Before reading on, pause and consider: What strategy would you use if you were playing this game?

The Optimal Group Strategy

If we’re working as a team to find the number as quickly as possible, what’s our best approach? We need a systematic way to eliminate as many possibilities as we can with each question.

The optimal strategy is binary search – repeatedly halving the possible range by asking if the number is in one half or the other. For a number between 1 and n, this takes approximately logâ‚‚(n) questions. This is provably optimal from an information theory perspective – each yes/no question can at best halve our uncertainty. (This approach might be familiar if you’ve ever studied or used binary sort algorithms in computer science – the same principle of dividing the search space in half each time.)

Why does this work so well? Each question eliminates half the possible numbers, regardless of the answer. After k questions, we’ve reduced the possible range by a factor of 2^k. For example, with a starting range of 1-1000:

• Question 1: >500? Reduces to 500 numbers
• Question 2: >750 or >250? Reduces to 250 numbers
• Question 3: >875 or >375? Reduces to 125 numbers
  And so on…

The Competitive Dynamic

But here’s where it gets interesting. The class isn’t actually trying to find the number as quickly as possible. Each child is competing to be the first to identify it. This transforms our optimization problem into a game theory challenge.

Consider child A’s decision when it’s their turn. They now face a trade-off:

1. Ask a “narrowing” question that helps everyone by reducing the possible range
2. Make a direct guess at the number

This becomes a competitive sequential search problem, touching on three related areas of study:

1. Competitive Search Problems: How do agents search through a solution space when competing to find something first? This appears in various contexts, from R&D races between firms to parallel search algorithms in computer science.
2. Information Cascade Theory: How do individuals make sequential decisions while observing others’ choices? This typically studies how public information influences private decisions, leading to potential herding behavior.
3. Strategic Information Revelation: How do agents decide what information to reveal when that information might help competitors? This is crucial in contexts like patent disclosures and auction bidding.

The optimal individual strategy shifts as the game progresses:

• Early game: The search space is too large for guessing to be rational
• Mid game: There’s a tipping point where direct guesses become more attractive
• Late game: Once the range is sufficiently narrow, direct guesses become optimal

For a rational player, the decision at each turn should be based on:

1. Calculate probability of winning with a direct guess: p = (current range size)^-1
2. Calculate probability of winning later after a narrowing question, accounting for:

• Reduced range size
• Number of other players who might guess correctly before your next turn
• Number of turns until you go again

1. Choose the action with higher expected value

Crucially, this assumes all other players are also playing rationally – a strong assumption that might not hold in practice, especially with younger players!

The Finite Guesses Twist

Here’s another interesting variation: what if players are limited to a fixed number of questions? This creates a fascinating optimization problem even in non-competitive scenarios.

At some point, if you have k questions left and n possible numbers, random guessing becomes better than binary search. The intuition is that binary search might narrow down the range significantly but leave you unable to identify the specific number.

For example, with just two questions left and a range of five numbers, you might be better off making two direct guesses (probability of success = 2/5) rather than two binary search questions that could narrow it to 2 numbers but not identify which one.

The optimal strategy becomes a dynamic programming problem, where at each stage you need to calculate the expected probability of success for:

1. Using a binary search question
2. Making a direct guess

The mathematics behind this optimization is fascinating, but I’ll leave that for another post!

An Interesting Parallel

This competitive dynamic reminds me of the board game Cluedo (Clue in North America). While the optimal strategy is usually to gather information methodically, the game dynamic shifts dramatically when you suspect another player is close to solving the mystery. At this point, making an educated guess at the solution, even with incomplete information, might be your best play despite the penalty for guessing incorrectly.

This perfectly mirrors our number game – when you suspect other players are close to identifying the number, the rational strategy might be to take a calculated risk with a direct guess, even if you’d normally prefer to gather more information.

Final Thoughts

What started as a simple classroom game reveals layers of mathematical and game theoretical complexity. It demonstrates how individual incentives can lead to strategies that are suboptimal for the group – a common theme in game theory.

Have you encountered similar games or puzzles that seem simple at first but reveal hidden complexity? I’d love to hear about them in the comments.

Queue management at the motor vehicle licence renewal office offers a window into a realm of complexity that extends far beyond the waiting room. Exploring priority queuing and weighted fair queuing (WFQ), we uncover a world of probabilistic models, dynamic rules, and real-world applications that challenge conventional wisdom.

Upon arrival, patrons encounter priority queuing, where elderly individuals receive absolute precedence to accommodate potential mobility challenges. While this prioritisation addresses (quite reasonably!) the needs of a vulnerable group, it can lead to excessively prolonged waits for others, depending on the arrival rate of elderly patrons and the dynamics of the queue. Consideration of different objective functions for various demographics, including the young and old, could unveil alternative optimal solutions.

Actuaries are used to implicitly allowing for objective functions that aim to minimise mean squared errors, but this isn’t the only possibility. Aside from different penalties for waiting times for all participants, explicitly considering different costs to different groups of individuals makes it more interesting.

In contrast to absolute priority queuing, WFQ introduces a probabilistic element, ensuring that even low-priority individuals have a chance of being served at every point in time. The weights can be tuned for those different groups, still achieving prioritisation for the elderly. While wait times for low-priority customers will still be longer than priority customers, they remain finite, underscoring the equitable nature of this approach. I can imagine an algorithm where the weight depends on the wait – the longer the duration of the existing wait the more an individual is prioritised.

The network packets to which WFQ most commonly applies probability aren’t conscious, and therefore their experience of the wait isn’t important. For humans, the conscious experience of knowing that you don’t have to wait until every individual in a priority group is served before you might make the wait more palatable. I’m not a fan of gamification; overall this seems to have result in evil outcomes, greedily addictive algorithms and a worse society overall. However, the introduction of a little randomness and the possibility of a serendipitous early shot at renewing a licence might be a net positive.

Beyond the motor vehicle licence renewal office, the principles of queue management find relevance in diverse fields, including insurance companies. By leveraging similar concepts, insurers can prioritise claims payment or customer service based on factors like severity or urgency, but also explicit objective functions to optimise for target response times within particular bands, aiming for prompt resolution while maintaining fairness and efficiency. Sometimes the maths will demonstrate that based on demand and supply of support calls, failing these objectives is inevitable. Modelling can determine how often that may occur and then you can weigh up the costs of more call centre agents with the unappealing customer experiences.

These waiting processes should ring bells from prior study of queuing processes, of exponential, poisson, negative binomial and gamma distributions. These provide a mathematical framework for understanding and optimising queue management, revealing the hidden complexities behind seemingly simple processes. While you may not think about claims processes as queuing processes, there are some surprising linkages there too.

For me, delving into the dynamics of network switching provided some fun insights. In network architecture, switches play a critical role in directing data packets efficiently across networks. However, the limited capacity of switches can lead to packet loss during times of congestion. Buffers within switches temporarily store packets to alleviate congestion, but if overwhelmed, they are forced to drop packets, resulting in packet loss and network underperformance.

A smarter (yet initially counterintuitive step) is to apply Random Early Detection (RED), a mechanism that deliberately drops packets in anticipation of the switch being overwhelmed. Under RED, these strategically dropped packets provide information to network endpoints to manage the speed of transmission to prevent larger problems down the line. To an extent, this can can be likened to passengers on an overbooked flight being paid to take a different flight. However, unlike RED, this approach does not alert endpoints or other passengers to slow transmission rates, highlighting the unique challenges and solutions in network management.

While queue management may appear remote from your regular work, some exploration might be worth the time spent. By embracing probabilistic models, dynamic rules, and real-world applications, we may see a way to apply our skills in new areas, or even learn from the significant advancements in network modelling in our financial services world.

It’s likely that several of your own service providers will roll this out in the next couple of years. Progress, right?

The problem is that in the short time since voice identification models have become mainstream and slick enough to be used in call centers, voice spoofing technology has exploded. It is now trivial to create voice tracks saying whatever you want them to say without human ears being able to tell any difference, and increasingly, fooling the voice authentication models too.

Worse, we are probably only a few years from trivial video spoofing with the same qualities.

I support the idea of moving away from passwords and improving security, but I struggle to understand how voice authentication is anything other than a tiny blip in the timeline before it becomes the easiest vector for fraud yet.

The security check point is a bottleneck. More specifically, the number of scanners open at certain rush times isn’t enough to cope with the arrival rate of passengers at the check point.

The bottleneck is the small number of open scanners. The bottleneck could be fixed and throughout increased by having more scanners. No amount of hurrying up the process to inspect tickets will change the bottleneck. No amount of directing people to stand in queues at each scanner will change the rate at which people go through the scanners (as long as there is always a queue at each of them). None of this changes the throughput of the sytem.

But it does mean that there are extra people employed to inspect tickets and extra handlers directing people to queues – these employees could presumably operate an extra scanner and massively increase throughput. This might in turn create a bottleneck somewhere else, but if that happens it shows that throughput has already been increased. It also presents an opportunity for the next stage of optimisation.

If you were university educated in South Africa, you are likely to be in the market at various times in your life for a large LED backlit LCD panel with a high refresh rate and more HDMI inputs than you will ever need. You will also quite likely have a market-oriented, Anglo-Saxon view of government’s role in industrial policy and international trade. Thus you would probably say “yes, import cheap TVs from China so I can buy a cheap TV and not pay for inefficient local firms to manufacturer expensive, inferior TVs.”

If you are a TV snob, you will still want free imports of Chinese TVs to keep the prices down of competing, but fancier Sony and LG models from Japan and Korea.

If you are a little cynical, you might say South Africa could never have the manufacturing capability and scale to produce all the components and assemble them into a modern LCD TV. That’s not actually the debate I ant to pursue now, so in that case let’s say the alternative would be to locally assemble sets made with significant local components, even if the LCD panel itself were imported. Of course, the reason South Africa doesn’t have the scale to produce the panels themselves at the moment is a function of industrial policy decisions decades go. There is no absolute reason we couldn’t have that capability. But, that debate is related but separate post.

If you never completed Grade 8 (that I still think of s Standard 6) then you probably would rather have a job than a Chinese TV and even a single HDMI input or output is one too many. Also, you don’t care about free international trade leading to optimal allocation of economic resources or the World Trade Association for that matter.

But you wouldn’t necessarily be wrong.

Reason #1
A large-screen LCD TV is not a factor of production of any industry that I’m aware of. Restricting cheap imports won’t increase the costs of a domestic industry an therefore won’t reduce the competitiveness of a local company or industry.

So far so good.

Reason #2
The Chinese will still buy or raw natural resources even if we don’t buy their TVs. We aren’t a sufficiently big market for them in the first place. We don’t reduce our exports but we do reduce our imports. This grows the local economy.

This does assume that over time, the higher prices of Imported TVs will discourage their purchase for other goods or encourage purchase of South African made TVs. It also assumes,NAND this is where things get more complex quickly, that our currency won’t strengthen to reflect the reduced imports. A strengthened currency would make all our exports less competitive and imports more competitive, leading to higher imports and lower exports across all industries and consumption.

Reason #3
Te burden if higher prices will fall on the wealthy, whereas the boom of greater employment will fall on the less wealthy, even increasing total demand for labour.

A critical question is what sort of labour.

If it is skilled labour, then our right skilled labour market will have wage pressure without increased employment and there will be a transfer of wealth from television set consumers to skilled labour and an increase in the cost of production of many other industries. All in all not a great outcome. We’ve damaged several industries and are reliant on the relative consumption habits of skilled labour versus television set consumers to see what this means for domestic versus import consumption.

If the demand is for unskilled labour, or separate but well-timed government education and training policies provide unskilled labour with the necessary skills, the result is far more appealing. Wages won’t increase as unemployed resources are put to work. So, no damage to other industries. The productive capacity of unemployed persons is put to work in an economically productive way. The country is producing more things domestically so all the additional money spent on more expensive TVs is employed back into the economy. The recently newly employed are likely to focus their purchases on domestically produced items (especially since TVs are now made locally)

Assuming the caloric requirements of these newly employed persons are not much different from during their unemployed years, the opportunity cost for the economy as a whole of paying them to make TVs less efficiently than the Chinese is zero. There is noting better they could have done with their time.

The arguments of comparative advantage totally break down when there is significant unemployment in one country. Free trade is centred on the idea that countries should focus on producing goods where they have a comparative advantage. Quotas and tariffs distort the true comparative advantage leading to suboptimal solutions.

There are other ways of achieving a similar result. Taxing TV set consumption to pay for skills training and subsidised public transport might lower some of the restrictions on supply and demand imbalances in the labour force, but without the focus on growing a particular industrial sector that will give rise to a cluster of related industries, developed scale to lower costs, eventually gain efficiencies and export to African markets with lower cost than the Chinese.

Why should the Chinese target attractive industries for growth through strong industrial policy and South Africa doesn’t? Why are the Chinese so uniquely well placed to target LG and Samsung?

There are a hundred problems with this scenario, including possibly the choice of TV as the product, but there is more to this discussion than the Washington Consensus would have you believe. Disagree, please comment below!

A MODEL ISN’T REALITY

Any simplified model I can imagine will also therefore not match reality exactly. The closer the model gets to the real world in more scenarios, the better it is.

Not all model parameters are created equal

Part of the approach to getting a model to match reality as closely as possible is calibration. Models will typically have a range of parameters. Some will be well-established and can be set confidently without much debate. Others will have a range of reasonable or possible values based on empirical research or theory. Yet others will be relatively arbitrary or unobservable.

We don’t have to guess these values, even for the unobservable parameters. Through the process of calibration, the outputs of our model can be matched as closely as possible to actual historical values by changing the input parameters. The more certain we are of the parameters a priori the less we vary the parameters to calibrate the model. The parameters with most uncertainty are free to move as much as possible to fit the desired outputs.

During this process, the more structure or relationships that can be specified the better. The danger is that with relatively few data points (typically) and relatively many parameters (again typically) there will be multiple parameter sets that fit the data with possibly only very limited difference in “goodness of fit” for the results. The more information we add to the calibration process (additional raw data, more narrowly constrained parameters based on other research, tighter relationships between parameters) the more likely we are to derive a useful, sensible model that not only fits out calibration data well but also will be useful for predictions of the future or different decisions.

How not to calibrate a model

Scientific American has a naive article outlining “why economic models are always wrong”. I have two major problems with the story:

1. All models are wrong. Some are useful (George Box). “wrongness” isn’t a problem with a model, but lack of usefulness is. The headline demonstrates a starting point poorly informed about the point of economic models.
2. The calibration approach criticised in the article is an extremely poor way to calibrate a model. No serious researcher thinks that is the right way to calibrate a model. So the article merely creates a straw man and then demonstrates how easy it is to knock the argument over.

Calibration and back-testing on separate data sets

The right way to calibrate a model is to separate the data-set into at least two independent subsets. Firstly, the “training set” or portion from which we will calibrate our parameters to get them to match as closely as possible the data. Again, this should make use of all information available and may give rise to several competing models that appear to fit the data similarly well.

The next step is crucial. We back-test the derived models against the second subset of data. This data comes from the same reality (perhaps a different time period) as used to calibrate the model, but the model won’t trivially match the data because none of that data was used to calibrate the model in the first place.

The importance of back-testing

Back-testing is critically important in the model building process, but back-testing against the same data used to calibrate the model is worth than useless (since it takes time and effort and an create a false sense of accuracy or reliability in the model.) Separating the data into two or more subsets is absolutely required, although it has the unfortunate side-effect of reducing the size of the data-set available for calibration.

Yes, it really matters.

A common example of the dangers of bad models fitting data well is with Economic Scenario Generators. These simulate economic scenarios to be used in valuing complex financial securities. If a model is properly calibrated, it will recreate the observable market prices of a wide range of instruments. However, the model could be a black-box neural network, a carefully constructed theoretical model with plausible relationships and constraints, or the proverbial ten thousand (possibly inebriated) monkeys. If all three models are perfectly calibrated to observable market prices, is any of the models inferior to any of the others?

Clearly the answer is yes, but only when it comes to extrapolation. I have far more confidence in the model’s ability to create “market consistent” valuations for instruments that do not have observable prices in the market if I understand how the mechanics of the model make sense on a level other than pure calibration.

Trivial 3 point example

The example above shows a perfect of a quadratic and cubic model to 3 data points. From this graph, both models appear exactly the same.

However, if we use the model to extrapolate to future time periods, the results are very different. Without additional data to back-test the results on, it’s not possible to tell whether either or any of these models is appropriate, but clearly both can’t be correct.

Bit by bit, for the past 40 years, the city of Copenhagen has done something revolutionary: The Danish capital has reduced its parking supply. Cutting the total number of parking spaces by a small percentage each year stands in stark contrast to the more common pattern of cities adding more and more parking to accommodate private cars.

It’s a longish article, but definitely worth reading. Hat-tip to Samora Adams of @CapeChat fame for the link.

I say “in large sectors of the economy” because it isn’t true to say that we have universal unemployment. In fact, a feature of structural unemployment is that it usually is not uniform throughout the economy (like cyclical unemployment often is).  I don’t know any actuaries or engineers who are unemployed for more than a brief period between jobs, and usually the jobs start and end back to back. There will be other examples too.

Unemployment is driven by education

Interesting that 75% of our unemployed are “unskilled”. (I heard this on the radio, so I don’t know that  the number is correct or it’s source, but it does map to my previous analysis based on census showing unemployment by education level attained.

• The unemployment rate for those with less than “matric with university exemption” is between 30% and 40%.
• Matric with university exemption unemployment is 23%
• The unemployment rate for this with better than “matric with university exemption” is on average below 10%.

Economic growth isn’t the only solution to unemployment; in fact it’s not even necessarily a solution.  Prior periods of strong economic growth added jobs only very slowly. We have massive, structural unemployment in this country. We are making some of the right noises with our government’s new jobs plan and jobs fund.

Education in South Africa is not performing as needed

However, given the obvious relationship to education, why don’t we take the problems of our education system seriously?

• Standards have dropped (although enrollment has increased, which is also critically important for improving average education attainment)
• While enrollment has increased, the percentage of GDP spent on education has dropped from 6.4% in 1994/1995 to 5.3% in 2006/2007. This is partly explained by an increase in government spending in total related to social programmes. The real expenditure per learner has increased by approximately 5% per year from 2000 to 2007. We still spend a high proportion of our budget on education, but with poor outcomes.
• We don’t seem to participate in as many international standardised tests as before (reflecting a lack of appreciation for wanting to know what our standards are actually like)
• Standards are too low so that passing isn’t sufficient for further education / job placement
• “Literacy” measures have improved, although this is measured as having completed 7 grades of education. Why is it so hard to appreciate the need to  use measures (or at least also use measures) independent of the system to measure the success of the system?
• The Department of Educations 2009 Macro report uses only 1999 and 2003 data from TIMSS.  It shows our results are very low, but have not decreased by much. We didn’t participate in 2007 (probably because it’s embarrassing coming bottom of the class), which is a shame. 2003 data is really not appropriate to be using in 2011 to evaluate performance of our education system. (It looks like we are at least signed up to participate in 2011, which is great news. We’ll have to wait till December 2012 to get the results though.)
• The Quality Improvement, Development, Support and Upliftment Programme has budget allocated per province to spend on improving education.  While the Free State managed to spend 99% of its budget and the Western Cape 96%, the Eastern Cape spent only 51%, Limpopo 36% and KZN an impressive 11%. This is also home to possibly the worst, most impossible to read, mixed-numbers, mixed-decimal point table I have ever seen.

International indicators of performance

TIMSS 1999 Rock-bottom in Maths and Science. 6 provinces results went backwards in Maths and Science. The DoE’s report indicates that our syllabus has the least overlap with the TIMSS assessment framework – this offered as an excuse for poor performance (which it is) but without even considering that maybe that is a problem in itself. Should our education system really be going about things in a different way to established international norms? Maybe, but I think that needs to be proved and the default position should be international standards unless otherwise proved.

PIRLS 2006 Rock-bottom in reading, although this is against primarily developed countries (and only above Kuwait and Qatar in terms of gender inequality of reading education) (Chapter 1 and full report) (Curiously, other measures including our own Department of Education and SACMEQ studies seem to show girls outperforming boys, rather than the other way around.)

South Africa’s SACMEQ scores didn’t  change significantly from 200 to 2007 – whereas only one peer country had a reading score decline and 2 had mathematics scores decline. and 14 countries’ scores increased across maths and science. The percentage of students not sharing a textbook declined from 45.5% in 2000 to 45.0% in 2007. We are not making progress.

The link between enrollment and attainment levels is clear, but more clearly than that South Africa is massively underperforming.Curiously, the DoE’s report seems to suggest that a reason for our low attainment is high enrollment, whereas the graph above shows that high enrollment is associated with high attainment for those enrolled. My guess at the cause of this strong correlation is that “countries that take education seriously focus on both enrollment and attainment”.

Some South African problems with matric pass rate measurement

The pass rate of matric exams is a function of the quality of education (higher means higher pass rate), standards (higher means lower pass rate) and selection of learners who write the exam (restricting it to only relatively strong students artificially increases the pass rate). We should not celebrate an increase in the pass rate unless we have shown that standards of the exam have not been decreased and that learners weren’t discouraged from writing the exam. I would suggest that the standard of the test is more likely to vary from one year to the next rather than miraculous changes in education quality levels for learners who have had 12 years (or more…) of education to get there.

For example, this graph worries me no end.

I’m not going to show the real/nominal expenditure per capita because the glaring errors in the “2006 real expenditure per capita column” suggests to me that the creator of the table needs to go back to school.

Conclusions?

This post has been more of a collection of observations and thoughts and extracts from other research. In summary:

1. Education is key to unemployment and growth. Unless we recognise that education is the most important long-term solution to structural unemployment we can’t take the problem seriously. Skilled citizens by and large are employed. Unskilled citizens are unemployed at 30% to 40% levels.
2. Government spends significant budget on education, but with poor outcomes. Some budget available is not being spent at all.
3. Measures of literacy and matric pass rates are fundamentally flawed. We need to place increased focus on outcomes compared to international standards if our measures are to be useful.
4. We need to participate more, and more regularly, in international comparatives, no matter how embarrassing the current results are.
5. Current performance against international (and other African country) performance is poor and not improving.
6. The quality of research and ability to recognise the problems in our education system by the Department of Education itself needs to be improved.