Valuation Uncertainty and Imperfect Introspection in Second-Price Auctions

Transcription

1 Valuation Uncertainty and Imperfect Introspection in Second-Price Auctions by David Robert Martin Thompson A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in The Faculty of Graduate Studies (Computer Science) The University of British Columbia July, 2007 c David Robert Martin Thompson 2007

2 ii Abstract In auction theory, agents are typically presumed to have perfect knowledge of their valuations. In practice, though, they may face barriers to this knowledge due to transaction costs or bounded rationality. Modeling and analyzing such settings has been the focus of much recent work, though a canonical model of such domains has not yet emerged. We begin by proposing a taxonomy of auction models with valuation uncertainty and showing how it categorizes previous work. We then restrict ourselves to single-good sealed-bid auctions, in which agents have (uncertain) independent private values and can introspect about their own (but not others ) valuations through possibly costly and imperfect queries. We investigate second-price auctions, performing equilibrium analysis for cases with both discrete and continuous valuation distributions. We identify cases where every equilibrium involves either randomized or asymmetric introspection. We contrast the revenue properties of different equilibria, discuss steps the seller can take to improve revenue, and identify a form of revenue equivalence across mechanisms. A version of this document has been accepted for publication. Thompson, D.R.M. and Leyton-Brown, K. (2007) Valuation Uncertainty and Imperfect Introspection in Second-Price Auctions. AAAI.

3 iii Table of Contents Abstract ii Table of Contents iii List of Figures v List of Tables vi Acknowledgements vii Dedication viii 1 Introduction Foundations: Decision Theory and Game Theory Decision Theory Normal Form Games Extensive Form Games Bayesian Games Foundations: Mechanism Design and Auction Theory Social Choice Mechanism Design Auction Theory Deliberative Agents Terminology Taxonomy Previous work Our Model Second price auctions Costly Introspection Limited Introspection Revenue relationships and bounds

4 Table of Contents iv 8 Conclusion Bibliography

5 v List of Figures 2.1 The probability of possible worlds in the umbrella game The utility of outcomes in the umbrella game Rock, Paper, Scissors as a normal form game The Prisoners Dilemma Two cars approaching a blind intersection as a normal form game The cake-cutting game Three-card Monte game The cake-cutting game in induced normal form Three-card Monte in induced normal form Stock market game Market spying game Stock market game in induced normal form Market spying game in induced normal form The revelation principle Induced valuation distribution of an introspection Induced normal-form game for two bidders who have valuations of 0 or Expected revenue under pure and mixed strategies Probability of ex post perfect efficiency under pure and mixed strategies Induced normal form of the 2-bidder limited deliberation case

6 vi List of Tables 4.1 The features of our taxonomy Classification of previous work on deliberative agents

7 vii Acknowledgments First and foremost, I am indebted to my supervisor, Kevin Leyton-Brown, for his teaching, inspired ideas and guidance, for pushing me when I would have stalled and for sticking around through the night for those early morning paper deadlines. My UBC professors and peers have taught me more than I ever thought I was capable of learning. Nando de Freitas (my second reader), Erik Zawadzki, Albert Xin Jiang, Baharak Rastegari and Jennifer Tillett all deserve special mention for their help with this work (and for listening to my same talk over and over again). Many people in the larger research community have given me support and constructive feedback, especially Jason Hartline, Michael Rothkopf, Eric Rasmusen, Kate Larson, Tuomas Sandholm and Liad Blumrosen. I am grateful to all of the professors and students at University of Guelph that helped to give me a start in A.I. and research: David Calvert, Deborah Stacey, Gary Grewal, Finnegan Southey and everyone else in GNCG and MPC 2. I want to thank all my friends and family, for their love and understanding, with special mention to everyone in and around Guelphtown, for making Vancouver feel like home: Tristram, Qixing, Dawn, McLean, Chau, Jason, Matt, Stacey, Sneha, Brook and Michelle. Finally, a big thank you to Fern, for being awesome.

8 viii Dedication To my father Bill, for all the years of unwavering, quiet support and to my mother Ruth, for the life-lessons that I ll always carry.

9 1 Chapter 1 Introduction Imagine deciding to bid for a particular car at an auction, and trying to determine what price you would be prepared to pay. You would probably start out very uncertain is the new car worth more than $10,000? $15,000? $18,526? You would probably have actions available to you that could increase your level of certainty, such as test driving the car, checking online reviews, and so on. But these actions would consume quite a lot of effort: pretty quickly you might feel compelled to make up your mind one way or the other, even if you were not yet entirely certain of your valuation. Similarly, imagine the problem of submitting a bid on behalf of your new startup company in an online advertisement auction. The value of a customer s click on your ad would not be obvious: at the moment of creating the ad you might not yet have accurate information about demographics, conversion rates, and so on. As before, it would be possible to collect additional information that would allow you to make a better decision; also as before, the cost of gathering such information would probably lead you to submit your bid even despite some residual uncertainty about your valuation. Both of these scenarios illustrate a fact that most canonical models of auctions fail to capture: bidders are often uncertain about their own valuations. While it is often possible for bidders to introspect, or otherwise gather information, in order to reduce this uncertainty, doing so is costly. Furthermore, such introspection is usually imperfect, in the sense that it does not entirely eliminate uncertainty. Bidders must be prepared to place bids despite residual uncertainty. In this paper, we explore the problem of building formal game-theoretic models to describe such settings. In what follows, we will begin by describing the essential concepts from game theory and auction theory that our results are based on. We define terminology and then offer a taxonomy of auction models in which bidders are uncertain about their valuations. We use this taxonomy to survey related work from the literature. We propose a novel auction model that aims to capture settings like those described above. Applying this model to the special case of second-price auctions, we offer several theoretical results, identifying equilibria and making revenue comparisons.

10 2 Chapter 2 Foundations: Decision Theory and Game Theory Decision theory and game theory are essential for discussing the behavior of agents in auctions. Decision theory is the study of how to make decisions optimally. Game theory extends this to strategic interactions (i.e. situations where there are multiple participants, each of which has an interest in and influence over the outcome). 2.1 Decision Theory Decision theory studies how to behave optimally in systems with only a single participant. We will describe these systems in the simplest terms possible while introducing the concepts of agents, utility, rationality, policy and value of information. Although there is no universal consensus on the exact definition of an agent, we will use a simple definition compatible with most work in game theory: Definition 1 (Agent) An agent (or player) is an entity that has preferences over the states of a system and can act to influence those states. Throughout decision theory and game theory, these preferences are defined in terms of von Neumann-Morgenstern utility theory. We will restrict ourselves to the case of finite decision systems (i.e. systems that are guaranteed to terminate and for which the end-state is a sufficient statistic for any sequence of intermediate states). Definition 2 (Outcome) An outcome is any end-state of the system. Definition 3 (Lottery) A lottery is a distribution over outcomes (and is itself an outcome for the purpose of preference modeling). Using outcomes and lotteries as our formalism for the events that the agent has preferences over, we can clarify how we model preferences: Definition 4 (Utility) Utility is a real-valued quantity measuring an agent s happiness. Definition 5 (Utility function) A utility function is a mapping from outcomes to utilities. The agent is indifferent between outcomes with equal utilities and strictly prefers outcomes with higher utilities.

11 Chapter 2. Foundations: Decision Theory and Game Theory 3 Weather Forecast Probability Rain Rain 0.4 Rain Clear 0.1 Clear Rain 0.1 Clear Clear 0.4 Figure 2.1: The probability of possible worlds in the umbrella game. Weather Umbrella Utility Rain True 2 Rain False 0 Clear True 4 Clear False 5 Figure 2.2: The utility of outcomes in the umbrella game. Von Neumann and Morgenstern showed that for any reasonable (subject to completeness, transitivity, substitutability, decomposability, monotonicity and continuity) set of preferences there is a utility function such that the agent s utility for any lottery is the expectation of their utility over the outcomes. Definition 6 (Rationality) A rational agent is an agent that acts to maximize its expected utility. Example 7 (Umbrella game) Consider the problem of whether to bring an umbrella out given the weather forecast. We can express the probability of possible worlds and the utility of different outcomes in tabular form. (See Figures 2.1 and 2.2.) The above example illustrates some important points about decision theory. The agent does not have absolute control or complete information. He must use the information available to respond to circumstances outside his control. Definition 8 (Policy) A policy is a mapping from available information to actions. Definition 9 (Optimal policy) An optimal policy is a policy that maximizes expected utility. Example 10 (Optimal policy of the umbrella game) The optimal policy of this game is to bring an umbrella only when the forecast calls for rain, which gives an expected utility of = 3.2. One other concept that is particularly relevant to our work is value of information. Definition 11 (Value of information) The value of a piece of information is the difference in expected utilities of the optimal policies with and without that piece of information.

12 Chapter 2. Foundations: Decision Theory and Game Theory 4 Example 12 (Umbrella game) The optimal policy given knowledge of the weather is to bring an umbrella when it is going to rain, which gives an expected utility of = 3.5. Thus, the value of information for the weather is = Normal Form Games Game theory extends decision theory to the case where there are multiple agents influencing the outcome. One fundamental assumption of game theory is that all agents are rational and aware of each other s rationality (and aware of each other s awareness of their rationality, and so on). Normal form games are a canonical representation of one of the simplest forms of interaction studied in game theory, where every agent acts simultaneously with no information about the other agents behaviors. Because every agent has an influence over the outcome, it is useful to talk about their behavior in aggregate: Definition 13 (Action profile) An action profile is a vector of actions, one per agent in the system. Definition 14 (Strategic equivalence) Two games are strategically equivalent if there is a mapping between pure strategies in each game where for any strategy profile in either game, there is a corresponding strategy profile in the other game that yields exactly the same expected utility for every player. Because of strategic equivalence, when analyzing games we frequently omit the outcomes and map directly from action profiles to utilities: Definition 15 (Game) A game is a system that has a set of agents, each of which has a set of actions and a utility function over action profiles. Definition 16 (Normal form game) A normal form game is a tabular representation of a game. In the two-dimensional case (i.e. for two agents) the first agent choses the row while the second choses the column. Each cell contains a vector of utility values, one per player. Two classic examples of normal form games are Rock, Paper, Scissors and the Prisoner s Dilemma: Example 17 (Rock, Paper, Scissors) Consider the game of rock, paper, scissors. Both players simultaneously choose rock, paper or scissors. Rock beats scissors, scissors beat paper and paper beats rock. The normal form game is in Figure 2.3. Example 18 (Prisoner s Dilemma) Two prisoners are being questioned separately for the same crime. If they cooperate with each other and say nothing to the police, each will receive a short sentence. If one criminal turns in the other, he will be set free (while the other gets the worst possible sentence). If they both turn in each other, they will each receive a moderate sentence. (See Figure 2.4.)

13 Chapter 2. Foundations: Decision Theory and Game Theory 5 Rock Paper Scissors Rock 0,0-1,1 1,-1 Paper 1,-1 0,0-1,1 Scissors -1,1 1,-1 0,0 Figure 2.3: Rock, Paper, Scissors as a normal form game Cooperate Defect Cooperate 3,3 0,5 Defect 5,0 1,1 Figure 2.4: The Prisoners Dilemma Intuitively, we would not want to deterministically play one action in rock, paper, scissors. We capture a richer set of options by allowing for strategies with an element of randomness. Definition 19 (Pure strategy (normal form game)) A pure strategy for a normal form game is a single action. Definition 20 (Mixed strategy) A mixed strategy is a probability distribution over pure strategies. Definition 21 (Support) The support of a mixed strategy is the set of pure strategies that have non-zero probability. As with actions, it is useful to talk about agents strategies in aggregate. Definition 22 (Strategy profile) A strategy profile is a vector of strategies, one per agent. Unlike decision theory, there is not necessarily an optimal policy. Whether or not an agent maximizes his utility by playing a particular strategy will depend on the behavior of the other agents. Definition 23 (Best response) A best response is a strategy (pure or mixed) which maximizes the agent s expected utility given the strategies of all the other agents.

14 Chapter 2. Foundations: Decision Theory and Game Theory 6 Best response can be weak or strict, depending on whether the expected utility is weakly or strictly maximized. (A mixed strategy cannot be a strict best response because it cannot be better than any action in its support.) Definition 24 (Dominant strategy) A dominant strategy is a strategy that is a best response regardless of the strategies of the other players. A dominant strategy is strict if it is always strictly a best response, semi-strict if it is sometimes a strict best response and weak otherwise. With these concepts, we can finally define some solution concepts for normal form games. These are stable configurations of the game. Definition 25 (Nash equilibrium) A Nash Equilibrium is a strategy profile where every agent is playing a best response to the actions of the others. We can characterize a Nash equilibrium as being a pure-strategy or mixed-strategy equilibrium. It is also useful to distinguish equilibria in dominant strategies. Definition 26 (Finite game) A finite game is a game where the sets of agents and actions are finite. Theorem 27 (Nash (1950)) Every finite game has at least one Nash equilibrium. Example 28 (Nash equilibrium of Rock, Paper, Scissors) The unique Nash equilibrium of rock, paper scissors is to play a mixed strategy which assigns each action equal probability. Example 29 (Nash equilibrium of Prisoner s Dilemma) The unique Nash equilibrium of the prisoner s dilemma is for both agents to defect. Definition 30 (Symmetric game) A symmetric game is one where all the agents have identical action sets, and for any strategy profile exchanging the actions of any two agents exchanges their expected utilities. Nash showed that every symmetric game has a symmetric equilibrium (i.e. one where every agent chooses the same strategy), though it can also have asymmetric ones. Definition 31 (Correlated equilibrium) In a correlated equilibrium, each agent gets a signal that tells them which pure strategy to play (with these signals being correlated). Given the joint distribution over signals, each agent s pure strategy is a best response to the distribution over other agents pure strategies. Correlated equilibria allow agents to coordinate to asymmetric Nash equilibria and to arrive at different distributions of expected utility among the agents. Example 32 (Traffic game) Consider the problem of two cars approaching a blind intersection. Drivers that stop will be slightly delayed. However, if they both enter the intersection without stopping, they will collide. (See Figure 2.5.) The only symmetric Nash equilibrium of the traffic game is for each agent to wait with probability 10/11, which only gives them an expected utility of 0. By a correlated signal (traffic lights), they can mix between (Go,Wait) and (Wait,Go) outcomes with no risk of miscoordinating and colliding.

15 Chapter 2. Foundations: Decision Theory and Game Theory 7 Go Wait Go -10,-10 1,0 Wait 0,1 0,0 Figure 2.5: Two cars approaching a blind intersection as a normal form game , , , big small big 2 small big small (0.25,0.75) (0.75,0.25) (0.67,0.33) (0.33,0.67) Figure 2.6: The cake-cutting game 2.3 Extensive Form Games (0.5,0.5) (0.5,0.5) Extensive form games allow us to model sequential settings. In many real-world strategic interactions, agents get to observe each other s behavior over time and condition their strategies on previous observations. Some examples include chess, tic-tac-toe and go. Definition 33 (Perfect information extensive form game) Perfect information extensive form games are games in tree representation where every inner node represents a choice by a single agent, with the out-edges of the node being the possible actions. Each leaf node is an outcome labeled by a vector of utilities (one per agent). Example 34 (Cake-cutting) Consider the problem of two agents trying to share a cake. First one agent choses what ratio of pieces to cut the cake into, then the second agent can choose which piece to take. (See Figure 2.6.) We can further generalize extensive form games to cover settings where the agents have less than perfect observations of each other s behavior. Definition 35 (Imperfect information extensive form game) Imperfect information extensive form games are extensive form games where the agent cannot distinguish two or more choice nodes (those nodes are said to belong to an equivalence set). Example 36 ( Three-card Monte ) Consider the problem of two agents playing a card game. The first agent arranges three cards face down (one of which is the queen). The second agent wins if he can guess which card is the queen. (See Figure 2.7.)

16 Chapter 2. Foundations: Decision Theory and Game Theory 8 1 Chapter 3. Foundations: l Decision Theory and Game Theory r 10 m l r l m 2 r l m l r m r m l r l r l r m m m ( 1,1) (1, 1) (1, 1) (1, 1) ( 1,1) (1, 1) (1, 1) ( 1,1) (1, 1) (1, 1) (1, 1) ( 1,1) (1, 1) (1, 1) (1, 1) ( 1,1) Figure 2.7: Three-card Monte game Figure 3.7: Three-card Monte game (1, 1) ( 1,1) b, b, b b, b, s b, s, b b, s, s s, b, b s, b, s s, s, b s, s, s 0.25, , , , , , , , , , , , , , , , , , , , , , , , , , ,0.5 Figure 3.8: The cake-cutting game in induced normal form Figure 2.8: The cake-cutting game in induced normal form Definition 36 (Pure strategy (extensive form game)) A pure strategy for an extensive form game is a mapping from nodes or equivalence sets to actions (analogous to a policy in decision theory.) Definition 37 (Pure strategy (extensive form game)) A pure strategy for an extensive form game is a mapping from nodes or equivalence sets to actions (analogous to a policy in decision theory.) Definition 37 (Induced normal form (extensive form game)) We can convert an extensive form game into a normal form game (called the induced normal form ) where the normal form game s actions are pure strategies in the extensive form game. The payoffs in the normal form game are taken from the leaf-node reached by following those strategies. Definition 38 (Induced normal form (extensive form game)) We can convert an extensive game, it form must have game a Nash into equilibrium a normal (by Theorem form game 26). The (called inducedthe normal induced forms of normal form ) where Because every extensive form game has a corresponding induced normal form the normal cake-cutting form gamegame s and Three-card actions Monteare in pure Figures strategies 3.3 and 3.3 respectively. in the extensive form game. The payoffs in the normal form game are taken from the leaf-node reached by following those 3.4 strategies. Bayesian Games Often, strategic interactions involve some notion of random events and private information. We model these using Bayesian games. Many card-games (for example, poker) are instances of this. Because every extensive form game has a corresponding induced normal form game, it must have a Nash equilibrium (by Theorem 27). The induced normal forms of the cake-cutting game and Three-card Monte are in Figures 2.8 and 2.9 respectively. Definition 38 (Type) An agent s type is a complete representation of their private information. 2.4 Bayesian Games Often, strategic interactions involve some notion of random events and private information. We model these using Bayesian games. Many card games (for example, poker) are instances of this type of interaction. Definition 39 (Type) An agent s type is a complete representation of his private information.

17 Chapter 2. Foundations: Decision Theory and Game Theory 9 l m r l -1,1 1,-1 1,-1 m 1,-1-1,1 1,-1 r -1,1-1,1-1,1 Figure 2.9: Three-card Monte in induced normal form Buy Sell Rise (p = 0.5) Buy 1,1 2,0 Sell 0,2-1,-1 Buy Sell Fall (p = 0.5) Buy -1,-1 0,2 Sell 2,0 1,1 Figure 2.10: Stock market game Definition 40 (Type profile) A type profile is a vector of types, one per agent. One simple but powerful way to represent Bayesian games is to say that the agents have different information about which normal form game they are actually playing. Definition 41 (Bayesian game) A Bayesian game is a set of players, each of which has a set of types, a set of actions and a utility function mapping action and type profiles to utility. A Bayesian game also includes a distribution over type profiles. Example 42 (Stock market game) Two investors must decide whether to sell off their stocks in a particular company or to buy more. Agent one has insider information and can correctly predict whether the price will rise or fall in the short term while agent two cannot. If both agents buy or sell it will have a significant, temporary impact on the stock s price. (See Figure 2.10.) As with decision theory and extensive form games, agents condition their behavior on the available information:

18 Chapter 2. Foundations: Decision Theory and Game Theory 10 Chapter 3. Foundations: Decision Theory and Game Theory 12 C Rise (p = 0.5) Fall (p = 0.5) 1 Buy Sell 1 Buy Sell Sell Buy Sell Buy Sell Buy Buy (1,1) (2,0) (0,2) ( 1, 1) Figure 3.11: Market spying game 2 Sell ( 1, 1) Figure 2.11: Market spying game Definition 42 (Pure strategy (Bayesian game)) A pure strategy in a Bayesian game is a function mapping from type to action. Definition 43 (Pure strategy (Bayesian game)) A pure strategy in a Bayesian game is a function mapping from type to action. An alternative way of representing Bayesian games is to model them as extensive form games where one of the agents is actually a random process (with all the real agents having full common knowledge of the probabilities): Definition An alternative 43 (Bayesianway game of (extensive representing form representation)) Bayesian games The extensive is to form model them as extensive representation of a Bayesian game augments a conventional imperfect information extensivegames form game form where with chance one ofnodes the agents where each is actually branch is chosen a random according process to (with all the real agents some (commonly having known) full common probability. knowledge of the probabilities): Extensive form representations of Bayesian games have the same pure strategy space as regular extensive form games. Definition 44 (Bayesian game (extensive form representation)) The extensive form representation of a Bayesian game augments a conventional imperfect information extensive form game with chance nodes where each branch is chosen according to some (commonly known) probability. Example 44 (Market spying game) Consider the same problem as the Market game, except that agent one cannot buy or sell stock without agent two finding out. (See Figure 3.4.) Extensive form representations of Bayesian games have the same pure strategy space as regular extensive form games. Definition 45 (Induced normal form (Bayesian game)) The induced normal form of a Bayesian game is a normal form game where each action is a pure strategy in the Bayesian game. The values in the cells of the table are filled in by computing the ex ante expected utility of each pure strategy profile. Example 45 (Market spying game) Consider the same problem as the Market game, except that agent one cannot buy or sell stock without agent two finding out. (See Figure 2.11.) When discussing the properties of Bayesian games, the amount of information available must be defined (particularly when taking expectations). We characterize this by the terms ex ante, ex interim and ex post. Definition 46 (Ex ante) Ex ante refers to situations where no agent s type is fixed or known. Definition 46 (Induced normal form (Bayesian game)) The induced normal form of adefinition Bayesian 47 (Ex game interim) is aex normal interim refers formtogame situations where whereeach exactlyaction one agent s is a pure strategy in the Bayesian type is known. game. The values in the cells of the table are filled in by computing the ex ante expected utility of each pure strategy profile. When discussing the properties of Bayesian games, the amount of information available must be defined (particularly when taking expectations). We characterize this by the terms ex ante, ex interim and ex post. Definition 47 (Ex ante) Ex ante refers to situations where no agent s type is fixed or known. Definition 48 (Ex interim) Ex interim refers to situations where exactly one agent s type is known. Definition 49 (Ex post) Ex ante refers to situations where all agent s types are known. (0,2) (2,0) (1,1)

19 Chapter 2. Foundations: Decision Theory and Game Theory 11 Buy Sell Buy, Buy 0,0 1,1 Buy, Sell 1.5, ,0.5 Sell, Buy -0.5, ,0.5 Sell, Sell 1,1 0,0 Figure 2.12: Stock market game in induced normal form Buy, Buy Buy, Sell Sell, Buy Sell, Sell Buy, Buy 0,0 0,0 1,1 1,1 Buy, Sell 1.5,0.5 1,1 2,0 1.5,0.5 Sell, Buy -0.5,-0.5-1,1 0,2-0.5,0.5 Sell, Sell 1,1 1,1 0,0 0,0 Figure 2.13: Market spying game in induced normal form

20 Chapter 2. Foundations: Decision Theory and Game Theory 12 Using these terms, we can discuss how strong an equilibrium is terms of available information: Definition 50 (Ex interim Bayes Nash equilibrium) An ex interim Bayes Nash equilibrium is a strategy profile where every agent s strategy maximizes his ex interim expected utility, for every type he can have. The Nash equilibrium of the induced normal form must be (at least) an ex interim Bayes Nash equilibrium in the corresponding Bayesian game. Because of this (and Theorem 27), at least an ex interim Bayes Nash equilibrium exists in any game. Definition 51 (Ex post Bayes Nash equilibrium) An ex post Bayes Nash equilibrium is a strategy profile where every agent s strategy maximizes his utility for any type profile. Ex post Bayes Nash equilibrium is a stronger condition than ex interim Bayes Nash equilibrium in that it requires that even if an agent were to learn the types of all the other agents, he would not prefer to change his strategy. Bayesian games can also have equilibria in dominant strategies.

21 13 Chapter 3 Foundations: Mechanism Design and Auction Theory 3.1 Social Choice Social choice is the study of preference aggregation: given that many agents have differing preferences over outcomes, how can we combine those preferences in a way that has desirable properties like fairness or optimality? Voting systems are the canonical example of this. Essentially, a social choice function picks an outcome given all the agents preferences. Definition 52 (Social choice function) A social choice function is a mapping from the preferences of a set of agents to an outcome. Although social choice is a rich field, social choice functions are the only concept we need to introduce from this field in order to discuss mechanism design. 3.2 Mechanism Design Mechanism design is the inverse problem to game theory. Rather than trying to analyze how agents will want to behave in a given system, mechanism design attempts to make systems where a desirable outcome is researched, given that agents will behave strategically and have types unknown to the mechanism designer. Definition 53 (Mechanism) A mechanism is a set of actions for each agent, and a mapping from action profiles to outcomes. A mechanism, combined with a set of agents with possible types and utility functions, and a distribution over type profiles, results in a Bayesian game. Definition 54 (Implementation) For every type profile, under equilibrium the agents will play some strategy profile, and given that strategy profile, the mechanism will choose the corresponding outcome. Thus, there is a mapping from type profile to outcome: a social choice function. We say that the mechanism implements this social choice function. We can characters the strength of an implementation, as Bayesian (ex interim), ex post or dominant strategies, depending on what kind of equilibria the induced Bayesian game has.

22 Chapter 3. Foundations: Mechanism Design and Auction Theory 14 Definition 55 (Mechanism design) Mechanism design is the study of how to design a mechanism that implements a desired social choice function. The space of possible mechanisms that can implement a given social choice function is extremely large. In mechanism design, we frequently ourselves to the space of direct, truthful mechanisms as this restriction is without loss of generality. Definition 56 (Direct) A mechanism is direct, if for every agent the set of actions is the set of types, i.e. the agents can directly reveal their types. Definition 57 (Truthful (or incentive compatible)) A direct mechanism is truthful, if in equilibrium, every agent prefers to reveal his true type. Theorem 58 (Revelation principle) If a social choice function can be implemented with a given strength, it can be implemented with that same strength by a direct truthful mechanism. Proof. Given any equilibrium of any mechanism, we can construct a set of proxies (as part of a new mechanism) that act on behalf of the agents. These proxies take their agent s type as an input and then follow the equilibrium strategy for that type. If some agent prefers to deviate from truthfully revealing his type, that implies that the original equilibrium was not actually an equilibrium. (See Figure 3.1.) There are many criteria that are desirable to meet when designing a mechanism. The designer frequently wants to maximize revenue or social welfare, or satisfy constraints such as budget balance and individual rationality. Definition 59 (Social welfare) Social welfare is the sum of all agents utilities for the outcome. Definition 60 ((Economic) efficiency) A mechanism is (economically) efficient if in equilibrium, social welfare is maximized. Definition 61 (Transferable utility) A setting supports transferable utility if there is some transferable, continuously-divisible commodity (for example, money) where for any transfers between a pair of agents, the sum of their utilities is preserved. Definition 62 (Revenue) Revenue is the amount of utility transfered to the mechanism. Definition 63 (Budget balance) A mechanism is budget balanced if expected revenue is zero, or weakly budget balanced if the expected revenue is weakly greater than zero. Budget balance can be ex ante or ex post, depending on how much information is known when computing expected revenue. Definition 64 (Individual rationality) A mechanism is individually rational if every agent s expected utility is weakly greater than zero. Individual rationality can be ex ante, ex interim or ex post, depending on how much information the agent has when computing his expected utility.

23 Chapter 3. Foundations: Mechanism Design and Auction Theory 15 Type Agent 1 Original Strategy 1: Action = f1(type) Action Original Mechanism Type Agent 2 Original Strategy 2: Action = f2(type) Action Direct Truthful Mechanism Agent 1 Proxy 1 Original Strategy 1: Type Type Strategy 1: Action Action = Type Action = f1(type) Original Mechanism Agent 2 Proxy 2 Original Strategy 2: Type Type Strategy 2: Action Action = Type Action = f2(type) Figure 3.1: The revelation principle

24 Chapter 3. Foundations: Mechanism Design and Auction Theory 16 Definition 65 (Groves mechanism) The Groves mechanism is a direct mechanism where an efficient outcome is chosen given the declared types of all the agents. Each agent also receives a transfer from the mechanism equal to the sum of the utilities of every other agent for the chosen outcome. Each agent can be charged a tax, but this tax cannot depend on his declared type. The Groves mechanism is efficient and dominant strategy truthful. However, it can involve the mechanism making large payouts. Definition 66 (Clarke tax) The Clarke tax is a tax that can be used in a Groves mechanism. With a Clarke tax, each agent must pay an amount equal to the social welfare generated by the mechanism if he had not been present, given the declared types of all other agents. The Clarke tax allows the Groves mechanism to be weakly budget balanced in cases where no agent has a negative utility for any outcome. The combination of a Groves mechanism with a Clarke tax is called the Vickrey-Clarke-Groves mechanism (or VCG), because it is a generalization of the Vickrey auction (see below in Definition 69). 3.3 Auction Theory Auction theory is the specialization of mechanism design to the problem of dividing up limited resources. In addition to studying conventional auctions, auction theory has been applied to problems like job markets, campaign finance and arms races. Throughout, we will make a number of assumptions that are common, but not universal in auction theory. First, we will assume that the auction is selling a single indivisible good. Second, we will assume that there are no externalities (i.e. no agent has any positive or negative utility for anything that happens to the other agents and that does not affect him directly). With these assumptions and the assumption of transferable utility, we can massively simplify our representation of an agents preferences. Definition 67 (Valuation) An agent s valuation is the maximum amount that he would be willing to pay for a good. Sealed-bid auctions are a special case of auctions where agents send a single private bid to the seller, and receive no information from the seller or the other agents. (Typically, an agent s type and bid can both be a single monetary value, making sealed bid auctions the equivalent of direct mechanisms.) This seems extremely limited, but there are a variety of sealed-bid auctions for single goods. Definition 68 (First-price auction) A first-price auction is a sealed-bid auction where the agent with the highest bid wins the good and must pay the amount he bid. Definition 69 (Vickrey (or second-price) auction) A Vickrey auction is a sealed-bid auction where the agent with the highest bid wins the good and must pay the amount of the second highest bid.

25 Chapter 3. Foundations: Mechanism Design and Auction Theory 17 Theorem 70 The Vickrey auction is truthful in dominant strategies. Proof. Let v be any agent s valuation and ˆv be his bid. Let b be the highest of the bids of the other agents. If v b, then any ˆv b yields v b utility (where v b 0), and any ˆv < b yields 0. Thus, ˆv = v is a best response. If v b, then any ˆv b yields v b utility (where v b 0), and any ˆv < b yields 0. Thus, ˆv = v is a best response. For any agent, for any b, ˆv = v is a (weakly) dominant strategy. Definition 71 (All-pay auction) An all-pay auction is a sealed-bid auction where the agent with the highest bid wins the good. Each agent pays the amount that he bid, regardless of whether or not he won. Intuitively, one would think that a seller would prefer a first-price or all-pay auction to a second-price as they get a higher revenue. However, this does not take into account the equilibrium bidding strategies of the agents. In fact, there is a theorem which proves that revenue is constant across these auctions for a reasonable set of assumptions: Theorem 72 (Revenue Equivalence) In any auction setting where all agents valuations are independent and identically distributed according to cumulative distribution F (v); the valuation distribution has bounded support (on [v, v]) and F (V ) is differentiable and increasing on that interval; there is an equilibrium that results in an efficient allocation; and any agent with the minimum possible valuation has an ex interim expected utility of zero in that equilibrium, any two auction types result in the same ex interim expected utility for every agent result in the same ex ante expected revenue for the seller under those equilibria. Proof. We can write an expression for the ex interim expected utility of an agent with valuation v bidding according to the equilibrium strategy for agents with valuation ˆv. Let γv be the ex interim probability of an agent with declared valuation v winning the good and p(v) be the ex interim expected payment and agent with declared valuation v must pay. u(v ˆv) = γ(ˆv)v p(ˆv) Because the agent is playing his equilibrium strategy, we know that that ˆv = v maximizes his expected utility: δu(v ˆv) δˆv = δ(γ(ˆv)v) δˆv δp(ˆv) δˆv = 0

26 Chapter 3. Foundations: Mechanism Design and Auction Theory 18 We can solve for the payment rule: p(v) = v v xγ(x)δx + c For any valuation distribution, and any fixed allocation rule there is a fixed γ(v). The above equation shows that for any fixed γ(v) in equilibrium, two auctions can only differ in p(v) by a constant. Since the auction is efficient, we have a fully specified γ(v): the probability of an agent with type v winning the good is the probability that all other n 1 agents have valuations less than v: γ(v) = F n 1 (v) Since an agent with v = v has no change of winning the auction, his payment must be fixed to zero (which implies that c = 0). Thus for any auction meeting these criteria, agents have the same ex interim expected payment and expected utility. Because the revenue is just the sum of payments, the ex ante expected revenue is also fixed.

27 19 Chapter 4 Deliberative Agents 4.1 Terminology It is common in auction theory to refer to an agent s type, meaning both the agent s private information (or signal) and the agent s valuation. Even in the classical literature there are settings in which this is not reasonable, such as common values: here agents all have the same valuation for the good, but they are uncertain about what it is and all have (potentially) different private signals. We will follow Bergemann and Morris (2006) in dividing type into belief type (private information) and payoff type (valuation). We must also be careful with the notions of ex ante, ex interim and ex post. Over the course of an auction, an agent might receive several signals, updating his beliefs after each. In such settings, it is not immediately clear which belief type ex interim should refer to. We will use ex interim to refer to all of them (so ex interim individually rational becomes a stronger condition: no sequence of signals should cause an agent to strictly prefer to drop out of the auction), while ex ante refers to taking an expectation with respect to an agent s beliefs given no private information. By ex post we refer to taking an expectation conditioned on all agents final belief types (so ex post efficiency means maximizing the expected social welfare). We will introduce ex interim perfect to refer to perfect knowledge of a single agent s valuation, and ex post perfect to refer to such knowledge of all agents valuations (so ex post perfect efficiency means maximizing the actual social welfare). Definition 73 (Deliberation) A deliberation is an action that updates an agent s beliefs. Because so-called deliberative agents must make decisions about how to deliberate and how to bid conditioned on the results of that deliberation, the underlying game is extensive-form. We distinguish two types of deliberation. Definition 74 (Introspection) An introspection is a deliberation where the signal provides information about the deliberator s valuation. Definition 75 (Strategic deliberation; Larson and Sandholm (2001b)) A strategic deliberation is a deliberation where the signal depends on another agent s valuation.

28 Chapter 4. Deliberative Agents 20 Parameter Valuation distribution: Secrecy: Perfection: Volatility: Costliness: Limitations: Separability: Possible values Independent, common, interdependent Secret, non-secret Perfect certainty, residual uncertainty Volatile, non-volatile Costly, free Limited, unlimited Separable, inseparable 4.2 Taxonomy Table 4.1: The features of our taxonomy Deliberating agents have been studied under a variety of assumptions, models and names. In order to survey the literature, we first introduce a unifying taxonomy with a number of free parameters (summarized in Table 4.1). Definition 76 (Valuation distribution) If all agents valuations are independent, we call the setting independent value. If every agent has the same valuation, we call the setting common value. We call all other cases interdependent value. Definition 77 (Secret) In secret settings, the only deliberations agents can perform are introspections. Notably, researchers have studied independent non-secret values (for example, Larson (2006)). Definition 78 (Perfection) A setting with perfect certainty requires agents to discover their exact valuation. Otherwise, the setting allows residual uncertainty. One example of a perfect setting is when agents must find a feasible solution to a constrained optimization problem (e.g., vehicle scheduling and routing), to discover how to use the good and thus its exact value. Without a feasible solution, the agent cannot consume the good. The seller can also impose perfection by requiring bidders to commit to such a solution while bidding. Definition 79 (Volatile) In volatile settings, deliberation can change an agent s underlying valuation. Like perfection, this can be motivated by the example of agents solving a constrained optimization problem. Feasible solutions tell the agent how to use the good and thus its value. By finding another solution, the agent might be able to update this value. Perfection is a special case of volatility where agents valuations are 0 until they perform a perfect introspection. Definition 80 (Costly) Performing costly deliberations costs an agent some utility. Costly settings involve costly deliberations.

29 Chapter 4. Deliberative Agents 21 We will model costs as another term in quasilinear utility: u i = v i (x) t i c i (q) where v i (x) is a value for outcome x, t i is a transfer from agent to seller and c i (q) is the cost of deliberation policy q. Costly deliberation is not equivalent to an entry fee because it is not transferred to the seller. Definition 81 (Limited) Limited settings have hard constraints about when any particular deliberation is possible. For example, agents might be limited as to how many deliberations are possible during a particular stage. Definition 82 (Separability) In separable settings, no deliberations are possible once the mechanism begins. (This is a special case of limited settings.) All sealed-bid mechanisms are trivially separable. Some staged auctions are also separable. (For example, in some auctions for antiques, bidders are only free to examine the goods until the English auction begins.) Because the problem of deliberation is equivalent whether the source of the information is internal or external, there is an equivalence between costly deliberation (with secrecy and non-volatility) and costly communication to a proxy. 4.3 Previous work The earliest work on modeling agents with valuation uncertainty is probably Wilson s (1967) work on common values. This model has been extensively studied since then, but primarily in settings in which agents are not deliberative. Another related literature considers tractably expressing bidder preferences in multi-good auctions (Ausubel & Milgrom, 2002; Blumrosen & Nisan, 2005; Nisan & Segal, 2005; Sandholm & Boutilier, 2006). We are specifically interested in deliberative agents. We will classify the existing work on this topic according to our taxonomy, though the classification is not always precise: some of the existing work spans and contrasts classes, and some results are more general than their authors state. To the extent that we are able, we classify work according to its most general results. A number of researchers have considered the problem of deliberative agents in settings requiring perfect certainty. Parkes (2005) stated that handling residual uncertainty appears beyond the scope of current methods (either analytic or computational), even for simple auctions such as an ascending-price auction for a single item. His paper focused on the dual problem to our own: auction and proxy design with costly communication. Parkes demonstrated that incremental revelation mechanisms can achieve the same allocations as direct mechanisms with lower communication costs. Cremer

30 Chapter 4. Deliberative Agents 22 Paper Values Secret Perfect Volatile Costly Limits Separable Cremer et al (2003) ind yes yes no yes no no Parkes (2005) all yes both no yes no no Larson & Sandholm (2005) ind no yes yes yes no yes Larson (2006) ind no yes no yes no no Larson & Sandholm (2001a) all no no yes no yes both Larson & Sandholm (2001b) all no no yes yes no both Blumrosen & Nisan (2002) ind yes no no no yes yes Bergemann & Valimaki (2002) all * no no yes no both Persico (2000) inter * no no yes yes yes Larson & Sandholm (2003) all no no yes yes yes yes Larson & Sandholm (2004) ind no no yes yes yes yes Compte & Jehiel (2001) ind yes no no yes no both Sandholm (2000) ind yes no no yes no yes Rasmusen (2006) ind yes no no yes yes no Our Paper ind yes no no yes yes yes Table 4.2: Classification of previous work on deliberative agents (Bold text indicates similar assumptions to our model. * indicates cases where values are secret but interdependent.)

31 Chapter 4. Deliberative Agents 23 et al. (2003) described how to extract full surplus from agents who commit to participating in the auction prior to deliberating, using a sequential auction based on optimal search. Larson and Sandholm (2005) proved that no (interesting) mechanism exists in which agents have no incentive to strategically deliberate or to mislead the seller and the mechanism does not depend on knowledge of agents deliberation costs and limits. The proof depends on costly deliberations, volatility (or non-volatile, perfect certainty), and non-secrecy. Larson (2006) described an optimal search auction in which agents never strategically deliberate or mislead the seller. Larson and Sandholm (2001a; 2001b) provided a very general model for costly and/or limited deliberations in auctions and showed that under costly deliberation models and in a number of common auction types (Vickrey, English, Dutch, first price and (for multiple goods) VCG) bidders perform strategic deliberation in equilibrium. There has been a small number of papers analyzing deliberative agents in settings that allow residual uncertainty. (Blumrosen & Nisan, 2002) showed that in auctions with severely limited communication (equivalent to severely limited deliberation), social welfare can be improved by using asymmetric proxies (equivalent to asymmetric deliberation strategies). Bergemann and Valimaki (2002) showed that in independent secret settings, VCG is ex post efficient and provides ex ante incentives to deliberate. For common value settings, they showed that no mechanism can achieve both properties. Larson and Sandholm (2003) demonstrated that in Vickrey auctions the ratio of computing costs between the social optimum and the worst case Nash equilibrium is unbounded, but that this can be improved by mechanism design. Larson and Sandholm (2004) described an experimental evaluation of deliberations on Vickrey auctions in non-secret settings. In simulation, no strategic deliberations occurred when agents had symmetric cost functions. Compte and Jehiel (2001) showed that sellers can get more revenue from Japanese-like ascending auctions than from second price auctions. Sandholm (2000) demonstrated that in second price auctions, bidding one s true expected valuation is a dominant strategy for risk neutral bidders but not for risk averse ones. He also demonstrated that second price auctions do not always have dominant deliberation strategies when deliberation is costly. Rasmusen (2006) presented residual uncertainty as a motivation for sniping on ebay auctions: buyers do not have time to deliberate in the last seconds of an auction, but do not deliberate earlier because of the cost. Persico (2000) demonstrated that when valuations are correlated (i.e. secrecy is not absolute), the value of information is greater in first price auctions than in second price auctions, because bidding strategies are conditioned on beliefs about opponents valuations.