$1.$ Two classmates from a BUS $190$ class, Nathan and Kelly, are interested in starting a new cybercaf$,$ hoping it will take off for people who are working remotely, but have too chaotic a home environment to work from there. The original thought was to have either a small or medium caf$,$ and the question was whether the future demand would be low, medium, or

high. Consider the following profit table they developed for the first $6$ months, with units in $$1000$s:

Low Medium High

Small $70110120$

Medium $60100150$

Lashon is also interested in investing in this caf$,$ but, while Nathan$$s and Kelly$$s decision criteria were respectively maximin and maximax, Lashon$$s decision$-$making criterion is $$minimax regret.$$

You are to help Lashon figure out what decision to make using this criterion.

The first step is to calculate the regret table, and then to take the minimum of the maximum regret for each decision $($hence minimax regret$).$

a$.$ What is the highest possible return for each state of nature in the table above? Show your work $(6$ pts$.)$

b$.$ Using your results from step a$,$ calculate the regret table below as shown in the video on Pages $($or on the powerpoint slides on minimax regret in Module $10).$ SHOW ALL WORK TO

GET FULL CREDIT.

c$.$ Calculate the minimax regret below. SHOW ALL WORK TO GET FULL CREDIT. $(3$ pts$.)$

d$.$ What decision should Lashon make? Why? $(2$ pts$.)$

$2.$ A startup tech company is planning to introduce a new device which it believes will be useful for both small and medium businesses. It has some novel features that might not be helpful to some larger$-$sized businesses, so whether to market the present design is still up in the air. After careful consideration of costs, $($e$.$g$.,$ there is a fixed cost of $$30$ million for

developing the present design$),$ the projected state of the economy, etc., Vansh, the product manager came up with the following payoff table $($in $millions$)$

Courses of action

Event Market present design Do not market present

design

Introduction successful $$450-$$$50$

Introduction not successful $-$$$50-$$$50$

a$.$ What is the highest possible return for each state of nature. Show all work to get the full credit for the problem $(4$ pts$.)$

b$.$ Show the regret table below. SHOW ALL WORK TO GET FULL CREDIT. $(6$ pts$.)$

c$.$ Calculate the minimax regret below. SHOW ALL WORK TO GET FULL POINTS. $(3$ pts$.)$

d$.$ What decision should be made? Why? $(2$ pts$.)$

$3.$ Elena$$s rich uncle died and left her a sizable inheritance. She thought about keeping his portfolio of investments just as he had left it since he ended up so rich, but she realized that since he was an old man, he invested in more conservative instruments as he got older.

Elena is young and optimistic, so she thought she might try a more aggressive portfolio, particularly investing in new technologies. Her decision alternatives at this point are to keep the portfolio as is $($D$1),$ or to invest more aggressively $($D$2),$ even though that has greater

risk, especially with inflation and the wars in Ukraine and Gaza. She developed the following payoff table $($in $$1000$s$),$ with S$1$ and S$2$ the state of new technological developments being strong $($S$1)$ or weak $($S$2)$ in the coming year.

Strong Weak

Keep portfolio as is $10010$

Invest aggressively $150-50$

a$.$ Find the highest return for each state of nature. Show your work. $(4$ pts$.)$

b$.$ Calculate the regret table below. SHOW ALL WORK TO GET FULL CREDIT. $(6$ pts$.)$

c$.$ Calculate the minimax regret below. SHOW ALL WORK TO GET FULL CREDIT. $(3$ pts$.)$

d$.$ What decision should be made? Why? $(2$ pts$.)$