November 14, 2011

Traffic Jam and the Prisoner's Dilemma

Traffic jams are an integral part of roads in India -- it does not matter if you are at Dalhousie Square in the heart of Calcutta or you are at Panagarh on the Grand Trunk Road -- and when you are stuck you have a lot of time to think on your hands. You begin with cursing people around about their lack of civic sense and road etiquette but eventually you resign yourself to the situation and start thinking about the problem analytically. Could there be a game theoretic approach to the problem ?

To begin with let us define a general traffic jam a little more precisely so that the problem is more tractable and amenable to a mathematical analysis. Anybody who has lived in IIT Kharagpur for any length of time knows that just outside the campus there is a railway level crossing (actually two) which are a reliable source of traffic jams on a regular and repeatable basis. Every day, 365 days a year, the gates close two or three times every hour and each time when they reopen there is a always traffic jam.

This is jam that can be modeled very easily. This is a two-lane road and if cars (and rickshaws, cycles, motorcycles ) stick to their designated left lane, both North and South bound traffic can move smoothly when the gates open. But this never happens because vehicles on both sides of the closed gate invariable occupy both lanes and when the gate finally open, both North and South bound traffic is blocked on both lanes leading to a jam. It takes quite some time (and a lot of shouting and cursing) before vehicles blocking the right lane manage to squeeze into the left lane and finally both lanes move again ... but after everyone has spent a considerable amount of time standing at the open gate !

Having suffered this N times ( with N tending towards infinity ) I keep wondering why people do not cooperate with each other, obey traffic rules and make it nice and easy for everyone. Then revelation dawned ! They never will ! and why not ? For an answer let us consider the classical game theory problem of the Prisoner's Dilemma.

Consider the following : Police investigating a crime have caught two suspects whom they suspect to have jointly committed the crime but they lack the evidence to prove the same. So they lock them up in two separate rooms and present them with three choices (a) if both maintain their innocence then both will be booked for a minor offense and imprisoned for 1 month (b) if one confesses to the crime and the other one does not the the confessor will be pardoned but the other will be imprisoned for 5 years (c) if both confess then both will be imprisoned for 12 months.

Ideally both should maintain their innocence ( cooperate with each other ) so that both can get away with a 1 month sentence but logically each suspect will try to minimise his penalty. Each argues ( to himself, since he cannot talk to the other suspect and cannot figure out what the other is doing ) that : "If I want to minimise my penalty I must confess. Then if the other fellow confesses we each get one year but if I do not confess and the other fellow does then I get 5 years. So I can minimise my penalty by confessing. Now the other suspect will also be thinking like me so he too will come to the same conclusion and confess. Hence if I do not confess, I will be in big trouble". Since both suspects will argue in this symmetric manner, both will logically come to the conclusion that they must confess.

So both will confess and get 12 months in jail whereas if both had maintained their innocence ( that is cooperated) they would have got only 1 month in jail.

Now let us apply this theory to the situation at the Puri Gate railway level crossing outside IIT Kharagpur.

If both the North bound traffic and the South bound traffic co-operate and stick to the designated left side lane (that is cooperate and maintain innocence) the road will clear in just about 1 min after the gate opens. But if traffic from one side crowds the right lane (confesses) but the other side obeys the law (maintains innocence) then the law-breaker (confessor) goes through instantly whereas the law-abider (one who claims innocence) will be stuck for ages. But if both sides break the law ( confess) then both are stuck in a jam for quite some time.

The analogy between the situation at the IIT Kharagpur Puri Gate level crossing ( and by extension to traffic jams ) in general and the Prisoner's Dilemma seems to be perfect. Given a chance people, on their own, will break the law and there will be traffic jams. Unless (a) there is an external agency to enforce the law or (b) people start to co-operate to each other.

But is the second option possible in India ? If you have an answer, please post a reply here ..

November 10, 2011

The Tunganath Trek

At 12,000+ feet above MSL, Tunganath is the highest Shiva Temple in the world.  On our way to Gangotri, we stopped at Guptakashi and made a detour to Tunganath. This is where we saw the first snow capped peaks.

Our trek started out from a tiny hamlet called Chopta, which is as far as our car could go. From here it is a 3 km, 3500 feet climb to Tunganath. 

We walked through some of very beautiful landscape and finally passed the tree line. Above this point ( shown in the picture above ) there were no more trees. 

and there was snow on the ground, left over from last winter ( 2010-2011). The summer of 2011 had not been warm enough to melt these snow deposits! 

This was the last sign post before Tunganath, and then we arrived !

The weather was turning chilly and with the mist and fog rolling in, it was darkness at noon.

Having reached Tunganath, I thought that I should say something profound !

and Indira also spoke her mind on the the charms and challenges of the Tunganath Trek

with the weather turning suddenly very cold, note the snow on the ground, we start our descent

in an earlier era, in 1997, we had visited Kedarnath but the lack of digital camera had severly limited our photographic ambitions. however these two images survive from the Kodakchrome era

you would notice that the Kedarnath temple is very similar to the Tunganath temple . After Tunganath, we drove back to Guptakashi and then on to Uttarkashi through some spectacular scenery

November 08, 2011

Gangotri 2011

Gangotri is the mythical fountainhead of the Ganga and the Gangetic civilisation around which the social, reliegios and cultural life of India has evolved. We visited Gangotri on Dipannita ( also known as Deepawali ) on 26th October and took a dip in the icy waters of the Bhagirathi, as the Ganga is known at its source. Here are the pictures.

We set our sights on the distant peaks and start from Rishikesh and follow the river upstream.

The first place of interest is Devaprayag where the Bhagirathi meets the Alakananda and is henceforth known as the Ganga. In these pictures we see the confluence of the two rivers. The one on the left is the Alakananda and the one on the right is the Bhagirathi that originates at Gangotri. While we intend to go to Gangotri, we follow the Alakananda because we have some of visiting Tunganath on the way.

We next touch Rudraprayag ( made famous by Jim Corbett and  his true story of the Man Eating Leopard of Rudrapryag). This is where the white waters of the Alakananda meet the sea green waters of the Mandakini.

The Mandakini, on the left, joins up with the Alakananda on the top

After Rudraprayag we reach Uttarkashi where we see the mighty Trishul or Trident said to be planted on the ground by Devi Durga herself.

note the tip of the giant trident sticking out of the top of the Shakti temple and compare it with the size of people who are entering the temple

This is the Kedar Ghat where we dipped our toes into the icy waters of the Bhagirathi for the first time.

From Uttarkashi we finally start for Gangotri

reflections !  on the Bhagirathi

traffic jam ! 

The Bhagirathi among the snow peaks

Finally we reach Gangotri !

finally the dip into the icy waters

and then on the back we passed, among others, the lake created by the Tehri Dam.