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September 22, 2004

Comments

Andrew B

Well I followed you here from Commentary hoping you'd be able to give a final answer to this. I'm sticking with 0 or very close.

The trick presumably is understanding that everybody second guesses all the time. Lets assume that we think the average of all the numbers will be 50 then we'd choose an answer of 34 (round up). But we know that everyone else is going to choose 34 too, so we go for 23 (round up again). But again we assume that everybody else goes through the same thought process so we choose 15. And so it goes on regressing until we end up with an answer close to 0. Is this right or am I missing something important?

Phil Hunt

One, surely. For the reasons Andrew gives.

Abiola Lapite

Andrew, you were on the right track, but Phil takes the prize, as the question explicitly states that 1 is the lower bound. The puzzle essentially comes down to asking for the lower limit on
1 + ((2/3)^n)*x
where [1 x 100] and [n {1, 2, 3, 4, ...}],
as n .

PS: The above has been edited slightly for added clarity.

Mats

Interesting! Now this has an application for market failure in the media industry: the Ice-Cream Vendors On The Beach Problem. Assume that the ice-cream vendors compete with their location to the beach and the customers always chose the closest vendor. As you see with the same reasoning as that above, glass vendors will suboptimally concentrate in the middle of the beach. Correspondingly (and annyoingly), most radio stations will end up playing music from the deep mainstream. There's your mathematical rationale for public service broadcasting!

Abiola Lapite

Mats,

The Hotelling "Vendors on the Beach" scenario is in fact the reasoning behind the Median Voter Theorem. One thing to keep in mind though - that problem makes the assumption that customers are evenly distributed throughout the beach strip, which needn't be the case where broadcasting is concerned, nor need it even be true that the public's tastes in viewing will be aligned along a simple 1-dimensional axis.

Also note that the vendors problem assumes there are only two vendors, and once you start to increase the vendor numbers things start to look rather different.

Finally, you make the implicit assumption that the public broadcaster will prove better at determining the "true" tastes of the public than the private alternatives, when in fact the lack of any feedback mechanism other than ratings - no price mechanism - suggests that it'll only end up chasing the same old bland middle when it isn't throwing money away on entertainment few people care for. That's just what's happened with the BBC, and from what I can make out of German and French public broadcasting, the situation in those two countries is no different.

In short, I don't think this makes much of an argument for public broadcasting at all.

bbartlog

While the answer of 1 may be correct in some mathematical sense, I think in a real life situation you would have to assume that many of the players will not play optimally (because they don't know math or are stupid or just didn't give it a lot of thought or are perverse...). I would want to know more about the hundred contestants before guessing just how much more than 1 I would choose, but unless they're all mathematicians I doubt choosing 1 will win you anything.

Mats

"Also note that the vendors problem assumes there are only two vendors, and once you start to increase the vendor numbers things start to look rather different."

The same probably goes for the puzzle in the post (if winners have to share the prize). But with perfectly rational agents in the puzzle, no relocation costs for the vendors, it feels like they should both produce the same result even for more then 2 participants.

I wouldn't really guess that moderate changes in customer distribution should qualitatively change the resuts. And what other parsimonous model would you suggest for the crowded mainstream even in media markets with many agents?

Mats

Odd fact: Representatives for Swedish Conservative Youth were very active in the liberalisation of Swedish radio. They actually thought it would lead to a very pluralistic consumer choice. Wrong they were, and one of them actually publically admitted it!

Abiola Lapite

"I wouldn't really guess that moderate changes in customer distribution should qualitatively change the resuts."

Why not, and what exactly do you mean by "moderate?" Intuition is often a poor guide in mathematics, so I wouldn't place too much faith on what I felt must be true if I were you.

"And what other parsimonous model would you suggest for the crowded mainstream even in media markets with many agents?"

I don't have the time right to provide them right now, but rest assured they're out there in quite some numbers. When I get a chance to look in my econ textbooks I'll get back to you on that.

"Odd fact: Representatives for Swedish Conservative Youth were very active in the liberalisation of Swedish radio. They actually thought it would lead to a very pluralistic consumer choice. Wrong they were, and one of them actually publically admitted it!"

But is it really a surprise to learn that the median Swedish radio listener, like the median Swedish voter, thinks the liberal welfare state just dandy? I credit the shock of the Conservative Youth to their having swallowed their own koolaid - members of marginalized parties everywhere have a tendency to believe that there's a "hidden majority" out there somewhere that subscribes to their views, but whose voice is stifled by "the biased media." At least with Swedish radio liberalized, that excuse becomes a bit harder to push, though the American Right still tries to do so anyway.

Mats

"Intuition is often a poor guide" - well, it might mislead you at times. So i looked it up. The median voter theorem requires that "preferences are single-peaked and in one direction". And for the ice-cream buyers, preferences are by definition single peaked (they don't want the ice-cream as close as it can be) and in one direction (along the beach). The theorem does however *not* require any specific distributions for preferences over voters - the median wins anyway.

http://www2.chass.ncsu.edu/garson/pa765/socialchoice.htm

Now, this theorem is in the textbooks, and it plays along quite well with what radio stations offer. So trying intuition again, I would guess that any competing theory is more likely to be found in a research paper, if at all.

Mats

Sorry, I was too fast here it seems. I'mean if there are two "population centra" on the beach, with a deserted stretch in between, ice cream vendors would lose if the move from a centra towards the middle. Populaton distribution have to be smoother than this for the theorem to work. My parallell between "median voter" and "vendors" is flawed.

Abiola Lapite

"The theorem does however *not* require any specific distributions for preferences over voters - the median wins anyway."

But it does require that the distribution be peaked, and at a single point at that. In short, the choice of distribution family does matter, as I pointed out.

"and it plays along quite well with what radio stations offer."

Which needn't mean anything at all. Where is your justification for the assumption that tastes fit along a unidimensional spectrum? How do you fit bebop, fusion jazz, African highlife, romantic classical music, baroque, 1970s punk, reggae and juju along one axis? You're grasping at straws here.

"So trying intuition again, I would guess that any competing theory is more likely to be found in a research paper, if at all."

And you'd be wrong (what did I tell you about intuition?) Do a search for "Hotelling model" and it shouldn't take you long to discover some of the alternatives that are out there. Here are just a few leads I've turned up with a few minutes of effort:
1 - Hotelling in the Classroom. Notice that it mentions (i) that according to d'Aspremont, an assumption of quadratic search costs flips the model over to one of overdifferentiation, indicating just how important the issue of distribution is, and (ii) that the conjecture that clustering would hold with more than two firms is false.

2 - Industrial Organization I - Stockholm School of Economics: has lots of references, including Tirole, Jean, 1989, "The theory of Industrial Organization", which is also mentioned in the previous link.

I'd bet that virtually any decent textbook on industrial organization would offer you as many alternatives as you could possibly desire.

The bottom line is that product differentiation can and does regularly occur along more than a single axis, and the only reason why the median voter theorem holds so well in America is because the first-past-the-post electoral system enshrines a duopoly which forces political differences into a unidimensional mould; there is no natural affinity between social conservatism and libertarian economics, nor is there between welfare statism and social liberalism. For the Hotelling model to apply to radio, certain assumptions would have to hold that are clearly breached in the real world, and as such you cannot conclude that it demonstrates that the output of radio stations isn't welfare-maximizing, let alone that public broadcasters could do any better.

Abiola Lapite

Ah! After responding to your previous comment, I see you've made another response in the interim.

Mats

Abiola,

Thanks for the links above, very interesting ones it seems. Before reading your latest comment, I was under the impression, from what you told me, that you were out to model why we have such a "crowded mainstream even in media markets with many agents" (my own phrase quoted once more). I was however wrong, you served me (thanks again) instead more general models.

Also, you now indicate that you don't really think that we get too little diversification (for the money no consumers really can avoid to spend) on media such as radio. In an effort to add clarity, here are my views written out explicitly, which I may have failed to do above:

I (subjectively) observe underdiversification. And I'm surprised to observe it given decent free market conditions. Then I find a simple Hotelling model the superior candidate for resolving the puzzle:

Firms (radiostations) move smooth without friction in (1-d) "taste space", but only in directions where their number of clients (listeners) strictly increase. Hence firms cluster upon some of the peaks of the client distribution (and miss the opportunity to spread out which sometimes is attained in the references you give, where the rules of the game is slightly different). (Rules are really important here, what would a few irrational agents do to the original puzzle?)

What is your standpoint, do you really feel we have the diversification most of us wish for in commercial radio?

Abiola Lapite

"do you really feel we have the diversification most of us wish for in commercial radio?"

I think that given the political and technological limitations faced by radio broadcasters in the past, we have had about as much diversity as could be expected in the circumstances. If politicians insist on handing out local oligopolies to broadcasters, then choice will obviously suffer, to which the answer isn't public broadcasting, but to lower the barriers to entry. The advent of spread-spectrum technology means that there really isn't an excuse any longer dishing out a limited number of licenses in each radio market, and even if that technology didn't exist, the radio-over-ip paradigm would also serve to lower the barriers, and we're already seeing the necessary infrastructure emerge with 2.5G and 3G networks.

In short, where radio is concerned, I don't see that choice is any longer much of a problem. From my desk I can listen to thousands of stations covering almost every niche from all over the world right at this moment, and I'll soon be able to do so from anywhere as long as I have a decent mobile phone. All it would take for me to become a broadcaster in my own turn would be to pay for the bandwidth and make whatever music licensing arrangements were necessary to start a station of my own. In such an environment, the last thing the world needs is another public broadcaster imposing onerous compulsory licensing fees and inefficiently utilizing precious bandwidth, simply in order to offer the same dull fare as I can find on thousands of other places already.

PS: You should find this Economist article on spectrum allocation interesting.

Daniel Waddell

Couldn't the regression also travel upwards towards 100?

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