Fancy Store-Bought Dirt

Pinker's NYT Magazine Article About Morality

We read the paper version yesterday; you can read the on-line version here.

At one point the article mentions three hypothetical situations that most people supposedly find immoral. My immediate reaction was a complete lack of objection to any of the three. (Admittedly, my eyes glazed over a critical detail in one of the three. I fervently defended my original position even after catching the important detail: That might be an example of some psychological effect whose name I can't think of (oversimplify a bit and call it stubbornness).)

Julia agreed with me about the lack-of-immorality in the flag case and the dog case. The flag case might have been a moral transgression depending on what standard of conduct the protagonist would have otherwise claimed to uphold.

(Reminds me of something our rabbi told us in Judaism classes over a year ago, that I'll do my best to recount adequately: It's wrong to violate certain Jewish laws/traditions, if and only if you believed in their validity to begin with. So if you assert from the outset, for example, "kosher dietary restrictions are silly"... that's your choice and life goes on. But if you believe that kosher is theoretically correct, yet fail to live up to it in practice, then you've transgressed. Does this make sense? But you could argue that this distinction doesn't apply to any moral rules about treatment of other people.)

I argued with her at length about the brother-sister incest case. It's unclear how much of this was devil's advocacy, how much was rationalizing a stake I'd taken, and how much is what I really believe. I think the two best arguments against brother-sister incest are the risk of pregnancy (you can't just hand-wave this risk down to zero in a hypothetical, no matter how much contraception is involved), and the premise that lots of childhood/adolescent sibling interaction would be really squicky if "these two people can never have sex ever" weren't a ground rule.

The article also mentions two competing flavors of the train problem, that most people will throw a switch to kill one person instead of five, but won't kill that one person with their bare hands (i.e. shove that person in front of the train) to spare five people.

I sort understand why people reflexively draw that distinction, but the way people pigeonhole things is ridiculous in light of simple physics. If you won't use your bare hands but will throw a switch, then what about the case where you save the five people by firing a pistol at the one person? (Railroad switches don't kill people, people kill people.)

[As you know from past moral philosophy posts, I'm in the distinct minority in claiming that you shouldn't even throw the switch. My defense of this is almost exactly the same as what I infer that people say in objection to killing someone with their bare hands. Despite what I claim in theory, in practice I'd probably throw the switch, pull the pistol shove the fat guy, or whatever it took. Unless I just froze.]

Fortunately, Scissors Has Developed an Immunity to Iocane Powder

WHY WHY WHY WHY WHY?

You have exactly three options for your starting move. As game structure goes they're all analogous: You could call them asphalt, bagpipe, and chrysanthemum and not change the game itself one bit.

It is, therefore, rather unlikely that any one of those three options merits the phrase "stalwart players have come up with a strategy," much less the label "how to win."

But hey, some company won a big business deal with scissors once. And now that I think of it, I once saw some guy win a huge pot with 5d 4s on televised World Series of Poker. (OK fine, I looked it up.)

My Favorite Dice Paradox

(Internal monologue in the car this morning: "Is it too soon to care about Week 15 fantasy football?" "Yes, because even aside from injuries you don't know who you're facing." "But why would that matter. Regardless of the opponent I'm just maximizing expected scoring.")

You probably already knew this but in a two-player game where margin of victory is unimportant, comparing the "expected" payouts tells you a lot less than you might think.

For example, consider three nonstandard six-sided dice. One has two 6's and four 2's on its faces; one has 3's on every face; one has four 4's and two 0's.

The first die has an expected roll of 3.3, the second 3 (obviously), the third 2.7. But if your opponent had the second die (the all 3's), and you wanted a higher number, for a quick ten points which one of the remaining dice would you choose?

You saw what happened there, right? Once you've convinced yourself that {4, 4, 0, 0, 0, 0} < {3, 3, 3, 3, 3, 3} < {6, 6, 2, 2, 2, 2} in the long run, add another die to the mix, this one with three 5's and three 1's. How do you expect that one to do against the others?

Paul gets comment of the month. That was quite the typo (and of course his main point is dead-on)

Word Stacks

An exercise for the reader is to tell me (through memory or research) what two seven-letter words were involved in the story, from Word Freak, of a Scrabble game whose first play was a seven-letter word and whose second play was a seven-letter word that also made seven two-letter words.

Earlier this week Julia and I finished a game in which I slipped AXONS under TOGAE. (For one of the ancillary words ET became ETA, but since TA itself is a word I think this still counts as 5/7 of the Word Freak thing.)

The Limits of Stipulations in Philosophical Problems

I think I might be gravitating to the "one box" camp for Newcomb's paradox. But I should tell you how I got there.

Julia, like nearly every intelligent person, disagrees with me about the track/switch "kill one man to save five" problem. She's surprised that once my own life is secure, I'm unwilling to be an agent of death. (I suppose anyone who knows that I'm not a pacifist -- e.g. that I continue to support what the U.S. is doing in Iraq -- should be many times as surprised.)

My immediate answer -- not my whole answer, but so instinctively first that the rest is just rationalization -- is that ground rules be damned, I reject the 100% certainty that it's a case of five innocent people dying versus one innocent person dying.

So what if the ground rules become absolutely ironclad?

Then my answer to the switch problem becomes really lame, though it continues to be my answer. (It's not nearly as lame as the people who'd let their entire basement hideout be massacred because they were too wussy to smother one baby.)

This got me to thinking about other ground rules that I find suspect. For example, even in the usual formulation of prisoner's dilemma (where players have a dominant strategy that works to the other player's disadvantage) I am the most outspoken advocate I know of the correctness of the "nice" option. No matter how hard you stipulate that this is a one-time game, in practice different flavors of it can and do come up frequently. The world as a whole is better off if everyone consistently cooperates, and if everyone realize that the world as a whole [etc. to infinity].

Although you'd think Newcomb's paradox stems from a game that you personally play just once (apparently billions of other people have played it though, for the Great Predictor to have amassed such a track record?), something similar might be at play here: Think of sustainable development, and of various free rider problems. (I wonder if they cover Newcomb's paradox in those "Green MBA" programs...)

One caveat: Several years ago at a New Year's Eve party thrown by geeks, for geeks, I overheard someone (whom I'd never met and have never seen since) mention that unlike his friends, he carries grudges from strategy game to strategy game. That is, if you screw him over in one game (think of Diplomacy and the like), he'll remember it. That guy apparently wished to use his openness about such grudges to his advantage, as a deterrent.

I think if you're playing a strategy game that you don't (can't) take payoff calculations personally. The best long-term reaction to that guy I overheard would be just to refuse to play games with him.

(So the paradox is that I advocate screwing people over when you're playing for fun, yet being "nice" when you play for money! This must be what I think given how fervently I advocated the "Friend" strategy in that old game show Friend or Foe.)

Time Morality Quiz

Take it (via Fark).

I find problems 1 and 2 to be slam-dunk easy (can you guess from my other political views how strongly I value the right of self-defense?).

Like everyone who's ever taken a college philosophy class I've had plenty of opportunity to think about (and discuss) the trolley problem and its variants; I tend to be one of the few people in the room (sometimes the only one) who opposes throwing the switch.

The Time quiz doesn't include the "famous violinist" problem (nine months of detaining one man so that another might live: one of the most thinly-veiled allegories in this realm); my answer to that one is consistent with pro-life principles.

(For what it's worth, related to practical sociopolitics rather than abstract philosophy, one of my five biggest regrets in life is having been so vocally anti-abortion over the years. Since I somehow doubt that I personally have talked anyone out of having an abortion, that outspokenness has easily done more harm than good.)

I Solved a Volokh Puzzle!

Got this one right away. (As did basically every commenter.)

Replay!

I just sent my wife a link to the news about baseball GMs voting for instant replay. That left stuck in my head the glorious sound a pinball machine makes when you hit the replay threshold (or succeed at the 10% chance of a freebie, on those machine gracious enough to include that option).

It's been so long since I played pinball. Even the concept of "extra ball" is fond memories.

377-377

Have you ever had a Scrabble game end in a dead draw? I don't think I ever had, but there's a first time for everything.

Also, that makes twice in one week where I needed the leftover tile bonus/penalty to avert a loss.

Thomas the Sudoku Engine

The identity of the U.S. sudoku champ (pictured in the linked-to article) won't surprise anyone who read Jason Z's travelogue e-mails from Prague a few months ago.

Mechanism Design

Today's exercise for the reader:

1. Read as much as you can about the work that won the 2007 Nobel Prize in Economics.

2. Think about eBay, and the last-second sniping everyone talks about.

In theory last-second snipers should not affect a second-place auction; more importantly, if you understand the structure of those auctions then those snipers shouldn't affect you. Yet last-second bids are not only prevalent but also part of the conventional wisdom. Why? ("Market inefficiency" is an easy answer but how likely is it to be true?)

The best answer I can come up with: If last-minute bids become the perceived norm then buyers collectively will come out ahead of where they would have been. eBay probably has more buyers than sellers. (Ironically, all the people who get outbid at the last minute and say "I would have bid more" have only themselves to blame for bad tactics, even if those bad tactics were part of a plausibly good overall strategy (of rent-seeking?).)

Mutually Assured Destruction was a Suck-Out Artist

Do you fully comprehend how incredibly lucky we are that the world as we know it still exists despite the nuclear arms race of the Cold War?

I probably mention this dozens of times but this is one of the many things that could have gone wrong but didn't. (If you know what was going on as of the instant the Soviet warning system reported a supposed U.S. missile launch, what odds would you lay that nuclear holocaust would ensue?)

Silly Math of the Day

(See, not all of my posts lately are about fantasy sports. (Oh wait, this one actually is if you notice the direct mapping.))

A science class has an even number of students (2n for some positive integer n) and a 50-50 male female ratio. Those students are assigned randomly into lab partnerships. (Process doesn't matter but if it helps you visualize: Pick two students at random - they're partners. Pick two more students at random from among the ones who are left. Repeat until the two remaining students become partners.)

As a function of n, how many male-female partnerships do we expect?

Silly Arithmetic Relic

Read my mind and tell me: In what context did I just solve for the fraction 161/180?

(Hint: Public education is involved but it has nothing to do with a 180-day school year. Don't think of the denominator as exactly 180. Instead, think of someone who's perfect at computation (i.e. they have a calculator) but inept at understanding certain mathematical principles, like rounding.)

it always does come full circle

Only five minutes instead of a day, but the joke in this Fark headline led me to this Wikipedia page, which refers to the same Fark thread.

The Difference Between an Angle-Shoot and a Cheat

Yesterday some co-workers were in a pool tournament and with the game on the line one of them called a table scratch on the other (it was indeed a table scratch), took ball in hand, and sank the 8-ball.

Earlier in the game, the scratcher's teammate had chosen not to enforce a table scratch. Whether that's relevant is an exercise for the reader.

Bathroom Tile Math

I believe this problem has long since been solved, and it came up several "sit-down meetings" ago, but Z's comment to a baseball post reminded me that we do have some cool math geek readers here.

Consider infinitely many rows of tiles in a triangle pattern: The first row has one tile, second has two tiles, nth has n tiles.

Which positive integers can('t) be formed as the sum of two or more consecutive rows of tiles?

(Henceforth "number" describes only positive integers.)

Two consecutive rows can form any odd number greater than 1. Any odd number is expressible as 2i+1 for some i, and if i > 0 then you use the ith row and the (i+1)th row.

Three consecutive rows can form any proper multiple of 3 because i + (i+1) + (i+2) = 3i + 3 = 3(i+1).

More generally any odd number of consecutive rows can form any multiple of that odd number greater than half the square of that odd number.

Four consecutive rows can form any 2mod4 number >= 10. Six consecutive rows can form any 3mod6 number >= 21, eight consecutive rows can form any 4mod8 number >=36.

More generally, any number that can be expressed as the product of an even number and an odd number can be represented by consecutive rows. If 2E > O then use O rows whose average is E. If 2E < O then use 2E rows whose average is O/2. (For a quick 10 points why will you never see 2E == O?)

We've covered every odd number greater than 1 and every even number expressible as the product of an odd and an even. The only even numbers not expressible as the product of an odd and an even are powers of 2.

An exercise for the reader is to prove that NO power of 2 can be expressed as the sum of N consecutive integers.

Probability Theory of the Day

Also feeling pangs of regret must be the San Francisco cab driver who drove Murphy and Kamal from the airport to their hotel. Hoping to get out of the $55 fare, they told the driver that if they caught Bonds' record home run ball, they'd give him a couple thousand dollars.

"He turned me down, man," Murphy said.
--Yahoo! Sports

I think the actual probability that Bonds would hit his record-breaking home run that night, AND that one of those guys would catch it, was a good deal less than 55 in 10,000.

"What do you do for a living?"

"I drive cabs... AND I chase inside straight draws."

Mathematical Misconception of the Day

Five friends gather for a marathon bridge session. They want to play ten rubbers, such that each person has each possible partner for two rubbers and sits out two rubbers. Moreover, the two times any given partners play together must NOT both involve the same pair of opponents.

I spent much of this morning convinced that these criteria were impossible to meet (but not sure how to prove it). But then I realized exactly how to do it and felt stupid.

(Hint: The more instinctively you know the best way to set up a five-person round robin, the more you're at risk of falling into a mental trap.)

A Devil's Advocate Defense of Point Spreads

In theory, gambling based on point spreads (as opposed to straight-outcome betting) should not only reduce the magnitude of harm of gambling-related "fixes" but also make certain kinds of cheating easier to detect if someone actually crunches the numbers.

Consider a corrupt referee who needs the home team to win by at least five points and who does something to cause the home team to win by six points instead of by four. In abstract, yes, this still ruins the integrity of the game, but on purely outcome-based analysis what has he really changed? (If, on the other hand, he needed the home team to win and did something corrupt to cause the home team not to lose...)

As for catching the cheats: You may remember the chapter in Freakonomics about suspicious results in sumo wrestling. Couldn't one design a similar study involving, say, NBA games with a final score within 2-3 points of the spread? (Or within 2-3 points of the over/under?)

Unless players, coaches, and refs were chronically acutely aware of the exact spread, you wouldn't expect much discontinuity in stat trends involving games barely on either side of the line. (You would expect bits and pieces of discontinuity between games some team barely lost and games it barely won.)

Some interesting questions:
1. For all NBA referees, what was the average distribution of fouls in games in which
a) a home underdog barely covered?
b) a home underdog barely failed to cover?
c) a home favorite barely covered?
d) a home favorite barely failed to cover?

2. For NBA referee X, what were those distributions?

3. For all NBA refs, on average how many fouls were called in games in which the teams {barely hit, barely missed} the over?

4. For NBA referee X [...]

Here's the only Freakonomics post I could find that mentioned the scandal. Somebody mentioned the difficulty of detecting a cheating ref whose actions could nudge the outcome to either side of the spread. Well... just look for refs with a relatively high frequency of "outlier" games (e.g. visiting team got more foul calls than usual) correlated to close calls where the outlying trend worked to the benefit of whichever team barely covered.

Meanwhile Mark Cuban is an outstanding business leader. I should have guessed he would write something like this (but both the position and the eloquence still surprised me).

Math of the Day

Combinatorics can involve some messy computation. All the more reason to keep it simple(r) whenever you have a chance.

Barry Bonds has not started 16 of the 91 games that the Giants have played so far this season. There are 260,462,895,672,871,000 (260 quadrillion) possible combinations of games that Bonds might have “chosen” to miss. For example, Bonds missing Giants games # 88, 4, 62, 18, 22, 23, 61, 2, 91, 54, 87, 10, 58, 19, 65 and 83 is one such combination.

Of those combinations, 1,200,635,647,008,340 (1.2 quadrillion) involve each of the three Giants games that have been nationally televised on ESPN this season, as well as any other set of 13 non-ESPN games. In other words, the probability that if Bonds were picking 16 out of 91 games at random to miss, it would so happen that all 3 of the ESPN games were included among those 16, is 1,200,635,647,008,340 divided into 260,462,895,672,871,000, or about 215-to-1 against.
--Nate Silver, Baseball Prospectus Unfiltered

Those numbers seem plausible but I don't think many humans have a good sense of whether hundreds of quadrillions are the right order of magnitude. In a perfect world you could solve this problem without numbers that large. In fact, you can solve the problem int his world without numbers that large.

This problem is equivalent to three friends each drawing (without replacing) a ball out of an urn that contains 91 balls, 16 black and 75 orange. The probability that each of the three friends draws a black ball is:

16/91 * 15/90 * 14/89

(If the first friend draws an orange ball then we know our condition is false. If the first friend draws a black ball then the number of balls and the number of black balls have each decreased by one, etc.)

According to Excel (don't bother with a calculator!) that's 3,360 out of 728,910. Six-digit numbers are still difficult for a human to put in perspective but at least they're not quadrillions.

(And yes, consistent with what Nate said, about 215.9 to 1 against.)

Oh, if you're being truly anal about problem equivalence, you can have 91 people in a line, each of whom will draw a marble without replacing. Three of those people are friends, and they happen to be at places within the line that correspond to the ESPN games on the Giants' schedule-to-date.

Poker Hand Flashback of the Day

"What's more annoying, hearing about somebody's fantasy team, or hearing about his bad poker beats?

Nate Silver: Fantasy Team. Surprisingly not close. With the bad beat story, you can usually just tune it out and be done with it, but with someone talking about his fantasy team, you're usually expected to produce some coherent reply."
--Baseball Prospectus all-star game live chat. Nate is known elsewhere as Nate Tha Great.

An old roommate is getting married in two days. Several friends are in town for this, including Chad (the best man at my own wedding). He and the Mrs. are staying at our place, just as they did before our wedding.

(Friday night Chad and I stayed in the same hotel where most wedding guests stayed. The previous two nights Julia stayed at her parents. Because we had a one-bedroom apartment then (now a two) I slept on the couch those nights.)

The nights Julia was away I ratcheted up the stakes a bit in the on-line poker. If this was all about the last days of bachelorhood then there are certainly worse ways to mark that passage.

Anyhow: No Limit cash game, [mumble] stakes. Hero had A6 suited in the big blind. Some people limped and/or folded. Button min-raised. Hero called, anyone else still in the hand called.

The ragged flop (let's say it was T63) gave Hero second pair and a backdoor flush draw. Hero led with a half-pot bet, Button min-raised again, Hero called. Two players left.

Turn was a 6. Hero check-pushed. Button thought about it a bit. Hero felt a rush of anticipation and turned his attention to other tables. A few seconds later Hero noticed a shocking lack of stack at the original table.

(Villain called with AA. Naturally the river was the other ace. [There's only a 33% chance the link is apt. My suit was red.])

One More Anniversary Oddity

This is (at least) the second year in a row that R. spent his June 16 involved in bachelor party festivities, among a group of friends for whom the highlight of a bachelor party is an especially intricate board game.

Do I Underrate How Surprising or Groundbreaking This Is?

Benford's Law

"In many data series a surprising number of entries begin with the number 1, and the number 2 is also more common than a random distribution might suggest." --Tyler Cowen

You might not have thought of this (I hadn't) but in hindsight it makes total sense. Pick a random integer and see what happens with the digits of the numbers between 1 and that number. For example, in the tens digit you see a 1 ten times before you see other number, a 2 ten times before [etc.].

The interesting question for any given set of data is whether the law is applicable. It probably is, but think about conditions (upper or lower bounds, whether the data in question is better thought of as a sequence of digits than as an n-digit number) that would make it inapplicable.

I hope everyone realizes that the second parenthetical reason above is why lottery numbers decidedly do not follow Benford's law.

Even if there was a lottery setup that sort of applied to Benford's law -- e.g, if a lotto drew just one ping-pong ball from balls numbered sequentially from 1 to some three-digit number and thus the first digit was more likely to be 1 than any other number -- this would tell you absolutely nothing about whether 101 was more likely to appear than 102, 102 than 103, etc.

While we're here: Take a lottery that involves drawing six balls (without replacing) numbered from 1 to 99. Assume the balls have been properly randomly shuffled. THINK QUICK: Which outcome is more likely, 1-2-3-4-5-6 or 48-40-1-79-72-93? (If you get this wrong I will sigh sadly and die a little inside.)

This is all very interesting (I claim) and very easy to understand if explained correctly but it doesn't get nearly enough coverage. I'm embarrassed to admit that until the Marginal Revolution entry I'd never heard of Benford's Law, nor intuited the same result.

Uh Oh

Poker-style betting rules for golf.

The funny thing is you can do this with just about any multi-round contest (c.f. the Backgammon doubling cube (shout-out to Dan Harrington)) but it's so perfect for golf on so many levels, I'm surprised nobody thought of it sooner.

Iron Dragon: Worth A Hot Poker to the Eye?

I'll admit I'd wanted to learn how to play Iron Dragon. That said... six players, including four of us newbies, made for five hours (even after reducing the victory condition from 7 major cities + 250gp to 7 + 125gp) worth of pain. The really frustrating part was the opportunity cost of playing this one thing instead using that same time to enjoy three or four games with the other half of the party.

(Have I mentioned how big a Trans[location] snob I've become? Simple and elegant tactical games are your friend.)

The rest of this post is basically for Greg (or any other Iron Dragon connoisseur out there).

So for someone who claims to be all about efficiency (I enjoy wonkish stuff about logistics or operations research, and despise manual processes that take linear time), it turns out the games at which I do the worst on my first play-through are the efficiency games -- the ones where your levels of production snowball out of past production.

I went third. Since my initial cards had some nice synergy involving Fish, I planned to start in the top left corner of the board (what I thought of as "northwest" until I noticed that the compass on the game board was a bit askew, so technically more like NNW). As it happens that's exactly where the first two players BOTH went. So I started out in Old World (bottom right of the board), with some high-risk, high-reward strategy involving Wands. I ended up having to cut all the way across the bottom of the board, and it was a long time before any other player's moves became remotely relevant to what I was doing.

(If it weren't for our "you can borrow anything from the bank as long as you eventually pay back double (and don't use the borrowed funds specifically to hose an opponent)" house rule, I'd have probably been dead in the water, reduced to several turns in a row turning in cards until I got something I was capable of doing.)

Eons later I actually got out of debt and ostensibly came in fourth (I had seven major cities plus 55 gp when the game ended, though I'd just made a huge delivery; fifth place had fewer gp but was about to pass me; sixth place was still in debt). Ironically the one city I didn't connect was the "K" city smack in the middle of the board.

(If it weren't for the Rainbow Bridge event I probably couldn't have made it to the top-right-corner city either.)

Anyhow, my post hoc understanding of the game is way, way different from what I assumed at the outset (though still quite different from what other players made it out to be). Unless I'm missing something huge, this is NOT a game of immediate adversarial maneuvering (yeah, sure, other people will hog the most direct routes to the places you get to last, but that zig-zag route costs you, what, about 6-8 extra gp? pshaw).

On the contrary, I wonder whether the best strategy (maybe even the only strategy if you don't have that "borrow from the bank" house rule) is to tacitly cooperate with one or two opponents: Build adjacent to each other (and rent each other's track) so that early on you have access to a whole bunch of cities without needing to burn precious capital laying track you can't afford yet.

Full circle: from the TransEuropa on the other side of the apartment I heard an otherwise very smart person ask, half-jokingly, why bother to build at all if everything you play helps everyone else as well as yourself. Very amusing way to miss the point (of Trans, of Settlers, and of a lot of similar games): OF COURSE you're helping [some] other people at the same time you help yourself. Part of the trick is to get yourself into a position where you're helped as much as possible by other people (read: fewer things FAIL to help you than fail to help other people).

What Am I Playing Lately?

Moai

Leap of Faith

In both cases the music is especially soothing.

Brain Teasers 101

Post a brain teaser whose answer is very simple (but not immediately obvious). Examples, which many of you have probably heard hundreds of times:

1. You're in a room with two doors. One leads to freedom, the other to torture. (Obviously you want freedom.) There are two guards here, one of whom always tells the truth, the other of whom always lies. You know that to be the case (it's unclear how you know) but you don't know which is which. You're allowed to ask either guard (but not the other) ONE question, and only one, after which you will choose a door. Devise a question from whose answer you can pick the correct door with 100% certainty.

2. You have three boxes of marbles. One contains only blue marbles, one contains only red marbles, and one contains both blue marbles and red marbles. One is labeled "BLUE MARBLES," one is labeled "RED MARBLES," and one is labeled "BLUE MARBLES & RED MARBLES." However, you happen to know that all three boxes are currently mislabeled. The good news is, you can figure out which label belongs on which box by drawing just ONE marble from one box. Explain how!

3. You and several other contestants are invited to a game show taped on the set of The Price is Right. However, this show consists entirely of spins of a wheel. Until the game is won, contestants will take turns, each getting one spin. If a spin lands on "$1.00" (assume that any given spin will hit that square with 1/20 probability) then that contestant wins $10,000 and the show ends immediately. Otherwise, that contestant goes to the back of the queue and the next person spins. (Assume the outcome of each spin is independent of the outcome of any other spin.) When the initial queue forms, what's the best spot in the line and why? (Hint: Your initial impression is probably exactly right but the trick is to see, mathematically, why it's right.)

4. (In theory this one is more difficult but I hope it becomes clear why I put it here.) Big Tex plays a lot of hold em poker: Hundreds of hands a day every day for the last several years (at a casino where the dealers are known to be 100% honest). Despite the reputation of some poker players, you know him to be 100% a man of his word. Big Tex wins a huge pot with pocket aces. He turns to you (watching on the rail) and says "Okay, that was hand zero. The next hand I'm dealt will be hand 1, then hand 2, etc. Now, you know the odds of getting pocket aces on a hand." [N.B. 4/52 x 3/51 = 1/13 x 1/17 = 1/221.] "I want you to write down on this post-it note the NUMBER of the hand on which I get my NEXT pocket aces. Fold it up and I won't open it until that hand. If you're right I'll give you $10,000." What hand number should you write and why?

False Epiphany of the Day

Wow. I'd been aware of "the long tail" as business jargon but
this post severely deflated my interest the main purveyor of that piece of jargon.

How best to simplify it's findings? Let's see...

If you choose a bunch of characters at random (one of which is a space) then the length of intervals between spaces will follow a power law. WHO KNEW?!?

(Hint: No matter what characters your monkey just typed, you have exponential waiting time until the next time the monkey hits the space bar.)

This Game Is So Wrong

I don't mean morally wrong: I have no objections to pretending to wipe out the human race with a viral pandemic. But this game has inaccuracies in multiple disciplines:

GEOGRAPHY - Western Europe has no airports? (I give a free pass to the low population levels and "one airport per continent" statuses because those are obvious simplifications.)

GEOPOLITICS - Countries wouldn't sealed off their borders until after 100,000 people [in this game, 1/6 of the human population] had already died over two months?

BIOLOGY - If thousands of people contract a virus, and several weeks later that virus mutates into something else, the virus's transformation would instantly affect the instances those people previously contracted?

COMPUTER INTERFACE DESIGN - The virus definition screen and the world map can't possibly be on the same screen at once?

It's a most annoying game, about which the most annoying thing of all is how much time I spent on it yesterday. (The only way I can "win" is to reduce the virus to zero lethality before everyone shuts down their transportation, then bide my time until it somehow migrates to Western Europe, at which point you drop the hammer on fatal symptoms. You have to get to Western Europe before it occurs to them to seal that border. This ends the world by about Day 90 but doesn't get me anywhere near the high scores.)

Hey, I Know That Guy!

The alphabetically last member of the U.S. Sudoku team, that is.

I don't think Jason Z. and Mike Develin have ever met each other, which is funny given their common interests and how many mutual friends they have. I'm jealous of both for playing particular mind games well enough to represent our country.

Speaking of the latter (by the way, doesn't Rex Grossman sort of look like him from the neck up?), his January 25 quick hit sounds like exactly the type of practical advice my father-in-law the engineer/inventor would give. Meanwhile, he's exactly right about his February 5 rambling.

I'm lucky to be a relatively healthy person so that I have the luxury of taking sick days on a somewhat lower threshold of actual sickness than most people have (yet still without using many). I can't stand it when sufficiently sick people are out in public (especially on airplanes!). But then I'm evolving into one of those people who thinks that people don't use Purel nearly frequently enough.

(I'm still a clutter-messy person, though I'd like to claim not a germ-messy person. But enough of this: How on EARTH did I get here all the way from congratulating Jason Z.?!)

The Princes of Florence

This game seems to combine the best elements of Funkenschlag with the best elements of Puerto Rico. At the moment I'm head-over-heels smitten with Trans Europa, which means that at the moment Rio games seem gratuitously complicated to me; that said, this one is worthwhile.

Once you know how to play this, it seems really straightforward, despite how fiendishly complicated it seems when encountering the rules cold. OVERSIMPLIFICATION of the rules (with some finer points left out):

Your goal is to get prestige points. The best way to get prestige points is for someone in your court to complete a work. (You also get prestige points from buildings, redundant landforms, and satisfying criteria on Prestige cards, though these are ancillary to the central thrust of the game.)

There are seven rounds; each round has an auction phase and an action phase.

Auction: you acquire a landform, a Builder, a Jester, a recruiter (reuse someone else's already-played artisan/scholar), or a Prestige card.

Action: you do up to two things from among completing a work, adding an artisan/scholar to your court, building something, adding a Freedom (makes works more valuable, see below), or getting a Bonus card that (when played with a work) might make that work more valuable.

Each artisan/scholar is associated with a particular building, a particular land form, and a particular freedom. When you complete a work with that artisan (by playing that card from your hand -- each person starts with three but you can only use any given artisan once), the value of that work depends on the answers to these questions:

1. Have you build that building?

2. Do you have that freedom?

3. Do you have that landform?

4. How many Jesters (if any) do you have [Jesters improve artisan morale: Jesters need to fetch a high price at auction to prevent a degenerate all-Jester strategy]?

5. How many artisans do you have (counting both the ones in your hand and the ones you've played)?

Based on the above, each work has a Work Value that you can claim in either Prestige Points or money. Prestige Points are how you win but if you go broke you'll end up s.o.l.

My first and only time playing, from memory:

1. Park; built a Workshop and completed a work (took the Prestige). I know now how shortsighted and pointless that was but at this point none of us really knew what we were doing.

2. Lake; took a freedom and another artisan.

3. Prestige card; another freedom and completed a work (took the money)

4. Builder; built a University and an Opera House

5. Jester; drew an artisan and completed a work

6. Another Park; took a freedom and completed a work

7. Another Lake; took a bonus card (very useful one as it turned out) and completed a work

Despite thinking I was losing badly I ended up winning.

We've Played So Much Trans Europa This Month

Also, we were in L.A. taking care of our niece and nephew as a birthday present to their mom (while their dad took their mom on vacation).

Our six-and-a-half year old nephew played two rounds of "the train game," one as Julia's teammate and then one holding his own cards (which he never did show anyone) and choosing moves completely for himself. He successfully connected some green (ex-Soviet) city with Budapest(?), got his tracks networked, and went to Bilbao all by himself. (We didn't announce any scoring but he'd have done especially well had his selfish Madrid-holding uncle actually helped him on the westward drive.)

The point is this is a very easy game for even a six-and-a-half year old to understand (and play a round), yet still engaging enough not to bore my science grad student and Ph.D. friends.

Oh, and our 10-year-old niece won both of the first two rounds she played, and had the patience to play one particular overarching game all the way to the rulesheet-specified end.

(The Trans Europa/America points system is similar to Hearts in that you're trying to avoid being charged points, indeed to have the fewest whenever someone hits a specified milestone.)

Trans Africa

Pro: Brazzaville and Kinshasa just a double-line track away from each other.

Con: The southern group of cities might be too bunched together. (Jo'burg, Capetown, Durban, [...]?) Mabe just give South Africa (and its surroundings) the Florida (non-Jacksonville) treatment and make that group be Luanda, Lusaka, Harare, Maputo, Dodoma, Gaborone?

Cecil Rhodes would surely disapprove.

Anyhow, we now own Trans Europa, and over the weekend I made a homemade map identical to the Trans America layout. As soon as I get around to finding playing cards w/five different color backs, we can replicate Trans America using tracing paper on that homemade board.

(And really once you have the concept down and an free easy source of printable triangular (NOT hex!) graph paper, you can design a map of anywhere you want.)

The Ultimatum Game

Have you read about this?

Two anonymous parties interact only once. One is told to split an amount of money; the other then either accepts or rejects that split. If the second player accepts the split then players divvy up that money in the specified ratio. If the second player rejects then both players get nothing.

I just took a sex of your brain test (apparently I'm quite masculine, including 11 out of 12 on a surprisingly difficult spatial test) that had the ultimatum game as part of it.

They asked how you'd propose to divide 50 pounds if you were the first player in the ultimatum game. I made a snap gut decision (incidentally, the dollar is weaker than I thought it was but not strikingly so).

So yeah: 50 pounds. Divide it up, knowing that an anonymous person would get to accept the offer (you get what you allocated to yourself, other person gets the rest) or reject the offer (you both get nothing). Make your call before you read on.

I demanded 35 pounds (thus offering 15 pounds). I wince at how much money I seemed to be "leaving on the table" but did feel somewhat risk-averse about whether an anonymous person would just hose me out of spite.

Two Games

Play within your browser. Requires Java but does not require any emulation.

Montezuma's Revenge (this site has another, vastly inferior, MR instance that unfortunately has a much higher Google PageRank).

JezzBall

The Year 2100 Is Not A Leap Year

Years divisible by 100 (but not by 400) are not Leap Years. Since 2000 is divisible by 400, it came off as an application of the Leap Year rule instead of an exception an exception.

This thwarts what I'd previously thought of as a nifty calendar fact: At least between 1901 and 2099 there's a repeating 28-year cycle of day to date.

Questions for astute readers (based on a trivia question I wrote that was returned to me, probably correctly):

1. Why is it a 28-year cycle, rather than 27 or 29 or 14 or some other number?

2. In particular, how many days over a period of exactly 28 years (some time between 1901 and 2099) will it be Friday the 13th?

The answers to both are fairly simple.

LOL Dimeaments

If you were at the deathbed of a woman you secretly loved all your life but whom you never had the courage to tell and then she tells you that she secretly loved you all these years, what a great opportunity that would be to practice your "poker face."
--Dan Liebert, Verbal Cartoonist

At the bank today I changed a twenty into two rolls of dimes and ten ones. No questions asked, though I can't immediately think of any legitimate, non-business use for two rolls of dimes. Even the illegitimate uses just scream out for better herbs and/or higher stakes.

Wikipedia used to have a donkament page; it now redirects to a list of poker jargon that doesn't even include the term. (The phrase "LOL donkaments" and variants used to be in vogue on certain 2+2 message boards. Right when I stopped reading those the meme was being run into the ground.)

Anyhow, I claim that even for home play ring games are better than tournaments. Your mileage may vary, but the only real hassle with a ring game is making change, while tournaments involve getting everyone to show up (sort of) on time, keeping track of when blinds should go up, reaching a consensus on whether to allow rebuys and when, prize structure, etc. Then the whole thing comes down to five people each having about ten times the big blind worth of chips, pushing if you're the first in with a decent hand, and hoping for the best.

(Note to the wife: Yes, that last sentence sounds like a claim that poker is about luck. Really it's that a particular structure of poker has decisive luck.)

One time I drove eight miles to play cards with friends. After I got there we spent 45 minues watching the Colts and Patriots (Thursday season opener) and waiting for critical mass. Very first hand was set over set. I calmly walked right out the door (they caught up with me to collect the buy-in) and went over to the office to write trivia questions. In a rink game that'd just be a rebuy (admittedly in a tournament with rebuys that would also just be a rebuy).

(I've often driven 35 miles to play cards with friends near Stanford. One of two things is true: Either that same hand would have given me a one-and-done after a 35 mile drive, or the distance I drove would have affected how I played a hand.)

I know someone who strongly prefers tournaments to ring games because he believes he's a much better tournament player than ring game player. Such people do exist (Phil H., for one) but I'm not convinced, hand-by-hand, that ring play is all that different from how you'd play the early stages of a rebuy tournament. That said, apparently (before the big U.S. crackdown on on-line poker), the tournaments at the leading on-line poker site were much softer than the cash tables.

Brain-Dead Story Problem of the Day

From one end of I-40 to the other troopers across eight states will be out in record numbers to keep the highway and drivers safe. [...] Under the program state troopers will be positioned every ten miles along I-40.
--Troops out in record numbers along I-40

Length: 2554 mi
Miles km state
155 249 California
359 578 Arizona
373 600 New Mexico
177 285 Texas
331 533 Oklahoma
284 457 Arkansas
455 733 Tennessee
420 675 North Carolina
2,554 4,110 Total
--Interstate 40 on Wikipedia

"The Highway Patrol said they couldn't say how many troopers would be on the roads [...]" --ibid

CAN YOU?

I did learn something from this exercise: The easternmost part of I-40 is so circuitous that the shortest distance from Barstow, CA, to Wilmington, NC, is actually 30 miles shorter than the length of I-40. It dips through South Carolina.

An Anecdote About Prodigy-Based Movies

Thinking of quasi-scholastic pursuits turned into movies reminded me: (I've probably told this a million times but to me it never gets old.)

1992 High School Chess Championship (Lexington, KY). [I played in the Open division despite being eligible for the Under 1600. They actually had trophies just for people like me, and I took home a trophy for having one of the N best results (4.5 of 7: D L W D W D W) among under-1600ers who "played up" into the Open.]

One of the championship contenders was Josh Waitzkin, whose dad had just written the book about him that would later become the movie. After Waitzkin lost badly in some climactic game, a joke circulated around the floor that the sequel would be called STILL Searching for Bobby Fischer.

Child's Games: A Strategy Guide

The rock-paper-scissors reference reminded me: I've been sitting on this one since our visit to the niece and nephew ten days ago.

Clue Mysteries (obviously not to be confused with the original Clue): My first strategic act was a clear mistake. Visiting the police inspector is a waste of time. Once you've visited enough people to unqiuely identify a suspect (if everyone you talked to were truthful), the probability that they're all telling the truth is way more than the probability that you'd beat your N opponents in a race.

Franklin Goes to School: When you answer a question correctly (or not), one square is at stake. Compare to the spinner variance and the card drawing variance. So much luck is involved that this is a flimsy framework within which to read elementary school questions off a card to kids who don't think of themselves as being quizzed.

Pet Detectives (card game): This is a great simpler case of Canadian fish. The best strategy just seems to be a good memory. If I knew more about what the different outcome cards did, I could recommend going for either the "cheaper" pets or the "costlier" pets.

Mario Party: The key to victory is stealing other players' stars, by an overwhelming margin. Coins are just the intermediate step of being able to afford to ride a vicious dog or whatever else it takes to steal those stars.

One somewhat strategically interesting game-within-a-game is "Same Is Lame." I assume table talk is discouraged, but the more obvious you can make it that you're always going to press the same button, the better you'll do. (If everyone knows "Matt is always pressing B" then nobody gains anything from also selecting B. To be sure, this would result in a standoff if four players settled on unique buttons, but hey.)

A Disappointing Meta-Disappointment

I was all set actually to solve the "Sudorku" in this past Sunday's Foxtrot, except that for some reason I was dead certain that 252/36 wasn't an integer for spurious reasons relating to the fact that 2 * 6 = 12. Then I actually bothered to type "=252/36" into the cell of a spreadsheet.

Apparently I'm the worst number theorist ever.

I missed the specific brainteaser but worked out the general case a day later

(But now I think the general case is actually easier to solve aside from the fact that it's so much more more confusing to explain. Anyhow, we had dinner Tuesday night with a family that has twins in eighth grade, one of whom stumped us. You've probably seen the specific version somewhere.)

N soldiers (but only one feeble lantern) are traveling at night when they reach a mountain bridge that all N of them (N > 2) need to get across. From past experience they happen to know with complete certainty that:

1. Each soldier can cross the bridge alone in a specific amount of time (either direction, always the same time interval). Without losing any generality you can sort those times from fast to slow and represent them as x[0], x[1], ..., x[n-1], where x[i] <= x[i+1].

2. If two soldiers cross the bridge together, they'll always take exactly as long as the slower one would have taken crossing alone.

3. The bridge cannot hold more than two people; it would fall apart and kill all three (or more).

4. Nobody can so much as stand on the bridge without being in immediate proximity of their light source (that one feeble lantern, which as you might guess will keep going back and forth) or else that person will misstep in the dark (or, if standing, lose their balance) and fall to a grisly death.

5. There is no hidden way to conjure up more light sources, circumvent the bridge, or anything else that would make this a lateral thinking puzzle rather than a math problem.

6. Luckily for them, the instant one or two people finish crossing the bridge, anyone making the opposite crossing can grab the light and immediately be recrossing with no time lost in the transfer.

Now, give me both the strategy that always minimizes their time to cross, and the formula that expresses (in terms of x[0], x[1], ..., x[n-1]) that time. It can be a conditional formula (it very much is conditional) but you have to describe the conditions precisely.

Some time soon I'll write up my answer (I'm confident I'm right; whether I succeed at a rigorous proof is to be seen) and put it in the extended entry.

UPDATE: Inelegant solution posted (I idly wonder how many points it would get on a Putnam problem - 6 of 10?). Oh, and if you just want a brainteaser that's tricky without being confusing, then here you go:

Same facts as above, but it happens to be four soldiers. A takes 1 minute, B takes 2 minutes, C takes 5 minutes, D takes 10 minutes.

Base cases:
N=0, time = 0 (no crossing is necessary)
N=1, time = x[0] (one crossing)
N=2, time = x[1] (one crossing)

N>2: One crossing is not enough, in fact the lantern now has go back (and be carried by at least one person), so there's a maximum net gain of one person for each pair of crossings until the final step. So for N>2, we need at least 2(N-2)+1 crossings [also known as "2N - 3"].

A reasonable plan for N>2 is for the fastest one to hold the lantern all the time and escort the others one by one. That strategy has time x[n-1] + ... + x[1] + (N-2)*x[0], where the last term is the sum of all the return trips and the other terms are the individual forward trips.

I believe it's easy to prove that any successful maneuver with more than 2N-3 crossings has some easily eliminated redundancies, though I'll "leave that as an exercise to the reader," with mild regret that I didn't find an elegant way to nail that part of it.

Anyhow, when N=3 the best you can do is x[2] + x[1] + x[0]; any possible set of crossings is exactly that fast, or slower. So we have well-known solutions for N=2 and N=3.

Now what if for a given value of N soldiers, we already knew the general solution for N-2 soldiers?

There are two ways to simplify the N problem to the N-2 problem by getting the two slowest soldiers across the bridge first:

Either 0 escorts (n-1) and (n-2) one by one -- taking time 2*x[0] + x[n-1] + x[n-2].

Or, 0 escorts 1; n-1 and n-2 cross together; 1 takes the lantern back -- taking time x[0] + 2*x[1] + x[n-1]. Whichever of those two is faster depends on whether the average of x[0] and x[n-2] is greater than x[1].

(Example: The canonical form of the brainteaser is four soldiers, where x = [1,2,5,10]. They can do it in 17 minutes, not 19, because 2 + 2 < 1 + 5.)

The proof that "move the two slowest guys first to simplify the problem" is efficient involves these particular points:
1. To pull this crossing off most efficiently, the slowest two guys only cross the bridge once each. Any other way of doing it could be rearranged more efficiently by having them switch roles with people who cross just once.

2. Thus there's no way for the existence of the two slowest guys to make the subset of the other N-2 guys crossing go any quicker.

So the strategy:
Until there fewer than four guys left, get the slowest two guys at a time across. In each case, have the fastest guy escort them both if he's fast enough to make up for the second-slowest guy, otherwise have the fastest guy escort the second-fastest guy (and that guy bring back the lantern) otherwise.

The formula:
If N is even,
min(2*x[0] + x[n-1] + x[n-2], x[0] + 2*x[1] + x[n-1]) + min(2*x[0] + x[n-3] + x[n-4], x[0] + 2*x[1] + x[n-3]) + ... + x[1]

If N is odd,
min(2*x[0] + x[n-1] + x[n-2], x[0] + 2*x[1] + x[n-1]) + min(2*x[0] + x[n-3] + x[n-4], x[0] + 2*x[1] + x[n-3]) + ... + x[2] + x[1] + x[0]

Ambiguity Aversion as a Built-In Swindle Protector

Read about the Ellsburg Paradox here (first post of four). The basic scenario: I have an empty urn and some red marbles, white marbles, and black marbles. First, I put a red marble in an urn. Second, I flip a fair coin; if it comes up heads I put a white marble in the urn, if it comes up tails I put a black marble in the urn. Third, I flip that fair coin again, same consequences. Finally, I give you a choice among these four games:

A1. Draw a marble from the urn. If it's red you get +2, else you get -1.
A2. Draw a marble from the urn. If it's black you get +2, else you get -1.

B1. Draw a marble from the urn. If it's red or white you get +1, else you get -2.
B2. Draw a marble from the urn. If it's black or white you get +1, else you get -2.

If you're astute at probability then you should notice that all four of these potential games have an expected payoff of 0, and moreover that A1 and A2 are functionally equivalent just as B1 and B2 are functionally equivalent. The "paradox" is that despite all that, most people polled will strongly prefer A1 over A2, and B2 over B1. (In both cases simplifying the problem to "red vs. not red" so that the outcome of the coin flips is irrelevant.)

Now, if you know what these probabilities are, and you're confident that your calculations are correct, of course you should have no preference. I assume that the people who do have a preference have this gut feeling that there's a trick involved, and so are gravitating to the simpler scenarios to minimize the chance of being fooled somehow.

Of course, some function of your risk aversion may cause you to prefer the A1/A2 odds over the B1/B2 odds (or vice verse).

Poker Ban Thoughts (By Request)

If you live in a U.S. state other than Nevada then depending on your jurisdiction on-line poker for money was probably already illegal. That said, I exceeded the speed limit over various stretches of 880 and 580 on my way to work, this morning and every morning.

A bill that specifically forbids using particular monetary sources to fund on-line gambling is a lot like (there are so many ways to mangle this metaphor, especially by holding onto it too long)... imagine that the technology existed physically to prevent cars from exceeding the speed limit of the road they were on. Requiring this technology would be some kind of dystopia.

There's a hardcore strain of political philosophy which insists that you should either enforce a law rigorously or repeal it. I'd be strongly in favor of repealing either speed limits or gambling bans but I'm not holding my breath for either one. In lieu of that, everyone should remember that there are only so many law enforcement resources available, and using them to go after Internet gamblers is farcical (unless we already won the Global War on Terror and I hadn't noticed).

As for me, it's been almost a month since the night that I uninstalled Party software at 5 a.m. I haven't looked back.

Funniest Phrase Ever

Courtesy of someone who sent e-mail to Bill Simmons:
Little League World Series of Poker

RuSTLING CARRIE(D) BEAtINGS

The card player in me can only dream of ever getting hands as sick as the tiles I got just now.

This Is Why A Formal Ban on On-Line Poker Will Pass Eventually

It would be a horrible law enacted for all the wrong reasons, but still:

"I just lost $600. My mom is going to kill me."
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