$(10$ points$)$ Worst$-$case

Consider an individual who is a computer scientist and is deciding between different algorithms,

all of which are known to be correct, for a programming problem. Let $Z=\{1,2,$dots,$K\}$ denotes

the set of all possible execution times when an arbitrary algorithm is run on a randomly

generated input case. An algorithm is then identified by the distribution $p$ it induces over

the execution times $Z.$ That is$,nL(Z).$ As a computer scientist, this individual evaluates

algorithms by their worst$-$case performance, which is simply the maximum possible execution

time they might take. Formally, let

$w(p)=$max$\{kinZ$:$p_k>0\}$

denote the maximum possible execution time of an algorithm $nL(Z).$ For instance, if

$Z=\{1,2,3,4\}$ and $p=(0\mathrm{.}9,0,0\mathrm{.}1,0),$ then the maximum possible execution time for the

algorithm is $w(p)=3.$

Given this, the individual's preference over algorithms is

$pqLongleftrightarroww(p)w(q).$

$($a$)$ Does $$ satisfy continuity$*(C^{**})?$

Note that a preference relation $$ on $L(Z)$ satisfies continuity ${?}^{**}(C^{**})$ if for any $p>-q>-r,$

there exists $in[0,1]$ such that

$qp+(1-)r.$

$($b$)$ Does $$ satisfy independence?

$(10$ points$)$ Insurance $($adapted from $Ch-3,$ Problem $8).$

Consider an individual who has wealth $w>0$ and a vNM utility function $u$:$ZR,$ where

$Z=R_{+}$denotes the set of possible wealth levels. Assume that the vNM utility function $u$ is

increasing The individual has $to$ choose $an$ amount, $x,he$ will pay for insurance

that will pay him $x(for$ some given $>1)if$ the accident occurs.

$(a)$ Notice that the insurer's expected profit $isx-px.$ Assume that $$ makes this profit zero,

$so$ that $=\frac{1}{p}.$ Show that the individual optimally chooses $x=pD,so$ that $heis$ fully

insured: his net wealth $is$ the same whether $or$ not $he$ has $an$ accident.

$(b)$ Assume now that the insurer's expected profit $is$ positive$).$ Show that the

individual optimally chooses partial insurance: $x.$