Why the example of a Nash equilibrium in the movie “A Beautiful Mind” is incorrect

I was reading an article today about the mathematician John Nash, whose life the movie “A Beautiful Mind” was based on.

The article contained a link to a clipping from the movie that, it said, explained the game theoretic concept of a Nash equilibrium.

In the clip, Nash and his three friends are at a bar and have to make a choice.

They can go and speak to the four brunettes at the bar, or they can all go to talk to the lone blonde, whom they all like better.

Nash explains to his friends that if they all went to speak to the blonde, she would be put off by all the attention and turn them all down.

But once the blonde turned them down, the brunettes would too, since no one wants to be someone’s second choice (and so they would all lose).

So, Nash convinces his friends to ignore the blonde and speak to one of the brunettes each (so that they would all win).

The strategy of ignoring the blonde, the movie suggests, results in a Nash equilibrium.

However, that turns out to be incorrect.

The strategies adopted by the four men do not result in a Nash equilibrium.

A Nash equilibrium is only obtained when all players adopt a strategy where no single player, by changing his strategy, can obtain a better outcome.

That is obviously not true in this case.

Any one of the four friends, by reneging on their deal, might get to go home with the blonde (a better deal).

nash_equilibrium_2

So, the strategy of going after the second choice does not satisfy the conditions for a Nash equilibrium.

A Nash equilibrium is really only obtained when all the men follow the strategy of going after the blonde (in vain).

The mistake has also been pointed out by others: http://math.stackexchange.com/questions/853988/is-the-nash-equilibrium-example-in-a-beautiful-mind-accurate

There is a better explanation of the Nash equilibrium in the video I shared in an earlier blog post:  https://aiaioo.wordpress.com/2013/01/26/what-game-theory-says-about-why-gas-stations-are-built-next-to-each-other/

Professor Nash passed away a few days ago in a car crash.

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Meet us at Barcamp Bangalore 2015: Fun With Text – Natural Language Processing For Hackers

We’re going to be at Barcamp Bangalore.  You can come and meet us at our session “Fun With Text” which is a workshop on text analytics for hackers.

We’re actually going to be trying something a bit crazy at this session.

We’ll start by going over an extremely simple machine learning algorithm.

And then we’re going to go about showing people how almost everything you could ever want to do with text can be done using only that algorithm.

All programmers welcome. Refresh your basic probability theory before you come!

Here’s the Barcamp link:  https://barcampbangalore.org/bcb/spring-2015/fun-with-text-natural-language-processing-for-hackers

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The purchase of nuclear reactors and the price

On 11th December, 2014 India agreed to buy 12 nuclear reactors from Russia for $40 billion.  India is also about to buy 6 reactors from the USA for $50 billion. The reactors referred to above are both 1 GW reactors.

So, each Russian 1 GW reactor costs about $3 billion.  Each American 1 GW reactor costs about $8 billion.

In comparison, Indian nuclear reactors cost between $360 million and $500 million each.

The two 540 MW reactors at Tarapur cost about $500 million each.  The four 220 MW Indian reactors at Kaiga cost $360 million each.

So, the cost for adding 1 GW of capacity is about $1.5 billion if you use Indian reactor technology.

It is between $3 billion and $7 billion if you buy reactors from Russia or from the USA.

If Indian reactors are of comparable price or cheaper, why then is India paying so much money to import reactors?

Are Indian Suppliers Unable to Build Reactors Fast Enough?

One possible answer is that the reactors (built by the Indian government’s nuclear agencies) are not being built fast enough.

However, it appears the Indian nuclear agency can build 540 MW reactors very fast indeed.  The one at Tarapur was apparently completed in 4 years and 10 months.

In comparison, the AP1000 reactors that Westinghouse designs seem to take up to 10 years to build.

Are Indian Reactors Unsafe?

Another possible reason for buying reactors from large external vendors could be that Indian reactors are not as safe as those from other suppliers.

However, the IAEA inspected the 220 MW reactors built in India (those costing $360 million each) and concluded that they were among the safest in the world and could withstand the type of natural disaster that caused the accident at Fukushima.

On the other hand, GE seems to have turned a blind eye to weaknesses in its containment structures at Fukushima even though the flaws had been identified 40 years ago.

If Indian reactors are cheaper, faster to build, and safer, then why exactly did India agree to purchase nuclear reactors from outside India at such a huge markup?

One possibility is that market forces are not the only factor driving the reactor purchases:

Possible Political Compulsions

Obama’s presidential campaign was possibly funded in part by energy firms.  So, it is possible that he is looking to help campaign donors.

But why would the Prime Minister of India play along?

It is possible that the same economic forces that come to bear on President Obama also play a part in Indian elections.  In the Indian election campaigns last year, the winning team spent twice what President Obama’s campaign spent and 75% of the money in the political parties’ war chests came from unknown sources.

So, there is a need for greater transparency and due diligence.

There is one more puzzling fact to consider.

Competitive bidding has not been used in the matter of nuclear reactor purchases.

The nuclear reactors have all been purchased in a manner reminiscent of the coal allocation scam – without any competitive bidding whatsoever.

So, who loses?

The Indian and American Taxpayers

I am going to go out on a limb and say that both Indian and American taxpayers stand to lose out in case this deal between the US and Indian governments has a corrupt angle to it.

How Indian Taxpayers Will Lose Out

Indian taxpayers will lose out because they will be paying approximately $100 billion for the 40 reactors that will be constructed.  $100 billion is about the size of the last bailout package for Greece.  It’s a large sum of money that the Indian government cannot afford.

An article on the Modi government’s purchase of 6 submarines last year for $12 billion hits the nail right on the head:

“According to the World Bank, India has the world’s largest share of people living on $1.25 a day or less. Currently, 400 million Indians live in extreme poverty, and that number will not decrease without prudent policy-making. Reducing poverty requires a degree of social spending and government intervention, and a government willing to spend billions on naval ships before addressing extreme poverty is telling of the government’s priorities.”

How American Taxpayers Might Lose Out

Trickle-down economics will have you believe that anything done to help large firms like GE and Westinghouse will also help the poorer sections of society in the USA.

But, if it is reasonable to suppose that wealth trickles down, it is also, I would argue, reasonable to suppose that poverty trickles up.

I am going to outline in the following paragraph one mechanism by which poverty might trickle up.

Can Poverty Trickle Up?

$100 billion is about half the size of the Indian central government’s income (tax revenue is about $180 billion annually).

If earnings in India drop because of decreased welfare spending, or a depreciating rupee, or lower salaries, more jobs could move to India and hurt American job seekers.

So, there is a way in which poverty can trickle across geopolitical boundaries.

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Is Kejriwal’s proposal to install 1.5 million CCTV cameras feasible?

“Surveillance video cameras, Gdynia” by Paweł Zdziarski – Own work. Licensed under CC BY 2.5 via Wikimedia Commons

A young political party that is contesting the Delhi elections next week (the Aam Aadmi Party, headed by Arvind Kejriwal) has made a promise to install 1 million to 1.5 million CCTV cameras all over Delhi to promote women’s security.

In comparison, the number of government-owned security cameras in the United Kingdom is only 70,000.

So, is the proposal a feasible one?

Well, let’s see.

How many people would be required to monitor 1.5 million cameras around the clock, 24 by 7?

Assuming that one person can monitor 100 cameras, 15,000 people would be needed to monitor the cameras at any given time.

But considering that a typical work day is 8 hours, 3 times that number would have to be employed, working in 3 shifts.

So, a total of 45,000 people would be needed to monitor the cameras.

In comparison, Delhi police only has a sanctioned strength of 80,000 personnel.

Well, is there a better solution?

Alternative 1 – Surveillance of Hotspots

It appears that analytics can be used to identify crime hotspots so that the hotspots alone can be monitored with a much smaller number of security cameras.

In her report titled ‘predictive policing’, Dr. Jennifer Bachner of John Hopkins writes about the Santa Cruz Police Department’s (SCPD) crime prevention program as follows:

The core of the SCPD program is the continuous identification of areas that are expected to experience increased levels of crime in a specified time-frame.  A computer algorithm draws upon a database of past criminal incidents to assign probabilities of crime occurring to 150 by 150 meter square cells on a map of Santa Cruz.  The database includes the time, location and type of each crime committed.

In the calculations of probabilities, more recent crimes are given greater weight.  The program then generates a map that highlights the 15 cells with the highest probabilities.  Prior to their shifts, officers are briefed on the locations of these 15 cells and encouraged to devote extra time to monitoring these areas.  During their shifts, officers can log into the web-based system to obtain updated, real-time, hot-spot maps.

So, by using analytics to calculate the suitable positioning of surveillance cameras, it might be possible to reduce the number of cameras required to a more manageable amount.

Alternative 2 – Self Surveillance

Phones with powerful cameras are available cheaply these days.  A watchdog app which lets a traveller at night register their source and destination addresses and to upload photographs of their conveyance would be a great way to promote safety.

The app could monitor a traveller’s route and alert the traveller if there was a serious deviation, especially towards any crime hotspots.

The app could let the user alert someone if something bad seemed about to happen.

The app could periodically check on the traveller till they reached the destination, and alert authorities if the traveller did not respond within a specified number of minutes.

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Kashmir Floods 2014: Designs For Simple Homemade Boats

I was watching reports on the Kashmir floods on television today.  Reporters in Srinagar went around a few neighborhoods that had been partially submerged in the waters of the Jhelum river and talked to people living there.

The impression I got was that there was no longer any serious danger to life and limb in Srinagar, and that the problem there was now merely one of logistics.

Many families (I believe 700,000 people) had decided to remain in their partially submerged homes, on higher floors that the waters wouldn’t reach, and now had no way to procure food and water for themselves because the roads were impassable and because the phone lines were down.

It seemed to me that these people could procure food and water if they could put together makeshift boats to get around in.

So, I started thinking of ways to build boats with household materials.

After thinking about the problem a little, I hit upon a simple boat design that anybody with a bed and some waterproofing material can construct.

Bed Frame Boat

Step 1:  Get hold of a light bed (a good option would be a steel-framed folding bed) such as the one in the image below:

beds_1

Step 2:  Turn the bed upside down as shown in the image below and if it is a folding bed, connect the legs with supporting rods.  If the bed is a stiff wooden bed, it might be possible to merely link the legs of the bed together with rope.

beds_2

Step 3:  Cover the bed with a waterproof tarpaulin sheet (such as the sheets you’d cover your car with).

beds_3

Step 4:  Optionally attach floats to the ends of the bed (to prevent sinking of the tarp tears).

beds_4

Each boat should easily carry one or two people.

The boats can be roped together to make larger craft that can act as taxis.

beds_5

We shall have to try this design out and make sure it works in practice because the forces on the tarpaulin could tear the fabric.  So the raft would work only if the tarps were strong enough to withstand the forces on the sides and bottom of the craft.

Another very simple design, that would make for a far more robust craft would be a barrel raft, shown in the image below.  It’s just a bunch of barrels tied together.

Barrel Raft

barrel_raftA barrel raft would be more resistant to debris and to sharp objects holing the bottom, but unless the barrels are broad enough to sit in, it would be less comfortable, and also less stable.

Barrels are going to be a bit unwieldy so the best design we came up with ultimately was the Jerry Can Raft.

Jerry Can Raft

The idea came from Dwiji Guru, a friend of mine who is a physicist and consults on design involving physics and on policy in Bangalore.  He figured out that a craft displacing 10 litres of water can carry 10 kilograms of weight on water.  So, ten 20 litre cans submerged to half their volume can carry a weight of 100 Kg.

Cans like these http://shardacontainers.com/narrow-mouth-rectangular.htm are pretty tough and have a handle at the top.

We figured that you can tie 12 cans with a long rope in pairs with their handles together along the center.  Then tie one stiffening rod along the outside edges and two spacers at the front and back, and you’ll have a roughly 1 m by 2 m raft.

This raft would be slim and light and almost impossible to sink.

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Languages and Numbers and Ways of Counting to 8 !

This article is about how small numbers are represented in various languages.
Acknowledgement: much of this article is taken from the Wikipedia page about positional notation.
Bases
The base is the mathematical term for the number of digits you would use to count in a language.
For example, if you used the fingers of both hands to count, you would be using a base of 10.
If you used the fingers of one hand to count, you would be using a base of 5.
If you used the fingers of both hands and the toes of both feet, you would be using a base of 20.
Base-20
Some languages have names for numbers that lead you to suspect that their users might have thought in terms of groups of 20.
French has an interesting way of describing numbers above 60.  In French, the word for 60 is “soixante”, the word for 75 is “soixante quinze” (sixty and fifteen) while 80 is “quatre-vingt” (four-twenties) and 95 is “quatre-vingt quinze” (four-twenties and fifteen).
And it is not just French.  English uses the word ‘score’ to describe a group of 20 things.  So, when we talk of “two score” we mean forty, and when we say “four score and seven” we mean 87.
The article also talks about Welsh and Irish and Maori:
The Irish language also used base-20 in the past, twenty being fichid, forty dhá fhichid, sixty trí fhichid and eighty ceithre fhichid. A remnant of this system may be seen in the modern word for 40, daoichead.
The Welsh language continues to use a base-20 counting system, particularly for the age of people, dates and in common phrases. 15 is also important, with 16–19 being “one on 15″, “two on 15″ etc. 18 is normally “two nines”. A decimal system is commonly used.
Danish numerals display a similar base-20 structure.
The Maori language of New Zealand also has evidence of an underlying base-20 system as seen in the terms Te Hokowhitu a Tu referring to a war party (literally “the seven 20s of Tu”) and Tama-hokotahi, referring to a great warrior (“the one man equal to 20″).
Base-12
Another interesting system is the base-12 system.
The Wikipedia article says:
Twelve is a useful base because it has many factors. It is the smallest common multiple of one, two, three, four and six. There is still a special word for “dozen” in English, and by analogy with the word for 102hundred, commerce developed a word for 122gross. The standard 12-hour clock and common use of 12 in English units emphasize the utility of the base. In addition, prior to its conversion to decimal, the old British currency Pound Sterling (GBP) partially used base-12; there were 12 pence (d) in a shilling (s), 20 shillings in a pound (£), and therefore 240 pence in a pound. Hence the term LSD or, more properly, £sd.
Base-2
There was even a language that made use of a base-2 (binary) system for counting.  Base-2 (binary) is mainly used in computers today (because switches can represent binary numbers – a switch that is off represents the 0 digit and a switch that is on represents the 1 digit).  But apparently, native Australian languages use binary too.
A number of Australian Aboriginal languages employ binary or binary-like counting systems. For example, in Kala Lagaw Ya, the numbers one through six are urapon,ukasarukasar-uraponukasar-ukasarukasar-ukasar-uraponukasar-ukasar-ukasar.
Base-8
The article also says that there is some evidence of the use of base-8 in language:
A base-8 system (octal) was devised by the Yuki tribe of Northern California, who used the spaces between the fingers to count, corresponding to the digits one through eight.[6] There is also linguistic evidence which suggests that the Bronze Age Proto-Indo Europeans (from whom most European and Indic languages descend) might have replaced a base-8 system (or a system which could only count up to 8) with a base-10 system. The evidence is that the word for 9, newm, is suggested by some to derive from the word for “new”, newo-, suggesting that the number 9 had been recently invented and called the “new number”.[7]
So much for bases.
Some languages have two sets of names for numerals!
Two Sets of Names for Numbers in Japanese and Korean
Japanese and Korean use two sets of names for numbers while counting.
In Japanese, there is a set of names that are typically used when small quantities are involved:
“hitotsu”, “futatsu”, “mittsu”, “yottsu”, “itsutsu“, “muttsu”, “nanatsu“, “yattsu“, “kokonotsu“, “to” (1 to 10).
But for larger numbers and for zero, the names used are ones derived from Chinese.
“ichi”, “ni”, “san”, “shi”, “go”, “roku”, “shichi”, “hachi”, “kyu”, “ju”.
These numbers correspond to the Chinese digits:
“yī”, “èr”, “sān”, “sì”, “wǔ”, “liù”, “qī”, “bā”, “jiǔ”, “shí”.
And similarly in Korean, you would use one set of names for small quantities (for example, hours in the day):
“hana”, “dul”, “seth”, “neth”, “thasoth”, “yosoth”, “ilgop”, “yodolp”, “ahop”, “yol”.
But to describe larger quantities, like minutes or the days in a month, you’d go with names based on Chinese:
“il”, “i”, “sam”, “sa”, “o”, “yug”, “chhil”, “phal”, “ku”, “ship”.
Finally, we come to some interesting irregularities in south Indian languages.
Irregular Numbering
In Tamil (a language spoken in south India), the word for 90 is “pre-hundred”.
The first ten numbers in Tamil go:
“ondru”, “irendu”, “muundru”, “naangu”, “aindhu”, “aaru”, “eelu”, “ettu”, “ombadhu”, “patthu”
But the word “ombadhu” which means 9 is not used in 90.
Tamil
In Tamil, the name for 80 is derived from the name for 8 by adding a suffix like in English.  Just as “eight” becomes “eight-y”, in Tamil, “ettu” becomes “embathu”.
But the name for 90 is not derived from the number for 9.  Instead,it is “pre-hundred”.  (In Tamil, 90 is “thonnuuru” – hundred being “nuuru”).  So, when counting from 90 to 99, you use the suffix one would normally associate with the hundred’s position.
So 91 is “pre-hundred and one”.  It is pronounced “thonnuutri-ondru” in Tamil.  92 is “pre-hundred and two”.  It is pronounced “thonnuutri-rendu” in Tamil.
I’ve not come across many languages in which 90 is described as pre-hundred.  But Hindi (a language from the north of India) has a similar feature.
Hindi
In many Indian languages spoken in the north of India, the names of the first ten numbers are similar to their names in Latin.  For example, Hindi has:
“ek”, “dho”, “thiin”, “chaar”, “paanch”, “che”, “saath”, “aaT”, “nov”, “dhas”
The Hindi names for various numbers are similar to the Sanskrit names of those numbers:
“ekam”, “dve”, “thriini”, “chathvaari”, “pancha”, “shath”, “saptha”, “ashta”, “nava”, “dhasha”
But when you get to 29 in Hindi, you say “pre-30″.  The word in Hindi is “unthees” (“thees” means 30 in Hindi).
Similarly, 39 is “pre-40″ (“unchaaliis” where “chaaliis” means 40).
This is different from how you count in Sanskrit.
Sanskrit
In Sanskrit, 39 is “navatrimshat” (nine and thirty) and 29 is “navavimshatihi” (nine and twenty).
Now the absence of a regular name for numbers with 9 in them supports a theory that Indic languages might once have used base-8 for counting.
I quote from the Wikipedia article again:
There is also linguistic evidence which suggests that the Bronze Age Proto-Indo Europeans (from whom most European and Indic languages descend) might have replaced a base-8 system (or a system which could only count up to 8) with a base-10 system. The evidence is that the word for 9, newm, is suggested by some to derive from the word for “new”, newo-, suggesting that the number 9 had been recently invented and called the “new number”.[7]
The assertion seems to have been made in an article titled ‘The Indo-European system of numerals from ‘1’ to ‘10’’ by Eugenio Ramón Luján Martínez.
Eugenio argues that each of the numerals in Indo-European languages gradually came into use when required by necessity, starting with the numbers 2 and 3 (which started as deictics – like in the words ‘duo’ and ‘trio’).
There’s an overview of his arguments in this article: http://smallislandnotesan.blogspot.in/2008/01/indo-european-numbers-1-10.html
Counting on the Fingers
To a twenty-first century human, a base-10 system of counting seems like the natural way to count.
But a base-8 system could have felt more natural than a base-10 system to early humans to count with.
This is because it is only possible to count to ten on the fingers of one’s hands if one has developed the technique of bending them to mark the number up to which one has counted.
If a person uses the technique of touching the thumb to a finger to mark a count, then one can only count up to 4 on each hand (and therefore only up to 8 on both hands).
Indian musicians still keep count of the rythmic patterns in music (the thaalas) by touching the tips of their fingers with the thumb (counting in multiples of 3 or 4).
So it is indeed possible that at some point in the distant past, speakers of Indo-European languages did indeed count in groups of 8.
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Funky language features – some things that you can never say in English and what that might tell us about human languages

Inexpressibility in English

There is a common expression that is widely used in South Indian languages that can’t be translated into English no matter how hard you try.  This post is about things that can’t be expressed in certain languages.  There are some things that cannot be expressed in even the most eclectic of languages though they can in others.

Now I have the unenviable task of trying to tell you in English what cannot be said in English!

Here goes.

Imagine two grown-up people A and B who meet on the street in South India.  B is with her son.  When A meets B, A feels that it would be impolite to not inquire about B’s son.

So, A asks B an open question about B’s son.

B replies, with a big smile and slow polite nods:  “This is my 2nd son.”

What is the question that A would have asked B, to elicit that response from B?

It is impossible to frame an open question in English that would elicit the answer that B gave.

But this exchange is something that South Indian parents have all the time.

When two South Indian parents run into each other, it is highly likely that one might ask the other (in their language) something like, “Oh, what a cute little boy/girl/child!  Whichth son of yours is this?”

The other parent would then reply very proudly: “This is my eldest son/daughter/child” or “This is my 2nd son/daughter/child”.

There is no way to ask someone in English that question because the word or even the concept of “whichth” doesn’t exist in English (and possible doesn’t exist in any European language).

Here’s how you would say that in Kannada (a language used in South India).

A:  Ivanu nimma yeshtaneya maga?  (This boy your whichth son?)

B:  Ivanu nanna eradaneya maga.  (This boy my 2nd son)

Acknowledgement:  This phrase was something I overheard someone discussing when I was a child.  I think it was someone working on translation theory.  I have no recollection of who it was.

Conditional Inexpressibility in South Indian languages

In South Indian languages, there are two ways of saying “and” / “or”.  One way is through a word meaning “and” or “or”.  In Kannada, the words would be “matthu” (means “and”) and “athava” (means “or”).

Another way is using a suffix.  In Kannada you can say something and add the suffix “aa” to indicate “or”.  You can add the suffix “uu” to indicate “and”.

You will find that in South Indian languages you can only express ORs of ANDs using the suffixes.  You cannot express ANDs of ORs.

So, using the suffix forms, we can say “A and B or C and D” but not “A or B and C or D”.

In Kannada, that would be “A-uu B-uu -aa, C-uu D-uu -aa”.  You cannot say “A-aa B-aa -uu, C-aa D-aa -uu”.

You will find a similar restriction in Japanese (though Japanese does not have a suffix form for AND).

Implications for Practical Linguistics

Years ago, we worked on a research project related to natural language programming.  We designed a programming language that would allow humans to program computers by saying things to them.  So, you could say things like: “x égale 2. Si x multiplié par 3 est moins que 5, dis “Salut” sinon dis “Ciao”!

The natural language programming system was designed to help students in rural India learn programming (they often don’t know English and so can’t use an English-based programming language).

It works only in the domain of numbers.  A Fibonacci number generator would looked like this in bad German: “z ist gleich 1. y ist gleich 1. x ist gleich 0. während x ist weniger als 13, z wird y plus x. Danach x wird y und y wird z. Danach schreib z.

(We didn’t put much work into it.  It’s just a research prototype.  But you can play with the technology yourself at http://www.aiaioo.com/cms).

Anyway, since South Indian languages and Japanese favour AND over OR, in this programming language, we specified that AND gets precedence over OR.

Implications for Universal Grammar

I recently read a small book on the latest efforts by Chomsky’s research group to find common grammatical frameworks that can be applied to all languages.

Personally, I do not much like the approach of using grammar to try to explain language.

People can speak a language even if they have only ever heard a few sentences in that language.

They would of course have to limit their use of the language to those few sentences and the variants thereof, but they are still generating language.

It is impossible to construct a grammar of a language from a few sentences.

So it is unlikely that the human language comprehension/generation system uses grammar as we formally understand the concept.

Chomsky believes that there is some language faculty that has a grammar of sorts that generates language and that the output of this faculty is transformed into Chinese or English as the case may be through the use of some simple transformation tools.

If this were true, than one can argue that what is expressible in one language must be expressible in another language.

This must be true at least for commonly used expressions.

But we find that it is not true.

The fact that obvious concepts can’t be expressed in a language with as large a vocabulary as English makes me wonder if there is a common universal grammar, and if languages are as comprehensive as we’d like to believe.

If all languages are derivable from a common grammar, then a concept such as “whichth” which is so common in Indian languages, should have been derivable from that common universal grammar in English just as it is in South Indian languages.

It seems more likely that languages evolve from societal and environmental needs (needs to express things from a cultural or practical perspective) and are nothing but a set of shared signals.

These shared signals eventually evolve to allow for the use of parameters, to allow for a fitting of expressions into slots recursively, that gives rise to an appearance of grammar.

Each language evolves that appearance of grammar independently and there’s nothing more to it.  Or at least, that’s someone’s pet theory.

For some other surprisingly non-universal language features, you might want to take a look at two of our articles on deictic references and ‘possessive verbs':

  1. Funky language features – the third spatial deictic reference in Japanese, Korean and Tamil
  2. Funky language features – the mystery of the missing possessive verb
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