Problem

AAA Co$.$ is considering its business overseas. The company has two choices: $(1)$ go to SE Asia; $(2)$ go to Africa. If the company chooses SE Asia, the chance of success is $\backslash (40\backslash \%\backslash )$ with a return of $\backslash (\backslash$$ $200\backslash$mathrm$\{$~m$\}\backslash )$ and the chance of failure is $\backslash (60\backslash \%\backslash )$ with a return of $\backslash (\backslash$$ $100\backslash$mathrm$\{$~m$\}\backslash ).$ If the company chooses Africa, the chance of overcoming cultural differences is $\backslash (30\backslash \%\backslash ),$ whereas the chance of not overcoming is $\backslash (70\backslash \%\backslash )$ with a loss of $\backslash (\backslash$$ $50\backslash$mathrm$\{$~m$\}\backslash ).$ When overcoming cultural differences, the company has two choices: $(1)$ business alone: either success $($with probability $\backslash (40\backslash \%\backslash )$ and a return of $\backslash (\backslash$$ $250\backslash$mathrm$\{$~m$\}\backslash ))$ or failure $($with probability $\backslash (60\backslash \%\backslash )$ and a return of $\backslash (\backslash$$ $80\backslash$mathrm$\{$~m$\}\backslash ))$; $(2)$ co$-$business with a local company: a return of $\backslash (\backslash$$ $120\backslash$mathrm$\{$~m$\}\backslash ).$

$($a$)$ Illustrate the above decision situation using a decision tree.

$($b$)$ You can identify three strategies from the problem. Explain these three strategies.

$($c$)$ For the following pairs of strategies, draw cumulative risk profiles separately $[3!]$ and explain whether there is deterministic dominance, $1$st order stochastic dominance, or $2$ nd order stochastic dominance.

$(1)$ Strategy $1$ vs$.$ Strategy $2$

$(2)$ Strategy $1$ vs$.$ Strategy $3$

$(3)$ Strategy $2$ vs$.$ Strategy $3$