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\lecture{Artificial Intelligence}{HW3: Game Playing}{CS5300, Spring 2009}

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\section{Non Zero Sum Games}

Alice is taking a class taught by Bob called ``Artificial
Intelligence.''  Bob has three ways he can teach the class: ``Hard,''
``Medium'' or ``Easy.''  Alice has three ways she can take the class:
``Hard Working,'' ``Working'' and ``Hardly Working.''  For each of
them, there are pros and cons.  For instance, it's easy for Bob to
each an Easy class or a Hard class, but hard to balance the two.
Obviously Alice doesn't like to work hard, but she realizes that she
might have to in order to learn something.  Bob is happy to teach a
hard class to students who are willing to work hard, but if the
students don't work hard, they punish Bob by giving him bad teaching
evals!\footnote{Yes, this implies that any bad teaching reviews
  \emph{must} be due to shortcomings of students, not of professors!}
All of these things considered gives rise to the following table of
rewards.  These are written as $(A,B)$ where $A$ is Alice's reward and
$B$ is Bob's reward:

\begin{tabular}{r|ccc}
              & {\bf Hard  } & {\bf Medium} & {\bf Easy  } \\
\hline
{\bf Hard   } & $(9,9)$      & $(6,6)$      & $(2,1)$      \\
{\bf Working} & $(5,8)$      & $(8,7)$      & $(4,2)$      \\
{\bf Hardly } & $(4,1)$      & $(5,2)$      & $(4,3)$      \\
\end{tabular}


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\bee
\i If Bob assumes that Alice will optimize her own reward (i.e., Bob
assume's Alice is an optimal agent), how should he teach the class,
supposing that Bob plays first?  If Alice assumes Bob is an optimal
agent, how hard should she work?

\i Draw a game tree for this problem supposing that Bob goes first.
Propagate values up through the tree using (the non-zero-sum variant
of) minimax search.

\i Alice is clearly a good student (see question one), but once in a
while we get students who aren't quite as dilligent {\tt :(}.  It
makes sense for Bob to model his class as a distribution over types of
students.  Suppose Bob believes that $30\%$ of his class will work
hard, $50\%$ will work, and $20\%$ will hardly work.  Draw the
expectimax tree for this setting, concentrating only on Bob's reward,
and compute expected node values.  What is Bob's expected reward for
this setting and which type of class should he teach?

\i Some faculty (Bob not included!) follow the model of trying to
scare away all students who won't work hard on the first day of
class.  The idea is that if they drop the class, then the professor
won't get a bad teaching review!  Suppose Bob decided to follow this
policy and all $20\%$ of the hardly working students dropped.  What is
Bob's new expected reward and what type of class should he teach?

\i {\bf (6300 only)} Suppose students were somewhat adversarial and
were only happy if Bob weren't happy.  Operationally, if $(A,B)$ was
the old reward pair, suppose the new reward pair is $(A-\be B,B)$, for
some $\be > 0$.  Is there a value of $\be$ that will make this a zero
sum game?  If so, what is it; if not, why not?  What is the
\emph{minimal} value of $\be$ that would cause the answer to problem
$1$ to change (if there is no such $\be$, why not?)?

\ene

\newpage

\section{Probability and Utility}

I don't have a cute story to wrap around the following questions, so
just answer them {\tt :)}.

\bee
\i I flip a fair coin but don't let you see how it came up.  I tell
you that if you guess right, I'll give you $\$10$.  What is your
expected reward (write out the computation!)?

\i Coins are boring.  Now I roll a fair six-sided die but don't let
you see how it came up.  I tell you that if you guess right, I'll give
you $\$10$.  What is your expected reward?

\i Let's say that now I tell you that the die isn't fair, but that the
probabilities are as follows: $p(1) = 0.3$, $p(2) = 0.1$, $p(3) = 0.1$,
$p(4) = 0.2$, $p(5) = 0.2$, $p(6) = 0.1$.  Again, I'll give you $\$10$
if you guess right.  For each of the six possible guesses you could
make, compute your expected reward.  Which would you guess?

\i Now, I make you the following offer.  Keep the same die as before.
But now, I tell you that if you guess right, I'll give you $\$10$
\emph{times} the number you guess.  I.e., if you guess $2$ and you're
right, I'll give you $\$20$.  Now what is your best option to guess?
Is this the same or different from the previous problem?  Explain why
or why not.

\i {\bf (6300 only)} Suppose now I roll two dice, each weighted as in
the previous questions.  You are allowed to make two guesses and if
\emph{either} one matches one of the die I rolled, you win $\$10$
times that guess.  (So, if I roll a $2$ and $6$ and you guess $2$ and
$3$, you get $\$20$; but if you have guessed $3$ and $6$ you would
have gotten $\$60$.)  If you guess both right, you get the larger
amount.  Rigorously compute your optimal action (i.e., guess).  (Hint:
if you get something that says you should guess the same number twice,
you've done something wrong!)  Next, suppose that we play the game
again but under slightly different rules.  You still win if you're
able to guess one of the die values, but now I give you $\$10$ times
your first guess times your second guess.  So if I roll a $2$ and $6$
and you guess $2$ and $4$, then you win $\$80$.  Now what is you best
action choice?

\ene


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