7-1 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.
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7-1 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.
7-2 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
7-3 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?
7-4 Auctions 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
7-5 Mechanism Design As an example, suppose a person has an estate auction. Desirable properties of mechanisms: Convergence/guaranteed success Maximizing social welfare Pareto efficiency: the item must sell to the buyer with the highest evaluation Individual rationality (encourages bidders to behave rationally) Stability (won’t desire to change mind once outcome is known) Simplicity, quick Distribution (no central control) Ability to set Reservation Price (a seller specified bid level below which no sale is made)
7-6 Pareto Efficient Solutions U1U1 U2U2 f 1 f 2 f 4 f 3
7-7 Pareto Efficient Solutions U1U1 U2U2 f 1 f 2 f 4 f 3 f 2 Pareto dominates f 3
7-8 Pareto Efficient Solutions U1U1 U2U2 f 1 f 2 f 4 f 3 The Pareto frontier
7-9 Auction Parameters Goods can have private value public/common value correlated value (partially private, partially values of others): consider the resale value Winner determination may be first 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
7-10 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 Mediator follows a strict policy for determining outcome based on messages Single seller auctions are a special case
7-11 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 or reauction it to others
7-12 English Auctions Most commonly known type of auction: first price open cry ascending Open exit (openly declare exit) 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 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.)
7-13 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.
7-14 Dutch Auctions Dutch auctions are examples of open-cry descending auctions: auctioneer starts by offering good at artificially high value auctioneer lowers offer price until some agent makes a bid equal to the current offer price the good is then allocated to the agent that made the offer
7-15 Auction protocols: Dutch (open-cry descending) 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 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
7-16 How do you counterspeculate? 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.
7-17 What is your expected profit? 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)
7-18 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. f(p) = 1/(b-a) * (vp -va -p 2 +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)
7-19 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. 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.
7-20 In general, with uniform distribution on range [0,Max] If there are |A| agents, each agent should bid (|A|-1)/|A| v i where vi is the evaluation for agent I When |A|=2, we get the results on the previous slide
7-21 What is a Rational Decision? 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
7-22 Lotteries A lottery is a combination of a probability and an outcome L = [p, A; 1 – p, B] L = [1, A] L = [p, A; q, B, 1 – p – q, C] Lotteries can be used to asses a human’s preference structure
7-26 Maximizing the Expected Payoff Maximize expected monetary value (EMV): EMV(guess) = p correct * U(guess correct) + p wrong * U(guess wrong) =1/4(1,000,000) + 3/4 (100,000) = 325,000 EMV(quit) = 500,000 What if you had narrowed the choice to two alternatives?
7-27 Properties of Preferences Orderability For any two states, either A > B, B > A, or A~B 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
7-28 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
7-29 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 yield 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.
7-30 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) 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.
7-32 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 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 & counterspeculation 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 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).
7-33 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 Revelation principle --- proxy bidder agents on www.ebay.com, www.webauction.com, www.onsale.com
7-34 Vickrey auctions used for allocating resources in operating systems allocate bandwidth computationally control heat
7-35 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. 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.
7-36 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. That's how eBay's proxy bidders work. Because they're unique to every user, proxy bidders will never reveal your maximum bid to anybody else. 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
7-37 Ten-Dollar Auction Object for sale: a $10 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
7-38 Other types of Auctions Continuous Double Auction (CDA) Multiple buyers and sellers Clears continuously Call Market Multiple buyers and sellers Clears periodically
7-39 Other types of Auctions Reverse Auction Single buyer Lowest seller gets to sell the object Used in many procurement situations Multi-item Auctions Single seller Multiple units for sale N highest bidders get objects and pay ?
7-41 Core Auction Activities (2) 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
7-43 Sealed-bid second-price auction (Vickrey) is Pareto efficient even though no buyer knows the other buyers’ valuations (more on this later). In SPSB – your bid is decoupled from what you pay if you win, so you have an incentive to bid your true valuation – but only if you trust that the auctioneer is not dishonest and pretends that the second bid was larger than it was. It might be best to reveal the second bidder. In the FPSB-type auction, your bid determines what you pay if you win. It also determines the probability of winning.
7-44 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 – 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.
7-45 Efficiency Is the SPSB auction efficient in allocating the goods? Yes, because the winner values the asset most. Note that, in contrast to FPSB auctions, both English and SPSB auctions elicit from all bidders the true willingness to pay out. And the price is the second largest valuation among all bidders. Isn’t it strange that two set-ups that are so different could give rise to the same outcome? The critical common feature is that both are structured to induce the bidder to reveal his/her true valuation (but the winner stops short of revealing his true valuation).
7-46 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)
7-47 To see that all auctions are the same with perfect information… Order the bids (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.
7-48 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
7-49 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.
7-50 Risk Let S = [1, x] L = [p, y; 1 – p, z] Where x =EMV(L) = py + (1 – p)z Risk averse: U(S) > U(L) Risk neutral: U(S) = U(L) Risk seeking: U(S) < U(L)
7-51 Risk Neutral Bidders with Private Independent Values Do not know the valuations of others, but do know the distribution (range and probability) If independent private values, expected revenue is invariant to type (requires much math to prove).
7-52 Risk Averse Unhappy if someone else gets bid at a price below your valuation. 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. Example: value an asset at 100K, but may let it get to 95K before bidding. If risk averse, may not let it go below 97K Expected revenues: Dutch = FPSB > English=SPSB
7-53 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. May even go to seller and say you should get it for less. Recognizing this, all bidders may adjust their bids downwards. The winner’s curse works against the seller, especially if the bidders are risk averse (don’t want to pay more than real value). It is then best to try to release as much information about true worth to bidders.
7-54 Strategic underbidding in first-price sealed-bid auction … Example 2 risk-neutral bidders: A and B A knows that B’s value is 0 or 100 with equal probability A’s value of 400 is common knowledge In Nash equilibrium, B bids either 0 or 100, and A bids 100 + (winning more important than low price as 100 is less than valuation)
7-55 Revenue from the auctions Among risk averse bidders, the Dutch and first-price sealed-bid auctions give higher revenue to the auctioneer. For auctioneer, risk is keeping the item when the high valuation was actually higher than the reservation price. A risk averse auctioneer achieves higher expected utility via the Vickrey or the English auction protocol. In practice, most auctions are not pure private value auctions – hearing other bids causes bidder to increase value thus, greater expected revenue in English & Vickrey protocols to the auctioneer.
7-56 Auction protocols: All-pay Protocol: Each bidder is free to raise his bid. When no bidder is willing to raise, the auction ends, and the highest bidder wins the item. All bidders have to pay their last bid Variations Each agent pays only part of his highest bid Each agent’s payment is a function of the highest bid of all agents Strategy: Series of bids as a function of agent’s private value, her prior estimates of others’ valuations, and past bids Best strategy: ? Can result in infinite escalation of price In private value settings it can be computed (low bids) Potentially long bidding process E.g. CS application: tool reallocation [Lenting&Braspenning ECAI-94]
7-58 Human Utility for Money The evidence suggests that humans do not have a linear utility for money We have regret Our utility seems to depend upon our existing wealth The first million has more of an effect on our lifestyle than the 100th million
7-59 St. Petersburg Paradox Bernoulli, 1738 Game: A fair coin is tossed If tails, you double the pot & flip again If heads, the game ends and you keep pot How much would you pay for a chance to play this game?
7-60 Expected (Monetary) Value EMV(St. P) = Sum i p(H on turn i) * (2 i ) = Sum i (1/ 2 i ) 2 i = ∞ If U EMV, then you should be willing to pay an infinite amount of money to play the game.
7-61 Expected Utility If utility for money has the form: U(m) = log 2 m Then EU(St. P) = Sum (1/ 2 i ) log 2 2 i = 2 So a player with this utility function should be willing to pay 2 utiles (= $4) to play
7-62 Revenue Equivalence MODEL (reasons are guesses)REVENUE RANKINGS Independent Private values Risk neutral bidders. V=E as bid true valuation. D=F will underbid as not worried about losing out, so second price bid would be achieved. D = F = V = E Independent (not tied to other bidder) private values Risk averse bidders D=F > as don’t dare underbid V=E < as others may not value it as much as you, so you don’t have to pay full price. D = F > V = E Note, V and E are not affect by risk. Affiliated or privately known values (not going to change valuation based on others) Risk neutral bidders, D = F < V = E Affiliated or private unknown values Risk neutral bidders. E gains more information than others in bidding and may adjust value upwards. V tends to overbid D = F < V < E
7-64 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 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)
7-65 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).
7-66 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.
7-67 Bayes-Nash equilibrium A vector of bidding strategies b* is in Bayes- Nash equilibrium if: for each agent i and each possible type, agent i cannot increase its expected utility by using an alternate bidding strategy b i ’, holding the bidding strategies for the other agents fixed.
7-68 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.
7-69 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.
7-70 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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