Building the Unsolvable Maze

I came across a tweet via Simon Singh, famous writer of books based on math-topics. I've read a couple of them: 'Fermat's last theorem' and 'The code book'. His tweet points to a picture of an amazing maze hand-drawn over 30 years ago in Japan. Although it is supposed to be 'unsolvable', some comments there claim that it could be solved very quickly if it was made publicly available. Among the very first papers I read after coming to the U.S to study traffic engineering (to understand the reasons for India's chaotic, maze like traffic) was about Moore's algorithm entitled "shortest path through a maze". Mathematically, the shortest path problem formulation has a couple of properties of small interest in the context of this discussion. It has no duality gap, and is totally unimodular: It is sufficient to solve the continuous 'relaxation' to recover an integral optimal solution to the primal or dual formulation. Wikipedia has a page on maze-solving algorithms.  Interesting as the optimization problem of finding an 'optimal route' to 'escape' this maze is, a more interesting question to me personally was: why would someone hand-build such an intricate maze over years; and then why not claim any credit for it? I have tried to interpret this based on my understanding of the Indian way.

(source link and main article at: http://imgur.com/gallery/4kyvVVb)

The intense concentration required for such a task is surely daunting: to at once elevate one's consciousness while also dissolving one's aham (ego) that hinders the mind from systematically growing a complex maze whose paths increase rapidly over time as more forks and merges are constructed. Paths that stop even as they begin, paths that ultimately lead nowhere, paths where you travel for a while, only to discover that you are back where you were before... and then after a lot of calm, refined, and introspective searching (not suffering), finding a path that leads one to satya (ultimate reality/truth) that transcends the maya of the maze that held us in its thrall. A path that dissolves the noisy duality between the world within the maze and without, uniting them harmoniously into a unified whole, even as the space enclosed within the maze maintains a provisional identity within this overall unity. And then perhaps a realization that there could be a pluralism of such (alternative optimal) transcendental paths to satya. A harmonious unity within multiplicity that celebrates its diversity, rather than a synthesized unity derived by optimizing the goal of orderly sameness. The latter produces an efficient monoculture, but one that invariably regards pluralism as a seed of chaos. The former, integral unity best represents the nature of the underlying philosophical unity of India that has continually preserved and refined its dharma civilization over several thousand years. This forms the dharmic basis for any reasonable 'idea of India'. I look forward to reading Rajiv Malhotra's new book on this subject.

Journeys that traverse such a path have led to amazing discoveries that enlightened the world, and will continue to do so. Perhaps it produced this captivating art that simultaneously appeals to the casual observer, the artist, the seeker, and the analyst alike; yet each of us seeing only a partial facet of its underlying truth. A work of art to which its 'creator' deliberately did not append a signature to, and claim ownership of, perhaps unwilling to disturb it's harmony. That is the way of the Yogi.

Dedicated to Rajiv Malhotra on the occasion of India's 65th republic day.Thank you.

Anatomy of an Online Debate

The Huffington Post ran a curious online debate a few days ago: Is Yoga a Hindu Practice? Let me state right of the bat that I thought the debate was utterly idiotic given that this question was like asking people to vote if baseball was quintessentially American, if the great pyramids were Egyptian, or if the great wall was built by the Chinese. After all, how the heck does a person debate against a fact? By twisting it into a silly insinuation about ownership. Nevertheless, let's look at the debate results measured by market-share for, against, and neutral to the topic, tracked before and after reading the debate. 

If this picture is unclear:

For, against, neutral (baseline) = (65%, 26%, 9%).

For, against, neutral (after reading debate) = (66%, 28%, 6%)

So the net result is that the 3% of the undecided split 1% toward 'for' and 2% toward 'against' after reading the debate.

Warning: If you are a predictive analytics connoisseur or swear by rigorous statistical methods, the rest of the post will be cringe inducing, so read on at your own risk.

Given that we have absolutely no data to use other than this pie chart, I tried to 'quick fit' a plausible Multinomial Logit choice model to these results with the aim of personally understanding how useful this debate really was. Toward this, I defined a utility function u(t) = exp (a0(t) + a1(t)), where:
a0 = baseline contribution for (t = aye, nay, neutral)
a1 = debate contribution
market share (t) = u(t) / (u1 + u2 + u3)

Using the 'before' pie chart, I obtained the following values:


where CONST = a0, and a1 = 0 at this point. Again, note that these are just plausible values and not statistically calibrated likelihood maximizing coefficients based on historical individual observations. Next, merrily using the  'after' pie chart to update the utility function taking the debate into account, I obtained the following values:



where DEBATE = a1 that is used to update the utility function. Note that the results don't really change dramatically.

Based on this plausible MNL model, we observe a positive value for a1 for both the 'for' and 'against' since their market-shares increase after the debate, and a negative value for 'neutral' since the debate forces a good chunk of the few fence sitters to switch. To measure the usefulness of the debate to each group, I looked at the ratio of "what was additionally useful" versus "prior understanding", i.e. the ratio utility_before/utility_after given in the last column. The results indicate that the debate itself was pretty close to useless to the overwhelming majority of the voters, i.e. for the 'aye' people (like me) and on the whole, the debate reinforced what they already knew, yielding a tiny usefulness change of about 0.4%. On the other hand, the debate was relatively more useful to the 'nay' people and it incrementally 'hardened their position' by about 6%. More than a third of the miniscule fence sitters actually took a stance and this debate appealed most to them.

Given that the voter response ('elasticity') for a particular choice-group to an event (debate) in an MNL model also depends on the incremental gain possible from their existing market-share, the degree of movement in market-shares for the three groups are not surprising, although the specific direction of the resultant net shift in market shares does indicate that the debate may have had something to do with it. Of course, one can game such online "changing minds" debates by entirely ignoring the debate and starting with a non-favorite position as the baseline and then simply selecting your most favorite position in the end.

On a side note, almost all the 'Yoga' practiced in the U.S and the west is really Yogasana, whose primary function is to help prep your mind and body for actual Yoga, which in turn has nothing to do with whether you can twist yourself into an exclusive USPO patent-protected double-pretzel or not, and everything to do with open-source inner-sciences that aim to rid a mind of ego, exclusivity, and dogma and reach higher levels of consciousness.

Book review: Choke

This post reviews Sian Beilock's recent book:  
"Choke: What the Secrets of the Brain Reveal About Getting It Right When You Have To"  from an O.R. perspective, as well as from the p.o.v of Dharmic philosophy that has some deep connections to some of the mind-training techniques mentioned in the book.

Scholastic/Intellectual test situations
In addition to aspiring sport stars and business leaders who certainly want to avoid 'choking' at any cost, this book can be particularly useful to students who plan to take competitive scholastic tests. Beilock characterizes 'choking' as suboptimal performance, which implies the existence of a clearly superior level that becomes feasible via practical mind-training techniques. When it comes to tests like SAT and GRE, 'worry' can negatively affect the parts of the brain that are most involved ("working memory" in the prefrontal cortex) in the Q&A process and this can directly result in choking due to a suboptimal allocation of brain power. In fact when constrained by 'worry', lab experiments show that smarter students who routinely ace practice tests are likely to drop to a more seriously suboptimal level relative to 'average' students who face a similar worry. In particular, the book presents evidence that shows that the effect of negative gender stereotyping (e.g. "girls can't do math") has been devastating in the US, inducing a lot of female exam takers to 'choke' in such situations. Subsequently, many of these candidates reconsider their original choice of a STEM career. The author presents a systematic rebuttal of Larry Summers' controversial gender-related remarks in Harvard a few years ago that looks compelling.

An important technique that is proven to minimize the chances of choking during intense time-constrained testing situations is the ancient Indian method of Yoga and meditation that is freely available to anybody (Vipasana in particular, is recommended by the author. Even three months of adopting such methods are known to have beneficial effects). Now if one were to, over an extended period of time, move along this positive Yogic meditation gradient to maximize its benefits, one can practically experience higher states of consciousness and self-realization. This is a central truth-claim of the Dharmic thought system (DTS) of India. The 2011 book 'Being Different' by Rajiv Malhotra is a scholarly and well-researched book that expounds on DTS and is particularly useful for western minds that seek to understand what Yoga and Sanskrit (the language of Yoga) truly mean.

Sports  
Finely honed motor-skills and fluid movements are critical to achieving optimal performance. Here, one can think of the task of winning a contest as constantly solving two nested decision optimization problems. The meta-problem is to manage tactics and overall strategy, while the inner problem is how best to 'operationalize' the chosen objectives in real-time. A conclusion in this book is that one must certainly think about the meta ("what") problem using working memory. On the other hand, it is better not to (like Yogi Berra said) intellectually analyze the "how" part where a player makes real-time play decisions and executes a sequence of precise movements since these have been optimized (objectified?) over years of careful practice and then 'outsourced' to the brain's 'procedural memory'. It's like trying to analyze your legs as you descend a staircase in a hurry.

When it comes to crunch free-throws in basketball, it appears that an important 'choke' statistic is the conditional probability that a player will make the shot given that his/her team is one point behind. Apparently, this conditional probability differs by about 7% on average from its unconditional counterpart. As far as crunch-time soccer penalty kicks, well-established European league stars are more likely to choke and have a success rate of 65%, which is much less than future stars, whose conversion rate was above 90%. In baseball, home teams that are a game away from winning a series, win that game only about 38% of the time. Clearly, heightened expectations from supporters increases the chances of choking.

 

A useful point to remember that will help minimize the chances of suboptimal performance in any situation is present in a Sanksrit Mantra that was uttered after what can be viewed as the world's first ever choke, when Arjuna, the hero of the Mahabharatha, on the eve of battle, is consumed by self-doubt and initially decides against fighting the good fight and plans on simply walking away, before Krishna who was selected by Arjuna to be his charioteer in this battle, reminds him of his Dharma (a Sanskrit untranslatable, roughly means 'fundamental duty') and says, among other things:  

 

Karmanyeva adhikaraste ma phaleshu kadachana
Ma karmaphalahetur bhurma te sangostvakarmani.


"Your attention must be directed toward the action alone, never with its fruits. Let not the fruits of action be your motive, neither should you be inclined toward inaction".

As we can see, even heroes can choke, but the truly great ones have a reliable 'corner man' like support system that helps them find a way to turn it around.

 

[update: fixed format]