Optimal Behavior in Auctions

Berufsakademie Eisenach Staatliche Studienakademie Thüringen Optimal Behavior in Auctions Prof. Dr. Stephan Rometsch

EXPERIMENT Auctions and the Theory of Demand • PART I • Second price closed seal bid auction • English auction •First price closed seal bid auction • Dutch auction • PART II • Demand Function and implications

Second price closed seal bid auction ASSUMPTIONS AND INTRODUCTION • A bidder knows not the bid of others • Person with the highest bid gets the item • But he/she pays the price of the second highest bid • Why is it sensefull to reveal the true willingness to pay? • Simple: Because there is no regret! • Every other decision doesn´t gain any advantage!

Background ? • I want to knowYour true willingness to pay! • Therefore, I want to know Your true evalution of the good to create Your true demand function • The true willingness to pay for the item of person i is zi

Example: • A person i evaluates the item with € 25,- (zi=25) cs=5 I gj=20 I zi=25 p= gi= • If person i bids gi=ziand no one bids more - then he/she gets the item and has to pay p=gj=20, the price of the second highest bid (of person j) and i gets a consumer surplus of 5.

Well ?1. • Would person i better off if he does everything else? cs=5 I gj=20 I zi=25 I gi=28 p= • The result is the same. He gets the item and has to pay p=gj=20, the price of the second highest bid and gets a true consumer surplus of 5. Therefore, gi=zi=25 was high enough.

Well ?2. • Would person i better off if he reduces the bid? cs=5 I p=gj=20 I gi=23 I zi=25 • The result is the same. He gets the item and has to pay p=gj=20, the price of the second highest bid and gets a true consumer surplus of 5. Therefore, gi=zi=25 was ok.

gi=zi=25 was high enough

But… • Be careful, if gi<gj, then person j gets the item… p= gk=18 I I gi=17 I gj=20 I zi=25 • Now person j gets the item and has to pay p=gk=18, the price of the second highest bid (of person k) and person i is very angry.

Person i is very angry, because he wants to have the item, because his true willingness to pay was zi=25.

Results • To higher the price over zi doesn´t gain any advantage. • To lower the price under zj doesn´t gain any advantage. • If you lower the price, be careful, there is an increasing probability that another person j with the bid gj has the highest bid and gets the item. • And then person i is very angry, because he wants to have the item, because his true willingness to pay was zi=25.

Results • Therefore, reveal your true willingness to pay with your true bid and you get the item for the price p, which is not higher then ziand you realize some positive consumer surplus. • Note: The probability that the price p equals your bid giand your willingness to pay zi is close to zero: • probability (p = zj = gi) = 0

Another person has a higher bid 1. zj=gj=28 • Would person i better off if he bids ĝi=30 and not gi=zi =25? cs= -3 I zi=25 I gj=28 I gi=30 p= • Person i gets the item and has to pay a price which is higher then his true willingness to pay zi=25 < p=gj=28. • Then he gets the item…. -- But, why ?

Another person gets the item 2. zj=gj=28 • Would person i better off if he does not bid gi=zi=25? I gi=27 I gj=28 I zi=25 p= • Person i doesn´t get the item and person j has to pay a higher price as before p=ĝi=27 < gj=28. Reduced cs for j - what is the advantage for person i?

Another person gets the item 3. zj=gj=28 • Would person i better off if he bids ği<ziand not gi=zi=25? large cs for j I ği=21 I gj=28 I zi=25 p= • Person i doesn´t get the item and person j has to pay a lower price as before p=ği=21 < zi=25 < gj=28. • Persons i and j are friends? - what is the advantage for i?

Another person gets the item • If your bid is higher then your true willingness to pay, there is an increasing probability that you get the item for the price p, which is higher then ziand you have to pay more then your true evaluation and you realize some negative consumer surplus. • If another guy gets the item, then he has to pay the price of the second highest bid. • Be careful and don´t try to influence the price of others! • Don´t try to influence the consumer surplus of others!

Second price auction is equal to an ”English auction” • The auctioneerstarts with a reserve priceand bidders successively offer higher prices - bids. Each bid exceed the previous by some minimal bid increment. • When no participant is willing to increase the bid further, then the item is awarded to the highest bidder. • Normally the auction stops below the willingness to pay of the highest bidder. • Therefore he pays the second highest bid!

Result is the same1. • Would person i better off if he does everything else? 20 I gi CS = 5 12 I gi 15 I gj 17 I gi 18 I gj I zi=25 p= • The result is the same. He gets the item and has to pay p=gi=20, the price of the second highest bid and gets a true consumer surplus of 5. Therefore, gi=zi=25 was high enough.

First price auction is equal to an ”Dutch auction” • The auctioneerstarts with a high priceand lowers the price successively until the first bidder rises his finger. • When no participant rises his finger he reduces the price more and more. • The auction stops until the first bidder shouts: “Mine!”. • Then he pays the price were the auction stops, when he rises his finger. • This is the first price!

Another Result 1. • The auctioneer begins with highest price, say 100. 90 I 60 100 I 60 I 70 I 80 I I MINE ! p=zi • Person i gets the itemand has to pay his bid! p=60=zi

Whats the problem for Person I ? 1. • The auctioneer begins with highest price, say 100. 90 I 60 I p=zi 100 I 40 I gj 70 I 80 I • Person i gets the itemand has to pay his bid! • But: What is in between the gap to the next highest bid? • Person I want to wait until the auctioneer achieves a border

Vielen Dank für Ihre Aufmerksamkeit Prof. Dr. Stephan M. Rometsch Berufsakademie Eisenach Staatliche Studienakademie Thüringen

Optimal Behavior in Auctions