Consider a first$-$price sealed$-$bid auction with three bidders: Bidder $1,$ bidder $2,$ bidder $3,$ who bid for

one object. A bidder$$s valuation can be $2$ with probability $1$

$3.$; $4$ with probability $1$

$4,$ or $6$ with a

probability of $5$

$12.$ Bidders$$ valuations are private and independently distributed. The auction rules

state that the minimum acceptable bid is $0.$

Let $1(1)$ be Bidder $1$s bidding strategy as a function of own valuation $1$;

Let $2(2)$ be Bidder $2$s bidding strategy as a function of own valuation $2$;

Let $3(3)$ be Bidder $3$s bidding strategy as a function of own valuation $3$;

We are looking at equilibrium strategies which have the following properties:

$1-$ All players$$ bidding strategies are the same: $1(2)=2(2)=3(2)=(2),$

$1(4)=2(4)=3(4)=(4),1(6)=2(6)=3(6)=(6)$

Bids are strictly increasing in valuation: $(2)<(4)<(6)$

$2-$ No bidder pays more than their valuation: $(2)2$; $(4)4$; $(6)6.$

$3-$ Bids can only be positive integers.

Answer Q$10-$Q$12$ based on the setup above.

Q$10.(9$ points$)$ Calculate the probability of receiving the object for a bidder with valuation $2?$

valuation $4?$

valuation $6?$