lab #3 worksheet
----------------

- make a new folder for week4

- files for this week:

  w4mRPS.m
  w4nm3.m
  w4game.p

- start matlab & change to the
  week 4 folder


warm-up 
-------

- you can rerun the modified RPS from
  class.  "w4mRPS", this is mostly a
  plotting script, but it does set the
  game matrix "Pmatrix".

- type "Pmatrix" to verify that it is
  modified RPS.

- "w4nm3" is the nelder-mead minimization
  algorithm to the matrix game defined by
  "Pmatrix".

- it should recover A's optimal system

  A = [3/12 5/12 4/12]
  expected winnings = 1/12

- redefine Pmatrix to be its negative
  transpose:

  Pmatrix = -transpose(Pmatrix)

- rerun "w4nm3" for the new "Pmatrix",
  surprisingly it returns the optimal
  system for B!

  B = [4/12 5/12 3/12]
  expected winnings = -1/12

- choose with your partner a 7-digit
  phone number - this will be your
  common seed.  (eg.  1234567)

  Pmatrix = w4game(3,1234567)   

  will give you a random 3x3 matrix
  whose entries are integers between
  -50 and 50.  it may make a lot of 
  noise before it stops.  also, this
  script is in pcode, which does not
  allow you to see what's inside
  (it actually checks that your matrix
  has a "good answer").

- use "w4nm3" to obtain the optimal
  systems & verify the minimax result.


INSTRUCTIONS
------------

* this is officially a programming 
  exercise.  you will have to figure
  out what the variables & code
  sections in "w4nm3" are doing.

- you will also have to debug your
  code modifications.

- remember: by putting "return"s in your 
  code, you can stop it before an error 
  so that you can query your variable 
  states.  staring at your code will 
  seldom fix your problem.

- note:

  help min
  help max

  will tell you that min & max can
  return two results, the second 
  being the index of the optimal
  vector element.


STUDY #1
-------- 

- using the same phone number, generate
  a 4x4 matrix game:

  Pmatrix = w4game(4,1234567)   

- this study will require you to modify
  the nelder-mead implementation of "w4nm3"
  to a 4x4 case "w4nm4".

- verify the minimax result.

$ try to explain why your optimal systems
  might be consistent with your matrix's 
  payoff scheme.


STUDY #2
-------- 

- using the same phone number, generate
  a 5x5 matrix game:

  Pmatrix = w4game(5,1234567)   

- this study will require you to modify
  the nelder-mead implementation of "w4nm3"
  to a 5x5 case "w4nm5".

- in this case, you are likely to find that
  your optimization code gets bogged down
  in "flat spots".  you will have find some
  way of modifying your initial simplex so
  that the nelder-mead finds the true optima.

- you will therefore have to "help" your code
  to find the optimum.  but you should in 
  short order be able to verify results 
  consistent with the minimax theorem.

$ try to explain why your optimal systems
  might be consistent with your matrix's
  payoff scheme.

- one of the purposes of this study is to 
  investigate one of the failure modes of
  the nelder-mead.  so, there is no reward
  for a "lucky draw" -- if your phone number
  just happens to generate a nicer than 
  average matrix.  i've tested several dozen
  5x5 matrices & they all fail.  if you 
  happen to get a lucky matrix, throw it back,
  choose a different phone number for this
  study & continue.  (i still can't believe
  someone got 10 digits last week.)