Friday, 27 November 2015

"Inequality - social evil or acceptable cost of free market capitalism?"

"Inequality – social evil or acceptable cost of free market capitalism?" was the title of a talk given recently at the University of Winchester Business School by the Director of The Equality Trust.

Their website states "UK income inequality is among the highest in the developed world and evidence shows that this is bad for almost everyone. The Equality Trust works to improve the quality of life in the UK by reducing economic inequality." so the talk clearly comes from a particular point of view.

It was really interesting to hear what he had to say and some of the conclusions arising from research in this area.  The presentation gave us some data without much definition and context but subsequently I have spent some time exploring their website  which I would commend to you as a source of summaries of findings and they provide plenty of links to research on the topic.  This page included an interesting contrast between income inequality and wealth inequality.   I knew from the talk that the UK was one of the most unequal countries on their list for income ( as measured by the Gini coefficient).  What surprised me in the data was how low we come in the ranking for wealth inequality.  Not only that but the most unequal country for wealth was the country that was shown in the talk as least unequal for income which was an interesting contrast.  I'll not spoil the surprise as to who it is...  Would be interesting to dig into this a bit more and understand what the boundaries are between "wealth" and "income" in this context.  Does, for example, the category of "income" include any returns that I receive from my "welath" - eg dividend payments if I had shares.

The other surprise to me from the talk was that the UK Gini coefficient actually declined a bit between 2010 and 2011.  With some bumps up and down in the intervening years we are tracking at the same level as it was in 1990,  all be it that this was sharply up over the preceding 2 decades. That's not to say it isn't a problem but my impression had been that we were in an era of rising inequality.

We saw how Income inequality and a general index of health and social problems seem to correlate - I did notice though that the data here for income inequality was actually not the same as the Gini coeficient data on the previous ranking chart so unsure exactly what this was measuring.  Need to dig into this a bit more to understand where this data is from - "The Spirit Level" now on the list of books I need to read someday.

Our speaker made a good point about use of language and how that influences our view.  Articles in business press for example will tend to refer to the high earners and their pay packages under heading of "talent management".  Discussions of lower earners will tend to be around "workforce costs" or "resource planning". Communicating a message of unequal valuing of the people perhaps.   The use of language to "guide" your audience is of course pervasive and often reflecting of the narrative that the writer is aligning with.   Looking at articles on CEO pay in the press there is usually some comment around the ratio of their pay to the median or lowest paid in the company.  Read about the income of a sportsman, and this language simply doesn't seem to be there.  To take just one example, I'm not sure I've ever seen a ranking of footballers' salaries that looked at how that compares as a multiple of the income for the lowest paid employee/worker at the club.

Discussion of inequality is often accompanied by a linked discussion of CEO pay and indeed this did come up during this talk.   What would happen I wondered if you looked at top FTSE 100 CEO salaries, which of itself has of course limited the view to leaders of public companies rather than privately held enterprises, and mixed in earnings from some other potentially high income groups.

To the extent that the reported incomes are actually reflective of reality ( sources noted at the end) here is what I found with a few minutes of web searching.

Top 3 FTSE 100 CEO earners

1 - £42.978m
2 - £19.51m
3 - £16.176m

Taking a look at top paid UK people on the "World's highest-paid Athletes" list that fall in this range we have

1 - £32.11m
2 - £25.93m
3 - £23.27m
4 - £17.88m

Adding in report of top paid actor

1 - £17m

So while a lot of  the heat and fury seems to be directed at the public company CEO pay, by the time we have worked our way down to third place we have already picked up 4 sportsmen and an actor. Our population of top earners is now only 37.5% CEO, and I'm sure there are other categories, such as the entire music industry, that I could have looked at to find other people that would come into this bracket.

I'm not saying here that there isn't an issue, what I'm wondering is why it is that the focus is so disproportionately on the minority of the top earners who happen to be FTSE 100 CEOs and the comparison of their salary as multiples of what others earn.   If the aim is to address inequality then why focus on this group?   I wonder perhaps if it is something to do with what skills we value as society?  Is it that we see a footballer/actor/singer/.... performing their art and we appreciate a skill in what they do but somehow we fail to recognise the skills and abilities of the CEO?

So, a really thought provoking evening and one which, as you will have noticed and perhaps unsurprisingly, leaves me with more questions than answers.

Sources of my data 

Huffington Post FTSE 100 Top Earners -

Forbes world's Highest paid athlete -   converting USD to UKP at current exchange rates on 27th Nov 2015 using search box.

Highest paid actor -

Saturday, 6 June 2015

What makes a problem difficult?

It's not every day that mathematics makes the main news programs but that is what happened yesterday.  A GCSE ( UK exams taken by 16 year olds) maths exam question made it to the news as being very hard and having stumped many.  ( A quick search for "Hannah's sweets" in your search engine of choice should show just how much coverage it got …) 

Here is the main question…

There are n sweets in a bag.  Six of the sweets are orange.  The rest of the sweets are yellow.

Hannah takes a sweet from the bag.  She eats the sweet.

Hannah then takes at random another sweet from the bag.  She eats the sweet.

The probability that Hannah eats two orange sweets is 1/3.

Show that n2-n-90=0

..and my solution 

Start by considering the first sweet she takes out.  The probability that it is orange is 
(number of orange sweets)/( total number of sweets) which is 6/n

Now consider the second sweet she draws from the bag and the chance that this one is also orange.  Again this is  ( number of orange sweets)/( total number of sweets) but we need to remember that we removed 1 sweet already and it was orange so we have 5/(n-1)

To get the probability of the two events both happening we need to multiply the individual probabilities so we have the chance of drawing out 2 sweets and them both being orange is (6/n) x (5/(n-1)) and that multiplies out to (6x5)/(n(n-1) = 30/(n2-n)

In the question however we are told that he probability is 1/3rd so we get

1/3 = 30/(n2-n)

We know we can multiply both sides of an equation by the same thing so multiply by 3 and then by (n2-n) to get

n2-n = 90

Subtract 90 from both sides and we have n2-n-90=0 so that's the first part done.

The second part of the question asked how many sweets there were in the bag.

To work this out we ideally need to know that (x+a)(x-b) multiplies out to x2 +(a-b)x -ab

So what we are looking for are 2 numbers which multiply together to give -90 and add together to give -1. Considering this for a moment leads us to -10 and 9 so we now know that 

n2-n-90 = (n-10)(n+9) = 0

There are hence 2 possible solutions , either n=10 or n=-9

Here we need to intersect our maths with the real world and realise that as we have a bag with actual sweets in it n has to be positive so there must be 10 sweets in the bag.

Always good practice to check your answer so if the are 10 probability of drawing 2 orange sweets is 6/10 x 5/9 = 30/90 = 1/3 so that looks good.

So back to my subject line "what makes a question difficult". To solve the mystery of Hannah's sugar rush we needed 4 pieces of mathematical knowledge: how to calculate a probability; how to combine probabilities; how to rearranging equations and how to do a simple factorisation of a quadratic equation.     If you don't know those then clearly you will have a problem but I suspect many of the people classing this problem as hard would have known those things so what made this question hard?

I've been reflecting on this and I think it's because to solve it you need to head out like an explorer down an uncertain path.  We know where we are and we know where we are trying to get to but as we start out we have no clue about the journey.  It isn't instantly obvious what the formula we are asked to derive comes from, this only becomes clear once we make a start and see where it takes us.

This ability to make a start and see what happens strikes me as a very useful skill to have.  With the world moving at en ever faster pace I think to ability to innovate and develop ideas as you go along is an important capability to have.  Challenge is how we cultivate this confidence and ability to head out down the unclear path and see what turns up and to use that to move us to our destination.

Sunday, 22 March 2015

Wadham Mathematics equinoctial subject reunion

This Saturday I joined fellow Wadham mathematicians from across the decades for a reunion in Oxford (these photos  were posted by Wadham after the event).  After lunch in college we walked to the new Andrew Wiles Building for a guided tour.

Inside the building Inside the building Interior detail

It's an impressive (and indeed award winning) building on a much bigger scale than the previous Institute and with lecture theatres big enough to hold the first year lectures so todays students are denied the regular site of dinosaurs that we enjoyed on our way to our lectures hosted in a the Natural History Museum.   As we passed through the common room we noticed an intriguing puzzle on one of the screens….. something to muse on.

Puzzle in common room

Settling into one of the lecture rooms we were treated to a delightful series of short talks from some of the Wadam Mathematics fellows.  First up we were introduced to "kiiking" which turns out to involve a large swing that you stand up on and aim to build up the required momentum to manage complete revolutions.  There are 2 records associated with this: the largest number of complete revolutions in 1 minute (video of record - recommend skip forward to just over 8 mins into the video); the greatest height of the swing (video of record).  Having introduced us to the sport Sam Howison then took us on a whistle stop tour of a mathematical model that suggests what the limits of these records will be.  More details - here

Next up Andrew Hodges ( most recently famed as the author of the book Alan Turing: The Enigma which inspired the film "The Imitation Game") spoke about "twistor" geometry.  Discovered back around 1970 by Roger Penrose these ideas were seen as what Andrew called "an Oxford eccentricity".  Work continued and 30 years later they came to be seen as fundamental in some areas of particle physics.  More details at

Nick Woodhouse spoke briefly about the Clay Mathematics Institute ( of which he is the President) and their PROMYS Maths Masterclass program before handing over to Alex Ritter for the last talk of the afternoon.  Alex treated us to a 30 minute tour of tilings.  More content from Roger Penrose but this time his work on non-periodic tiling - fascinating stuff.   Full details in the notes from a masterclass that Alex ran here

With our last lecture of the day complete it was time for a drink and to say goodbye.

It turned out that the 21st was also the day for the opening of the Weston Library so we took the chance for a quick look in there - impressive transformation.

In an action packed day for Oxford it was also the start of the Oxford Literary Festival (maybe the date for the opening of the library across the street wasn't a complete coincidence) so we rounded off our day with a talk in the Sheldonian by Tony Hawks on his new book.