Monday, November 18, 2013

Darwin inside the machine: A brief history of digital life

In 1863, the British writer Samuel Butler wrote a letter to the newspaper The Press entitled "Darwin Among the Machines". In this letter, Butler argued that machines had some sort of "mechanical life", and that machines would some day evolve to be more complex than people, and ultimately bring humanity into extinction:

Samuel Butler (1835-1902)
Source: Wikimedia 
"Day by day, however, the machines are gaining ground upon us; day by day  we are becoming more subservient to them; more men are daily bound down as slaves to tend them, more men are daily devoting the energies of their whole lives to the development of mechanical life. The upshot is simply a question of time, but that the time will come when the machines will hold the real supremacy over the world and its inhabitants is what no person of a truly philosophic mind can for a moment question."   
(S. Butler, 1863)

While futurist and doomsday prognosticator Ray Kurzweil would probably agree, I think that the realm of the machines is still far in the future. Here I would like to argue that Darwin isn't among the machines just yet, but he is certainly inside the machines.

The realization that you could observe life inside a computer was fairly big news in the early 1990s. The history of digital life has been chronicled before, but perhaps it is time for an update, because a lot has happened in twenty years. I will try to be brief: A Brief History of Digital Life. But you know how I have a tendency to fail in this department.

Who came up with the idea that life could be abstracted to such a degree that it could be implemented (mind you, this is not the same thing as simulated) inside a computer? 

Why, that would be just the same guy who actually invented the computer architecture we all use! You all know who this is, but here's a pic anyway:

John von Neumann (Source: Wikimedia)
I don't have many scientific heroes, but he is perhaps my number one. He was first and foremost a mathematician (who also made fundamental contributions to theoretical physics). After he invented the von Neumann architecture of modern computers, he asked himself: could I create life in it? 

Who would ask himself such a question? Well, Johnny did! He asked: if I could program an entity that contained the code that would create a copy of itself, would I have created life? Then he proceeded to try to program just such an entity, in terms of a cellular automaton (CA) that would self-replicate. 

Maybe you thought that Stephen Wolfram invented CAs? He might want to convince you of that, but the root of CAs goes back to Stanislaw Ulam, and indeed our hero Johnny. (If Johnny isn't your hero yet, I'll try to convince you that he should be. Did you know he invented Game Theory?) Johnny actually wrote a book called "Theory of Self-Reproducing Automata" that was only published after von Neumann's death. He died comparatively young, at age 54. Johnny was deeply involved in the Los Alamos effort to build an atomic bomb, and was present at the 1946 Bikini nuclear tests.  He may have paid the ultimate prize for his service, dying of cancer likely due to radiation exposure. Incidentally Richard Feynman also succumbed to cancer from radiation, but he enjoyed a much longer life. We can only imagine what von Neumann could have given us had he had the privilege of living into his 90s, as for example Hans Bethe did. And just like that, I listed two other scientists on my hero list! They both are in my top 5. 

All right, let's get back to terra firma. Johnny invented Cellular Automata just so that he can study self-replicating machines. What he did was create the universal constructor, albeit completely in theory. But he designed it in all detail: a 29-state cellular automaton that would (when executed) literally construct itself. It was a brave (and intricately detailed) construction, but he never got to implement it on an actual computer. This was done almost fifty years later by Nobili and Pesavento, who used a 32-state CA. They were able to show that von Neumann's construction  ultimately was able to self-reproduce, and even led to the inheritance of mutations.

The Nobili-Pesavento implementation of von Neumann's self-replicating CA, with two copies visible.
Source: Wikimedia 
von Neumann used information coded in binary in a "tape" of cells to encode the actions of the automaton, which is quite remarkable given that DNA had yet to be discovered. 

Perhaps because of von Neumann's untimely death, or perhaps because computers would soon be used for more "serious" applications than making self-replicating patterns, this work was not continued. It was only in the late 1980s that Artificial Life (in this case, creating a form of life inside of a computer) became  fashionable again, when Chris Langton started the Artificial Life conferences in Santa Fe, New Mexico. While Langton's work focused also on implementations using CA, Steen Rasmussen at Los Alamos National Laboratory tried another approach: take the idea of computer viruses as a form of life seriously, and create life by giving self-replicating computer programs the ability to mutate. To do this, he created self-replicating computer programs out of a computer language that was known to support self-replication: "Redcode", the language used in the computer game Core War.  In this game that was popular in the late 80s, the object is to force the opposing player's programs to terminate. One way to do this is to write a program that self-replicates.
Screen shot of a Core War game, showing the programs of two opposing players in red and green. Source: Wkimedia
Rasmussen created a simulated computer within a standard desktop, provided the self-replicator with a mutation instruction, and let it loose. What he saw was first of all quick proliferation of the self-replicator, followed by mutated programs that not only replicated inaccurately, but also wrote over the code of un-mutated copies. Soon enough no self-replicator with perfect fidelity would survive, and the entire population died out, inevitably. The experiment was in itself a failure, but it ultimately led to the emergence of digital life as we know it, because when Rasmussen demonstrated his "VENUS" simulator at the Santa Fe Institute, a young tropical ecologist was watching over his shoulder: Tom Ray of the University of Oklahoma. Tom quickly understood what he had to do in order to make the programs survive. First, he needed to give each program a write-protected space. Then, in order to make the programs evolvable, he needed to modify the programming language so that instructions did not refer directly to addresses in memory, simply because such a language turns out to be very fragile under mutations. 

Ray went ahead and wrote his own simulator to implement these ideas, and called it tierra.  Within the simulated world that Ray had created inside the computer, real information self-replicated, and mutated. I am writing "real" because clearly, the self-replicating programs are not simulated. They exist in the same manner as any computer program exists within a computer's memory: as instructions encoded in bits that themselves have a physical basis: different charge states of a capacitor. 
Screen shot of an early tierra simulator, showing the abundance distribution of individual genotypes in real time. Source: Wikimedia. 
The evolutionary dynamics that Ray observed was quite intricate. First, the 80 instruction-long self-replicator that Ray had painstakingly written himself started to evolve towards smaller sizes, shrinking, so to speak. And while Ray suspected that no program could self-replicate that was smaller than, say, 40 instructions long, he witnessed the sudden emergence of an organism that was only 20 lines long. These programs turned out to be parasites that "stole" the replication program of a "host" (while the programs were write-protected, Ray did not think he should insist on execution protection). Because the parasites did not need the "replication gene" they could be much smaller, and because the time it takes to make a copy is linear in the length of the program, these parasites replicated like crazy, and would threaten to drive the host programs to extinction. 

But of course that wouldn't work, because the parasites relied on those hosts! Even better, before the parasites could wipe out the hosts, a new host emerged that could not be exploited by the parasite: the evolution of resistance. In fact, a very classic evolutionary arms race ensued, leading ultimately to a mutualistic society. 

While what Ray observed was incontrovertibly evolution, the outcome of most experiments ended up being much of the same: shrinking program, evolution of parasites, an arms race, and ultimately coexistence. When I read that seminal 1992 paper [1] on a plane from Los Angeles to New York shortly after moving to the California Institute of Technology, I immediately started thinking about what one would need to do in order to make the programs do something useful. The tierran critters were getting energy for free, so they simply tried to replicate as fast as possible. But energy isn't free: programs should be doing work to gain that energy. And inside a computer, work is a calculation. 

After my postdoctoral advior Steve Koonin (a nuclear physicist, because the lab I moved to at Caltech was a nuclear theory lab) asked me (with a smirk) if I had liked any of the papers he had given me to read on the plane, I did not point to any of the light-cone QCD papers, I told him I liked that evolutionary one. He then asked: "Do you want to work on it?", and that was that.

I started to rewrite tierra so that programs had to do math in order to get energy. The result was this paper, but I wasn't quite happy with tierra. I wanted it to do much more: I wanted the digital critters to grow in a true 2D space (like, say, on a Petri dish)
A Petri dish with competing E. coli bacteria. Source: BEACON (MSU).
and I wanted them to evolve a complex metabolism based on computations. But writing computer code in C wasn't my forte: I was an old-school Fortran programmer. So I figured I'd pawn the task off to some summer undergraduate students. Two were visiting Caltech that summer: C. Titus Brown who was an undergrad in Mathematics at Reed College, and Charles Ofria, a computer science undergrad at Stony Brook University, where I had gotten my Ph.D. a few years earlier. I knew both of them because Titus is the son of theoretical physicist Gerry Brown, in whose lab I obtained my Ph.D, and Charles used to go to high school with Titus. 
From left: Chris Adami, Charles Ofria, Cliff Bohm, C. Titus Brown (Summer 1993)
Above is a photo taken during the summer when the first version of Avida was written, and if you clicked on any of the links above then you know that Titus and Charles have moved on from being undergrads. In fact, as fate would have it we are all back together here at Michigan State University, as the photo below documents. Where we were attempting to recreate that old polaroid!
Same cast of characters, 20 years later. This one was taken not in my rented Caltech appartment, but in CTB's sprawling Michigan mansion.  
The version of the Avida program that ultimately led to 20 years of research in digital evolution (and utlimately became one of the cornerstones of the BEACON Center) was the one written by Charles. (Whenever I asked any of the two to just modify Tom Ray's tierra, they invariably proceeded by rewriting everything from scratch. I clearly had a lot to learn about programming.)

So what became of this digital revolution of digital evolution? Besides germinating the BEACON Center for the Study of Evolution in Action, Avida has been used for more and more sophisticated experiments in evolution, and we think that we aren't done by a long shot. Avida is also used to teach evolution, in high-school and college class rooms.

Avida: the digital life simulator developed at Caltech and now under active development at Michigan State University, exists as a research as well as an educational version. Source: BEACON Institute. 
Whenever I give a talk or class on the history of digital life (or even its future), I seem to invariably get one question that wonders whether revealing the power that is immanent in evolving computer viruses is, to some extent, reckless.

You see, while the path to the digital critters that we call "avidians" was never really inspired by real computer viruses, you had to be daft not to notice the parallel. 

What if real computer viruses could mutate and adapt to their environment, almost instantly negating any and all design efforts of the anti-malware industry? Was this a real possibility?

Whenever I was asked this question, in a public talk or privately, I would equivocate. I would waffle. I had no idea. 

After a while I told myself: "Shouldn't we know the answer to this question? Is it possible to create a new class of computer viruses that would make all existing cyber-security efforts obsolete?"

Because if you think about it, computer viruses (the kind that infect your computer once in a while if you're not careful) already displays some signs of life. I'll show you here that one of the earliest computer viruses (known as the "Stoned" family) displayed one sign, namely the tell-tale waxing and waning of infection rate as a function of time.
Incidents of infection with the "Stoned" virus over time, courtesy of [2].
Why does the infection rate rise and fall? Well, because the designers of the operating system (the Stoned virus infected other computers only by direct contact: an infected floppy disk) were furiously working on thwarting this threat. But the virus designers (well, nobody called them that, really--they were called "hackers") were just as furiously working on defeating any and all countermeasures. A  real co-evolutionary arms race ensued, and the result was that the different types of Stoned viruses created in response to the selective pressure imparted by operating system designers could be rendered in terms of a phylogeny of viruses that is very reminiscent of the phylogeny of actual biochemical viruses (think influenza, see below).
Phylogeny of  Stoned computer viruses (credit: D.H. Hull)
What if these viruse could mutate autonomously (like real biochemical viruses) rather than wait for the "intelligent design" of hackers? Is this possible?

I did not know the answer to this question, but in 2006 I decided to find out.

And to find out, I had to try as hard as I could to achieve the dreaded outcome. The thinking was: If my lab, trained in all ways to make things evolve, cannot succeed in creating the next-generation malware threat, then perhaps no-one can. Yes, I realize that this is nowhere near a proof. But we had to start somewhere. But if we were able to do this, then we would know the vulnerabilities of our current cyber-infrastructure long before the hackers did. We would be playing white hat vs. black hat, for real. But we would do this completely secretly.

In order to do this, I talked to a private foundation, which agreed to provide funds for my lab to investigate the question, provided we kept strict security protocols. No work is to be carried out on computers connected to the Internet. All notebooks are to be kept locked up in a safe. Virus code is only transferred from computer to computer via CD-ROM, also to be stored in a safe. There were several other protocol guidelines, which I will spare you. The day after I received notice that I was awarded the grant, I went and purchased a safe.

To cut a too long story into a caricature of "short": it turned out to be exceedingly difficult to create evolving computer viruses. I could devote an entire blog post to outline all the failed approached that we took (and I suspect that such a post would be useful for some segments of my readership). My graduate student Dimitris Iliopoulos set up a computer (disconnected, of course)  with a split brain: one where the virus evolution would take place, and one that monitored the population of viruses that replicated--not in a simulated environment--but rather in the brutal reality of a real computer's operating system.

Dimitris discovered that the viruses did not evolve to be more virulent. They became as tame as possible. Because we had a "watcher" program monitoring the virus population, (culling individuals in order to keep the population size constant) programs evolved to escape the atttention of this program. Because being noticed by said program would spell death, ultimately.

This strategy of "hiding" turns out to be fairly well-known amongst biochemial viruses, of course. But our work was not all in vane. We contacted one of the leading experts in computer security at the time, Hungarian-born Péter Ször, who worked at the computer security company Symantec and wrote the book on computer viruses. He literally wrote it: you can buy it on Amazon here.

When we first discussed the idea with Péter, he was skeptical. But he soon warmed up to the idea, and provided us with countless examples of how computer viruses adapt--sometimes by accident, sometimes by design. We ended up writing a paper together on the subject, which was all the rage at the Virus Bulletin conference in Ottawa, in 2008 [3]. You can read our paper by clicking on this link here.

Which bring me, finally, to the primary reason why I am reminiscing about the history of digital life, and my collaboration with Péter Ször in particular. Péter passed away suddenly just a few days ago. He was 43 years old. He worked at Symantec for the majority of his career, but later switched to McAfee Labs as Senior Director of Malware Research. Péter kept your PC (if you choose to use such a machine) relatively free from this artfully engineered brand of viruses for decades. He worried whether evolution could ultimately outsmart his defenses and, at least for this brief moment in time, we thought we could.

Peter Szor (1970-2013)
[1] T.S. Ray, An approach to the synthesis of life, In : Langton, C., C. Taylor, J. D. Farmer, & S. Rasmussen [eds], Artificial Life II, Santa Fe Institute Studies in the Sciences of Complexity, vol. XI, 371-408. Redwood City, CA: Addison-Wesley.

[2] S. White, J. Kephart, D. Chess.  Computer Viruses: A Global Perspective. In: Proceedings of the 5th Virus Bulletin International Conference, Boston, September 20-22, 1995, Virus Bulletin Ltd, Abingdon, England, pp. 165-181. September 1995

[3] D. Iliopoulos, C. Adami, and P. Ször. Darwin Inside the Machines: Malware Evolution and the Consequences for Computer Security (2009). In: Proceedings of VB2008 (Ottawa), H. Martin ed., pp. 187-194

Saturday, November 2, 2013

Black holes and the fate of quantum information

I have written about the fate of classical information interacting with black holes fairly extensively on this blog (see Part 1, Part 2, and Part 3). Reviewers of the article describing those results nearly always respond that I should be considering the fate of quantum, not classical information. 

In particular, they ask me to comment on what all this means in the light of more modern controversies, such as black hole complementarity and firewalls. As if solving the riddle of what happens to classical information is not nearly good enough. 

I should first state that I disagree with the idea that it is necessary to discuss the fate of quantum information in an article that discusses what happens to classical information. I'll point out the differences between those two concepts here, and hopefully I'll convince you that it is perfectly reasonable to discuss these independently. However, I have given in to these requests, and now written an article (together with my colleague Kamil Bradler at St. Mary's University in Halifax, Canada) that studies the fate of quantum information that interacts with a black hole. Work on this manuscript explains (in part) my prolonged absence from blogging.

The results we obtained, it turns out, do indeed shed new light on these more modern controversies, so I'm grateful for the reviewer's requests after all. The firewalls have "set the physics world ablaze", as one blogger writes.  These firewalls (that are suggested to surround a black hole) have been invented to correct a perceived flaw in another widely discussed theory, namely the theory of black hole complementarity due to the theoretical physicist Leonard Susskind.  I will briefly describe these in more detail below, but before I can do this, I have to define for you the concept of quantum entanglement.

Quantum entanglement lies at the very heart of quantum mechanics, and it is what makes quantum physics different from classical physics. It is clear, as a consequence, that I won't be able to make you understand quantum entanglement if you have never studied quantum mechanics. If this is truly your first exposure, you should probably consult the Wiki page about quantum entanglement, which is quite good in my view. 

Quantum entanglement is an interaction between two quantum states that leaves them in a joint state that cannot be described in terms of the properties of the original states. So, for example, two quantum states $\psi_A$ and $\psi_B$ may have separate properties before entanglement, but after they interact they will be governed by a single wavefunction $\psi_{AB}$ (there are exceptions). So for example, if I imagine a wavefunction $\psi_A=\sigma|0\rangle +\tau|1\rangle$ (assuming the state to be correctly normalized) and a quantum state B simply given by $|0\rangle$, then a typical entangling operation $U$ will leave the joint state entangled:

      $U(\sigma|0\rangle +\tau|1\rangle)|0\rangle=\sigma|00\rangle +\tau|11\rangle$.    (1)

The wavefunction on the right hand side is not a product of the two initial wavefunctions, and in any case classical systems can never be brought into such a superposition of states in the first place. Another interesting aspect of quantum entanglement is that it is non-local. If A and B represent particles, you can still move one of the particles far away (say, to another part in the galaxy). They will still remain entangled. Classical interactions are not like that. At all.

A well-known entangled wavefunction is that of the Einstein-Podolsky-Rosen pair, or EPR pair. This is a wavefunction just like (1), but with $\sigma=\tau=1/\sqrt{2}$. The '0' vs '1' state can be realized via any physical quantum two-state system, such as a spin 1/2-particle or a photon carrying a horizontal or vertical polarization. 

What does it mean to send quantum information? Well, it just means sending quantum entanglement! Let us imagine a sender Alice, who controls a two-state quantum system that is entangled with another system (let us call it $R$ for "reference"), this means that her quantum wavefunction (with respect to $R$) can be written as 

                $|\Psi\rangle_{AR}=\sigma|00\rangle_{AR} +\tau|11\rangle_{AR}$    (2)

where the subscripts $AR$ refer to the fact that the wavefunction now "lives" in the joint space $AR$. $A$ and $R$ (after entanglement) do not have individual states any longer.

Now, Alice herself should be unaware of the nature of the entanglement between $A$ and $R$ (meaning, Alice does not know the values of the complex constants $\sigma$ and $\tau$). She is not allowed to know them, because if she did, then the quantum information she would send would become classical. Indeed, Alice can turn any quantum information into classical information by measuring the quantum state before sending it. So let's assume Alice does not do this. She can still try to send the arbitrary quantum state that she controls to Bob, so that after the transmittal her quantum state is unentangled with $R$, but it is now Bob's wavefunction that reads

           $|\Psi\rangle_{BR}=\sigma|00\rangle_{BR} +\tau|11\rangle_{BR}$    (3). 

In this manner, entanglement was transferred from $A$ to $B$. That is a quantum communication channel.

Of course, lots of things could happen to the quantum entanglement on its way to Bob. For example, it could be waylayed by a black hole. If Alice sends her quantum entanglement into a black hole, can Bob retrieve it? Can Bob perform some sort of magic that will leave the black hole unentangled with $A$ (or $R$), while he himself is entangled as in (3)?

Whether or not Bob can do this depends on whether the quantum channel capacity of the black hole is finite, or whether it vanishes. If the capacity is zero, then Bob is out of luck. The best he can do is to attempt to reconstruct Alice's quantum state using classical state estimation techniques. That's not nothing by the way, but the "fidelity" of the state reconstruction is at most 2/3. But I'm getting ahead of myself.

Let's first take a look at this channel. I'll picture this in a schematic manner where the outside of the black hole is at the bottom, and the inside of the black hole is at the top, separated by the event horizon. Imagine Alice sending her quantum state in from below. Now, black holes (as all realistic black bodies) don't just absorb stuff: they reflect stuff too. How much is reflected depends on the momentum and angular momentum of the particle, but in general we can say that a black hole has an absorption coefficient $0\leq\alpha\leq1$, so that $\alpha^2$ is just the probability that a particle that is incident on the black hole is absorbed.

So we see that if $n$ particles are incident on a black hole (in the form of entangled quantum states $|\psi\rangle_{\rm in}$), then $(1-\alpha^2)n$ come out because they are reflected at the horizon. Except as we'll see, they are in general not the pure quantum states Alice sent in anymore, but rather a mixture $\rho_{\rm out}$. This is (as I'll show you) because the black hole isn't just a partially silvered mirror. Other things happen, like Hawking radiation. Hawking radiation is the result of quantum vacuum fluctuations at the horizon, which constantly create particle-antiparticle pairs. If this happened anywhere but at the event horizon, the pairs would annihilate back, and nobody would be the wiser. Indeed, such vacuum fluctuations happen constantly everywhere in space. But if it happens at the horizon, then one of the particles could cross the horizon, while the other (that has equal and opposite momentum), speeds away from it. That now looks like the black hole radiates. And it happens at a fixed rate that is determined by the mass of the black hole. Let's just call this rate $\beta^2$.

As you can see, the rate of spontaneous emission does not depend on how many particles Alice has sent in. In fact, you get this radiation whether or not you throw in a quantum state. These fluctuations go on before you send in particles, and after. They have absolutely nothing to do with $|\psi\rangle_{\rm in}$. They are just noise. But they are (in part) responsible for the fact that the reflected quantum state $\rho_{\rm out}$ is not pure anymore. 

But I can tell you that if this were the whole story, then physics would be in deep deep trouble. This is because you cannot recover even classical information from this channel if $\alpha=1$. Never mind quantum. In fact, you could not recover quantum information even in the limit $\alpha=0$, a perfectly reflecting black hole! (I have not shown you this yet, but I will). 

This is not the whole story, because a certain gentleman in 1917 wrote a paper about what happens when radiation is incident on a quantum mechanical black body. Here is a picture of this gentleman, along with the first paragraph of the 1917 paper:

Albert Einstein in 1921                        Einstein's 1917 article
              (source: Wikimedia)            "On the quantum theory of radiation"
What Einstein discovered in that paper is that you can derive Planck's Law (about the distribution of radiation emanating from a black body) using just the quantum mechanics of absorption, spontaneous emission, and stimulated emission of radiation. Stimulated emission is by now familar to everybody, because it is the principle upon which Lasers are built. What Einstein showed in that paper is that stimulated emission is an inevitable consequence of absorption: if a black body absorbs radiation, it also stimulates the emission of radiation, with the same exact quantum numbers as the incoming radiation.

Here's the figure from the Wiki page that shows how stimulated emission makes "two out of one":

Quantum "copying" during stimulated emission from an atom (source: Wikimedia)

In other words, all black bodies are quantum copying machines!

"But isn't quantum copying against the law?"

Actually, now that you mention it, yes it is, and the law is much more stringent than the law against classical copying (of copy-righted information, that is). The law (called the no-cloning theorem) is such that it cannot--ever--be broken, by anyone or anything. 

The reason why black bodies can be quantum copying machines is that they don't make perfect copies, and the reason the copies aren't perfect is the presence of spontaneous emission, which muddies up the copies. This has been known for 30 years mind you, and was first pointed out by the German-American physicist Leonard Mandel. Indeed, only perfect copying is disallowed. There is a whole literature on what is now known as "quantum cloning machines" and it is possible to calculate what the highest allowed fidelity of cloning is. When making two copies from one, the highest possible fidelity is $F$=5/6. That's an optimal 1->2 quantum cloner. And it turns out that in a particular limit (as I point out in this paper from 2006) black holes can actually achieve that limit!  I'll point out what kind of black holes are optimal cloners further below.

All right, so now we have seen that black holes must stimulate the emission of particles in response to incoming radiation. Because Einstein said they must. The channel thus looks like this:

In addition to the absorbed/reflected radiation, there is spontaneous emission (in red), and there is stimulated emission (in blue). There is something interesting about the stimulated "clones". (I will refer to the quantum copies as clones even though they are not perfect clones, of course. How good they are is central to what follows). 

Note that the clone behind the horizon has a bar over it, which denotes "anti". Indeed, the stimulated stuff beyond the horizon consists of anti-particles, and they are referred to in the literature as anti-clones, because the relationship between $\rho_{\rm out}$ and $\bar \rho_{\rm out}$ is a quantum mechanical NOT operation. (Or, to be technically accurate, the best NOT you can do without breaking quantum mechanics.) That the stimulated stuff inside and outside the horizon must be particles and anti-particles is clear, because the process must conserve particle number. We should keep in mind that the Hawking radiation also conserves particle number. The total number of particles going in is $n$, which is also the total number of particles going out (adding up stuff inside and outside the horizon). I checked. 

Now that we know that there are a bunch of clones and anti-clones hanging around, how do we use them to transfer the quantum entanglement? Actually, we are not interested here in a particular protocol, we are much more interested in whether this can be done at all. If we would like to know whether a quantum state can be reconstructed (by Bob) perfectly, then we must calculate the quantum capacity of the channel. While how to do this (and whether this calculation can be done at all) is technical, one thing is not: If the quantum capacity is non-zero then, yes, Bob can reconstruct Alice's state perfectly (that is, he will be entangled with $R$ exactly like Alice was when he's done). If it is zero, then there is no way to do this, period.

In the paper I'm blogging about, Kamil and I did in fact calculate the capacity of the black hole channel, but only for two special cases: $\alpha=0$ (a perfectly reflecting black hole), and $\alpha=1$ (a black hole that reflects nothing). The reason we did not tackle the general case is that at the time of this writing, you can only calculate the quantum capacity of the black hole channel exactly for these two limiting cases. For a general $\alpha$, the best you can do is give a lower and an upper bound, and we have that calculation planned for the future. But the two limiting cases are actually quite interesting.

[Note: Kamil informed me that for channels that are sufficiently "depolarizing", the capacity can in fact be calculated, and then it is zero. I will comment on this below.]

First: $\alpha=0$. In that case the black hole isn't really a black hole at all, because it swallows nothing. Check the figure up there, and you'll see that in the absorption/reflection column, you have nothing in black behind the horizon. Everything will be in front. How much is reflected and how much is absorbed doesn't affect anything in the other two columns, though. So this black hole really looks more like a "white hole", which in itself is still a very interesting quantum object. Objects like that have been discussed in the literature (but it is generally believed that they cannot actually form from gravitational collapse). But this is immaterial for our purposes: we are just investigating here the quantum capacity of such an object in some extreme cases. For the white hole, you now have two clones outside, and a single anticlone inside (if you would send in one particle). 

Technical comment for experts: 
A quick caveat: Even though I write that there are two clones and a single anti-clone after I send in one particle, this does not mean that this is the actual number of particles that I will measure if I stick out my particle detector, dangling out there off of the horizon. This number is the mean expected number of particles. Because of vacuum fluctuations, there is a non-zero probability of measuring a hundred million particles. Or any other number.  The quantum channel is really a superposition of infinitely many cloning machines, with the 1-> 2 cloner the most important. This fundamental and far-reaching result is due to Kamil. 

So what is the capacity of the channel? It's actually relatively easy to calculate because the channel is already well-known: it is the so-called Unruh channel that also appears in a quantum communication problem where the receiver is accelerated, constantly. The capacity looks like this:

Quantum capacity of the white hole channel as a function of z
In that figure, I show you the capacity as a function of $z=e^{-\omega/T}$, where $T$ is the temperature of the black hole and $\omega$ is the frequency (or energy) of that mode. For a very large black hole the temperature is very low and, as a consequence, the channel isn't very noisy at all (low $z$). The capacity therefore is nearly perfect (close to 1 bit decoded for every bit sent). When black holes evaporate, they become hotter, and the channel becomes noisier (higher $z$). For infinitely small black holes ($z=1$) the capacity finally vanishes. But so does our understanding of physics, of course, so this is no big deal. 

What this plot implies is that you can perfectly reconstruct the quantum state that Alice daringly sent into the white hole as long as the capacity $Q$ is larger than zero. (If the capacity is small, it would just take you longer to do the perfect reconstructing.). I want to make one thing clear here: the white hole is indeed an optimal cloning machine (the fidelity of cloning 1->2 is actually 5/6, for each of the two clones). But to recreate the quantum state perfectly, you have to do some more work, and that work requires both clones. But after you finished, the reconstructed state has fidelity $F=1$.) 

"Big deal" you might say, "after all the white hole is a reflector!"

Actually, it is a somewhat big deal, because I can tell you that if it wasn't for that blue stimulated bit of radiation in that figure above, you couldn't do the reconstruction at all! 

"But hold on hold on", I hear someone mutter, from far away. "There is an anti-clone behind the horizon! What do you make of that? Can you, like, reconstruct another perfect copy behind the horizon? And then have TWO?"

So, now we come to the second result of the paper. You actually cannot. The capacity of the channel into the black hole (what is known as the complementary channel) is actually zero because (and this is technical speak) the channel into the black hole is entanglement breaking. You can't reconstruct perfectly from a single clone or anti-clone, it turns out. So, the no-cloning theorem is saved. 

Now let's come to the arguably more interesting bit: a perfectly absorbing black hole ($\alpha$=1). By inspecting the figure, you see that now I have a clone and an anti-clone behind the horizon, and a single clone outside (if I send in one particle). Nothing changes in the blue and red lines. But everything changes for the quantum channel. Now I can perfectly reconstruct the quantum state behind the horizon (as calculating the quantum capacity will reveal), but the capacity in front vanishes! Zero bits, nada, zilch. If $\alpha=1$, the channel from Alice to Bob is entanglement breaking.  

It is as if somebody had switched the two sides of the black hole! 
Inside becomes outside, and outside becomes inside!

Now let's calm down and ponder what this means. First: Bob is out of luck. Try as he might, he cannot have what Alice had: the same entanglement with $R$ that she enjoyed. Quantum entanglement is lost when the black hole is perfectly absorbing. We have to face this truth.  I'll try to convince you later that this isn't really terrible. In fact it is all for the good. But right now you may not feel so good about it.

But there is some really good news. To really appreciate this good news, I have to introduce you to a celebrated law of gravity, the equivalence principle

The principle, due to the fellow whose pic I have a little higher up in this post, is actually fairly profound. The general idea is that an observer should not be able to figure out whether she is, say, on Earth being glued to the surface by 1g, or whether she is really in a spaceship that accelerates at the rate of 1g (g being the constant of gravitational acceleration on Earth, you know: 9.81 m/sec$^2$). The equivalence principle has far reaching consequences. It also implies that an observer (called, say, Alice), who falls towards (and ultimately into) a black hole, should not be able to figure out when and where she passed the point of no return. 

The horizon, in other words, should not appear as a special place to Alice at all. But if something dramatic would happen to quantum states that cross this boundary, Alice would have a sure-fire way to notice this change: she could just keep the quantum state in a protected manner at her disposition, and constantly probe this state to find out if anything happened to it. That's actually possible using so-called "non-demolition" experiments. So, unless you feel like violating another one of Einstein's edicts (and, frankly, the odds are against you if you do), you better hope nothing happens to a quantum state that crosses from the outside to the inside of a black hole in the perfect absorption case ($\alpha=1$). 

Fortunately, we proved (result No. 3) that you can perfectly reconstruct the state behind the horizon when $\alpha=1$, that this capacity is non-zero. And that as a consequence, the equivalence principle is upheld. 

This may not appear to you as much of a big deal when you read this, but many many researchers have been worried sick about this, that the dynamics they expect in black holes would spell curtains for the equivalence principle. I'll get back to this point, I promise. But before I do so, I should address a more pressing question.

"If Alice's quantum information can be perfectly reconstructed behind the horizon, 
what happens to it in the long run?"

This is a very serious question. Surely we would like Bob to be able to "read" Alice's quantum message (meaning he yearns to be entangled just like she was). But this message is now hidden behind the black hole event horizon. Bob is a patient man, but he'd like to know: "Will I ever receive this quantum info?"

The truth is, today we don't know how to answer this question. We understand that Alice's quantum state is safe and sound behind the horizon--for now. There is also no reason to think that the ongoing process of Hawking radiation (that leads to the evaporation of the black hole) should affect the absorbed quantum state. But at some point or other, the quantum black hole will become microscopic, so that our cherished laws of physics may lose their validity. At that point, all bets are off. We simply do not understand today what happens to quantum information hidden behind the horizon of a black hole, because we do not know how to calculate all the way to very small black holes. 

Having said this, it is not inconceivable that at the end of a black hole's long long life, the only thing that happens is the disappearance of the horizon. If this happens, two clones are immediately available to an observer (the one that used to be on the outside, and the one that used to be inside), and Alice's quantum state could finally be resurrected by Bob, a person that no doubt would merit to be called the most patient quantum physicist in the history of all time. 

Now what does this all mean for black hole physics? I have previously shown that classical information is just fine, and that the universe remains predictable for all times. This is because to reconstruct classical information, a single stimulated clone is enough. It does not matter what $\alpha$ is, it could even be one. Quantum information can be conveyed accurately if the black hole is actually a white hole, but if it is utterly black then quantum information is stuck behind the horizon, even though we have a piece of it (a single clone) outside of the horizon. But that's not enough, and that's a good thing too, because we need the quantum state to be fully reconstructable inside of the black hole, otherwise the equivalence principle is hosed.  And if it reconstructable inside, then you better hope it is not reconstructable outside, because otherwise the no-cloning theorem would be toast. 

So everything turns out to be peachy, as long as nothing drastic happens to the quantum state inside the black hole. We have no evidence of something so drastic, but at this point we simply do not know (1). 

Now what are the implications for black hole complementarity? The black hole complementarity principle was created from the notion (perhaps a little bit vague) that, somehow, quantum information is both reflected and absorbed by the black hole channel at the same time. Now, given that you have read this far in this seemingly interminable post, you know that this is not allowed. It really isn't. What Susskind, Thorlacius, and 't Hooft  argued for, however, is that it is OK as long as you won't be caught. Because, they argued, nobody will be able to measure the quantum state on both sides of the horizon at the same time anyway!

Now I don't know about you, but I was raised believing that just because you can't be caught it doesn't make it alright to break the rules.  And what our more careful analysis of quantum information interacting with a black hole has shown, is that you do not break the quantum cloning laws at all. Both the equivalence principle and the no-cloning theorem are perfectly fine. Nature just likes these laws, and black holes are no outlaws.

Adventurous Alice encounters a firewall? Credit: Nature.com
What about firewalls then? Quantum firewalls were proposed to address a perceived inconsistency in the black hole complementarity picture. But you now already know that that picture was inconsistent to begin with. Violating no-cloning laws brings with it all kinds of paradoxes. Unfortunately, the firewall hypothesis just heaped paradoxes upon paradoxes, because it proposed that you have to violate the equivalence principle as well. This is because that hypothesis assumes that all the information was really stored in the Hawking radiation (the red stuff in the figures above). But there is really nothing in there, so that the entire question of whether transmitting quantum information from Alice to Bob violates the concept of "monogamy of entanglement" is moot. The Hawking radiation can be entangled with the black hole, but it is no skin off of Alice or Bob, that entanglement is totally separate. 

So, all is well, it seems, with black holes, information, and the universe. We don't need firewalls, and we do not have to invoke a "complementarity principle". Black hole complementarity is automatic, because even though you do not have transmission when you have perfect reflection, a stimulated clone does make it past the horizon. And when you have perfect transmission ($\alpha$=1) a stimulated clone still comes back at you. So it is stimulated emission that makes black hole complementarity possible, without breaking any rules. 

Of course we would like to know the quantum capacity for an arbitrary $\alpha$, which we are working on. One result is already clear: if the transmission coefficient $\alpha$ is high enough that not enough of the second clone is left outside of the horizon, then the capacity abruptly vanishes. Because the black hole channel is a particular case of a "quantum depolarizing channel", discovering what this critical $\alpha$ is only requires mapping the channel's error rate $p$ to $\alpha$. 

I leave you with an interesting observation. Imagine a black hole channel with perfect absorption, and put yourself into the black hole. Then, call yourself "Complementary Alice", and try to send a quantum state across the horizon. You immediately realize that you can't: the quantum state will be reflected. The capacity to transmit quantum information out of the black hole vanishes, while you can perfectly communicate quantum entanglement with "Complementary Bob". Thus, from the inside of the perfectly absorbing black hole it looks just like the white hole channel (and of course the reverse is true for the perfectly reflecting case). Thus, the two channels are really the same, just viewed from different sides! 

This becomes even more amusing if you keep in mind that (eternal) black holes have white holes in their past, and white holes have black holes in their future. 

Note added: This paper appeared in the Journal of High Energy Physics: JHEP 05(2014) 95.

(1) Note added: But see my later post on this exact issue!

Thursday, August 1, 2013

Survival of the nicest: Why it does not pay to be mean

According to the "MIT Technology Review", the world of game theory is "currently on fire". To be sure, I am not the arsonist. But, judging by the interest a paper by Bill Press and Freeman Dyson that was published a year ago in the Proceedings of the National Academy of Sciences has created, the flamboyant description is not completely off the mark. Before I describe the contents of the Press and Dyson paper (which has been done, to be sure, by many a blogger before me, see here or here), let me briefly tell you about the authors.

I know both of them in different ways. Bill Press is a Professor of Computer Science as well as Professor of Integrative Biology at UT Austin. Incidentally, UT Austin is a sister campus within the BEACON Center for the Study of Evolution in Action, which I also belong to here at Michigan State University. Bill Press is an astrophysicist by training, but perhaps most famous for a book on scientific computing, called "Numerical Recipes". That's where I know him from. If you've used scientific computing in the 90s and 00s, say, then you know this book. Bill Press is also the President of the American Association for the Advancement of Science (AAAS), which you may know as the publisher of Science Magazine. AAAS elected me a fellow in 2011, so I'm perfectly happy with this organization, as you can imagine. Press was also a "Prize Postdoctoral Fellow" at Caltech (incidentally as was I), and actually grew up in Pasadena (I spent almost 20 years there).  So, besides the fact that he is way more famous than I am, there are some interesting parallels in our careers. I was once offered a Director's postdoc fellowship at Los Alamos National Laboratory that I declined. He accepted the Deputy Laboratory Director position there in 1998. He worked on black hole physics and is kinda famous for his work with Teukolsky. I worked on black hole physics… and if you read this blog then you know that story. So the parallels are, at best, marginal. He now works in computational biology, and I work there too, and that's where this story really has its beginning. 

I've known of Freeman Dyson ever since I took Quantum Field Theory in grad school at SUNY Stony Brook. I can't and won't sing his praises here because my blog posts tend to be on the long side to begin with. I will probably always be able to write down the Schwinger-Dyson equation, perhaps even when somewhat inebriated (but I've never tried this). I've met him a couple of times but I do not expect that he would remember me. He is 89 now, so many people were surprised to hear that the mathematics of the PNAS paper mentioned above were entirely his (as recounted here). I'm not that surprised, because I've known the theoretical physicist Hans Bethe quite well in the last thirteen years of his life, and he was still able to calculate. And Bethe was 85 when I first met him. So I know that this is possible as long as you do not stop.  

So, what did Press and Dyson do? To begin to understand this, you have to have heard of the "Prisoner's Dilemma". 

The Prisoner's Dilemma (PD for short), is a simple game. Two players, two moves. The back story is a bit contrived: Two shady characters commit a crime, but are discovered as they flee the scene, speedily. 

The authorities nab both of them, but are unsure which of the two is the actual perpetrator. They are ready to slap a "speeding while leaving a crime scene" conviction on both of them ("What, no grand-jury indictment? Have you even heard of "trial by jury of peers"?), when they are interrogated one more time, separately. 

"Listen", the grand interrogator whispers to one of them, "you can go scot-free if you tell me it was the other guy". "Whoa. An incentive.", thinks perp. No. 1. "I get one year for accelerated haste, but I get 5 years if they figure we both were in on the heist".   "If I say nothing we both get a year, and are out by Christmas on good behavior. But if they offer the same deal to my buddy, I bet he squeals like a pig. He gets out, and I go in for 20. I better rat him out". As a consequence, they rat each other out and spend 5 years in the slammer. They could have both spent Christmas at home. But instead, they chose to "defect" rather than "cooperate". How lame! 

Theory says, actually, not lame at all. Theory says: "Under these circumstances, thou shalt betray your fellow man!"

That theory, by the way, is due to no other than John Forbes Nash, who even got a Nobel for it. Well, John von Neumann, the all-around genius and originator of Game Theory would have had a few choice words for that prize, but that's for another blog. "A Nobel for applying Brouwer's fixed point theorem? Are you f***ing kidding me?", is what Johnny might have said. (von Neumann's friends called him Johnny. I doubt anybody called Nash 'Johnny'). But let's agree not to digress. 

So, Nash says: "Screwing your adversary is the rational choice in this situation". And that's certainly correct. The aforementioned fixed point theorem guarantees it. Nowadays, you can derive the result in a single line. And you thought Nobel prizes are hard to get? Well, today you may have to work a bit harder than that.

Here then, is the dilemma in a nut shell. "If we both cooperate by not ratting each other out, we go home by Christmas". But there is this temptation: "I rat out the other guy and I go free, like, now (and the other guy won't smell flowers until the fourth Clinton administration). But if we both rat each other out, then we may regret that for a long time."

You can view the dilemma in terms of a payoff matrix as follows. A matrix? Don't worry: it's just a way to arrange numbers. We both keep silent: one year for each of us. We both rat each other out: 3 years for each of us, because now the cops are certain we did it. But if I rat you out while you're being saintly…. then I go free and you spend serious time (like, 20 years). Of course, the converse holds true too: If I hold my breath while you squeal, then and I am caught holding the bag.

We can translate the "time-in-the-slammer" into a payoff-to-the-individual instead like this: hard time=low payoff. Scot-free=high payoff. You get the drift. We can then write a payoff matrix for the four possible plays. Staying silent is a form of cooperation, so we say this move is "C". Ratting out is a form of defection, so we call this move "D'.

The four possible plays are now: we both cooperate: "CC" (payoff=3), we both defect: "DD" (payoff=1), and we play unequally: "CD" or "DC". If I cooperate while you defect, I get nada and you get 5. Of course, the converse is true if I defect and you cooperate.  So let's arrange that into a matrix:

The matrix entry contains the payoff to the "row player", that is, the player on "the left of the matrix", as opposed to the top. Two players could do well by cooperating (and earning 3s), while the rational choice is both players defect, earning 1s each. What gives?

What gives is, the perps are not allowed to talk to each other! If they could, they'd certainly conspire to not rat each other out. I've made this point in a paper three years ago that was summarily ignored by the community.

(I concede that, perhaps, I didn't choose a very catchy title.) 

Still, I'm bitter. Moving on.

How could players, engaged in this dilemma, communicate? Communication in this case entails that players recognize each other. "Are you one of these cheaters, or else a bleeding-heart do-gooder?" is what every player may want to know before engaging the adversary. How could you find out? Well, for example, by observing your opponent's past moves! This will quickly tell you what kind of an animal you're dealing with. So a communicator's strategy should be to make her moves conditional on the opponent's past moves (while not forgetting what you yourself did last time you played).

So let's imagine I (and my opponent) make our moves conditional on what we just did the last time we played. Because there are four possible histories (CC, CD, DC, and DD, meaning, the last time we played, I played C and you played C, or I played C and you played D, etc.) Now instead of a strategy such as "I play C no matter what you do!" I could have a more nuanced response, such as: "If I played C (believing in the good of mankind) and you played D (ripping me off, darn you) then I will play D, thank you very much!". Such nuanced strategies are clearly far more effective than unconditional strategies, because they are responsive to the context of the opponent's play, given your own.

Strategies that make their choice dependent on previous choices are communicating. Such communicating strategies are a relatively new addition to the canon of evolutionary game theory (even though one of the "standard" strategies called "Tit-for-Tat", is actually a strategy that communicates). Because this strategy ("TFT" for short) will come up later, let me quickly introduce you to it:

TFT does to its opponent what the opponent just did to TFT 

So, TFT is sort of an automaton replaying the Golden Rule, isn't it? "I'll do unto you what you just did to me!" Indeed, it kinda is, and and it is quite a successful strategy all in all. Let me introduce a notation to keep track of strategies, here. It will look like math, but you really should think of it as a way to keep track of stuff. 

So, I can either cooperate or defect. That means, I either play option "C" or option "D". But you know, I reserve the right to simply have a bias for one or the other choice. So my strategy should really be a probability to play C or D. Call this probability $p$. Keep in mind that this probability is the probability to cooperate. The probability to defect will be $1-p$.

If communication is involved, this probability should be dependent on what I just learned (my opponent's last play). The last exchange is four possible situations: CC, CD, DC, an DD: the first: I and my opponent just played "C", the last one: I and my opponent just played "D". A strategy then is a set of four probabilities

$$ p(CC), p(CD), p(DC), p(DD). $$
Here, $p(CC)$ is the probability to cooperate if my previous move was 'C' and my opponent's was 'C' also, and so on. Clearly, armed with such a set of probabilities, you can adjust your probability to cooperate to what your opponent plays. The TFT strategy, by the way, would be given in terms of this nomenclature as 

TFT=(1,0,1,0) ,

that is, TFT cooperates if the opponent's last move was "C", and defects otherwise. What other strategies are out there, if the probabilities can be arbitrary, not just zero and one?

That's the question Bill Press asked himself. Being a computational guy, he figured he'd write a program that would have all possible strategies play against all other possible strategies. Good idea, until the supercomputers he enlisted for this task started to crash. The reason why Press's computers were crashing (according to him, in an interview at Edge.org), was that he assumed that the payoff that a strategy would receive would depend on the way it played (that is, the four probabilities written above). 

Well, you would think that, wouldn't you? I certainly would have. Me and my opponent both have four probabilities that define the way we play. I would assume that my average payoff (assuming that we play against each other a really long time), as well as my opponent's average payoff, will depend on these eight probabilities. 

That is where you would be wrong, and Freeman Dyson (according to Press) sent Press a proof that this was wrong the next day. The proof is actually fairly elegant. What Dyson proved is that one player (let us call her "ZD") can choose her four probabilities judiciously in such a manner that the opponent will receive a payoff that only depends on ZD's strategy, and nothing else.

ZD, by the way, does not stand for Zooey Deschanel, but rather "Zero Determinant", which is a key trick in Dyson's proof. I thought I might clarify that.

The opponent (let's call him "O") can do whatever he wants. He can change strategy. He can wail. He can appeal to a higher authority. He can start a petition on Change.org. It won't make a dime of  a difference. ZD can change her strategy (while still staying within the realm of judiciously chosen "ZD strategies", and change O's payoff. Unilaterally. Manipulatively. Deviously.

To do this, all that ZD has to do ist to set two of her probabilities to be a very specific function of the two others. I can show you exactly how to do this, but the formula is unlikely to enlighten you. 

OK, fine, you can see it, but I'll just give you the simplified version that uses the payoffs in the matrix I wrote above (you know, the one with 5,3,1, and 0 in it), rather than the general case described by Press and Dyson. Let's also rename the clunkish $$p(CC), p(CD), p(DC), p(DD)$$ as $$p_1,p_2,p_3,p_4$$
Then, all ZD has to do is to choose
See, I told you that this isn't particularly enlightening. But you can see that there is a whole set of these strategies, defined by a pair of probabilities $p_1,p_4$. These can be arbitrary within bounds: they have to be chosen such that $p_2$ and $p_3$ are between zero and one.

So let's take a step back. ZD can be mean and force O to accept a payoff he does not care for, and cannot change. Does ZD now hold the One Ring that makes the Golden Rule a quaint reminder of an antiquated and childish belief in the greater good? 

Not so fast. First,  ZD's payoff under the strategy she plays isn't exactly the cat's meow. For example, if ZD's opponent plays "All-D" (that is, the strategy (0,0,0,0) that is the game-theoretic equivalent of the current Republican-dominated congress:  "No,no.no,no") ZD forces All-D to take a constant payoff alright, but this is higher than what ZD receives herself! This is because ZD's payoffs do depend on all eight probabilities (well, six, because two have been fixed by playing a ZD strategy), and playing All-D does not make you rich, if you play ZD or not. So you better not play ZD against All-D, but you might be lucky against some others.

Take, for example, one of the best communicating strategies ever designed, one called either "Win-Stay-Lose-Shift" (WSLS), or alternatively "Pavlov". WSLS was invented by Martin Nowak and Karl Sigmund in a fairly famous Nature paper. This strategy is simple: continue doing what you're doing when you're winning, but change what you do when you lose. So, for example, if you and your opponent cooperate, then you should be cooperating next time ('cause getting a '3' is winning). Hence, $p_1=1$. On the contrary, if you just played 'D' and so did your opponent, this would not be called winning, so you should switch from defection to cooperation. Thus, $p_4=1$. If you played 'C' and your opponent suckered you with a 'D', then that is definitely not called winning, so you should switch to 'D' next: $p_2=0$. If you instead played 'D' to the hapless opponent's 'C', well that's a major win, and you should continue what you're doing: $p_3=0$.  

How does WSLS fare against ZD? Well, WSLS gets handed his lunch! ZD sweeps the floor with WSLS. I can give you the mean expected payoff that WSLS gets when playing a perfectly vanilla ZD with $p_1=0.99$ and $p_4=0.01$. (That's a ZD strategy that likes to cooperate,  but is fairly unforgiving.) WSLS get an average of 2 against this strategy, just as much as any other strategy will gain. If ZD is playing those numbers, it doesn't matter what you do. You get two and like it. That's the whole point. ZD is a bully.

What does ZD get against WSLS? What ZD receives generally (against any opposing strategy) is a more complicated calculation. It's not terribly hard but it requires pencil and paper. A lot of pencil and paper, to be exact. (OK, fine, maybe not for everyone. I required a lot of pencil and paper. And eraser. A lot of eraser.) The result is too large to show on this blog, so: sorry!

Alright, this is not the days of Fermat because we can link stuff, so there it is, linked. You learned nothing from that. If you plug in the WSLS strategy (1,0,0,1) into that unwieldy formula you just not looked at, you get

The mean payoff a ZD strategy defined by $p_1=0.99, p_4=0.01$ receives from WSLS is $11/27\approx 2.455$. That's a lot more than what WSLS gets (namely 2). That's Grand Larceny, isn't it? ZD the victorious bully, while WSLS is the victim?

Not so fast, again.

You see, in real situations, we don't necessarily know who we play. We're just a bunch of strategies playing games. I may figure out "what kind of animal" you are after a couple of moves. But within a population of coexisting agents, you know, the kind that evolves according to the rules that Charles Darwin discovered, you play with those you meet.

And in that case, you, the domineering ZD strategist, may just as well run into another ZD strategist! For all you know, that ZD player may be one of your offspring!

What does ZD get against another ZD? I'll let you think this over for a few seconds.





Keep in mind, ZD forces ANY strategy to accept a fixed payoff.



ZD must accept the same payoff she forces on others when playing against other ZDs!

But that is terrible! Because once ZD has gained a foothold in a population that ZD can rip off, that means that ZD has achieved a sizable population fraction. Which means that ZD is going to have to play against versions of itself more and more. And fare terribly, just like all the other schmuck strategies she rips off.

But, what if the schmuck strategy plays really well against itself? Like, cooperates and reaps a mean payoff of 3?

You know, like, for example, WSLS?

That would mean that ZD would be doomed even though it wins every single friggin' battle with WSLS! Every one! But at some point, it becomes clear that winning isn't everything. You've got to be able to play with your own kind. WSLS does that well. ZD... not so much. ZD will take your lunch money, and ends up eating its own lunch too. How perfectly, weirdly, appropriate.

I'll leave you with the plot that proves this point, which is almost the one that appeared on the MIT Technology Review post that stated that the world of Game Theory is on fire. In that plot, the green lines show the fraction of the population that is WSLS, while the blue lines show the ZD fraction for different starting fractions. For example, if ZD (blue line) starts at 0.9, it competes with a green line that starts at 0.1. And green wins, no matter how many ZDs there are at the outset. The bully gets her come-uppance!

The plot looks the same as what appeared on MIT Technology Review, but on the original blog post they used a figure from the first version of the paper, where we showed that ZD would lose against All-D in a population setting. We realized soon after that using All-D is not such a great example, because All-D wins against ZD in direct competition too. So we chose WSLS to make our point instead. But it doesn't really matter. ZD sucks because it is so mean, and has no choice to also be mean against "its own kind". That's what takes it down.

If I was naive, I'd draw parallels to politics right here at the conclusion. But I'm not that naive. These are games that we investigated. Games that agents play. There, winning isn't everything, because karma is a bitch, so to speak.  In real life, we are still searching for definitive answers.

These results (and a whole lot more) are described in more detail in the publication, available via open access:

C. Adami and A. Hintze: Evolutionary instability of zero-determinant strategies demonstrates that winning is not everything: Nature Communications 4 (2013) 2193.