Imagine that Eva, who has a current wealth of $$15,000,$ receives an unexpected opportunity to invest in one of two startup companies. Her utility function over the wealth she will have after the investment, $Z($which equals her current wealth plus or minus any amount she gains or loses from the investment$),$ is $U(Z)=\sqrt[\phantom{2}]{Z}.$ Investment $1$: She can invest $$2,000$ in Startup A$.$ If the startup succeeds, which happens with a probability of $\frac{25}{100},$ she triples her investment; otherwise, she loses her $$2,000$ investment.

Investment $2$: She can invest $$1,000$ in Startup B$.$ If this startup succeeds, which happens with a probability of $\frac{10}{100},$ she gets ten times her investment; otherwise, she loses her $$1,000$ investment.

Eva is required to choose one of the two investments. Which would she prefer according to her expected utility?

a$)$ Eva would prefer Investment $1,$ and it's expected utility from investment $1$ is approximately $127\mathrm{.}97.$

b$)$ Eva would prefer Investment $1,$ and it's expected utility from investment $1$ is approximately $137\mathrm{.}97.$

c$)$ Eva would prefer Investment $2,$ and it's expected utility from investment $2$ is approximately $121\mathrm{.}98.$

d$)$ Eva would prefer Investment $2,$ and it's expected utility from investment $2$ is approximately $131\mathrm{.}98.$

Consider the theoretical concept of the risk$-$return trade$-$off in financial markets. Which of the following statements best captures the generally accepted relationship between risk and expected return $(E(R))?$

a$)$ The relationship between risk and expected returns is ambiguous, and thus the concept of a risk$-$return trade$-$off is not considered by cautious investors.

b$)$ There is a positive correlation between risk and expected return, meaning higher risk is usually accompanied by higher expected returns.

c$)$ The levels of risk and expected returns are generally not factors in investment decisions; asset selection is random.

d$)$ Increasing risk typically leads to decreasing expected returns, which is why lower$-$risk assets are generally preferred.

In the theory of game design, which of the following is NOT considered one of the basic elements of a game?

a$)$ Structures

b$)$ Players

c$)$ Feedbacks

d$)$ Payoffs

TechA and TechB are deciding on a new technology standard. The payoff matrix in millions of dollars is $($TechA$\text{'}$s profits are listed first, followed by TechB's$)$:

$\backslash$table$[[$TechA $/$ TechB,Adopt,Don't Adopt$],[$Adopt$,-1,-1,-1,10$