Gladiators out on the field,
Tuesday, March 29, 2011
Gladiators out on the field,
Wednesday, March 23, 2011
An exercise in probability theory
You are of course familiar with the Kithne Aadhmi The scene in Sholay. For those of you who are not, this is how it goes.
Kalia and two others (they are not named in the film - let’s call them Amar and Akbar) are sent by Gabbar on a special assignment to liquidate Jay and Veeru. The three brigands are thrashed by our heroes and return empty handed. Gabbar is not amused. He lines them up in front of him and starts loading his revolver.
It’s the famous Hindi film revolver with six chambers. Gabbar loads three bullets in three adjacent chambers. He rotates the chambers effectively randomizing them. He turns and faces Amar, Akbar and Kalia who are standing in that order respectively.
The plan is simple. He will point the gun at Amar’s head and pull the trigger. He will then proceed to Akbar and do the same. Kalia is the last in line, and he will do the same to Kalia. He will not randomize the chambers between shots. You are now at the point where he is about to start this famous Bollywood version of Russian roulette. You are part of Gabbar’s gang and you are also the bookie. You would like to offer odds to the rest of the gang. Before that, however, you need to calculate the odds. So you take out your pad and proceed to calculate probabilities for the following possibilities. You had better be right, since your money is on the line.
- Before the shooting starts: What is the probability of Kalia dying?
- After the first shot: if Amar is dead, what is the probability of Kalia dying?
- After the second shot: If Akbar is dead, what is the probability of Kalia dying? (you don’t know Amar’s fate)
- After the second shot: If Amar and Akbar are both dead, what is the probability of Kalia dying?
You also know that Gabbar is an avid gambler and likes to play cat and mouse with his victims. He has already told you his alternate game plan. After the first shot, Gabbar might give an option to Akbar to choose whether he wants Gabbar to randomize the chambers or not. In such a case:
5) In case Amar is dead, what would Akbar choose?
6) In case Amar is alive, what would Akbar choose?
Akbar happens to have an M.Sc. in statistics and hence is likely to be very logical in his choice. He also wants to live if he can help it.
Now that you have calculated all the above, imagine another scenario, where Gabbar is going to randomize after every shot. In such a scenario:
7) Before the shooting starts: What is the probability of Kalia dying?
8) After the first shot: if Amar is dead, what is the probability of Kalia dying?
9) After the second shot: If Akbar is dead, what is the probability of Kalia dying? (you don’t know Amar’s fate)
10) After the second shot: If Amar and Akbar are both dead, what is the probability of Kalia dying?
You can post your solutions in the “comments” section against this article.
(don’t try googling – you won’t find the answers since I just made all this up)
Tuesday, March 15, 2011
“The crisis in Japan is going to cause huge losses to insurance firms.” That much is obvious, but the point to be noted is that it seems to have caught both the insurance companies and the power utilities running the plants off guard. One thought the business of insurance firms as well as people running nuclear plants is to be prepared for unpredictable risks. As one of the officials of the firm running the nuclear plants was quoted saying “we were prepared for disasters such as Tsunamis, but we were unprepared for an earthquake and Tsunami of this magnitude causing this extent of damage”. The whole world is concerned with the extent of damage wrought in Japan. The tentacles of the financial system extend across the world, and any incident anywhere impacts the rest of the world in myriad ways.
Rise and fall in stock market indices are normal. What is not normal is the extent of crash that sometimes happens – a case to point being the worldwide fall in the aftermath of the Global Financial Crisis. The reason was not just sub-prime, but a host of other global economic factors, which again somehow seemed to be linked to sub-prime; no one really knows or can unravel the full extent of these linkages.
We are shocked and dismayed at such meltdowns. When the entire system suffers a shock like this, our faith in our infallibility is shaken. We believe we have built systems and processes to take care of any shocks and that the even tenor of our lives will not be shaken. Both at an individual level, as well as a systemic level, we convince ourselves, we have the ability to withstand shocks of any kind. How valid is this belief and in reality is this kind of insulation achievable?
Historically, humankind has always been exposed to unpredictable events. One king used to invade another and there used to be plunder and pillage as a natural consequence. Natural calamities would destroy or carry away lifetimes of effort and destroy entire towns or villages. Lack of communication systems would keep people cut off. A flood or a drought would destroy means of livelihood with monotonous, but unpredictable, regularity. There were no insurance mechanisms in place. People had to rely on their own ingenuity, and their personal and social networks to get back on their feet and continue with their lives. This has always been true and it has been accepted as part of the human condition.
What has also been true is that we always recognized the unpredictability of human affairs, and accepted a lot of things as the workings of fate. Every culture has a strong thread of resilience running through its philosophy and way of life to enable its people to cope with unpredictability. Human ingenuity has evolved over time to absorb these shocks and emerge stronger for it. To that extent things are the same. What is different, however, is our increasing belief that we have built, or shall be able to build, systems of increasing sophistication to mitigate these risks. I am now referring specifically to financial mitigation – we feel that our stock markets, banks, insurance companies, global tie-ups, and other constructs will somehow eliminate these risks.
We have built complicated statistical models with all kinds of distributions and sigma limits to contain the world within bounds of predictability. We build such models based on past experiences after ignoring outliers as not fitting the model; make a range of assumptions on the future which, being a range, will be restricted to one; model the future into one of our known theoretical constructs; make our decisions based on that, and then rest easy on the assurance that we have managed to control the world as known to us!
It is not any different from what we have historically done; humans have always insulated themselves into a cocoon of comforting predictability, whether it be in terms of geography, community, religion, culture, business networks, or martial alliances – all to make life more predictable. Invariably, some outside influences have broken into these cocoons, shattering all illusions, and exposed the vulnerability of human life and human endeavor. Our statistical models are nothing but an extension of this search for predictability and security.
What is different now is the interlinkages across the world which makes more and more people depend on the same models and the same constructs, and the forces that compel entire systems to make decisions based on the same set of assumptions. When money is pulled out of the Indian stock market, a lot of money is pulled out because all FII’s think alike; they think alike since they follow the same accepted wisdom; they follow the same accepted wisdom since everyone is interlinked and believes in the same infallible reality on a far greater scale. Insurance companies compete with each other to offer lower premiums – for doing this I am sure they have to ignore some of the “outliers” in terms of risk; other insurance companies have to follow suit since if they do not, they will be out of business. All hedge fund managers think alike. So do all fund managers, actuaries in insurance companies, central banks worldwide, presidents and prime ministers,… the list goes on. Even if they don’t think alike they are forced to act alike. The interlinkability of the modern world ensures that.
When problems are at a retail level, the impact is restricted. More and more problems today reverberate globally in terms of impact. Paradoxically, as we strive to control our own destiny more and more through these global systemic constructs, we get more and more impacted through remote events happening elsewhere and are less and less in control of our own destiny!