Imagine there’s a completely random event with two outcomes, say flipping a coin. Each flip has an equal probability of landing heads or tails. Now imagine that we’re interested in seeing how long it takes to get a certain sequence of outcomes.

Pattern 1

Tails, Heads, Tails

Pattern 2

Tails, Heads, Heads

Now, suppose we flip a coin until Pattern 1 is reached, note how many coin flips it took, and then we repeat the process many times and average how many flips it takes to get a tails-heads-tails sequence . After that, we go through the same process to see how many flips it takes to get Pattern 2, a tails-heads-heads sequence. For example if we start flipping a coin for pattern 1 and we see:

tails, heads, heads, tails, heads, tails

Then we reached Pattern 1 after only six coin tosses. Sometimes it will take as few as three coin tosses, but other times it will take many more. If we were to repeat this test thousands of times and calculate the average number of tosses it takes to get Pattern 1 and compare it to the average number of tosses it takes to get Pattern 2, which be the bigger number?

Sorry, there are no polls available at the moment.

The first correct answer with a valid explanation wins a beer (if you can make it to Taipei to collect).
Update: Two correct answers are in! Ray Myers, with some lisp code to brute force the answer, and Robin with a clear explanation of why. When and if you make it out to collect, drinks at the Taiwan Beer Factory are on me.

Related Posts:
Even Simple Probabilty Puzzles Can Be Tricky
Game Theory and Bluffing
Good God are There a Lot of Morons on Digg

Milton Friedman (July 31, 1912 – November 16, 2006) was an American Nobel Laureate economist and public intellectual. An advocate of laissez-faire capitalism, Friedman made major contributions to the fields of macroeconomics, microeconomics, economic history and statistics. In 1976, he was awarded the Nobel Prize in Economics for his achievements in the fields of consumption analysis, monetary history and theory and for his demonstration of the complexity of stabilization policy.[1]

According to The Economist, Friedman “was the most influential economist of the second half of the 20th century…possibly of all of it.”[2] Alan Greenspan stated “There are very few people over the generations who have ideas that are sufficiently original to materially alter the direction of civilization. Milton is one of those very few people.”[3] In his 1962 book Capitalism and Freedom, Friedman advocated minimizing the role of government in a free market as a means of creating political and social freedom.

Wikipedia: Milton Friedman

Friedman’s words about the War on Drugs are just as relevant today as they were two decades ago. Currently, America leads the entire world in prison population, both in total number, and on a per-capita basis.

The proper role of the government is to prevent other people from harming an individual. Government never has any right to interfere with an individual for that individual’s own good. The case for prohibiting drugs is exactly as strong and as weak as the case for prohibiting people from over eating.
We all know that over-eating causes more deaths than drugs do.”

-Milton Friedman

The following video is about Milton’s idea about the limited role of government:
Milton’s definitive work on the subject, considered by many to be amongst the 100 most influential post WWII books ever written, is Capitalism and Freedom.

study stuff Hosted on Zooomr

There are several components in Farber’s system, but the one that has helped me the most is the use of what he calls “hidden moments”. The idea is nothing new, but I’ve found it incredibly effective. The premise is simple. Forget all of those over hyped language programs claiming that you can learn a language in 20 minutes a day. It’s just not that simple. Learning a language is a gigantic undertaking and it takes time. The trick, is to free up time you didn’t know you had.

Flashcards

I’ve used flashcards many times, for more than one language, and I’ve always had mixed feelings towards them. One one hand, they work. If I had to memorize 30 Chinese characters for a test in 30 minutes, flashcards are definitely the way I’d go. They also served me very well when I did an intensive first year Japanese class over a summer in college. On the other hand, flashcards are about the most boring, decontextualized vocabulary-learning method I can think of. Relying on them over the long run has always been demotivating for me.

The solution is to use flashcards only during the times that you’d be bored anyway. Buy a set of flashcards on a ring, and make sure it’s small enough that you can carry it all the time. Then, rather than using them during a block of time you could be using for something else, use them during all of those brief moments when you have nothing else to do. It used to be that every time I took the subway anywhere, I’d sit down and basically just zone out for five minutes. Now I spend that time to review 30 words on my flashcards. Similarly, when I’m waiting in line at the grocery store, that’s another chance to review several cards. If I call a friend on the phone, I usually have time to scan over three or four words while I’m waiting for him or her to pick up the phone.

All throughout the day, there are bits and pieces of time that are perfect for flashcard review. Sure, flipping through flashcards is boring compared to other things I could do at home, but it’s not boring compared to doing nothing while sitting on a subway! In fact, during times that I’d otherwise have nothing to do, looking at my flashcards is fun.

MP3 recordings

Obviously, if one wants to get good at speaking a language, rather than just writing it, listening is an important component to any language study course. For beginners and intermediate level students, target language media isn’t that comprehensible, but there are a number of learner resources available. Most modern textbooks are accompanied by CDs, and there are numerous podcasts and other recordings to download, as long as the target language isn’t too obscure.

I’ve had the opposite problems with listening to recordings as I’ve had with flashcards. As long as I can understand them, they’re fairly interesting, but I just don’t learn that many new words. For the most part, they’re a re-enforcement tool. There are other ways to build up my core language proficiency faster, such as working through a textbook or finding opportunities for a two-way conversation. It’s always felt like a waste.

Fortunately, the prudent use of “hidden moments” solves these problems as well. I recently bought a very cheap MP3 player (about $25US), and I carry it with me all the time. I downloaded 50 Chinese Pod lessons, and 50 Japanese Pod 101 lessons. Whenever there’s nothing better to be doing, I listen to them. In general, I put the flashcards at a higher level of priority than the MP3 recordings. If I’m in a situation when I can look at the flashcards, I do. But when my eyes are occupied, such as while I’m walking to work, I listen to the MP3s. This way, I get the benefits of more listening practice, and it’s all without having to sacrifice any time I could use on more productive activities.

Authentic reading materials

Authentic reading materials are hard. Especially if you’re studying a language such as Chinese or Japanese, it takes a long, long time before authentic materials can be read with ease. Until that time, your choice in extensive reading materials will be limited to graded readers designed for language learners. Unfortunately, Chinese learners don’t have anything even remotely like the selection that exists for English learners. That leaves a more intensive approach.

Reading material that’s too hard is frustrating. However, by supporting it with the above flashcard method, it becomes much more palatable, and even motivational. For this exercise, I bought a copy of the Guoyu Ribao. I’m not a big fan of the contents of the paper, since it’s always crammed full of articles about various new regulations the Ministry of education is implementing, but there are a few interesting stories. More importantly, zhuyin is printed next to every single Chinese character. That makes looking them up a breeze compared to the nightmare it would be to look up every character I didn’t know by radical and stroke number.

Putting it all together

After buying a paper, I take it home and try to read an article. Anytime I see a word that I don’t know I highlight it. If I think I know what it means but I’m not completely sure, I still highlight it. Then after reading a paragraph or two, I quickly look up the words and make flash cards. I write the first word at the top of the card, flip it end over end (the long way), write the English translation at the top of the back of the card, flip it back over, write the second word beneath the first, and so on. I usually put about five words on a card. If there’s one that I can’t find in the dictionary, such as “資優” in article in the photo above, I just write it down and leave a space on the back for when I find out what it means. This whole process took me about 20 minutes the first time I did it, but it’s much faster now.

After marking up the first paragraph and making flashcards, I stop working on the article and move onto other things. The next day, when I come back to the article and try reading the first paragraph again, it’s much, much easier. Why? Well, it’s because I’ve made flashcards of every word I didn’t and I always carry those flashcards with me. I’ve probably reviewed them ten times each, and done all the review during time I otherwise would have wasted. So with what feels like zero effort, I can suddenly read a paragraph that was difficult just one day earlier!

After the first paragraph is done, I go on to the second paragraph and then the rest of the article. The rest of the article is always easier. Some of those words I had to look up in the first paragraph inevitably appear in later parts of the article. As you can see from the photo above, the beginning of the article (the upper right-hand side) is highlighted all over the place, and the highlighted sections decrease throughout the article, until the bottom, where it’s mostly un-marked. That in itself is very motivating. Better still, each time I buy a paper, the first paragraph I take a stab at is easier and easier.

Conclusions

I like Farber’s suggestions. I suddenly feel like I’m making more progress than I have in ages, and I’m doing the majority of it during time that I would just be wasting otherwise.

“Writing by hand is useless; I can just type everything at my computer,” explains another.

I disagree both arguments. While it is possible to get by without the ability to write by hand, it’s also possible to get by without learning Chinese at all. In fact, I know foreigners who have lived here for nearly two decades and who speak less Chinese than most students do after a single year. One of them was even my former boss. He got by just fine. The existence of people such as him is evidence that having fully functional language skills means something more than just being able to survive in a Chinese speaking city. Being fully functional in a language means using it to accomplish whatever daily tasks one chooses, not choosing daily activities based upon the limits of one’s language skills.

It is conceivable, that the day will come when computers are so ubiquitous and so convenient to use that the pen and paper will no longer serve any utilitarian purpose. Like the calligrapher’s brush and ink, the pen and paper will be cast aside, with only a small number of people of an artistic bent or with an interest in history using them at all. Has that day come yet? The best indicator is to look at what people who don’t have any interest in history or art do. Look at what those who just want to get on with their lives do, or better yet, look at what businesses do. Businesses aren’t run for aesthetics or culture. Businesses are all about utility. See if businesses still bother stocking those outmoded and archaic pens and pencils. I suspect that any business you might inspect would still use them, regardless of whether it were a Chinese or western company, and regardless of how high or low-tech their operations might be.

When is the last time you, yourself, used a pen or paper? Unless you are an exception, you used one yesterday. Odds are also overwhelmingly high that you didn’t go a single week in all of last year without using a pen or paper. This also suggests that the pen and paper still have some utility that computers haven’t yet been able to match. One significant feature, is that they are portable and don’t require a power source. Taking notes on a laptop during a meeting might not be too bad, but for writing down what toppings your friends want on a pizza, using computer would likely be slower than the traditional approach. Another advantage of the pen and paper is that they are less restrictive: while typing this essay, I have no choice but to make each letter the same size, and to put them into neat rows. While it would be possible use different font sizes or format a web page to display my writing a bit differently, it would be very difficult to circle various words, draw arrows from one word to another, or to draw a brainstorming diagram. Each of those techniques is both useful and common for pencil and paper users.

Why is it then, that so many people would claim that being able to write Chinese doesn’t matter? My guess is that it’s just because learning how to write in Chinese is a pain in the ass and people want a reason not to do it. That’s fine, and I can understand the desire completely. There’s nothing ignoble at all about not learning how to write Chinese characters. For that matter, there’s nothing ignoble about not learning Chinese at all.

Mark’s Opening Argument:

As any student of Chinese knows, learning enough characters to really become literate is no small task. For some, such as myself, it’s proving to be a sisyphean ordeal. Fortunately for me, and everyone else who isn’t fully literate in Chinese characters yet, street signs and subway signs are often romanized. Operating under the assumption that signs are indeed romanized for the benefit of the people who need them, as opposed to purely for political ends, and drawing on my own experiences, I’ll explain why it would serve the best interest of the most people to include tone marks on the signs.

Like many westerners, I arrived in Taiwan knowing a only a little bit of Mandarin. Communication of any sort was difficult. At the beginning, I had to stay close to other foreigners, my English speaking co-teachers, and I had to approach things with quite a bit of patience. After a while, though, my spoken Chinese improved and I was able to do things like ask people for directions and take taxis with confidence. Now, I very rarely have any serious communication difficulties, but my reading is still pretty weak.

I know I’m not alone in this. I know many, many westerners here who can speak well enough, but only read 1000-2000 characters. That’s enough to get by in day to day, life, but there are many, many street signs that my western friends mentioned above can’t read. We can all read pinyin, though. In fact, just about anybody who’s taken even a single semester of Chinese can read pinyin and at least sound it out. Unfortunately, we don’t get nearly the mileage out of this pinyin ability that we could.

As we all know, the state of romanization in Taiwan is less than ideal. I remember wasting an hour once asking people where “tingshi” MRT station was before realizing that what I was really looking for was dĭngxī (頂溪). In the end, I was frustrated, I was pissed off, and I’d missed an interview as a result of getting lost. A few months later, I found Mark S.’s useful site, Romanization.com, and never had a problem with saying the names of MRT stations again.

In the three years that have gone by since then, most of Taibei city’s street signs and MRT stations have adopted standard (i.e. hanyu pinyin) spellings, and the whole nightmare of newcomers not knowing if they’re looking for “kuting”, “guting”, “kuding”, or “guding” MRT are over. However, I feel that there’s one more step to be taken that’s both inexpensive and useful to a large number of people for whom the romanized signs are intended, i.e., the people who can’t read Chinese characters with full literacy- street signs should have tone marks above the pinyin.

Even with the current, standardized pinyin, there’s no way for a person who reads “Guting” to know if the word is gŭtíng”, gūtìng, gùtíng, or one of thirteen other possible tone combinations. In the case of the Guting MRT, this isn’t a big problem. Even though character-illiterate foreigners will have no way of knowing how to pronounce the stop, there aren’t any other stops with ambiguous names. Sometimes, though, we aren’t so lucky. There are many streets, such as TóngĀn Jiē and TōngĀn Jiē[1], which share identical romaniations unless tone marks are used. In fact, I’ve personally wasted cab fares as a result of not being quite sure of the tone of a street name. While I haven’t had this problem in quite a while, I don’t have any sort of desire to see every new crop of beginning Chinese students here suffer all the same misfortunes I did.

The Costs of Adding Tone Marks

• Remaking road signs would be expensive. However, for new signs being made, adding tone marks would incur very few additional costs. The government is already paying consultants to romanize the signs to begin with. For anybody qualified to work as a “pinyin consultant”, converting characters into romanized text with tone marks should be no difficulty at all.
• Adding tone marks above each syllable could increase paint costs by a small amount.
• People who don’t know any Chinese whatsoever could just ignore the tone marks. However some may find the tone marks visually unappealing. Conversely, those who didn’t ignore the tones might even pick up a little bit more Chinese through regularly seeing romanized Chinese words that they hear frequently.

The Benefits of Adding Tone Marks

• With tone marks, pinyin is a fully-functional writing system. Anybody who has learned pinyin, which takes orders of magnitude less effort than becoming character-literate, can read any Mandarin word written in pinyin.
• Many, many foreigners here would have a smoother time asking for directions and getting around the city.
• Tone marks on street signs would be a boon to students hoping to learn Chinese from the living environment here.
• Identical street names would be disambiguated.

Closing

In my opinion, writing Chinese words with the Latin character set and neglecting tone marks is the equivalent of writing English words with dots in place of the vowels. Yes, it’s possible to guess the missing information from context, but it’s not a very complete writing system.

Be sure to read Prince Roy argument in favor of not putting tone marks on road signs!

Tone Marks on Roadsigns Part II- Prince Roy’s opening argument.

Notes
[1]ICompletelyAgreeWithMarkSThatInnerCapItalizationIsAVeryVeryBadThing. CursesUponJavaForEverPopularizingSuchAnAbomination.

Me: Hi. We want to go to the night market.
Driver: Oh! Can you speak Chinese!!?
Nathan: Uh…. yeah.
Driver: You guys are Americans, right? Right?
Me: Yep. We live here, though.
Driver: What do you do? Are you teachers?
Nathan: He is, and I’m a volunteer worker.
Driver: What do you mean? What do you do?
Nathan: I do work at hospitals and juvenile reform centers…
Driver: Do they pay you?
Nathan: No, it’s all volun…
Driver: They don’t PAY you? Why do you do it?
Nathan: To help people. It’s…
Driver: No salary? I wouldn’t do it!

I’m sure a lot of westerners secretly think the same way. I’ve never heard any say it so bluntly, though. Even if it’s only lip-service and they can’t really relate to volunteerism or charity, they’re at least familiar with what would motivate other people to engage in those activities.

From my experiences in first grade onward, it has been clear to me that learning as much as possible was rarely a goal in school. The point was driven home over the years as I endured being forced to do hours of boring coloring worksheets as a young child, being told to take high school math and science courses after having already completed three courses in calculus, one in linear algebra as well as various other engineering coursework, and having to take zealously left-wing courses in which dissenting political views were barely tolerated when I went back to college. Truth be told, there have been periods spanning for years in which I have felt an aversion towards education, just because I associated it so strongly with schooling.

The cold unyielding truth of the matter is that the biggest benefit most people receive from their schooling isn’t the education, it’s the diploma. People in virtually every field have attained excellent levels of education outside the traditional school system. Unfortunately, many people seem unaware of the ample evidence of this fact and tend to feel that a person isn’t “qualified” to do things without the appropriate academic credentials. Before I’m misinterpreted, I should say that professional certifications really are necessary in fields in which unpredictabilities must be minimized, such as medicine, or structural engineering. However, I’ve seen many, many cases in which companies, human resource departments, and even governments are so blinded by academic qualifications that they forget what the qualifications have to do with the job at hand.

Living in Taiwan, I’ve met many people in the ESL world who feel that those without education degrees cannot teach. Interestingly, none of the most effective teachers I’ve seen hold them. Most new TESL and BA Education guys I see come here are woefully unprepared to teach at a Taiwanese school. Due to a lack of Chinese skills, 99% would be incapable of working at a school such a my own. Before, when I was in the software industry, I encountered many a manager who felt that they were better programmers, or “developers” as they like to style themselves, than other people just because they had M.S. degrees in computer science. Once again, the truth is that in the real world, some top programmers have M.S. degrees in computer science, some have music degrees, and others have none at all. Worst of all is the idea that a candidate just needs to have “any degree”. I’ve seen companies pass up potential employees with exactly what they need due to such narrow-minded thinking. In fact, I myself was unable to take a job offered to me by EA Games in Vancouver due to being denied a Canadian work visa. I had just the skills they needed, but they couldn’t hire me. Now, years later, my skills aren’t as relevant to EA Games. I could get a work visa, though, thanks to my degree… in Japanese literature. Before deciding that the 75% of Americans who don’t have degrees just couldn’t amount to much, consider these examples from Mr. Gatto’s article:

Do we really need school? I don’t mean education, just forced schooling: six classes a day, five days a week, nine months a year, for twelve years. Is this deadly routine really necessary? And if so, for what? Don’t hide behind reading, writing, and arithmetic as a rationale, because 2 million happy homeschoolers have surely put that banal justification to rest. Even if they hadn’t, a considerable number of well-known Americans never went through the twelve-year wringer our kids currently go through, and they turned out all right. George Washington, Benjamin Franklin, Thomas Jefferson, Abraham Lincoln? Someone taught them, to be sure, but they were not products of a school system, and not one of them was ever “graduated” from a secondary school. Throughout most of American history, kids generally didn’t go to high school, yet the unschooled rose to be admirals, like Farragut; inventors, like Edison; captains of industry, like Carnegie and Rockefeller; writers, like Melville and Twain and Conrad; and even scholars, like Margaret Mead. In fact, until pretty recently people who reached the age of thirteen weren’t looked upon as children at all. Ariel Durant, who co-wrote an enormous, and very good, multivolume history of the world with her husband, Will, was happily married at fifteen, and who could reasonably claim that Ariel Durant was an uneducated person? Unschooled, perhaps, but not uneducated.

One person I’m surprised was not included, was Einstein. He had completed a college degree (after having dropped out of secondary school at Luitpold Gymnasium, failing his entrance to get into Eidgenössische Technische Hochschule the first time, returning to secondary school and trying again). However, during his amazing Annus Mirabilis, in which he wrote four papers that laid the foundations for relativity, quantum mechanics, and modern physics itself, he could sign his name as nothing other than “Mr. Einstein”. Considering this, it’s nothing short of amazing how elitist so many people in academia are about who has a P.H.D. and who doesn’t.

At the higher levels, I do see some advantages in attending formal schools, however I don’t think the “university system” is what makes them valuable. What makes them valuable is that they tend to concentrate people with similar abilities and interests. In other words, going to MIT really would be great for a budding young bio-tech engineer. It probably wouldn’t be a great as working next to Craig Venter at Google, though. At least in my own humble opinion, education is one of the most important things in life. Schooling isn’t.

One of the most grating things, is constantly being told I’m fat because “American food is unhealthy”. Sometimes I feel like wringing the necks of my usually well-meaning acquaintances and telling them I was in great shape until moving to Taiwan, and that I ate healthier food back home than I do here. “KFC” is unhealthy, they say. Well, I can agree with that. The KFCs here are certainly a lot less healthy than the ones at home were, though. In KFCs in Colorado, Texas and California, I’ve eaten spinach, red beans and rice, carrots, potatoes, corn, and a variety of healthy foods. That’s not even including the salad bar, either. The fried chicken may not be too healthy, but all the sides were. Here, none of those sides are sold. You can eat fried chicken or fries, and little else. McDonald’s is the same. All of my beloved salad-shakers are gone. True whole-wheat bread isn’t even sold in this town. Likewise, I can’t find water-packed tuna. The grocery store only sells oil-packed tuna. I haven’t seen baked chicken (or any low-fat meat) offered more than a handful of times in the entire time I’ve lived here. Sure, people here don’t weigh as much as back home, but I’m not sure if they’re doing any better in terms of body composition. Barring the capital city, I don’t think I’ve ever seen anybody on this entire island with an “athletic” physique.

I used to tell people here about this stuff, whenever they tried how terrible American food is. I’d tell them how not a damned bit of western food that I saw them eat was healthy and how only the junk food was popular. I told them about the wonders of Whole Foods and Trader Joe’s. I told them how places selling healthy western food here would go out of business, because Chinese people just want white-bread (probably pastries), fried chicken, chips, soda and all the crap, but have no interest in our healthier, more traditional fare. Most people just didn’t believe me. They’d already heard about US food on TV and from all their friends. They “knew”.

I couldn’t see how people could be so blind and stuck in their own ways of thinking… until I went home. I’d forgotten the sesame chicken factor. At every single Chinese restaurant I visited, I was greeted by sesame chicken with so much sugar that it coagulates with a bit of stirring. I had crunchy “Kung-Pao” chicken soaked in so much grease it could give a small rhino heart disease. I had fried tofu sweeter than most deserts. Then it dawned on me. Nobody bothers that much with other cultures’ health foods. We all import the nastiest, most unhealthy stuff we can get our hands on and corrupt the rest of it in that direction quickly as possible. If and when we do eat healthier foods, they’re invariably either unprocessed whole foods, such as fruits, or staples basic to our own cultures, things that we grew up eating. Now, when people tell me how unhealthy “American food” is, I never tell them about the healthy foods I used to eat back home. I just ask them if they’ve ever eaten sesame chicken.

This can be illustrated with a few examples. Let’s say that you are a wealthy investor and that company A sells children’s books. You feel that company A’s books are educational, wholesome, and deserving of your financial support. They earn $7 million per year, and are currently valued at $100 million dollars. Seeing some growth potential in the company and wanting to support their cause, you buy $1 million of company A’s stock. The question is, how much did company A earn from your million dollar purchase? The answer is nothing. They still earned $7 million dollars this year; the only difference is that you now own 1% of company A and they will pay you next time they issue dividends.

If company A’s business improves, you will make money from your investment; if it deteriorates, you will lose money. Either way, your effect on them is non-existent unless you buy the stock directly from them (e.g. in an IPO), or they decide to sell more shares after you’ve made your purchase. Had you simply bought a million dollars worth of books, you would have increased their earnings from $7 million to $8 million and had a huge effect on their business for the year. Investing though, won’t help them a bit. If their earnings falter, you’ll lose your money, but it won’t have done them any good.

Likewise, if you don’t want to support company B which sells oil refinery equipment, you’ll have to find a better way than just not buying their stock. In fact, even selling their stock would do nothing to them in the long run. Suppose Bill Gates, Warren Buffet, the Vatican, Opera Winfrey, and the estate of Colonel Sanders managed to secretly buy all of company B’s stock and then agreed to sell all of it on the open market at a prearranged time.

The stock price would plummet, yes, but company B’s earnings and business prospects would stay the same. Opportunistic people, such as myself, would recognize how under-valued the stock was and buy it at a fraction of its true worth. Very quickly, the stock price would climb back to its actual value, give or take a factor of two, and our sneaky “pentavirate” would have accomplished nothing more than throwing away their own money.

When it comes to ethics and corporations, I have to make rational choices. That doesn’t mean I don’t think the effect of my actions is too small to matter. It just means that making investments based on ethical evaluations of companies doesn’t make much sense. If there’s a company I want to support, I’ll buy its products. If I don’t want to support a company, I won’t buy its products. If it’s a really evil organization, such as the RIAA, I might even try to get others to boycott it or write to my congressional representative. However, barring an IPO or an SPO, I’d buy stock in any of the RIAA’s member companies if I found the valuation attractive. Heck, I might even be able to influence them with the voting privilages that came with the stock…

Game Theory

Then game theory and optimal bluffing came up. Sometimes, there are situations in which a consistent strategy will always fail, and yet a somewhat random strategy, or a mixed strategy will prevail. It sounds irrational, but it is true. Mike asked me to explain it, and I wasn’t able to do so very clearly. So, I’ll have another go at it here. First, there are a few important distinctions to make, though.

Optimal Strategies vs. Exploiting Strategies

There are two kinds of strategies used in poker. In optimal strategies, the opponent is assumed to be strong and adaptive. Optimal strategies are evaluated based on how well they would fare against an optimal opponent. Exploiting strategies are designed to exploit a weak opponent as fully as possible. For example, if you play against a timid opponent who never calls, you can win money from him more quickly by bluffing every time (a strategy designed to exploit his weakness) than you could by using an optimal strategy. The bluffing strategy I’m about to describe is an optimal strategy that will work even if opponents know you do it.

An Example in Which Random Betting is Optimal

Imagine that you are playing a poker game in which the first four cards are dealt out face up, and the last card is dealt face down. After each new card is dealt, each player may bet and raise once. In this game, you and your opponent have just been dealt your final cards, there are $40 in the pot and he has bet $10. The maximum bet is $10.

There are 8 cards on the table. Your last card, which your opponent has not seen, is one of the 44 remaining cards in the deck. Of those remaining cards, any of the Kings, Jacks, Queens or Tens would give you a bigger pair than your opponent’s threes, and any of the Aces or nines would give you a straight. In other words, of the remaining 44 cards, 20 will give you a winning hand and 24 will give you a losing hand. However, you are still in the stronger position if you use an optimal bluffing strategy. Consider these three cases:

You Never Bluff

In 24 cases out of 44, you have the weaker hand and you fold. In the other 20 cases you bet $10. Your opponent, being a strong player, recognized that you do not bluff and never calls your bet. You win the $40 pot in 20 games out of 44. Since half of the money in the pot was yours to begin with, you earn $20×20 = $400. In the 24 games in which you fold, you lose $20×24 = $480. In the long run, if you employ this strategy, you’ll lose $80 every 44 times you play this way.

You Always Bluff

In 20 cases out of 44, you have the stronger hand and bet $10. Your opponent knows you always bluff, so he calls. You win $40 from the pot, plus his $10 from calling. Since $20 of the pot is your money to begin with, you win $30×20 = $600. In the 24 games you lose, you lose your $20 in the pot, plus a $10 bet each time. That’s a $30×24 = $720 loss. In the long run, if you employ this strategy, you’ll lose $120 every 44 times you play this way.

You Use Game Theory to Bluff an Optimal Amount

It is possible to use your opponent’s pot odds to determine how often to bluff. In this case, always bet on the 20 winning cards plus four of the 24 losing cards, and you’ll have the edge. It doesn’t matter how you determine when to bluff, as long as it’s random (at least to your opponent’s perspective). You could say, I’ll bet if I draw a winning card OR a two of any suit. You could ask your friend to give you a random number and then divide it by 24 and only bet on losing cards if the remainder were under 4. Anything random will work. Bet on the 20 winning cards, plus 4 losers. Unless you give tells or your opponent can crack your “randomization” scheme, there is no strategy he can employ that will give him the edge.

Your Opponent Folds When You Bet

You bet on your 20 winners, plus 4 of the losers. Your opponent folds every time, so you win $20 from the pot 24 times for a total of $480. You fold on 20 of the 24 hands in which your last card was a loser, losing $20×20 = $400. In the long run you’ll win $80 every 44 times this situation comes up against an opponent who folds.

Your Opponent Calls You

You bet on your 20 winners, plus 4 of the losers. Your opponent calls each time. On the 20 hands in which you really do have the stronger hand, you win the $20 he put in the pot, plus the $10 call. That’s a total of $30 x $20 = $600. On the four hands in which you bluff and lose, you
lose $30 x 4 = 120. On the 20 hands you fold, you lose $20 x 20 = $400. In the long run, you’ll win $80 every 44 times this situation comes up against an opponent who calls.

Conclusion

Don’t underestimate math geeks! By utilizing game theory, it is possible to construct a mixed strategy that can win in some situations when any consistent strategy would fail.