Paradoxes and Dilemmas

This section is still in work and will likely remain so!

"There is a place where Contrarieties are equally True... ."
- William Blake (1757-1827) - from "Milton" Book the Second, Plate 30

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  A definition of paradox appropriate for this essay is that given by the Random House Unabridged Dictionary; 'any person, thing, or situation exhibiting an apparently contradictory nature'. A concept can appear to be a paradox due to our lack of understanding or the inadequacies of language. While such paradoxes may be resolved in time with better understanding, it is unlikely that the paradoxes mentioned here will be so easily resolved.

  • The Voting Paradox

    I should make it clear that what I am referring to here is the apparent paradox for the individual voting in a general election. The term, "Voter's Paradox" as often used in the scholarly Social Choice publications refers to a situation in which the personal preference ordering of three or more alternatives can result in a group ordering that is not transitive. I will describe this paradox further below. But my subject here is the "insignificant vote" situation in general elections.

    This paradox is quite representative of the general problem of the Social Dilemmas which I discuss here and has to do with the fact that an individual's vote has no significant impact on the outcome of an election. The Voter's Paradox seems to be characterized by two paradoxes, not one. The first is common to the Prisoner's Dilemma, described here. Briefly, the first paradox results from the individual following rational decisions. However, if everyone follows the path of rationality (defection), the result for the group is inferior to what would be achieved if everyone acted irrationally (i.e., cooperate).

    A second paradox results from the apparent insignificance on any particular individual's input. The actual voting situation best illustrates this paradox. In a national election, one vote will not make any difference in the result, yet the accumulation of all the individual votes does, in fact, decide the election. One vote will only impact the results if there is a tie which is incredibly unlikely in a national election. (But what if there was a tie and your vote was the tie breaker? For an amusing fictional account of this situation, see "RESPUBLICA" by GNN of the ezine, UXU).

  • Prisoner's Dilemma

    This famous dilemma is discussed in detail in my section on Social Dilemmas. The dilemma results from the fact that a rational choice by the participants in this non-zero-sum game results in an inferior outcome for all participants.

  • Newcomb's Paradox

    This paradox appears, at first glance, to be based on the concept of omniscience. If you do not believe in omniscience then the discussion may appear pointless. But the problem is deeper than that and seems to get right to the heart of what "rationality" is all about. And that is, we are not sure!

    The paradox is important in theory since some philosophers claim that Newcomb's Paradox and the Prisoner's Dilemma are essentially the same phenomenon. Details on Newcomb's Paradox are here.

  • The Unexpected Execution (Hanging) Paradox

    This paradox goes by a variety of names: "Unexpected Hanging", "Unexpected Examination", "Swedish Drill", and possibly more. Basically a prisoner is told that he will be hanged next week but the day of the hanging will be a surprise. The prisoner realizes that if he wakes up Saturday morning and finds himself not dead, then he can't be hanged that day because it would not be a surprise. By induction, he then eliminates Friday and so on for every day of the week. But come Wednesday he was hanged -- much to his surprise -- as the judge promised. I think the puzzle is important since it seems to have the same characteristics of certain aspects of the Social Dilemmas and the Vagueness Problem. In particular, it is often claimed that in an Iterated Prisoner's Dilemma game if the number of plays is finite then rational players must defect on the first play due to the same backwards inductive logic as used in the Unexpected Execution Paradox. I'm thinking that the Unexpected Execution Paradox creates some doubt in that method.

    An alternate formulation of this problem might help. Let us take this approach.

    Consider a stack of 7 playing cards, all of which are red except one which is black. It is your job to assemble the cards in a stack face down with the black one in some position. It is my job to turn the cards over one at a time until I get to the black one.

    Can you arrange the cards in the deck in such a way that at every position, I will not be able to deduce that the next card is a black one before I turn it over? That is, as I go through the stack, one at a time, I will not be able to correctly deduce that the next card is black.

    You cannot put it in the bottom, 7th, position, for I can certainly deduce that it is black if I get down to the last card and I haven't seen a black one. So that rules out the 7th position. There seems to be no doubt about that. (It would seem even that the 7th card is useless and we might as well play the game with 6, but I will let that pass.)

    What about the 6th position? Well when I get down to the 6th card, I can deduce that the it must be black since we have already eliminated the 7th position. So you can't use the 6th position either.

    Now, I say the 5th position has exactly the same problem. We have eliminated the 6th and 7th haven't we? So if I get to the 5th card can I not deduce that it is black?

    What do you think? Does this clarify the problem? Does it change the problem?

    I have extracted a fairly thorough summary of the various analyses of the Unexpected Execution Paradox from a puzzles archive formerly on the net and posted it here.

  • "Take it or Leave it" (or The Ultimatum game)

    (This game is discussed further, along with similar games over in "Social Dilemma Games and Puzzles".)

    The following situation is described in the book, Bargaining Games, by J. Keith Murnighan:

    You and an acquaintance, "Pat", are walking down the street when you meet an older couple with a bag of money. The older couple makes the following offer: We wish to give the two of you $100,000 if you can decide how it should be decided between the two of you in the next 3 minutes. You say, "So, Pat, what do you say? How about fifty thousand dollars each". To your dismay, Pat answers, "Gee, I'm really sorry, but my mother needs an expensive operation. So, I'll take eighty thousand dollars and you can have twenty thousand. I won't settle for anything less!"

    What do you do? Insist on an even split and get nothing or take the $20,000 and be happy?

    (Another version of this game is called the "Ultimatum Game", in which two people play, with one getting to chose how the gift is to be divided and the other gets to decide whether to accept the division on nothing. A description can be found in one of Jon Elster's essays called "Doing our level best". Unfortunately, this essay seems to be offline at this time.)

    I would appreciate hearing from you on this puzzle. I am particularly interested in knowing whether you would accept the division proposed by Pat.

  • The Voter's Paradox (Social Choice Version)

    This summary is based on the description of the Voter's Paradox provided in the book, Game Theory in the Social Sciences by Martin Shubic (MIT Press, 1982). Several sources are online that describe this paradox: Pat Gunning's "More on Voting and Elections", Lorrie Faith Cranor's "Declared-Strategy Voting: An Instrument", and Donald G. Saari's papers are the best I found.

    Three individuals have the following personal preference orderings for options A, B and C.

    (I will use the symbology, X > Y > Z to mean Y is preferred over Z and X is preferred over Y.)
    
    • Individual 1:
      A > B > C

    • Individual 2:
      C > A > B

    • Individual 3:
      B > C > A

      Now if these individuals were asked to make a group choice (majority vote) between A and B, they would chose A; if asked to make a group choice between B and C, they would chose B; if asked to make a group choice between C and A, they would chose C.

      So for the group A is preferred to B, B is preferred to C, but C is preferred to A! This is not transitive which certainly goes against what we would logically expect.

  • The Vagueness and Ambiguity Dilemmas

    This problem results from the apparent necessity of having to quantize many naturally smooth, continuous parameters of nature just to make it easier for the bureaucrats to do their jobs and to make it easier for other humans to settle (or avoid) disputes. Things that are by nature continuous -- such as age, driving speed, income, pain, racial make-up, performance, and alcohol blood content are broken up into discrete levels. Since the parameters in question are continuous, the "break-points" are purely arbitrary and cannot be defended. Therein lies much difficulty and the potential for much abuse. For more information on this subject, click here.

  • The "A Penny Saved is a Penny Earned" Dilemma

    We've all heard the old dictum that "A Penny Saved is a Penny Earned". Makes sense, but if it is true, we have some difficulties.

    Let us say you had a chance to buy a new MP3 music player on sale for $200 that normally costs $300. OK, you buy it and you have saved $100, so chalk up $100. But what if you later found out you could have bought if for $150 at another merchant? Now what? What if you didn't buy it at all, did you save $300?

    To get a handle on this, let us look at your year end balance sheet. If you went out an earned a hundred extra bucks and then paid $300 for the player, you would have at the end of the year, let us say X dollars. On the other hand, if you had not earned the $100 and bought the player for $200, then you would also have had X dollars at the end of the year. So, it appears that a savings of $100 is the same as and earning of an extra $100.

    But let us not just limit ourselves to this one item. What about all the other things that you elected not to buy at all. Now we are talking thousands, millions, BILLIONs that you saved. Were those earnings?

    We need to be careful here. Let us take a good example and examine it carefully. How about investment in stocks? Let us say I own 100 shares of IBM that I paid $100 a share for. It goes up $10 and then I sell. I have earned $1,000. Now consider an alternative scenario -- it went down $10. But just before it went down, I sold and then I repurchased after at $90. Have I earned anything? Some say no. Well what if it had gone down $10, and I had not sold, would I have lost a thousand dollars? Well, yes, you say. So, it seems to me that by selling high and buying back at $90, I have saved a loss of $1,000 and therefore have earned $1,000.

    But it gets messier! Suppose an insider calls me and says that stock is going to jump from $40 a share to $70 a share today. I have the necessary cash to buy 200 shares but decline to do so because it looks risky. At the end of the day, I notice the stock has in fact jumped to over $70 a share. Have I lost $6,000 (200*$30)? Surely I have for it impacts me in the same way as a loss would -- which is I can now buy less than I could have had I acted appropriately.

    But surely this can't be true for there are thousands of stocks I may have lost money on by not investing today.

    This disturbing problem is actually a representation of a more fundamental problem -- the problem of inaction. Do the results of inaction count the same as the results of action? If I see that my house is catching afire and I do nothing to stop it, am I responsible if it burns down? If I see that a person is about to be hit by an auto and I can easily pull her out of the way, but I don't, have I caused this person to be hurt? If I hear a person express a belief that I know to be untrue, is my remaining silent the same as lying?

    We know that law recognizes negligence and does hold individuals responsible for "not taking action". Surely it is reasonable to accept that no action can have results as good or as bad as a positive action.

    But if we accept that, we have a huge problem. I shudder to think about being responsible for all the bad things that happen that maybe, in some way, I could have prevented. It is too much of a burden!

  • The Ethics Paradox

    The great dispute in ethical theory as to whether ethics are relative or absolute results in logical difficulties for either choice.

    The problems with absolute ethics are many, chief of which is "what is the basis of the ethical rules?". Since we cannot derive these morals on a scientific, logical basis, we have to conclude that they are either religious based or simply a set of rules that the community agrees to.

    Religion as the basis for morality presents major problems in that not everyone who would like to be moral is ready to accept a religious life style. And even for those who would accept religion have to admit that there are other religions in the world and they don't all share the same ethical rule set.

    Of course if we concede that morals are just a set of rules that the community agrees to, we are admitting that morals are not absolute.

    Relative morality -- more commonly referred to as "moral relativism" or "ethical relativism" -- fares no better. Since the concept is relative, it is inherently vague and suffers from all the problems I outline in my essay on vagueness.

    Please go to my page, "Common Sense" for a continuation of this discussion.

  • The Paradox of Self-Amendment

    Constitutions and their implementation are a lot more complex than most people realize. While there are many complexities that would be very interesting, let us examine just one here now to illustrate the point.

    It is generally agreed that if a constitution is created, there needs to be a way to amend it. Since the Constitution is the highest law of the land, a clause for amending the Constitution must be within the Constitution itself. Article V of our Constitution defines how the Constitution may be amended.

    Now consider this: what if an amendment was proposed that modified Article 5 itself? It is apparent that having such a clause would allow for self suicide by the Constitution. For an amendment could be proposed that eliminated the contents of Article 5 altogether and replaced it with a statement that the first 10 Amendments were null and void! Not a happy situation, for now we have lost our freedom and the amendment process. There is an on-line book that discusses this paradox in detail, The Paradox of Self-Amendment.

  • The Bureaucrat's Dilemma

    We all like to pick on the bureaucrats. In particular, we find it most annoying that they make decisions on some arbitrary fixed criteria without using any judgment. For example, a family of four is poor in the USA if they make only $15,569. If they make $15,570 they are not poor! Doesn't matter where they live. We conclude that to make decisions that way, bureaucrats must be lazy or afraid of criticism.

    Another criticism we have of some bureaucrats is that they make decisions that seem arbitrary where there is vague criteria. My personnel manager rejected some of my travel expenses as being unreasonable. What is reasonable when you are stuck in a hick town, no TV worth watching, and the only relief from boredom is the Holiday Inn bar?

    These two criticisms define the Bureaucrat's Dilemma. Most conditions for making a decision in life are vague. When is a person poor? It is important to know because if you are "poor" you are eligible for lots of benefits from the government. Obviously there is no precise point in which a person really is poor if below and not poor if above. There are at least two complications. One is that being poor is a continuum like being bald or tall or rich. The other problem is that there are many other factors, such as what area of the country you are in, your health, access to provisions, etc., that would determine whether you really are poor. So setting a fixed point is obviously ridiculous.

    But the bureaucrat knows that she will be subjected to an even greater amount of hassling if she uses her on judgment in determining whether a person is poor or not. The customers will raise an incredible amount of hell when they are rejected based on a purely subjective judgment. A professor who assigns grades based on her subjective judgment will not last long before she is tied up in litigation. Instead, she must go through the charade of "testing" the students, assigning test scores and then determining the grade. The only problem is that the so-called "testing" is, for many subjects, is a purely subjective selection of a few questions that is highly unlikely to reflect whether this student will in fact be a good lawyer, let us say.

    The Bureaucrat's Dilemma is characterized, then, by the impossibility of assigning a logical and defensible breakpoint to a continuous function yet the breakpoint must be assigned to be fair to those who's lives are affected by the value of the breakpoint. See my essay on Vagueness and Ambiguity for further elaboration on the problem of breakpoints.

  • The Honest Politician's Dilemma

    If she is honest she cannot get into office and therefore can do no good.

  • The Two Envelopes Paradox

    I first discovered this paradox in Roy Sorensen's book, A Brief History of the Paradox, but according to the background given on Wikipedia, "Two Envelopes Problem", it was first reported in 1953 by the Belgian mathematician Maurice Kraitchik.

    Sorensen states the problem like this:

    You have a choice between two envelopes that contain money. You are allowed to look at one before you chose. You are told that one envelope contains twice as much as the other. You pick one (let's say A). You find $10. So, the other envelope (B ) must have $5 or $20. Do you keep this one or go select B? Well look at the expected value of the other one. It is (1/2 x $5) + (1/2 x $20) = $12.50. Hmm, looks like you need to pick B. But wait, what if you had picked it first? The same analysis would have caused you to conclude that A was the better one. How come?

    I leave it to you to consult the references for the various proposed solutions:
    1. Another Wiki article, The Exchange Paradox
    2. "The Two Envelopes Paradox" by Keith Devlin
    3. "The Two-Envelope Paradox: A Complete Analysis?" by David Chalmers
    4. A long, difficult to read article, "THE TWO-ENVELOPE PARADOX AND THE FOUNDATIONS OF RATIONAL DECISION THEORY" by Terry Horgan
    5. An easy to read essay, "Envelopes and Indifference" by Graham Priest and Greg Restall.

• Links to Other Sites with Related or Similar Material

  • An essay on an example of a self-referential paradox, The Berry Paradox.

  • "The paradox of the question" by Ned Markosian.
    What would you ask if the genie allowed you to ask any question -- but only one? (Also, see "On the Paradox of the Question" by Ted Sider.)

  • A variety of puzzles and paradoxes are here.

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Comments from you on these issues are very welcome. You may send email to leonf@perspicuity.net.

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