But just because those are the most frequent letters doesn’t mean they are the best letters to call. For example, many words take the form of _ _ _ _ N _; Wheel fans know that an N in the penultimate space is a very good indicator that the word ends in I-N-G. As a result, calling G is not very useful in this situation, since you can guess it is there already anyway.
Then what are the best letters? Unfortunately, this post will not provide an answer. However, it will show one wrong way of deriving an answer, which serves as a perfect example of post treatment bias.
First, let’s look at the data. Here is a list of non-RSTLNE letters, the frequency contestants call them, and how often they win if they called them. Keep in mind that data consists of every bonus round from 2007 to 2012, or 1166 total puzzles.
The results seem surprising. H, G, D, and O aren’t anywhere near the top. Instead, V, U, J, and Y rule the day. (Yes, yes, small sample sizes.) Maybe we all should start calling V J Y U!
Let’s talk about post-treatment bias for a moment. In the absence of a controlled, randomized experiment, our goal here was to take existing data on games played and infer which letters are the best. Our treatment, of course, is the selected letters. But this causes a problem. From Gary King’s slides, post-treatment bias occurs when:
- when controlling away for the consequences of treatment
- when causal ordering among predictors is ambiguous or wrong
The latter is problematic in this case. What if the letters weren’t helping players solve the puzzle but rather solving the puzzle was helping players pick the letters?
To see what I mean, let’s look through that list again. Before, I secretly hid Q, Z, and X from you. Now let’s throw them back in:
Q and Z end up on top, albeit with three observations total. Why are players undefeated when they select these letters?
Looking through the data gives the obvious answers: players already solved the puzzle before picking their letters and selected Q and Z appropriately. For example, in one of the cases, the player called Z W J I on JIGSAW PUZZLE. To say the least, I don’t think that person was randomly guessing letters.
I found a bunch of cases where this happened with V. Calling G V W I on GIVE IT A WHIRL strikes me as more than just a coincidence.
Consequently, those tables don’t tell us much. Post-treatment bias sucks, and I will have to figure out a different way to infer the best letters to call.
Sorry this post didn’t end on a happier note. Here’s a picture of some puppies trending on /r/aww to make up for it:
The United States, the rest of the permanent members of the United Nations Security Council, Germany, and Iran are in Kazakhstan this week, negotiating over the Iranian nuclear program. The West wants it to stop, fearing it will eventually yield a nuclear weapon. Iran continues to claim that its program is purely for scientific and energy purposes. No one believes that. The United States’ focus has therefore been figuring out how to convince Iran to let it go.
Unfortunately, the dialogue in the U.S. has been hopelessly misguided. In this post, I will make two claims: (1) the United States can always offer sufficient concessions to induce Iran to end its program; (2) Iran does not believe the United States can credibly commit to these concessions over the long term, thus explaining Iran’s obstinate behavior. The policy implication is obvious: if we want Iran to stop proliferating, we need to stop pretending that it is difficult to buy Iranian compliance and start seriously questioning our own commitment to giving Iran a good deal.
I discuss claim (1) in the main theoretical chapter of my dissertation. Many years ago, I saw President Obama making a speech about getting Iran to join “the community of nations” and give up its nuclear program. From my knowledge of the existing political science literature, I figured it would be easy to show that no such agreement would work. After all, if you were Iran, why would accept concessions and not build when you could accept concessions and build anyway? The temptation to shift power seemingly destroys the possibility of negotiated settlements.
However, I was unequivocally wrong. The conversation about Iranian duplicity (and my own initial intuition) fails to properly analyze Iranian incentives. Nuclear weapons are costly. If Iran is offered most of the concessions it expects to receive if it were to proliferate, it has no further incentive to develop a bomb. Sure, Iran could continue proliferating, but it ultimately will not receive any more than the United States is already giving it. However, it will have to pay the cost of the weapons, which is a complete waste at that point.
Meanwhile, the United States has incentive to make this sort of “butter-for-bombs” offer. Although the U.S. would like to offer no concessions, such a strategy is naive, since this would induce Iran proliferate. On the other hand, offering the butter-for-bombs deal ensures that Iran will not build nuclear weapons. Moreover, since Iran is not paying the cost of proliferation, there is extra pie to go around. The United States can extract it.
The paper linked above gives the details. Surprisingly, these agreements work under very loose conditions. They work in an infinitely repeated interaction; they work when Iran could freely renege on the deal without any recourse from the United States; and they work even when the United States is completely incapable of observing Iranian nuclear progress. Butter-for-bombs is all about getting the incentives right. Offer Iran enough, and you do not have to worry about the deal falling through under any circumstance.
One important assumption of the above model is that the United States’ ability to launch preventive war remains constant through time. What happens if we relax this assumption? Another chapter from my dissertation asks this exact question. I show that if the United States goes from being unable to credibly threaten preventive war one day to being able to credibly threaten preventive war the next day, bargaining breaks down, and Iran develops nuclear weapons. But the blame goes squarely on the United States.
To understand why, suppose we reach the future time when the United States can credibly threaten preventive war. At this point, Iran cannot develop nuclear weapons. If it were to, the United States would respond by launching preventive war. Iran would not receive the benefits of nuclear proliferation but still waste the cost it paid to develop nuclear technology. Internalizing this, Iran will not proliferate. But because Iran will not proliferate, the United States has no reason to make those butter-for-bombs offers from before. Instead, it can offer no concessions and still induce Iranian compliance, as Iran does not have any better options.
Now think about how this endgame affects bargaining today. The United States would like Iran to accept a butter-for-bombs agreement and avoid proliferation. But consider the problem from the Iranian perspective. If Iran accepts those concessions today, then it advances to that future world where the United States can effectively leverage preventive war. At that point, the concessions stop. Alternatively, Iran could pay a cost upfront, proliferate today, and leverage the additional power to force the United States to continue giving concessions into the future. Needless to say, the second option looks a lot more attractive. Thus, bargaining breaks down, and Iran proliferates.
The problem here is not Iran’s stubbornness. Rather, it is the United States’ inability to credibly commit to continue providing concessions in the future. If the U.S. could, then Iran would have no reason to proliferate. (This was shown in claim 1.) However, Iran expects the United States to renege on the concessions, which in turn causes the proliferation behavior.
One might object that the United States would never take advantage of its strength in that manner. To anyone who doubts that, I point to May of 2003. This was the perhaps the height of American power. Things were going well in Afghanistan, we had just run over Saddam Hussein’s army, and the Iraqi insurgency had not yet begun. Iran felt enclosed. Rather than panicking, Iran extended an olive branch. Tehran sent the Swedish ambassador (who takes care of American interests in Iran) over Washington with a sweeping offer. Iran essentially waved a white flag and put everything on the table. Their demands in return were minimal: they wanted a prison swap and normalized relations. If this type of proposal arrived today, it would be magical Christmas land in DC.
I’d like to say that the Bush administration gleefully accepted the offer and sent a warm reply. But they didn’t. In fact, they sent no reply at all. They simply ignored it and chastised the Swedish ambassador for bringing it to their attention.
The domestic political consequences in Iran were bad. Moderates held the presidency at that point and pushed for the deal with the Ayatollah’s blessing. After their failure, they were pushed out of government. Mahmound Ahmadinejad’s administration replaced them.
As we all know, America’s position of strength evaporated, leading to the drawn-out insurgency. Iran knows that the decade-plus of war has left us exhausted from conflict. Preventive war is unlikely today. But given enough time, our war-weariness will fade away. Iran is concerned that the United States will immediately switch back to the firm fist of the Bush administration’s years. They see our temporary weakness as now-or-never opportunity to proliferate. And they are going for it.
How do we get out of this mess? It’s possible that we cannot, and we just have to suffer the consequences of another poor foreign policy decision from the Bush years. But if there is any hope of reaching an agreement with Iran, it must come through the United States demonstrating its commitment to ongoing concessions that will not instantly disappear at some later date. Unfortunately, the domestic political dialogue in the U.S. focuses entirely on the credibility Iranian commitments while treating our own as the word of God. This is misguided. And until we can have a serious conversation about our own credibility, we will not make any progress with Iran.
C M D A? Try H G D O.
I have been watching Wheel of Fortune for more than 20 years now–my parents even tell me that the game taught me how to read. And all the while I have unquestionably thought that the best letters to call during the bonus round are C M D and A. But watching the program last night, I realized I had no factual basis for that. It was a belief. It was not science.
So I figured I would do some quick Googling and find out what the best letters actually were. Turns out, it seems no one has figured this out yet. (The best result was some dude on Yahoo! Answers, which wasn’t exactly reassuring.)
No problem. I found this website, which archives Wheel of Fortune bonus round puzzles and other associated information. It has a complete record from 2007-2012, or 1166 total puzzles. I scrapped the data and began my analysis. Here are some of the important findings:
1) I am not a lone in my belief: C M D A are the four most frequently called letters at 64.6%, 59.9%, 57.9%, and 48.3%, respectively.
2) P H O G are the next four in order at 38.2%, 34.5%, 31.1%, and 21.0%.
3) O is the most common letter to appear in puzzles, consuming 9.5% of all letters. This just goes to show you that the bonus round puzzles are not a random sample of words from the English language–in real life, O is the fourth most common letter after E, T, and A.
4) Despite being the most common letter in English, E is the fourth most common letter in the puzzles after O, I, and A. Ostensibly, they give you R S T L N E for free because they are common letters. However, the producers intentionally pick puzzles where those letters don’t show up. Like cake, the value of R S T L N E is a lie.
5) M is an awful pick, ranking 21st on the list. It only accounts for 2.1% of the letters. Only V, J, Q, Z, and X are less frequent. No one ever calls V, J, Q, Z, or X unless they already know the answer to the puzzle and want to show off. Yet 57.9% of players pick M. Go figure.
6) H is a great selection. It has a frequency of 4.6%, placing the highest among non-R S T L N consonants. It ranks just slightly below the least frequent vowel (U, 4.7%) but higher than N (4.5%), S (3.8%) and L (3.7%).
7) If you solely want to maximize the number of letters that are revealed, H G D O is the best selection. D (3.5%) is very close to P and B (both 3.4%), so there is some wiggle room here.
To hammer home the point, the plot below shows the frequency of called letters versus what appears on the board (click to enlarge):
The mess on the bottom left corner is the V, J, Q, Z, X trash.
A couple of notes before I wrap this up. First, I want to emphasize the distinction between “most frequent letters” versus “best letters.” What shows up most frequently might not be the most useful in terms of actually solving the puzzle. G’s frequency might be overrated since a lot of those come from -ING suffixes, which you could reasonably guess if you see a word like _ _ _ _ _ N _. Letters like C, B, or P might have an advantage in that they could appear at the beginning of words more frequently and are thus more valuable. This is something I could check on later.
This segues to the second point nicely. There are a bunch of interesting questions we can now answer now that I have this dataset. Expect more investigative posts like this in the future.
 The category What Are You Doing? only appears 9 out of 1166 times. Since this category always begins with a word ending in -ING, having the G be revealed in that slot is worthless to a contestant. But even if you remove those puzzles from the sample, G ranks much higher than the nearest alternatives.
The Walking Dead is cable’s most successful TV show, ever. I’m writing this after “Home,” and I’m going to assume you know what is going on by and large.
Here’s what’s important. As far as we care, there are only two groups of humans left alive. One, the good guys, have fortified themselves inside an abandon jail. The other lives in a walled town called Woodbury. They became aware of each other a few episodes ago, and they have various reasons to dislike each other.
War appears likely and will be devastating to both parties, likely leaving them in a position worse than if they pretended the other simply did not exist. For example, in “Home,” the Woodbury group packs a courier van full of zombies, breaches the jail’s walls, and opens the van for an undead delivery. Now a bunch of flesh-eaters are wandering around the previously secure prison.
Meanwhile, the jail’s de facto leader went on a mysterious shopping spree and came back with a truck full of unknown supplies. I suspect next episode will feature the jail group bombing a hole in Woodbury’s city walls.
All this leads to an important question: why can’t they all just get along? It’s the end of the world for goodness sake!
As someone who studies war, I am sympathetic to the problem. Woodbury and the jail group are capable of imposing great costs on one another merely by allowing zombies to penetrate the other’s camp. The situation seems ripe for a peaceful settlement, since there appear to be agreements both parties prefer to continued conflict.
This is the crux of James Fearon’s Rationalist Explanations for War, one of the most important articles in international relations in the last twenty years. Fearon shows that as long as war is costly and the sides have a rough understanding of how war will play out, then both parties should be willing to sit down at the bargaining table and negotiate a settlement.
However, Fearon notes that first strike advantages kill the attractiveness of such bargains. From the article:
Consider the problem faced by two gunslingers with the following preferences. Each would most prefer to kill the other by stealth, facing no risk of retaliation, but each prefers that both live in peace to a gunfight in which each risks death. There is a bargain here that both sides prefer to “war”…[but] given their preferences, neither person can credibly commit not to defect from the bargain by trying to shoot the other in the back.
The jail birds and Woodbury are in a similar position:
This is a prisoner’s dilemma. Both parties prefer peace to mutual war. But peace is unsustainable because, given that I believe you are going to act peacefully, I prefer taking advantage of you and attacking. This leads to continued conflict until one side has been destroyed (or, in this case, eaten by zombies), leaving both worse off. We call this preemptive war, as the sides are attempting to preempt the other’s attack.
In the real world, countries have tried to reduce the attractiveness of a first strike by creating demilitarized zones between disputed territory, like the one in Korea. But such buffers require manpower to patrol to verify the other party’s compliance. Unfortunately, the zombie apocalypse has left the world short of people–Woodbury has fewer than a hundred, and the jail birds have fewer than ten. As a result, I believe we be witnessing preemptive war for the rest of this season.
 Get it? They live in a jail, and they are in a prisoner’s dilemma!
 I’m lame.
In case you missed it, last night’s Final Jeopardy was flat terrible. This was the semifinal game in the teen tournament; only the top (strictly positive) scorer advanced to the next round, and no one keeps any money. The scores were $16,400, $12,000, and $1,200. The Final Jeopardy category was capital cities. Pretend you are the leader and place your wager.
Ready? Cue music:
It’s criss-crossed by dozens of “peace walls” that separate its Catholic & Protestant neighborhoods
Was your response Dublin? Mine was, as was all of the contestants’. Dublin is also wrong. The correct response was Belfast.
Nothing wrong with a triple stumper, though. The wagering strategies, on the other hand, were horrible. Every contestant wagered everything. With no one coming up with the correct response, no one had any money and thus no one qualified for the finals.
This made me go insane. The leader had no reason to wager more than $7,601; such a wager ensures that the leader wins with certainty if he receives the correct response and also gives him a win against a wider variety of opposing bids, including the set of bids from the game. In game theory terms, wagering $7,601 weakly dominates wagering everything.
I then vented in YouTube form:
Here’s a comment from the YouTube view page:
This is why the idea that people are intelligent self interested agents makes me laugh. People do this kind of thing ALL THE TIME, and it’s why economic theories that don’t account for this can’t predict [stuff].
Only he didn’t say stuff.
There are two big problems with this logic. First, rational self-interest is an assumption. We use assumptions to build theories not for their accuracy but for their usefulness. The better metric for modeling is a simple question: is this model more useful than the alternative? If yes, the model is satisfactory. If not, then use the alternative. We could discard certain reality and instead use some probability distribution over rational agents and automaton agents. While this would certainly be a more realistic model, it would come at the expense of being substantially more computationally intensive without much obvious reward. We should find no inherent shame in simplicity.
Second, a good theory explains and predicts behavior. Theories are not laws–we should not require a theory to hold 100% of the time for us to find a theory useful. Contrary to what the commentator wrote, we can use “intelligent, self-interested agents” as an assumption and predict quite a lot. In fact, the reason Final Jeopardy last night caused such a stir is because it egregiously violated what intelligent individuals should do. Intelligent individuals make up about 99.9% of the Jeopardy players, which is what made last night so extraordinary.
If models are useless because of the .1%, then all of academia–hard and soft science alike–needs to close up shop immediately.
Leon Panetta, the outgoing U.S. Secretary of Defense, made an interesting claim during his exit interview. Automatic defense budget cuts will take place on March 1 unless Congress reworks the Pentagon’s allotment. Panetta urged Congress to act, saying that if the ten year, $500 billion cuts take effect, the U.S will become a “second-rate power” (his words).
$500 billion is a lot of money, so you may be inclined to agree. But anyone who has spent more than ten seconds looking at world defense spending would know the absurdity of Panetta’s claim.
Take a look at this Wikipedia article on defense spending. SIPRI keeps good data on defense expenditures around the world, and the Wiki gives a nice comparative visualization. Last year, the world’s militaries consumed about $1.7 trillion.
The U.S.’s share? $711 billion.
That’s approximately 41% of the entire world’s spending.
It’s more than Chinese (8.2%), Russian (4.1%), British (3.6), and French (3.6%) spending…combined.
It’s more than every non-permanent member of the United Nations Security Council…combined.
So, the U.S. isn’t exactly at a crossroads when it comes to military spending.
Moreover, $500 billion over the course of ten years comes out to $50 billion a year. That would still give the United States $661 billion in defense spending, still 39% of the new world’s spending and more than 4.5 times as large as China’s receipts. Thus, even if you think of China as having a second-rate military, ours would still be substantially better.
I understand that $50 billion will result in meaningful cuts to our military. But the United States needs to reduce spending somehow. Among all the alternatives, this one seems relatively painless.
 SIPRI has to estimate expenditures from a lot of countries, notably China and Russia. While their figures may be underestimates, SIPRI would have to have made massive mistakes for the effects to be relevant to the argument.