Topics in Algorithmic Game Theory and Economics

(1)

Topics in Algorithmic Game Theory and Economics

Pieter Kleer

Max Planck Institute for Informatics (D1) Saarland Informatics Campus

January 13, 2020

Lecture 8

Some Mechanism Design

(2)

Mechanism Design

Mechanism design is a form of reversed game theory:

Given a (desired) outcome, how should we design the game to obtain that outcome as a result of strategic behaviour?

Examples: Auctions

Sponsored search auctions (e.g., Google) Online selling platforms (e.g., eBay) (Stable) matching problems

Matching children to schools

Matching medical students to hospitals Kidney exchange markets

We focus mostly on (online) auctions.

(3)

Mechanism Design

Mechanism design is a form of reversed game theory:

Given a (desired) outcome, how should we design the game to obtain that outcome as a result of strategic behaviour?

Examples: Auctions

Sponsored search auctions (e.g., Google) Online selling platforms (e.g., eBay) (Stable) matching problems

Matching children to schools

Matching medical students to hospitals Kidney exchange markets

We focus mostly on (online) auctions.

(4)

Mechanism Design

Mechanism design is a form of reversed game theory:

Given a (desired) outcome, how should we design the game to obtain that outcome as a result of strategic behaviour?

Examples:

Auctions

Sponsored search auctions (e.g., Google) Online selling platforms (e.g., eBay) (Stable) matching problems

Matching children to schools

Matching medical students to hospitals Kidney exchange markets

We focus mostly on (online) auctions.

(5)

Mechanism Design

Mechanism design is a form of reversed game theory:

Given a (desired) outcome, how should we design the game to obtain that outcome as a result of strategic behaviour?

Examples:

Auctions

Sponsored search auctions (e.g., Google) Online selling platforms (e.g., eBay) (Stable) matching problems

Matching children to schools

Matching medical students to hospitals Kidney exchange markets

We focus mostly on (online) auctions.

(6)

Mechanism Design

Mechanism design is a form of reversed game theory:

Given a (desired) outcome, how should we design the game to obtain that outcome as a result of strategic behaviour?

Examples:

Auctions

Online selling platforms (e.g., eBay) (Stable) matching problems

Matching children to schools

Matching medical students to hospitals Kidney exchange markets

We focus mostly on (online) auctions.

(7)

Mechanism Design

Mechanism design is a form of reversed game theory:

Given a (desired) outcome, how should we design the game to obtain that outcome as a result of strategic behaviour?

Examples:

Auctions

(Stable) matching problems Matching children to schools

Matching medical students to hospitals Kidney exchange markets

We focus mostly on (online) auctions.

(8)

Mechanism Design

Mechanism design is a form of reversed game theory:

Given a (desired) outcome, how should we design the game to obtain that outcome as a result of strategic behaviour?

Examples:

Auctions

Sponsored search auctions (e.g., Google) Online selling platforms (e.g., eBay) (Stable) matching problems

Matching children to schools

Matching medical students to hospitals Kidney exchange markets

We focus mostly on (online) auctions.

(9)

Mechanism Design

Mechanism design is a form of reversed game theory:

Given a (desired) outcome, how should we design the game to obtain that outcome as a result of strategic behaviour?

Examples:

Auctions

Sponsored search auctions (e.g., Google) Online selling platforms (e.g., eBay) (Stable) matching problems

Matching children to schools

Matching medical students to hospitals Kidney exchange markets

We focus mostly on (online) auctions.

(10)

Mechanism Design

Mechanism design is a form of reversed game theory:

Given a (desired) outcome, how should we design the game to obtain that outcome as a result of strategic behaviour?

Examples:

Auctions

Sponsored search auctions (e.g., Google) Online selling platforms (e.g., eBay) (Stable) matching problems

Matching children to schools

Matching medical students to hospitals

Kidney exchange markets

We focus mostly on (online) auctions.

(11)

Mechanism Design

Mechanism design is a form of reversed game theory:

Given a (desired) outcome, how should we design the game to obtain that outcome as a result of strategic behaviour?

Examples:

Auctions

Sponsored search auctions (e.g., Google) Online selling platforms (e.g., eBay) (Stable) matching problems

Matching children to schools

Matching medical students to hospitals Kidney exchange markets

We focus mostly on (online) auctions.

(12)

Mechanism Design

Mechanism design is a form of reversed game theory:

Given a (desired) outcome, how should we design the game to obtain that outcome as a result of strategic behaviour?

Examples:

Auctions

Sponsored search auctions (e.g., Google) Online selling platforms (e.g., eBay) (Stable) matching problems

Matching children to schools

Matching medical students to hospitals Kidney exchange markets

We focus mostly on (online) auctions.

(13)

Selling one item

(14)

Selling one item

Bidders:

Set of bidders {1, . . . , n} and one item. Bidder i has valuation v

_i

for the item.

Maximum amount she is willing to pay for it.

Private information: v

_i

not known to other players or seller. Bidder submits bid b

_i

.

Vector of all bids denoted by b = (b

1

, . . . , b

n

). Seller: Collects (sealed) bids.

Gives item to some bidder (if any).

Allocation rule x = x (b) = (x

1

, . . . , x

n

), with x

i

=

1 if i gets the item, 0 otherwise. Charges price of p to bidder i

^∗

receiving item.

Pricing rule p = p(b). Utility of bidder i:

u

_i

(b) = x

_i

(b)(v

_i

− p(b)) =

v

_i

− p(b) if i gets the item,

0 otherwise.

(15)

Selling one item

Bidders:

Set of bidders {1, . . . , n} and one item. Bidder i has valuation v

_i

for the item.

Maximum amount she is willing to pay for it.

Private information: v

_i

not known to other players or seller. Bidder submits bid b

_i

.

Vector of all bids denoted by b = (b

1

, . . . , b

n

). Seller: Collects (sealed) bids.

Gives item to some bidder (if any).

Allocation rule x = x (b) = (x

1

, . . . , x

n

), with x

i

=

1 if i gets the item, 0 otherwise. Charges price of p to bidder i

^∗

receiving item.

Pricing rule p = p(b). Utility of bidder i:

u

_i

(b) = x

_i

(b)(v

_i

− p(b)) =

v

_i

− p(b) if i gets the item,

0 otherwise.

(16)

Selling one item

Bidders:

Set of bidders {1, . . . , n} and one item.

Bidder i has valuation v

_i

for the item.

Maximum amount she is willing to pay for it.

Private information: v

_i

not known to other players or seller. Bidder submits bid b

_i

.

Vector of all bids denoted by b = (b

1

, . . . , b

n

). Seller: Collects (sealed) bids.

Gives item to some bidder (if any).

Allocation rule x = x (b) = (x

1

, . . . , x

n

), with x

i

=

1 if i gets the item, 0 otherwise. Charges price of p to bidder i

^∗

receiving item.

Pricing rule p = p(b). Utility of bidder i:

u

_i

(b) = x

_i

(b)(v

_i

− p(b)) =

v

_i

− p(b) if i gets the item,

0 otherwise.

(17)

Selling one item

Bidders:

Set of bidders {1, . . . , n} and one item.

Bidder i has valuation v

_i

for the item.

Maximum amount she is willing to pay for it.

Private information: v

_i

not known to other players or seller. Bidder submits bid b

_i

.

Vector of all bids denoted by b = (b

1

, . . . , b

n

). Seller: Collects (sealed) bids.

Gives item to some bidder (if any).

Allocation rule x = x (b) = (x

1

, . . . , x

n

), with x

i

=

1 if i gets the item, 0 otherwise. Charges price of p to bidder i

^∗

receiving item.

Pricing rule p = p(b). Utility of bidder i:

u

_i

(b) = x

_i

(b)(v

_i

− p(b)) =

v

_i

− p(b) if i gets the item,

0 otherwise.

(18)

Selling one item

Bidders:

Set of bidders {1, . . . , n} and one item.

Bidder i has valuation v

_i

for the item.

Maximum amount she is willing to pay for it.

Private information: v

_i

not known to other players or seller. Bidder submits bid b

_i

.

Vector of all bids denoted by b = (b

1

, . . . , b

n

). Seller: Collects (sealed) bids.

Gives item to some bidder (if any).

Allocation rule x = x (b) = (x

1

, . . . , x

n

), with x

i

=

1 if i gets the item, 0 otherwise. Charges price of p to bidder i

^∗

receiving item.

Pricing rule p = p(b). Utility of bidder i:

u

_i

(b) = x

_i

(b)(v

_i

− p(b)) =

v

_i

− p(b) if i gets the item,

0 otherwise.

(19)

Selling one item

Bidders:

Set of bidders {1, . . . , n} and one item.

Bidder i has valuation v

_i

for the item.

Maximum amount she is willing to pay for it.

Private information: v

_i

not known to other players or seller.

Bidder submits bid b

_i

.

Vector of all bids denoted by b = (b

1

, . . . , b

n

). Seller: Collects (sealed) bids.

Gives item to some bidder (if any).

Allocation rule x = x (b) = (x

1

, . . . , x

n

), with x

i

=

1 if i gets the item, 0 otherwise. Charges price of p to bidder i

^∗

receiving item.

Pricing rule p = p(b). Utility of bidder i:

u

_i

(b) = x

_i

(b)(v

_i

− p(b)) =

v

_i

− p(b) if i gets the item,

0 otherwise.

(20)

Selling one item

Bidders:

Set of bidders {1, . . . , n} and one item.

Bidder i has valuation v

_i

for the item.

Maximum amount she is willing to pay for it.

Private information: v

_i

not known to other players or seller.

Bidder submits bid b

_i

.

Vector of all bids denoted by b = (b

1

, . . . , b

n

). Seller: Collects (sealed) bids.

Gives item to some bidder (if any).

Allocation rule x = x (b) = (x

1

, . . . , x

n

), with x

i

=

1 if i gets the item, 0 otherwise. Charges price of p to bidder i

^∗

receiving item.

Pricing rule p = p(b). Utility of bidder i:

u

_i

(b) = x

_i

(b)(v

_i

− p(b)) =

v

_i

− p(b) if i gets the item,

0 otherwise.

(21)

Selling one item

Bidders:

Set of bidders {1, . . . , n} and one item.

Bidder i has valuation v

_i

for the item.

Maximum amount she is willing to pay for it.

Private information: v

_i

not known to other players or seller.

Bidder submits bid b

_i

.

Vector of all bids denoted by b = (b

1

, . . . , b

n

).

Seller: Collects (sealed) bids.

Gives item to some bidder (if any).

Allocation rule x = x (b) = (x

1

, . . . , x

n

), with x

i

=

1 if i gets the item, 0 otherwise. Charges price of p to bidder i

^∗

receiving item.

Pricing rule p = p(b). Utility of bidder i:

u

_i

(b) = x

_i

(b)(v

_i

− p(b)) =

v

_i

− p(b) if i gets the item,

0 otherwise.

(22)

Selling one item

Bidders:

Set of bidders {1, . . . , n} and one item.

Bidder i has valuation v

_i

for the item.

Maximum amount she is willing to pay for it.

Private information: v

_i

not known to other players or seller.

Bidder submits bid b

_i

.

Vector of all bids denoted by b = (b

1

, . . . , b

n

).

Seller:

Collects (sealed) bids.

Gives item to some bidder (if any).

Allocation rule x = x (b) = (x

1

, . . . , x

n

), with x

i

=

1 if i gets the item, 0 otherwise. Charges price of p to bidder i

^∗

receiving item.

Pricing rule p = p(b). Utility of bidder i:

u

_i

(b) = x

_i

(b)(v

_i

− p(b)) =

v

_i

− p(b) if i gets the item,

0 otherwise.

(23)

Selling one item

Bidders:

Set of bidders {1, . . . , n} and one item.

Bidder i has valuation v

_i

for the item.

Maximum amount she is willing to pay for it.

Private information: v

_i

not known to other players or seller.

Bidder submits bid b

_i

.

Vector of all bids denoted by b = (b

1

, . . . , b

n

).

Seller: Collects (sealed) bids.

Gives item to some bidder (if any).

Allocation rule x = x (b) = (x

1

, . . . , x

n

), with x

i

=

1 if i gets the item, 0 otherwise. Charges price of p to bidder i

^∗

receiving item.

Pricing rule p = p(b). Utility of bidder i:

u

_i

(b) = x

_i

(b)(v

_i

− p(b)) =

v

_i

− p(b) if i gets the item,

0 otherwise.

(24)

Selling one item

Bidders:

Set of bidders {1, . . . , n} and one item.

Bidder i has valuation v

_i

for the item.

Maximum amount she is willing to pay for it.

Private information: v

_i

not known to other players or seller.

Bidder submits bid b

_i

.

Vector of all bids denoted by b = (b

1

, . . . , b

n

).

Seller: Collects (sealed) bids.

Gives item to some bidder (if any).

Allocation rule x = x (b) = (x

1

, . . . , x

n

), with x

i

=

1 if i gets the item, 0 otherwise. Charges price of p to bidder i

^∗

receiving item.

Pricing rule p = p(b). Utility of bidder i:

u

_i

(b) = x

_i

(b)(v

_i

− p(b)) =

v

_i

− p(b) if i gets the item,

0 otherwise.

(25)

Selling one item

Bidders:

Set of bidders {1, . . . , n} and one item.

Bidder i has valuation v

_i

for the item.

Maximum amount she is willing to pay for it.

Private information: v

_i

not known to other players or seller.

Bidder submits bid b

_i

.

Vector of all bids denoted by b = (b

1

, . . . , b

n

).

Seller: Collects (sealed) bids.

Gives item to some bidder (if any).

Allocation rule x = x (b) = (x

1

, . . . , x

n

), with x

_i

=

1 if i gets the item, 0 otherwise.

Charges price of p to bidder i

^∗

receiving item. Pricing rule p = p(b).

Utility of bidder i:

u

_i

(b) = x

_i

(b)(v

_i

− p(b)) =

v

_i

− p(b) if i gets the item,

0 otherwise.

(26)

Selling one item

Bidders:

Set of bidders {1, . . . , n} and one item.

Bidder i has valuation v

_i

for the item.

Maximum amount she is willing to pay for it.

Private information: v

_i

not known to other players or seller.

Bidder submits bid b

_i

.

Vector of all bids denoted by b = (b

1

, . . . , b

n

).

Seller: Collects (sealed) bids.

Gives item to some bidder (if any).

Allocation rule x = x (b) = (x

1

, . . . , x

n

), with x

_i

=

1 if i gets the item, 0 otherwise.

Charges price of p to bidder i

^∗

receiving item.

Pricing rule p = p(b). Utility of bidder i:

u

_i

(b) = x

_i

(b)(v

_i

− p(b)) =

v

_i

− p(b) if i gets the item,

0 otherwise.

(27)

Selling one item

Bidders:

Set of bidders {1, . . . , n} and one item.

Bidder i has valuation v

_i

for the item.

Maximum amount she is willing to pay for it.

Private information: v

_i

not known to other players or seller.

Bidder submits bid b

_i

.

Vector of all bids denoted by b = (b

1

, . . . , b

n

).

Seller: Collects (sealed) bids.

Gives item to some bidder (if any).

Allocation rule x = x (b) = (x

1

, . . . , x

n

), with x

_i

=

1 if i gets the item, 0 otherwise.

Charges price of p to bidder i

^∗

receiving item.

Pricing rule p = p(b).

Utility of bidder i:

u

_i

(b) = x

_i

(b)(v

_i

− p(b)) =

v

_i

− p(b) if i gets the item,

0 otherwise.

(28)

Selling one item

Bidders:

Set of bidders {1, . . . , n} and one item.

Bidder i has valuation v

_i

for the item.

Maximum amount she is willing to pay for it.

Private information: v

_i

not known to other players or seller.

Bidder submits bid b

_i

.

Vector of all bids denoted by b = (b

1

, . . . , b

n

).

Seller: Collects (sealed) bids.

Gives item to some bidder (if any).

Allocation rule x = x (b) = (x

1

, . . . , x

n

), with x

_i

=

1 if i gets the item, 0 otherwise.

Charges price of p to bidder i

^∗

receiving item.

Pricing rule p = p(b).

Utility of bidder i:

u

_i

(b) = x

_i

(b)(v

_i

− p(b)) =

v

_i

− p(b) if i gets the item,

0 otherwise.

(29)

Selling one item

Bidders:

Set of bidders {1, . . . , n} and one item.

Bidder i has valuation v

_i

for the item.

Maximum amount she is willing to pay for it.

Private information: v

_i

not known to other players or seller.

Bidder submits bid b

_i

.

Vector of all bids denoted by b = (b

1

, . . . , b

n

).

Seller: Collects (sealed) bids.

Gives item to some bidder (if any).

Allocation rule x = x (b) = (x

1

, . . . , x

n

), with x

_i

=

1 if i gets the item, 0 otherwise.

Charges price of p to bidder i

^∗

receiving item.

Pricing rule p = p(b).

Utility of bidder i:

(30)

Selling one item

Bidders:

Set of bidders {1, . . . , n} and one item.

Bidder i has valuation v

_i

for the item.

Maximum amount she is willing to pay for it.

Private information: v

_i

not known to other players or seller.

Bidder submits bid b

_i

.

Vector of all bids denoted by b = (b

1

, . . . , b

n

).

Seller: Collects (sealed) bids.

Gives item to some bidder (if any).

Allocation rule x = x (b) = (x

1

, . . . , x

n

), with x

_i

=

1 if i gets the item, 0 otherwise.

Charges price of p to bidder i

^∗

receiving item.

Pricing rule p = p(b).

Utility of bidder i:

v − p(b) if i gets the item,

(31)

We have

Bidders with valuations v = (v

₁

, . . . , v

_n

) and bids b = (b

₁

, . . . , b

_n

). Seller with allocation rule x (b) and pricing rule p(b).

Utility of player given by u

_i

(b) = x

_i

(b)(v

_i

− p(b)). Revenue of seller is p if item is sold.

Definition

A (deterministic) mechanism (x , p) for selling an item to one of n bidders is given by an allocation rule x : R

ⁿ

→ {0, 1}

ⁿ

with P

i

x

_i

≤ 1, and pricing rule p : R

ⁿ

→ R .

Goal of bidder i is to maximize utility given mechanism (x , p). Bidders will try to bid strategically.

How should we design auction to prevent undesirable outcomes?

(32)

We have

Bidders with valuations v = (v

₁

, . . . , v

_n

) and

bids b = (b

₁

, . . . , b

_n

). Seller with allocation rule x (b) and pricing rule p(b).

Utility of player given by u

_i

(b) = x

_i

(b)(v

_i

− p(b)). Revenue of seller is p if item is sold.

Definition

A (deterministic) mechanism (x , p) for selling an item to one of n bidders is given by an allocation rule x : R

ⁿ

→ {0, 1}

ⁿ

with P

i

x

_i

≤ 1, and pricing rule p : R

ⁿ

→ R .

Goal of bidder i is to maximize utility given mechanism (x , p). Bidders will try to bid strategically.

How should we design auction to prevent undesirable outcomes?

(33)

We have

Bidders with valuations v = (v

₁

, . . . , v

_n

) and bids b = (b

₁

, . . . , b

_n

).

Seller with allocation rule x (b) and pricing rule p(b). Utility of player given by u

_i

(b) = x

_i

(b)(v

_i

− p(b)). Revenue of seller is p if item is sold.

Definition

A (deterministic) mechanism (x , p) for selling an item to one of n bidders is given by an allocation rule x : R

ⁿ

→ {0, 1}

ⁿ

with P

i

x

_i

≤ 1, and pricing rule p : R

ⁿ

→ R .

Goal of bidder i is to maximize utility given mechanism (x , p). Bidders will try to bid strategically.

How should we design auction to prevent undesirable outcomes?

(34)

We have

Bidders with valuations v = (v

₁

, . . . , v

_n

) and bids b = (b

₁

, . . . , b

_n

).

Seller with allocation rule x (b) and pricing rule p(b).

Utility of player given by u

_i

(b) = x

_i

(b)(v

_i

− p(b)). Revenue of seller is p if item is sold.

Definition

A (deterministic) mechanism (x , p) for selling an item to one of n bidders is given by an allocation rule x : R

ⁿ

→ {0, 1}

ⁿ

with P

i

x

_i

≤ 1, and pricing rule p : R

ⁿ

→ R .

Goal of bidder i is to maximize utility given mechanism (x , p). Bidders will try to bid strategically.

How should we design auction to prevent undesirable outcomes?

(35)

We have

Bidders with valuations v = (v

₁

, . . . , v

_n

) and bids b = (b

₁

, . . . , b

_n

).

Seller with allocation rule x (b) and pricing rule p(b).

Utility of player given by u

_i

(b) = x

_i

(b)(v

_i

− p(b)).

Revenue of seller is p if item is sold.

Definition

A (deterministic) mechanism (x , p) for selling an item to one of n bidders is given by an allocation rule x : R

ⁿ

→ {0, 1}

ⁿ

with P

i

x

_i

≤ 1, and pricing rule p : R

ⁿ

→ R .

Goal of bidder i is to maximize utility given mechanism (x , p). Bidders will try to bid strategically.

How should we design auction to prevent undesirable outcomes?

(36)

We have

Bidders with valuations v = (v

₁

, . . . , v

_n

) and bids b = (b

₁

, . . . , b

_n

).

Seller with allocation rule x (b) and pricing rule p(b).

Utility of player given by u

_i

(b) = x

_i

(b)(v

_i

− p(b)).

Revenue of seller is p if item is sold.

Definition

A (deterministic) mechanism (x , p) for selling an item to one of n bidders is given by an allocation rule x : R

ⁿ

→ {0, 1}

ⁿ

with P

i

x

_i

≤ 1, and pricing rule p : R

ⁿ

→ R .

Goal of bidder i is to maximize utility given mechanism (x , p). Bidders will try to bid strategically.

How should we design auction to prevent undesirable outcomes?

(37)

We have

Bidders with valuations v = (v

₁

, . . . , v

_n

) and bids b = (b

₁

, . . . , b

_n

).

Seller with allocation rule x (b) and pricing rule p(b).

Utility of player given by u

_i

(b) = x

_i

(b)(v

_i

− p(b)).

Revenue of seller is p if item is sold.

Definition

A (deterministic) mechanism (x , p) for selling an item to one of n bidders is given by an allocation rule x : R

ⁿ

→ {0, 1}

ⁿ

with P

i

x

_i

≤ 1, and pricing rule p : R

ⁿ

→ R .

Goal of bidder i is to maximize utility given mechanism (x , p). Bidders will try to bid strategically.

How should we design auction to prevent undesirable outcomes?

(38)

We have

Bidders with valuations v = (v

₁

, . . . , v

_n

) and bids b = (b

₁

, . . . , b

_n

).

Seller with allocation rule x (b) and pricing rule p(b).

Utility of player given by u

_i

(b) = x

_i

(b)(v

_i

− p(b)).

Revenue of seller is p if item is sold.

Definition

A (deterministic) mechanism (x , p) for selling an item to one of n bidders is given by an allocation rule x : R

ⁿ

→ {0, 1}

ⁿ

with P

i

x

_i

≤ 1, and pricing rule p : R

ⁿ

→ R .

Goal of bidder i is to maximize utility given mechanism (x , p). Bidders will try to bid strategically.

How should we design auction to prevent undesirable outcomes?

(39)

We have

Bidders with valuations v = (v

₁

, . . . , v

_n

) and bids b = (b

₁

, . . . , b

_n

).

Seller with allocation rule x (b) and pricing rule p(b).

Utility of player given by u

_i

(b) = x

_i

(b)(v

_i

− p(b)).

Revenue of seller is p if item is sold.

Definition

A (deterministic) mechanism (x , p) for selling an item to one of n bidders is given by an allocation rule x : R

ⁿ

→ {0, 1}

ⁿ

with P

i

x

_i

≤ 1, and pricing rule p : R

ⁿ

→ R .

Goal of bidder i is to maximize utility given mechanism (x , p).

Bidders will try to bid strategically.

How should we design auction to prevent undesirable outcomes?

(40)

We have

Bidders with valuations v = (v

₁

, . . . , v

_n

) and bids b = (b

₁

, . . . , b

_n

).

Seller with allocation rule x (b) and pricing rule p(b).

Utility of player given by u

_i

(b) = x

_i

(b)(v

_i

− p(b)).

Revenue of seller is p if item is sold.

Definition

A (deterministic) mechanism (x , p) for selling an item to one of n bidders is given by an allocation rule x : R

ⁿ

→ {0, 1}

ⁿ

with P

i

x

_i

≤ 1, and pricing rule p : R

ⁿ

→ R .

Goal of bidder i is to maximize utility given mechanism (x , p).

Bidders will try to bid strategically.

How should we design auction to prevent undesirable outcomes?

(41)

We have

Bidders with valuations v = (v

₁

, . . . , v

_n

) and bids b = (b

₁

, . . . , b

_n

).

Seller with allocation rule x (b) and pricing rule p(b).

Utility of player given by u

_i

(b) = x

_i

(b)(v

_i

− p(b)).

Revenue of seller is p if item is sold.

Definition

A (deterministic) mechanism (x , p) for selling an item to one of n bidders is given by an allocation rule x : R

ⁿ

→ {0, 1}

ⁿ

with P

i

x

_i

≤ 1, and pricing rule p : R

ⁿ

→ R .

Goal of bidder i is to maximize utility given mechanism (x , p).

Bidders will try to bid strategically.

How should we design auction to prevent undesirable outcomes?

(42)

First price auction

Bidders report bids b = (b

₁

, . . . , b

_n

).

Item is given to i

^∗

= argmax

_i

b

_i

and price p = max

i

b

_i

is charged.

Example

Suppose there are three bidders

Valuations (v

₁

, v

₂

, v

₃

) = (10, 30, 25). Bids (b

₁

, b

₂

, b

₃

) = (5, 22, 23).

Winner is bidder i

^∗

= 3, with price p = 23. Utilities are u = (0, 0, 2). Is this a good auction format?

Does not incentivize truthful bidding.

Bidders have incentive to lie (i.e., not report true valuation v

i

). Bidder 2 values item the most, but does not get it.

Allocation rule does not maximize social welfare objective

“Revenue for seller” + “Player utilities” = P

i

v

i

x

i

(b) = v

i^∗

(43)

First price auction

Bidders report bids b = (b

₁

, . . . , b

_n

). Item is given to i

^∗

= argmax

_i

b

_i

and price p = max

i

b

_i

is charged. Example

Suppose there are three bidders

Valuations (v

₁

, v

₂

, v

₃

) = (10, 30, 25). Bids (b

₁

, b

₂

, b

₃

) = (5, 22, 23).

Winner is bidder i

^∗

= 3, with price p = 23. Utilities are u = (0, 0, 2). Is this a good auction format?

Does not incentivize truthful bidding.

Bidders have incentive to lie (i.e., not report true valuation v

i

). Bidder 2 values item the most, but does not get it.

Allocation rule does not maximize social welfare objective

“Revenue for seller” + “Player utilities” = P

i

v

i

x

i

(b) = v

i^∗

(44)

First price auction

Bidders report bids b = (b

₁

, . . . , b

_n

). Item is given to i

^∗

= argmax

_i

b

_i

and price p = max

i

b

_i

is charged.

Example

Suppose there are three bidders

Valuations (v

₁

, v

₂

, v

₃

) = (10, 30, 25). Bids (b

₁

, b

₂

, b

₃

) = (5, 22, 23).

Winner is bidder i

^∗

= 3, with price p = 23. Utilities are u = (0, 0, 2). Is this a good auction format?

Does not incentivize truthful bidding.

Bidders have incentive to lie (i.e., not report true valuation v

i

). Bidder 2 values item the most, but does not get it.

Allocation rule does not maximize social welfare objective

“Revenue for seller” + “Player utilities” = P

i

v

i

x

i

(b) = v

i^∗

(45)

First price auction

Bidders report bids b = (b

₁

, . . . , b

_n

). Item is given to i

^∗

= argmax

_i

b

_i

and price p = max

i

b

_i

is charged.

Example

Suppose there are three bidders

Valuations (v

₁

, v

₂

, v

₃

) = (10, 30, 25). Bids (b

₁

, b

₂

, b

₃

) = (5, 22, 23).

Winner is bidder i

^∗

= 3, with price p = 23. Utilities are u = (0, 0, 2). Is this a good auction format?

Does not incentivize truthful bidding.

Bidders have incentive to lie (i.e., not report true valuation v

i

). Bidder 2 values item the most, but does not get it.

Allocation rule does not maximize social welfare objective

“Revenue for seller” + “Player utilities” = P

i

v

i

x

i

(b) = v

i^∗

(46)

First price auction

Bidders report bids b = (b

₁

, . . . , b

_n

). Item is given to i

^∗

= argmax

_i

b

_i

and price p = max

i

b

_i

is charged.

Example

Suppose there are three bidders

Valuations (v

₁

, v

₂

, v

₃

) = (10, 30, 25).

Bids (b

₁

, b

₂

, b

₃

) = (5, 22, 23).

Winner is bidder i

^∗

= 3, with price p = 23. Utilities are u = (0, 0, 2). Is this a good auction format?

Does not incentivize truthful bidding.

Bidders have incentive to lie (i.e., not report true valuation v

i

). Bidder 2 values item the most, but does not get it.

Allocation rule does not maximize social welfare objective

“Revenue for seller” + “Player utilities” = P

i

v

i

x

i

(b) = v

i^∗

(47)

First price auction

Bidders report bids b = (b

₁

, . . . , b

_n

). Item is given to i

^∗

= argmax

_i

b

_i

and price p = max

i

b

_i

is charged.

Example

Suppose there are three bidders

Valuations (v

₁

, v

₂

, v

₃

) = (10, 30, 25).

Bids (b

₁

, b

₂

, b

₃

) = (5, 22, 23).

Winner is bidder i

^∗

= 3, with price p = 23. Utilities are u = (0, 0, 2). Is this a good auction format?

Does not incentivize truthful bidding.

Bidders have incentive to lie (i.e., not report true valuation v

i

). Bidder 2 values item the most, but does not get it.

Allocation rule does not maximize social welfare objective

“Revenue for seller” + “Player utilities” = P

i

v

i

x

i

(b) = v

i^∗

(48)

First price auction

Bidders report bids b = (b

₁

, . . . , b

_n

). Item is given to i

^∗

= argmax

_i

b

_i

and price p = max

i

b

_i

is charged.

Example

Suppose there are three bidders

Valuations (v

₁

, v

₂

, v

₃

) = (10, 30, 25).

Bids (b

₁

, b

₂

, b

₃

) = (5, 22, 23).

Winner is bidder i

^∗

= 3, with price p = 23. Utilities are u = (0, 0, 2).

Is this a good auction format?

Does not incentivize truthful bidding.

Bidders have incentive to lie (i.e., not report true valuation v

i

). Bidder 2 values item the most, but does not get it.

Allocation rule does not maximize social welfare objective

“Revenue for seller” + “Player utilities” = P

i

v

i

x

i

(b) = v

i^∗

(49)

First price auction

Bidders report bids b = (b

₁

, . . . , b

_n

). Item is given to i

^∗

= argmax

_i

b

_i

and price p = max

i

b

_i

is charged.

Example

Suppose there are three bidders

Valuations (v

₁

, v

₂

, v

₃

) = (10, 30, 25).

Bids (b

₁

, b

₂

, b

₃

) = (5, 22, 23).

Winner is bidder i

^∗

= 3, with price p = 23. Utilities are u = (0, 0, 2).

Is this a good auction format?

Does not incentivize truthful bidding.

Bidders have incentive to lie (i.e., not report true valuation v

i

). Bidder 2 values item the most, but does not get it.

Allocation rule does not maximize social welfare objective

“Revenue for seller” + “Player utilities” = P

i

v

i

x

i

(b) = v

i^∗

(50)

First price auction

Bidders report bids b = (b

₁

, . . . , b

_n

). Item is given to i

^∗

= argmax

_i

b

_i

and price p = max

i

b

_i

is charged.

Example

Suppose there are three bidders

Valuations (v

₁

, v

₂

, v

₃

) = (10, 30, 25).

Bids (b

₁

, b

₂

, b

₃

) = (5, 22, 23).

Winner is bidder i

^∗

= 3, with price p = 23. Utilities are u = (0, 0, 2).

Is this a good auction format?

Does not incentivize truthful bidding.

Bidders have incentive to lie (i.e., not report true valuation v

i

). Bidder 2 values item the most, but does not get it.

Allocation rule does not maximize social welfare objective

“Revenue for seller” + “Player utilities” = P

i

v

i

x

i

(b) = v

i^∗

(51)

First price auction

Bidders report bids b = (b

₁

, . . . , b

_n

). Item is given to i

^∗

= argmax

_i

b

_i

and price p = max

i

b

_i

is charged.

Example

Suppose there are three bidders

Valuations (v

₁

, v

₂

, v

₃

) = (10, 30, 25).

Bids (b

₁

, b

₂

, b

₃

) = (5, 22, 23).

Winner is bidder i

^∗

= 3, with price p = 23. Utilities are u = (0, 0, 2).

Is this a good auction format?

Does not incentivize truthful bidding.

Bidders have incentive to lie (i.e., not report true valuation v

i

).

Bidder 2 values item the most, but does not get it.

Allocation rule does not maximize social welfare objective

“Revenue for seller” + “Player utilities” = P

i

v

i

x

i

(b) = v

i^∗

(52)

First price auction

Bidders report bids b = (b

₁

, . . . , b

_n

). Item is given to i

^∗

= argmax

_i

b

_i

and price p = max

i

b

_i

is charged.

Example

Suppose there are three bidders

Valuations (v

₁

, v

₂

, v

₃

) = (10, 30, 25).

Bids (b

₁

, b

₂

, b

₃

) = (5, 22, 23).

Winner is bidder i

^∗

= 3, with price p = 23. Utilities are u = (0, 0, 2).

Is this a good auction format?

Does not incentivize truthful bidding.

Bidders have incentive to lie (i.e., not report true valuation v

i

).

Bidder 2 values item the most, but does not get it.

Allocation rule does not maximize social welfare objective

“Revenue for seller” + “Player utilities” = P

i

v

i

x

i

(b) = v

i^∗

(53)

First price auction

Bidders report bids b = (b

₁

, . . . , b

_n

). Item is given to i

^∗

= argmax

_i

b

_i

and price p = max

i

b

_i

is charged.

Example

Suppose there are three bidders

Valuations (v

₁

, v

₂

, v

₃

) = (10, 30, 25).

Bids (b

₁

, b

₂

, b

₃

) = (5, 22, 23).

Winner is bidder i

^∗

= 3, with price p = 23. Utilities are u = (0, 0, 2).

Is this a good auction format?

Does not incentivize truthful bidding.

Bidders have incentive to lie (i.e., not report true valuation v

i

).

Bidder 2 values item the most, but does not get it.

Allocation rule does not maximize social welfare objective

“Revenue for seller” + “Player utilities” = P

i

v

i

x

i

(b) = v

i^∗

(54)

First price auction

Bidders report bids b = (b

₁

, . . . , b

_n

). Item is given to i

^∗

= argmax

_i

b

_i

and price p = max

i

b

_i

is charged.

Example

Suppose there are three bidders

Valuations (v

₁

, v

₂

, v

₃

) = (10, 30, 25).

Bids (b

₁

, b

₂

, b

₃

) = (5, 22, 23).

Winner is bidder i

^∗

= 3, with price p = 23. Utilities are u = (0, 0, 2).

Is this a good auction format?

Does not incentivize truthful bidding.

Bidders have incentive to lie (i.e., not report true valuation v

i

).

Bidder 2 values item the most, but does not get it.

Allocation rule does not maximize social welfare objective

(55)

Selling one item

Second price auction

(56)

Second price auction

Given bids b = (b

₁

, . . . , b

_n

):

Item is allocated to highest bidder i

^∗

= argmax

_i

b

_i

. Price charged is second-highest bid p = max

j6=i^∗

b

_j

. Ties are broken according to some fixed tie-breaking rule. Example

Suppose we have three bidders.

Valuations (v

₁

, v

₂

, v

₃

) = (10, 30, 25). Bids (b

₁

, b

₂

, b

₃

) = (10, 30, 22).

Winner is bidder i

^∗

= 2 and pays p = 22. Utilities are u = (0, 8, 0).

Second price auction has many desirable properties.

(57)

Second price auction

Second price auction Given bids b = (b

₁

, . . . , b

_n

):

Item is allocated to highest bidder i

^∗

= argmax

_i

b

_i

.

Price charged is second-highest bid p = max

j6=i^∗

b

_j

. Ties are broken according to some fixed tie-breaking rule. Example

Suppose we have three bidders.

Valuations (v

₁

, v

₂

, v

₃

) = (10, 30, 25). Bids (b

₁

, b

₂

, b

₃

) = (10, 30, 22).

Winner is bidder i

^∗

= 2 and pays p = 22. Utilities are u = (0, 8, 0).

Second price auction has many desirable properties.

(58)

Second price auction

Second price auction Given bids b = (b

₁

, . . . , b

_n

):

Item is allocated to highest bidder i

^∗

= argmax

_i

b

_i

. Price charged is second-highest bid p = max

j6=i^∗

b

_j

.

Ties are broken according to some fixed tie-breaking rule. Example

Suppose we have three bidders.

Valuations (v

₁

, v

₂

, v

₃

) = (10, 30, 25). Bids (b

₁

, b

₂

, b

₃

) = (10, 30, 22).

Winner is bidder i

^∗

= 2 and pays p = 22. Utilities are u = (0, 8, 0).

Second price auction has many desirable properties.

(59)

Second price auction

Second price auction Given bids b = (b

₁

, . . . , b

_n

):

Item is allocated to highest bidder i

^∗

= argmax

_i

b

_i

. Price charged is second-highest bid p = max

j6=i^∗

b

_j

. Ties are broken according to some fixed tie-breaking rule.

Example

Suppose we have three bidders.

Valuations (v

₁

, v

₂

, v

₃

) = (10, 30, 25). Bids (b

₁

, b

₂

, b

₃

) = (10, 30, 22).

Winner is bidder i

^∗

= 2 and pays p = 22. Utilities are u = (0, 8, 0).

Second price auction has many desirable properties.

(60)

Second price auction

Second price auction Given bids b = (b

₁

, . . . , b

_n

):

Item is allocated to highest bidder i

^∗

= argmax

_i

b

_i

. Price charged is second-highest bid p = max

j6=i^∗

b

_j

. Ties are broken according to some fixed tie-breaking rule.

Example

Suppose we have three bidders.

Valuations (v

₁

, v

₂

, v

₃

) = (10, 30, 25).

Bids (b

₁

, b

₂

, b

₃

) = (10, 30, 22).

Winner is bidder i

^∗

= 2 and pays p = 22. Utilities are u = (0, 8, 0).

Second price auction has many desirable properties.

(61)

Second price auction

Second price auction Given bids b = (b

₁

, . . . , b

_n

):

Item is allocated to highest bidder i

^∗

= argmax

_i

b

_i

. Price charged is second-highest bid p = max

j6=i^∗

b

_j

. Ties are broken according to some fixed tie-breaking rule.

Example

Suppose we have three bidders.

Valuations (v

₁

, v

₂

, v

₃

) = (10, 30, 25).

Bids (b

₁

, b

₂

, b

₃

) = (10, 30, 22).

Winner is bidder i

^∗

= 2 and pays p = 22. Utilities are u = (0, 8, 0).

Second price auction has many desirable properties.

(62)

Second price auction

Second price auction Given bids b = (b

₁

, . . . , b

_n

):

Item is allocated to highest bidder i

^∗

= argmax

_i

b

_i

. Price charged is second-highest bid p = max

j6=i^∗

b

_j

. Ties are broken according to some fixed tie-breaking rule.

Example

Suppose we have three bidders.

Valuations (v

₁

, v

₂

, v

₃

) = (10, 30, 25).

Bids (b

₁

, b

₂

, b

₃

) = (10, 30, 22).

Winner is bidder i

^∗

= 2 and pays p = 22. Utilities are u = (0, 8, 0).

Second price auction has many desirable properties.

(63)

Second price auction

Second price auction Given bids b = (b

₁

, . . . , b

_n

):

Item is allocated to highest bidder i

^∗

= argmax

_i

b

_i

. Price charged is second-highest bid p = max

j6=i^∗

b

_j

. Ties are broken according to some fixed tie-breaking rule.

Example

Suppose we have three bidders.

Valuations (v

₁

, v

₂

, v

₃

) = (10, 30, 25).

Bids (b

₁

, b

₂

, b

₃

) = (10, 30, 22).

Winner is bidder i

^∗

= 2 and pays p = 22. Utilities are u = (0, 8, 0).

(64)

Desired properties

Bidders have incentive to be truthful: Reporting v

_i

is dominant strategy. Definition (Strategyproof)

Mechanism (x , p) incentivizes truthful bidding if for every bidder i , alternative bid b

⁰_i

, and bids b

_−i

= (b

₁

, . . . , b

_i−1

, b

_i+1

, b

_n

) of other bidders, it holds that

u

_i

(b

−i

, v

_i

) ≥ u

_i

(b

−i

, b

⁰_i

),

where u

_i

(b) = x

_i

(b)(v

_i

− p(b)).

Bidders have non-negative utility (when reporting truthfully). Definition (Individually rational)

Mechanism (x , p) is individually rational if for every bidder i it holds u

_i

(b) ≥ 0

for every bid vector b = (b

₁

, . . . , b

_i−1

, v

_i

, b

_i+1

, . . . , b

_n

).

(65)

Desired properties

Bidders have incentive to be truthful: Reporting v

_i

is dominant strategy.

Definition (Strategyproof)

Mechanism (x , p) incentivizes truthful bidding if for every bidder i , alternative bid b

⁰_i

, and bids b

_−i

= (b

₁

, . . . , b

_i−1

, b

_i+1

, b

_n

) of other bidders, it holds that

u

_i

(b

−i

, v

_i

) ≥ u

_i

(b

−i

, b

⁰_i

),

where u

_i

(b) = x

_i

(b)(v

_i

− p(b)).

Bidders have non-negative utility (when reporting truthfully). Definition (Individually rational)

Mechanism (x , p) is individually rational if for every bidder i it holds u

_i

(b) ≥ 0

for every bid vector b = (b

₁

, . . . , b

_i−1

, v

_i

, b

_i+1

, . . . , b

_n

).

(66)

Desired properties

Bidders have incentive to be truthful: Reporting v

_i

is dominant strategy.

Definition (Strategyproof)

Mechanism (x , p) incentivizes truthful bidding if for every bidder i, alternative bid b

⁰_i

, and bids b

_−i

= (b

₁

, . . . , b

_i−1

, b

_i+1

, b

_n

) of other bidders, it holds that

u

_i

(b

−i

, v

_i

) ≥ u

_i

(b

−i

, b

⁰_i

), where u

_i

(b) = x

_i

(b)(v

_i

− p(b)).

Bidders have non-negative utility (when reporting truthfully). Definition (Individually rational)

Mechanism (x , p) is individually rational if for every bidder i it holds u

_i

(b) ≥ 0

for every bid vector b = (b

₁

, . . . , b

_i−1

, v

_i

, b

_i+1

, . . . , b

_n

).

(67)

Desired properties

Bidders have incentive to be truthful: Reporting v

_i

is dominant strategy.

Definition (Strategyproof)

Mechanism (x , p) incentivizes truthful bidding if for every bidder i, alternative bid b

⁰_i

, and bids b

_−i

= (b

₁

, . . . , b

_i−1

, b

_i+1

, b

_n

) of other bidders, it holds that

u

_i

(b

−i

, v

_i

) ≥ u

_i

(b

−i

, b

⁰_i

),

where u

_i

(b) = x

_i

(b)(v

_i

− p(b)).

Bidders have non-negative utility (when reporting truthfully).

Definition (Individually rational)

Mechanism (x , p) is individually rational if for every bidder i it holds u

_i

(b) ≥ 0

for every bid vector b = (b

₁

, . . . , b

_i−1

, v

_i

, b

_i+1

, . . . , b

_n

).

(68)

Desired properties

Bidders have incentive to be truthful: Reporting v

_i

is dominant strategy.

Definition (Strategyproof)

Mechanism (x , p) incentivizes truthful bidding if for every bidder i, alternative bid b

⁰_i

, and bids b

_−i

= (b

₁

, . . . , b

_i−1

, b

_i+1

, b

_n

) of other bidders, it holds that

u

_i

(b

−i

, v

_i

) ≥ u

_i

(b

−i

, b

⁰_i

),

where u

_i

(b) = x

_i

(b)(v

_i

− p(b)).

Bidders have non-negative utility (when reporting truthfully).

Definition (Individually rational)

Mechanism (x , p) is individually rational if for every bidder i it holds u

_i

(b) ≥ 0

for every bid vector b = (b b v b b ).

(69)

Mechanism has good performance guarantee.

Definition (Welfare maximization)

Mechanism (x , p) is welfare maximizer if it maximizes P

i

v

_i

x

_i

(b) = “Revenue for seller” + “Player utilities” assuming that bidders are truthful.

For now this just means we want to allocate item to a bidder with highest (true) valuation v

^∗

= max

_i

v

_i

.

(In online setting, we are content with approximation.) Definition (Computational efficiency)

Mechanism (x , p) should be implementable in polynomial time, i.e.,

compute allocation x and price p in polynomial time.

(70)

Mechanism has good performance guarantee.

Definition (Welfare maximization)

Mechanism (x , p) is welfare maximizer if it maximizes P

i

v

_i

x

_i

(b) = “Revenue for seller” + “Player utilities”

assuming that bidders are truthful.

For now this just means we want to allocate item to a bidder with highest (true) valuation v

^∗

= max

_i

v

_i

.

(In online setting, we are content with approximation.) Definition (Computational efficiency)

Mechanism (x , p) should be implementable in polynomial time, i.e.,

compute allocation x and price p in polynomial time.

(71)

Mechanism has good performance guarantee.

Definition (Welfare maximization)

Mechanism (x , p) is welfare maximizer if it maximizes P

i

v

_i

x

_i

(b) = “Revenue for seller” + “Player utilities”

assuming that bidders are truthful.

For now this just means we want to allocate item to a bidder with highest (true) valuation v

^∗

= max

_i

v

_i

.

(In online setting, we are content with approximation.) Definition (Computational efficiency)

Mechanism (x , p) should be implementable in polynomial time, i.e.,

compute allocation x and price p in polynomial time.

(72)

Mechanism has good performance guarantee.

Definition (Welfare maximization)

Mechanism (x , p) is welfare maximizer if it maximizes P

i

v

_i

x

_i

(b) = “Revenue for seller” + “Player utilities”

assuming that bidders are truthful.

For now this just means we want to allocate item to a bidder with highest (true) valuation v

^∗

= max

_i

v

_i

.

(In online setting, we are content with approximation.)

Definition (Computational efficiency)

Mechanism (x , p) should be implementable in polynomial time, i.e.,

compute allocation x and price p in polynomial time.

(73)

Mechanism has good performance guarantee.

Definition (Welfare maximization)

Mechanism (x , p) is welfare maximizer if it maximizes P

i

v

_i

x

_i

(b) = “Revenue for seller” + “Player utilities”

assuming that bidders are truthful.

For now this just means we want to allocate item to a bidder with highest (true) valuation v

^∗

= max

_i

v

_i

.

(In online setting, we are content with approximation.) Definition (Computational efficiency)

Mechanism (x , p) should be implementable in polynomial time, i.e.,

compute allocation x and price p in polynomial time.

(74)

Proof strategyproofness (second price auction):

Mechanism (x , p) incentivizes truthful bidding if for every i, alter- native bid b

⁰_i

, and b

−i

= (b

₁

, . . . , b

i−1

, b

_i+1

, . . . , b

_n

), it holds that

u

_i

(b

₁

, . . . , v

_i

, . . . , b

_n

) ≥ u

_i

(b

₁

, . . . , b

⁰_i

, . . . , b

_n

).

Fix i and b

−i

. Let p(b) = s be second-highest bid. We compare v

_i

, b

⁰_i

and s (using case distinction).

Assume v

i

6= s for simplicity.

Case s > v

_i

:

Bidder i would only win if b

_i⁰

≥ b

_max

, but then u

_i

= v

_i

− p < 0. For any bid b

⁰_i

< b

_max

(then i does not get item), we have u

_i

= 0. Case s < v

_i

:

Bidder i wins. Charged price s same for all b

⁰_i

> s. For b

⁰_i

< s, we

have u

_i

= 0. Hence, bidding v

_i

is an optimal choice.

(75)

Proof strategyproofness (second price auction):

Mechanism (x , p) incentivizes truthful bidding if for every i, alter- native bid b

⁰_i

, and b

−i

= (b

₁

, . . . , b

i−1

, b

_i+1

, . . . , b

_n

), it holds that

u

_i

(b

₁

, . . . , v

_i

, . . . , b

_n

) ≥ u

_i

(b

₁

, . . . , b

⁰_i

, . . . , b

_n

).

Fix i and b

−i

. Let p(b) = s be second-highest bid.

We compare v

_i

, b

⁰_i

and s (using case distinction). Assume v

i

6= s for simplicity.

Case s > v

_i

:

Bidder i would only win if b

_i⁰

≥ b

_max

, but then u

_i

= v

_i

− p < 0. For any bid b

⁰_i

< b

_max

(then i does not get item), we have u

_i

= 0. Case s < v

_i

:

Bidder i wins. Charged price s same for all b

⁰_i

> s. For b

⁰_i

< s, we

have u

_i

= 0. Hence, bidding v

_i