07.22.10
Posted in General Relativity, biology, evolution, fitness, measurement, ontology, philosophy, science at 6:39 pm by nogre
4.2 Relativistic Evolution
4.2.1 Two Kinds of Fitness
To understand Natural Selection we need to understand fitness and how to calculate its value. One way the fitness of an organism can be understood is in terms of how well it will be able to interact with its ecology to acquire what it needs to live and reproduce. The traits of the organism will be crucial as it struggles to survive: every little adaptation or edge that the organism has can be the difference between survival and death. Therefore the traits of the organism determine its fitness.
However, the fitness of an organism is dependent upon its environment. The different situations an organism finds itself in, which are determined by the ecology and chance, will determine its ability to reproduce. For example being fast is meaningless if there is no secure footing to run on. Therefore it is the situation that determines the traits that matter and hence fitness is a function of environmental selection.
At this point it can look as if there are two distinct and incompatible methods for calculating the fitness of an organism: trait based selection and environmental selection.
4.2.2 The Equality of Trait Selection and Environmental Selection
Imagine a jaguar out in the jungle. Unbeknownst to anyone, however, his welfare is being carefully monitored by stealthy scientists. Any time the jaguar might be in trouble, be it a lack of food or an unfriendly competitor, the scientists step in and protect the jaguar from harm and do it without being seen.
An independent observer, someone who doesn’t know about the scientists watching over the jaguar, might think that the jaguar has an uncanny ability to find food and avoid dangerous situations. He might suspect that the jaguar has excellent ears that can hear danger from very far away and a nose that can smell even the faintest waft of food. He would believe that in the struggle for survival the jaguar was incredibly well adapted.
Ought we to smile at the man and say that he errs in his conclusion? I do not believe we should. We could be in the very same position as the jaguar. We like to think that we have evolved the way we have by struggling and adapting. However, we may have just as easily been assisted by some benevolent but reclusive extraterrestrials. They could be the reason our species has been able to accomplish all that we have, and we would not know.
Regardless of the existence of any such extraterrestrials, the example shows that we cannot tell the difference between struggling and surviving based upon traits, and nature conforming (or disconforming) to our adaptations. It is a matter of perspective to believe either that our adaptations were the cause of our success or if it was the environment that happened to favor us.
4.2.3 The Natural Selection Field
Instead of switching back and forth between environmental and trait selection, we can say that both kinds of selection create a field. This field is ontologically as basic as the two kinds of selection and it is what interacts with the individual organisms and environment. The interactions of an organism and the field determines the course of the organism’s life, and an ecology’s total field is determined by everything in it.
Although every organism and each ecology is unique, none are alien. By looking at similar organisms and similar ecologies, we can use natural history to determine important adaptations and key environmental features. Taken together these features specify the shape of the Natural Selection field of that ecology, which informs us on how an organism or species will interact with their environment.
An organism’s overall fitness will determine how great its effect will be in the Natural Selection field. Introducing a species with high fitness into a new ecosystem can cause great changes, whereas introducing a species into an environment that it cannot survive in will barely create a change at all. For example, when humans, with our high fitness, move into a new area, we will profoundly alter that ecology. However, if we bring a flower with us that can’t survive the cold nights in our new home, then the flower will die, barely registering any change in the Natural Selection field.
4.2.4 General Relativistic Natural Selection
With the existence of the field we can say how evolution acts upon a species. At every moment an organism interacts with a natural selection field created by its surrounding ecology. The constant interaction with the field will gradually modify the species by benefiting certain individuals and by putting others at a disadvantage.
Insofar as the natural selection field is indistinguishable from the struggle for survival, we will not be able to further analyze why species change: this theory is terminal in the same way as General Relativity. If we could show that the way organisms and species benefitted or were put at a disadvantaged by the environment, without regard to the individual adaptations of the organisms, or conversely show how an adaptation increased an organism’s fitness without regard to the environment, then an investigation into these specific phenomena could yield insight into why a species changes. However, since we cannot make this distinction, the natural selection field is the final answer as to why a species changes.
Unlike the previous theory, general relativistic natural selection is wider because it is applicable during rapid ecological changes. The prior theory of natural selection relied upon trait based analysis to determine future reproductive success and hence was unable to accurately predict success during rapid change. Relativized natural selection can say that the organisms and species experiencing a disaster (or utopia) are experiencing a change in the natural selection field. This change in the natural selection field manifests as a rapid change in the lives of the organisms. Once the ecological change is finished, then we can revert back to the old notion of natural selection.
[this is an excerpt from a longer paper, which can be found here]
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07.13.10
Posted in General Relativity, Relativity, biology, epistemology, evolution, fitness, measurement, philosophy, physics, science at 5:18 pm by nogre
New theory of evolution! Hooray!
Patched a bunch of things together to make a nice story. Fixed the little issue about fitness being circular. Expanded natural selection to apply more generally. Causal structure. Epistemological foundations. ooOoOO0Ooooooo.
And it’s good fun. I swear. Epistemology, history of physics, evolution… makes me happy. You should really read it.
Download here. [pdf, 304kb]
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12.31.09
Posted in Relativity, biology, evolution, measurement, philosophy, physics, science at 4:48 pm by nogre
I have previously argued that the history of species must be treated like a evolutionary trajectory: we can only appreciate a species in a relative sense, just as we must evaluate physical trajectories relative to our own motion.
But what happens when we try to measure the very small in physics? We find there is a limit to the precision at which we can measure, as given by the uncertainty principle.
This suggests that there may be some similar limit when it comes to measuring small changes in species. The more we try to pin down exactly what a species is, the less sure we will be about its future and the more we measure the direction the species is heading, the less sure we will be about exactly what constitutes that species.
If genetic drift is just another way of saying that we cannot pin down the exact genetic make-up of a species then drift may be considered to be an instance of the uncertainty principle.
and HAPPY NEW YEAR
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03.20.09
Posted in General Relativity, biology, evolution, fitness, measurement, philosophy, physics, science at 1:21 pm by nogre
Fodor argued that the theory of evolution is not a legitimate theory of science because it is either vacuously true or wrong. He accused Darwin of committing the intentional fallacy. (synopsis here)
Insofar as he made no logical mistakes in his reasoning, we need a different strategy to defend the theory of evolution. In this post I will argue that his argument is an instance of gerneral underdetermination, and hence not a problem of evolution but of philosophy of science.
Underdetermination means that we can’t specifically identify the exact cause of scientific phenomena. For example, given some phenomenon, say darkness during the day, there can be many possible explanations: an eclipse, an exploding volcano shooting ash into the air, the sun has gone out, the electric company has blocked the sun to make more money, it was the work of Claw Vipers, etc. The exact cause of the darkness is underdetermined; sure we can research the problem and eliminate some of the possible explanations, but because of our limitations we will never be able to check everything. So the cause of the darkness can be said to be underdetermined, i.e. there is just not enough determining evidence.
Fodor argues that the theory of evolution is vacuous becuase given any trait we identify as benficial to the fitness of the organism is arbitrarily selected. Since there are too many factors to identify within an ecosystem or organism acting within that ecosystem, any hypothesis we propose about the fitness of that organism in that ecosystem will be trivially compatible with evolution.
For example assume there is an argument that having a certain trait, say longer legs, increases a zebra’s fitness. We can recognize that this argument could be unfounded because it might not be the longer legs but something else that increases the zebra’s fitness. It just happened that increased leg length was a harmless side affect of this truly beneficial trait. Either way, if it is the longer legs or some other unidentified trait, evolution is always compatible with our theories, and so it is trivially vacuously true.
In short I would say that he is arguing the cause of natural selection is underdetermined. The task is to identify whether this is a unique case of underdetermination or an instance of general underdetermination. I will now show that this sort of underdetermination can exist in physics*:
Imagine we are doing physics and we want to know which of two metal ingots is the more massive. We pull out our scale, place each object on one of the trays and wait for the scale to indicate which is the more massive.
Why does the scale tip in the direction of object A? We could argue that object A has a trait, it is composed of iron, and that trait makes it more massive than some other object. However, maybe object B is connected to a helium balloon. Maybe there is a gravitational anomaly in the location where we are doing our experiment. Maybe the iron is magnetized and there is another ingot with the opposite polarity under the table. Maybe a God is tampering with our experiment with a noodly appendage. Feel free to make up as many of these as you want. There any number of reasons why one object could tip the scale in its favor, and being more massive is among them, though selecting this as the reason is arbitrary.
(One of the things that is wrong here is that we don’t expect General Relativity to predict which objects are more massive. The mass of an object is the result of the history of its creation and ‘life’ up till the point we measure it. We do expect Relativity to suggest methods for testing such claims, which it does. Likewise Evolution should not be expected to predict which organism is fitter, but to suggest methods for testing fitness.)
If I now recast Fodor’s criticism into physical terms, in reference to the above thought experiment, this is the result: The theory of General Relativity (gravity) is vacuous because any given trait we identify as increasing the mass of an object is arbitrarily selected. Since there are too many factors to identify within a physical system, any hypothesis we propose about mass of the object in that physical system will be trivially compatible with General Relativity.
Therefore physics suffers from the same kind of underdetermination that Fodor accused of evolution. Anyone who persists in disbelieving evolution on these grounds should also deny General Relativity. Of course this is excessive: since the underdetermination criticism goes to the heart of our scientific theories in general, it is a problem of philosophy of science and not a problem of biology or physics specifically. Insofar as underdetermination remains an issue within the philosophy of science we still have to take it into consideration, but this should not be seen as a reason to think our current scientific theories are wrong.
.
[EDIT: I've put up a new analysis (24 March 2010) of Fodor's argument here: Hypotheses Natura Non Fingo]
See a continuation of the argument against Fodor in What Fodor Got Wrong, and in Fodor’s Intensional Criticism of Evolution.
.
* This is the argument I presented Fodor with during our brief conversation after his talk at CUNY. He tried to block it by saying that Natural Selection is statistical, whereas General Relativity is not. In my previous post, What Fodor Got Wrong, I argued that this position begs the question or is just wrong.
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02.22.09
Posted in Relativity, measurement, ontology, philosophy, physics, science, time at 11:17 pm by nogre
Measurement takes time; measurement is a process. So the measurement of time immediately yields this theoretical issue:
Since measurement takes time, our ability to break time into ever smaller pieces will always be proportional to the method of measurement used. The faster our measurement device that measures time, the more divisible time will be. Insofar as there are limits to how fast a measurement process can occur (relativistic or other), there will be limits on the lengths of time we can measure. From this perspective, time is discontinuous: there will be a point at which we can no longer split time into smaller pieces.
From a different perspective, time must be continuous: we can start our measurement of time whenever. Since there are no restrictions on when our measurement may begin, each and every instant must be just as good as every other instant, hence time is continuous.
So which is it: Is time continuous or discontinuous?
Or is the question badly formed? The discontinuity argument is based upon the ideas of measurement and relativity. The latter argument, for continuity, is based upon what might be considered a fact of modal reality. Perhaps the two arguments are not talking about the same thing.
I can’t give an end-all be-all answer to the questions of time, but here is my opinion: Time is continuous, but when we start to do scientific activities, time can and will only be able to be measured discretely. Therefore the two arguments are not using one word to describe two different phenomena.
The question then becomes how doing science limits what we can observe.
This might sound like an extremely unlikely situation, but consider the case of organized sports. When playing a sport or game you are bound, restricted, to following certain rules. However, by following these rules, you and the other players can demonstrate skills and abilities that you otherwise would not have been able to observe: Lots of people may be in shape, but only a small fraction of those people are professional athletes. Those athlete demonstrate their superior physical and mental prowess by performing on the game field by being restricted by the official rules.
Getting back to science, does it now seem so unlikely that we restrict ourselves in certain ways in order to accomplish other tasks? For time to be scientifically useful, we need to have some sort process that has a fixed point from which to start counting from, and a unit to count. Then we can compare an unknown process to this known process, and we have done so with much success.
This comparison could not have occurred without the introduction of an arbitrary fixed point and unit of measurement: by restricting our concept of time to these particular processes we enable ourselves to perform scientific research. Research is not possible if we use the unrestricted modal notion: no comparison can be made because there is no inter-modal process to compare a worldly (intra-modal) phenomenon to. But with the use of fixed points, units and processes, we also become subject to relativistic limitations. It seems like a very small price to pay considering the success of science.
To sum up: time is subject to modal considerations, which gives it special properties such as being continuous. Once we start to do science, though, we restrict ourselves to the non-modal aspects of time, which allows us to use it as a tool in scientific research. This also makes time appear to have different properties, but upon closer study, these properties are artifacts of the measurement process and not time itself.
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10.18.08
Posted in Relativity, measurement, science, time at 9:32 pm by nogre
There is something about time that I can’t seem to stop thinking about.
We measure time by agreeing upon an event and then counting from that point onward. Today is October 17, 2008 AD. It is this AD that keeps my attention. It has been 2008 years, ten months and seventeen days since the birth of Jesus of Nazareth: AD stands for Anno Domini, or year of our lord. Those not wanting to be explicitly Christian use CE, which stands for Common Era, which is just a nice way of saying the same thing without recognizing Jesus as the lord. Wikipedia dates the use of this term to 525 AD, though this is how everyone has been measuring time forever. AD began to be used in 525, but before that people just used other events (like natural disasters, battles worn or lost, etc.) as starting points to count the date from.
The only result is that time is not universal but relative to whenever people agree to start counting from. This is nothing new, but maybe like The Ring, if I pass it along, then it won’t bother me any more. If you become similarly afflicted, I apologize, but you know what to do.
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08.18.08
Posted in Relativity, Special Relativity, biology, epistemology, evolution, fitness, independence friendly logic, logic, measurement, mind, philosophy, physics, science, technology at 1:26 pm by nogre
I just returned from a cruise to Alaska. It is a wonderful, beautiful place. I zip-lined in a rain forest canopy, hiked above a glacier, kayaked coastal Canada and was pulled by sled-dogs. Anywho, as on many cruises, there was a casino, which is an excellent excuse for me to discuss probability.
What is probability and where does it come from? Definitions are easy enough to find. Google returns:
a measure of how likely it is that some event will occur; a number expressing the ratio of favorable cases to the whole number of cases possible …
So it’s a measure of likelihood. What’s likelihood? Google returns:
The probability of a specified outcome.
Awesome. So ‘probability as likelihood’ is non-explanatory. What about this ‘ratio of favorable cases to the whole number of cases possible’? I’m pretty wary about the word favorable. Let’s modify this definition to read:
a number expressing the ratio of certain cases to the whole number of cases possible.
Nor do I like ‘a number expressing…’ This refers to a particular probability, not probability at large, so let’s go back to using ‘measure’:
a measure of certain cases to the whole number of cases possible.
We need to be a bit more explicit about what we are measuring:
a measure of the frequency of certain cases to the whole number of cases possible.
OK. I think this isn’t that bad. When we flip a fair coin the probability is the frequency of landing on heads compared to the total cases possible, heads + tails, so 1 out of 2. Pretty good.
But notice the addition of the word fair. Where did it come from, what’s it doing there? Something is said to be fair if that thing shows no favoritism to any person or process. In terms of things that act randomly, this means that the thing acts in a consistently random way. Being consistently random means it is always random, not sometimes random and other times not random. This means that fairness has to do with the distribution of the instances of the cases we are studying. What governs this distribution?
In the case of of a coin, the shape of the coin and the conditions under which it is measured make all the difference in the distribution of heads and tails. The two sides, heads and tails, must be distinguishable, but the coin must be flipped in a way such that no one can know which side will land facing up. The shape of the coin, even with uniform mass distribution, cannot preclude this previous condition. Therefore the source of probability is the interdependence of physical conditions (shape and motion of the coin) and an epistemic notion (independence of knowledge of which side will land up). When the physical conditions and our knowledge of the conditions are dependent upon each other then the situation becomes probabilistic because the conditions preclude our knowing the exact outcome of the situation.
It is now time to recall that people cheat at gambling all the time. A trio of people in March 2004 used a computer and lasers to successfully predict the decaying orbit of a ball spinning on a roulette wheel (and walked out with £1.3 million). This indicates that after a certain point it is possible to predict the outcome of a coin flipping or a roulette ball spinning, so the dependence mentioned above is eventually broken. However this is only possible once the coin is flipping or the roulette ball is rolling, not before the person releases the roulette ball or flips the coin.
With the suggestion that it is the person that determines the outcome we can expand the physical-epistemic dependence to an physical-epistemic-performative one. If I know that I, nor anyone else, can predict the outcome until after I perform a task, then the knowledge of the outcome is dependent upon how I perform that task.
This makes sense because magicians and scam artists train themselves to be able to perform tasks like shuffling and dealing cards in ways that most of us think is random but are not. The rest of us believe that there is a dependence between the physical setup and the outcome that precludes knowing the results, but this is merely an illusion that is exploited.
What about instances in which special training or equipment is unavailable; can we guarantee everyone’s ability to measure the thing in question to be equal? We can: light. Anyone who can see at all sees light that is indistinguishable from the light everyone else sees: it has no haecceity.
This lack of distinguishability, lack of haecceity (thisness), is not merely a property of the photon but a physical characteristic of humans. We have no biology that can distinguish one photon from another of equivalent wavelength. To distinguish something we have to use a smaller feature of the thing to tell it apart from its compatriots. Since we cannot see anything smaller, this is impossible. Nor is there a technology that we could use to augment our abilities: for us to have a technology that would see something smaller than a photon would require us to know that the technology interacted at a deeper level with reality than photons do. But we cannot know that because we are physically limited to using the photon as our minimal measurement device. The act of sight is foundational: we cannot see anything smaller than a photon nor can anything smaller exist in our world.
The way we perceive photons will always be inherently distributed because of this too. We cannot uniquely identify a single photon, and hence we can’t come back and measure the properties of a photon we have previously studied. Therefore the best we will be able to accomplish when studying photons is to measure a group of photons and use a distribution of their properties, making photons inherently probabilistic. Since the act of seeing light is a biological feature of humans, we all have equal epistemological footing in this instance. This means that the epistemic dependence mentioned above can be ignored because it adds nothing to the current discussion. Therefore we can eliminate the epistemic notion from our above dependence, reducing it to a physical-performative interdependence.
Since it is a historical/ evolutionary accident that the photon is the smallest object we can perceive, the photon really is not fundamental to this discussion. Therefore, the interdependence of the physical properties of the smallest things we can perceive and our inherent inability to tell them apart is a source of probability in nature.
This is a source of natural randomness as well: once we know the probability of some property that we cannot measure directly, the lack of haecceity means that we will not be able to predict when we will measure an individual with said property. Therefore the order in which we measure the property will inherently be random. [Assume the contradiction: the order in which we measure the property is not random, but follows some pattern. Then there exists some underlying structure that governs the appearance of the property. However, since we are already at the limit of what can be measured, no such thing can exist. Hence the order in which we measure the property is random.]
————–
If I were Wittgenstein I might have said:
Consider a situation in which someone asks, “How much light could you see?” Perhaps a detective is asking a hostage about where he was held. But then the answer is, “I didn’t look.” —— And this would make no sense.
hmmmm…. I did really mean to get back to gambling.
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07.29.08
Posted in Relativity, independence friendly logic, logic, measurement, philosophy, physics, science at 8:40 pm by nogre
Ever have the experience of sitting in traffic and believe that you are moving in reverse, only to realize a second later that you were fooled by the vehicle next to you moving forward? You were sitting still, but because you saw something moving away, you mistakenly thought you started to move in the opposite direction.
Two different senses may be at work here: your sight and your balance. Lets assume that your balance did not play any role in this little experiment (you would have been moving too slowly to feel a jolt). Your sight told you that you were moving in a certain direction (backwards) because of something you saw, say a bus pulling forward. Then you saw something other than the bus, say the ground, and you realized that your initial appraisal of the situation was incorrect.
At the point when you look away from the bus, you believe that you are moving backwards. Then when you see the ground, you believe that you are not moving backwards. You reconcile these two contradictory beliefs by deciding that it was not you who were moving backwards but the bus that was moving forwards.
What this illustrates is that objects require something other than themselves to be considered in motion. Without the ability to reference a ‘stationary’ system (the ground), it is impossible to make a determination who is moving and who is staying still.
Now imagine this situation was taking place in a very gray place. The only things visible are yourself and the bus on a gray background. Then you notice that the bus is getting smaller. There is nothing for you to use as a reference (no stars, no ground, no nothing) to decide if it is you who is moving away from the bus or if it is the bus moving away from you, or both*. The only thing you have is the information that you and the bus are moving away from each other.
I refer to the statement that you and the bus are moving away from each other as information and not a belief because it is much more certain than what I called beliefs above, namely that you were in a certain kind of motion, which quickly turned out to be questionable.
The information that you and the bus are moving away from each other is not your everyday sort of information. It would be inaccurate to reduce this statement to a conjunction (you and the bus are moving), which is incorrect, or a disjunction (you or the bus is moving) because you are only moving with regard to the bus. By claiming that either you or the bus is moving, it makes it seem that the motion of one has nothing to do with the other. The motion of you and the bus need to be mutually dependent upon each other, and a mutual interdependence is not reducible.
If we return to the everyday, we can say that you have the information that you and the bus are moving away from each other and you and the bit of ground you are on are not moving away from each other. Since the bit of ground we initially selected was arbitrary (we could have chosen anything, like another bus) it is subject to the same issues as the bus; we merely take the ground to be stationary for most purposes, but this is a pragmatic concern. Hence all determinations of motion (or non-motion) are instances of informational interdependence.
The result that relativity is part of a larger class of mutually interdependent structures is non-trivial. Minimally this formalism will allow us to specify exactly when the use of relativity is warranted, but more importantly it will allow us to identify and provide insight into other situations of informational interdependence. Cases of mutual interdependence are relatively rare as far instances of logic go (they can’t even be described in first order logic) and having such a well studied example gives us a head start on this phenomenon.
—————————————-
* or if the bus is shrinking, or you are growing, or all of the above, but lets assume no Alice in Wonderland scenarios.
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03.30.08
Posted in game theory, independence friendly logic, logic, measurement, philosophy at 8:44 pm by nogre
The Monty Hall Problem illustrates an unusual phenomenon of changing probabilities based upon someone else’s knowledge. On the game-show Let’s Make a Deal the host, Monty Hall, asks the contestant to choose one of three possibilities – Door One, Two or Three – with one door leading to a prize and the other two leading to goats. After the contestant selects a door, another door is opened, one with a goat behind it. At this point the contestant is allowed to switch the previously selected door with the remaining (unopened) door.
Common intuition is that this choice does not present any advantage because the probability of selecting the correct door is set at 1/3 at the beginning. Each door has this 1 out of 3 chance of having a prize behind it, so changing which door you select has no effect on the outcome.
In hindsight, this intuition is wrong. If you initially selected the first goat and then switch when you get a chance, you win. If you selected the second goat and switch, you win. If you selected the prize and switch, you lose. Therefore if you switch, you win 2 out of 3, whereas if you do not switch you win only 1/3 of the time.
So what has gone horribly wrong here:
- Why is most everyone’s intuition faulty in this situation?
- How does switching doors make any difference?
- When did the 1/3 probability turn into a 2/3 probability?
At the beginning of the game you have a 2 out of 3 chance of losing. Likewise the game show has a 2 out of 3 chance of winning (not giving you a prize) at the beginning of the game. Both of these probabilities do not depend upon which door the prize is behind, but only upon the set-up of a prize behind only one of three doors. For instance, an outside service (not the game show) could have set everything up such that both you and the game show would be kept in the dark: there would still be 2 goats and a prize, but neither you nor the game show would know which door led to the prize.
Now imagine that it is the game show that is playing the game. The game show is trying to win by selecting a goat. From this perspective, whichever door that was chosen is good: this door has a 2 out of 3 probability of being a winner (being a goat). Therefore when given the opportunity to change (after the outside service opens a door and shows a goat), there is no reason to do so.
Of course you, the contestant, are the one making the selection, and you do not want a goat. However, if you imagined yourself in the position of the game show at the beginning, as trying to select a goat, you would reasonably assume that, just as the game show did, you were successful in choosing a goat. When given the choice to switch, now that the other goat has been removed, it seemingly makes sense to change your selection.
In this case the easiest way to view the situation is in terms of how to lose, or by considering all the possible outcomes (as mentioned above). Though this is a guess, it seems that our first blush reaction to this problem is always to view it in terms of winning and this is the reason we do not immediately recognize the benefit in switching. We start out with a 1/3 chance of winning and switching doors doesn’t immediately seem to increase this percentage.
To answer how switching doors makes a difference we need to look more closely at the doors. The door that was initially selected has a 1 out of 3 chance of being a prize, and this does not change. If you were to play many times and ignore changing doors, then you would win 33.3% of the time. At the outset the other two doors each have the exact same chance of being a winner, 1 out of 3. So the other two doors combined have a 2 out of 3 chance of containing a winning door.
Now the game show changes the number of doors available from 3 to 2, with one door guaranteed to contain a prize. If you were presented this situation without knowledge of the previous process, then you would rightly put the chance of selecting the prize at 1 out of 2, 50%.
However, you know something about the setup: The door that was initially selected had a probability of having a prize behind it set at 1 out of 3. The thing behind the other door, though, has been selected from a stacked deck: Whatever is behind the door was selected from a group of objects with a 2 out of 3 chance of containing a prize (1/3 + 1/3). You know that the odds on this door are stacked in your favor because the game show knowingly reveals the goat: In the 2/3 case in which you have previously selected a goat, the prize is behind one of the other two doors. When the game-show reveals (and removes) a goat, it guarantees that the prize is behind the last door. Therefore switching doors at the end is equivalent to combining and selecting the probability associated with the two doors not initially selected.
If the game show did not knowingly reveal the goat, you would not be able to take advantage of the stacked deck. Imagine that you select the first door and then another door is opened randomly, revealing a goat. By randomly eliminating this door (and not looking behind the unselected doors) the door that was initially selected becomes unrelated to the present choice: Only by looking behind the unselected doors does the initial selection become fixed in reference to the other doors. Since no one looked behind the doors, some bored, but not malicious, demon could have come and switched whatever was behind the selected and remaining door and neither you nor the game-show would be able to tell. Therefore switching doors when a goat is randomly revealed provides no advantage because the initial selection cannot be related to the probable location of the prize.
Only when the contestant can fix the probable locations of the prize because the location of the prize is known by the game-show, is it possible to assign interdependent probabilities on the location of the prize and the previous selection made. The odds are then tilted in the contestant’s favor by switching away from the low probability initial selection to the door that has the combination of remaining probabilities.
The logic of this needs to be represented game-theoretically with the different quantifiers representing different players of a game of incomplete information. The game would run
* like this:
Domain={prize, goat, goat}
| |
Contestant |
Game Show |
| 1. |
– |
∃x∃y∃z∀a/x,y,z∃b∀c/x,y,z(a=x & b=y & c=z) |
| 2. |
– |
∃y∃z∀a/x,y,z∃b∀c/x,y,z(a=g & b=y & c=z) |
| 3. |
– |
∃z∀a/x,y,z∃b∀c/x,y,z(a=g & b=g & c=z) |
| 4. |
∀a/x,y,z∃b∀c/x,y,z(a=g & b=g & c=p) |
– |
| 5. |
– |
∃b∀c/x,y,z(p=g & b=g & c=p) |
| 6. |
∀c/x,y,z(p=g & g=g & c=p) |
– |
| 7. |
∀d∀c/x,y,z(d=g & g=g & c=p) |
– |
| 8. |
∀c/x,y,z(g=g & g=g & c=p) |
– |
| 9. |
(g=g & g=g & p=p) |
– |
Line 1 is the initial setup of the prize game: the goal is for the contestant to make his or her placement of the prize and goats match the game show’s placement. Whatever is on the left side of an = will be what the contestant thinks is behind a door and what is on the right of an = will be what the game show puts behind the door, such that each = represents a door. If the formula is satisfied then the contestant will have successfully guessed the location of the prize.
Lines 2, 3 and 4 represent the results of the Game Show placing the prize and goats. Line 5 is the result of the first move of the contestant choosing where he or she thinks the prize is: the ‘a/x,y,z’ means that whatever placed in spot a has to be done independently, i.e. without knowledge, of what x or y or z is. Then the game show reveals a goat behind one of the doors not selected by the contestant. Line 7 represents the choice that is given to the contestant to switch his or her initial placement of where the prize is. Line 8 is the important step: since the contestant does not know what is behind the doors (c/x,y,z) it looks as if there is no advantage to switching. However, the contestant does know that when making a choice to reveal a goat in line 6 that at this point the game show had to know what was behind every door. This means that c is dependent upon b which was depended upon x, y, and z. With this knowledge the contestant can figure out that there is an advantage to switching because the selection of b in line 6 fixed the locations of the prize & goats and in doing so fixed the odds. Since the odds were intially stacked against the contestant, switching to the only remaining door flips the odds in the contestant’s favor, and is done so in this example. Line 9 shows that all the contestant’s choices match up with what the game show has placed behind the doors and hence she or he has won the prize.
* To do a better representation would require keeping the gameshow from not placing a prize anywhere by using a line like ‘x≠y or x≠z’. For graphical brevity I left it out.
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