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it’s over. sleep now.

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Tick tock
There’s no way to stop the clock

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Oh dear, how did it get to here.

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Just list­ing some recent files:

  • Poster: pdf
  • Thes­is present­a­tion: slides pdf, notes for first minute pdf docx

The thes­is present­a­tion was my first attempt at doing a Lessig Meth­od present­a­tion — I know it’s not entirely ori­gin­al, but I wanted to do some­thing to make people wake up after half a dozen or so present­a­tions. It wasn’t entirely smooth, but with prac­tice, it should get bet­ter (I did it again at the SUITS AGM — slides later).

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I haven’t blogged for over an entire month, and I’ve been want­ing to blog, because I love blog­ging but things in RL have con­spired to take me away from here. I have a whole list of things I want to blog about, and I have a few posts queued up wait­ing to be fin­ished, but I think they’ll have to wait until after everything is done for this semester.


What an awful concept. They’re hard to write.

If I were a world dic­tat­or, I’d change the onus from writers to read­ers for con­struct­ing coher­ent argu­ments and struc­tur­ing prose. 🙂 We’re already doing you the favour of find­ing all the mater­i­al to talk about — why should we also both­er to present it nicely?

Sigh. Well, it’s been an inter­est­ing year, and I have no regrets tak­ing a year off law to do hon­ours. I’ll blog more about that later, when I have the pleas­ure of reflect­ing back in a stress-free envir­on­ment.

Sign­ing off for prob­ably anoth­er couple of weeks…

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This semester, we’ve been tak­ing a course called “Algorithmic Game The­ory”, which is the broad area that my thes­is top­ic belongs in. Although Tasos is the course coordin­at­or, and lec­tured the first couple of lec­tures, the bulk of the “lec­tur­ing” has fallen to the stu­dents in the course.

Last week was my turn, and I did my talk on evol­u­tion­ary game the­ory. I had been inter­ested in that ever since I read Dawkin’s The Selfish Gene, where he makes use of evol­u­tion­ary game the­ory, albeit in a non-math­em­at­ic­al way, to explain his ideas for the evol­u­tion of genes. In a nut­shell, evol­u­tion­ary game the­ory allows you look at the evol­u­tion of strategies/​genes/​behaviours in a large pop­u­la­tion of organ­isms. For example, can a mutant gene over­take an incum­bent gene? See the link before for more inform­a­tion, or read my lec­ture slides: evolutionary.pptx, evolutionary.pdf.

Now, onto the second half of the post’s title: why I’d hes­it­ate to use Power­Point again. I’ll begin with a cla­ri­fic­a­tion: why I’d hes­it­ate to use Power­Point again where I need to use equa­tions at all. (If you’re an Open­Of­fice fan and you’re begin­ning to smirk, here’s some­thing to wipe your smirk off: Open­Of­fice Impress fails to impress me even more dra­mat­ic­ally. Sorry.)

I’ve been using LaTeX with Beam­er for my present­a­tions this year, and I’ve had a good exper­i­ence with it so far. Why did I use Power­Point? Mainly because I haven’t used Power­Point 2007 for any real pur­pose so far, and secondly, because I saw that Word 2007 had a new flashy equa­tion edit­or that’s kind of nice. It was a bit of a dis­ap­point­ment for me when I had fin­ished writ­ing all the slides with no maths to find that Power­Point some­how failed to inher­it this. Back to old Equa­tion Edit­or. I hate it, so I took to doing the equa­tions in Word and then copy­ing them over as pic­tures. The main prob­lem with all this is that, for a math­em­at­ic­al present­a­tion, equa­tions should not be treated as pic­tures. Power­Point and Open­Of­fice both lack the abil­ity to insert equa­tions as inline text, and that frus­trates me to no end. Anoth­er minor little gripe is that there’s no in-built way to have nav­ig­a­tion bars like you do in Beam­er.

The shock­ing thing is that most lec­tur­ers in aca­demia, such as the School of IT, con­tin­ue to use Power­Point even though the set of tools it provides for tech­nic­al present­a­tions is min­im­al. (If you’re doing a sales pitch with pie charts and dot points, it’s fine.) Unfor­tu­nately, this just means there’s little incent­ive for Microsoft to go and improve the tools for this import­ant mar­ket seg­ment.

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Braess’ para­dox is a counter-intu­it­ive res­ult in trans­port­a­tion sci­ence, which in essence states that if you add addi­tion­al capa­city to a net­work, over­all net­work per­form­ance may in fact decrease, if agents are free to (greed­ily) choose a path that min­im­ises their own latency. In this post, I’ll explain Braess’ para­dox, and provide a new example that uses only non-increas­ing cost func­tions.

Let’s con­sider the fol­low­ing net­work setup:

Braess’ paradox - initial setup

We have four towns, s, v, w and t, and one unit of traffic that needs to get from s to t using this net­work. The cost func­tions c(x) on the links indic­ate how long it takes to use that route as a func­tion of the frac­tion of the traffic that will take that link (in some unit of time, say hours). For example, if the link’s cost func­tion is c(x) = x and half the traffic goes along that link, then it will take every­one half an hour to tra­verse that link. You can think of these roads as nar­row roads that are rather sus­cept­ible to con­ges­tion. The links with cost func­tions c(x) = 1 are not affected by the traffic flow at all; these are ana­log­ous to long, but very wide, high­ways.

In the above case, because the top route (s-v-t) is identic­al to the bot­tom route (s-w-t), traffic will spread itself out between the two routes: half will go up the top and the oth­er half will go down the bot­tom. The aver­age time taken is 1.5 hours. But what if the loc­al gov­ern­ment decided to improve the net­work by adding a new super-fast high­way from v to w that takes abso­lutely no time to tra­verse?

Braess’ paradox - augmented network

The people tak­ing the s-w link will think, “Hey, going via s-v-w is no worse than tak­ing s-w, and it can actu­ally be quick­er. I’ll take that instead!” The same logic goes for why every­one will end up tak­ing v-w-t instead of v-t, because v-w-t can­not be worse, but it can actu­ally be faster, if a small frac­tion of people don’t take it (note the cost func­tion on w-t). So every­one ends up on s-v-w-t, which means an aver­age time of 2 hours. Oops.

If you think back to the real world, that’s why build­ing a new high­way doesn’t solve all our traffic night­mares, and can, in some cases, make the prob­lem worse. People using sides roads or tak­ing the train to work con­verge on the new high­way, and you’re back to square one.

Braess’ para­dox doesn’t just occur for net­works where the cost of a link goes up as the num­ber of people use it. For net­works with non-increas­ing cost func­tions (e.g. the cost of a link goes down as the num­ber of people use it goes up), where con­ges­tion is a good thing, we can find an ana­log­ous situ­ation. Con­sider the fol­low­ing net­work, where 1 unit of traffic needs to be routed from s1 to t1, and anoth­er unit from s2 to t2:

Decreasing case - initial setup

You’ll find that the total cost is 6 (3 each for the s1-t1 and s2-t2 guys). Now, someone with deep pock­ets builds a new link to shave a little off the s1-t1 route:

Decreasing case - augmented network

The dir­ect s1-t1 route is more attract­ive than the 3 hour hike via u and v, so every­one who wants to go from s1 to t1 takes that route instead. Because the u-v link isn’t shared any­more, its cost goes up quite a bit, and the total cost now becomes 6.5. Shar­ing is caring.

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So I did a bit of AMPL mod­el­ling today, now that I’ve read the AMPL book from cov­er to cov­er. (AMPL lacks online doc­u­ment­a­tion. The book is the doc­u­ment­a­tion!) I just did the basics… did up Pigou’s example — it’s inter­est­ing how you think you have all the tools for the job and then you try the simplest of things and it falls apart. The first stum­bling block is that there’s no way you can rep­res­ent mul­tiedges in AMPL, because sets can­not have repeated ele­ments. So, if you’ve got two s-t edges, you’ll need to break one up into s-u-t.

The next weird thing is that once you move bey­ond the tra­di­tion­al examples where you have nice pretty con­vex func­tions, CPLEX goes belly up (the School has a license for AMPL and CPLEX), so I’m forced to go back to the free stu­dent ver­sion, which has a lar­ger vari­ety of solv­ers (but all lim­ited in a mul­ti­tude of ways). Even then, you run into the guess and check game with non-lin­ear optim­isa­tion, because maths and com­puter sci­ence haven’t pro­gressed far enough to make non-lin­ear optim­isa­tion just work. I guess you could sum­mar­ise the book’s recom­mend­a­tions: guess and check, but guess and check intel­li­gently.

So, where I’ve pro­gressed since I last blogged about hon­ours a while back is that I’ve gone through some more papers (Tasos poin­ted me to the com­ple­ment­ar­ity proof by Aash­tiani et al), and learnt AMPL. Next sta­tion: install some for loops (like wow!) to do the stu­pid guess and check and build some­thing that’s a little less fra­gile to play with.

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Yes­ter­day, I did my hon­ours research pro­pos­al present­a­tion: the slides can be down­loaded here. Des­pite the very math­sy top­ic that I have, I’ve man­aged to make it pretty much maths free (except for the last slide 😛 ) and just about any­one should be able to make it through at the least the intro­duc­tion — so take a look if you’re won­der­ing just what on earth I’m doing.

In oth­er news, some (obvi­ously very bored) people have applied game the­ory to the prob­lem of cohab­it­ing males and females shar­ing a toi­let: the cooper­at­ive and non-cooper­at­ive cases are ana­lysed.

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Yes­ter­day, I com­pleted the Research Pro­pos­al for the Research Meth­ods course. Research pro­pos­als. Yawn.

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