Bertrand's Box Paradox

The Setup: There are three boxes: a box containing two gold coins, a box with two silver coins, and a box with one of each.

The Problem: After choosing a box at random and withdrawing one coin at random that happens to be a gold coin, what is the probability the remaining coin is gold?

The Solution: The correct answer is two-thirds. It may seem that the probability that the remaining coin is gold has a probability of 1⁄2; in fact, the probability is actually 2⁄3. This is because of the six coins on the table, there are only three that could have been drawn. We know all three are gold and of those three, two are in the box with another gold coin. The coin seen is equally likely to be any of the three gold coins, only one of which is opposite a silver coin

Back to the Future Time Travel

As Marty McFly explores the world of 1955, he is confronted with the fact that the history of his parent's life is changing before his eyes, and his own existence is in jeopardy. He has interfered with the meeting of his parents, and must correct the situation before it's too late, or he will cease to exist. Back to the Future is essentially the classic grandfather paradox: a man goes back in time and undoes his own birth. However, Back to the Future's theory of time travel suggests a different resolution than a never-ending loop.

Like the Terminator series, Back to the Future insists there can only be one timeline and rejects a many worlds interpretation. Both series also allow 'temporal refugees,' visitors from possible future timelines who by virtue of their presence in the past, nullify their own creations. In T2 we have the T800 that comes from a world where Cyberdyne created Skynet, In BttF we have Marty. The difference between the two theories of time travel, however, is that Marty immeadiately feels the consequnces of his temporal mischief and begins to dissappear whereas the terminator persists even after destroying Cyberdyne.

Unlike the traditional grandfather paradox solution, where a time traveler who undoes their own existence thereby nullifies their own trip, in Back to the Future, the consequences of Marty's time travel aren't felt until after he meddles with his parents meeting. Marty exists long enough to mess up the past, but its not until the moment of divergence between his past and the new past that his continued existence is put in jeopardy. Of course, Marty is given a chance to redeem his timeline by getting his parents back together before he blinks out of existance (imagine his increasingly vanishing presence as representing the diminishing probability of his conception). This is what allows for the suspense in the Back to the Future movies, Marty could concievably nullfy his own existance without nullifying his trip altogether--the universe will only reconcile his anarchorismic presence after the point he meddled with his parent's meeting.

In the Back to the Future theory of time travel, a grandfather paradox would be resolved like this: one day, Papy Joe would be walking home, when someone who sorta looks like him pops out of thin air, murders him and then proceeds to fade out of existance. In a Terminator universe, the assailiant wouldn't dissappear but could linger around the timeline indefinetly, even though his origin had been nullified. If Terminator gives refugees a passport, Back to the Future only gives them a travel visa.

Terminator Time Travel

In the Terminator films, Skynet, a computer program that controls nearly the whole world in the future, sends a machine to the past in order to kill John Connor, the future leader of the human resistance, at different points of his life: once before he is conceived (by killing his mother, Sarah Connor). In The Terminator, the machines send the T-101 and the humans send Kyle Reese: the first will give the people in the past the necessary components that will end up starting the Skynet project, and Kyle will be John Connor's father (that is, if the time travel hadn't happened then Skynet wouldn't have been created and John Connor wouldn't have been born).

A predestination paradox (also called causal loop, causality loop, and (less frequently) closed loop or closed time loop) is a paradox of time travel. It exists when a time traveler is caught in a loop of events that "predestines" or "predates" them to travel back in time. Because of the possibility of influencing the past while time traveling, one way of explaining why history does not change is by saying that whatever has happened must happen. A time traveler attempting to alter the past in this model, intentionally or not, would only be fulfilling their role in creating history as we know it, not changing it. Or that the time-traveler's personal knowledge of history already includes their future travels to their own experience of the past.

Terminator 2: Judgment Day, commonly abbreviated as T2, is set ten years after the events of The Terminator, it follows Sarah Connor, her 10-year-old son John, and a reprogrammed Terminator from the future as they defend themselves from a T-1000 and attempt to prevent Judgment Day, a future event in which machines will begin to exterminate humanity. The surviving arm and CPU chip of the original Terminator was analyzed and found that the technology was so advanced, they (humans) would have never invented the technology themselves and was used to create Skynet in the first place, suggesting a second predestination paradox.

Terminator 3: Rise of the Machines does somewhat negate any paradox by showing that the American military have secretly developed the technology independently, suggesting that Skynet was always developed this way and not from the arm of the T-101.

The predestination paradox of the Terminator movies neatly resolve the grandfather paradox by suggesting that time lines inevitably collapse into one sustainable sequence. Unlike a grandfather paradox, where each event negates the other, in a predestination paradox each event is dependent on the other. This avoids the logical pitfalls of the grandfather paradox because once both events occur, the timeline is complete.

An interesting aspect of the "loop" is that the initial events that gave rise to the loop need not be in the final loop. For instance, maybe the original Terminator timeline had no John Connor at all--instead Earth's savior was a man named "Ron Steadman" and initially the Terminators were sent back to kill him. The humans send Kyle Reese back to save Ron, and along the way he impregnates Sarah Connor and she gives birth to John Connor--who now becomes a greater hero than Ron and the target of Skynet's attack in the new timeline. Thus, Kyle Reese is now sent back to save John instead of Ron (who is forgotton) and the 'movie timeline' is born despite its seemingly trigerless series of events.

Another interesting possibility is that somethings have to happen--like the birth of Skynet. Since Skynet can manipulate time, it can send parts of itself back in time and thus will its own creation into a past where it otherwise would not exist. The terrifying implication of this is that if Skynet exists in one timeline it can put itself in all timelines; making it a truly unbeatable opponent. Alternatively, T3 suggests that Skynet must exist not because it sent itself back in time but because its creation is a necessary product of human nature--an invention as inevitable as the wheel. Either way, The Terminator series suggests a creative solution to the grandfather paradox: a 'guided' time travel where some events must or must not occur to create a cohesive timeline.

Welcome to Time Travel Week

Check in everyday this week for new posts on time travel and its perils...

Time Travel 101
Physicists take for granted that if one were to move away from the Earth at relativistic velocities and return, more time would have passed on Earth than for the traveler, so in this sense it is accepted that relativity allows "travel into the future." Any theory which would allow time travel into the past would require that issues of causality be resolved. The classic example of a problem involving causality is the "grandfather paradox."

The grandfather paradox is this: suppose a man traveled back in time and killed his biological grandfather before the latter met the traveler's grandmother. As a result, one of the traveler's parents (and by extension the traveler himself) would never have been conceived. This would imply that he could not have traveled back in time after all, which in turn implies the grandfather would still be alive, and the traveler would have been conceived allowing him to travel back in time and kill his grandfather Thus each possibility seems to imply its own negation a type of logical paradox.

Check back tomorrow for more time travel.

Schrödinger's Cat

This is a follow up to the Double Slit experiment

Schrödinger's cat is a thought experiment devised by Austrian physicist Erwin Schrödinger in 1935. It illustrates what he saw as the problem of quantum mechanics being applied to everyday objects. The thought experiment presents a cat that might be alive or dead, depending on an earlier random event.

Schrödinger wrote: "One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts."

The purpose of the thought experiment is to illustrate an apparent paradox: our intuition says that no observer can be in a mixture of states, yet the cat, it seems from the thought experiment, can be such a mixture. Is the cat required to be an observer, or does its existence in a single well-defined classical state require another external observer?

The Dollar Auction: Irrational Escalation of Commitment

The dollar auction is a game designed by economist Martin Shubik to illustrate a paradox brought about by traditional rational choice theory in which players with perfect information are compelled to make irrational decisions.

The setup involves an auction for a one dollar bill with the following rule: the dollar goes to the highest bidder, who pays the amount he bids. The second-highest bidder also must pay the highest amount that he bid, but gets nothing in return. The second highest bidder might not have to pay on eBay, but in many real contests, both sides end up paying but only one gets the prize--like lawsuits, sports competitions, gambling and political campaigns.

Bidding when the price is below fifty cents or so seems harmless because it’s an obvious deal to buy a dollar for any amount less. The twist becomes clear about when the high bid is 80 cents. People start to think about how the second rule–the one requiring the loser to pay–would affect incentives. What might the second highest bidder think at this stage? He is offering 70 cents but being outbid. There are two choices he could make:

• do nothing and lose 70 cents if the auction ends
• bid up to 90 cents, and if the auction ends, win the dollar, and profit 10 cents

But this action has an effect on the person bidding 80 cent, who is now the second highest bidder. This person will now make a similar calculation. He can either do nothing and lose 80 cents if the auction ends, or he can raise the bid to a dollar and have a chance of breaking even. Again, bidding higher makes sense. Thinking more generally, it always make sense for the second highest bidder to increase the bid.

Soon people will bid more than one dollar and fight over who will lose less money. It is the incentives that dictate this weird outcome. Consider an example when the highest bid is $1.50. Since the high bid is above the prize of $1, it is clear no new bidder will enter. Hence, the second bidder faces the two choices of doing nothing and losing $1.40, or raising the bid to $1.60 to lose only 60 cents if the auction ends.

In this case, it makes just as much sense to limit loss as it does to seek profit. The second highest bidder will raise the bid. In turn, the other bidder will perform a similar calculation and again raise the top bid. This bidding war can theoretically continue indefinitely. In practical situations, it ends when someone chooses to fold. This game is played at Stanford in economics classes, and its not uncommon to see the game end anywhere between five and ten dollars.

Here are some other real life examples of the irrational escalation of commitment:

• After a heated and aggressive bidding war, Robert Campeau ended up buying Bloomingdale's for an estimated 600 million dollars more than it was worth. The Wall Street Journal noted that "we're not dealing in price anymore but egos." Campeau was forced to declare bankruptcy soon afterwards.
• Supporters of the Iraq War have used the casualties of the conflict in Iraq since 2003 to justify years of further military commitment. This rationale was also used during the sixteen-year Vietnam War, another military example of the logical fallacy.
• Two competing brands often end up spending money on advertising wars without either increasing market share in a significant manner. Though the most commonly cited examples of this are Maxwell House and Folgers in the early 1990s, this has also been seen between Coke and Pepsi, and Kodak and Polaroid.
  
• Shakespeare's Macbeth comments, "I am in blood stepped in so far that, should I wade no more, returning were as tedious as go o'er." The metaphor represents Macbeth's crimes and rather than stop committing crimes (presumably, for fear of damnation) Macbeth says that he has "passed the point of no return" and might as well continue, even though it will inevitably lead to his downfall.
  

The Two Envelopes Paradox

The Setup:
You are given two indistinguishable envelopes, each of which contains a positive sum of money. One envelope contains twice as much as the other (i.e., ten dollars and twenty dollars). The player may select one envelope and keep whatever amount it contains, but upon selection, is offered the possibility to take the other envelope instead.

The Problem:
I open up enevelope #1 and it contains ten dollars. There is a 50% chance the other envelope contains twenty dollars, 50% it contains five. The average value of the other envelope is twelve and a half dollars ((20 + 5) / 2) so it is always worth it to swap--I don't even have to open my envelope to know this. In fact, if I don't open my envelopes its always +ev (expected value) to keep swapping envelopes back and forth forever.

The logician Raymond Smullyan expressed the problem in a way which doesn't involve probabilities. The following plainly logical arguments lead to conflicting conclusions:
1. Let the amount in the envelope chosen by the player be A. By swapping, the player may gain A or lose A/2. So the potential gain is strictly greater than the potential loss.
2. Let the amounts in the envelopes be Y and 2Y. Now by swapping, the player may gain Y or lose Y. So the potential gain is equal to the potential loss.

Status: unsolved.