Can anyone please read the pics attached below and answer this question by SETTI
ID: 3143427 • Letter: C
Question
Can anyone please read the pics attached below and answer this question by SETTING UP A DIFFERENTIAL EQUATION AND PLEASE SOLVE IT ! It has to be solved through differential equation A painting is auctioned off by an art gallery. You and several other art collectors can place a bid into an envelope, and the highest bidder wins. This is a silent auction so no one knows how much other art collectors value the artwork. How should you decide how much to bid? In part, t depends on how much you want the artwork. Each art collector gets some amount of happiness from owning this painting, and so each bidder i attributes some value v to the artwork. Each bidder i submits a bid b, which is a function of how much they want the artwork. If you win, your payoff is the amount of value you receive from the artwork minus how much you paid for it. So if you win, the utility or happiness they gain is v, b. A tie is decided by coin toss, so the utility in this scenario is (v,- b)/2 since you have half the chance of winning in a tie If you lose the auction, you don't pay anything, so your utility gained is zero Let's formalize this. The utility U for a bidder i who is in an auction against bidder j is The expected utility multiplies each pay-off by the probability they occur: Our goal is to maximize the expected utility gained. The fact that this formula depends on bidder j's bid hints at something important: the optimal bid doesn't just depend on how much you value the artwork; it also depends on how much other people value the artwork and, consequently, how high their bids are Each art collector wants to submit the lowest possible bid that will win them the artwork. So how do you know how much other people will bid? Some assumptions are in order. Let's assume that each bidder is rational and wants to maximize how much utility (or happiness) they get from the auction. Because of this, the optimal bid for bidder i is the same bid anyone would make if they valued that artwork as much:Explanation / Answer
Let's say there are 2 bidder only, the person with the highest bid will take the art.
For bidder1, the utility function is profit it gets out of art which is difference of value he assigns to the art and the amount he pays for the art (the bid). there can be 3 cases for bidder1.
1. his bid is highest {probability of that happening is let's say Pr (b1(v1) > b2(v2))}
in this case he get profit as v1 - b1
2. his bid is not highest {probability of that happening is let's say Pr (b1(v1) < b2(v2))}
profit = 0
3. in case of a tie {probability of that happening is let's say Pr (b1(v1) = b2(v2))}
in this case he get profit as 0.5 (v1 - b1)
utility function is = sum (Pi x profiti) for i = 1,2,3....
E (v1, v2, b1, b2) = (v1-b1) Pr (b1(v1) > b2(v2)) + 0.5 (v1-b1) Pr (b1(v1) = b2(v2))
since we have considered, values randomly distributed over [0,1] and the probability of 2 art collectors valuing it exactly same is 0 so the second part of the equation becomes 0.
E (v1, v2, b1, b2) = (v1- b1) Pr (b1(v1) > b2(v2))
E (v1, v2, b1, b2) = (v1- b1) Pr (b1(v1) > max( bj(vj)) ;where max( bj(vj) = max ({b2, b3,...})
E (v1, vj, b1, bj) = (v1- b1) Pr (b1> bj(vj))
E (v1, b1) = (v1- b1) b-1(b1)
E (v1, b1) = (v1- b (v1)) v1
dE/dv1 = (v1- b(v1)) +v1 (1- db(v1)/ d(v1)) = 0
let's write v1 = v and b ( v1) = b for simpicity sake
(v- b) +v (1- db/ dv) = 0
vb' + b - 2v = 0
b' + b/v - 2 = 0
b =v + c/v
so the bid value function would be of the v + c/v