Introduction to Auctions Protocols Revenue and Optimal Auctions Common Value Auctions

Auctions

Kate Larson

Introduction

Auction Protocols Common Auction Introduction to Auctions Protocols Revenue and Optimal Auctions Common Value Auctions

Vulnerabilities Kate Larson in Auctions

Beyond Single Cheriton School of Computer Science Item Auctions University of Waterloo Summary Outline

Auctions

Kate Larson 1 Introduction Introduction

Auction Protocols 2 Auction Protocols Common Auction Protocols Common Auction Protocols Revenue and Optimal Auctions Common Value Revenue and Optimal Auctions Auctions Common Value Auctions Vulnerabilities in Auctions Beyond Single 3 Vulnerabilities in Auctions Item Auctions

Summary 4 Beyond Single Item Auctions

5 Summary Auctions

Auctions Kate Larson Methods for allocating goods, tasks, resources,... Introduction Participants Auction auctioneer Protocols Common Auction bidders Protocols Revenue and Optimal Auctions Enforced agreement between auctioneer and the Common Value Auctions winning bidder(s) Vulnerabilities in Auctions Easily implementable (e.g. over the Internet) Beyond Single Conventions Item Auctions

Summary Auction: one seller and multiple buyers Reverse auction: one buyer and multiple sellers Todays lecture will discuss the theory in the context of auctions, but this applies to reverce auctions as well (at least in 1-item settings). Auction Settings

Auctions

Kate Larson

Introduction Private value: the value of the good depends only on

Auction the agent’s own preferences Protocols e.g a cake that is not resold of showed off Common Auction Protocols Revenue and Common value: an agent’s value of an item is Optimal Auctions Common Value determined entirely by others’ values (valuation of the Auctions item is identical for all agents) Vulnerabilities in Auctions e.g. treasury bills Beyond Single Item Auctions Correlated value (interdependent value): agent’s

Summary value for an item dpends partly on its own preferences and partly on others’ value for it e.g. auctioning a transportation task when bidders can handle it or reauction it to others Outline

Auctions

Kate Larson 1 Introduction Introduction

Summary 4 Beyond Single Item Auctions

5 Summary Four Common Auctions

Auctions

Kate Larson

Introduction

Auction Protocols Common Auction Protocols English auction Revenue and Optimal Auctions Common Value First-price, sealed-bid auction Auctions Vulnerabilities Dutch auction in Auctions

Beyond Single Vickrey auction Item Auctions

Summary English auction aka ﬁrst-price open-cry auction

Auctions

Kate Larson

Introduction Protocol: Each bidder is free to raise their bid. When

Auction no bidder is willing to raise, the auction ends and the Protocols Common Auction highest bidder wins. Highest bidder pays its last bid. Protocols Revenue and Optimal Auctions Strategy: Series of bids as a function of agent’s private Common Value Auctions value, prior estimates of others’ valuations, and past Vulnerabilities bids in Auctions

Beyond Single Best strategy: Item Auctions Variations: Summary Auctioneer controls the rate of increase Open-exit: Bidders have to openly declare exit with no re-entering possibilities First-price sealed-bid auction

Auctions

Kate Larson

Introduction

Auction Protocols Common Auction Protocol: Each bidder submits one bid without Protocols Revenue and knowing others’ bids. The highest bidder wins the item Optimal Auctions Common Value at the price of it’s bid Auctions Vulnerabilities Strategy: Bid as a function of agent’s private value and in Auctions its prior estimates of others’ valuations Beyond Single Item Auctions Best strategy: Summary Example

Auctions

Kate Larson Assume there are 2 agents (1 and 2) with values v1, v2 Introduction drawn uniformly from [0, 1]. Utility of agent i if it bids bi and Auction Protocols wins is ui = vi − bi . Common Auction Protocols How should 1 bid? (i.e. what is b(v ) = z?). Revenue and 1 Optimal Auctions Common Value Auctions Z 2z 2 Vulnerabilities U1 = (v1 − z)dz = (v1 − z)2z = 2zv1 − 2z in Auctions z=0 Beyond Single Item Auctions Note: given z = b2(v2) = v2/2, 1 only wins if v2 < 2z

Summary Therefore, 2 arg max[2zv1 − 2z ] = v1/2 z

Similar arguement for agent 2, assuming b1(v1) = v1/2. Example

Auctions

Kate Larson

Introduction

Auction Protocols Common Auction Assume that there are 2 risk-neutral bidders, 1 and 2. Protocols Revenue and Optimal Auctions Agent 1 knows that 2’s value is 0 or 100 with equal Common Value Auctions probability Vulnerabilities in Auctions 1’s value of 400 is common knowledge Beyond Single Item Auctions What is a Nash equilibrium?

Summary Dutch auction Descending auction

Auctions

Kate Larson

Introduction

Auction Protocols Common Auction Protocols Protocol: Auctioneer continuously lowers the price Revenue and Optimal Auctions until a bidder takes the item at the current price Common Value Auctions Strategy: Bid as a function of agent’s private value and Vulnerabilities in Auctions prior estimates of others’ valuations Beyond Single Item Auctions Best strategy:

Summary Dutch (Aalsmeer) ﬂower auction

Auctions

Kate Larson

Introduction

Auction Protocols Common Auction Protocols Revenue and Optimal Auctions Common Value Auctions

Vulnerabilities in Auctions

Beyond Single Item Auctions

Summary Dutch (Aalsmeer) ﬂower auction

Auctions

Kate Larson

Introduction

Auction Protocols Common Auction Protocols Revenue and Optimal Auctions Common Value Auctions

Vulnerabilities in Auctions

Beyond Single Item Auctions

Summary Vickrey Auction aka Second price, sealed bid auction

Auctions

Kate Larson

Introduction

Auction Protocol: Each bidder submits one bid without Protocols Common Auction knowing the others’ bids. The highest bidder wins and Protocols Revenue and pays an amount equal to the second highest bid. Optimal Auctions Common Value Auctions Strategy: Bid as a function of agent’s private value and Vulnerabilities its prior estimates of others’ valuations. in Auctions

Beyond Single Best strategy: Item Auctions Widely advocated for computational multiagent systems Summary Old (Vickrey 1961) but not widely used by humans Results for Private Value Auctions

Auctions

Kate Larson

Introduction

Auction Protocols Dutch and ﬁrst-price sealed-bid auctions are Common Auction Protocols strategically equivalent Revenue and Optimal Auctions Common Value For risk neutral agents, Vickrey and English auctions Auctions are strategically equivalent Vulnerabilities in Auctions Dominant strategies Beyond Single Item Auctions All four auctions allocate item efﬁciently

Summary Assuming no reservation price for the auctioneer Outline

Auctions

Kate Larson 1 Introduction Introduction

Summary 4 Beyond Single Item Auctions

5 Summary Revenue

Auctions Theorem (Revenue Equivalence) Kate Larson Suppose that Introduction

Auction values are independently and identically distributed and Protocols Common Auction all bidders are risk neutral. Protocols Revenue and Optimal Auctions Then any symmetric and increasing equilibrium of any Common Value Auctions standard auction, such that the expected payment of a Vulnerabilities bidder with value zero is zero, yields the same expected in Auctions revenue. Beyond Single Item Auctions

Summary Revenue equivalence fails to hold if agents are not risk neutral. Risk averse bidders: Dutch, ﬁrst-price ≥ Vickrey, English Risk seeking bidders: Dutch, ﬁrst-price ≤ Vickrey, English Optimal Auctions

Auctions

Kate Larson

Introduction

Auction Protocols Common Auction Protocols Revenue and Optimal Auctions Common Value Auctions

Vulnerabilities in Auctions

Beyond Single Item Auctions

Summary Outline

Auctions

Kate Larson 1 Introduction Introduction

Summary 4 Beyond Single Item Auctions

5 Summary Common Value Auctions

Auctions

Kate Larson

Introduction In a common value auction, the item has some unknown Auction value and each agent has some partial information about Protocols Common Auction the value. Each agent i has signal Xi ∈ [0, ωi ]. The value V Protocols Revenue and of the item is Optimal Auctions Common Value Auctions V = v(X1,..., Xn) Vulnerabilities in Auctions Examples Beyond Single Item Auctions Art auctions and resale

Summary Construction companies effected by common events (e.g. weather) Oil drilling Common Value Auctions

Auctions

Kate Larson

Introduction

Auction Protocols Common Auction Protocols Revenue and At time of bidding the common value is unknown Optimal Auctions Common Value Auctions Bidders may have imperfect estimates about the value Vulnerabilities True value only observed after the auction has taken in Auctions place Beyond Single Item Auctions

Summary Winner’s Curse

Auctions

Kate Larson

Introduction

Auction Protocols No agent knows for sure the true value of the item Common Auction Protocols The winner is the agent who made the highest guess Revenue and Optimal Auctions Common Value If bidders all had “reasonable” information about the Auctions value, then the average of all guesses should be Vulnerabilities in Auctions correct

Beyond Single i.e. the winner has overbid! Item Auctions

Summary Agents should shade their bids downward (even in English and Vicrey auctions). Results for Non-Private Value Auctions

Auctions

Kate Larson

Introduction Dutch and ﬁrst-price sealed-bid are strategically Auction Protocols equivalent Common Auction Protocols Vickrey and English are not strategically equivalent Revenue and Optimal Auctions Common Value All four auctions are efﬁcient Auctions

Vulnerabilities in Auctions Theorem (Revenue Non-Equivalence) Beyond Single Item Auctions With more than 2 bidders, the expected revenues are not

Summary the same:

English ≥ Vickrey ≥ Dutch = ﬁrst-price sealed-bid Bidder Collusion

Auctions

Kate Larson

Introduction

Auction Example: v1 = 20 and vi = 18 for other bidders. Protocols Common Auction Collusive agreement for English auction: 1 bids 6 and Protocols Revenue and others bid 5. This is self-enforcing Optimal Auctions Common Value Auctions Collusive agreement for Vickrey auction: 1 bids 20 and Vulnerabilities others bid 5. This is self-enforcing in Auctions

Beyond Single In ﬁrst-price or Dutch auction, if 1 bids below 18, others Item Auctions are motivated to break the collusion Summary Need to identify coalition parties Misbehaving Auctioneers

Auctions

Kate Larson

Introduction Shill bidding is bidding to artiﬁcally increase an item’s Auction Protocols price. Common Auction Protocols In theory, only a problem in non-private value auctions Revenue and Optimal Auctions English and all-pay auctions are vulnerable Common Value Auctions Classic analysis ignores the possibility of shills Vulnerabilities Vickrey, ﬁrst-price, and Dutch are not vulnerable in Auctions

Beyond Single In Vickrey auction, auctioneer can overstate 2nd Item Auctions highest bid Summary Auctioneer can refuse to sell once the auction has closed Undesirable Information Revelation

Auctions

Kate Larson

Introduction

Auction Protocols Vickrey and English auctions reveal agents’ strategic Common Auction Protocols marginal cost information since truthful bidding is a Revenue and Optimal Auctions dominant strategy Common Value Auctions Observed problems with subcontractors Vulnerabilities in Auctions First-price and Dutch may not reveal this information as accurately Beyond Single Item Auctions No dominant strategy and bidding decisions depend on Summary beliefs of others Beyond Single Item Auctions

Auctions

Kate Larson

Introduction

Auction Protocols Common Auction Protocols Revenue and Optimal Auctions Common Value Multi-Item Auctions (Combinatorial Auctions) Auctions

Vulnerabilities Position Auctions in Auctions

Beyond Single Item Auctions

Summary Summary

Auctions

Kate Larson

Introduction

Auction Protocols Common Auction Auctions are nontrivial but often analyzable Protocols Revenue and Important to understand merits and limitations Optimal Auctions Common Value Unintuitive auctions may have better properties (i.e. Auctions

Vulnerabilities Vickrey auction) in Auctions Choice of a good auction depends on the setting in Beyond Single Item Auctions which the protocol is used Summary