Consider a game G $=$N$,($Si$)\{$i in N$\}$Consider a game $G=($:$N,(S_i{)}_{\{iinN\}},(u_i{)}_{\{iinN\}}$:$)$ where $S_i$ is finite for all iinN. Suppose the

players are pessimistic and the utility of player i for a mixed strategy profile $$ is defined as

follows:

hat$(u{)}_i()=$min$\{u_i(s)|sinSupp()\}$

where Supp$()=\{\begin{array}{ccc}(s_1,& \text{d}\text{o}\text{t}\text{s}\text{,}{s}_n)in{S}_1\text{c}\text{d}\text{o}\text{t}\text{s}& S_n|{}_i(s_i)>0\end{array}$ for all $\{$:iinN$\}.$ Either prove or

disprove the following: The game $G$ has a mixed strategy NE when players are pessimistic.

$,($ui$)\{$i in N$\}$

$$ where Si

is finite for all i in N$.$ Suppose the

players are pessimistic and the utility of player i for a mixed strategy profile $\backslash$sigma is defined as

follows:

u$$i$(\backslash$sigma $)=$ min$\{$ui$($s$)|$ s in Supp$(\backslash$sigma $)\}$

where Supp$(\backslash$sigma $)=\{($s$1,...,$sn$)$ in S$1\backslash$times $\backslash$times Sn $|\backslash$sigma i$($si$)>0$ for all i in N$\}.$ Either prove or

disprove the following: The game G has a mixed strategy NE when players are pessimistic.