follow individual birds or migratory birds throughout their lives.
Childhood experiences
Do young birds accompanied by older birds on their first migration south do better than young birds who are too late to profit from the experience of older birds? Piersma's ambition is to first interpret migratory behaviour in terms of the interaction between hereditary information and contextual information (about food availability, natural enemies and routines learned from parents and other birds of the same species). To what extent do childhood experiences – and thus not hereditary disposition – determine later migration routines? To what extent are migratory birds able to cope with the massive loss of habitat and the swift climate changes they are exposed to? Is their evolutionary flexibility the result of their ability to learn? And finally, why does one species or individual find it much easier to adapt to environmental changes than another?
'We will concentrate on three iconic Dutch species – knots, spoonbills and black-tailed godwits – so we can build on a lot of essential basic knowledge. We have been registering these species for years and closely observing their habitats in the Netherlands and along their migration routes', says the winner of the Spinoza Prize.
Theunis Piersma (1958) is professor in the Centre for Ecological and Evolutionary Studies (CEES) of the University of Groningen. He studied biology in Groningen, and lead his first research expedition, to Mauretania, in 1980 while still a student. In 1994 he gained his PhD from the same university. Since then he has worked as a researcher both at the University of Groningen and at the NIOZ Royal Netherlands Institute for Sea Research. He was appointed professor of Animal Ecology in 2003. He exchanged this chair for the new chair in Global Flyway Ecology in 2012, sponsored by a collaboration between the University of Groningen, the Worldwide Fund for Nature and Vogelbescherming. He has been a member of the Royal Netherlands Academy of Arts and Sciences (KNAW) since 2009.
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text:
\section{Introduction}\label{sec:Intro}

Evolutionary game dynamics results from the transfer of economic ideas to biology \cite{maynard-smith:1973to,maynard-smith:1982to,nowak:2004aa,nowak:2006bo}.
In economics, rational players try to find the best strategy to maximize their payoffs.
In biology, those individuals who use the best strategy obtain the highest reproductive fitness and spread in the population.

Traditionally, evolutionary game dynamics is considered in infinitely large, well--mixed populations.
This typically leads to the replicator dynamics, a system of nonlinear differential equations governing the evolutionary dynamics \citep{taylor:1978wv,zeeman:1980ze,weibull:1995hp,hofbauer:1998mm}.
For any composition of the population, the replicator dynamics determines deterministically the direction and velocity of evolutionary dynamics.
The replicator dynamics can be derived from microscopic models of strategy spreading, which are typically
stochastic \cite{helbing:1993aa,helbing:1996aa,schlag:1998aa,traulsen:2005hp,traulsen:2006ab}.
The precise definition of strategy spreading between individuals can have decisive consequences for the dynamics, in particular in structured populations \cite{szabo:2007aa,abramson:2001nx,vainstein:2001hp,ebel:2002aa,santos:2005pm,pacheco:2006pb,ohtsuki:2007pr,poncela:2008aa}.

Since microscopic models of strategy spreading are typically stochastic,
evolutionary game dynamics in finite populations can only be characterized in a probabilistic way.
The most important quantities are the probability that a mutant takes over a population and the average time for this process \cite{nowak:2004pw,taylor:2004wv,antal:2006aa,taylor:2006jt}.
Different models for strategy spreading have been proposed.
A popular model is to choose two players, Harry and Sally, at random and to let Harry adopt the strategy of Sally with a probability given by the Fermi function, $(1+\exp[-\beta (\pi^\text{H} - \pi^\text{S})])^{-1}$, where $\pi^\text{H}$ is the payoff of Harry and $\pi^\text{S}$ is the payoff of Sally \cite{blume:1993jf,szabo:1998wv,traulsen:2006bb,traulsen:2007cc}.
The parameter $\beta$ measures the intensity of selection.
For $\beta \ll 1$, selection is weak and strategy spreading is essentially random.
For $\beta \gg 1$, selection is strong and only strategies that are more successful will be
imitated.
For $\beta \to \infty$, the direction of the process for two strategies becomes deterministic and thus
the fixation probability is either $0$ or $1$.
However, even in this case, the process is only semi-deterministic, as the time of fixation remains stochastic \cite{traulsen:2007cc}.

Here, we introduce a variant of the Moran process which leads to a fully deterministic evolutionary process in finite populations under strong selection.
For weak selection, we essentially recover the transition probabilities of the standard frequency dependent Moran process under weak selection.

We describe evolutionary game dynamics in symmetric $2 \times2$ games defined by the general payoff matrix
\begin{align}\label{eq:Pmatrix}
\bordermatrix{
  & A & B \cr
A & a & b \cr
B & c & d \cr}.
\end{align}
An $A$ player will obtain $a$ when playing against another $A$ or $b$ when playing against $B$.
Choosing strategy $B$ results in either obtaining $c$ (against $A$) or $d$ (against $B$).

The average payoffs are obtained from pairwise interactions with all other individuals in the population of size $N$.
This is the standard assumption and refers to the fact that the population is well--mixed, i.~e.\  there is no explicit population structure.
Excluding self interactions, this leads to
\begin{align}
\label{eq:payoffA}
\pi^A_i &= \frac{i-1}{N-1} \,a + \frac{N-i}{N-1}\,b
\\
\label{eq:payoffB}
\pi^B_i &= \frac{i}{N-1} \,c + \frac{N-i-1}{N-1} \,d,
\end{align}
where $i$ is the current number of $A$ players in the population.
Individuals with higher average payoffs produce offspring (or are imitated) with a higher probability.
Thus, reproductive success is based on the payoff from the game.
The intensity of selection $\beta$ controls the importance of success in the game for reproductive success.
The larger the intensity of selection, the stronger the influence of the average payoff difference on reproductive fitness.

The paper is organized in the following way.
In \refS{EvDyn} we introduce the birth--death process as a general framework of evolutionary dynamics between two types in finite, well--mixed populations.
In particular, we address the probability and times of absorption.
In \refS{Moran} we give three explicit analytical forms for the microscopic dynamics and we discuss the possibility to analyze strong selection in each case.
We show that the standard Moran process and a previous generalization do not allow a fully deterministic strong selection limit and propose a new generalization of the Moran process with selection at birth as well as selection at death.
In \refS{Strong} we perform the strong selection limit analytically for the new process.
In \refS{RPS} we consider the process with selection at birth and death for two player games with three strategies, namely the the Rock--Paper--Scissors game.
Finally, in \refS{Disc} we conclude and discuss our findings.

\section{Evolutionary Game Dynamics in finite populations}\label{sec:EvDyn}

In this section, we recall some important properties of stochastic evolutionary game
dynamics in finite populations. For simplicity, we restrict ourselves to birth--death processes in which the number of $A$ players
can change at most by $\pm 1$ in each update step.

Let $i$ be the number of $A$  individuals players
in a population of size $N > 2$.
The number of $B$ players
is given by $N - i$.
The transition probabilities to move from $i$ to $i + 1$ and to $i - 1$ are denoted by $T^+_i(\beta)$ and $T^-_i(\beta)$, respectively.
The probability to stay in the current state is thus $1 - T^+_i(\beta) - T^-_i(\beta)$.
These microscopic details do not have to be specified further at this point.
The only requirement is that the expressions for $T^\pm_i(\beta) \neq 0$ are analytic in system size, payoffs and intensity of selection.
It is also assumed that the population size remains constant.
Mutations are excluded, such that a strategy
that is lost will not re-appear in the system.

In the continuous limit $N\to\infty$, the state of the system $x=i/N$ becomes a continuous variable, strategy $A$ can have any abundance and we recover a deterministic differential equation \cite{helbing:1993aa,helbing:1996aa,schlag:1998aa,traulsen:2005hp,traulsen:2006ab}.
This allows to compute the fixed points of the system.
There are always fixed points at $x=0$ and $x=1$.
In addition, there can be a third fixed point at $x^\ast=(d-b)/(a-b-c+d)$, which is unstable when $a>c$ and $b<d$ and stable when $a<c$ and $b>d$.

In finite population models, stochasticity does not allow the definition
of fixed points.
However, the boundaries $i=0$ and $i=N$ are absorbing due to the absence of mutations,
$T^+_0(\beta) = 0$ and $T^-_N(\beta) = 0$.
For recurrent Markov chains ($T^\pm_i(\beta) > 0$ for $0<i<N$), the system will eventually be absorbed at the boundaries.
The probability $\phi^A_i(\beta)$ that a given number of $i$ $A$ players will reach the absorbing boundary at $N$ is an important quantity to describe the process.
In addition to this fixation probability, the unconditional and conditional fixation times, $t_i(\beta)$ and $t^A_i(\beta)$, characterize the stochastic process \cite{antal:2006aa,altrock:2008nj,traulsen:2009bb,cremer:2008aa}.
These two average times are the expectation values of the number of time steps it needs either to reach any homogenous state (all $A$ or all $B$) or
to reach fixation at all $A$ under the condition that this event occurs.
In the following we recall recursions for these three quantities which can be solved regardless of the details of the birth--death process.
Each solution is only based on the microscopic transition probabilities $T^\pm_i(\beta)$.

\subsection{Fixation probability}

The probability of fixation $\phi^A_i(\beta)$ describes the probability that a given
number $i$ of new $A$ mutants in a population of $B$ will reach fixation at all $A$.
Since the homogenous states are absorbing, we have
$\phi^{A}_0(\beta) = 0$ as well as $\phi^A_{N}(\beta) = 1$.
For all the intermediate states we can write a balance equation
for the probability to fixate at all $A$ ($i=N$),
\begin{align}\label{eq:FixProbRec}
\begin{split}
	0 &= \,T^-_i(\beta)\,\left(\phi_{i-1}^{A}(\beta)-\phi^A_i(\beta)\right)\\
	&+  \, T^+_i(\beta)\,\left(\phi_{i+1}^{A}(\beta)-\phi^A_i(\beta)\right).
\end{split}
\end{align}
With the boundary conditions $\phi_0^{A}(\beta)=0$ and $\phi^A_{N}(\beta)=1$, this can be solved recursively \cite{nowak:2006bo,antal:2006aa}.
We obtain
\begin{align}\label{eq:FixProbGeneral}
	\phi^A_i(\beta) =\frac
	{1+\sum_{k=1}^{i-1}\prod_{m=1}^{k} \frac{T^-_m(\beta)}{T^+_m(\beta)}}
	{1+\sum_{k=1}^{N-1}\prod_{m=1}^{k}\frac{T^-_m(\beta)}{T^+_m(\beta)}}.
\end{align}
The probability to fixate at the pure state all $B$ starting from $i$ $A$ individuals is given by $\phi^B_i(\beta)=1-\phi^A_i(\beta)$.

\subsection{Unconditional average fixation time}

The average time (measured in elementary time steps) it needs to reach fixation at one of the homogenous states ($i = 0$ or $i = N$) starting with $i$  players of type $A$ is denoted by $t_i(\beta)$.
Obviously, $t_0(\beta) = 0$ and $t_N(\beta) = 0$.
The unconditional fixation times also fulfill a balance equation with the transition probabilities describing the rate of change,
\begin{align}\label{eq:UFixTRec}
	t_i(\beta) =1 &+ T^-_i(\beta)\,t_{i-1}(\beta)\nonumber\\
	&+\left( 1 - T ^+_{i}(\beta) - T^-_i(\beta) \right)\,t_i(\beta)\nonumber\\
	&+T^+_i(\beta)\,t_{i+1}(\beta).
\end{align}
This is a recursion equation for the unconditional mean exit times or average times of fixation.
Its solution for the unconditional average fixation time reads \cite{traulsen:2009bb,goel:1974aa}
\begin{align}\label{eq:UfixTIME}
	t_i(\beta) =& \,\sum\limits_{k=i}^{N-1}\sum\limits_{l=1}^k\frac{1}{T^+_l(\beta)}\prod\limits_{m=l+1}^{k}\frac{T^-_m(\beta)}{T^+_m(\beta)}\nonumber\\
	&- t_1(\beta)\sum\limits_{k=i}^{N-1}\prod\limits_{m=1}^{k} \frac{T^-_m(\beta)}{T^+_m(\beta)},\\
	t_1(\beta)=&\phi^A_1(\beta)\sum\limits_{k=1}^{N-1}\sum\limits_{l=1}^k\frac{1}{T^+_l(\beta)}\prod\limits_{m=l+1}^{k}\frac{T^-_m(\beta)}{T^+_m(\beta)}.
\end{align}
Next, we address the time it takes to reach a particular absorbing state.

\subsection{Conditional average fixation time}

Under the condition that the process reaches the absorbing state at all $A$,
$i=N$, the average time of fixation starting from $i$, is $t^A_i(\beta)$.
Following \citep{antal:2006aa}, we start from the recursion
\begin{align}\label{eq:CFixTRec}
	\phi^A_i(\beta)\,t^A_i(\beta) =\,\,& T^-_i(\beta)\,\phi^A_{i-1}(\beta)(t^A_{i-1}(\beta)+1)\nonumber\\
	&+\left( 1 - T ^+_{i}(\beta) - T^-_i(\beta) \right)\,\phi^A_i(\beta)(t^A_i(\beta)+1)\nonumber\\
	&+T^+_i(\beta)\,\phi^A_{i+1}(\beta)(t^A_{i+1}(\beta)+1).
\end{align}
The conditional average fixation time is conditioned upon fixation at all $A$, which occurs with probability $\phi^A_i(\beta)$.
Thus, the product of probability and conditional time of fixation appears in the recursion.
There can be a finite expectation value even for vanishing fixation probability, because $\phi^A_i(\beta)t^A_i(\beta)\to0$ does not imply $t^A_i(\beta)\to0$.
For the average fixation time under the condition of absorption at all $A$ we find \citep{antal:2006aa,kampen:1997xg,Fisher1988aa,traulsen:2009bb},
\begin{align}\label{eq:CfixTIME}
	t^A_i(\beta) =& \frac{1}{\phi^A_i(\beta)}\sum\limits_{k=i}^{N-1}\sum\limits_{l=1}^k\frac{\phi^A_l(\beta)}{T^+_l(\beta)}\prod\limits_{m=l+1}^{k}\frac{T^-_m(\beta)}{T^+_m(\beta)}\nonumber\\
			&- t^A_1(\beta)\frac{\phi^A_1(\beta)}{\phi^A_i(\beta)}\sum\limits_{k=i}^{N-1}\prod\limits_{m=1}^{k}\frac{T^-_m(\beta)}{T^+_m(\beta)}, \\
	t^A_1(\beta) =&  \sum\limits_{k=1}^{N-1}\sum\limits_{l=1}^k\frac{\phi^A_l(\beta)}{T^+_l(\beta)}\prod\limits_{m=l+1}^{k}\frac{T^-_m(\beta)}{T^+_m(\beta)}.
\end{align}
There is an analogous expression for the fixation time under the condition that strategy $B$ gets fixed in the population, $t^B_i(\beta)$, \citep{traulsen:2009bb}.

As the expressions \eq{FixProbGeneral}, \eq{UfixTIME} and \eq{CfixTIME} are functions of the $T^{\pm}_{i}(\beta)$, the study of a strong selection limit has to be performed in the transition probabilities.
In the next section, we introduce a process with the analytical strong selection limit  $T^+_i(\beta \to \infty)\to1$ and $T^-_i(\beta \to \infty )\to0$ (or the other way around),
and we show that the resulting dynamics is fully deterministic in this limit.

\section{The Moran process}
\label{sec:Moran}
\subsection{Selection at birth and random death}

A standard model for evolutionary dynamics in finite populations is the frequency
dependent Moran process \citep{nowak:2004pw}. This process incorporates the following
steps:
An individual is selected at random, but proportional to its fitness.
This individual produces identical offspring.
The offspring replaces an individual randomly selected for death.
Fitness $f$ is a convex combination of the average payoffs from the game, $\pi_i^A$ and $\pi_i^B$, and a background fitness, which is usually set to one.
Thus, we have $f_i^A(\beta) = 1 - \beta + \beta \,\pi_i^A$ and $f^{B}_{i}(\beta) = 1 - \beta + \beta\,\pi_i^B$.
The quantity $0\leq\beta\leq\beta_{\text{max}}$ determines the intensity of selection.
The transition probabilities of the Moran process are thus given by
\begin{subequations}
\begin{align}
T^+_i(\beta)
&=
\underbrace{
\frac{i \, f_i^A(\beta) }{i f_i^A(\beta) +(N-i) f_i^B(\beta)}
}_{\hbox{\footnotesize Selection at birth}}
\times
\underbrace{
\frac{N-i }{N},
}_{\hbox{\footnotesize Random death}}\label{eq:TMoranPlus}
\\
T^-_i(\beta)
&=
\underbrace{
\frac{(N-i) f_i^B(\beta) }{i f_i^A(\beta) +(N-i) f_i^B(\beta)}
}_{\hbox{\footnotesize Selection at birth}}
\times
\underbrace{
\frac{i }{N}.
}_{\hbox{\footnotesize Random death}} \label{eq:TMoranMinus}
\end{align}
\end{subequations}
Selecting proportional to fitness implies that fitness is positive.
Thus, for payoff matrices with negative entries, the intensity of selection $\beta$ cannot exceed a threshold $\beta_{\text{max}}$.
This process does not have a generic deterministic limit with arbitrarily strong selection intensity and remains stochastic with random death.

A possibility to extend the Moran process to higher intensities of selection is to
choose fitness as an exponential function of the payoff, i.e.\ $f_i^A(\beta) = \exp(+\beta\,\pi_i^A)$
and $f_i^B(\beta) =\exp(+\beta\,\pi_i^B)$ \citep{prugel:1994hb,traulsen:2008aa}.
Now, the intensity of selection $\beta$ can be any positive number.
For $\beta\to\infty$, the fitter individual is always selected for reproduction, compare equations \eq{TMoranPlus} and \eq{TMoranMinus}.
The direction of the process becomes deterministic.
But due to random death, the system can remain longer or shorter in a particular state.
Thus, the process remains stochastic in what concerns the times to fixation.
With two strategies, we have a semi deterministic process with deterministic direction and stochastic speed \cite{traulsen:2007cc}.
If there are more than two strategies, random death can also change the composition of the less fit types in the population.
This can affect the direction of selection, as the fittest type can change due to frequency dependent selection.

\subsection{Selection at birth and death}\label{sec:BDprocess}

Here, we introduce a birth--death process that recovers the usual results for weak selection,
but also leads to fully deterministic asymptotic behavior for strong selection.
The process has deterministic microscopic dynamics if the $T_i^\pm(\beta)$ are zero or one.
As in the standard Moran process, we assume that selection at birth is proportional to fitness.
In addition to producing less offspring, individuals with a lower fitness now have a higher probability to die.
A simple way to incorporate this is to select at death proportional to inverse fitness.
To ensure that fitness is a positive number, we follow the approach discussed above and
define fitness as an exponential function of the payoff.
This leads to the transition probabilities
\begin{subequations}
\begin{align}
T^+_i(\beta)
=\, &
\underbrace{
\frac{i\mathrm e^{+\beta_b\,\pi_i^A}}{i\mathrm e^{+\beta_b \pi_i^A}+(N-i)\mathrm e^{+\beta_b\, \pi_i^B}}
}_{\hbox{\footnotesize Selection at birth}}\nonumber\\
&\times
\underbrace{
\frac{(N-i)\mathrm e^{-\beta_d \,\pi_i^B}}{i\mathrm e^{-\beta_d\, \pi_i^A}+(N-i)\mathrm e^{-\beta_d\, \pi_i^B}}
}_{\hbox{\footnotesize Selection at death}}\label{eq:TBDPlus},
\\
T^-_i(\beta)
= \, &
\underbrace{
\frac{(N-i)\mathrm e^{+\beta_b\,\pi_i^B}}{i\mathrm e^{+\beta_b \,\pi_i^A}+(N-i)\mathrm e^{+\beta_b \,\pi_i^B}}
}_{\hbox{\footnotesize Selection at birth}}\nonumber\\
&\times
\underbrace{
\frac{ie^{-\beta_d \,\pi_i^A}}{i\mathrm e^{-\beta_d \,\pi_i^A}+(N-i)\mathrm e^{-\beta_d \,\pi_i^B}}
}_{\hbox{\footnotesize Selection at death}}\label{eq:TBDMinus}.
\end{align}
\end{subequations}
$\beta_b$ is the intensity of selection at birth and $\beta_d$ is the intensity of selection at death.
For $\beta_d=0$, we recover the process discussed in \refS{Moran}.
It is known that under weak selection many birth--death processes have the same general properties \citep{fudenberg:2006fu,lessard:2007aa, taylor:2006jt}.
Especially, for $\beta_{b,d}\ll1$ the behavior of the Moran process is recovered \citep{traulsen:2007cc, altrock:2008nj}.

For simplicity, we assume $\beta=\beta_{b}=\beta_{d}$ in the following.
The transition probabilities can be written as
\begin{subequations}
\begin{align}
T^+_i(\beta)
&=
\frac{i}{i+(N-i)\mathrm e^{-\beta\, \Delta\pi_i}}
\frac{N-i}{i\,\mathrm e^{-\beta\, \Delta\pi_i}+N-i}\label{eq:Tstrong01a},
\\
T^-_i(\beta)
&=
\frac{N-i}{i\,\mathrm e^{+\beta\, \Delta\pi_i}+N-i}
\frac{i}{i+(N-i)\mathrm e^{+\beta\, \Delta\pi_i}}\label{eq:Tstrong01b}.
\end{align}
\end{subequations}
Thus, as far as the payoffs are concerned, the transition probabilities only depend on the difference between the frequency dependent average payoffs, $\Delta\pi_i=\pi_i^A-\pi_i^B$.

The case of $\Delta\pi_i=0$ is a form of neutral selection.
In this case, the transition probabilities are $T_i^\pm(\beta)\equiv(N-i)/N^2$, for arbitrary $\beta$.
Note that for neutral selection, moving into one direction is equally probable as moving into the other $T^+_i(0)=T^-_i(0)$.
But the probability to leave a given interior state changes with $i$,
$T_i^\pm(0)\neq T_j^\pm(0)$ for $ i\neq j $.

For arbitrary $\beta$, the ratio of the transition probabilities reduces to an exponential function
of the payoff difference,
\begin{align}
\frac{T^-_i(\beta)}{T^+_i(\beta)} = e^{- 2 \beta \Delta\pi_i}.
\end{align}
Hence, the fixation probabilities of the process can be approximated with the closed
expressions derived in \cite{traulsen:2006bb}, after rescaling the intensity of selection
by a factor of 2. From this, it is clear that the usual weak selection behavior is recovered.

\section{Strong selection}
\label{sec:Strong}

For strong selection, $\beta \to \infty$, the asymptotic behavior of the transition probabilities depends only on the sign of the payoff difference.
We focus on the generic cases $\Delta\pi_i\lessgtr0$ to discuss this limit.
The limiting cases can be obtained from equations \eq{Tstrong01a} and \eq{Tstrong01b} and yield
\begin{align}\label{eq:Tstrong02a}
	\lim_{\beta\to\infty}\,T^+_i(\beta) =
	\begin{cases}
			\,0 \hspace{0.2cm}\,&\text{for}\,\,\,\Delta\pi_i<0\vspace{0.25cm}\\
			\,1\hspace{0.2cm}\,&\text{for}\,\,\,\Delta\pi_i>0
	\end{cases},
\end{align}
as well as
\begin{align}\label{eq:Tstrong02b}
	\lim_{\beta\to\infty}\,T^-_i(\beta) =
	\begin{cases}
			\,1 \hspace{0.2cm}\,&\text{for}\,\,\,\Delta\pi_i<0\vspace{0.25cm}\\
			\,0\hspace{0.2cm}\,&\text{for}\,\,\,\Delta\pi_i>0
	\end{cases}.
\end{align}
Since $\lim_{\beta\to\infty}(T_i^+(\beta)+T_i^-(\beta))=1$, the probability to stay in the state $i$ ($0<i<N$) vanishes for $\beta\to\infty$ and nontrivial payoff  difference, $\Delta\pi_i\neq0$.
Thus, we have a fully deterministic process for arbitrary population size.
With this, we consider the fixation probability $\phi^A_i(\beta)$ and average fixation times $t_i(\beta)$ and $t^A_i(\beta)$ in the limiting case of strong selection.
In the following let $\phi_i^A(\infty)$ as well as $t_i(\infty)$ and $t_i^A(\infty)$ denote the finite asymptotic ($\beta\to\infty$) values of the fixation probability and times.
We identify them in terms of the initial frequency $i$, depending on the average payoff difference for the process introduced above under strong selection.

\subsection{Fixation Probability}\label{ssec:Strong01}

Starting with equation \eq{FixProbRec} and inserting the limiting cases of $T_i^\pm(\beta)$ leads to
\begin{align}\label{eq:FixProbL01}
	\phi_i^A(\infty)=&\,\left(\lim_{\beta\to\infty}T^-_i(\beta)\right) \phi^A_{i-1}(\infty)\nonumber\\
	&+\left(\lim_{\beta\to\infty}T^+_i(\beta)\right) \phi^A_{i+1}(\infty).
\end{align}
That is, in the strong selection limit we have a very simple recursion for the asymptotic value of the fixation probability,
depending on the sign of the payoff difference $\Delta\pi_i$.
When strategy $A$ dominates strategy $B$ ($a>c$ and $b>d$), we have $\Delta\pi_i>0$.
For  $0 < i < N$, this yields $\lim_{\beta\to\infty}T^-_i(\beta) = 0$, and
$\lim_{\beta\to\infty}T^+_i(\beta) = 1$,
which results in $\phi_i^A(\infty) = 1-\delta_{0,i}$.
In other words, the probability to reach the state with all $B$ individuals is zero, except if there are no $A$ individuals initially.
Equivalently, for dominance of strategy $B$ we obtain  $\phi_i^A(\infty) = \delta_{N,i}$.
More interesting cases are found when $\Delta\pi_i$ changes its sign, which occurs at $i^\ast=(N(d-b)+a-d)/(a-b-c+d)$ when $a>c$ and $d>b$ or when $a<c$ and $d<b$.
These are two important classes of games: coordination games and coexistence games.

Let us first focus on coordination games ($a>c$ and $d>b$).
In these games, the threshold value $i^\ast$ cannot be crossed for infinitely large systems with deterministic dynamics \cite{skyrms:2003aa}.
For finite systems under strong selection, we observe something similar.
The payoff difference $\Delta\pi_i$ changes sign from negative to positive and selection points always away from $i^\ast$ toward the boundaries.
Depending on the initial condition $i\gtrless i^\ast$, the fixation probability $\phi^A_i(\infty)$ is one or zero.
\begin{itemize}
\item[(i)] If $i<i^\ast$, $\Delta\pi_i$ is negative and with Eqs.\ \eq{Tstrong02a} and \eq{Tstrong02b} we have $\phi_i^A(\infty)= \phi^A_{i-1}(\infty)$.
We start the recursion for the fixation probabilities with $\phi^A_1(\infty)=\phi^A_0(\infty)=0$.
This yields $\phi^A_{i<i^\ast}(\infty)=0$.
\item[(ii)] For $i>i^\ast$, $\Delta\pi_i$ is positive and Eqs.\ \eq{Tstrong02a} and \eq{Tstrong02b} yield the recursion $\phi_i^A(\infty)=\phi^A_{i+1}(\infty)$.
Starting with the maximal $i$ we obtain $\phi^A_{N-1}(\infty)=\phi^A_N(\infty)=1$
and thus $\phi^A_{i>i^\ast}(\infty)=1$.
\item[(iii)] If $i^\ast$ happens to be an integer value and the system starts there, the first step has equal probabilities, $T^+_{i^\ast}(\beta)=T^-_{i^\ast}(\beta)=\frac{1}{2}$.
This leads to $\phi^A_{i^\ast}(\infty)=\frac{1}{2}$.
\end{itemize}
In summary, for the fixation probability we find
\begin{align}\label{eq:FixProbL02}
	\,\phi_i^A(\infty) =
	\begin{cases}
		\,0\hspace{0.2cm}\,&\text{for}\,\,\,i<i^\ast \vspace{0.25cm}\\
		\,\frac{1}{2}\hspace{0.2cm}\,&\text{for}\,\,\,i=i^\ast \vspace{0.25cm}\\
		\,1\hspace{0.2cm}\,&\text{for}\,\,\,i>i^\ast
	\end{cases}.
\end{align}
This is clearly what is to be expected because $A$ is selected for $i> i^{\ast}$ and $B$ is selected for $i< i^{\ast}$, \fig{FIG1}.
Dominance of strategy $A$ can be seen as a special case of coexistence with $i^{\ast}<0$.
\begin{figure}[h]
	\includegraphics[width=0.475\textwidth,angle=0]{FIG1.pdf}
\caption{
The fixation probability as a function of selection intensity for a coordination game ($a>c$, $b<d$), given by Eq.~\eq{FixProbGeneral}, with selection at birth and death, Eqs.~\eq{Tstrong01a} and \eq{Tstrong01b}.
From the payoff matrix given in the figure, we obtain $i^\ast = \frac{17}{22} + \frac{5}{11}N$, which gives $i^\ast \approx5.32$ for our numerical example with $N=10$.
With increasing selection intensity $\beta$, for any $l<i^\ast$ we have $\phi_l^A(\beta)\to0$ (straight lines),
whereas for any $m>i^\ast$ we have $\phi_m^A(\beta)\to1$ (dotted lines).
The $\phi^A_i(\beta)$ for each $i$ can be identified via its neutral value $\phi^{A}_i(0)=\frac{i}{N}$.
}
\label{fig:FIG1}
\end{figure}

Next, we consider coexistence games with $a<c$ and $d<b$.
In this case, $\Delta\pi_i$ changes sign from positive to negative and selection points always away from the boundaries toward $i^\ast$. For strong selection, the system gets trapped and fixation never occurs.
\begin{itemize}
\item[(i)] If $i^\ast$ is an integer, the system switches from $i^\ast$ to $i^\ast \pm 1$ with equal probability.
From $i^\ast \pm 1$, it always returns to $i^{\ast}$.
\item[(ii)] If $i^\ast$ is not an integer, we observe deterministic flipping between the two neighboring states states $i_1<i^\ast$, $i_2>i^\ast$.
\end{itemize}
Since fixation never occurs in coexistence games, it does not make sense to compute the asymptotic value of the fixation probability.
Formally, the probability to get absorbed in all $A$ converges to $1$ if $i^{\ast}>N/2$ and to $0$ otherwise.
However, it turns out that the fixation times diverge.

\subsection{Unconditional average fixation time}\label{ssec:Strong02}

The average time it takes for $i$ $A$ players to either become extinct or take over the population, $t_i(\beta)$, can be found by solving equation \eq{UFixTRec} recursively.
To examine the limit of strong selection, we perform the limit on both sides of the balance equation, assuming that there exists an asymptotic value $t_i(\infty)$ of the unconditional fixation time.
With the previous analysis of the transition probabilities, this leads to
\begin{align}\label{eq:UFixTL01}
	t_i(\infty) = 1&+\left( \lim_{\beta\to\infty}T^-_i(\beta)  \right)\,t_{i-1}(\infty)\nonumber\\
	&+\left( \lim_{\beta\to\infty}T^+_i(\beta)  \right)\,t_{i+1}(\infty).
\end{align}
This strong selection recursion has to be analyzed for the two different cases of behavior at the threshold $i^\ast$,
coordination and coexistence.
Again, we first examine the coordination game ($a>c$ and $d>b$).
\begin{itemize}
\item[(i)] For $i<i^\ast$, the payoff difference is negative, $\Delta\pi_i<0$.
The recursion amounts to $t_i(\infty)=1+t_{i-1}(\infty)$, because $T^+(i)\to0$ whereas $T_i^-(\beta)\to1$.
With $t_0(\infty)=0$ at the boundary we have $t_1(\infty)=1$, $t_2(\infty)=1+t_1(\infty)=2$, and eventually $t_i(\infty)=i$.
\item[(ii)] For $i>i^\ast$ the payoff difference is $\Delta\pi_i>0$. The recursion from \eq{UFixTL01} is $t_i(\infty)=1+t_{i+1}(\infty)$.
The transition probabilities $T^{\pm}(i)$ behave exactly in the opposite way as before.
The time starting from next to the absorbing boundary is $t_{N-1}(\infty)=1+t_N(\infty)=1$.
Hence, we have $t_{N-k}(\infty)=k$, or with $k=N-i$, the uncondtional averag time is $t_i(\infty)=N-i$.
\item[(iii)] If the threshold is an integer and the system is initiated there the number of steps needed to fixate is $i^\ast$ or $N-i^\ast$ with equal probability.
Thus, we can compute the average time with the previous findings, $t_{i^\ast}(\infty)=1/2(i^\ast+N-i^\ast)=N/2$.
\end{itemize}
In summary, depending on the the starting point $i^{\ast}$, the asymptotic value for the unconditional average fixation time in a coordination game is
\begin{align}\label{eq:UFixTL02}
	t_i(\infty) =
	\begin{cases}
		\,i\hspace{0.2cm}\,&\text{for}\,\,\,i<i^\ast,\vspace{0.25cm}\\
		\,\frac{N}{2}\hspace{0.2cm}\,&\text{for}\,\,\,i=i^\ast,\vspace{0.25cm}\\
		\,N-i\hspace{0.2cm}\,&\text{for}\,\,\,i>i^\ast.
	\end{cases}
\end{align}
In the strong selection limit $t_i(\infty)$ converges to the distance between initial state and the final state, as expected from deterministic motion.

We can also infer the dynamics for games in which one strategy dominates.
When strategy $A$ dominates, we can formally set $i^{\ast} <0$ and obtain $t_i(\infty)=N-i$,
cf.~\fig{FIG2}.
When strategy $B$ dominates, the equivalent procedure yields $t_i(\infty)=i$.

In case of a coexistence game, the system gets trapped around $i^\ast$ and cannot reach the absorbing boundaries. As expected, the recursions lead to $t_i(\infty)\to\infty$.
\begin{figure}[h,t]
	\includegraphics[width=0.475\textwidth,angle=0]{FIG2.pdf}
\caption{
Probability distribution of the unconditional fixation time  (measured in elementary time steps) of a single $A$ player in a population of $N-1$ dominated $B$ players.
For $\beta=1$ and $\beta=2$, the distribution has two peaks corresponding to the two absorbing boundaries.
For stronger selection (inset), the probability that the advantageous $A$ individual goes extinct becomes small and
fixation takes at least $N-1$ time steps. In this case, the distribution becomes single peaked.
For $\beta \to \infty$, the distribution converges to a delta peak at $t_1(\infty)=N-1$ (payoff matrix $a=2.2$, $b=1.5$, $c=2$, $d=0.5$, population size $N=100$,
histograms obtained from $10^7$ realizations.
Lines are guides to the eye).
}
\label{fig:FIG2}
\end{figure}

\subsection{Conditional average fixation time}\label{ssec:Strong03}

If $i$ $A$ players take over the population, the asymptotic fixation time under this condition, $t_i^A(\infty)$, can be obtained by solving the balance equation \eq{CFixTRec} recursively.
However, this situation is more complex as we have to consider the
fixation probability and the conditional fixation time in a combined way.
Introducing the asymptotic value $\theta_i^A(\infty)=\phi_i^A(\infty)\,t_i^A(\infty)$, the recursion \eq{CFixTRec} yields
\begin{align}\label{eq:CFixTL01}
	\theta_i^A(\infty) =\,& \left( \lim_{\beta\to\infty}T^-_i(\beta) \right)(\theta_{i-1}^A(\infty)+\phi^A_{i-1}(\infty))\nonumber\\
	&+\left( \lim_{\beta\to\infty}T^+_i(\beta) \right)(\theta_{i+1}^A(\infty)
	+\phi^A_{i+1}(\infty)).
\end{align}
The formulation of a similar equation for $\theta_i^B(\infty)=\left(1-\phi_i^A(\infty)\right)\,t^{B}_i(\infty)$ is straightforward.
Both are analyzed regarding the different behavior at either side of the threshold $i^\ast$.

For the coordination game the system reaches the absorbing boundaries after a finite time.
\begin{itemize}
\item[(i)] If $i>i^\ast$,  the system fixates at $i=N$ with probability $\phi^A_i(\infty)=1$.
Thus, $\theta_i^A(\infty)=t_i^A(\infty)$ and we recover the same recursion as for the unconditional
fixation time.
This yields $ t_i^A(\infty) = t_i(\infty) = N-i$, see \fig{FIG3}.
\item[(ii)] If $i<i^\ast$, the system fixates at $i=0$, $\phi^A_i(\infty)=0$.
Thus, we cannot formulate a meaningful recursion for $t_i^A(\infty)$.
In this case we observe $T^-_i(\beta)\to1$, $T^+_i(\beta)\to0$ and we can only make a statement for $t_i^B(\infty)$,
which results in $t_i^{B}(\beta) = i$.
\item[(iii)] If $i=i^\ast$ is an integer, the system is not fully deterministic as fixation of $A$ and fixation of $B$ are observed with equal probability $\frac{1}{2}$.
In this case, we obtain
$t_{i^\ast}^A(\infty) = N-i^{\ast}$
and
$t_{i^\ast}^B(\infty) = i^{\ast}$.
\end{itemize}
In a regime where $A$ always performs better than $B$ the unconditional fixation time $t_i^A(\infty)$ is equal to the conditional fixation time $t_i(\infty)$, see \fig{FIG3}.
Equivalently, when $B$ always performs better, we have $t^B_i(\infty)=t_i(\infty)$.
\begin{figure}[h]
	\includegraphics[width=0.475\textwidth,angle=0]{FIG3.pdf}
\caption{
Unconditional (red) and conditional (black) average fixation times  (measured in elementary time steps) as a function of the intensity of selection for a coordination game (top) and a game in which $A$ dominates (bottom).
The payoff matrices of the games are given in the figures, the population size is $N=10$.
Top: In a coordination game ($a>c$, $b<d$), the conditional and unconditional fixation times, $t_i(\beta)$ and $t_i^A(\beta)$ converge to $N-i$ for $\beta \to \infty$ if initially  more than $i^{\ast} =117/22\approx5.32$ individuals play $A$, compare Eq.~\eq{UFixTL02}.
The lines show the initials states $i=6$ (dotted lines) and $i=9$ (full lines).
Bottom: When $A$ dominates $B$ ($a>c$, $b>d$), the unconditional fixation time $t_1(\beta)$ first increases with $\beta$.
For any initial condition $i$, $t_i(\beta)$ and $t_i^A(\beta)$ converge to $N-i$ in the limit of strong selection.
The initials states are $i=1$ (dotted lines) and $i=9$ (straight lines).
}\label{fig:FIG3}
\end{figure}

For a coexistence game, the system does not reach any of the boundaries but is always dragged toward $i^\ast$, as discussed before.
The recursion \eq{CFixTL01} for $t_i^A(\infty)$ or its equivalent for $t_i^{B}(\infty)$
are not meaningful here because they contain the fixation probabilities.
However, in this case all fixation times diverge with $\beta$.

In this section, we have derived asymptotic values for the birth--death process with selection at birth and selection at death.
We have identified the underlying games that lead to a fully deterministic process in the limit of strong selection.
As we have seen, the difference of the average payoffs $\Delta\pi_i$ as a function of the the relative abundance of type $A$ plays an important role.

\section{Games with three strategies}\label{sec:RPS}

Here, we demonstrate that the process we have introduced above
 leads to rather simple and often deterministic dynamics.
We focus on games with two players and three strategies with cyclic dominance \cite{sinervo:1996le,kerr:2002xg,czaran:2002ya,szabo:2004ww,szolnoki:2005wq,reichenbach:2006aa,perc:2007bb,sandholm:2007bo,claussen:2008aa,peltomaki:2008ls}.
Cyclic dominance among three strategies corresponds to Rock--Paper--Scissors games, where each strategy can be beaten by another one:
Rock crushes Scissors, Scissors cut Paper and Paper wraps rock.
In general, the payoff for winning does not have to be equal to the payoff for losing,
which leads to non--zero--sum games.
For simplicity, we set the payoff for a tie to zero.
Setting the winners payoff to one and the losers payoff to $-s \leq 0$, the $3\times3$ payoff matrix reads
\begin{align}\label{eq:RPSmatrix}
\bordermatrix{
  & R & P & S \cr
R & 0 & -s & 1\cr
P & 1 & 0 & -s \cr
S & -s & 1 & 0 \cr}.
\end{align}
For infinite populations, the state of the system is defined by the frequencies of the three strategies, $x_R$, $x_P$, and $x_S$.
Thus, the state space is the simplex  $S_3\subset\mathbb R^2$, an equilateral triangle between the three states all $R$, all $P$, and all $S$.
Apart from the three trivial equilibria, the replicator dynamics has an
interior equilibrium at $ (x_R, x_P, x_S)^{\ast}=(\frac{1}{3}, \frac{1}{3},\frac{1}{3})$, which follows from the symmetry of the system.
The quantity $s$ determines wether the interior equilibrium is asymptotically stable (the system spirals inwards towards the interior fixed point for $s<1$) or unstable (the system spirals out toward a heteroclinic cycle along the boundaries for $s>1$).
In the zero--sum game with $s=1$, the system oscillates around the interior equilibrium with the Hamiltonian $-x_Rx_Px_S$ being a constant of motion \citep{hofbauer:1998mm, nowak:2006bo}.
\begin{figure}[h]
\begin{center}
	\includegraphics[width=0.45\textwidth,angle=0]{FIG4.pdf}
	\end{center}
\caption{
The discrete simplex of the Rock--Paper--Scissors game for the different ranges of $s$.
The three strategies are arranged in such a way that cyclic dominance is counter clockwise.
For $N=33$ and $s=1$ (top left), the three lines of equal payoffs defined by Eqs. \eq{RPSPayEq}
are parallel to the boundaries of the simplex. Changing the population size to $N=11$ (top right) does not
change these lines, but only the state space of the system indicated by dots. For example, the system can no longer access the center of the simplex.
With decreasing $s$, the three lines of equal payoffs are rotated clockwise (bottom left, $s=0.5$).
With increasing $s$, these lines are rotated counterclockwise (bottom right, $s=2.5$).
Only in special cases, the three lines of equal payoffs defined by Eqs. \eq{RPSPayEq}
intersect with the possible states of the system.
The arrows in the upper left figure indicate the direction of selection as it is induced by the cyclic dominance of the three strategies.
}\label{fig:FIG4}
\end{figure}
In finite, well--mixed populations, the state space is only a subset of $(N+1)(N+2)/2$ states within the simplex  $S_3$, cf. \fig{FIG4}.
Moreover, the dynamics is typically stochastic.
Properties such as the average drift for a Moran process or the average time to reach the absorbing the boundaries are attainable for weak selection \citep{reichenbach:2006aa,claussen:2008aa}.
To analyze the strong selection limit, we adopt the evolutionary process with selection at birth and death discussed above for $2\times2$ games.
Starting with the payoffs from \eq{RPSmatrix} in a well--mixed population the average payoffs read
\begin{align}
	\pi_
	&= i_S - i_P\,s\label{eq:RPS01},\\
	\pi_
	&= i_R -i_S\,s\label{eq:RPS02},\\
	\pi_
	&= i_P -i_R\,s\label{eq:RPS03},
\end{align}
where $i_R$, $i_P$, and $i_S=N-i_R-i_P$ are the number of individuals playing Rock, Paper, or Scissors in a population of size $N$, respectively.
Individuals are selected proportional to fitness at birth and proportional to inverse fitness at death,
both with the intensity of selection $\beta$.
The fitness of strategy $X$ is given by $f_X=\exp [ + \beta\,\pi_X ]$.
The dynamics on the discrete finite set of states is governed by six transition probabilities in each state.
The transition probabilities in each state $(i_R,i_P,i_S)$ to change to one of the six neighboring states are thus given by
\begin{align}\label{eq:RPST01}
T_{Y\to X}(i_R,i_P,i_S) \,=\,
\underbrace{
	\frac{i_X \,f_X}{\sum_{Z} i_{Z}f_{Z}}
}_{\hbox{\footnotesize birth}}
\times
\underbrace{
	\frac{i_Y \,f_Y^{-1}}{\sum_{Z} i_{Z}f_{Z}^{-1}}
\,
}_{\hbox{\footnotesize death}},
\end{align}
where $X$, $Y$, and $Z$ stand for $R$, $P$, or $S$.
Note that the probability to stay in the given state is given by $T_{R\to R}(i_R,i_P,i_S)+T_{P\to P}(i_R,i_P,i_S)+T_{S\to S}(i_R,i_P,i_S)$.
For strong selection, $\beta\to\infty$, the system moves from each state into one direction with probability one unless two payoffs are identical.
In the following we address how this direction depends on the payoffs in the state and on the paprameter $s$.

Let us first assume that for given state $(i_R,i_P,i_S)$, we have the unique ordering of the average payoffs from \eq{RPS01}--\eq{RPS03}.
Let $\pi_1$ denote the largest and $\pi_3$ denote the lowest value, i.~e.\  $\pi_1>\pi_2>\pi_3$.
The number of individuals playing the according strategies
can be denoted as $i_1$, $i_2$, and $i_3$.
For $T_{3\to 1}$, we have
\begin{align}
	T_{3\to 1} =\,& \frac{i_1\mathrm e^{\beta\pi_1}}{i_1\mathrm e^{\beta\pi_1}+i_2\mathrm e^{\beta\pi_2}+i_3\mathrm e^{\beta\pi_3}}\nonumber\\
	&\times\frac{i_3\mathrm e^{-\beta\pi_3}}{i_1\mathrm e^{-\beta\pi_1}+i_2\mathrm e^{-\beta\pi_2}+i_3\mathrm e^{-\beta\pi_3}}\label{eq:RPSTd}\\
				=\,&\frac{i_1}{i_1+i_2\mathrm e^{-\beta(\pi_1-\pi_2)}+i_3\mathrm e^{-\beta(\pi_1-\pi_3)}}\nonumber\\
	&\times\frac{i_3}{i_1\mathrm e^{-\beta(\pi_1-\pi_3)}+i_2\mathrm e^{-\beta(\pi_2-\pi_3)}+i_3}\label{eq:RPSTe},
\end{align}
where $\pi_1-\pi_2 > 0$,
$\pi_1-\pi_3 >0$, as well as
$\pi_2-\pi_3>0$.
For $\beta\to\infty$, this leads to
\begin{align}\label{eq:RPSLim01}
	\lim_{\beta\to\infty}T_{3\to 1} = 1.
\end{align}
All the other transition probabilities vanish.
In each reproductive event, an individual with the largest payoff replaces an individual with the smallest payoff.
This holds for any unique ordering of the three payoffs.

If the payoffs are not in unique order, that is, if two or more payoffs are equal, at least two probabilities become non--trivial.
This yields the following three scenarios:
\begin{itemize}
\item[(i)] For $\pi_1>\pi_2=\pi_3$, the individual with the highest average payoff is certainly selected at birth.
But selection at death will remove an individual carrying one of the two remaining strategies
with probability given by their abundance.
We find
\begin{subequations}
\label{eq:RPSLim02}
\begin{align}
&\,\lim_{\beta\to\infty}T_{2 \to 1} = \frac{i_{2}}{i_2+i_3},\\
\; \;
&\hbox{and}\nonumber\\
\; \;
&\,\lim_{\beta\to\infty}T_{3\to 1}  = \frac{i_{3}}{i_2+i_3}.
\end{align}
\end{subequations}
Obviously, we have $T_{2\to 1}+T_{3\to 1}=1$.
\item[(ii)] For $\pi_1=\pi_2>\pi_3$, the individual with the lowest payoff is selected for death with certainty, but selection at birth is still probabilistic.
It is easy to see that
\begin{subequations}
\label{eq:RPSLim03}
\begin{align}
&\,\lim_{\beta\to\infty}T_{3 \to 1} = \frac{i_{1}}{i_1+i_2},\\
\; \;
&\hbox{and}\nonumber\\
\; \;
&\,\lim_{\beta\to\infty}T_{3\to 2}  = \frac{i_{2}}{i_1+i_2}.
\end{align}
\end{subequations}
Thus, $T_{3\to 1}+T_{3\to 2}=1$.
\item[(iii)] For $\pi_1=\pi_2=\pi_3$, selection is stochastic at birth and at death. In this case, we find
\begin{align}\label{eq:RPSLim04}
	\lim_{\beta\to\infty}T_{X \to Y} = \frac{i_X}{N} \frac{i_Y}{N}.
\end{align}
As expected, we have $\sum_{X,Y} T_{X \to Y} =1$.
\end{itemize}
These results are valid if two or more payoffs are equal at a given lattice site,
which is only obvious for $s=1$ and might not occur for any lattice site at all in the general case of $s\neq1$.

With \eq{RPS01}--\eq{RPS03}, we can compute the point sets in the simplex where two or all three average payoffs are equal, depending on the value of $s$.
For $\pi_R=\pi_P=\pi_S$ this is only the center of the simplex, independent of $s$.
For two payoffs being equal, we obtain the three linear equations
\begin{align}
\nonumber
	\pi_R=\pi_S \quad & \hbox{at}\quad  i_P(s+2)  = {N-i_R(1-s)}{},\\
\label{eq:RPSPayEq}
	\pi_S=\pi_P \quad & \hbox{at}\quad  i_P(s-1)  = Ns-i_R(1+2s),\\
	\pi_R=\pi_P \quad & \hbox{at}\quad  i_P(2s+1)  =N(1+s)-i_R(2+s).\nonumber
\end{align}
However, it is not obvious at which of the discrete states $(i_R,i_P,i_S)$ we can observe equal average payoffs of two strategies if the losers payoff is not equal to $-1$, and especially if the system size is not a multiple of three, compare \fig{FIG4}.

\begin{figure}[h]
\begin{center}
	\includegraphics[width=0.4\textwidth,angle=0]{FIG5.pdf}
\end{center}
\caption{
The probability that the system fixates at the lower left corner, e.~g.\,\, $\phi^R$, in a system with $N=33$, and $s=1$ for strong selection $\beta\to\infty$, depending on the initial state $(i_R, i_P, i_S)$.
As discussed in the main text, fixation is stochastic.
Near the corners and along the edges fixation is deterministic.
In a central area of the simplex, the system spirals out in a probabalistic fashion.
The closer the initial condition is to the center, the closer the probability to get absorbed in a given state is to one third (fixation probabilities obtained from numerical simulations over $10^4$ realizations).
}\label{fig:FIG5}
\end{figure}

For simplicity
we thus concentrate on the case of $N$ being a multiple of three.
In general, the strong selection behavior of the system is determined by the transition probabilities near the lines of equal average payoffs.
When the population size $N$ is a multiple of three, the dynamics for the three different cases of $s=1$, $s<1$, or $s>1$, is as follows:

For $s=1$, we have
$\pi_R=\pi_S$ at $i_P=N/3$,
$\pi_S=\pi_P$ at $i_R=N/3$, and
$\pi_R=\pi_P$ at $i_S=N/3$.
Hence, there is always stochastic movement induced by Eqs. \eq{RPSLim02} and \eq{RPSLim03} as well as \eq{RPSLim04}.
Apart from these points, the direction of selection is indeed deterministic, which means that an individual with higher payoff always replaces an individual with a lower one.
In certain regimes, near the corners of the simplex and along the edges the initial condition determines the final state where the system fixates.
In a much larger area, however, the system fixates stochastically.
In \fig{FIG5} we illustrate this by showing one fixation probability obtained from numerical simulations of the birth--death process.
Due to the symmetry of the system, this fixation probability can be either $\phi^R$, $\phi^P$, or $\phi^S$.

For $s<1$, the lines where payoffs are equal rotate clockwise in our setup of cyclic dominance.
In general, no states of the finite population system coincide with the lines of equal payoffs
(except for special cases), see \fig{FIG4}. But as soon as the process crosses these lines, it changes direction.
Near the corners and on the edges the system fixates deterministically, but in a central area the process spirals inwards if it does not hit the boundary of the system.
However, it turns out that there is a largest limit cycle (LLC) depending on $s$ and $N$ and that there can be several other limit cycles (OLC) inside the LLC.
We demonstrate this finding as it is obtained from numerical simulations in \fig{FIG6}, showing one sample trajectory that ends as the LLC.
Note that the term cycle actually refers to a hexagon.
\begin{figure}[h]
\begin{center}
	\includegraphics[width=0.4\textwidth,angle=0]{FIG6.pdf}
	\end{center}
\caption{
For a population size of $N=33$ and $s=0.5$, the dynamics is such, that fixation does not occur in a central area, buy it is deterministic near the corners.
For instance, any initial condition in the dark blue area will lead to the pure state at the lower left.
The white area marks the sites from which, chosen as initial condition, the system does not fixate, but approaches a closed cycle.
We show an example trajectory of the system that ends on the largest of these cycles (red hexagon) starting from the initial state $(i_R,i_P,i_S)=(1,9,23)$.
Inside this largest cycle, there are other cycles such that every site except the center is already on such a cycle.
}\label{fig:FIG6}
\end{figure}

For $s>1$, as the lines of equal payoffs rotate counter clockwise (\fig{FIG4}) the system spirals outwards in a deterministic fashion.
The movement is no longer deterministic only near the corners and along the edges but everywhere (except for the center), compare \fig{FIG7}.
This means that if
the modulus of
the payoff for losing in our cyclic game, $s$, is larger than the payoff for winning, $1$, we observe strictly deterministic fixation depending on the initial condition.

In this section, we have shown that a birth--death process with exponential payoff to fitness mapping and selection at birth and death is able to induce deterministic movement in the strong selection limit
even in $3 \times 3$ games.
For the transition probabilities $T_{Y\to X}$ this limit can be performed analytically.
It turns out that the microscopic dynamics is dependent only on the hierarchy of the average payoffs.
As for finite systems an analytic description of the fixation probabilities (and times) is lacking, further examination of this system has to be numerical.
Under strong selection the patterns that emerge show a very interesting regularity; it turns out that apart from system size, the results are dependent on the parameter $s$.
\begin{figure}[h]
\begin{center}
	\includegraphics[width=0.45\textwidth,angle=0]{FIG7.pdf}
	\end{center}
\caption{
Deterministic fixation for a population of size $N=33$ and different $s>1$.
The dynamics is deterministic everywhere, except in the center (white dot), where all three payoffs are the same.
Depending on the initial state, the system fixates to a given pure state.
This is indicated by the three different colors at each state.
For instance, when the system is initially at a site in dark blue, the system will fixate to the pure state at the lower left with probability one.
We show one example trajectory (orange arrow) starting from the initial state $(i_R,i_P,i_S)=(10,13,10)$.
By increasing $s$, we can vary the state that is ultimately reached by the system.
}\label{fig:FIG7}
\end{figure}

\section{Discussion}\label{sec:Disc}

The standard approaches to evolutionary game dynamics such as the Moran process or
pairwise comparison based on the Fermi rule lead to stochastic dynamics in finite populations \cite{traulsen:2005hp}.
Even if the direction of selection becomes deterministic, the time scale typically remains stochastic and leads to a
distribution of the average fixation or mean exit times \cite{traulsen:2007cc}.
Moreover, these standard approaches do not lead to a deterministic
direction of selection in games with more than two different strategies \cite{traulsen:2009aa}.
Here, we have introduced a process with selection at birth and at death.
This process allows to interpolate between weak selection, usually considered in evolutionary
biology, and arbitrary strong selection, such that in the extreme case the worst performing individual
is always replaced by a copy of the best performing individual. This kind of selection is sometimes
used in evolutionary optimization \cite{prugel:1994hb,mitchell:1996ef}.
While the limiting case itself may not be of most interest for real biological or social systems, which
are always subject to stochastic noise, we discussed the most important
features of this limit. In particular, it reveals speed limits of evolutionary dynamics in $2 \times 2$ games that stochastic dynamics cannot cross and shows that
in games with more than two strategies, the limiting deterministic dynamics can have a crucial
dependence on the initial conditions.

\section*{Acknowledgement}
	Financial support by the Emmy-Noether program of the DFG is gratefully acknowledged.

redpajama_set_name: RedPajamaArXiv


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text:
Subtotal, Disaster Payments
Pick a county Utah State Total Beaver County, Utah Box Elder County, Utah Cache County, Utah Carbon County, Utah Daggett County, Utah Davis County, Utah Duchesne County, Utah Emery County, Utah Garfield County, Utah Grand County, Utah Iron County, Utah Juab County, Utah Kane County, Utah Millard County, Utah Morgan County, Utah Piute County, Utah Rich County, Utah Salt Lake County, Utah San Juan County, Utah Sanpete County, Utah Sevier County, Utah Summit County, Utah Tooele County, Utah Uintah County, Utah Utah County, Utah Wasatch County, Utah Washington County, Utah Wayne County, Utah Weber County, Utah Utah NRCS
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Counties in Utah Receiving Subtotal, Disaster Payments payments, 1995-2019‡
Pct of
1 Box Elder County, Utah $40,207,019 17.3% 17.3%
2 Sanpete County, Utah $20,216,744 8.7% 25.9%
3 Millard County, Utah $18,539,213 8.0% 33.9%
4 Juab County, Utah $14,122,845 6.1% 40.0%
5 Utah County, Utah $12,455,331 5.3% 45.3%
6 San Juan County, Utah $12,220,652 5.2% 50.6%
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8 Emery County, Utah $10,289,810 4.4% 60.0%
9 Duchesne County, Utah $9,716,874 4.2% 64.1%
10 Iron County, Utah $8,288,876 3.6% 67.7%
11 Washington County, Utah $7,905,322 3.4% 71.1%
12 Rich County, Utah $7,860,117 3.4% 74.5%
13 Beaver County, Utah $7,594,994 3.3% 77.7%
14 Cache County, Utah $7,093,464 3.0% 80.8%
15 Wayne County, Utah $6,245,304 2.7% 83.5%
16 Kane County, Utah $5,828,465 2.5% 86.0%
17 Garfield County, Utah $5,717,793 2.5% 88.4%
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19 Sevier County, Utah $4,307,198 1.8% 92.1%
20 Carbon County, Utah $3,665,407 1.6% 93.7%
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22 Summit County, Utah $3,422,990 1.5% 96.6%
23 Wasatch County, Utah $1,820,983 0.8% 97.4%
24 Daggett County, Utah $1,359,686 0.6% 98.0%
25 Morgan County, Utah $1,315,775 0.6% 98.6%
26 Grand County, Utah $1,136,022 0.5% 99.1%
27 Weber County, Utah $1,066,674 0.5% 99.5%
28 Davis County, Utah $724,913 0.3% 99.8%
29 Salt Lake County, Utah $387,973 0.2% 100.0%
Source: Environmental Working Group. Compiled from USDA data.


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The most ambitious research project on entrepreneurship globally
The Global Entrepreneurship Monitor (GEM) is a global consortium that conducts research on 66 global economies, covering 82% of the world's GDP and 71% of its population. GEM brings together over 400 researchers from across the globe and includes more than 100 institutions every year. The involvement of all these people and institutions makes GEM the largest study on entrepreneurship and entrepreneurial activities in the world.
2018/19 GEM Saudi Arabia Entrepreneurship Monitor
The 2018/19 GEM Saudi Arabia survey represents the fifth year that GEM has tracked entrepreneurship across multiple phases and assessed the characteristics, motivations, and ambitions of entrepreneurs. The subsection below discuss selected major findings from the report. The GEM Saudi Arabian national team also collected opinions about components of the entrepreneurship ecosystem through a national experts' survey (NES).
Here are some of the key findings of the 2018/19 GEM Saudi report:
Among Saudi Arabia's working-age population (18-64 years), the majority of people (78%) believe that those who successfully start a new business enjoy a high level of social status and respect. Two-thirds consider starting a business a desirable career choice.
Supporting these positive attitudes are the ease of starting a business (65%) and frequent representations of successful entrepreneurs in the media (71%), both of which increased by over 4 percentage points from the prior year. Media attention for entrepreneurs in Saudi Arabia is among the highest in the MENA region.
Most people in Saudi Arabia (79%) personally know an entrepreneur, and a majority see good opportunities around them for starting a business (76%), the highest rates on these indicators in the MENA region.
Want to know more ?
Receive the 2018 GEM Report


Chicago Central and Pacific Railroad Company (CCP) voluntarily performed repairs on a drainage improvement. When its request for reimbursement for those repairs was denied by the Calhoun County Board of Supervisors, the party responsible for keeping drainage district improvements in repair, CCP filed a petition for an appeal. The district court dismissed CCP's petition and entered judgment in favor of the Board, concluding that the CCP was liable for the costs of repair. The court of appeals affirmed. the Supreme Court vacated the court of appeals and affirmed the district court, albeit on a different basis, holding (1) under the statutory scheme enacted by the legislature, the Board has the duty to keep improvements in repair, but it also has the discretion to decide how to fulfill that duty; and (2) because CCP's suit essentially asked the Court to remove the Board's discretion while leaving its responsibilities intact, the suit was not permitted by law, and the district court properly dismissed the action.


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