Question $2$

A machine sharing game is hk$,$ M$,\{\backslash$Sigma iik$,\{$cm$\}$m in Mi$,$ where k is the number of players, M is a collection

of m machines, for each i k$,\backslash$Sigma i $$ M is the set of machines that Player i can choose, and cm is the

cost of Machine m$.$ A profile is P $=$ hm$1,...,$ mki meaning that mi in $\backslash$Sigma i

is the machine that Player i chooses.

The cost of Player i in P is as in cost$-$sharing games, the players who choose a machine split its cost. Formally,

costi$($P$)=$ cmi $/|\{$j : mi $=$ mj$\}|.$

$$ Describe a polynomial$-$time algorithm to find an NE$.($Hint: act greedy$)$

$$ Show that the profile that the profile that the algorithm outputs is in fact a strong equilibrium.Question $2$

A machine sharing game is hk$,$ M$,\{\backslash$Sigma iik$,\{$cm$\}$m in Mi$,$ where k is the number of players, M is a collection

of m machines, for each i k$,\backslash$Sigma i $$ M is the set of machines that Player i can choose, and cm is the

cost of Machine m$.$ A profile is P $=$ hm$1,...,$ mki meaning that mi in $\backslash$Sigma i

is the machine that Player i chooses.

The cost of Player i in P is as in cost$-$sharing games, the players who choose a machine split its cost. Formally,

costi$($P$)=$ cmi $/|\{$j : mi $=$ mj$\}|.$

$$ Describe a polynomial$-$time algorithm to find an NE$.($Hint: act greedy$)$

$$ Show that the profile that the profile that the algorithm outputs is in fact a strong equilibrium.Question $2$

A machine sharing game is $($:$k,M,\{{}_i{\}}_{1}ik,\{c_m{\}}_{\text{m}\text{i}\text{n}\text{M}}$:$),$ where $k$ is the number of players, $M$ is a collection

of $m$ machines, for each $1ik,{}_i$subeM is the set of machines that Player i can choose, and $c_m$ is the

cost of Machine $m.$ A profile is $\frac{?}{\text{b}\text{a}\text{r}}(P)=($:$m_1,$dots,$m_k$:$)$ meaning that $m_{i}in{}_i$ is the machine that Player i chooses.

The cost of Player $i$ in $P$ is as in cost$-$sharing games, the players who choose a machine split its cost. Formally,

$cos{t}_i(P)=\frac{c_{m_i}}{|}\{j$:$m_i=m_j\}|.$

Describe a polynomial$-$time algorithm to find an NE$.($Hint: act greedy$)$

Show that the profile that the profile that the algorithm outputs is in fact a strong equilibrium.