1.Consider the following three player parametric game in normal form:
\begin{tabular}{cc}
Strategy triple & Payoff \
\hline$(1,1,1)$ & $(1,2,3)$ \
$(1,1,2)$ & $(2,1,3)$ \
$(1,2,1)$ & $(1,3,2)$ \
$(1,2,2)$ & $(3,2,1)$ \
$(2,1,1)$ & $(2,3,1)$ \
$(2,1,2)$ & $(3,1,2)$ \
$(2,2,1)$ & $(1,2,3)$ \
$(2,2,2)$ & $(x, y, z)$
\end{tabular}
where $x, y, z$ are arbitrary real numbers satisfying $x \geq 1, y \geq 1, z \geq 1$ and $x+y+z \leq 5$. For this game:
(i) Write out the characteristic function of this game, giving the details of your computations. Note that some of the coalition values may appear as functions of (some or all of) the parameters. [15]
(ii) Find out if for some values of the parameters $x, y, z$ (where $x \geq 1, y \geq 1, z \geq 1$ and $x+y+z \leq 5$ ) the game in CFF found in (i) is inessential, constant-sum or has a non-empty core. Give the details of your arguments.
(iii) Using the strategic equivalence find the $(0,1)$-reduced form of the characteristic function written out in (i). Note that some of the coalition values may appear as functions of (some or all of) the parameters.
(iv) Assuming $x=y=z=1$ compute the Shapley value of the game in characteristic function form, which you wrote out in (i). Give the details of your computation.

2. Suppose that we have a set $G$ of gin merchants and a set $W$ of tonic water merchants. For $i \in G$, the $i$ th gin merchant has $\alpha_{i}$ litres of gin, and for $j \in W$, the $j$ th tonic water merchant has $\beta_{j}$ litres of tonic water. Assume that $\alpha_{i}$ and $\beta_{j}$ are real positive numbers for each $i$ and $j$ (also mind that, in general, $\alpha_{i}$ ‘s are different for different $i$ and $\beta_{j}$ ‘s are different for different $j$ ).

The merchants can form any coalition (i.e., any subset of $G \cup W$ ) and any coalition has to use 2 volume units of tonic water and 1 volume unit of gin to produce 3 volume units of gin and tonic cocktail (in other words, the ratio of tonic water to gin has to be $2: 1$ ).

Suppose $S \subseteq G \cup W$ is a coalition of merchants. Let the value of $S$ be defined as the (maximal) volume of gin and tonic cocktail which this coalition $S$ can produce. Write out the formula for the value of $S$. Does this formula define a game in characteristic function form? If so then check if this game is essential and if it is constant-sum. Prove all your answers.
[20]
Hint: You may use the following algebraic property: $a+\min (b, c)=\min (a+b, a+c)$ for any real numbers $a, b, c$.

Consider the cones
$$
\begin{aligned}
&K=\left{\left(x_{1}, x_{2}, x_{3}\right)^{\top} \in \mathbb{R}^{3}: x_{3} \geq \sqrt{2 x_{1}^{2}+3 x_{2}^{2}}\right} \
&M=\left{\left(x_{1}, x_{2}, x_{3}\right)^{\top} \in \mathbb{R}^{3}: x_{3} \geq \sqrt{\frac{x_{1}^{2}}{2}+\frac{x_{2}^{2}}{3}}\right}
\end{aligned}
$$
Prove that $M=K^{}$. Hint: Adapt the proof from the lecture notes for finding the dual of the Lorentz cone. Alternatively, prove the formula $(A L)^{}=\left(A^{\top}\right)^{-1} L^{*}$, for any cone $L \subseteq \mathbb{R}^{3}$ and any $A \in \mathbb{R}^{3 \times 3}$ nonsingular matrix, where $A L=\left{A x \in \mathbb{R}^{3}: x \in L\right}$, and apply it to the 3-dimensional Lorentz cone with an appropriately chosen matrix $A$.

Let
$$
\begin{aligned}
M_{1} &=\left{\left(x_{1}, x_{2}\right)^{\top} \in \mathbb{R}^{2}: 2 \leq x_{1} \leq 8,2 \leq x_{2} \leq 8\right}, \
M_{2} &=\left{\left(x_{1}, x_{2}\right)^{\top} \in \mathbb{R}^{2}: 4 \leq x_{1} \leq 6,0 \leq x_{2} \leq 10\right}, \
M_{3} &=\left{\left(x_{1}, x_{2}\right)^{\top} \in \mathbb{R}^{2}: 0 \leq x_{1} \leq 10,4 \leq x_{2} \leq 6\right}, \
M=M_{1} \cup M_{2} \cup M_{3} \text { and } K &=\text { cone }\left{(1,2)^{\top},(1,3)^{\top}\right} \subseteq \mathbb{R}^{2} .
\end{aligned}
$$
(a) Determine the set $E(M, K)$ of efficient points of $M$ with respect to $K$.
$[20]$
(b) Determine the set $P(M, K)$ of properly efficient points of $M$ with respect to $K$.
[20]