Chen is participating in an auction for a given object and he estimates that the highest bid of his competitors can be $0,2$ or $4,$ with equal probabilities.

Chen$$s $($monetary$)$ value of the object is v and he wins the object if his bid, b$,$ is greater or equal to the highest bid of his competitors. If Chen wins, his payoff is v $-$ b$.$ If he does not win, his payoff is $0.$ Chen is risk$-$neutral and his bid can be b $=0,$ b $=2$ or b $=4.$

a$.$ What is the probability that Chen wins if his bid is b $=0?$ What is the

probability that he wins if his bid is b $=2?$ What is the probability that he wins if his

bid is b $=4?$ Explain your work.

b$.$ What is Chen$$s best bid if v $=0?$ Show and explain your work.

c$.$ What is Chen$$s best bid if v $=2?$ Show and explain your work.

d$.$ What is Chen$$s best bid if v $=4?$ Show and explain your work.

e$.$ Now assume that Chen is risk averse and his utility function is U$($i$)1005+$i$.$ Does this affect any of your previous answers? Show and explain your

work.