Paradoxes and Dilemmas

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

"The Paradox frog of the Amazon starts out as a 9-inch tadpole and ends up as a 3-inch frog. That's life." L. M. Boyd (for more see his column on the Web)

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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 on 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, at 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)

    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".)

    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)

    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 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 fares no better.

  • 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.

• Links to Other Sites with Related or Similar Material

  • An essay on an example of a self-reverential 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?

  • 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 [email protected].

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