Game Theory: Part IX: Coordination and Anti-Coordination

In the last game, we assumed that the players are able to coordinate, in order to avoid a mutual mistake. But we cannot always confer with others before making decisions. And so, we must decide: how much is it worth to confer with someone else?

The following material is flagged Green Level. It is intended to reflect material that the author believes to be a matter of consensus among experts in the field. This belief may be incorrect, however; and as the author is not an expert and does not have an expert fact-checking the article, errors may creep in.

So, let us look at two classic games, the Coordination Game and the Anti-Coordination Game. These are very common; one example of the Coordination Game might be deciding which side of the road to drive on.
The Coordination Game:

	X	Y
X	(A,A)	(B,B)
Y	(B,B)	(A,A)

The Anti-Coordination Game:

	X	Y
X	(B,B)	(A,A)
Y	(A,A)	(B,B)

where in all cases A>B.

So, as you can see, in the coordination game, the players do better if they play the same move. In the anti-coordination game, the players do better if they play different moves. And in neither case is one move objectively better for one player than its alternative. So, how do the players resolve this?
The solution is obvious. The players must arrange ahead of time what move they will make.

(The players gain nothing from deviating from their arrangement, so they gain no benefit from the ability to make binding promises. In effect, any promise either player makes will punish them if they break it. This will be important later.)

Now, let us say that the players cannot get this for free. Maybe the players are playing via the postal service, and need to pay for stamps. How much should the players pay, maximum, for the ability to coordinate their moves? (As mentioned in Utility Functions, we are assuming that there is some conversion function between the payment and what the players are rewarded in. )

Let's look at what will happen if the players cannot coordinate. Since neither move is dominant, and in fact no move offers any advantage over the other, the players have no choice but to act randomly. Each move will be taken 50% of the time. (It occurs to me that perhaps I should explain how to calculate the odds with which you should decide your moves. Later. For now, just accept that the moves should be played with even odds.) So, the value of this game to each player is:

A/2+B/2.

So what is the value of the game where the players are allowed to communicate? Well, that should be obvious. Neither player stands to gain anything by going against their agreement, so the value of that game is A. So, the value of the ability to communicate is, in this case, A-(A/2+B/2)=A-A/2-B/2=A/2-B/2.

<<Metagames: The Battle of the Sexes | Game Theory | Metagames: Extortion>>

Evolution: Part VI: Cooperation III: Kin Selection

In a number of species, humans included, individuals will often aid others, at their own detriment, when there is no possibility of reciprocation. In extreme examples, this can even extend to total self-sacrifice. How can evolution possibly explain this?

The following material is flagged Green Level. It is intended to reflect material that the author believes to be a matter of consensus among experts in the field. This belief may be incorrect, however; and as the author is not an expert and does not have an expert fact-checking the article, errors may creep in.

It is time we moved beyond the idea of evolution as purely selfish. It is time we looked at it in terms of genes, and how they move through populations. What matters is not the survival of the self, but rather the survival of as much of the self's genome as possible. This is why reproduction happens: in order to preserve the genome.

So, let's look at a population. We will think of this population as a flock of birds. When a predator approaches, a bird has two options: it can sneak away, making its own survival more likely; or it can call out, letting the rest of the flock know a predator approaches but dooming itself. Why would a bird ever cry out?

From a perspective saying that evolution is purely selfish, this would never happen. But it happens all the same, so that must not be strictly true.

But remember: one of the core ideas that absolutely must be true for evolution to work is that descendants must share traits with ancestors, and vice versa. And likeliness to alert the flock at the expense of oneself is a trait. If this trait is inherited, than any given bird's offspring are more or less likely to have it, and whether this is "more" or "less" depends on the parent itself.

So let's look at the probabilities. The "cry out" gene is known to be possessed by this one bird. That bird's parents, offspring, and full siblings each have a one-half chance of having it. Grandparents, grandchildren, aunts, uncles, nephews, nieces, first cousins, and half-siblings have a one-fourth chance. Great-grandparents, great-grandchildren, great-aunts, great-uncles, great-nephews, great-nieces, half-aunts, half-uncles, half-nephews, half-nieces, first-half-cousins, second cousins, and first-cousins-once-removed (whew!) have a one-eighth chance. And so on through the entire flock.

Parenthetical Note:
Anyone familiar with the work of Gregor Mendel will realize that I am absolutely butchering it. For instance, these exact probabilities only work with genomes that have exactly one copy of each trait. If the trait is a dominant and this is a diploid species, as most birds are, the odds of any parent having at least one copy are slightly better than one in two. If it's a recessive, the odds of any parent having at least one rises to one in one. So these exact numbers work for some strange hypothetical haploid species where each individual has two parents, but not for birds or peas, and especially not for bacteria or potatoes. And for that matter, the statistics are leaving out mutations, because the odds of an individual's germ-line mutations affecting a specific gene are pretty small. As in 1 in 100,000, or thereabouts. But it's still pretty close to correct. It's a heck of a lot more accurate than the "electrons are particles whizzing very fast around a nucleus, like planets around a star" Lie To Children in chemistry, or the "centrifugal and Coriolis forces are illusions" one in physics, and those can still make useful predictions. So yeah.

Also, anyone familiar with Bayes' Theorem will notice that it's getting the same treatment. The listed probabilities don't take the background incidence of the trait into consideration. If something's universal, the odds of it being in a given member of the species go way up.

We now return you to your regularly scheduled Topic.

 So, the odds are pretty good that, if you take half of one bird's surviving parents, and half of its offspring, and half of its siblings, and a quarter of its slightly more distant relatives, and an eighth of its extended family, and so on, they add up to more than one bird. So, as far as the gene is concerned, there is more of it outside this bird than inside it, and sacrificing less than half of all existing copies leaves it better off than sacrificing more than half.

This leads to an interesting idea that has shown up in evolutionary thought that, oddly, itself keeps evolving, called group selection. Essentially, this is the idea that, just as individuals compete, so do groups of individuals. To put it in a human perspective: individuals compete, but so do their clans, tribes, nations, and treaty organizations, and cooperation between the individuals of a group leads to more effective functioning as a group. The idea was discredited in its earliest stages (since the death of all individuals leads to the extinction of the group), but the modern and widely-accepted gene-centric view reaches conclusions that are shockingly similar.

Nothing in here is to say, of course, that evolution, competition, and outright conflict never happen between relatives. But it does point to a "no one hits my brother but me" type of attitude between them: that is, they will still compete, but will stand together against an outside threat.

<<Cooperation II: Tit-For-Tat|Evolution|Common Criticisms>>

Evolution: Part V: Cooperation II: Tit-Fot-Tat

We have seen that sometimes, organisms cooperate because, for each one, it is better than the alternative. But this is not always the case. Sometimes, two organisms are placed in direct competition.

The following material is flagged Green Level. It is intended to reflect material that the author believes to be a matter of consensus among experts in the field. This belief may be incorrect, however; and as the author is not an expert and does not have an expert fact-checking the article, errors may creep in.

So, last time we looked at a population in which cooperation was an undeniable advantage. But what about most other forms of cooperation listed in the article last time? What about times when not cooperating is better for the individual than cooperating?

Let's look at how that works.

• All other things being equal, it is better for the individual if they do not cooperate.
• All other things being equal, it is better for the individual if others cooperate with them.
• All other things being equal, it is better for the individual if all individuals cooperate than no individuals cooperate.
• If it is possible for individuals to meet more than once, it is better for each for all to cooperate twice than for each to cooperate once and be cooperated with once.

That's right, those of you who were following the game theory Topic. This is the Prisoner's Dilemma. And, if you recall, we have already shown that, in the case where the Prisoner's Dilemma is repeated and no player knows when the final iteration will be, it is to the advantage of each player to cooperate until their opponent fails to cooperate, and then respond in kind.

So what about when each player knows when the game will end?  Well, the behavior described there is consistent with what is observed in nature. If two organisms are cooperating, they will tend to break off their cooperation as soon as one is likely to have been mortally injured.

So, if a population is full of sociopathic individuals, how does this arise? Let's look at how the Tit-For-Tat strategy handles such a population.
In the first generation, Tit-For-Tat plays cooperate on the first turn, and defect on each turn afterward. Thus, Tit-For-Tat is at a marginal disadvantage. But, the selective pressure against a trait at a disadvantage is proportional to how much of a disadvantage it is. Which, in this case, is "not that much", so there's a decent chance that Tit-For-Tat makes it into the next generation.
In the second generation, it is possible for two individuals playing Tit-For-Tat to meet. In this case, these individuals will play cooperate on one another, and react to other individuals as described above. But since each is meeting an individual playing cooperate, the selective pressure is lessened.
In other words, the selective pressure against Tit-For-Tat is inversely proportional to (a strictly-increasing function of) the number of individuals already playing Tit-For-Tat.

<<Cooperation I: Bootstrapped Cooperation|Evolution|Cooperation III: Kin Selection>>

Evolution: Part IV: Cooperation I: Bootstrapped Cooperation

So, if evolution means that only the fittest survive; that passing on one's genes is impossible unless those genes make one claw one's way to the top of the heap, why is it that living things can work together? Why is it that a cell can have multiple parts (each descended from a separate living thing) that work together instead of tearing itself apart that way? Why do wolves work together to bring down their prey, instead of each going it alone and letting its competitors for food and mates get skewered by moose? Why is it that humans have an instinct to help one another in times of need, instead of (Objectivists and other sociopaths aside) saying "Screw you, I got mine"?
This is something of a complex question with multiple answers, each completely true and therefore none the complete truth.

The following material is flagged Green Level. It is intended to reflect material that the author believes to be a matter of consensus among experts in the field. This belief may be incorrect, however; and as the author is not an expert and does not have an expert fact-checking the article, errors may creep in.

 First, let us consider a simple organism. Let us consider the simplest of organisms, a single cell. Each of these cells can either photosynthesize and eat nutrients (say, sulfates) that bubble up from a volcanic vent far below.
So, all of these cells are coexisting happily. Well, not really. There's a limited amount of light, and a limited amount of nutrients, so the cells are competing for these resources, and frequently a cell is outcompeted and starves to death. (Not to mention that, lacking nervous systems, it isn't exactly easy for a cell to be happy anyway.)
Now, let us suppose that the area covered by the cells increases.
So, it seems as though nothing would happen here, right?
Not quite.
In the future, suppose that the descendants of these cells are neighbors. In the interface between the areas occupied by the two strains, interactions between the cells show up. Sometimes, the two strains fight one another. Sometimes, they ignore one another.
And, once in a great while, they cooperate. Sometimes, the two cells will grow together, starting to share resources. Sometimes, the cells will join, sharing resources so that when one does poorly, the other covers its needs.
Of course, in the environment described, the joined-cell pair must still compete with single cells. Its resource requirements are twice those of a single cell, but so is its resource production.
Usually, anyway. Suppose that the environment is not constant. Sometimes, a cloudy day means that there is less light. Sometimes, changes in the volcanic vent's output change the nutrients available. So, sometimes the ratio of light to nutrients changes. Sometimes, light-specialized cells outcompete nutrient-specialized cells. Sometimes, the opposite happens. But which is the joined-cell pair?
Both, obviously. In a completely dark area, the pair will do half as well as a nutrient-focused cell, and in nutrient-less water, the pair will do half as well as a light-focused cell. But, in each case, it will do infinitely better than a cell focused on what is absent. So, the cell pair is better able to handle changing environments. So, with the changing environment, eventually the cell-pairs outcompete the single cells.
We can even apply this (we can apply it better, in fact) using the assumption that in the beginning, each cell is able to handle both resources. In any population of cells, there will be mutations. In this case, some of the mutants are better able to handle one resource than the general population. So, sometimes one mutant will join to a mutant able to handle a different resource better than itself. And remember, each cell pair must compete with single cells.
Now, suppose that there is a limit on "safe resource absorption". There is a certain level at which more or better resource absorption "crowds out" some other mechanism. For instance, suppose that each type of resource-absorber takes up a certain amount of space in the cell, and a better absorber takes more space in the cell. In a normal cell, one absorber eventually collides with another, so past a certain point, the cell must "decide" between not increasing its efficiency or making itself more vulnerable to changes in its environment.
A cell-pair, though, need not make that decision. Since there are two cells, each can specialize in a different resource.

(Now, on to blog stuff. Unless I get comments telling me to continue evolution, next week it's back to game theory. Depending on what you tell me I ought to be doing, I'll be alternating between Topics each week.)

<<Speciation|Evolution|Cooperation II: Tit-For-Tat>>