discontinuity – Pertinent Observations

Goalkeeper Mishmash

So one of the comments on my previous post about goalkeepers talked about how the relegated teams (Wolves, Bolton and Blackburn) had the worst keepers. So I wondered how they would have done had they had better goalies. I’ve still not figured out how to correlate a goalie’s distribution success to goals scored and so I’ll simply stick to shot stopping criteria.

I use the ratio of big chances to goals in each game to figure out how a different goalkeeper would have reacted. So if I have a goalie with a 90% shot-stopping ability and the opposing team has 10 big chances in the game, then I concede 1 goal. However, if my goalie has a 50% stopping ability I let in 5.

Based on the shot-stopping success ratio of each goalkeeper and the number of big chances faced by each team in each game, I have estimated the number of goals the team would have let in in each game. Comparing this against goals scored, I have come up with a hypothetical points tally for the season.

I know I abuse excel graphics a lot but I couldn’t think of any non-excel method to present the data here. I paired each goalie who played at least 1000 minutes during the season with each team and estimated how many points the team would have raked up.

Some pertinent observations.

1. The teams on whom the quality of goalie had the most impact are Arsenal, Blackburn, Wigan and Wolves. This goes to show how much Arsenal have to credit Sczsesny for their ability to reach the Champions’ League.

2. Everton is the team where the maximum and minimum possible points due to change in goalie is minimum (4, opposed to 14 for Arsenal). Shows that they have a pretty compact and tight defence, and what stops them from a top four slot is the quality of attack.

3. Due to the low number of big chances that occur in each game and due to rounding of goals conceded, you see some kind of a discontinuity in scores as you go down the list, as well as lots of ties. There is no mistake in the data or the calculations.

4. Manchester United has a much lower “goalkeeper impact” than Manchester City. With a lesser goalie than Joe Hart, it is unlikely City would have won the title.

5. Since we use overall averages of a goalie’s shot stopping ability, these simulations show different numbers for “real” goalie-team pairs than what the teams actually achieved.

6. The difference in maximum and minimum possible points as a function of a goalkeeper is a good indication of the overall quality of a team’s defense. The table below ranks the teams as per quality of defense.

7. While Blackburn and Wolves both had poor defence, part of Bolton’s relegation blame can be attributed to the quality (or otherwise) of their goalkeepers (Adam Bogdan and Juusi Jaaskaleinen). Which makes it even more surprising that West Ham (upon re-entry to the Premier League) sold Robert Green (to QPR, where he warms the bench) and recruited Jaaskaleinen in his place.

8. Last season, Liverpool had a pretty good defence (especially their first-choice back four of Johnson-Skrtel-Agger-Enrique). Their attacking ability (and especially their finishing – same story this season) let them down badly.

Teerth Yatre

The Yatre (journey) took us through four different worlds. All at the same time. We kept flipping from one world to another. Each of us were going through the worlds independently, yet we seemed to meet in one of the worlds (let’s call this one “reality”) once in a while. Time moved extremely slowly. It was like TDMA (time division multiple access) was going through our minds, as we went through the four or five worlds simultaneously.

Parts of the human brain are sequential and parts are parallel. I discovered during the course of the Yatre that our minds are equipped with a parallel, maybe even superscalar, processor. However, certain features such as context switches are not very well developed in human minds so this capacity is seldom used. The human mind prefers linear processing and thus most of the time, all but one processor is shut. And there is a continuous stream of thought that allows us to “execute”.

Those like me with ADHD seem to have an easier time in context switching. While this results in a generally higher level of mental output, it also means that there is greater discontinuity in thought. This discontinuity in thought leads to what psychiatrists term as ‘lack of executive functioning’. “Executive functioning” as us humans have defined it depends on a single train of thought working continuously to get things done.

The Teertha (holy water) however ensures that all human beings, ADHD or the lack of it, become equal, and opens up the superscalar processes in people’s heads. It is like everyone who imbibes it reaches a state that is an advanced level of ADHD. Four or five streams of thought. Parallel inhabition of four or five different worlds. And constant switches between the worlds. One moment you have a sense of achievement. The next you are paranoid. Paranoid about getting through the madding crowd and back safely to the dirty hotel room.

Fifteen minutes past six, we are on the way to the river bank to watch the world-famous aarati. An eternity later (but with the watch only showing six twenty) we see a chaat shop on the way and decide to imbibe some chaat. Another eternity later, some parallel thoughts drive us to the aarati, the rest recommend stopping at the restaurant for an early dinner.

I order pav bhaji. Four pieces arrive. In my journeys through the various worlds, I think I’ve spent an enormous amount of time eating it. During fleeting visits to “reality”, though, less than one piece has been eaten. The rest of the table also consists of plates with a lot of leftover food. I break a piece of pav. By the time I bring it to my mouth I’m in another world. And when i return to “reality” the piece of pav is still in my hand, uneaten.

The final bit of the teerth yatre is the most surreal, when we have to get back to the hotel. We are in no mental state to tell our driver where to reach us. We decide to take cycle rickshaws. To get to a cycle rickshaw, though, we need to go through a sea of humanity.

We don’t know where we are going. We hold hands. In the moments when we are in “reality” we check if we are still together. In the fleeting between-moments, we worry about losing each other. We do our independent trips of the other worlds (I think we have our own set of worlds and the only intersection is “reality”). We independently worry where we are going. The sea of humanity means that traffic is rather slow and there is little chance of being run over. Yet, we worry.

The teertha in question is a product of this tiny store at Godowlia Chowk called “Mishrambu”. It came highly recommended by a friend who had studied at the Banaras Hindu University. It was sweet, laced with dry fruits and nuts, and dollops of butter. “Shall I put a little or more?” asked the kindly shopkeeper as he displayed a dirty-looking green paste from a small stainless steel box. In one of those collective fleeting moments of bravado we asked him to put ‘lots’. Maybe our inexperienced showed up there.

So what if we had gone to Varanasi and not seen the famous Ganga Arati? So what if we didn’t take the boat-ride to see the various famous ghats, and instead settled ourselves in a rooftop cafe on the banks of the Ganges (we were the only Indians there)? So what if we went all the way to the Kumbh Mela and spent our time mostly clicking photos and walking around, and didn’t venture close to the river?

We’ve undergone the most exhilarating Teerth Yatre ever. I’m not sure any of the religious experiences could match this parallel journey across four worlds.

What Should Mexico Do?

If Mexico and Uruguay draw their last league game, then both of them go through to the second round irrespective of what France and South Africa do. However, on account of a better goal difference, Uruguay will qualify as group winners and face the second-placed team from Group B, while Mexico will qualify second and meet the Group B winners, likely to be Argentina.

Uruguay’s option is clear. Play for a draw. If Mexico go for a win, Uruguay should just sit back and try hit back on the counterattack (and in terms of players and style, they are very well equipped for that). Simple case of getting men behind the ball and putting gaaji.

Mexico’s strategy is not so straightforward. The “greedy” thing to do would be to play for a draw, in which case they will most likely end up facing Argentina in the second round (if you remember, Mexico went out last World Cup by losing to Argentina at the same stage). On the other hand, if Mexico beat Uruguay, they will top Group A and meet a potentially inferior team (Korea or Greece) in the second round.

As mentioned earlier, if Mexico go for a win, Uruguay will simply defend and play a counterattacking game which they are good at, so I don’t know if they are going to go for it.

Thinking about it, it comes down to Mexico’s payoff function. I’m sure their payoff is an increasing function of how far they progress in the tournament. However, we should be able to identify one particular “jump” in payoff – some kind of a discontinuity, where the payoff increases considerably for one additional round of progress in the tournament.

If this “jump” is for the second round, Mexico can afford to put Ranatunga Principle, get a peaceful draw against Uruguay and claim their “jump reward”.

If the “jump” is for the quarter-finals, however, then Mexico will want to take the risk at this round in order to get themselves easier opponents in the round of 16 (I’m assuming here that Mexico consider Korea or Greece as much easier opponents than Argentina).

If the “jump” occurs further down in the tournament, I think there is way too much randomness about their potential quarterfinal opponents (especially given the fuzzy results in Groups C and D) and opponents as hard as (or harder than ) Argentina cannot be ruled out. Nevertheless, they would rather face one tough opponent than two (and I’m assuming here that no team is significantly “harder” for Mexico than Argentina) and so they should put fight to avoid Argentina and thus go for a win.

Considering that they have reached the Round of 16 with reasonable regularity in the last few World Cups, I presume that their “payoff jump” will occur later in the tournament. And based on the above reasoning that means they should go for a win against Uruguay, so that they can try avoid Argentina.

And it is on this thin thread that the French are hanging their hopes (though first they need to thulp South Africa, no easy task).

Discontinuous Yield Curves

I think that the equity markets have topped out and have cashed out all my equity and equity mutual fund holdings, and am thus sitting on a pile of cash, which I’m looking to invest in debt. Happened to check out the websites of a few banks where I hold accounts and what caught my eye was the discontinuity in the yield curves.

Here is HDFC Bank:

1 year 1 day – 1 year 15 days	Below Rs.15 Lacs	6.00%	6.50%	May 18, 2009
1 year 16 days	Below Rs.15 Lacs	6.50%	7.00%	August 03, 2009
1 year 17 days – 2 years	Below Rs.15 Lacs	6.00%	6.50%	May 18, 2009
2 years 1 day – 2 years 15 days	Below Rs.15 Lacs	6.00%	6.50%	May 18, 2009
2 years 16 days	Below Rs.15 Lacs	7.00%	7.50%	August 03, 2009
2 years 17 days – 3 years	Below Rs.15 Lacs	6.00%	6.50%	May 18, 2009
3 years 1 day – 5 years	Below Rs.15 Lacs	6.00%	6.50%	May 18, 2009

Notice the discontinuity? About how for a couple of randomly chosen dates the interest rates suddenly shoot up?

Similarly with ICICI Bank:

391 days to 589 days	6.25	6.25
590 days	6.25	6.25
591 days to less than 2 years	6.25	6.25
2 years to 789 days	7.00	7.00
790 days	7.00	7.00
791days to 989 days	7.00	7.00
990 days	7.25	7.25
991 days to less than 3 years	7.00	7.00

Again same story. On certain “magical” days, interest rates shoot up. The degree of increase in rates here is much less dramatic, though. Nevertheless this is extremely interesting, and I wonder why. I remember last year going to Karnataka Bank and asking for a 1 year deposit, and they asked me to make one for 400 days saying that I’ll get 0.5% per annum better for that.

This morning I went to State Bank of India and found that they don’t offer these special rates. I had a friend check at another nationalized bank and found that they too don’t offer special rates. Wonder why the private banks are offering it, though. Why it makes that big a difference to them that the deposit is for 990 days as against 991 or 889. Or is it some way to prevent early closure?

In other news, SBI is offering teaser rates for home and auto loans. Their ads have been there all over the airwaves for the last few weeks. They offer 8% for first year, 8.5% for second and third years and then what they call as “normal rates” after that. If SBI is getting into teaser rates, god only save Indian finance.