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**Chapter 14: Allocating Scarce Resources**

With two player games, we had a “one shot” decision with no way of binding the negotiation. In this chapter, we rethink those decisions.

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Reaching Agreements How do agents reach agreements when they are self interested? In an extreme case (zero sum encounter) no agreement is possible in which both win — but in most scenarios, there is potential for mutually beneficial agreement on matters of common interest Example: handling an estate. How do you decide who gets what? There is only one item than many may want.

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**Mechanisms, Protocols, and Strategies**

Negotiation is governed by a particular mechanism, or protocol. We have seen mechanisms for making a decision on candidates – various voting protocols. The mechanism defines the “rules of encounter” between agents Mechanism design is designing mechanisms so that they have certain desirable properties Overview – auctions we are familiar with At seats, what would desirable properties of the mechanism be? Given a particular protocol, how can a particular strategy be designed that individual agents can use?

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New problems Auctions have been around for a long time (500 BC) – but the web has changed: ease of finding a buyer cost of conducting auction is small the additional problems created (sniping, shill bidding, etc)

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Example Suppose we want to design a mechanism for dividing up items in an estate. How do we frame the problem? <Item:DesirabiltyRating> for each person for each item? What is goal? fair division? maximize happiness? person who likes item the most gets it? What is the protocol? Desirable attributes: Do our previous desirable attributes make sense? individual rationality stable pareto optimal resistant to manipulation

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Mechanism Design As an example, suppose a person has an estate auction. Desirable properties of mechanisms: Convergence/guaranteed success Maximizing social welfare: auctioneers sell all items if anyone wants them. Maximize total happy people. Pareto efficiency: the item must sell to the buyer with the highest evaluation as profit to auctioneer is best and bidder is happiest. Individual rationality (encourages bidders to behave rationally) Stability (won’t desire to change mind in stating preferences once outcome is known) Simple, quick Distributed (no central control) Ability to set Reservation Price (a seller specified bid level , below which no sale is made)

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**Auctions – a specific form of negotiation**

An auction takes place between an agent known as the auctioneer and a collection of agents known as the bidders The goal of the auction is for the auctioneer to allocate the goods to the bidders In most settings the auctioneer desires to maximize the price; bidders desire to minimize price

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History of Auctions Some point to Babylon in 500 B.C. as the origin of the auction. In these early auctions, women, sought after as brides, were the commodities offered for sale. Attractive brides would command a great price; unfortunately, less desirable women would have to be accompanied by a dowry, making the winning bid price negative. In other words, the winning bidder would actually have to be paid to marry the woman! Some early auctions had incredibly high stakes. In 193 A.D., the entire Roman Empire was actually auctioned off! The highest bidder, Didius Julianus, won the bidding at a price of 6,250 drachmas for each Roman guard. He didn’t get to enjoy his purchase for long though, since he was beheaded two months later by Septimius Severus during his conquest of Rome.

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Assumption is – If you buy the item for exactly what it is worth, there is no utility. If you sell the item for exactly what is it worth, there is no utility. We are trying to do better than value – a good deal for both.

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**Auction Parameters Goods can have private value (Aunt Bessie’s Broach)**

public/common value (oil field to oil companies) correlated value (partially private, partially values of others): consider the resale value Winner pays first price (highest bidder wins, pays highest price) second price (to person who bids highest, but pay value of second price) Bids may be open cry sealed bid Bidding may be one shot ascending descending

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Reservation Price A reserve price auction operates in the same manner as a straight auction, with bidders trying to outbid each other for the item. The major difference is that the seller establishes a (possibly) secret “reserve price,” the lowest prices that he will accept for the item offered. If the bidding does not reach the reserve price, the seller is not obligated to sell. What would the advantages/disadvantages be of a reserve price?

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What is an Auction? Auctions are mediated negotiation mechanisms in which one negotiable parameter is price Note: Mediated implies messages are sent to mediator, not directly between participants Is the mediator is important? Why? Mediator follows a strict policy for determining outcome based on messages Single seller auctions are a special case

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**English Auctions Most commonly known type of auction: first price**

open cry ascending Open exit (openly declare exit, cannot re-enter) Real time Dominant strategy is for agent to successively bid a small amount more than the current highest bid until it reaches their valuation, then withdraw Efficient (pareto sense) as person who values item most gets it If you win in an English auction, you know that no one else valued it as much as you. Are you happy you have won?

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**English Auctions winner’s curse – get excited and bid too much**

Susceptible to: winner’s curse – get excited and bid too much shills (no intention of buying. Bid up the price. Work for auctioneer on commission. Illegal in most cases.) The earliest use of the word 'shill' actually dates back to Elizabethan England when theatre owners would pay a 'shilling' to a theatre goer who would applaud and cheer loudly at the end of a performance. Since applause is contagious, this would help ensure the success of a production.

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The key point is that the winner pays no more than the highest price that the second-highest bidder remaining would pay (note, there is an assumption that the bid increases in quite small intervals so that the last bidder recognizes when the second-last bidder drops out of the auction.) From a pareto efficiency standpoint, the bidder that values the item the most ends up with the item. This makes economists happy. Not all auctions are efficient in that sense.

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**Dutch (Aalsmeer) flower auction**

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**Auction protocols: Dutch (open-cry descending)**

The hand of the clock starts always on the top. The hand of the clock runs anti clockwise, from left to right. The price drops from high to low. The first person, out of the 300 buyers, who pushes the button first is the buyer of the goods

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Protocol: Auctioneer continuously lowers the price until a bidder takes the item at the current price Strategically equivalent to first-price sealed-bid protocol in all auction settings (we will discuss this option shortly) Time efficient (real-time) Strategy: Bid as a function of agent’s private value and his prior estimates of others’ valuations Best strategy: No dominant strategy in general (without more info) Lying (down-biasing bids) & counter-speculation Possible to determine Nash equilibrium strategies via common knowledge assumptions regarding the probability distributions of others’ values Requires multiple rounds of posting current price Dutch flower market, Ontario tobacco auction, Filene’s basement, Waldenbooks

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Questions What are advantages and disadvantages of such an auction? Why would you choose such a mechanism? Private: Learn only who values it the most, not the values that others have Quick

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Dictionary Moment What Does Dutch Auction Mean? (from Investopedia) 1. A public offering auction structure in which the price of the offering is set after taking in all bids and determining the highest price at which the total offering can be sold. In this type of auction, investors place a bid for the amount they are willing to buy in terms of quantity and price. 2. A type of auction in which the price on an item is lowered until it gets a bid. The first bid made is the winning bid and results in a sale, assuming that the price is above the reserve price. This is in contrast to typical options, where the price rises as bidders compete.

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**Ebay Multiple Item Dutch Auction - Ascending**

Multiple identical items are for sale Winning bids are selected in order of bid price per item. You cannot lower your “total bid value” (your bid price per item times the number of items on which you’re bidding) in raising your bid in this type of Multiple Item Auction. Example: For a listing with 10 available items and 4 bidders: Bidder A bid for 3 items at $3 each Bidder B bid for 8 items at $5 each Bidder A bid for 2 items at $4.50 each Bidder C bid for 3 items at $6 each Bidder D bid for 2 items at $7 each

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**Therefore, the outcome of this listing is: **

Bidder D wins 2 items at $5 each. Bidder C wins 3 items at $5 each. Bidder B wins 5 items at $5 each. Bidder A wins no items. The ranking of the bids affects the allocation of the items. Winning bidders have the right to refuse partial quantities. This means that if you win some, but not all, of the quantity you bid for, you don't have to buy any of them. In the above example, Bidder B bid on 8 items, but won only 5 of them. Bidder B can refuse to complete the purchase, because Bidder B did not win the quantity he or she bid on.

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The Yankee auction Same as before – but actually pay what you bid: So for the previous example: Bidder D wins 2 items at $7 each. Bidder C wins 3 items at $6 each. Bidder B wins 5 items at $5 each. Bidder A wins no items.

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**Auction protocols: First-price sealed-bid (FPSB)**

Protocol: Each bidder submits one bid without knowing others’ bids. The highest bidder wins the item at the price of his bid Single round of bidding (once you bid, there are no counter offers) Strategy: Bid as a function of agent’s private value and his prior estimates of others’ valuations Best strategy: No dominant strategy in general Strategic underbidding & counter speculation Can determine Nash equilibrium (not do anything different, knowing what others would do) strategies via common knowledge assumptions about the probability distributions from which valuations are drawn Goal is to try to maximize the expected profit. No relevant information is revealed – not even price or buyer (if you aren’t the winner) Bidder uncertainty of valuation is a factor No dominant strategy – as may not be pareto optimal with the “best strategy”. Efficient in real time as each person takes minimal time (as bidding happens in parallel) – but not too satisfying (no feedback).

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**Vickrey Auctions one shot second-price sealed-bid**

Vickrey auctions are: one shot second-price sealed-bid Good is awarded to the agent that made the highest bid; at the price of the second highest bid Bidding to your true valuation is dominant strategy in Vickrey auctions. Why? But in practice, you bid less due to the winner’s curse. Not prone to strategic manipulation. Vickrey auctions susceptible to antisocial behavior (bid really high to guarantee win, someone else bids somewhat high to stick you with it) Effort not wasted in counter-speculation as just bid true value. Widely advocated for computational multiagent systems Old method [Vickrey 1961], but not widely used among humans

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**What is learned in each auction?**

Dutch auctions are tense events, but are not very informative. You do learn that the winner values the item at no less than the price bid. But you learn nothing about how others value the item. As with the FPSB auction, you learn only that you are (are not) the high bidder. Even the SPSB (Vickrey) auction yields more information as it reveals to the winner the second highest bid Why do we care that the bidder’s don’t learn anything? If the bidders are unhappy (always lose the bid but don’t know why) or win (but suffer winner’s remorse), they may not choose to frequent your auction. That could be bad for you.

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**How do you counterspeculate?**

Consider a Dutch auction While you don’t know what the other’s valuation is, you know a range and guess at a distribution (uniform, normal, etc.) For example, suppose there is a single other bidder whose valuation lies in the range [a,b] with a uniform distribution. If your valuation of the item is v, what price should you bid? Thinking about this logically, if you bid above your valuation, you lose. If you bid lower than your valuation, you increase profit. If you bid very low, you lower the probability that you will ever get it.

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**Try counter-speculating**

The camera is for sale again. There are only two bidders. The other bidder will bid a value that you guess is uniformly distributed on the interval from You value it for $70. What should you bid?

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**Try counter-speculating**

The camera is for sale again. There are only two bidders. The other bidder will bid a value that you guess is uniformly distributed on the interval from You value it for $70. What should you bid? profit Chance of winning 50 20 0 55 15 .1 60 10 .2 65 5 .3 70 .4

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**What is your expected profit (Dutch auction)? (assume only two bidders)**

It seems natural to try to maximize your expected profit. Expected profit (as a function of a specific bid) is the probability that you will win the bid times the amount of your profit at that price. Let p be the price you bid for an item. v be your valuation. [a,b] be the uniform range of others bid. The probability that you win the bid at this price is the fraction of the time that the other person bids lower than p. (p-a)/(b-a) The profit you make at p is v-p Expected profit as a function of p is the function = (v-p)*(p-a)/(b-a) + 0*(1- (p-a)/(b-a))

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**Finding maximum profit is a simple calculus problem**

Expected profit as a function of p is the function (v-p)*(p-a)/(b-a) Take the derivative with respect to p and set that value to zero. Where the slope is zero, is the maximum value. (as second derivative is negative) f(p) = 1/(b-a) * (vp -va -p2+pa) f’(p) = 1/(b-a) (v-2p+a) = 0 p=(a+v)/2 (half the distance between your valuation and the minimum of the range of values)

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Are you surprised? The results make sense. You never bid higher than your valuation. You can’t win these cases, so we’ll ignore them. Of the remaining cases, if you bid halfway between the low evaluation and your valuation, you expect to win half the time and lose half the time [in the cases where you have a chance]. When you do win, you pay considerably less than your valuation. You have to bid more often as you won’t get everything you bid on – but this is a good plan. As there are more bidders, how would that affect your decision?

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**In general, with uniform distribution on range [0,Max]**

If there are n total agents, what should each agent bid? For simplicity, lets assume a=0. In order to win the bid at price p,every other person would need to bid below p. What is the chance of that? Since we want bidder 1 to bid below p and bidder 2 (and so forth), we multiply the probabilities for each of the other n-1 bidders: Expected valued = (p/b)n-1 * (p-v) = (1/bn-1)*(p(n) – vpn-1) derivative wrt p = (1/bn-1*((n)p(n-1) – (n-1)vp(n-2)) = 0 (n)p(n-1) – nvp(n-2) +vp(n-2) = p(n-2)(pn –nv + v) so either p = 0 (obviously a minimum) or p=((n-1)/n) * v When n=2, we get the results on the previous slide. Idea is that with more bidders (randomly bidding within the range), you have to bid closer to your valuation to win; the idea of competition.

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**Revenue for the Auctioneer**

Which protocol is best for the auctioneer? A friend says: All four protocols give the same expected revenue for private value auctions amongst risk-neutral bidders with valuations independently drawn from a uniform distribution. Argue for or against this result

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**Revenue for the Auctioneer**

Which protocol is best for the auctioneer? • Revenue-equivalence Theorem (Vickrey, 1961): All four protocols give the same expected revenue for private value auctions amongst risk-neutral bidders with valuations independently drawn from a uniform distribution. • Intuition: revenue second highest valuation: – Vickrey: clear – English: bidding stops just after second highest valuation – Dutch/FPSB: because of the uniform value distribution and counter speculation, top bid second highest valuation But: this applies only to an artificial and rather idealized situation; in reality there are many exceptions.

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**RISK - preliminary information What is a Rational Decision?**

We introduce notation to talk about preferences over choices. We assume that agents have preferences over states of the world A > B A is strictly preferred to B A ~ B agent is indifferent between A & B A ≥ B A is weakly preferred to B (could be equal)

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Lotteries A lottery is a combination of a probability and an outcome (like considering a mixed choice) L = [p, A; 1 – p, B] (p% chance of receiving A) L = [1, A] L = [p, A; q, B, 1 – p – q, C] (percent chance of receiving each of A, B or C) Lotteries can be used to represent a human’s preference structure

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**“Want to be a Millionaire” Scenario**

You have just achieved $500,000 You have no idea on the last question If you guess [3/4, $100,000; 1/4, $1,000,000] If you quit [1, $500,000] What do you do? What is expected utility in each case? If you pick something with larger expected utility, that is “rational”. If you pick something of lesser expected value (but higher possible value), you are risk seeking.

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Risk We want to capture how we feel about the possibility of getting a large return. Just looking at expected utility doesn’t capture the possibility of a better return [3/4, $100; 1/4, $1,000] [1, $500] [1, $325] If we are risk seeking, we don’t just go for less money. We go for less expected money for the possibility of more. This can be accomplished by valuing money u(w) = w2 (risk seeking) With this lens, we value the choices as: .75* *1000*1000 = 1*500*500 = 325*325 =

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**Maximizing the Expected Payoff**

Maximize expected value (EV): EV(guess) = pcorrect * U(guess correct) + pwrong * U(guess wrong) =1/4(1,000,000) + 3/4 (100,000) = 325,000 EV(quit) = 500,000 What if you had narrowed the choice to two alternatives?

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**Risk Averse: Which would you choose?**

Gamble 1: win $500 with a probability ½ and nothing with probability ½ Gamble 2: win $1000 with a probability of .05 and win $50 with probability .95 Gamble 3: win $10,000 with a probability of .05 and win $0 with probability .95 Gamble 4: win $100 guaranteed Even a risk averse person wouldn’t prefer any kind of safe situation. prob choice 1 choice 2 Expected Revenue 0.5 500 250 0.05 1000 50 98 10000 1 100

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**Risk Averse: Which would you choose?**

Gamble 1: win $500 with a probability ½ and nothing with probability ½ Gamble 2: win $1000 with a probability of .05 and win $50 with probability .95 Gamble 3: win $10,000 with a probability of .05 and win $0 with probability .95 Gamble 4: win $100 guaranteed If our utility over wealth: u(w) = Even a risk averse person wouldn’t take any kind of safe situation. prob choice 1 choice 2 Expected Revenue Expected Utility 0.5 500 250 11 0.05 1000 50 98 8 10000 5 1 100 10

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**Risk Seeking: Which would you choose?**

Gamble 1: win $500 with a probability ½ and nothing with probability ½ Gamble 2: win $1000 with a probability of .05 and win $50 with probability .95 Gamble 3: win $10,000 with a probability of .05 and win $0 with probability .95 Gamble 4: win $100 guaranteed If our utility over wealth: u(w) = Even a risk averse person wouldn’t take any kind of safe situation. prob choice 1 choice 2 Expected Revenue Expected Utility 0.5 500 250 125000 0.05 1000 50 98 52375 10000 1 100

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**Individuals have different tolerance for risk**

Individuals have different tolerance for risk. An individual who ranks lotteries according to their expected value (rather than expected utility) is said to be risk neutral. In other words, an risk neutral individual who is offered $100 outright or a 50% chance of winning $200 will value the choices EQUALLY!

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If the utility function over wealth is linear u(w) = aw + b the person is risk neutral If the utility function is concave(a line between points is under curve), the individual is risk averse. If the utility function is convex(line between points is above curve), the individual is risk seeking. Note, gambling is like staying on the line as the two endpoints are picked with probability p or (1-p).

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**Expected Utility Theory**

describes behavior under uncertainty If people are risk neutral or risk averse, they would never play the lottery or gamble via slot machines (as return there is usually negative) The expected value of Powerball lottery (if tickets cost $1 and jackpot is 7 million) is * 1/ *( / ) =

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**But people do play powerball - Why?**

Loss is so small, people often ignore it. If losses were larger, people may behave very differently. People who buy lottery tickets may behave in very risk averse manner in other situation

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**How does risk affect revenue? Risk Averse**

With English and SPSB, risk aversion makes no difference – as you pay second price. Thus, you automatically have potential for profit. With Dutch and FPSB, to generate profit, you must take the risk of bidding below your valuation. If you are risk averse, you would rather win a little money than run the risk of making nothing. So you bid higher and revenues for auctioneer increase. Expected revenues for risk averse bidders: Dutch = FPSB > English=SPSB Since the risk averse bidder values a little money more than the potential of more, he will be happier with a sure thing at less profit. For risk seeking, Dutch = FPSB < English=SPSB Since the risk seeking bidder values a big profit more than the expected value of less, he will be happier with the gamble.

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Revelation Principle You can transform any auction into an “equivalent” auction which is direct (simply announce your valuation) and incentive compatible (i.e., bidder will bid the true valuation) Do you believe that? That is quite a statement. Rather than lie (bid less than your true valuation), the mechanism will “lie” for you Example: assume two bidders (with valuations drawn from a uniform distribution on a fixed interval [0,max]). The optimal strategy is to bid ½ your true value. But if the rule is changed so that the winner only pays half his bid, it is optimal to bid your true value. Example: You have to pay an entrance fee and then the it is FPSB. Thus, you have to figure out when to enter the bidding and what to bid. How would you transform it?

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**Other types of Auctions Continuous Double Auction (CDA)**

Multiple buyers and sellers Clears continuously (determines who trades and at what price) A double auction market can be carried out by open outcry: buyers and sellers call out prices they are willing to buy and sell at, and a match is made if a buyer and seller call out the same price. NYSE, Pit (the game). Under double auction rules, any buyer who makes a bid must raise his/her hand and be recognized. All bids and offers are written on the blackboard as they are made. Only most attractive bid/offer has "standing" or can be accepted. Any buyer is free at any time to accept a standing offer, and any seller can accept a standing bid. It is common practice to add an "improvement rule"; that is, that a new bid be greater than the standing bid and that a new offer be lower than the standing offer. This is a double auction in a sense that bids rise and offers fall at the same time. Like Haggling: each suggest price Trading does not stop as each auction is concluded Try it!

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**Other types of Auctions**

Reverse Auction (different from a descending auction) Single buyer Lowest seller gets to sell the object Used in many procurement situations

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Japanese auction Similar to an English auction in which an auctioneer regularly raises the current price. Participants must signal at every price level their willingness to stay in the auction and pay the current price. The auction concludes when only one bidder indicates his willingness to stay in. also known as the button auction.

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**Strategic EQUIVALENCE**

Same expected revenue for the auctioneer Same bidding strategy for the bidder English and Vickrey auctions have same strategic equivalence if we have independent values (meaning my valuation doesn’t increase by hearing your bid) – though they are constructed differently. With a common value, English and Vickrey are no longer equivalent as information is gained through the open cry bidding that is not obtained from Vickrey. First-price sealed-bid and Dutch auctions are strategically equivalent. The differences are superficial. The essential features (you pay the price you bid, and you have no information about others when you bid) are the same. Therefore, you are gaining no real advantage from observing the auctioneer’s price fall in a Dutch auction. The optimum bid strategies for bidders are the same in each.

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Perfect Information Perfect information occurs when each bidder knows the value of an asset to himself and to the other bidders. With perfect information, all auctions have the same results. The second highest valuation (or an infinitesimal bit above it)

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**To see that all auctions are the same with perfect information…**

Order the valuations (high to low): v1, v2, v3, … In English, the winning bid is v2+ In FPSB, all know the values, so v2+ wins here also. In Dutch auction, the bidder knows v2+ is the price that should be held out for In Vickrey, just bid your true evaluation, and you’ll get it for the second price.

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**Results for private value auctions**

English and Vickrey auctions - Most (pareto) efficient (as those that value item the most, get item) All four protocols allocate item efficiently (assuming no reservation price for the auctioneer) English & Vickrey have dominant strategies => no effort wasted in counterspeculation

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**On cheating in sealed-bid auctions**

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Bidder Collusion Collusion: bidders agreeing together to control bids to their advantage. (Let’s keep price low and split profit.) Which is best if we want to prevent collusion?

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Bidder Collusion Collusion: bidders agreeing together to control bids to their advantage. (Let’s keep price low and split profit.) None of the four is collusion-proof May be hard to identify other bidders May be hard to punish defectors

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**With Vickrey, if collude**

Say one bidder values item at 20, the rest value it at 18. They agree to have one bid 20 and the rest bid 5. The high bid gets it for 5. No reason for anyone to bid any higher as they wouldn’t get it anyway (and they wouldn’t want it for over 20).

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**How could auctioneer “lie” in each case?**

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Lying Auctioneer Problem in Vickrey auction – auctioneer overstates second bid – electronic signatures (or have trusted third party handle bids) Non-private value auctions – English auction – auctioneer’s shills (someone who bids up the price to increase perceived value, but never wants to take it home. Works for auctioneer - illegal) Overstated reservation price (minimum price that the auctioneer will accept) – sometimes risky to the auctioneer as he may not sell it No risk in first price sealed bid, as know how much you offered and are not affected by other bidders.

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**First-price sealed bid auction, cheating bidder**

Consider the case where the seller is honest, but there is a chance the other agents will look at the bids before submitting their own. Notice that this kind of cheating is pointless in second-price auctions.

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Combinatorial Auction: Related Item Bidding What if we are interested in sets of items (chairs, china, sports memorabilia)? What if we are interested in play tickets for a specific evening and several choices are liked?

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**Complements and Substitutes**

The value an agent assigns to a bundle of goods may relate to the value it assigns to the individual goods in a variety of ways • Complements: The value assigned to a set is greater than the sum of the values assigns to its elements. Ex: a left shoe and a right shoe or GI Joe will all the original clothing • Substitutes: The value assigned to a set is lower than the sum of the values assigned to its elements. Ex: ticket to the theatre and another one to a football match for the same night. In such cases an auction mechanism allocating one item at a time is problematic as the best bidding strategy in one auction may depend on whether the agent can expect to win certain future auctions.

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**Combinatorial Auction Protocol**

• Setting: one seller (auctioneer) and several potential buyers (bidders); many goods to be sold • Bidding: the bidders bid by submitting their valuations to the auctioneer (not necessarily truthfully) • Clearing: the auctioneer announces a number of winning bids The winning bids determine which bidder obtains which good, and how much each bidder has to pay. No good may be allocated to more than one bidder.

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**Interrelated auctions**

Strategies might be different when interrelated items are auctioned at a time instead of each item separately. Say – bid for two tasks, but second is cheaper if already doing the first. Lookahead is a key feature in auctions of multiple interrelated items. Auctioneers often allow bidders to pool all of the interrelated items under one bid.

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**Interrelated auctions (cont.)**

Sometimes auctioneers allow bidders to backtrack from commitments by paying penalties. This is helpful if you win bid on one item hoping to get interrelated item (but don’t get it). Different kind of speculations: trying to guess what items will be auctioned in the future, and which agents are going to win in those auctions. Trade-off: (partial) lookahead vs. cost.

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Bidding Languages • As there are 2n − 1 non-empty bundles for n goods, submitting complete valuations may not be feasible. (similar to problem of stating value of each coalition) • We assume that each bidder submits a number of atomic bids (unable to be split) (Bi, pi) specifying the price pi the bidder is prepared to pay for a particular bundle Bi. • The bidding language rules specify what combinations of individual bids may be accepted. Today, we (mostly) assume that only one bid of each bidder can be accepted. • In general, we may think of the bidding language as determining a conflict graph: bids are vertices and edges connect bids that cannot be accepted together. • The bidding language also determines how to compute the overall price

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**The Winner Determination Problem**

The winner determination problem (WDP) is the problem of finding a set of winning bids (1) that is feasible and (2) that will maximize the sum of the prices offered.

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Example Each bidder submits a number of bids describing their valuation. Each bid (Bi, pi) specifies which price pi the bidder is prepared to pay for a particular bundle Bi. The auctioneer may accept at most one atomic bid per bidder (other bidding languages are possible). Agent 1: ({a, b}, 5), ({b, c}, 7), ({c, d}, 6) Agent 2: ({a, d}, 7), ({a, c, d}, 8) Agent 3: ({b}, 5), ({a, b, c, d}, 12) What would be the optimal solution?

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Example Each bidder submits a number of bids describing their valuation. Each bid (Bi, pi) specifies which price pi the bidder is prepared to pay for a particular bundle Bi. The auctioneer may accept at most one atomic bid per bidder (other bidding languages are possible). In this example, we can remove clearly inferior bids ({a, b}, 5) is inferior to ({b}, 5), ({a, b, c, d}, 12) is inferior to ({a, d}, 7) plus ({b, c}, 7) Agent 1: ({a, b}, 5), ({b, c}, 7), ({c, d}, 6) Agent 2: ({a, d}, 7), ({a, c, d}, 8) Agent 3: ({b}, 5), ({a, b, c, d}, 12) What would be the optimal solution?

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**Complexity of Winner Determination**

The decision problem underlying the WDP is NP-complete: Theorem 1 Let K Z. The problem of checking whether there exists a solution to a given combinatorial auction instance generating a revenue exceeding K is NP-complete. (Note, they have changed the optimization problem into a simpler boolean form of the problem.) M.H. Rothkopf, A. Peke˘c, and R.M. Harstad. Computationally Manageable Combinational Auctions. Management Science, 44(8):1131–1147, 1998.

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**What should each pay? Vickrey-Clarke-Groves Mechanism**

The bidder isn’t penalized by its own bid. For example, suppose that we want to auction two apples, and we have three bidders. Bidder A wants one apple and bids $5 for that apple. Bidder B wants one apple and is willing to pay $2 for it. Bidder C wants two apples and is willing to pay $6 to have both of them, but is uninterested in buying only one without the other. A and B should have the apples – as together they value them the most. . W. Vickrey. Counterspeculation, Auctions, and Competitive Sealed Tenders. Journal of Finance, 16(1):8–37, E.H. Clarke. Multipart Pricing of Public Goods. Public Choice, 11(1):17–33, 1971. T. Groves. Incentives in Teams. Econometrica, 41(4):617–631, 1973. How Many Total Price A 1 5 B 2 C 6

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**Vickrey-Clarke-Groves Mechanism**

The bidder isn’t penalized by its own bid. What should be paid? But both A and B are thinking, the seller thinks the apples are worth more BECAUSE OF ME. The price shouldn’t be higher for me because of me. Currently, B has a payment of $2. If bidder A had not been present, C would have won, and had a utility of $6, so A pays $6-$2 = $4. (The real cost to the seller of A’s getting the apple.) For the payment of bidder B: currently A has a utility of $5 and C has a utility of 0. If bidder B had been absent, C would have won and had a utility of $6, so B pays $6-$5 = $1. C does not need to pay anything because he doesn’t get anything. W. Vickrey. Counterspeculation, Auctions, and Competitive Sealed Tenders. Journal of Finance, 16(1):8–37, E.H. Clarke. Multipart Pricing of Public Goods. Public Choice, 11(1):17–33, T. Groves. Incentives in Teams. Econometrica, 41(4):617–631, 1973.

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**VCG Mechanism for Combinatorial Auctions**

The value of the best allocation possible without me minus the value to everyone else of the best allocation with me Consider the following example: Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 9) Agent 2: ({a}, 4), ({b}, 0), ({a, b}, 0) Agent 3: ({a}, 0), ({b}, 2), ({a, b}, 0) Agent 4: ({a}, 0), ({b}, 2), ({a, b}, 2) Clearly agent 1 should win BUT he feels he shouldn’t be bidding against himself. He would like to pay only what it is worth to others: without agent 1 – worth 6 with agent 1 – nobody else gets any value so worth = 0 Agent 1 is happy to pay 6

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Another Example The value of the best allocation possible without me minus the value to everyone else of the best allocation with me Consider the following example: Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 19) Agent 2: ({a}, 3), ({b}, 0), ({c}, 3) Agent 3: ({a}, 0), ({b}, 2), ({c}, 2) Agent 4: ({a}, 0), ({b}, 2), ({a, b, c},13) Who wins and what do they pay? Agent 2 = 21 – 19 = 2

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Another Example The value of the best allocation possible without me minus the value to everyone else of the best allocation with me Consider the following example: Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 19) Agent 2: ({a}, 3), ({b}, 0), ({c}, 3) Agent 3: ({a}, 0), ({b}, 2), ({c}, 2) Agent 4: ({a}, 0), ({b}, 2), ({a, b,c},13) Clearly agent 1 should win BUT he feels he shouldn’t be bidding against himself. He would like to pay only what it is worth to others: without agent 1 bidding – worth 13 with agent 1 – Agent 2 still gets 3 Agent 1 is happy to pay 10 What should agent 2 pay? Agent 2 = 21 – 19 = 2

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Strategy-Proofness In the VCG mechanism, “reporting their true valuation” is a dominant strategy for each bidder – similar to the reasons that truthfulness was dominant with Vickrey

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**So, does it work well? Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 4)**

Consider the following example: Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 4) Agent 2: ({a}, 1), ({b}, 0), ({a, b}, 0) Agent 3: ({a}, 0), ({b}, 2), ({a, b}, 0) Agent 4: ({a}, 0), ({b}, 2), ({a, b}, 2) Best allocation is agent 1 ({a, b}, 4) Agent 1 pays: 3 – 0 = 3 So it works just like we thought it would Lots of the problems come from very few bidders – which would always be a problem for Vickrey auctions.

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**Another Example Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 4)**

Consider the following example: Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 4) Agent 2: ({a}, 2), ({b}, 0), ({a, b}, 0) Agent 3: ({a}, 0), ({b}, 2), ({a, b}, 0) Agent 4: ({a}, 0), ({b}, 2), ({a, b}, 2) A best allocation is agent 1 ({a, b}, 4) (assume handle ties some reasonable way) Agent 1 pays: 4 – 0 = 4 So it works just like we thought it would Second price bid is really the same as the first price bid

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Collusion The VCG mechanism is not collusion-proof: if bidders work together they can manipulate the mechanism. Consider the following example: Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 4) Agent 2: ({a}, 1), ({b}, 0), ({a, b}, 0) Agent 3: ({a}, 0), ({b}, 1), ({a, b}, 0) Who wins? What do they pay? But if the two losing bidders collude and increase their two bids to ({a}, 4) and ({b}, 4), respectively, they can obtain the items for free. Agent 2: ({a}, 4), ({b}, 0), ({a, b}, 0) Agent 3: ({a}, 0), ({b}, 4), ({a, b}, 0) Agent 1 wins and pays 2 (2-0) IN second case: Agents 2 and 3 win, but pay nothing.

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Collusion ANSWERS The VCG mechanism is not collusion-proof: if bidders work together they can manipulate the mechanism. Consider the following example: Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 4) Agent 2: ({a}, 1), ({b}, 0), ({a, b}, 0) Agent 3: ({a}, 0), ({b}, 1), ({a, b}, 0) Agent 1 wins and pays 2-0= 2. But if the two losing bidders collude and increase their two bids to ({a}, 4) and ({b}, 4), respectively, they can obtain the items for free. Agent 2: ({a}, 4), ({b}, 0), ({a, b}, 0) Agent 3: ({a}, 0), ({b}, 4), ({a, b}, 0) Agents 2 and 3 win and pay 4-4= 0. Can you explain the logic?

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**Problems with the VCG Mechanism**

Despite their nice game-theoretical properties, combinatorial auctions using VCG to determine payments have several problems: Low (and possibly even zero) revenue for the auctioneer Non-monotonicity: “better” bids don’t entail higher revenue Collusion amongst (losing) bidders False-name bidding: bidders may benefit from submitting bids using multiple identities. Can you give an example? The following examples illustrating these problems are adapted from Asubel and Milgrom (2006). L.M. Asubel and P. Milgrom. The Lovely but Lonely Vickrey Auction. In P. Cramton et al. (eds.), Combinatorial Auction, MIT Press, 2006.

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Zero Revenue There are cases where the VCG mechanism gives zero revenue: Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 2) Agent 2: ({a}, 2), ({b}, 0), ({a, b}, 0) Agent 3: ({a}, 0), ({b}, 2), ({a, b}, 0) Payments are computed as follows: Agent 1: 0 Agent 2: 2 − (2) = 0 (the best without my bid is 2, the value to others of the best allocation computed with my bid is 2) Agent 3: 2 − (2) = 0 (the best without my bid is 2, the value to others of the best allocation computed with my bid is 2)

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**Non-monotonicity Try it!**

Example 1: Revenue is not necessarily monotonic in the set of bids or the amounts that are being bid. Consider again the following example: Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 2) Agent 2: ({a}, 2), ({b}, 0), ({a, b}, 0) Agent 3: ({a}, 0), ({b}, 2), ({a, b}, 0) As seen before, revenue for this example is 0. If we either remove agent 3 or decrease the amount agent 3 is offering for item b, then revenue will increase. Try it!

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Non-monotonicity Example 1: Revenue is not necessarily monotonic in the set of bids or the amounts that are being bid. Consider again the following example: Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 2) Agent 2: ({a}, 2), ({b}, 0), ({a, b}, 0) Agent 3: ({a}, 0), ({b}, 2), ({a, b}, 0) As seen before, revenue for this example is 0. If we either remove agent 3 or decrease the amount agent 3 is offering for item b, then revenue will increase. Example 2: Agent 3: ({a}, 0), ({b},1), ({a, b}, 0) Revenue for this example Agent 2 pays = 2-1 = 1 (the best without my bid is 2, the value to others of the best allocation computed with my bid is 1) Agent 3 pays = 2-2 = 0 ( the best without my bid is 2, the value to others of the best allocation computed with my bid is 2) Example 3: Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 2) Agent 1 pays 0 Agent 2: ({a}, 2), ({b}, 0), ({a, b}, 0) Agent 2 pays 2 -0 = 2

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False-name Bidding False-name bidding (aka. shill or pseudonymous bidding) is yet another form of manipulation the VCG mechanism is exposed to. Example: Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 4) Agent 2: ({a}, 1), ({b}, 1), ({a, b}, 2) Agent 1 wins. But agent 2 could instead submit bids under two names: Agent 2: ({a}, 4), ({b}, 0), ({a, b}, 0) Agent 2’: ({a}, 0), ({b}, 4), ({a, b}, 0) Agent(s) 2 (and 2’) will win and not pay anything! This form of manipulation is particularly critical for electronic auctions, as it is easier to create multiple identities online than it is in real life. M. Yokoo. Pseudonymous Bidding in Combinatorial Auctions. In P. Cramton et al. (eds.), Combinatorial Auction, MIT Press, 2006.

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Computational Issues Observe that computing VCG requires solving an additional n winner determination problems (who wins with me and who wins without me). • That means, the auctioneer has to solve n + 1 NP-hard optimization problems. • If allocations and prices are not being computed according to the optimal solutions to these problems, then we cannot guarantee strategy-proof-ness anymore.

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Conclusions Pros VCG processes have great theoretical appeal. They are a dominant strategy mechanism. This means that, in theory, a bidder’s decision to use the strategy they call for does not depend on what the bidder thinks her competitors’ strategies are, and she need spend no effort in trying to find them out or to keep her competitors from learning her strategy. In some circumstances, they produce, in theory, expected revenue equivalent to other common auction forms. Cons However, VCG processes are just not practical. They do not work the way the (simple) theory says they should. So Why do we study VCG processes ? Because finding equilibrium strategies in combinatorial auctions is extraordinarily difficult except in VCG processes, there may well be useful insights to be had from such knowledge. For example, Mishra and Parkes (2007) analyze an iterative version of the VCG process. In addition, computerized bidding agents may be able to be programmed to avoid some of the 13 problems discussed here. Rothkopf: Thirteen Reasons Why the Vickrey-Clarke-Groves Process Is Not Practical Operations Research 55(2), pp. 191–197, ©2007 INFORMS

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