The Impossibility of Democracy I: Condorcet’s Paradox

I recently noted that political scientists tend to argue that democracy is impossible without political parties, and expressed my doubt about the empirical truth of that statement. As a follow-up, here’s one theoretical problem with democracy, Condorcet’s Paradox, and an explanation of how parties help resolve it. In the spirit of full disclosure, while I studied the type of positive political theory discussed here, I never studied parties and elections, so I’m not actually certain that this is in fact what most political scientists think of when they talk about parties being necessary for democracy. But I won’t let that stop me, because the paradox itself is pretty cool.

Imagine three friends trying to decide what kind of joint entertainment they’re going to engage in. One wants to go to (A) a football game, would settle for going to (B) the park, but really prefers not to go to (C) the museum. Another prefers going to (B) the park, would accept going to (C) the museum, but hates (A) football. The third loves (C) the museum, likes (A) football, and loathes (B) the park. To make this clear, we can list their preference orders in a simple grid.

Friend 1 Friend 2 Friend 2
A B C
B C A
C A B

Naively, the friends decide to vote on what to do, and end up with a three way tie.  So they try again, and pit each alternative against each other alternative, first A-B, then B-C, then C-A.  But as it turns out, A beats B 2-1, B beats C 2-1, and C beats A 2-1.  There is no clear winner.

This is crucially important because it reveals the important distinction between an individual and a group.  An individual  can have a coherent, transitive, preference order.  You learned about transitivity in elementary school, but may have forgotten if you’re not a math person.  It simply means that if A > B, and B > C, then A > C.  Or in other words, if I like pizza more than hamburgers, and hamburgers more than hot dogs, I should like pizza more than hot dogs. Imagine the poor person with an intransitive preference order.  He’d starve to death while thinking to himself, “Pizza’s better than hamburgers, I should get a pizza. But hot dogs are better than pizza, so I should get a hot dog. But hamburgers are better than hot dogs, so I should get a hot dog. But pizza’s better than hamburgers, so I should get a pizza.”  It’s impossible to engage in rational decision-making if you have an intransitive preference order (the best thing to do then is select randomly). We would, rightly, say this person has a serious problem.

So individuals are assumed to have intransitive* preference orders, but as the example shows, that doesn’t necessarily mean that a group has a transitive preference order.  Our three friends are stuck mumbling, “football beats the park, which beats the museum, which beats football, which beats the park, which beats the museum, which beats football….” Their best option is to choose by playing rock, paper, scissors–i.e., randomly.  We call that round-and-round problem “cycling.”

This is bad news for democracy, because democracy is supposed to tell us what the public’s preference is.  We always hear the talking heads pompously proclaim, “the public has chosen…” and every politician tries to tell us “the public wants X.”  But what this shows is that we can’t say that the public has a true preference order–the public’s preference order is not necessarily transitive, so we can’t say that it has a top choice.  What it selected could be just a matter of which two alternatives happened to be placed before it.  It was given a choice of football and the park, so it chose football, but if it was given a choice of the park and the museum it would have chosen the park, and if ti was given a choice of the museum and football, it would have chosen the museum.  The public’s electoral choice could  cycle, and our policy outcomes could mutate endlessly.

So how realistic is this?  After all, the grid above reveals a carefully stylized model, and perhaps in real life nobody really likes the museum.  Here’s the bad news.  That’s the simplest model of cycling possible, 3 people with 3 options, and the larger either of the numbers becomes, the greater the potential for cycling.  (One reason I’m not a big fan of very multi-party systems–think Italy post WWII.)  In a large public with a large number of choices, cycling is probably an inevitable outcome, making democratic politics wholly irrational.

This is where parties come in. By proposing certain policies and pushing others off the agenda, parties help eliminate–or at least mitigate–cycling. This agenda-setting function is hugely important. If you know what the alternatives are, and know what other people’s preferences are, you can sometimes get to your own preferred outcome by manipulating the agenda.

In our example above, let’s say Friend 1 is cleverer than the other two. He proposes that they pit two options against each other, then pit the winner against the third. Hey, that sounds fair! So Friend 1, the football fan, proposes that first they vote between options B (the park) and C (the museum), with the winner facing off against A (football). He knows the park will beat the museum, and he know his preference, football, will beat the park. He has manipulated the agenda to resolve the cycling problem, in favor of his own preference. What seemed, on the surface, to be fair, may not have been fair at all. It’s certainly not what we think we want from democracy. But that’s what makes democracy actually functional.

In my next post on this subject, discussing Arrow’s Impossibility Theorem, I’ll show why your intuition that this solution isn’t quite right is correct. It violates one of the basic principles of democracy. But disturbingly, democracy can’t fulfill its own basic principles while successfully aggregating preferences into a group preference. We’ll save that one for next week, though.

_________________________
* Post-hoc correction.

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About J@m3z Aitch

J@m3z Aitch is a two-bit college professor who'd rather be canoeing.
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45 Responses to The Impossibility of Democracy I: Condorcet’s Paradox

  1. thoreau says:

    It is easy to show that as long as everybody evaluates all of the candidates along the same left-right spectrum, there will not be a Condorcet paradox. I think this result is called Black’s Single Peakedness. (I might be misremembering.) However, the second we step beyond left vs. right, and go 2D, paradoxes. If, say, social issues and economic issues are on separate axes, and so stances are points in a plane rather than positions along a line, you get paradoxes.

    Suppose candidate A is fiscally conservative and socially liberal, candidate B is fiscally liberal and socially liberal, and candidate C is fiscally liberal and socially conservative (e.g. a Catholic candidate) then it is rational for one person to prefer A>B>C (e.g. a libertarian who values social freedoms over economic freedoms), another to prefer B>C>A (e.g. a liberal whose paramount concern is economic), and another to prefer C>A>B (e.g. a social and economic conservative who values social issues over economic issues, but would take a libertarian over a liberal because at least the libertarian agrees with him on something while the liberal agrees with him on nothing).

    The point of this example is that even if everybody is using similar, objective criteria (i.e. comparing candidates along the same 2 directions instead of 1 direction), if they are using multiple criteria and have rational reasons for giving more weight to one criterion or another (but still giving at least some weight to the other) then paradoxes result in this very rational electorate.

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  5. stuartl says:

    The point of this example is that even if everybody is using similar, objective criteria (i.e. comparing candidates along the same 2 directions instead of 1 direction), if they are using multiple criteria and have rational reasons for giving more weight to one criterion or another (but still giving at least some weight to the other) then paradoxes result in this very rational electorate.

    Objective criteria? Rational reasons? Rational electorate?

    This reminds me of the joke about a certain kind of scholar: “First, we assume a spherical cow…”

  6. James Hanley says:

    Thoreau and StuartL,

    Precisely.

  7. thoreau says:

    Hey, I’m not saying rationality will actually prevail, I’m just saying that paradoxes can arise even with rationality. Clearly, with irrationality paradoxes will be even easier.

    I’ve done a bit of work on the mathematics of voting, so I’m looking forward to your discussion of Arrow’s Theorem.

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  9. D.A. Ridgely says:

    I assume “So individuals are assumed to have intransitive preference orders…” should read “transitive preference order.”

  10. James K says:

    You know, right up until the last paragraph I was planning a reply about how parties don’t help in quite the manner you were suggesting due to Arrow’s Impossibility Theorem, but since you clearly have it covered, I’ll skip it 😉

    For me, the principal benefit of parties is informational, like the branding of commercial products. For comparison I look at the difference between local and national elections in New Zealand. In both cases, the speeches of politicians are little more than bundles of platitudes duck-taped together. But because there are parties (and strong ones too) at the national level, I can just read the party’s manifesto and make my judgement from there. But at the local level, there’s no party structure and so I can’t extract any signal from the noise.

    I would predict that a party-less democracy would be even more of a personality contests than it is now. Also the party mechanism should impose some discipline on political dishonesty , since a bad politician damages the party. This effect will be stronger in places where party nominations are controlled more centrally.

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  13. AMW says:

    Is there an impossibility theorem if we have bids instead of votes? Say preferences (expressed as willingness-to-pay) is arranged as follows

    Friend 1 Friend 2 Friend 3
    A $150 $25 $50
    B $80 $100 $20
    C $10 $50 $90

    Note that the ordinal rankings are the same above as in the table that Hanley posts. But now, A has $225 of support, B has $200 and C has $150. Now, instead of voting, allow the friends to cast bids for the three options (you can bid on as many options as you like). Highest sum of bids wins, and whoever put in a bid on that option pays his bid. (You don’t have to pay bids on options that lose.) Ties are broken randomly.

    It seems like your odds of transitive group preferences are much higher in this framework. Tied group preferences are still possible, but if bid functions are linear in willingness to pay, the bidding system would return a random outcome, as is optimal.

  14. AMW says:

    WordPress cuts out duplicate spaces? That’s ridiculous. I’ll try posting my table slightly differently:

    ……….Friend 1……….Friend 2……….Friend 3
    A……..$150……………..$25……………….$50
    B……..$80……………….$100……………..$20
    C……..$10……………….$50………………..$90

  15. James Hanley says:

    DAR–Thanks.

    AMW–Interesting question, that I hadn’t thought about. I think this relates back to what I said on the other thread about expressing intensity of preferences. The paradox as I described it occurs in the context of a forced equality of the intensity of the expression of the preference. But I think your case depends upon an assumption about how much the expression of intensity of preference (let’s call that EIP to make it easier) varies. If their EIP doesn’t vary much, you’d still have the paradox, right? So the ability to fully express intensity of preference may mitigate the likelihood of the paradox occurring, and mitigate the frequency of cycling, but won’t eliminate the potential for it.

  16. D. C. Sessions says:

    OK, so I plead guilty: http://xkcd.com/793/

    Making the usual physicist’s fudge of approximating the statistical behavior of a large number of discrete entities using continuous mathematics, I’m suspecting that this problem may be isomorphic to the one we have with discrete sampling of a continuous variable: provably not reliable. For a monotonic independent variable, resolution time is unbounded, for a nonmonotonic independent variable (and keeping in mind that humans in aggregate are chaotic) the result is indeterminate.

    You get really weird ideas from the collision of a background in physics and engineering with a daughter in the social sciences. I keep seeing isomorphisms after conversations with her.

  17. James Hanley says:

    Are you sure that cartoon was about physicists, and not us social scientists? Jeez, here I’ve been suffering from physics envy for years, and it turns out that all along they were doing exactly what I was doing (except they tended to be better at obscuring their fudging with math).

  18. D. C. Sessions says:

    Absolutely about physicists. Their only serious competition for brutal application of simplifying assumptions are engineers, and for engineers it’s because our job is “close enough to meet spec” rather than correctness.

    Everyone else has the perfect excuse that the reality they’re modeling is already hopelessly fudged by the physicists first 🙂

    Which still doesn’t keep me from suspecting that the same math that works for thermodynamics, information theory, and nuclear chain reactions might be useful for sufficiently large societies. Thereby proving that the cartoon is about me.

  19. James Hanley says:

    Which still doesn’t keep me from suspecting that the same math that works for thermodynamics, information theory, and nuclear chain reactions might be useful for sufficiently large societies.

    Well, I agree, but then I was taught by people who believe this. Not all of my natural scientist friends agree, and I have far too many social scientist acquaintances who don’t agree. With the latter, I’m never sure if they’re actually acting on a firm belief that human behavior non-measurable or if they’re just math-phobes. There’s probably a strong correlation between the two, and possibly a causal one.

  20. Heidegger says:

    Mr. Thoreau, what a great honor and pleasure to meet you! Curious, was life in that 10′ by 15′ cabin ever claustrophobic? Did it prepare you for your painfully long incarceration for tax evasion? Whoops, must be thinking of someone else. I think your time in clink was one day, but hey, everything is relative and you made your point even if Emerson did paid your taxes. But this Nat Hawthorne fellow—is he your, “with friends like you, who needs enemies” kind of guy?

    NH–(Henry David Thoreau) is a singular character — a young man with much of wild original nature still remaining in him; and so far as he is sophisticated, it is in a way and method of his own. He is as ugly as sin, long-nosed, queer-mouthed, and with uncouth and somewhat rustic, although courteous manners, corresponding very well with such an exterior. But his ugliness is of an honest and agreeable fashion, and becomes him much better than beauty.”

  21. AMW says:

    The paradox as I described it occurs in the context of a forced equality of the intensity of the expression of the preference. But I think your case depends upon an assumption about how much the expression of intensity of preference (let’s call that EIP to make it easier) varies. If their EIP doesn’t vary much, you’d still have the paradox, right? So the ability to fully express intensity of preference may mitigate the likelihood of the paradox occurring, and mitigate the frequency of cycling, but won’t eliminate the potential for it.

    No, I don’t think that’s right. I think the crux of the issue is cardinal versus ordinal rankings. (Naive) Votes are based on individuals’ ordinal rankings of the alternatives. So it’s possible for two alternatives to have the same sum of ordinal rankings, but for one to beat the other in a vote.

    In your example above, give an ordinal ranking of 3 for one’s most preferred alternative, 2 for one’s second most preferred alternative, and 1 for one’s least preferred alternative. If you look at the sum of ordinal rankings for each alternative, it’s six in each case. But A can beat B, B can beat C, and C can beat A.

    Now suppose you’ve got three alternatives where the sum of willingness to pay is the same for each. Assuming agents have bid functions that are linear,*,** it is not possible for any alternative to systematically beat another in a head-to-head auction. So you wouldn’t get cycling.

    Moreover, since one can meaningfully sum willingness to pay across individuals, I think I’m on solid ground saying that if a group would pay more for A than B, and more for B than C, it is not possible that they would be willing to pay more for C than A. Find a counterexample and prove me wrong. (It will probably involve an imaginary number.)

    *By linear bid function, I mean that if someone’s willingness to pay for an alternative is x, their bid would take the form a + bx. If bid functions are concave, then between two alternatives with identical aggregate willingness to pay, the one with more supporters would win. If they are convex, then the one with fewer supporters would win. I’m willing to rule out convexity based on introspection alone, though I admit that concavity is a real possibility.

    **In addition to linearity, I’m implicitly assuming that all agents’ bid functions are identical. But I believe the thrust of my argument would still hold if the a and b terms in the bid functions are distributed symmetrically among agents.

  22. Heidegger says:

    Speaking of…. “Because there is a law such as gravity, the universe can and will create itself from nothing. Spontaneous creation is the reason there is something rather than nothing, why the universe exists, why we exist,” Hawking writes.”

    Is gravity “nothing”?

  23. James K says:

    DC Sessions:

    Which still doesn’t keep me from suspecting that the same math that works for thermodynamics, information theory, and nuclear chain reactions might be useful for sufficiently large societies. Thereby proving that the cartoon is about me.

    A number of physicists defected to economics in the 20th Century and brought their math with them. After all Von Neumann was a physicist and he invented game theory, along with Schelling. Also, the Black-Scholes Option Pricing Model (which is used to calculate the value of some derivatives and gave birth to the Hedge Fund) is based on Ito’s Lemma, a formula invented to track the position of a rocket in flight.

  24. ppnl says:

    I have not had time to read carefully here but the first thing that occurs to me is that maximizing democracy should not be the prime objective. We have a constitution that limits the power of the majority and divides the government into three different branches. This is done to protect government from the inherent instability of democracy.

    Get rid of party primaries and institute a sequential runoff system and I think the problems with intransitive preference order is minimized.

    I think the role of parties should be minimized. Party branding seems inherently irrational. You see fanboy behavior in all things from coke vs pepsi to Intel vs AMD. These things tend to be transitory and mostly irrelevant to rational policy. But they can be destructive in the short term.

  25. James K says:

    But without the party structure, elections degenerate into meaninglessness. You can’t tell anything useful about a candidate’s position based on what they say, the signal to noise ratio is too low.

    If anything, I think the problem the US has is the parties are too weak. In New Zealand if a politician goes too far off the reservation their party can reign them in by threatening not to nominate them. The result are stable policy platforms, eliminating the need to try and work out what each politician claims to stand for today.

  26. mcmillan says:

    “If anything, I think the problem the US has is the parties are too weak. ”

    I think it’s both weak in some ways and strong in others, and tend to be reinforced as well. They’re strong in that it’s pretty much impossible to be taken seriously as a candidate without the support of one of the two major parties. But since those two parties have to be able to include a wide range of views and can be significantly different in different parts of the country it’s difficult for them to be able to actually enforce a concrete platform.

    I’ve generally thought having an extra party or two that is a serious contender would be good for allowing the parties to actually stand for something as a group, increasing the signal to noise as you put it.

  27. Matty says:

    I’d go so far as to say that the US doesn’t have political parties in the same sense as other countries. Instead you have two big coalitions, one for everyone left of the median, the other for everyone to the right.

    I’d also suggest that they are effectively branches of the government given way you involve public officials in organising party primaries and identifying party members through the idea of ‘registered’ whatever.

  28. ppnl says:

    James K,

    I think James Hanley made the point that the point that the democratization of the primaries has made the parties unable to enforce any discipline on the candidates. The candidate that gets the vote controls the party. Look at the Christine O’Donnell race to see how this plays out.

    A sequential runoff system would give the party power over the candidate again while simultaneously reducing the power of the two party duopoly. The parties are free to pursue their agenda rather than simply chase victory at any cost. The candidates are in turn freed to pursue a wider agenda rather than being forced to commit to the narrow agenda of the party’s most radical elements. Again the Christine O’Donnell race in Delaware shows how this works.

    In your country the party is an explicit part of the government. I don’t think we can get there from here. The constitution clearly gives the office to a person and not a party.

  29. James K says:

    mcmillan:
    I agree with you there, I strongly believe that ease of entry and exit are the most important factor in determining how competitive a market is. The nature of your campaign finance restrictions and the difficulty people who aren’t Democrats or Republicans have getting onto the ballot bodes ill for the prospect of additional parties arising in US politics.

    ppnl:
    It seems to me that parties are organs of the state in your country too. After all, the manner in which they can nominate their candidates is strictly regulated, where down here it’s entirely left up to the party.

    I think getting the government out of the party nomination process would help, but I’m worried about the effect of the cultural conditions those regulations have created.

  30. ppnl says:

    It seems to me that parties are organs of the state in your country too. After all, the manner in which they can nominate their candidates is strictly regulated, where down here it’s entirely left up to the party.

    Well see that is the problem. Where your parties pick candidates internally and the party itself is what holds power we pick the party candidate publicly and then the individual holds the power. Parties are not so much organs of the government as grafted on extra appendages. This is what makes them weak and unable to stand for their supposed principles.

    Now maybe in principle you could fix this by making parties an integral part of government. But you would have to do open heart surgery on the constitution to accomplish this. We simply can’t get there from here.

    The simpler and I think better solution is to end government involvement with the primary process. There is no constitutional or legal barrier to doing this. In fact the states could do this on an individual basis without involving the federal government at all. Our primaries are regulated but not strictly. States are allowed a wide latitude in how to do it and don’t have to do it at all.

    I think getting the government out of the party nomination process would help, but I’m worried about the effect of the cultural conditions those regulations have created.

    Those cultural conditions will continue until the government is out of the party nomination process. The cure may be bad but the underlying disease is fatal in the long run.

  31. James K says:

    I agree, my concern is that simply eliminating the regulations will do nothing because of cultural inertia. but still, it can’t hurt.

  32. James Hanley says:

    AMW wrote;

    I think the crux of the issue is cardinal versus ordinal rankings.

    OK, maybe I just don’t get where you’re going with cardinality, but if you just mean the ordinary concept of “counting numbers,” I’m still not seeing how it necessarily avoids cycling. Let’s use an example that instead of simple ordinal ranking shows how much each person is willing to pay (the number in parentheses) for the particular outcomes A, B, and C. It still looks like cycling to me.

    ____1_____2_____3___
    ……A(21)….B(20)….C(19)
    ……B(15)….C(14)….A(13)
    ……C(5)……A(4)……B(3)

  33. AMW says:

    Hanley,

    Those values give you cycling if everybody’s voting. But if they’re casting bids, and the highest sum wins, then cycling may not be possible. As I pointed out above, linear bid functions would mean you get a tie in bids, and you pick randomly. Nonlinear bid functions would tip the process in favor of either majorities or minorities. This wouldn’t cause cycling, though it could potentially cause inefficient outcomes.

    As for cardinality, I just mean you can sum the values in a meaningful way. A tie in values means the group would be willing to pay as much for either option. You can’t sum ordinal rankings in a meaningful way. A tie in summed rankings doesn’t mean the group is indifferent between the two.

  34. James Hanley says:

    OK, I think I get you. If they’re casting bids, then the person/group with the greatest collective intensity of preference wins, yes? Which could mean minority rule at times, as you note. Potentially it could mean one person, solely, selects the winner, if they are willing to bid enough. Generally that wouldn’t be considered an acceptably democratic outcome, but if we’re measuring not agglomerations of individuals mere preferences, but agglomerations of individuals’ intensity of preferences, the process is measuring something valid about public opinion, at least from a utilitarian perspective.

    And I think you’re right that it would eliminate cycling (unless people’s intensity of preferences fluctuated, which they probably do, but which the horror of prevents me from thinking too much about).

  35. thoreau says:

    If you ask everybody to supply absolute information rather than relative information, then you can certainly avoid paradoxes in comparisons. Paradoxes arise from the type of information used to make the comparisons.

    As to minority rule, though, in practice people with access to polling data are likely to vote strategically if you use a point system (e.g. everybody rate candidates on a scale of 0 to 10, or whatever). If a majority of the public sincerely believes that candidate A is best, then irrespective of what they think about the other candidates they should all give A as many points as possible (depending on the rules of the game). They should give zero points to A’s leading opponent. And if there are other candidates whom they prefer to A’s leading opponent, they can give those candidates as many points as they like, to send whatever sort of messages that they like, knowing that A will get full points from a majority and nobody else will, and hence that A will win.

    Even if there is no majority favorite, suppose that people have access to polls. Initially, everybody responds to the polls by saying how many points they’ll give to candidates A, B, and C, and the points they assign may be based on a mixture of strategic calculation and sincere support. If they are at all strategic, they will indicate a desire to give full points to their favorite, zero points to their least favorite, and something in between to their middle candidate.

    Once that info is out there, though, they might realize that their second choice is in a close race with their first choice, and at that point they’ll respond to follow-up polls with ballots that give full points to a contender, zero points to the other contender, and something else to the third candidate. Unless they respond to polls with a Machiavellian “I want them to think that…” response.

  36. D.A. Ridgely says:

    Unless they respond to polls with a Machiavellian “I want them to think that…” response.

    “But it’s so simple. All I have to do is divine from what I know of you: are you the sort of man who would put the poison into his own goblet or his enemy’s? Now, a clever man would put the poison into his own goblet, because he would know that only a great fool would reach for what he was given. I am not a great fool, so I can clearly not choose the wine in front of you. But you must have known I was not a great fool, you would have counted on it, so I can clearly not choose the wine in front of me…. Because iocane comes from Australia, as everyone knows, and Australia is entirely peopled with criminals, and criminals are used to having people not trust them, as you are not trusted by me, so I can clearly not choose the wine in front of you. … And you must have suspected I would have known the powder’s origin, so I can clearly not choose the wine in front of me. … You’ve beaten my giant, which means you’re exceptionally strong, so you could’ve put the poison in your own goblet, trusting on your strength to save you, so I can clearly not choose the wine in front of you. But, you’ve also bested my Spaniard, which means you must have studied, and in studying you must have learned that man is mortal, so you would have put the poison as far from yourself as possible, so I can clearly not choose the wine in front of me. … You only think I guessed wrong! That’s what’s so funny! I switched glasses when your back was turned! Ha ha! You fool! You fell victim to one of the classic blunders – The most famous of which is “never get involved in a land war in Asia” – but only slightly less well-known is this: “Never go against a Sicilian when death is on the line”! Ha ha ha ha ha ha ha! Ha ha ha ha ha ha ha! Ha ha ha… ” [falls over dead as a doornail]

  37. AMW says:

    Hanley,

    I think you’ve got the basic idea. Minority rule is possible if the minority wants something very badly relative to the majority. One way of making the process more democratic is to give the bids of the winners to the losers.

    But my main point about majorities & minorities was with regards to ties. Say you’ve got one guy who values A at $1,000,000, and 10 guys who value B at $100,000 each. So each option is valued at a million bucks in aggregate. Imagine three possibilities for bid functions:

    Linear Bid Function: Everyone just bids a fraction of his value; let’s say half. So the guy who values A bids $500,000, and each of the 10 guys who value B bid $50,000, and you’ve got a tie. It is broken by a coin flip.

    Concave Bid Function: The fraction of your value that you bid falls with the size of your value. So let’s say everyone has a bid function defined by b = log(x), where b is your bid and x is your value for your preferred option. The guy who prefers A bids log(1,000,000) = $6 for A. Each of the 10 guys who prefer B bid log(100,000) = $5 for B. So the final bid is $6 for A, $50 for B, even though both sides value their options equally. Majority wins.

    Convex Bid Function: The fraction of your value that you bid increases with the size of your value. So let’s say everyone has a bid function defined by b = x^((x-1000)/x). The guy who prefers A bids roughly $986,000. Each of the 10 other guys bid roughly $89,000. So you have $986,000 for A versus $890,000 for B, even though both sides value their options equally. Minority wins.

    See? It’s all so simple.

  38. James Hanley says:

    See? It’s all so simple.

    And how does saying that affect your student evaluations? *grin*

    Yes, I get it now. I’ll have to chew on it for a while to work out what it means, in that sense of, “how do I fit this new concept in with this other concept and use it to help make sense of the world.”

  39. AMW says:

    I teach MBA students about economics. You should have seen my mid-semester evaluations the first time I taught the course.

    I still wake up in cold sweats.

  40. James K says:

    DAR:
    The funny thing about that scene (well, apart form the scene itself which is hilarious without over-thinking it) is that there’s a pretty simple equilibrium in game theory. The optimal strategy for Westley (assuming he was playing the game fairly) is to poison a random goblet. This means that Vizzini should have realised this and simply chosen a goblet at random.

    The there’s the higher level point about the wisdom of playing your opponent’s game …

  41. James Hanley says:

    Heh, there’s another video clip to add to my strategic behavior class.

  42. AMW says:

    Quoth James K:

    The optimal strategy for Westley (assuming he was playing the game fairly) is to poison a random goblet. This means that Vizzini should have realised this and simply chosen a goblet at random.

    If people had good internal random number generators, that would be true. But given that we’re pretty lousy at it, there might be some sense to Vizzini’s ramblings. He also shows some nice meta-rationality in switching the glasses. If Westley (assuming initial fair play) realized that Vizzini guessed right, there’s nothing to stop him foregoing the wine-tasting game altogether, and just lunging across the table at his opponent, since he has a clear advantage in that game. By switching glasses and making his stronger opponent think that he had chosen wrong, Vizzini was (in his own mind) neatly avoiding that possibility. In fact, by the same logic, not seeing Westley lunge over the table at him would look like a pretty clear signal that he had guessed right.

    Further quoth James K:

    Then there’s the higher level point about the wisdom of playing your opponent’s game

    Or, as Guys & Dolls puts it:

    One of these days in your travels, a guy is going to show you a brand-new deck of cards on which the seal is not yet broken. Then this guy is going to offer to bet you that he can make the jack of spades jump out of this brand-new deck of cards and squirt cider in your ear. But, son, do not accept this bet, because as sure as you stand there, you’re going to wind up with an ear full of cider.

  43. Pingback: Book Review: Good Strategy, Bad Strategy « Srini's Corner

  44. Matthew says:

    I think this post somewhat mis-characterizes the Condorcet paradox as it relates to the arrow impossibility theorem. Arrow’s theorem says that a polity cannot have a complete and transitive choice structure unless there is a “dictator” (by “dictator” we mean a “swing voter” like Kennedy on the supreme court). The Condorcet paradox arises whenever there is no dictator. Here’s what all that means: the outcome of the election will depend on how you design the voting system. Moreover, the choice of voting system (and hence, the outcome of the election), is arbitrary–there is no way to “fix” the problem of the Condorcet paradox.

    For example, our system of holding primary elections for each party and then a general election between the winners of the primaries will result in a different candidate winning than if we abolished the primaries and pooled all the candidates in the general election.

    Since the public’s choice structure is intransitive, there is no “fair” way to design the election: the outcome is ultimately arbitrary. Hence, voting is really a non-trivial game in which participants try to strategically use their votes and change the voting rules in order to influence the result of the election. This is why it is often sub-optimal to vote for your favorite candidate (if have a preference between Obama and Romney, you definitely don’t want to vote for Gary Johnson, no matter how much you like him!), and why people often complain about third-party candidates spoiling the general election.

  45. Pingback: How to Rig a Majority Vote - The Liberty Caucus

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