In this lab you will play a game against the computer and try to
develop a winning strategy.

Before You Get Started
  
  Here are some cool sights for getting a new name or nickname
  for you or your friends:
    Hobbit Name
    Prof. Poopy-Pants Name
    Anagrams
    Tough Name
    Pokemon Name

For you RISK fanatics:

   Here's the table of probabilities I got for up to 
   10 Attackers vs. 8 Defenders: risk2.out
 
Gamebot

  Save this MATLAB function gamebot.m in
  the MATLAB work folder.

  To play a game using Gamebot, you just type gamebot
  and then it will ask you for a payoff matrix and a
  strategy vector for Player 2.  You will play as Player 1.
  Then you get to play the game repeatedly, and Gamebot keeps
  track of the payoffs and other statistics.  Use 0 as your
  choice to end the game.

  Example 

  gamebot
  Enter payout matrix: [2 -2;-2 4]
  Strategy vector for Player 2: [1/2 1/2]
  Enter 0 for your choice to end the game.
  Your choice: 1
  Computer chooses 1 for a payout of 2.
  Number of games: 1  Avg. Payout: 2
  .
  .
  Your choice: 2
  Computer chooses 1 for a payout of -2.
  Number of games: 50  Avg. Payout: -0.16
  Your choice: 0
  Strategy used by Player 1: 0.7  0.3

  Note the strategy reported is the probability for me
  choosing each choice, i.e. my strategy vector
 
Mora

   For Mora the payout matrix is [2 -2;-2 4]

   The plays for Player 1 and for Player 2 are:  1  or 2

   Play against the gamebot, with Player 2 having the strategy:

   (a) [1/2, 1/2]
   (b) [3/4, 1/4]
   (c) [3/5, 2/5]

   Play at least 20 games against each strategy and try to develop a
   winning strategy for Player 1.

   E-mail me your best strategy in each case. ccollins@math.utk.edu


Mountain City Siege

   For Mountain City Siege the payout matrix is 
   [3 1 0 -1;0 2 -1 -2; -3 1 1 -3; -2 -1 2 0; -1 0 1 3]

   The plays for Player 1 are: 1=(4,0), 2=(3,1), 3=(2,2), 4=(1,3), 5=(0,4)
   and for Player 2: 1=(3,0), 2=(2,1), 3=(1,2), 4=(0,3)

   Play against the gamebot, with Player 2 having the strategy:

   (a) [1/4 1/4 1/4 1/4]
   (b) [1/18, 4/9, 4/9, 1/18]

   Play at least 20 games against each strategy and try to develop a
   winning strategy for Player 1.

   E-mail me your best strategy in each case. ccollins@math.utk.edu


Challenge

   Write a MATLAB program which accepts the payout matrix and
   the strategy vectors for Players 1 and 2 and determines the
   average payoff (called the value).