Math 339
News and Updates
Final Problems, V3 10 April, 2012 Thanks to your feedback, here is the latest version of the final problems.
The Tragedy of the Commons 02 April, 2012 Today we played a fishing game that mimics the "tragedy of the commons". This term first appeared in Garrett Hardin's article. Managing the commons is a notoriously difficult problem, one for which no general solution exists. Privatization, reward/punishment and regulation schemes have all been suggested and sometimes commons are managed quite effectively. But what about the oceans? The atmosphere? There's still a good deal of work to do.... In the mean time, play the Tragedy of the Bunnies.
The end is near! 30 March, 2012 We are almost at the end of our time together, sadly.
Of Cakes and Bonuses 23 March, 2012 As it turns out, cutting cake can be a difficult task. Is it possible to satisfy everyone? Indeed it is!. For an interesting discussion on the problem, take a look at this article.
The executive problem is also intriguing. For low numbers of executives, it is easy. But what if we exceed 200? The solution is quite interesting.
Fair(?) Division 19 March, 2012 How should a homogeneous entity be divided equitably? What is "equitable"?
- The Talmud, a book of laws, provides some interesting examples of fair division. One of which was not understood for two millenia. It was by using game theory that the solution was finally found. Here's a blog that introduces the surprisingly complex and powerful solution.
- Dividing a cake is also a surprisingly difficult problem. It is easy to divide a cake into an equal number of pieces. This is known as proportional division. When distributing the cake, however, it does not matter if the slices are actually of the same size, but if each recipient perceives that they have received an equitable portion. A division that satisfies a perceived equity is called envy-free; no recipient is envious of another's share. To find an envy-free division—which turns out to be proportional, as well—for two people, one cake, is to have one person cut it into two and the other to choose their own piece. This is a game!
Resistance and Relatedness on an Evolutionary Graph 16 March, 2012 Time to abandon inclusive fitness theory in this course. It is a topic seldom seen in undergraduate courses, partly because the calculations can be daunting. We did learn one main point: the effect of an individual's action on a population can be decomposed into the effects on each individual, weighted by relatedness. The difficult bit of any inclusive fitness calculation is in the calculation of relatedness. But that doesn't mean we can't find a better way.
Inclusive Fitness, my old friend 12 March, 2012 Today we began to work on our lone inclusive fitness example. Here are the first part of the notes.
Notes, etc. 9 March, 2012 One of your colleagues has been feverishly typing up notes for this class.
Games, Price, and Hamilton 7 March, 2012 We saw how the Price equation can be applied to the framework we know and love. We also began our look at the evolution of altruism and stumbled upon Hamilton's rule. Hamilton's rule has been to hold in actual populations of bacteria and robots! Take a look at the video.
The Price Equation 5 March, 2012 Today we learned about the Price Equation, a foundational result in evolutionary theory. I've created some notes on its derivation.
Axelrod's Tournaments 2 March, 2012 We continued our investigation of repeated prisoner's dilemma games. We found that it is very difficult to find a provably "best" strategy. Robert Axelrod determined that some time ago and decided to hold a tournament. He invited some of the best minds working in the field to find a best strategy for the repeated PD game. The winner? Tit-for-tat. As some of you mentioned after class, care needs to be taken when claiming tit-for-tat is "best". It can still be invaded by some strategies, but, on average, is better than most (all....?).
The Maintenance of Cooperation 29 February, 2012 Today we begun our exploration of the emergence and maintenance of cooperation. How can a population of cooperators survive, when defection seems like a better strategy? Let's find out!
Replicator Equation Extravaganza! 27 February, 2012 One of your classmates was kind enough to type up some notes on the replicator equation. Thanks, Shahrokh! Today we expanded the scope of the replicator equation to include mutation. We ended up with the replicator-mutator equation. Mutation can knock a system out of the equilibria you'd expect in the absence of mutation. Here is a little animation that illustrates the changing replicator dynamic with increasing mutation. This from an example in the replicator-mutator notes.
Nim FTW 17 February, 2012 We had fun today playing the exceptionally cruel game, nim. We learned that there is a deviously creative strategy that guarantees the first player a win, provided the initial numbers satisfy a certain condition. This was discovered over a hundred years ago by some guy at Harvard. Another look at the solution can be found on wikipedia. This is the perfect game to win you free drinks at the Brass. Or play it with your students; you can't lose!
Maple code, for the one guy in class that uses Maple 15 February, 2012 Here's the Maple code for the mind-blowing demonstrations we had in class today on the side-blotched lizards and hawk-dove-retaliator game. For those of you that do not use Maple, you can check out these screen shots of the hawk-dove-retaliator dynamics, lizards on a cycle, and lizards converging to an equilibrium. Also: the command is "linecolor=blue".
Replicator Equations 13 February, 2012 The replicator equations describe how a population of game-players evolves. The replicator equations were originally formulated here at Queen's by Peter Taylor and Leo Jonker. Neato!
Equilibrium stability analysis. 10 February, 2012 Two things:
- Carly was very generous and typed out her notes on stability analysis and syphilis. She also has some Matlab simulations that you may want to run: Model.m, phseSI.mw, RunMyModel.m.
- Some students took an interest in the highest unique number game. Here is the general solution. It is phrased in terms of the lowest unique number, but is entirely equivalent to the problem we considered.
Evolutionary Stable Strategies. 5 February, 2012 Check out these notes on evolutionary stable strategies. More notes like this can be found on the previous Math 339 course page.
Evolutionary Games, Maynard Smith and Price style. 30 January, 2012 Today marks the start of our adventure into evolutionary games. We studied the first model of games in evolution, the Hawk/Dove game.
TWL 27 January, 2012 Today we learned that common side-blotched lizards are rock-paper-scissor playing machines. And they do pushups.
In solidarity with the SOPA/PIPA protests 20 January, 2012 Here is the solution to our file sharing example.
A challenged day; notationally and technologically 18 January, 2012 Some interesting plots:
- I plotted the expected payoff function for our group forming game. The surface looks curved, and it is, but it is actually linear for any fixed p or q. This is interesting. If we look at the graph from the side, we can see that, indeed, if q=1/2, then p receives the same payoff no matter what p is. Amazing.
- I also did a little simulation (here's the octave code) for the group forming game. The plot is of the average payoff a player receives over time. I've plotted two players, red and blue, that are playing in an entire sea of other players. Notice how the plots are jumping around, but slowly converges to 1/2. This is what one would expect...
Grading groups 16 January, 2012 The groups are out!
Nash equilibrium 13 January, 2012 Today we learned
- Nash's Ph.D. thesis is boss.
- Due to the assumptions that are sufficient for a player to adopt the Nash equilibrium, the Nash equilibrium seldom predicts human behaviour.
Big Monkey, Little Monkey 11 January, 2012 Here is the the Little Monkey, Big Monkey game from Herbert Gintis's Game Theory Evolving that we covered in class today.
Another year, another math course! Someday in January, 2012 Hi everyone. I'm Wes Maciejewski, I'll be guiding you through this course. You can reach me at wes@mast.queensu.ca. My office hours are Monday to Friday, 9:00 to 12:00. I'll let you know if I can't make any of those.