My friend Professor Jayakrishna Ambati of Louisville, KY, USA, alerted me about a recent paper by Dr Derek Abbott and his group from the University of Adelaide, Australia, with the title: “Too good to be true: when overwhelming evidence fails to convince” (which appears in <http://arxiv.org/abs/1601.00900>). They ask the following: “A small amount of ‘bad’ can produce a ‘good’ outcome. (Incidentally, this is referred to as Parrondo’s Paradox). But, can too much ‘good’ produce a ‘bad’ outcome?” Or, is an inverse Parrando’s Paradox possible? And they show that indeed it is possible. This inverse is termed the Braess Paradox. In order to understand the whole thing, I went over to Wikipedia and found some nuggets.
Parrondo's paradox (named after its creator Juan Parrondo) says that acombination of losing strategies becomes a winning strategy. Wikipedia cites an excellent example of the Parrondo’s Paradox in gambling. I quote: consider two games Game A and Game B, with the following rules: 1. In Game A, you simply lose $1 every time you play. 2. In Game B, you count how much money you have left. If it is an even number, you win $3. Otherwise you lose $5.
Say you begin with $100 in your pocket. If you start playing Game A exclusively, you will obviously lose all your money in 100 rounds. Similarly, if you decide to play Game B exclusively, you will also lose all your money in 100 rounds. However, consider playing the games alternatively, starting with Game B, followed by A, then by B, and so on (BABABA...). Now, you will steadily earn a total of $2 for every two games.
Thus, even though each game is a losing proposition if played alone, because the results of Game B are affected by Game A, the sequence in which the games are played can affect how often Game B earns you money, and subsequently the result is different from the case where either game is played by itself. Parrondo's paradox is used extensively in game theory, and its application in engineering, population dynamics, financial risk and other areas.
What then is the Braess Paradox? Wikipedia states the following. This is the inverse of the Parrondo paradox, discovered by Dr. Dietrich Braess, a mathematician at Ruhr University, Germany. He cites the following example. The traffic police in a city decide to construct a bypass road in order to ease the flow of vehicles in a crowded road. Now, each driver decides to take the bypass in order to optimize his time. Soon enough, the bypass road is also crowded and traffic jammed. Good intention has led to bad result! Alas, we see this all too often in cities across India. This is but one example of the Braess paradox.
The paper by Abbott and colleagues wants us to imagine a court case where witness after witness is lined up, in an identity parade, to testify whether the suspect defendant is actually the same person as the perpetrator of the crime. Now, human memory is inherently uncertain, and thus the process would include random error. The more witnesses there are the better (say about 13 of them). If the identity parade is biased, intentionally or not, say, because (i) the suspect is somehow conspicuous, or (ii) the staff conducting the parade directs the witness towards him, or (iii) the suspect by chance resembles the actual perpetrator, or (iv) the witness holds a bias, say, they have previously seen the suspect, then an innocent suspect may be selected with the probability greater than chance. The sheer number of witnesses can now influence the judgment.
Readers may recall the case of the murder of Miss Jessica Lal in April 1999, who was shot dead because she refused to serve drinks to the criminal, stating that it was past the closing time of the bar. As many as nine eyewitnesses to the event, all friends and acquaintances of the criminal, refused to collaborate with the trail judge, the police botched up the report and the trial court had to let the culprit free. Too much of a ‘good’ thing — manipulated overwhelming evidence (lack of it here) failed to convict the murderer — leading to a ‘bad’ result. (The case had to go up to higher courts before he was convicted years later).
The famous line “lies, damned lies and statistics” was attributed by Mark Twain to Disraeli, but curiously even here there is no conclusive evidence. Some say it was Bagehot who said it, some attribute it to Leonor Courtney in 1895, but the journal Nature in its November 26, 1885 issue quotes a lawyer as saying that there are simple liars, damned liars and experts.
D. Balasubramanian
dbala@lvpei.org
