Auction Design for Atypical Situations

Overview General review of common auctions General review of common auctions Auction design for agents with hard valuation problems Auction design for agents with hard valuation problems Auction design for goods in unlimited supply Auction design for goods in unlimited supply

Auction Design Many different protocols Many different protocols Major auction types: Major auction types: Ascending Price (English) Ascending Price (English) Descending Price (Dutch) Descending Price (Dutch) First Price, Sealed Bid First Price, Sealed Bid Second Price, Sealed Bid (Vickrey) Second Price, Sealed Bid (Vickrey) Essentially mixing and matching certain properties, but some combinations work better than others Essentially mixing and matching certain properties, but some combinations work better than others

Auction Evaluation Revenue for the sellers Revenue for the sellers Profit for the bidders Profit for the bidders Avoidance of “winner’s curse” helps both Avoidance of “winner’s curse” helps both Winner in most auction protocols is the participant who made the biggest overvaluation mistake Winner in most auction protocols is the participant who made the biggest overvaluation mistake Auctions that decouple an agent’s bid from the actual price paid encourage higher bids Auctions that decouple an agent’s bid from the actual price paid encourage higher bids

Ascending Price (English) Most commonly known protocol Most commonly known protocol Auctioneer starts with an opening bid and successively raises the price as participants are willing Auctioneer starts with an opening bid and successively raises the price as participants are willing Allows for dynamic adjustment of bidders’ valuations by giving information about other bidders Allows for dynamic adjustment of bidders’ valuations by giving information about other bidders Buyers bid at most their utility, which may be adjusted on the fly Buyers bid at most their utility, which may be adjusted on the fly

Descending Price (Dutch) Biddings starts at an extremely high price and descends until someone claims the item Biddings starts at an extremely high price and descends until someone claims the item High degree of Winner’s Curse High degree of Winner’s Curse Intuitively, raises seller’s revenue in cases when the high bidder wants an item badly Intuitively, raises seller’s revenue in cases when the high bidder wants an item badly

First Price, Sealed Bid Buyers submit bids once Buyers submit bids once No knowledge of one another’s bids No knowledge of one another’s bids Auctioneer opens bids and sells the item to the highest bidder at the price he submitted Auctioneer opens bids and sells the item to the highest bidder at the price he submitted Encourages buyers to bid conservatively (shade down from utility) to maximize profit versus probability of winning Encourages buyers to bid conservatively (shade down from utility) to maximize profit versus probability of winning

Second Price, Sealed Bid (Vickrey) Designed to alleviate Winner’s Curse in the First Price, Sealed Bid protocol Designed to alleviate Winner’s Curse in the First Price, Sealed Bid protocol Same sealed bid format, but item is awarded to the highest bidder at the second highest price Same sealed bid format, but item is awarded to the highest bidder at the second highest price Buyers can bid their utility to increase probability of winning, but are guaranteed a price closer to market consensus Buyers can bid their utility to increase probability of winning, but are guaranteed a price closer to market consensus

Optimal Auction Design for Agents with Hard Valuation Problems David C Parkes

Motivation Standard auction theory assumes that either agents know own their utility for an item (private value) or that there is a common utility that is unknown to the agents (common value) Standard auction theory assumes that either agents know own their utility for an item (private value) or that there is a common utility that is unknown to the agents (common value) As transactions are increasingly turned over to software agents, the cost of obtaining this utility value may be significant As transactions are increasingly turned over to software agents, the cost of obtaining this utility value may be significant Certain auction designs can simplify this problem Certain auction designs can simplify this problem

Paper Goals Compare the performance of agents with hard valuation problems within three auction designs Compare the performance of agents with hard valuation problems within three auction designs Posted Price, sequential Posted Price, sequential Second Price, sealed bid Second Price, sealed bid Ascending Price Ascending Price What the paper isn’t about: Actual methods for refining beliefs about values What the paper isn’t about: Actual methods for refining beliefs about values

Examples of Hard Valuation Agents bidding for components on behalf of a manufacturer Agents bidding for components on behalf of a manufacturer Agents bidding for collectibles or other rarities Agents bidding for collectibles or other rarities

Problem Formulation Agents with hard valuation problems operate in three phases: Agents with hard valuation problems operate in three phases: Metadeliberation: Decide how much effort to spend refining the valuation of the item Metadeliberation: Decide how much effort to spend refining the valuation of the item Valuation: The refinement process – solve an optimization problem, interact with a human expert, etc Valuation: The refinement process – solve an optimization problem, interact with a human expert, etc Bidding Bidding

Agent Model Each agent has an unknown true value for an item Each agent has an unknown true value for an item Each agent maintains an upper and lower bound, in which the true value is assumed to be somewhere, uniformly distributed, in between Each agent maintains an upper and lower bound, in which the true value is assumed to be somewhere, uniformly distributed, in between Expected true value is then the average of the upper and lower bounds Expected true value is then the average of the upper and lower bounds The deliberation process refines the upper and lower bounds The deliberation process refines the upper and lower bounds

Agent Model  Deliberation  Incurs cost C Upper Bound Lower Bound New Upper Bound New Lower Bound Expected True Value New Expected True Value

Parameters Cost of Deliberation (C) Cost of Deliberation (C) Computational Effectiveness of Deliberation (1-  ) Computational Effectiveness of Deliberation (1-  )

Metadeliberation Solve the tradeoff between reducing uncertainty and incurring the cost of deliberation Solve the tradeoff between reducing uncertainty and incurring the cost of deliberation Deliberation is only worthwhile when: Deliberation is only worthwhile when: It changes the agent’s bid It changes the agent’s bid The new bid has a greater expected utility than the old bid The new bid has a greater expected utility than the old bid Metadeliberation strategies vary by auction type Metadeliberation strategies vary by auction type

Metadeliberation Strategies Vickrey Auction Need distributional information about the bids of other agents Need distributional information about the bids of other agents Paper assumes such information is somehow obtained by the agents (eg learning) Paper assumes such information is somehow obtained by the agents (eg learning) Metadeliberation strategy: Metadeliberation strategy: Deliberate until utility of placing a bid now is greater than the estimated utility of placing a bid after another deliberation step Deliberate until utility of placing a bid now is greater than the estimated utility of placing a bid after another deliberation step Bid utility is a nonlinear function of expected value Bid utility is a nonlinear function of expected value Higher bid decreases profit but raises probability of winning Higher bid decreases profit but raises probability of winning

Metadeliberation Strategies Vickrey Auction Length of time that an agent spends deliberating depends on: Length of time that an agent spends deliberating depends on: The number of agents in the auction The number of agents in the auction The agent’s current upper and lower bounds on the value of the item The agent’s current upper and lower bounds on the value of the item The computational effectiveness of deliberation (1-  ) The computational effectiveness of deliberation (1-  ) The cost of deliberation (C) The cost of deliberation (C)

Metadeliberation Strategies Posted Price Sequential No uncertainty about the actions of other agents No uncertainty about the actions of other agents Need only to worry about the cost of the good Need only to worry about the cost of the good Deliberate only when the ask price is within the bounds of some threshold function  ( , C,  ) Deliberate only when the ask price is within the bounds of some threshold function  ( , C,  ) ** Accept Price Deliberate Reject Price v v

Metadeliberation Strategy Ascending Price Third action available in addition to bid and deliberate: wait Third action available in addition to bid and deliberate: wait Agents that wait benefit from the deliberation of others Agents that wait benefit from the deliberation of others Optimal Strategy: Optimal Strategy: Bid Wait, or Deliberate if auction will close, with probability 1/(Na -1) Leave auction v v

Evaluation Metrics Efficiency: True Value for the Good for the Winning Agent / Maximum True Value over All Agents Efficiency: True Value for the Good for the Winning Agent / Maximum True Value over All Agents Revenue: Price Paid for the Good / Maximum True Value over All Agents Revenue: Price Paid for the Good / Maximum True Value over All Agents Utility of Participation: (Surplus to Winning Agent – Total Deliberation Cost for All Agents) / Number of Agents Utility of Participation: (Surplus to Winning Agent – Total Deliberation Cost for All Agents) / Number of Agents

Evaluation Arms Variance of: Variance of: Number of Agents Number of Agents Computational Effectiveness (1-  ) Computational Effectiveness (1-  ) Cost of Deliberation (C) Cost of Deliberation (C) Agent “experience” – Adjust C and  for fractions of the agent population Agent “experience” – Adjust C and  for fractions of the agent population

Varying the Number of Agents + Ascending Price X Sealed Bid O Posted Price Sequential

Varying the Bidding Increment + Ascending Price

Varying the Computational Effectiveness of Deliberation + Ascending Price X Sealed Bid O Posted Price Sequential

Varying Agent Experience + Ascending Price X Sealed Bid O Posted Price Sequential

Competitive Auctions and Digital Goods Andrew Goldberg Jason D Hartline Andrew Wright

Motivation Looking for an optimal way to sell goods in unlimited supply Looking for an optimal way to sell goods in unlimited supply Downloadable music Downloadable music Pay per view movies Pay per view movies Examines auctions as an alternative to fixed pricing, which requires expensive and probably inaccurate market research Examines auctions as an alternative to fixed pricing, which requires expensive and probably inaccurate market research Scary implication: Charge more for media created by entities with small, rabid followings? Scary implication: Charge more for media created by entities with small, rabid followings?

The Optimal Threshold Function Given a set of bids, determine the single price that maximizes revenue Given a set of bids, determine the single price that maximizes revenue at 3 (9) 5 at 2 (10) 6 at 1 (6) Sell…

Paper Goals Examine classes of single round, sealed bid auctions for products with no marginal cost of reproduction Examine classes of single round, sealed bid auctions for products with no marginal cost of reproduction Need to solve tradeoff between selling a lot at a low price versus a few at a high price Need to solve tradeoff between selling a lot at a low price versus a few at a high price Need to ensure that participants bid their utilities Need to ensure that participants bid their utilities Would like an auction mechanism that compares well to optimal fixed pricing Would like an auction mechanism that compares well to optimal fixed pricing

Terminology Truthful auctions Truthful auctions Encourage participants to bid their utility Encourage participants to bid their utility More formally: More formally: Bidder’s profit (bid – price if wins, or 0 otherwise) is maximized when bid is the same as utility for any fixed values for the bids of other participants Bidder’s profit (bid – price if wins, or 0 otherwise) is maximized when bid is the same as utility for any fixed values for the bids of other participants Example: Vickrey Example: Vickrey Counterexample: First price sealed bid Counterexample: First price sealed bid Why is this important? Why is this important? Revenue is maximized in truthful auctions Revenue is maximized in truthful auctions

Truthful Auction Example Imagine participating in an auction for a jar of 100 pennies that you (and only you) have counted Imagine participating in an auction for a jar of 100 pennies that you (and only you) have counted In a first price sealed bid auction, you cannot hope to profit by bidding 100 In a first price sealed bid auction, you cannot hope to profit by bidding 100 In a Vickrey auction, by bidding 100 you will at worst break even, but most likely pay less In a Vickrey auction, by bidding 100 you will at worst break even, but most likely pay less

Terminology Competitive auctions Competitive auctions Produce revenue within a constant factor of optimal fixed pricing Produce revenue within a constant factor of optimal fixed pricing Must vary the number of items sold based on the bids received Must vary the number of items sold based on the bids received Why is this important? Why is this important? Matching optimal fixed pricing is the best possible result Matching optimal fixed pricing is the best possible result Being within a constant factor is a reasonable approximation Being within a constant factor is a reasonable approximation

Evaluated Auction Designs All auctions in this paper are single round, sealed bid All auctions in this paper are single round, sealed bid Auction mechanisms studied: Auction mechanisms studied: Deterministic Deterministic Deterministic Optimal Threshold Deterministic Optimal Threshold Randomized Randomized Single Price Single Price Dual Price Dual Price Weighted Pairing Weighted Pairing

Bid Independence Agent’s bid determines whether or not he wins the auction, but not the price Agent’s bid determines whether or not he wins the auction, but not the price Typically multi-price, but not always Typically multi-price, but not always Example: Vickrey Example: Vickrey Why is this important? Why is this important? Bid independence allows for one to bid her utility and still hope for a profit Bid independence allows for one to bid her utility and still hope for a profit Deterministic auctions must be bid independent in order to be truthful Deterministic auctions must be bid independent in order to be truthful

Truthful Deterministic Auctions Consider the deterministic optimal threshold auction Consider the deterministic optimal threshold auction To determine if bid b i wins To determine if bid b i wins Remove b i from the set of bids (ensure bid independence/truthfulness) Remove b i from the set of bids (ensure bid independence/truthfulness) Determine the threshold price at which maximal revenue is attained in the remaining set of bids Determine the threshold price at which maximal revenue is attained in the remaining set of bids If b i >= this threshold, accept b i at the threshold price If b i >= this threshold, accept b i at the threshold price Note that this removal of bi is the only thing that differentiates this auction from optimal fixed pricing Note that this removal of bi is the only thing that differentiates this auction from optimal fixed pricing

Deterministic Optimal Threshold Auction Example Step 1: Remove the bid to be evaluated: 3 Step 2: Compute the optimal threshold on the remaining bids: Sell1 at 5: 5 3 at 3: 9 5 at 2: 10 6 at 1: 6 Optimal threshold is 2 Step 3: Compare the removed bid to the optimal threshold 3 > 2 Step 4: Accept bid 3 at price 2

Truthful Deterministic Auctions Theoretical Results Truthful Deterministic Bid-Independent Auctions are not Competitive in the worst case Truthful Deterministic Bid-Independent Auctions are not Competitive in the worst case Removing b i causes bigger problems than one would expect Removing b i causes bigger problems than one would expect Consider an input set where the high bid h occurs r times, and there are (h – 1) (r – 1) other bids at 1 Consider an input set where the high bid h occurs r times, and there are (h – 1) (r – 1) other bids at 1

Proof Sketch Example: h = 5, r = 3 Example: h = 5, r = Sell… 11 at price 1 (11 revenue) 3 at price 5 (15 revenue)

Proof Sketch, Continued Deterministic Optimal Threshold Auction on this input: Deterministic Optimal Threshold Auction on this input: To determine if b1 wins, remove it and compute the threshold on the rest of the input To determine if b1 wins, remove it and compute the threshold on the rest of the input Sell 2 units at 5, or 10 units at 1 Threshold falls to 1, so end up selling to the high bids at the low price Threshold falls to 1, so end up selling to the high bids at the low price

More Results This result generalizes to all Deterministic Auctions due to the theorem that all Truthful Deterministic Auctions are Bid Independent This result generalizes to all Deterministic Auctions due to the theorem that all Truthful Deterministic Auctions are Bid Independent Proof intuition: Proof intuition: Can’t profit by bidding utility if you’ll pay it upon winning Can’t profit by bidding utility if you’ll pay it upon winning Paper goes on to present empirical evidence that on realistic data, the worst case for these auctions doesn’t come up often Paper goes on to present empirical evidence that on realistic data, the worst case for these auctions doesn’t come up often

Random Sampling Auctions Motivation Auction mechanisms need not only be resistant to bad inputs, but also to attack Auction mechanisms need not only be resistant to bad inputs, but also to attack By using some nondeterminism in deciding who wins and at what price, we can avoid dominance by worst case inputs By using some nondeterminism in deciding who wins and at what price, we can avoid dominance by worst case inputs

Random Sampling Auctions Choose a random sample from the set of bids Choose a random sample from the set of bids Run the optimal threshold function on the sample and use the result on the bids not in the sample Run the optimal threshold function on the sample and use the result on the bids not in the sample Single price Single price Nondeterministic Nondeterministic Dual price variant: Dual price variant: Choose roughly half of the bids for the sample Choose roughly half of the bids for the sample Calculate thresholds on both sets Calculate thresholds on both sets Use the threshold from one set on the other, and vice versa Use the threshold from one set on the other, and vice versa Avoids having to throw out bids from the sample Avoids having to throw out bids from the sample

Random Sampling Example Step 1: Choose subset at random: Step 2: Compute optimal threshold: 322 Sell 1 at 3: 3 Sell 3 at 2: 6 Optimal threshold is 2 Step 3: Apply optimal threshold to those not in the sample: 3315 Step 4: Accept 335 at price 2

Random Sampling Auctions Theoretical Results Random sampling auctions are competitive Random sampling auctions are competitive Revenue generated is within a constant factor of optimal fixed pricing with arbitrarily high (but not 1.0) probability Revenue generated is within a constant factor of optimal fixed pricing with arbitrarily high (but not 1.0) probability The higher the probability, the lower the constant The higher the probability, the lower the constant General flavor of the proof: General flavor of the proof: The chosen subset is a good representation of the whole most of the time The chosen subset is a good representation of the whole most of the time The revenue lost from losing the sampled bids is a constant factor of the whole The revenue lost from losing the sampled bids is a constant factor of the whole The dual price version performs even better The dual price version performs even better

Weighted Pairing Auctions To determine if a particular bidder i wins with bid bi: To determine if a particular bidder i wins with bid bi: Choose another bid b with probability proportional to its value –higher bids get picked more often Choose another bid b with probability proportional to its value –higher bids get picked more often Compare b with bi. If b i >= b, bidder i wins and pays b Compare b with bi. If b i >= b, bidder i wins and pays b Multi-price Multi-price Nondeterministic Nondeterministic High bidders likely to win and pay high prices, but some low bidders sneak in as well High bidders likely to win and pay high prices, but some low bidders sneak in as well

Weighted Pairing Example Step 1: Choose a bid 3 Step 2: Choose another bid with probability proportional to its value 2 Step 3: Compare 3 > Step 4: Bid 3 wins and pays price 2 2

Weighted Pairing Auctions Theoretical Results Weighted Pairing auctions are not quite competitive Weighted Pairing auctions are not quite competitive Within a logarithmic factor of fixed pricing, so not bad either Within a logarithmic factor of fixed pricing, so not bad either Perform well on inputs that random sampling does not Perform well on inputs that random sampling does not