Pirates of Squarrr Island

Due Friday January 29, 2010.

To play the game you must submit a rectangle in the unit square.  That is four numbers between 0 and 1.  The first two numbers are the x and y coordinates respectively for the bottom left corner of your rectangle and the next two numbers are x and y coordinates respectively for the upper right corner of your rectangle.

If someone plays the exact same numbers as you then you are automatically on the same “crew.”

The mark you receive on this assignment will be proportional to the area successfully claimed by your crew divided by 1.2^(n-1) where n is the number of people in your crew.

If your crew’s rectangle overlaps the rectangle of another crew then that overlapped area is disputed.  To resolve how much of the disputed area your crew receives the following rule is used.  The length of the perimeter of your undisputed territory is calculated. Say it is L.  Then the number of people in your crew is counted up.  Say it is N.  Then N/L is the density of crew folks along the perimeter of your undisputed area.  There is a border between your crew’s undisputed territory and the disputed territory. Say that border has length B.  Then the force which your crew puts into the disputed area is

force into disputed area = B * N/L

The other crews that are disputing this area with you do a similar calculation and which ever crew applies the greater force to the disputed area counts the disputed area as part of their total area and everyone else does not get to count it.  In the event of a tie, the area is split up evenly between the tied crews. Note that this resolution method is not sensitive to the order in which the conflicts are resolved.

Three weird cases, (and hopefully no more but if you find one let me know, and include it in your analysis, if you do one, for bonus marks)

The first weird case is when a crew has no undisputed area. (The simplest way for this to happen is for one crews rectangle to be completely enclosed within another’s). In this case the enclosed crew’s original perimeter is treated as their undisputed perimeter and the calculations are then made as above.

The second weird case is when there is a disputed area, and all the other area’s that border on it are also disputed.  In this case the conflict is temporarily unresolvable, so it is left along with any other unresolvable situations, until all the resolvable disputes have been resolved by the above rules.  The formerly disputed areas are now treated as undisputed areas making some of the formerly unresolvable conflicts resolvable.  This process is iterated through until all conflicts are resolved.

The third weird case is when two crews share some but not all of their borders.  Say Crew 1 has chosen the rectangle with lower left corner (0.1,0.1) and upper right corner (0.3,0.3), and Crew 2 has chosen the rectangle with lower left corner (0.1,0.1), and upper right corner (0.2,0.2).  The total perimeter of Crew 2 is the same as the perimeter applied to their disputed area which is 0.4 (four sides of length 0.1 each).  Crew 1 on the other hand has a total perimeter of 1 (four sides of length 0.2 each plus two sides of “interior” perimeter each of length 0.1). Crew 1 is considered to have 0.4 units of perimeter bordering the disputed area.

There is a fair socially optimal strategy in this game, but there may be temptations to deviate from this socially optimal strategy.  The scaling constant for your marks will be chosen so that if everyone plays the socially optimal fair strategy then everyone will receive a 17/20 on the assignment.

As will always be the case in this course, if you would rather not leave your marks in this class up to the outcome of a game (which depends on others actions not just your own) then in addition to playing the game you may do an analysis of the game.  Your mark on the assignment will be the better of your analysis mark and your game score mark.  Even if you do an analysis you should still play.  An analysis of this game will include,

1. Finding the fair socially optimal strategy profile.
2. Determining whether or not the fair socially optimal strategy profile is a Nash equilibrium.
3. Determining whether or not this game has any(other?) Nash equilibria
4. Some reasoning about how you think the class will play and why.

Each of those points is worth five marks.

One last note, it is possible while playing the game to get a mark greater than 20/20.  I will put a ceiling on this just in case something ridiculous happens so that the best anyone can get on this assignment is 25/20.  Such a mark will really help drag up your average assignment grade.  Similarly since this such a tricky game, if anyone has some mind-blowing insights into the game, include those in the analysis for some bonus marks.

Good Luck!

Rules for your final project

Here are the rules for the final project tournament that we will be working on throughout the year.  You can also find them on the resources page (formerly the course notes page.) By next Friday I will also have posted matlab code for how the simulation will actually be run, (It might change a little over the course of the semester as I debug it), and some basic example strategies.

Rules

Assignment 1

Due Wednesday January 20th in class.

You need to find a game in your life and write it up.

You should say who the players are (2 marks), what the strategy set of each player is (3 marks), and what the payoff function for each player is (3 marks).  You can do this by drawing a game tree or by drawing up a grid of strategies and payoffs.  Note that drawing a grid for more than 2 players will be tricky so lists work fine as well.

Try and have it so that the game is “gamey” ie. the decisions of each player interact to determine the outcome for each player.  I don’t want examples of several people playing solitaire beside each other even though that is still technically a game (2 marks).

There are another 2 marks for the originality of the game, and 4 marks for the brief well written paragraphs linking your real life experience to this game model in a reasonable way.  I don’t want you to just draw a game tree and throw some numbers on it, I want a story explaining where the numbers came from.

Lastly you need to say how you think this game situation will play out according to your model and compare this to how this game actually plays out in your life, and draw some conclusions about your model (4 marks).

Good luck!

Hello 2010 math 339 people!

As you can see I’ve left all the posts from last year’s class up.  This is intentional! The old assignments, the discussion around them and the solutions are a lovely resource and I hope that you use them.

Tex no dance

protofinalsol4

Here are the solutions to the practice problems I gave (except number six since a solution wouldn’t really help, although there are some hints on proving something is an ESS)

What you missed in the review session

So we had an excellent review session this Thursday April 2nd evening.  I’ll fill you in on the highlights in tomorrow’s class.  But some super important things that I could get in trouble with the fairness police for not sharing are that

1. Calculators of the Casio 991 type (gold sticker) will be allowed on the exam.
2. There will be three questions on the exam, one will be some sort of variant on the who called the cops problem, the focus of which is turning a story into a sensible payoff function and finding Nash Equilibria both pure and MSNE, one will test basic definitions, and one will involve proving that an MSNE is an ESS and finding replicator equations, and using Jacobians to asses stability and using deductive reasoning  to draw sensible trajectories.

Here are the cheat sheet and the cover page of the final. 2009finalfrontandback

If there are any grievous concerns let me know soon before I send the final into be printed and don’t see it until after you’ve written it.

Marks and Exam Review Session

I forgot to mention in class, in case you are worried about there not being enough review going on there is a help session this Thursday from 4pm until 7pm I will leave at 6pm if no one is around asking me questions. We will meet in the undergrad resource room and migrate to a larger venue if lots of people show up.  Also here are your marks including assignment 10. If you’ve handed things in late then you’re just going to have to wait, till later. I have three assignment 10’s with no name so come and claim them if you think they are yours.

math339marks

As for exam prep, I can’t stress enough, if you can do the proto-final then the real final will be relatively easy. I’ll try and post solutions to the proto-final well I’m away, hopefully by the tenth.

Also here is an old final from when the course was 239. This final is reported to have caused student weeping, and my final will be nothing like it, but if you can do this old 2007 final you’ll be peachy keen for the this final.

final_exam_2007

Proto Final

So I’ve looked over all your suggestions and marked them and here are the final candidates (pun intended.)  You should look them over soon if you get a chance.  This is two or three too many questions to be the actual final.  The actual final will have variants of these problems on it.  I will certainly take your feedback as too which problems are better into account, so post that here.

2009final

Just to be clear

I’m leaving for Scotland on April 3rd and won’t be back until well after your exam, so I won’t be able to do any last minute study sessions, or be there when you write the exam to answer questions. To compensate I will try to be reasonably responsive to comments and emails, and I will do a study session sometime next week if that is what is wanted and also all three of the remaining classes will be direct preparation for the final.  That said the next few classes may not entirely be “review” in the strictest sense of the word although everything will be familiar.

Duck Battle! Spring 2009 (Bonus 3)

The third bonus assignment is to hand in a decent submission to the duck battle tournament.  The deadline for this assignment will essentially be two days before I have to have your marks submitted, which is some time near the end of April.  Remember this  is an optional bonus assignment, and you should really only do it if you enjoy this kind of problem solving, however anyone who puts effort into a tournament entry will receive 20/20.  I recommend that you work in groups of two or three.  There is also a small cash prize for the winner of this tournament, just to sweeten the pot.  Here is a rough first draft of the tournament rules which gives the basic gist of the problem.  Hopefully we will work together over the coming weeks and refine the rules to something we all agree is fair and clear.

ducktournamentrules