PROJECTon

GAME THEORYin

AUCTION

Prepared for: Assist . Prof. Dr.Uğur SOYTAŞ

Prepared by: Ş. Özlem ELBEYLİ 1331297 Hüsnüye KAMANLI 1331487 Artem MYAKİSHEV 1367333

İpek ÖZKAN 1331677

Table of Content

1. Introduction…………………………………………………………..........1

2. General Preview of Issue……………………………………………......... 1

o Second-Price Sealed-Bid Auction……………………….…..........1

Nash equilibria of second-price sealed-bid auction…………….........2

o First-Price Sealed-Bid Auction………………………….……......3

Nash equilibria of first-price sealed-bid auction…………………..... 4

3. “eBay”…………………………………………………………………......5

4. Modeling Auction on eBay……………………………………………......5

o First Price Sealed-Bid Auctions…………………………….…......5

o Second Price Sealed-Bid Auctions……………………………...... 6

o Example for Second Price Sealed-Bid Auction……………...........9

5. Cheating with False Bids……………………………………………......... 11

6. Conclusion................................................................................................... 12

7. References………………………………………………………..…..........13

8. Text Related Questions…………………………………………...….........14

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GAME THEORY in AUCTION

Introduction

An auction is the process of buying and selling things by offering them up for bid, taking

bids, and then selling the item to the highest bidder. In economic theory, an auction is a

method for determining the value of a commodity that has an undetermined or variable price.

Auctions can be with reserve or minimum, or without minimums, or absolute or no reserve. In

reserve auctions, there is a minimum bid or reserve price; if the bidding does not reach the

minimum, there is no sale (but the person who puts the item up for auction may still owe a fee

to the auctioneer or auction company). In absolute or no reserve auctions, the sale is

guaranteed, with only the price left to be determined. In the context of auctions, a bid is an

offered price.

For the project, we analysed a game theory in auctions, eBay. Today, these kinds of activities

are very popular, and there are some kinds of web sites such as www.eBay.com, worldwide,

and www.gittigidiyor.com, in Turkey. Most of the people prefer to buy products from these

sites. Therefore, using this type of auction with game theory serves a better understanding

perspective. The most important point is to see the usage and importance of game theory in

daily life with conscious or without conscious.

General Preview of Issue

Second-Price Sealed-Bid Auction

Assume every bidder knows her own valuation and every other bidder’s valuation for the

good being sold

Model

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• Each person decides, before auction begins, maximum amount she is willing to bid

• Person who bids most wins

• Person who wins pays the second highest bid.

Strategic game:

• Players: bidders

• Set of actions of each player: set of possible bids (nonnegative numbers)

• Preferences of Player i: represented by a payoff function that gives Player i vi − p if

she wins (where vi is her valuation and p is the second-highest bid) and 0 otherwise.

Simple (but arbitrary) tie-breaking rule: number players 1, . . . , n and make the winner the

player with the lowest number among those that submit the highest bid.

Assume that v1 > v2 > · · · > vn.

Nash equilibria of second-price sealed-bid auction

One Nash equilibrium

(b1, . . . , bn) = (v1, . . . , vn)

Outcome: Player 1 obtains the object at price v2; her payoff is v1 − v2 and every other

player’s payoff is zero.

Reason:

• Player 1:

– If she changes her bid to some x ≥ b2 the outcome does not change (remember

she pays the second highest bid)

– If she lowers her bid below b2 she loses and gets a payoff of 0 (instead of v1 −

b2 > 0).

• Players 2, . . . , n:

– If she lowers her bid she still loses

– If she raises her bid to x ≤ b1 she still loses

– If she raises her bid above b1 she wins, but gets a payoff vi − v1 < 0.

Another Nash equilibrium

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(v1, 0, . . . , 0) is also a Nash equilibrium

Outcome: Player 1 obtains the object at price 0; her payoff is v1 and every other player’s

payoff is zero.

Reason:

• Player 1:

– any change in her bid has no effect on the outcome

• Players 2, . . . , n:

– if she raises her bid to x ≥ v1 she still loses

– if she raises her bid above v1 she wins, but gets a negative payoff vi − v1.

• For each Player i the action vi weakly dominates all her other actions

• That is, Player i can do no better than bid vi no matter what the other players bid.

Table 1.

First-Price Sealed-Bid Auction

Strategic game:

• Players: bidders

• Actions of each player: set of possible bids (nonnegative numbers)

• Preferences of Player i: represented by a payoff function that gives Player i vi − p if

she wins (where vi is her valuation and p is her bid) and 0 otherwise.

b’i < vi B’i = vi b’’i > vi

I Payoff if

b^-i ≤ b’i

vi - b^-i

+

vi - b^-i

+

vi - b^-i

+

II Payoff if

b’i < b^-i ≤ vi

0 vi - b^-i

+

vi - b^-i

+

III Payoff if

vi < b^-i ≤ b’’i

0 0 vi - b^-i

-

IV Payoff if

b^-i > b’’i

0 0 0

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Nash Equilibria of First-Price Sealed-Bid Auction

(b1, . . . , bn) = (v2, v2, v3, . . . , vn) is a Nash equilibrium

Reason:

• If Player 1 raises her bid she still wins, but pays a higher price and hence obtains a

lower payoff. If Player 1 lowers her bid then she loses, and obtains the payoff of 0.

• If any other player changes her bid to any price at most equal to v2 the outcome does

not change. If she raises her bid above v2 she wins, but obtains a negative payoff.

Nash Equilibria of First-Price Sealed-Bid Auction

Property of all equilibria

• In all equilibria the object is obtained by the player who values it most highly

(Player 1)

Argument:

• If Player i ≠ 1 obtains the object then we must have bi > b1.

• But there is no equilibrium in which bi > b1 :

• If bi > v2 then i’s payoff is negative, so she can do better by reducing her bid to 0

• If bi ≤ v2 then Player 1 can increase her payoff from 0 to v1 − bi by bidding bi.

• As in a second-price auction, any Player i’s action of bidding bi > vi is weakly

dominated by the action of bidding vi :

1. If the other players’ bids are such that Player i loses when she bids bi, then it

makes no difference to her whether she bids bi or vi

2. If the other players’ bids are such that Player i wins when she bids bi, then she

gets a negative payoff bidding bi and a payoff of 0 when she bids vi

• Unlike a second-price auction, a bid bi < vi of Player i is NOT weakly dominated by

any bid.

– It is not weakly dominated by a bid bi’ < bi

– Neither by a bid bi’ > bi

Revenue Equivalence

The price at which the object is sold, and hence the auctioneer’s revenue, is the same in the

equilibrium (v1, v2, . . . , vn) of the second-price auction as it is in the equilibrium (v2, v2, v3,

. . . , vn) of the first-price auction.

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eBay

eBay is The World's Online Marketplace®, enabling trade on a local, national and

international basis. With a diverse and passionate community of individuals and small

businesses, eBay offers an online platform where millions of items are traded each day.

Founded in 1995, eBay created a powerful platform for the sale of goods and services by a

passionate community of individuals and businesses. On any given day, there are millions of

items across thousands of categories for sale on eBay. Through an array of services, such as

its payment solution provider PayPal, eBay is enabling global e-commerce for an ever-

growing online community.

How to bid on eBay:

1. When the bidders place a bid, they enter the maximum amount they would be willing

to pay for the item. Their maximum amount is kept confidential from other bidders

and the seller.

2. The eBay system compares their bid to those of the other bidders.

3. The system places bids on the bidders behalf, using only as much of their bid as is

necessary to maintain their high bid position (or to meet the reserve price). The system

will bid up to your maximum amount.

4. If a bidder has a higher maximum, others will be outbid and the highest maximum

wins the item. S/he could pay significantly less than his/her maximum price! This

means the bidders don't have to keep coming back to re-bid every time another bid is

placed.

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Modeling Auction on eBay

First Price Sealed-Bid Auctions

In a first price sealed-bid auction, the buyers simultaneously submit bids for the good being

sold. The person who bids the most wins and gets the good at the amount he bid. All bids are

made simultaneously and secretly. For example, assume that the following amounts are bid:

Table 2.

BIDDER AMOUNT BID

Player 1 $100

Player 2 $90

Player 3 $30

Player 1 wins the auction and pays $100.

In a first price sealed-bid auction, the players should always bid less than what the good is

worth to them. If they bid more and win, they are worse off. If players bid what the good is

worth to them and win, they really get nothing, because what they paid was exactly equal to

what they got. In first price sealed-bid auctions, the players need to decide how much to risk.

The lower the players bid, the greater the benefit of winning. However, the lower the chance

that they will actually win. Ideally, they should attempt to estimate what other people will bid

when formulating their own bidding strategy.

Second Price Sealed-Bid Auctions

Second price sealed-bid auctions include the type of auction effectively used on eBay. In

these auctions each player secretly makes a bid, the person who bids the most wins. The

winner, however, pays what the second highest person bid.

In a second price sealed-bid auction, the amount that the player bid determines whether s/he

wins; it does not determine how much the player pays in the case of win. In these auctions

player wants to bid only when the amount s/he would have to pay if s/he won is less than the

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item’s value. If player bids what the item is worth to , s/he will win if and only if the second

highest bid (the amount s/he pay if victorious) is less than what the item is worth.

If the player knows the value of the good, and believes the auction to be honest, the optimal

strategy on eBay is to bid what the good is worth to him/her and never raises his/her price. If

the player bids what the item is worth, and someone bids more, s/he should never outbid her

because then if player wins, s/he will necessarily pay more than the item’s value. Many

people on eBay seem to follow the strategy of bidding small amounts and then raising their

bid amount if outbid. This strategy wastes time, because his/her bid determines when s/he

wins, not how much the player pay. To see this, assume again that the good is worth $100 to

the player. Imagine that there will be many other bids, and the highest is X. If the player just

bids $100 from the beginning, s/he will win the auction when X < $100 and will pay X if s/he

wins. If player initially bids $20, keep raising the bid when someone else outbids him/her, but

never bid more than $100, then the rival gets the same result as if s/he just bids $100 from the

beginning. The player wins whenever X < $100, and when s/he wins, then pays X.

If eBay were a first price auction, where the winner always paid what he bid, then the player

would always want to just barely outbid the second highest bidder. In a first price eBay

auction, player would want to visit the auction just before it ended and bid just a little bit more

than the second highest bidder. In fact, if everyone were rational, in a first price eBay auction,

everyone would wait until the end of the auction to bid. It would be stupid to bid before the

end; doing so would give the players no advantage and might even hurt them because they

could end up bidding far more than everyone else. Remember, in a second price auction it

doesn’t matter by how much the player outbids everyone, since if s/he wins, player always

pays the second highest bid. In a first price auction, by contrast, the winner always wants to

just barely outbid the second highest bidder, because the winner pays what he bids. Thus, in

first price auctions, the player wants to wait until the end, when s/he has as much information

as possible about the other bids.

Table 3.

BIDDER AMOUNT BID

Player 1 $X

Player 2 $Y

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Player 3 $Z

In the following figure the values are assumed:

0<X<A 0<Y<A 0<Z<A

Figure1.

There are three players in the game and player one bids for $X, Player 2 bids for $Y and

Player 3 bids for $ Z, respectively. The entire players have just one chance to bid and all of

them can buy that good immediately if they pay $A. The first player bids for a value $X

which is smaller than $A and the second player bids for $Y smaller than $A. Then, the value

of Y can be smaller or larger than X. That is, there are two branches for player Y. The last

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player, Player 3, can see the price which is smaller than the highest bid. So, in the first branch

where the player two plays Y>X, Player 3 can see only X and decide a bid price Z which can

be smaller or greater than X. If Z is greater than Y, then player 3 wins and pays Y. Otherwise;

if Z is smaller than Y, then player 2 wins the game by paying $Z. In the second branch where

the two player plays X>Y, Player 3 can see the second highest price which is Y. If Z is greater

than X, Player 3 wins and pays X. Otherwise; if Z is smaller than X, Player 1 wins and pays

$Z for that good.

Imagine that the item being bid on is worth $A to player 1. Let’s first examine why bidding

$A-1 is always better than bidding some amount less than $A, such as $A-3. If the other

players buy the product immediately, it doesn’t matter whether the first player bid $A-1 or

$A-3 because s/he lose the auction either way. If the highest bid other than him/her is below

$A-3, it again doesn’t matter whether player 1 bid $A-1 or $A-3. In both cases s/he wins the

auction and pays the amount of the second highest bid. The only time there is a substantive

difference between bidding $A-1 or $A-3 occurs when the highest bid made by other players

else is between $A-1 and $A-3. If the highest bid made by others else is $A-2, player 1 lose

the auction if s/he bid $A-3 and win the auction if s/he bid $A-1. If first player bid $A-1, s/he

would get the good for $A-2, which is beneficial since the item is worth $A to him/her.

Consequently, if the highest bid other than player 1’s falls between $A-1 and $A-3,player 1 is

always better off bidding $A-1 if the good is worth $A to him/her. Thus, bidding $A-1 is

either the same or better than bidding $A-3 or any other amount below $A.

Example for Second Price Sealed-Bid Auction.

i. All players can buy the good without bidding at the immediate buy price which is

$100.

ii. Player 1 bids for that good $70.

iii. Player 2 bids for that product maximum $80 or minimum $60.

iv. Player 3 has a strategy which is increasing the second highest bid $20 or $5.

Draw the gametree in the light of given information and find who wins the game if Player

2 can afford only $60 and Player 3 can afford $65.

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Table 4.

BIDDER AMOUNT BID

Player 1 $70

Player 2 $60 or $80

Player 3 If $60 then play min $65 or max $80

If $70 then play min $75 or max $90

If $75 Then play min $80 or max $95

Figure 2.

So, the payoff matrix of the game is as follows:

If Player 1 plays $70;

Table 5.

Player 3

Minimum Bid Maximum Bid

Player 2 Minimum Bid (0, 0, 20) (0, 25, 0)

Maximum Bid (0, 0, 30) (35, 0, 0)

Cheating with False Bids

A dishonest seller could use false bids to increase the amount he receives in a second price

auction. To see this, assume again that the following bids are made:

Table 6.

BİDDER AMOUNT BİD

Player 1 $100

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Player 2 $90

Player 3 $30

In the light of these values, Player 1 will win the auction and pay $90. If the seller knew the

bid amounts, however, she should bid $99 herself. Such a bid would not prevent Player 1

from winning, but it would cause Player 1 to pay the seller $99 rather than $90. The seller, of

course, would not want to bid more than $100, for this would cause the seller to win her own

auction.

On eBay, a seller would not know all the amounts bid. If the above three bids were made,

eBay would show that Player 1 was the high bidder but would indicate that the current price

was $90. Consequently, a new bidder wouldn’t know how much he would have to bid to

become the high bidder. Thus, a seller making false bids would be taking a chance if he bid

more than $90. He might get lucky and bid less than Player 1, but he also could bid too much

and win the auction himself. If the seller won, he could auction off his good again or

withdraw his bid. Both strategies have some drawbacks since sellers must pay eBay a fee for

every completed auction and eBay imposes limitations on when a buyer can withdraw a bid.

If eBay allowed it, however, a seller could greatly benefit from making false bids and then

withdrawing them if they were too high. A seller could keep raising her false bid by $1 until

she won. She could then withdraw her final bid and thus get the maximum possible amount

for her product. Making false bids constitutes criminal fraud, however, so a seller engaged in

such a scheme would have to be careful about withdrawing too many bids, or eBay might

catch on. Presumably eBay understands how sellers could use false bids to fraudulently

increase their take, so eBay probably investigates at least some withdrawn bids. If you won an

auction in which a bid was withdrawn, however, you should beware that a seller might have

been trying to illegally increase the amount you had to pay.

If you fear that a seller might make false bids to increase his price, you should consider

bidding only at the conclusion of an auction. This way a seller won’t have the opportunity to

test you with false bids.

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CONCLUSION

As explained by the real life situations and general rules for auctions, game theory support an

important perspective for them. Not just in first price sealed-bid auctions, but also in second

price sealed-bid auctions, there can be played games between bidders.

In this project, we played a game which includes 3 bidders who can only play once. They bid

once, and can not react after. However, most of the real life situations is not so simple as it is.

Players can bid more than once, and, as a result they can change their payoffs (more chance to

win).

Moreover; the variable price may change while the game (bidding) continues in eBay. A

person who wants to maximize his/her payoff not only thinks about the risk h/she gets in, but

also thinks on time h/she waits for to buy the product. Immediately buy perception do not

always resulted in higher payoffs.

On the other hand, the type of the auction is important. First price or second price sealed-bid

auctions differs on the basic information for players. There seems to have complete

information for players, but actually it is complete and imperfect information. Because

players actions are secret most of the time, and the game is playing simultaneously.

By the way, based on the project, we claim that everybody should consider such probabilities

which were explained. Consequently, they can make different strategic decisions.

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References (in alphabetical order)

1. econlinks.com/glossary/ second_price_sealed_bid_auction

2. ‘Game Theory at Work: How to use game theory to outthink and outmaneuver your

competition’ by James Miller, McGraw Hill, 2003

3. http://en.wikipedia.org/wiki/Auction

4. www.eBay.com

5. www.econ.umd.edu/~herrera/lectures

6. http://ist-socrates.berkeley.edu/~villas/winnerc

7. www.answers.com/topic/winner-s-curse

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Text Related Questions

Q1. What is the bid amount should the winner pay for the good in the first and

second price sealed-bid auctions?

In the first price sealed-bid auctions the winner should pay the highest amount what

they are bid for the good. In the second price sealed-bid auctions the winner pays the

second highest bid to buy the product.

Q2. What is the best strategy for the bidder in the second price sealed-bid auctions?

The best strategy for the bidder in the second price sealed-bid auctions is offering the

value that the good is maximum worth for the bidder.

Q3. What is “to win” the auction?

To win an auction means that buying the good at equal or less than the maximum of

what the good is worth for the bidder.

Q4. What is “winner’s curse”?

Winner's curse is a tendency for the winning bid in an auction to exceed the intrinsic

value of the item purchased on auction sites like eBay's. Because of incomplete

information, emotions or any other number of factors regarding the item being

auctioned, bidders can have a difficult time determining the item's intrinsic value. It's

what people suffer when they win an auction by overestimating how much something

is worth and therefore bidding too much. As a result, the largest overestimation of an

item's value ends up winning the auction.

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