Project: Theory of Games
Due: November, 22nd
Solve the following questions:

  1. Let a game have the payoff matrix

    \begin{displaymath}A = \left( \matrix{ 3&5 &-2&-1 \cr -2 & 4 &-3& -4 \cr 6& -5& 0& 3 \cr } \right) .\end{displaymath}

    1. If the players R and C use strategies

      \begin{displaymath}p = \left(\matrix {\frac {1}{6} &\frac {1}{3} & \frac {1}{2} ... ...c {1}{4}\cr \cr \frac {1}{3}\cr \cr \frac {1}{6}\cr } \right) ,\end{displaymath}

      respectively, what is the expected payoff of the game ?

    2. If player $C$ keeps his strategy fixed as in part (a), what strategy should player $R$ choose to maximize his expected payoff ?

    3. If player $R$ keeps his strategy fixed as in part (b), what strategy should player $C$ choose to minimize the expected payoff to player $R$.

    4. The word Expected is used above. Can you come up with a rigourous set up in which the above ``expected payoff'' is indeed an expectation of some function of a random variable ? or another way to phrase the question: provide a rigourous set up using the probability learnt in class for the above model.

  2. The Federal Government desires to inoculate its citizens against a certain flu virus. The virus has two strains, and it is not known in what proportions of the two strains occur in the virus population. Two vaccines have been developed with different effectiveness against the two strains. Vaccine 1 is $85 \%$ is effective against strain $1$ and $70 \%$ effective against strain $2$. Vaccine $2$ is $60 \%$ effective against strain $1$ and $90 \%$ effective against strain $2$. What inoculation policy should the government adopt ?


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