An introduction to Auctions: Single Item Auctions Maria Serna Fall - - PowerPoint PPT Presentation

Context and Definitions Some classic auctions Analyzing auctions

An introduction to Auctions: Single Item Auctions

Maria Serna Fall 2016

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

1 Context and Definitions 2 Some classic auctions 3 Analyzing auctions

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Prices

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Prices

What is the right price for objects?

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Prices

What is the right price for objects? The government wants to sell a big building.

Buyer 1: willing to pay 200 Billion. Buyer 2: willing to pay 100 Billion. Buyer 3: does not need the building, but wants to buy it if he can resale it with a profit.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Prices

What is the right price for objects? The government wants to sell a big building.

Buyer 1: willing to pay 200 Billion. Buyer 2: willing to pay 100 Billion. Buyer 3: does not need the building, but wants to buy it if he can resale it with a profit.

But, all the values are private.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Prices

What is the right price for objects? The government wants to sell a big building.

Buyer 1: willing to pay 200 Billion. Buyer 2: willing to pay 100 Billion. Buyer 3: does not need the building, but wants to buy it if he can resale it with a profit.

But, all the values are private. Each value is known only to the potential buyers.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Prices

What is the right price for objects? The government wants to sell a big building.

Buyer 1: willing to pay 200 Billion. Buyer 2: willing to pay 100 Billion. Buyer 3: does not need the building, but wants to buy it if he can resale it with a profit.

But, all the values are private. Each value is known only to the potential buyers. You work for the government,

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Prices

What is the right price for objects? The government wants to sell a big building.

Buyer 1: willing to pay 200 Billion. Buyer 2: willing to pay 100 Billion. Buyer 3: does not need the building, but wants to buy it if he can resale it with a profit.

But, all the values are private. Each value is known only to the potential buyers. You work for the government, how to get the building worth?

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Prices

Even when the true preferences of the buyers are known it is not clear how to price the objects and how to allocate them.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Prices

Even when the true preferences of the buyers are known it is not clear how to price the objects and how to allocate them. But we don’t know the preferences . . .

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Prices

Even when the true preferences of the buyers are known it is not clear how to price the objects and how to allocate them. But we don’t know the preferences . . . Buyers will lie and manipulate to get better prices and better allocation.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Prices

Even when the true preferences of the buyers are known it is not clear how to price the objects and how to allocate them. But we don’t know the preferences . . . Buyers will lie and manipulate to get better prices and better allocation. How can the true preferences be revealed?

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Prices

Even when the true preferences of the buyers are known it is not clear how to price the objects and how to allocate them. But we don’t know the preferences . . . Buyers will lie and manipulate to get better prices and better allocation. How can the true preferences be revealed? At which cost?

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Prices and allocations: Auction theory

Auction theory is a sub-field of Mechanism Design.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Prices and allocations: Auction theory

Auction theory is a sub-field of Mechanism Design. Aim: Design and analyze the rules and properties of an auction.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Prices and allocations: Auction theory

Auction theory is a sub-field of Mechanism Design. Aim: Design and analyze the rules and properties of an auction. Goal:

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Prices and allocations: Auction theory

Auction theory is a sub-field of Mechanism Design. Aim: Design and analyze the rules and properties of an auction. Goal: Design an auction so that in equilibrium we get the results we want.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Prices and allocations: Auction theory

Auction theory is a sub-field of Mechanism Design. Aim: Design and analyze the rules and properties of an auction. Goal: Design an auction so that in equilibrium we get the results we want. As in Game theory we rely on rationality.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

What is an Auction?

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

What is an Auction?

An auction is a mechanism to allocate resources among a group of bidders.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

What is an Auction?

An auction is a mechanism to allocate resources among a group of bidders. An auction model includes three major parts:

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

What is an Auction?

An auction is a mechanism to allocate resources among a group of bidders. An auction model includes three major parts:

The set of possible resource allocations.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

What is an Auction?

An auction is a mechanism to allocate resources among a group of bidders. An auction model includes three major parts:

The set of possible resource allocations. The number (or portion) of goods of each type including legal

• r other restrictions on how the goods may be allocated.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

What is an Auction?

An auction is a mechanism to allocate resources among a group of bidders. An auction model includes three major parts:

The set of possible resource allocations. The number (or portion) of goods of each type including legal

• r other restrictions on how the goods may be allocated.

The values of those resource allocations to each participant.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

What is an Auction?

An auction is a mechanism to allocate resources among a group of bidders. An auction model includes three major parts:

The set of possible resource allocations. The number (or portion) of goods of each type including legal

• r other restrictions on how the goods may be allocated.

The values of those resource allocations to each participant. A procedure to determine who wins what and how much pays

• n the basis of the received information.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Single item auctions

For today’s lecture assume that we have a single item or good to sell. An auction is defined by three kinds of rules:

Rules for bidding. Rules for what information is revealed. Rules for clearing.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Rules for Bidding

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Rules for Bidding

Who can bid?, when?

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Rules for Bidding

Who can bid?, when? What is the form of a bid?

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Rules for Bidding

Who can bid?, when? What is the form of a bid? Set restrictions on offers, as a function of:

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Rules for Bidding

Who can bid?, when? What is the form of a bid? Set restrictions on offers, as a function of:

bidder’s own previous bid auction state (others’ bids) eligibility (e.g., budget constraints) expiration, withdrawal, replacement

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Rules for what information is revealed

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Rules for what information is revealed

Here is the strategic component!

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Rules for what information is revealed

Here is the strategic component! When to reveal? What information? To whom?

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Rules for clearing

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Rules for clearing

When to clear?

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Rules for clearing

When to clear?

At intervals? On each bid? After a period of inactivity? At a prefixed time?

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Rules for clearing

When to clear?

At intervals? On each bid? After a period of inactivity? At a prefixed time?

Allocation: Who gets the object?

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Prices Auctions Single item auctions

Rules for clearing

When to clear?

At intervals? On each bid? After a period of inactivity? At a prefixed time?

Allocation: Who gets the object? Payment: Who pays? How much?

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

1 Context and Definitions 2 Some classic auctions 3 Analyzing auctions

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

Open auction

The auctioneer and the bidders interact physically.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

English Auction

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

English Auction

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

English Auction

The auctioneer starts the bidding at some reservation price.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

English Auction

The auctioneer starts the bidding at some reservation price. The bidders then shout out ascending prices.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

English Auction

The auctioneer starts the bidding at some reservation price. The bidders then shout out ascending prices. Once bidders stop shouting, the highest bidder gets the good at the declared price.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

Japanese Auction

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

Japanese Auction

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

Japanese Auction

The auctioneer calls out ascending prices.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

Japanese Auction

The auctioneer calls out ascending prices. All bidders start out standing, when the price reaches a level that a bidder is not willing to pay, that bidder sits down.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

Japanese Auction

The auctioneer calls out ascending prices. All bidders start out standing, when the price reaches a level that a bidder is not willing to pay, that bidder sits down. Once a bidder sits down, they can’t get back up

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

Japanese Auction

The auctioneer calls out ascending prices. All bidders start out standing, when the price reaches a level that a bidder is not willing to pay, that bidder sits down. Once a bidder sits down, they can’t get back up The only action that a bidder may take is to drop out of the auction

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

Japanese Auction

The auctioneer calls out ascending prices. All bidders start out standing, when the price reaches a level that a bidder is not willing to pay, that bidder sits down. Once a bidder sits down, they can’t get back up The only action that a bidder may take is to drop out of the auction The last person standing gets the good at the last price shouted.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

Japanese Auction

The auctioneer calls out ascending prices. All bidders start out standing, when the price reaches a level that a bidder is not willing to pay, that bidder sits down. Once a bidder sits down, they can’t get back up The only action that a bidder may take is to drop out of the auction The last person standing gets the good at the last price shouted. analytically more tractable than English because jump bidding can’t occur.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

Dutch Auction

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

Sealed bid auctions

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

Sealed bid auctions

The auctioneer starts a clock at some high value; it descends

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

Sealed bid auctions

The auctioneer starts a clock at some high value; it descends At some point, a bidder shouts mine! and gets the good at the price shown on the clock.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

Dutch Auction

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

Sealed bid auctions

The auctioneer and the bidders do not interact physically. The bidders submit their bid privately to the auctioneer. The bidder on the basis of the bid sets allocation and price.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

First price (FP) Auction

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

First price (FP) Auction

The bidders write down a price and send it to the auctioneer.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

First price (FP) Auction

The bidders write down a price and send it to the auctioneer. The auctioneer awards the good to the bidder with the highest bid.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

First price (FP) Auction

The bidders write down a price and send it to the auctioneer. The auctioneer awards the good to the bidder with the highest bid. The winner pays the amount of his bid.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

Second price (SP) Auction

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

Second price (SP) Auction

The bidders write down a price and send it to the auctioneer.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

Second price (SP) Auction

The bidders write down a price and send it to the auctioneer. The auctioneer awards the good to the bidder with the highest bid.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

Second price (SP) Auction

The bidders write down a price and send it to the auctioneer. The auctioneer awards the good to the bidder with the highest bid. The winner pays the amount bid by the second-highest bidder.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

Second price (SP) Auction

The bidders write down a price and send it to the auctioneer. The auctioneer awards the good to the bidder with the highest bid. The winner pays the amount bid by the second-highest bidder. Second price auctions are also known as Vickrey auctions. Were defined by William Vickrey in 1961. Who won the Nobel prize in economics in 1996.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

All-Pay Auction

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

All-Pay Auction

The bidders write down a price and send it to the auctioneer.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

All-Pay Auction

The bidders write down a price and send it to the auctioneer. The auctioneer awards the good to the bidder with the highest bid.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Open auctions Sealed-Bid auctions

All-Pay Auction

The bidders write down a price and send it to the auctioneer. The auctioneer awards the good to the bidder with the highest bid. Everyone pays the amount of their bid regardless of whether

• r not they get the good.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

1 Context and Definitions 2 Some classic auctions 3 Analyzing auctions

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

Auctioneer goals

A seller (“auctioneer”) may have several goals. Revenue: maximize profit.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

Auctioneer goals

A seller (“auctioneer”) may have several goals. Revenue: maximize profit. Efficiency: maximize social welfare:

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

Auctioneer goals

A seller (“auctioneer”) may have several goals. Revenue: maximize profit. Efficiency: maximize social welfare: Give the item to the buyer that wants it the most. (regardless

• f payments.)

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

Auctioneer goals

A seller (“auctioneer”) may have several goals. Revenue: maximize profit. Efficiency: maximize social welfare: Give the item to the buyer that wants it the most. (regardless

• f payments.)

Fairness: An auction that greedily maximizes the total utility of all users (i.e., the social welfare) in each round could lead to a subset

• f secondary users starving for products.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

Bidder goal

Auctions are used precisely because the seller is unsure about the values that bidders attach to the object being sold. A valuation is the maximum amount each bidder is willing to pay.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

Bidder goal

Auctions are used precisely because the seller is unsure about the values that bidders attach to the object being sold. A valuation is the maximum amount each bidder is willing to pay. If each bidder knows the value of the object to himself at the time of bidding, the situation is called one of private values.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

Bidder goal

Auctions are used precisely because the seller is unsure about the values that bidders attach to the object being sold. A valuation is the maximum amount each bidder is willing to pay. If each bidder knows the value of the object to himself at the time of bidding, the situation is called one of private values. No bidder knows with certainty the valuations of other bidders and knowledge of other bidders’ values would not affect how much the object is worth to a particular bidder.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

Bidder goal

Auctions are used precisely because the seller is unsure about the values that bidders attach to the object being sold. A valuation is the maximum amount each bidder is willing to pay. If each bidder knows the value of the object to himself at the time of bidding, the situation is called one of private values. No bidder knows with certainty the valuations of other bidders and knowledge of other bidders’ values would not affect how much the object is worth to a particular bidder. A bidder wants to get the object as cheap as possible.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

Strategic equivalence

Definition Two games with the same set of players and the same strategy space are said to be strategically equivalent if each player’s expected profits under one of the games are identical to his expected profits in the other game.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

SP-Auctions: strategic equivalences

Lemma SP-auctions are equivalent to Japanese auctions.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

SP-Auctions: strategic equivalences

Lemma SP-auctions are equivalent to Japanese auctions. Given that bidders bid truthfully, the outcomes in the two auctions are the same. Actually, in Japanese auctions bidders observe additional information: valuations of other players. This might create a herd phenomena.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

SP-Auctions: strategic equivalences

Lemma SP-auctions are equivalent to Japanese auctions. Given that bidders bid truthfully, the outcomes in the two auctions are the same. Actually, in Japanese auctions bidders observe additional information: valuations of other players. This might create a herd phenomena. But do bidders bid truthfully in SP-auctions?

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

SP-Auction: Modeling

n bidders Each bidder has value vi for the item willingness to pay. Known only to him – private value. If Bidder i wins and pays pi, his utility is vi–pi. Her utility is 0 when she loses.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

SP-Auction: Modeling

n bidders Each bidder has value vi for the item willingness to pay. Known only to him – private value. If Bidder i wins and pays pi, his utility is vi–pi. Her utility is 0 when she loses. Bidders prefer losing than paying more than their value.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

SP-Auction: Strategies

A strategy for each bidder:

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

SP-Auction: Strategies

A strategy for each bidder: how to bid given your value?

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

SP-Auction: Strategies

A strategy for each bidder: how to bid given your value? Examples for strategies:

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

SP-Auction: Strategies

A strategy for each bidder: how to bid given your value? Examples for strategies:

bi(vi) = vi truthful! bi(vi) = vi/2 bi(vi) = vi/n if you have information on the number of bidders. If vi < 50, bi(vi) = vi; otherwise, bi(vi) = vi + 17.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

SP-Auction: Strategies

A strategy for each bidder: how to bid given your value? Examples for strategies:

bi(vi) = vi truthful! bi(vi) = vi/2 bi(vi) = vi/n if you have information on the number of bidders. If vi < 50, bi(vi) = vi; otherwise, bi(vi) = vi + 17.

The auction is a strategic game, where these strategies are the pure strategies (infinitely many).

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

SP-Auctions: Equilibrium behaviour

Theorem In SP-price auctions truth-telling is a dominant strategy. In Japanese auctions with private values too.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

SP-Auctions: Equilibrium behaviour

Theorem In SP-price auctions truth-telling is a dominant strategy. In Japanese auctions with private values too. Let’s prove now that truthfulness is a dominant strategy. The proof is by case analysis. We have to show that Bidder 1 will never benefit from bidding a bid that is not v1.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

SP-Auctions: Equilibrium behaviour

Case 1: Bidder 1 wins when bidding v1.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

SP-Auctions: Equilibrium behaviour

Case 1: Bidder 1 wins when bidding v1. v1 is the highest bid and b2 is the 2nd highest. Bidder 1 utility is v1 − b2 > 0.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

SP-Auctions: Equilibrium behaviour

Case 1: Bidder 1 wins when bidding v1. v1 is the highest bid and b2 is the 2nd highest. Bidder 1 utility is v1 − b2 > 0. Bidding above b2 will not change anything. Bidding less than b2 will turn him into a loser. From positive utility to zero!

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

SP-Auctions: Equilibrium behaviour

Case 1: Bidder 1 wins when bidding v1. v1 is the highest bid and b2 is the 2nd highest. Bidder 1 utility is v1 − b2 > 0. Bidding above b2 will not change anything. Bidding less than b2 will turn him into a loser. From positive utility to zero! No gain from lying!.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

SP-Auctions: Equilibrium behaviour

Case 2: Bidder 1 loses when bidding v1.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

SP-Auctions: Equilibrium behaviour

Case 2: Bidder 1 loses when bidding v1. Let b2 be the 2nd highest bid now. Bidder 1 utility is 0.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

SP-Auctions: Equilibrium behaviour

Case 2: Bidder 1 loses when bidding v1. Let b2 be the 2nd highest bid now. Bidder 1 utility is 0. Any bid below b2 will gain him zero utility. Any bid above b2 will gain him either 0 (still not wining)

• r an utility smaller that v1 − b2 < 0 (losing is better).

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

SP-Auctions: Equilibrium behaviour

Case 2: Bidder 1 loses when bidding v1. Let b2 be the 2nd highest bid now. Bidder 1 utility is 0. Any bid below b2 will gain him zero utility. Any bid above b2 will gain him either 0 (still not wining)

• r an utility smaller that v1 − b2 < 0 (losing is better).

Again, no gain from lying!.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

SP-Auctions: Efficiency

Since SP-auction is truthful, we can conclude it is efficient. That is, in equilibrium,

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

SP-Auctions: Efficiency

Since SP-auction is truthful, we can conclude it is efficient. That is, in equilibrium, the auctioneer allocates the item to the bidder with the highest value.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

SP-Auctions: Efficiency

Since SP-auction is truthful, we can conclude it is efficient. That is, in equilibrium, the auctioneer allocates the item to the bidder with the highest value.

With the actual highest value, not just the highest bid. Without assuming anything on the values.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

FP-Auctions Strategic equivalences

Lemma FP-price auctions are strategically equivalent to Dutch auctions.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

FP-Auctions Strategic equivalences

Lemma FP-price auctions are strategically equivalent to Dutch auctions. Strategies: FP: Given that no one has a higher bid, what is the maximum I am willing to pay? Dutch: Given that no body has raised their hand, when should I raise mine?

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

FP-Auctions Strategic equivalences

Lemma FP-price auctions are strategically equivalent to Dutch auctions. Strategies: FP: Given that no one has a higher bid, what is the maximum I am willing to pay? Dutch: Given that no body has raised their hand, when should I raise mine? No new information is revealed during the auction!

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

FP-Auctions: properties

FP and Dutch auctions are Equivalent.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

FP-Auctions: properties

FP and Dutch auctions are Equivalent. Efficient? Yes, in equilibrium the good will be allocated to the player with a higher valuation. Truthful?

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

FP-Auctions: properties

FP and Dutch auctions are Equivalent. Efficient? Yes, in equilibrium the good will be allocated to the player with a higher valuation. Truthful? v1 = 100 and other’s highest bid b2 = 30.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

FP-Auctions: properties

FP and Dutch auctions are Equivalent. Efficient? Yes, in equilibrium the good will be allocated to the player with a higher valuation. Truthful? v1 = 100 and other’s highest bid b2 = 30. Player 1 by bidding 31

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

FP-Auctions: properties

FP and Dutch auctions are Equivalent. Efficient? Yes, in equilibrium the good will be allocated to the player with a higher valuation. Truthful? v1 = 100 and other’s highest bid b2 = 30. Player 1 by bidding 31 gets the good and a positive benefit.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

FP-Auctions: properties

FP and Dutch auctions are Equivalent. Efficient? Yes, in equilibrium the good will be allocated to the player with a higher valuation. Truthful? v1 = 100 and other’s highest bid b2 = 30. Player 1 by bidding 31 gets the good and a positive benefit. No truthfulness on the strategic setting.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

FP-Auctions: properties

FP and Dutch auctions are Equivalent. Efficient? Yes, in equilibrium the good will be allocated to the player with a higher valuation. Truthful? v1 = 100 and other’s highest bid b2 = 30. Player 1 by bidding 31 gets the good and a positive benefit. No truthfulness on the strategic setting. To analyze further we have to take into account the beliefs of the players on the preferences of the other players.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

FP-Auctions: properties

FP and Dutch auctions are Equivalent. Efficient? Yes, in equilibrium the good will be allocated to the player with a higher valuation. Truthful? v1 = 100 and other’s highest bid b2 = 30. Player 1 by bidding 31 gets the good and a positive benefit. No truthfulness on the strategic setting. To analyze further we have to take into account the beliefs of the players on the preferences of the other players. We take a detour on Bayesian games.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

Revenue

At equilibrium neither SP nor FP lead the maximum possible benefit to the seller.

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

Revenue

At equilibrium neither SP nor FP lead the maximum possible benefit to the seller. Can we get a better understanding of revenue?

AGT-MIRI Single item auctions

Context and Definitions Some classic auctions Analyzing auctions Goals SP Auctions FP Auctions

Revenue

At equilibrium neither SP nor FP lead the maximum possible benefit to the seller. Can we get a better understanding of revenue? Can we have a truthful auction giving maximum revenue?

AGT-MIRI Single item auctions