Chapter 7: Reaching Agreements In chapter six, we had a “one shot” decision with no way of binding the negotiation. In this chapter, we rethink those decisions. 7-1 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 The capabilities of negotiation and argumentation are central to the ability of an agent to reach such agreements 2 Mechanisms, Protocols, and Strategies Negotiation is governed by a particular mechanism, or protocol 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? 3 Example Suppose we want to design a mechanism for dividing up items in an estate. What is the protocol? Could protocol have some negative ramifications? 2007 Nobel Prize in Economics – for mechanism design the market, under ideal conditions, ensures an efficient allocation of scarce resources. But in practice conditions are usually not ideal; for example, competition is not completely free, consumers are not perfectly informed and privately desirable production and consumption may generate social costs and benefits 4 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 5 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 women! 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. 6 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 once outcome is known) – Simple, quick – Distribution (no central control) – Ability to set Reservation Price (a seller specified bid level below which no sale is made) 7 Pareto Efficient Solutions: f represents possible solutions for two players. Assuming utility is amount more than minimal sale price for seller and less that maximum purchase price for buyer. U2 f1 f2 f3 f4 U1 8 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. 9 Pareto Efficient Solutions U2 f1 f 2 Pareto dominates f 3 f2 f3 f4 U1 10 Pareto Efficient Solutions U2 f1 f2 The Pareto frontier f3 f4 U1 11 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 12 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 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. Example: Cy knows that he could sell his grandfather’s antique dresser for $2,000 to a neighbor; however, he hopes to make more than that at an auction. So, he tells the auctioneer that the lowest he will accept for the dresser is $2,000. Cy asks the auctioneer not to announce this price to the bidders because he wants them to bid as high as they think the dresser is worth, potentially much more than $2,000. If the bidding comes in lower than $2,000, Cy will not sell the dresser at the auction, but instead to his neighbor. If the bidding ends up being higher than $2,000, Cy will sell the dresser at the auction. 13 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 14 Auction settings Private value : value of the good depends only on the agent’s own preferences – E.g. cake which is not resold or showed off Common value : agent’s value of an item determined entirely by others’ values – E.g. treasury bills Correlated value (Affiliated value): agent’s value of an item depends partly on its own preferences & partly on others’ values for it – E.g. painting when bidders can keep it and like the colors or reauction it to others 15 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 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. 16 The key point is that the winner pays no more than the highest price that the second-last 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. 17 Dutch (Aalsmeer) flower auction 18 Dutch Clock 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 man, out of the 300 buyers, who pushes the button first is the buyer of the goods 19 Auction protocols: Dutch (open-cry descending) 20 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) & counterspeculation – 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 21 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 22 How do you recommend the auction is handled? Erin opens her birthday presents and realizes that she has received six Yoda figurines. She only wants one, so decides to auction off the other five in one auction. 23 Descending price Multiple item A descending price auction is a type of auction used to sell multiple units of the same item at the same time. In a descending price auction, multiple items are offered at an opening bid price. A potential buyer can bid on one or more units of the item. The price is then lowered in successive decrements until the entire lot has been purchased. Erin opens her birthday presents and realizes that she has received six Yoda figurines. She only wants one, so decides to auction off the other five in one auction. $80, but lower price by $10 each hour. • 3 bidders at $60 • 1 bidder at $50 • Many bidders at $40, first wins 24 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) if you raise your bid in this type of Multiple Item Auction. Example: For a listing with 10 available items and 4 bidders: Bidder A bid for 2 items at $4 each Bidder B bid for 8 items at $5 each Bidder C bid for 3 items at $6 each Bidder D bid for 2 items at $7 each 25 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. 26 The Yankee auction This is another type of auction that is used to sell multiple units of the same item at the same time. It operates identically to the ascending multiple-item auction except that the winning bidders each pay their winning bid price instead of the lowest winning bid price. In other words, in a Yankee auction it is possible to have buyers that pay different prices; whereas in a Dutch auction, everybody pays the same price. 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. 27 Auction protocols: First-price sealed-bid Example – Outdoor rec - kayac 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). 28 Vickrey Auctions 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. 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 29 What is learned? 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. 30 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. 31 What is your expected profit (dutch auction)? 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)) 32 Finding maximum profit is a simple calculus problem Expected profit as a function of p is the function (v-p)*(pa)/(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 bid and the min range value) 33 Are you surprized? 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, and hence make a handsome profit. 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? 34 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. 35 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 riskneutral 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. 36 How does this happen Consider buying something on Craig’s list. The Craigslist auction strategy is not as easy as EBay because you can’t view other bids. A starting point is realizing that you don’t want to bid higher than your value. If your value is $100, it does not make sense to bid $101. You might not win, but if you do, you are sure to overpay. The next step is realizing you don’t want to bid exactly your value. If you do that, then you’ll only break even when you win. So the idea is to bid some amount lower than your value. How much lower? In this auction, you know that you only win if you’re the highest value, and you ideally would want to pay the least amount—one bid above the maximum for the second highest person. It’s impossible to know where your relative value is, so what you do is the following. You guess that you have the highest value, and then estimate the second highest value and bid just higher. 37 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) 38 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 picking A) – L = [1, A] – L = [p, A; q, B, 1 – p – q, C] (percent chance of picking each of A, B or C) Lotteries can be used to represent a human’s preference structure 39 Millionaire Scenario You have just achieved $500,000 You have 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? 40 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? 41 Properties of Preferences Orderability – For any two states, either A > B, B > A, or A~B (equivalent) Transitivity – If A > B and B > C, then A > C Continuity – If A > B > C, then there is some p, s.t. [p, A; (1-p) C] ~ B 42 More Properties Substitutability – If A~B, then [p, A; (1-p) C] ~ [p, B; (1-p) C] for any value of p Monotonicity – If A > B and p ≥ q then [p, A; (1-p) B] ≥ [q, A; (1-q) B] Decomposibility – Compound lotteries can be reduced to simpler ones using laws of probability 43 Revelation Principle You can transform any auction into an “equivalent” one which is direct 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. 44 As an example of the revelation principle Proxy Bidding on ebay Once you have found an item you want to buy on eBay and decided you are willing to pay £25 for it but the current price is £2.20, what should you do? Well, you could bid just £2.40 and probably be the high bidder... but what happens when you leave your computer and someone else comes in and bids £2.60? Do you have to sit in front of your computer day and night until the auction ends to make sure you win? Thanks to eBay's automatic proxy bidding tool, the answer is no. You allow the system to increase you bid up to a max. How does Proxy bidding work? Here's a step by step guide: An Auction is listed that starts at £1. I come along and submit a maximum bid of £100. The proxy server executes this bid, and as there are no other bids yet (mine is the first), the bid is on me for £1. Now you come along and see that the current bid is £1 and you decide to bid £5. You enter the bid and then you get an immediate outbid notice. Why? Because the proxy system has my £100 maximum bid to execute while keeping the bid at the lowest possible amount. So you see an outbid notice, and the bid goes up to £5.50, with me as the high bidder. 45 You rebid £10 and the same thing happens. I am still the high bidder at £10.50 and will remain so until you or someone else surpasses my initial maximum bid. Now, say you bid £150. The highest bid then falls to you at £101, as you have passed my maximum bid (I'm now out of the running until I place a new maximum bid). If someone should come by and decide to bid £103, then they would get an immediate outbid notice because the proxy system automatically outbid the new bidder…and so on until the new bidder gives up or places a maximum bid which outstrips your £150. Once your maximum bid is reached, you receive an outbid notification by email. You can decide then whether or not you want to increase your maximum bid. Thus, you can tell the truth (your real valuation) and the mechanism lies for you (bids a competitive bid without overpaying) Revelation principle – by changing the mechanism, we can convince bidders to reveal their true valuation 46 Twenty-Dollar Auction Object for sale: a $20 bill Rules – Highest bidder gets it – Highest bidder and the second highest bidder pay their bids – New bids must beat old bids by 50¢. – Bidding starts at $1 – What would your strategy be? 47 This is sometimes called the entrapment game--that is particularly nasty. Suppose that anyone who bids at the auction of our $20 bill must pay the amount of the bid whether he wins or not. Someone will open the bidding low at $.50 in hopes of getting a real bargain. Someone else will top the bid with a $1 bid. Bidding will usually proceed up to about $10 and then pause. The second bidder must now decide whether to lose his $8 or $9 bid, or continue. If he continues, the bidding will usually advance up to $20 and then pause again. The second highest bidder now realizes that he is not going to gain anything on this auction, but has the potential for a substantial loss, so he has a strong temptation to up his bid beyond $20. 48 Here is how Frank and Cook describe this game: "One might be tempted to think that any intelligent, well-informed person would know better than to become involved in an auction whose incentives so strongly favor costly escalation. But many of the subjects in these auctions have been experienced business professionals; many others have had formal training in the theory of games and strategic interaction. For example, psychologist Max Bazerman reports that during the past ten years he has earned more than $17,000 by auctioning $20 bills to his MBA students at Northwestern University.... In the course of almost two hundred of his actions, the top two bids never totaled less than $39, and in one instance totaled $407." 49 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! 50 Other types of Auctions Reverse Auction – Single buyer – Lowest seller gets to sell the object – Used in many procurement situations 51 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. 52 Call auction Investors can place orders with their financial intermediary throughout the day until 4.30 pm. orders dispatched to the central order book for a 30-minute accumulation phase. this phase ends at 5.00 pm; match buy and sell orders anonymously and centrally in order to establish a price. The price determined by the auction procedure is the reference price and serves as opening price for the following day’s auction. 53 Taxonomy http://gsb.haifa.ac.il/~sheizaf/ecommerce/auction.types.html 54 Core Auction Activities Receive bids – Enforce any bidding rules Release intermediate information (optional) – Produce quotes – List of winning bidders Clear – Determine who trades with who and at what price 55 PROTOCOL EQUIVALENCE 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. 56 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) 57 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. 58 Results for private value auctions English and Vickrey auctions - Most 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 59 Bidders may desire private valuation to remain private Reminder: the Vickrey auction’s dominant strategy in private-value auctions is bidding truthfully. May reveal sensitive information as identity of first and second bidders and price of second bidder may be known (a main reason why the Vickrey auction protocol is not widely used). Doesn’t occur in first-price sealed-bid auctions, as no one even needs to know who won the bid. 60 RISK: For each of the three deals which choice would you pick? Deal 1 – I give you $1 OR – We pick a random integer 1..100 and if it comes out 1 you get $100 otherwise you get nothing. Deal 2 – I give you $10 OR – We flip a coin and if it is heads, you get $20, otherwise you get nothing. Deal 3 – I give you $1 – We pick a random integer 1..100 and if it comes out 1 you get $1000 otherwise you get nothing. 61 Risk Let S = [1, x] get x with probability of 1 Where x =ExpectedValue(L) = py + (1 – p)z L = [p, y; 1 – p, z] Suppose the expected values are the same! Risk averse: Utility(S) > Utility(L) (a sure thing is valued more than choice with same expected value) Risk neutral: Utility(S) = Utility(L) (a sure thing is valued the same as a choice with same exp value) Risk seeking: Utility(S) < Utility(L) (a sure thing is valued less than a chance at more) 62 Risk Averse: Which would you choose? Gamble 1: win $400 with a probability ½ and nothing with probabiltiy ½ Gamble 2: win $225 with a probability of ½ and win $36 with probability ½ Gamble 3: win $200 guaranteed If our u(w) = w The expected utility of gamble 1 is ½ sqrt(400) + 1/2(0) = ½ 20 = 10 The expected utility of gamble 2 = ½*sqrt(36) + ½ sqrt(225) = ½(6+15) =10.5 The expected utility of gamble 3=sqrt(200) = 14.14 63 Risk SeekingWhich would you choose? Gamble 1: win $400 with a probability ½ and nothing with probabiltiy ½ Gamble 2: win $225 with a probability of ½ and win $36 with probability ½ Game 3: win $200 guaranteed If our u(w) = x2 The expected utility of gamble 1 is ½(400)2 + 1/2(0) = ½ 20 = 80,000 The expected utility of gamble 2 = ½*(36)2 + ½(225)2 = ½(51921) =25,960 The expected utiltiy of gamble 3=(200)2 = 40,000 64 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! 65 If the utility function over wealth is linear u(w) = aw + b the person is risk neutral If the utility function is concave(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). 66 So u(w) = w is risk neutral u(w) = is risk averse u(w) = w2 is risk seeking (as large amount of money is worth much more than small amounts) w 67 Expected Utility Theory describes behavior under uncertainty If people are risk neutral or risk averse, they would never play the lottery or gamble (as return there is usually negative) The expected value of Powerball lottery (if tickets cost $1 and jackpot is 7 million) is 7000000 * 1/85000000 -1(84999999/85000000) = -.917647 68 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 69 Risk Averse With English and SPSB, risk aversion makes no difference – as you bid your true valuation, but may not pay that much. Thus, you automatically have potential for profit. With Dutch and FPSB, to generate profit, you must take the risk of letting the price go below your valuation. If you are risk averse, you would rather win a little money than run the risk of making nothing or making a lot. 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 vaue of less, he will be happier with the gamble. 70 Winner’s Curse In auctions where bidders have common values, the winner tends to have overestimated asset value. He/she may come to regret the bid (curse). You know you’ve bid too much as others bid less. Recognizing this, all bidders may adjust their bids downwards. The winner’s curse works against the seller, especially if the bidders are risk seeking. It is then best to try to release as much information about true worth to bidders. 71 On cheating in sealed-bid auctions 7-72 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 – First-price sealed-bid and Dutch auctions make it harder to conspire against auctioneer (as hard to know who is going to bid) – Bidders need not identify each other ahead of time to collude in English auctions, unlike in the others. Could have a way of signalling or contacting the active bidders to arrange something. – Also, can enforce collusion in English auction as can respond to defector (who bids higher than agreed upon) 73 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). 74 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. 75 First-price sealed bid auction, cheating bidder Consider the case 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. 76 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. A standard example for complements would be a pair of shoes (a left shoe and a right shoe). • Substitutes: The value assigned to a set is lower than the sum of the values assigned to its elements. A standard example for substitutes would be a 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. 77 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. 78 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. 79 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. 80 Bidding Languages • As there are 2n − 1 non-empty bundles for n goods, submitting complete valuations may not be feasible. • We assume that each bidder submits a number of atomic bids (Bi, pi) specifying the price pi the bidder is prepared to pay for a particular bundle Bi. • The bidding language determines what combinations of individual bids may be accepted. Today, we (mostly) assume that at most 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 81 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. The sum of prices can be given different interpretations: (1) If the simple pricing rule is used where bidders pay what they offered, then it is the revenue of the auctioneer. (2) If the prices offered are interpreted as individual utilities, then it is the utilitarian social welfare of the selected allocation. 82 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? 83 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. 84 Solving the Winner Determination Problem We have seen that the WDP is intractable (NP-complete) in its general form. Nevertheless, sophisticated search algorithms often manage to solve even large CA instances in practice. There are two types of approaches to optimal winner determination in the general case: • Use powerful general-purpose mathematical programming software (next slide) • Develop search algorithms specifically for winner determination, combining general AI search techniques and domain-specific heuristics (rest of this lecture) Other options include developing special-purpose algorithms for tractable subclasses and approximation algorithms for the general case 85 Integer Programming Approach Suppose bidders submit n bids as bundle/price pairs (Bi, pi) with the implicit understanding that the auctioneer may accept any combination of non-conflicting bids and charge the sum of the associated prices (this is the so-called OR bidding language). Introduce a decision variable xi {0, 1} for each bid (Bi, pi). The WDP becomes the following Integer Programming problem: Maximize pi · xi subject to each item only being in one bid that is accepted. Highly optimized software packages for mathematical programming (such as CPLEX/ILOG) can often solve such problems efficiently. 86 Search for an Optimal Solution Next we are going to see how to customise well-known search techniques developed in AI so as to solve the WDP. This part of the lecture will largely follow the survey article by Sandholm (2006). T. Sandholm. Optimal Winner Determination Algorithms. In P. Cramton et al. (eds.), Combinatorial Auctions, MIT Press, 2006. 87 Search Techniques in AI A generic approach to search uses the state-space representation: • Represent the problem as a set of states and define moves between states. Given an initial state, this defines a search tree. • The goal states are states that correspond to valid solutions. • Each move is associated with a cost (or a payoff ). • A solution is a sequence of moves from the initial state to a goal state with minimum cost (maximum payoff ). • Example: route finding (states are cities and moves are directly connecting roads), but it also applies to CAs . . . A search algorithm defines the order in which to traverse the tree: • Uninformed search: breadth-first, depth-first, iterative deepening • Heuristic-guided search: branch-and-bound, A* 88 State Space and Moves There are (at least) two natural ways of representing the state space and moves between states: • Either: Define a state as a set of goods for which an allocation decision has already been made. Then making a move in the state space amounts to making a decision for a further good. • Or: Define a state as a set of bids for which an acceptance decision has already been made. In this case, a move amounts to making a decision for a further bid. What is the initial state? What are the goal states? According to Sandholm (2006), the bid-oriented approach tends to give better performance in practice. 89 Moves in Bid-oriented Search We represent bids as triples (ai,Bi, pi): agent ai is offering to buy the bundle Bi for a price of pi. The initial state is when no decisions on bids have been made. A move amounts to making a decision (accept/reject) for a new bid. The bidding language specifies which bids (if any) must be accepted/rejected given earlier decisions. We are in a goal state once a decision for every bid has been made (some of which will be consequences of the explicit choices). Observe that that the search tree will be binary (accept or reject?). 90 Problem Suppose we have the following independent bids in a combinatorial auction: {1,2} 10 {3} 4 {2,3} 12 {1,3} 12 {4,5,6} 12 {4,5} 9 {5,6} 10 {3,5} 6 91 Example from Sandholm (2006) shows a PARTIAL conflict graph for our example Conflict graph Use it (in) or Don’t use it (out) Show the conflict graph at this point. 92 Applying g(n) +h(n) as the heuristic for A* search 93 Applying g(n) +h(n) as the heuristic for A* search 94 Uninformed Search Uninformed search algorithms (in particular depth-first search) can be used to find a solution with a given minimum revenue: traverse the tree and keep the best solution encountered so far in memory. Optimality can only be guaranteed if we traverse the entire search tree (not feasible for interesting problem instances). 95 Uniform Cost Strategy: expand lowest path cost The good: UCS is complete and optimal! The bad: Explores options in every “direction” No information about goal location 96 Best First • Strategy: expand nodes which appear closest to goal Heuristic: function which maps states to distance • A common case: Best-first takes you straight to the (wrong) goal • Worst-case: like a badlyguided DFS • In our case, explore the bids with the most items. 97 Heuristic-guided Search In the worst case, any algorithm may have to search the entire search tree. But good heuristics, that tell us which part of the tree to explore next, often allow us to avoid this in practice. For any node N in the search tree, let g(N) be the revenue generated by accepting (only) the bids accepted according to N. g(N) is the revenue earned from previous decisions. We are going to need a heuristic that allows us to estimate for every node N how much revenue over and above g(N) can be expected if we pursue the path through N. This will be denoted as h(N). h(N) is future possible revenue. The more accurate the estimate, the better — but the only strict requirement is that h never underestimates the true revenue. We are going to describe two algorithms using such heuristics: • depth-first branch-and-bound • the A* algorithm 98 Slides from: Dan Klein – UC Berkeley Many slides over the course adapted from either Stuart Russell or Andrew Moore 99 Heuristic Upper Bounds on Revenue Sandholm (2006) discusses several ways of defining a heuristic function h such that g(N) + h(N) is guaranteed to be an upper bound on revenue for any path through node N. Here is one such heuristic function: • For each good g, compute its maximum contribution as: c(g) = max{p/|B| | (B, p) Bids and g B} • Then define h(N) as the sum of all c(g) for those goods g that have not yet been allocated in N. This is indeed an upper bound (why?). This assumes the revenue from a bundle is credited equally. The only way this wouldn’t be true is if a partner could get more, and then the one in question gets less. The quality of this heuristic can be improved by recomputing c(g) after every step (need to balance accuracy and computing time). Bids is the set of Bids not yet ruled out. 100 Depth-first Branch-and-Bound This algorithm works like basic (uninformed) depth-first search, except that branches that have no chance of developing into an optimal solution get pruned on the fly: • Traverse the search tree in depth-first order. • Keep track of which of the nodes encountered so far would generate maximum guaranteed revenue . Call that node N*. • If a node N with g(N) + h(N) <=g(N*) is encountered, remove that node and all its offspring from the search tree. This is correct (guarantees that the optimal solution does not get removed) whenever the heuristic function h is guaranteed 101 The A* Algorithm The A* algorithm (Hart et al., 1968) is probably the most famous search algorithm in AI. It works as follows: • The fringe is the set of leaf nodes of the subtree visited so far (initially just the root node). • Compute f(N) = g(N) + h(N) for every node N in the fringe. • Expand the node N with the largest f(N); that is, remove it from the fringe and add its (two) immediate children instead to the visited subtree. By a standard result in AI, A* with an admissible heuristic function (here: h never underestimates marginal revenue) is optimal: the first solution found (when no bids are left) will generate maximum revenue. P. Hart, N. Nilsson, and B. Raphael. A Formal Basis for the Heuristic Determination of Minimum Cost Paths. IEEE Transactions on Systems Science and Cybernetics, 4(2):100–107, 1968. 102 Branching Heuristics So far, we have not specified which bid to select for branching in each round (for any of our algorithms). This choice does not affect correctness, but it may affect speed. There are two basic heuristics for bid selection: • Select a bid with a high price and a low number of items. • Select a bid that would decompose the conflict graph of the remaining bids (if available). 103 Tractable Subproblems As a final example for possible fine-tuning of the algorithm, we can try to identify tractable subproblems at nodes and solve them using special-purpose algorithms. Here are two very simple examples: • If the bid conflict graph is complete, i.e. any pair of remaining bids is in conflict, then only one of them can be accepted. Simply pick the one with the highest price. • If the bid conflict graph has no edges, then there is no conflict between any of the remaining bids. Accept all remaining bids (assuming positive prices). 104 Mechanism Design Mechanism design is concerned with the design of mechanisms for collective decision making that favor particular outcomes despite of agents pursuing their individual interests. Mechanism design is sometimes referred to as reverse game theory. While game theory analyses the strategic behavior of rational agents in a given game, mechanism design uses these insights to design games inducing certain strategies (and hence outcomes). We are going to concentrate on mechanism design questions in the context of (private value) combinatorial auctions. 105 Revelation Principle This is somewhat simplified and informal: Theorem 1 Any outcome that can be implemented through some indirect mechanism with dominant strategies can also be implemented by means of a direct mechanism (where agents simply reveal their preferences) that makes truth-telling a dominant strategy. Intuition: Whatever the agents are doing in the indirect mechanism to transform their true preferences into a strategy, we can use as a “filter” in the corresponding direct mechanism. So, first apply this filter to whatever the agents are reporting and then simulate the indirect mechanism with the filtered input. The outcome will be the same as the outcome we’d get with the indirect mechanism iff the agents report their true preferences. Discussion: we can concentrate on searching for a one-step mechanism Example: the (direct) Vickrey auction may be regarded as a direct implementation of the (indirect) English auction 106 Quasi-linear Utilities • Each agent i has a valuation function vi mapping agreements x (e.g. allocations) to the reals. This could be any such function. • The actual utility ui of agent i is a function of its valuation vi(x) for agreement x and a possible price p the agent may have to pay in case x is chosen. In principle, this could be any such function. • However, we make the (common) assumption that utility functions are quasilinear (linear in one of the parameters): ui(x, p) = vi(x) − p That is, utility is linear in both valuation and price paid. 107 Reminder: Vickrey Auction Dominant strategy: bid your true valuation – if you bid more, you risk paying too much – if you bid less, you lower your chances of winning while still having to pay the same price in case you do win • How can we generalize this idea to combinatorial auctions? 108 Reinterpreting the Vickrey Pricing Rule for Combinatorial Auction • Distinguish allocation rule and pricing rule • Allocation rule: highest bid wins • Pricing rule: winner pays price offered, but gets a discount • The amount of the discount granted reflects the contribution to overall value made by the winner. How can we compute this? – Without the winner’s bid, the second highest bid would have won. So the contribution of the winner is equal to the difference between the winning and the second highest bid. – Subtracting this contribution from the winning bid yields the second highest bid (the Vickrey price). 109 Vickrey-Clarke-Groves Mechanism This idea is used in the so-called Vickrey-Clarke-Groves mechanism, which we are going to introduce next. We are going to concentrate on the variant introduced by Edward H. Clarke (for combinatorial auctions), but also mention the more general form of the mechanism as put forward by Theodore Groves. 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. First, we decide the outcome of the auction by maximizing bids: the apples go to bidder A and bidder B. Next, to decide payments, we consider the opportunity cost that each bidder imposed on the rest of the bidders. Currently, B has a utility 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. 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, 1961. 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. 110 Notation Set of bidders: A = {1, . . . , n} • Set of possible agreements (allocations): X • (True) valuation function of bidder i A: vi : X R • Valuation function reported by bidder i A: ˆvi : X R • Top allocation as chosen by the auctioneer: x* arg maxxX n vj^ ( x) j 1 Allocation that would be chosen if agent i were not to bid: x *i arg maxxX n vj^ ( x) j 1 111 VCG Mechanism for Combinatorial Auctions Allocation rule: solve the WDP and allocate goods accordingly • Pricing rule: Again, the idea is to give each winner a discount reflecting its contribution to overall value. In short, bidder i should pay the following amount: bidi − (max-value − max-value−i) The same more formally: The value of the best allocation possible without me minus the value to everyone else of the best allocation with me 112 Strategy-Proofness Theorem 2 In the VCG mechanism, reporting their true valuation is a dominant strategy for each bidder. Proof: Consider the situation of bidder i. Remark: Contrast this with the Gibbard-Satterthwaite Theorem, which (roughly) says that in the context of voting there is no such strategy-proof mechanism. The crucial difference is that here we can use money to affect people’s incentives. 113 114 Efficiency By construction, if all bidders submit true valuations (dominant strategy), then the outcome maximizes utilitarian social welfare (the benefit to society in general in terms of their utility): • payments (including the auctioneer’s) sum up to 0; and • the sum of valuations is being maximized. But note that this does not mean that revenue gets maximized as well (unlike for the basic combinatorial auction without special pricing rules). 115 So, does it ever work well? 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. 116 Another Example 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) Best allocation is agent 1 ({a, b}, 4) 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 Lots of the problems come from very few bidders – which would always be a problem for Vickrey auctions. 117 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 1: ({a}, 0), ({b}, 0), ({a, b}, 4) Agent 2: ({a}, 4), ({b}, 0), ({a, b}, 0) Agent 3: ({a}, 0), ({b}, 4), ({a, b}, 0) Who wins? What do they pay? 118 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 1: ({a}, 0), ({b}, 0), ({a, b}, 4) 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? 119 Example The following example shows problems if we drop the free disposal assumption: Agent 1: accept one of ({a}, 90), ({b}, 10), ({a, b}, 10) Agent 2: accept one of ({a}, 20), ({b}, 30), ({a, b}, 50) We end up with the following payments in all items must be disposed of: Agent 1: 50 -30 = +20(the best without my bid is 50, the value to others of the best allocation computed with my bid is 30) Agent 2: 10-90 = −80 (the best without my bid is 10, the value to others of the best allocation computed with my bid is 90) That is, agent 2 should receive money from the auctioneer! 120 Problems with the VCG Mechanism Despite their nice game-theoretical properties, CAs using the Clarke tax 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 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. 121 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 − (4 − 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 − (4 − 2) = 0 (the best without my bid is 2, the value to others of the best allocation computed with my bid is 2) Note that this problem is independent from whether or not we admit free disposal. 122 Non-monotonicity 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. Agent 1: ({a}, 0), ({b}, 0), ({a, b}, 2) Agent 2: ({a}, 2), ({b}, 0), ({a, b}, 0) 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) New Example 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 = 2 123 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 1: ({a}, 0), ({b}, 0), ({a, b}, 4) 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. 124 Computational Issues Observe that computing the Clarke tax requires solving an additional n winner determination problems. • 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. 125 Summary We have introduced the Vickrey-Clarke-Groves mechanism, a mechanism for collective decision making that makes truth-telling the dominant strategy. • Distinguish most general form of the VCG mechanism and the variant where the Clarke tax is used to determine payments • Additional properties: efficiency and weak budget balance (the latter under suitable conditions) • Drawbacks: high complexity, potential for low revenue, manipulation through collusion or use of false-name bids, . . . • Restriction: applies to agents with quasi-linear utilities only 126 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 127

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# Lecture 7: Reaching Agreements