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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions An introduction to Auctions: Single Item Auctions (2) Maria Serna Fall 2016 AGT-MIRI Single item auctions English and Japanese auctions Prices FP auctions:

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

An introduction to Auctions: Single Item Auctions (2)

Maria Serna Fall 2016

AGT-MIRI Single item auctions

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Prices

1 English and Japanese auctions 2 FP auctions: Bayesian analysis 3 Revenue in auctions

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Prices

English and Japanese auctions

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Prices

English and Japanese auctions

A much more complicated strategy space than sealed bid auctions

AGT-MIRI Single item auctions

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Prices

English and Japanese auctions

A much more complicated strategy space than sealed bid auctions

extensive form game bidders are able to condition their bids on information revealed by others in the case of English auctions, the ability to place jump bids

AGT-MIRI Single item auctions

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Prices

English and Japanese auctions

A much more complicated strategy space than sealed bid auctions

extensive form game bidders are able to condition their bids on information revealed by others in the case of English auctions, the ability to place jump bids

Nevertheless, the revealed information doesn’t make any difference in the independent private values setting.

AGT-MIRI Single item auctions

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Prices

English and Japanese auctions

A much more complicated strategy space than sealed bid auctions

extensive form game bidders are able to condition their bids on information revealed by others in the case of English auctions, the ability to place jump bids

Nevertheless, the revealed information doesn’t make any difference in the independent private values setting. Theorem Under the independent private values model (IPV), it is a dominant strategy for bidders to bid up to (and not beyond) their valuations in both Japanese and English auctions.

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

1 English and Japanese auctions 2 FP auctions: Bayesian analysis 3 Revenue in auctions

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

FP auctions: Bayesian analysis

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

FP auctions: Bayesian analysis

There is not dominant strategy in FP auctions.

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

FP auctions: Bayesian analysis

There is not dominant strategy in FP auctions. How do people behave?

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

FP auctions: Bayesian analysis

There is not dominant strategy in FP auctions. How do people behave? They have beliefs on the preferences of the other players!

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

FP auctions: Bayesian analysis

There is not dominant strategy in FP auctions. How do people behave? They have beliefs on the preferences of the other players! As usual beliefs are modeled with probability distributions.

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

FP auctions: Bayesian analysis

There is not dominant strategy in FP auctions. How do people behave? They have beliefs on the preferences of the other players! As usual beliefs are modeled with probability distributions. Bidders do not know their opponent’s values, i.e., there is incomplete information.

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

FP auctions: Bayesian analysis

There is not dominant strategy in FP auctions. How do people behave? They have beliefs on the preferences of the other players! As usual beliefs are modeled with probability distributions. Bidders do not know their opponent’s values, i.e., there is incomplete information. Each bidder’s strategy must maximize her expected payoff accounting for the uncertainty about opponent values.

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

Auctions with uniform distributions

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

Auctions with uniform distributions

A simple Bayesian auction model:

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

Auctions with uniform distributions

A simple Bayesian auction model:

2 buyers Values are between 0 and 1. Values are distributed uniformly on [0, 1]

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

Auctions with uniform distributions

A simple Bayesian auction model:

2 buyers Values are between 0 and 1. Values are distributed uniformly on [0, 1]

What is the equilibrium in this game of incomplete information?

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

Simple FP: Equilibrium

2 bidders uniform distribution Bidding b(v) = v/2 is an equilibrium

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

Simple FP: Equilibrium

2 bidders uniform distribution Bidding b(v) = v/2 is an equilibrium Assume that Bidder 2’s strategy is b2(v) = v2/2.

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

Simple FP: Equilibrium

2 bidders uniform distribution Bidding b(v) = v/2 is an equilibrium Assume that Bidder 2’s strategy is b2(v) = v2/2. Let us show that b1(v) = v1/2 is a best response to Bidder 2. (clearly, no need to bid above v1).

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

Simple FP: Equilibrium

2 bidders uniform distribution Bidding b(v) = v/2 is an equilibrium Assume that Bidder 2’s strategy is b2(v) = v2/2. Let us show that b1(v) = v1/2 is a best response to Bidder 2. (clearly, no need to bid above v1). Bidder 1’s utility is:

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

Simple FP: Equilibrium

2 bidders uniform distribution Bidding b(v) = v/2 is an equilibrium Assume that Bidder 2’s strategy is b2(v) = v2/2. Let us show that b1(v) = v1/2 is a best response to Bidder 2. (clearly, no need to bid above v1). Bidder 1’s utility is: Prob[b1 > b2] × (v1 − b1) = =Prob[b1 > v2/2] × (v1 − b1) =2b1 ∗ (v1 − b1)

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

Simple FP: Equilibrium

2 bidders uniform distribution Bidding b(v) = v/2 is an equilibrium Assume that Bidder 2’s strategy is b2(v) = v2/2. Let us show that b1(v) = v1/2 is a best response to Bidder 2. (clearly, no need to bid above v1). Bidder 1’s utility is: Prob[b1 > b2] × (v1 − b1) = =Prob[b1 > v2/2] × (v1 − b1) =2b1 ∗ (v1 − b1) maximizing for b1 we have:

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

Simple FP: Equilibrium

2 bidders uniform distribution Bidding b(v) = v/2 is an equilibrium Assume that Bidder 2’s strategy is b2(v) = v2/2. Let us show that b1(v) = v1/2 is a best response to Bidder 2. (clearly, no need to bid above v1). Bidder 1’s utility is: Prob[b1 > b2] × (v1 − b1) = =Prob[b1 > v2/2] × (v1 − b1) =2b1 ∗ (v1 − b1) maximizing for b1 we have: [2b1 ∗ (v1 − b1)]′ = 2v1 − 4b1 = 0 which gives b1 = v1/2

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

FP: uniform values

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

FP: uniform values

We consider the simple Bayesian model

n bidders Values drawn uniformly form [0, 1]

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

FP: uniform values

We consider the simple Bayesian model

n bidders Values drawn uniformly form [0, 1]

Theorem In a FP auction, under the uniform values model, it is a Bayesian Nash equilibrium when, for any i, bidder i bids n−1

n vi.

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

FP uniform values: Efficiency

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

FP uniform values: Efficiency

An auction is efficient, if in a Bayesian Nash equilibrium the bidder with the highest value always wins.

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions

FP uniform values: Efficiency

An auction is efficient, if in a Bayesian Nash equilibrium the bidder with the highest value always wins. Thus, in the uniform value model FP is efficient.

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

1 English and Japanese auctions 2 FP auctions: Bayesian analysis 3 Revenue in auctions

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Optimal auctions

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Optimal auctions

Usually the term optimal auctions stands for revenue maximization.

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Optimal auctions

Usually the term optimal auctions stands for revenue maximization. What is maximal revenue?

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Optimal auctions

Usually the term optimal auctions stands for revenue maximization. What is maximal revenue?

We can always charge the winner his value, but ...

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Optimal auctions

Usually the term optimal auctions stands for revenue maximization. What is maximal revenue?

We can always charge the winner his value, but ...

Maximal revenue: optimal expected revenue in equilibrium.

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Optimal auctions

Usually the term optimal auctions stands for revenue maximization. What is maximal revenue?

We can always charge the winner his value, but ...

Maximal revenue: optimal expected revenue in equilibrium.

Assuming a probability distribution on the values. Over all the possible mechanisms. Under individual-rationality constraints.

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Simple auctions with uniform distributions

We consider the simple Bayesian model

2 bidders Values drawn uniformly form [0, 1] x, y

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Simple auctions with uniform distributions

We consider the simple Bayesian model

2 bidders Values drawn uniformly form [0, 1] x, y

What is the expected revenue gained by SP and FP auctions?

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue in SP Simple auctions with uniform distributions

In a SP auction, the payment is the minimum of the two values. E[revenue] = E[min{x, y}]

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue in SP Simple auctions with uniform distributions

In a SP auction, the payment is the minimum of the two values. E[revenue] = E[min{x, y}] Claim: When x, y ≡ U[0, 1] we have E[min{x, y}] = 1/3

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Claim’s proof

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Claim’s proof

Assume that v1 = x. Then, the expected revenue is: x x 2 + (1 − x)x = x − x2 2 . The expectation over all possible x is: E[min{x, y}] = 1 (x − x2 2 )dx = x2 2 − x3 6 1 = 1 3.

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Order statistics

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Order statistics

Let v1, . . . , vn be n random variables

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Order statistics

Let v1, . . . , vn be n random variables

The highest realization is called the 1st-order statistic. The second highest is the called 2nd-order statistic. . . . The smallest is the n-th-order statistic.

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Order statistics

Let v1, . . . , vn be n random variables

The highest realization is called the 1st-order statistic. The second highest is the called 2nd-order statistic. . . . The smallest is the n-th-order statistic.

Example: the uniform distribution, 2 samples.

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Order statistics

Let v1, . . . , vn be n random variables

The highest realization is called the 1st-order statistic. The second highest is the called 2nd-order statistic. . . . The smallest is the n-th-order statistic.

Example: the uniform distribution, 2 samples.

The expected 1st-order statistic: 2/3 Expected efficiency.

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Order statistics

Let v1, . . . , vn be n random variables

The highest realization is called the 1st-order statistic. The second highest is the called 2nd-order statistic. . . . The smallest is the n-th-order statistic.

Example: the uniform distribution, 2 samples.

The expected 1st-order statistic: 2/3 Expected efficiency. The expected 2nd-order statistic: 1/3 Expected revenue.

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Order statistics

In general, for the uniform distribution with n samples:

k-th order statistic of n variables is (n + 1 − k)/(n + 1) 1st-order statistic: n/(n + 1).

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue in FP Simple auctions with uniform distributions

In a FP auction, bidders bid vi/2 bayesian Nash Revenue is the highest bid. Thus, expected revenue is

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue in FP Simple auctions with uniform distributions

In a FP auction, bidders bid vi/2 bayesian Nash Revenue is the highest bid. Thus, expected revenue is E[revenue] =E[max{v1/2, v2/2}] = 1 2E[max{v1, v2}] =1 2 × 2 3 = 1 3

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue in FP Simple auctions with uniform distributions

In a FP auction, bidders bid vi/2 bayesian Nash Revenue is the highest bid. Thus, expected revenue is E[revenue] =E[max{v1/2, v2/2}] = 1 2E[max{v1, v2}] =1 2 × 2 3 = 1 3 Same revenue as in SP auctions!

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue: FP vs. SP auctions with uniform distributions

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue: FP vs. SP auctions with uniform distributions

Revenue in SP:

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue: FP vs. SP auctions with uniform distributions

Revenue in SP:

Bidders bid truthfully. Revenue is 2nd highest bid: E[revenue] = n − 1 n + 1

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue: FP vs. SP auctions with uniform distributions

Revenue in SP:

Bidders bid truthfully. Revenue is 2nd highest bid: E[revenue] = n − 1 n + 1

Revenue in FP:

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue: FP vs. SP auctions with uniform distributions

Revenue in SP:

Bidders bid truthfully. Revenue is 2nd highest bid: E[revenue] = n − 1 n + 1

Revenue in FP:

Bidders bid. Revenue is highest bid:

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue: FP vs. SP auctions with uniform distributions

Revenue in SP:

Bidders bid truthfully. Revenue is 2nd highest bid: E[revenue] = n − 1 n + 1

Revenue in FP:

Bidders bid. Revenue is highest bid: E[revenue] =E

n − 1 n v1, . . . , n − 1 n vn

=n − 1

n E[max{v1, . . . , vn}] = n − 1 n n n + 1 = n − 1 n + 1

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue Equivalence Theorem

Assumptions:

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue Equivalence Theorem

Assumptions:

vi‘s are drawn independently from some F on [a, b]. F is continuous and strictly increasing. Bidders are risk neutral: utility is a linear function of hos wealth.

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue Equivalence Theorem

Assumptions:

vi‘s are drawn independently from some F on [a, b]. F is continuous and strictly increasing. Bidders are risk neutral: utility is a linear function of hos wealth.

Theorem (The Revenue Equivalence Theorem) Consider two auction such that: (same allocation) When player i bids v his probability to win is the same in the two auctions (for all i and v) in equilibrium. (normalization) If a player bids a (the lowest possible value) he will pay the same amount in both auctions. Then, in equilibrium, the two auctions earn the same revenue.

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue Equivalence Theorem

Assumptions:

vi‘s are drawn independently from some F on [a, b]. F is continuous and strictly increasing. Bidders are risk neutral: utility is a linear function of hos wealth.

Theorem (The Revenue Equivalence Theorem) Consider two auction such that: (same allocation) When player i bids v his probability to win is the same in the two auctions (for all i and v) in equilibrium. (normalization) If a player bids a (the lowest possible value) he will pay the same amount in both auctions. Then, in equilibrium, the two auctions earn the same revenue. Revenue depends most on allocation than on valuations!

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue in All-pay auction

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue in All-pay auction

Rules

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue in All-pay auction

Rules

Sealed bid Highest bid wins Everyone pay their bid

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue in All-pay auction

Rules

Sealed bid Highest bid wins Everyone pay their bid

Equilibrium with the uniform distribution is b(v) = n − 1 n vn

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue in All-pay auction

Rules

Sealed bid Highest bid wins Everyone pay their bid

Equilibrium with the uniform distribution is b(v) = n − 1 n vn Revenue?

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue in All-pay auction

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue in All-pay auction

Expected payment per each player: her bid

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue in All-pay auction

Expected payment per each player: her bid Each bidder bids b(v) = n−1

n vn.

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue in All-pay auction

Expected payment per each player: her bid Each bidder bids b(v) = n−1

n vn.

Expected payment for each bidder: 1 n − 1 n vndv = n − 1 n vn+1 n + 1 1 = 1 n n − 1 n + 1

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue in All-pay auction

Expected payment per each player: her bid Each bidder bids b(v) = n−1

n vn.

Expected payment for each bidder: 1 n − 1 n vndv = n − 1 n vn+1 n + 1 1 = 1 n n − 1 n + 1 Revenue for n bidders E[revenue] = n − 1 n + 1.

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

Revenue in All-pay auction

Expected payment per each player: her bid Each bidder bids b(v) = n−1

n vn.

Expected payment for each bidder: 1 n − 1 n vndv = n − 1 n vn+1 n + 1 1 = 1 n n − 1 n + 1 Revenue for n bidders E[revenue] = n − 1 n + 1. Again revenue equivalence!

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

All-pay auctions: examples

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

All-pay auctions: examples

crowdsourcing over the internet:

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

All-pay auctions: examples

crowdsourcing over the internet:

First person to complete a task for me gets a reward. A group of people invest time in the task. (=payment) Only the winner gets the reward.

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

All-pay auctions: examples

crowdsourcing over the internet:

First person to complete a task for me gets a reward. A group of people invest time in the task. (=payment) Only the winner gets the reward.

Advertising auction:

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

All-pay auctions: examples

crowdsourcing over the internet:

First person to complete a task for me gets a reward. A group of people invest time in the task. (=payment) Only the winner gets the reward.

Advertising auction:

Collect suggestion for campaigns, choose a winner. All advertiser incur cost of preparing the campaign. Only one wins.

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English and Japanese auctions FP auctions: Bayesian analysis Revenue in auctions Maximal revenue SP auctions FP auctions Revenue equivalence All-pay auctions

All-pay auctions: examples

crowdsourcing over the internet:

First person to complete a task for me gets a reward. A group of people invest time in the task. (=payment) Only the winner gets the reward.

Advertising auction:

Collect suggestion for campaigns, choose a winner. All advertiser incur cost of preparing the campaign. Only one wins.

War of attrition

Animals invest (b1,b2) in fighting. Maynard Smith, J. (1974) Theory of games and the evolution

f animal conflicts.

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