Consider 4 boys: a tug. d and 4 girls: A BC D. The preference ordering for boys

Question

Consider 4 boys: a tug. d and 4 girls: A BC D. The preference ordering for boys and girls is described as below. Find a stable matching using boy-proposing algorithm. Find a stable matching using girl-proposing algorithm. Private value auction Consider 2 bidders: player 1 and 2. You are player 1 and your private value of the object is 0.6. You believe that the value of player 2 is uniformly distributed between [0, 1]. If you win by bidding b, your payoff is 0.6-b. Consider the first price auction. Suppose your opponent bids av_2 with v_2 her valuation. Find your NE bidding strategy and put down your procedure.

Explanation / Answer

1)Girls:
A a c d b
B b c a d
C c a d b
D b d a c

Boys:
a B C A D
b D B C A
c D C A B
d B A C D

a) Boy-proposing:
on day 1: a proposes B, b proposes D, c proposes D, d proposes B
   B will be married to a(B prefers a to d)
   D will be married to b(D prefers b to c)
   A->*
   B->a
   C->*
   D->b

on day 2: c proposes C , d proposes A
   C will be married to c
   A will be married to d
   A->d
   B->a
   C->c
   D->b
Algo ends here as no boy is left to get married.
stable matching for boy-proposing is
   A->d
   B->a
   C->c
   D->b  
      
b) Girl-proposing:
on day 1: A proposes a, B proposes b, C proposes c, D proposes b
   a will be married to A
   b will be married to D(b prefers D to B)
   c will be married to C
   a->A
   b->D
   c->C
   d->*
on day 2: B proposes to c
   c rejects B as he prefers C to B.
on day 3: B proposes to a
   a will be married to B and a rejects A(a prefers B to A)
   a->B
   b->D
   c->C
   d->*
on day 4: A proposes b.
   b rejects A.
   d will marry A
   a->B
   b->D
   c->C
   d->A

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