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A seller with one item runs an auction with three bidders, where each bidder$$s valuation

$1,$ $2,$ $3$ is independently drawn from the uniform distribution on $[0,1].$

Note that truthful bidding si$($i$)=$ i is a dominant strategy for each bidder in the

second$-$price sealed$-$bid auction.

Find the symmetric Bayes$-$Nash equilibrium for:

$($a$)$ the first$-$price sealed$-$bid auction,

$($b$)$ the all$-$pay auction,

$($c$)$ the third$-$price sealed$-$bid auction $($where the highest bidder wins but pays the

third$-$highest bid$).$