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Duck Battle Winners!

The duck battle is done!

Most people where able to cobble together a decent strategy so almost everyone who participated will be getting a mark somewhere in the high 80% mid 90% range.  After running the tournament eight entries were close enough to warrant further examination and testing.  They were in no particular order, superduck, simple_duck, SmartDuck, ThomasWagner, Nikki, marshallduck, Fythian, and finalbamm.  After further testing it became clear that although all these strategies are close, Fythian and marshallduck are the superior two.  Unfortunately these two strategies were so close and the time was too short to determine if they were actually significantly different and if so which was better.

The final exam is also marked and your marks are submitted so they should be showing up on qcard any day now.  The class average was a little high this year 79, and I’m happy to report that no one who put effort into the class failed.  I agree question two was hard if you got confused by two variable algebra, but there were lots of opportunities for part marks.

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Last Minute Exam Study info

Here are some things to know.
The box on isoclines remains hidden. When finding the equilibria of the dynamical system associated with the replicator equations, it is sufficient to confirm that the “obvious” equilibria are indeed equilibria.  Since the the rate of change of the frequencies of the strategy types are proportional to the frequencies of the strategy types themselves, we would expect that any point where there is only a single strategy type present would be an equilibrium.  Hence in the example in the practice final you would check that (0,0), (0,1), and (1,0) are indeed equilibria of your dynamical system by plugging those points into your replicator equations and checking that they are indeed zero.  The other obvious equilibrium to check is the one corresponding to your mixed strategy nash equilibrium.  In the case of the practice final that would be (1/2,1/2).
I’ll be at the exam tomorrow and you can just hand in your bonus assignment before you go into your exam.
Someone asked some good questions and I answered them:
1. We talked about dominated strategies, and how they could be “crossed out” or ignored for strong nash.  But can a dominated strategy enter into a mixed Nash?  Or is it always irrelevant to the game analysis entirely?
A dominated strategy will not be part of mixed strategy Nash equilibrium, it is essentially irrelevant.
2. For mixed strategy nash, we sometimes set payoffs equal for each strategy.  Besides assuming the population is large, this also assumes a symmetric game.  Is that right?
No, this technique works for asymmetric games as well.  Setting the payoffs of player one’s actions equal to eachother allows you to find the probabiliites with which player two should choose their strategies in order to make player one indifferent to their own choice of action.  And vice versa, Setting the payoffs of player two’s actions equal to eachother allows you to find the probabiliites with which player one should choose their strategies in order to make player two indifferent to their own choice of action
3. For the technique of setting payoffs equal to find mixed nashes, if there are 3+ strategies, do all three have to be equal, or just any two and both larger than the third?
Just any two and both that are equal larger than the other strategies.
4. In an evolutionary game, we have symmetry, but there can still be off-diagonal nash equilibria.  They will not correspond to ESS since they are impossible strategies for a population.  They also will not be fixed points for the replicator equations for the same reason.  Is that true?  Nash <==> fixed point otherwise though, right?
Yes this is true.
5. Is it true that ESS <==> stable fixed point?  Or is that just a one way (ESS==> stable fixed point)?
Just one way ESS==> stable fixed point.  (the double implication holds in most cases but not all)

6. For section 3 of question 2 in this year’s practice final, the online solutions says that the only nash is a (0,0), but in class when we drew it, there were clearly at least 3 points of intersection. Can you please clarify on which one is the correct solution?

There were actually only two points of intersection of the best response function (x=-3/sq3, y = 3) and
(x=2, y=1).  The three intersections I believe you are referring to are from when we were comparing the payoff to player one at their two distinct local maxima to determine the global maxima.

7. Also, if the the one with 3 equilibria is correct, how do we know which one is the best for either player? just plug in an estimated value and check?

You plug in each equilibrium point into the original payoff functions and see which equilibrium is best for each player.

Practice Final Solutions And Help Session Times

Here are some solutions to the practice final.

I’ll be in that room behind the math help centre tomorrow (Wednesday) from 2 till 5 answering questions and working through examples.  And then again Saturday evening, 5:30 until people want to go home or I’m exhausted.

Practice Final 2010

Here is a practice final.  It is dangerously similar to your real final. If you only do one practice final do this one.

I’m aiming to have solutions up for this by Monday.

Final Exam help sessions

The final exam is written.  I’m not a believer in big surprises on final exams, so here is the breakdown.

The exam consists of three questions, each with lots of subparts.

There is one question about taking a 3 by 3 evolutionary matrix game, finding the Nash equilibria both mixed and pure, determining which strategies are ESS, writing out the replicator equations, finding the isoclines, finding the equilibria, computing the Jacobian, using the Jacobian to evaluate the stability of the equilibria, and drawing a picture to summarize your results.

There is one question where you are given the payoff functions of two players.  There payoff functions are polynomials in x and y.  One player has control over x, one player has control over y.  One of the polynomials will be cubic.  You will have to find the best response functions of each player, and then use those best response functions to find the Nash equilibria.

One of the question is kind of riddle like.  It is very similar the last few games we played as a class.

The exam takes me a little under two hours to complete, so hopefully three hours will be plenty of time for you.  I am prepared to run several help sessions over the next week, where we will just walk through examples of the sort of problems that will be on the final.  Please post comments suggesting times that work.

I’ll aim to have a practice final up by Saturday, and solutions for it and assignment 8 by Monday.

Duckbattle Help TimeSlots

Someone suggested that it would be more efficient if people who wanted help with code were to book time with me since I was sometimes spending really long chunks of time with one group, and others were left frustrated and waiting. Good idea.

Post a comment to book timeslot. (You’ll have to identify yourselves somehow.)

Slots are half an hour long on starting on the half hour going from 2:30-5:30 on Friday
and 10:00 till 5:30 on Monday. I’ll be taking a break in the last unbooked slot between 12:00 and 3:00 for food.

I’ll just float around helping in the unbooked chunks. Heads up, be sure to redownload the duckcode from the resources page when you start tweaking your ducks. The code on your computer may be from before the parameters were updated.

Bonus #2 The Hermits of Diamond Lake

Due April 26

This Bonus is actually several bonus assignments rolled into one.  Each question can replace one of your previous assignments. Be warned, every question but the first is quite hard.

There are three mystic hermits, Hassan, John, and Faxian, living in a hut on one of the corners of diamond lake.  Each morning they get up and find a place along the shores of diamond lake to meditate.  Hassan goes out first, then John, then Faxian.  The payoff to a mystic increases with their distance from the nearest other mystic.  All that chanting, and locust munching of another mystic can really distract from mediation.  Payoff decreases with the distance a monk walks, since time spent walking is time not spent meditating.  Hermits are mortified at the thought of interrupting another hermits mediation and so they would rather walk the long way around or end up very near another hermit than actually walk past a meditating hermit. Diamond lake is perfectly square with sides of length 1km.  The payoff for a hermit is 2D – W.  Where D is the shortest distance from the nearest other Hermit, and W is the distance the hermit has walked along the shore of the lake.

Question 1

Suppose that Hassan is sick today and now this is simply a two player game, between John and Faxian.  What are the Nash equilibrium strategies for each player.  (Figure out Faxian’s best response to what John does. Suppose that John knows that Faxian will play the best response to John.  How does John maximize his payoff.)

Question 2

Suppose Hassan is no longer sick and that this is now a three player game.  What are the Nash equilibrium strategies for each player.  (Figure out Faxian’s best response to what John and Hassan have done.  Supposing that John knows that Faxian will play his best response what is John’s best response to Hassan.  Supposing that Hassan knows that John and Faxian will play their best responses, what choice of actions should Hassan take to maximize his payoff.

Question 3

Faxian has transcended the material world, and now pays no attention to where he is walking.  He is equally likely to go counterclockwise or clockwise around the lake and walks a random distance drawn from a uniform distribution on zero to two.  Faxian has transcended payoff functions.  The payoff functions of John and Hassan are as before, but now they have the possibility of crazy Faxian walking past them and interrupting their meditation which causes them to lose 2 satisfaction points.

Question 4

Faxian has passed from crazy to “enlightened”.  He  can now walk on water, although he still needs to sit on the shore of the lake to meditate. (Everyone else still has to walk along the shore.)  Everyone’s payoffs are as in part one, what is the Nash equilibrium strategy of each player.