[PPT] - A Unifying Model of Winner-takes-all Contests Julio Gonz alez-D az PowerPoint Presentation

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A Unifying Model of Winner-takes-all Contests

Julio Gonz´ alez-D´ ıaz

Kellogg School of Management (CMS-EMS) Northwestern University and Research Group in Economic Analysis Universidad de Vigo

March 13th, 2007

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Motivation Winner-takes-all Contests Various Models of Contests Results

Outline

1

Motivation

2

Winner-takes-all Contests

3

Various Models of Contests

4

Results

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 1/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

Outline

1

Motivation

2

Winner-takes-all Contests

3

Various Models of Contests

4

Results

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 2/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

Motivation

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 3/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

Motivation

Contests:

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 3/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

Motivation

Contests: Auctions, Bertrand Competition,. . .

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 3/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

Motivation

Contests: Auctions, Bertrand Competition,. . . Similar results across models and further strategic connections

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 3/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

Motivation

Contests: Auctions, Bertrand Competition,. . . Similar results across models and further strategic connections Well Known!

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 3/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

Motivation

Contests: Auctions, Bertrand Competition,. . . Similar results across models and further strategic connections Unifying model

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 3/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

Motivation

Contests: Auctions, Bertrand Competition,. . . Similar results across models and further strategic connections Unifying model Complete information

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 3/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

First Example: Sharing a Cake

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 4/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

First Example: Sharing a Cake

Mommy&Daddy

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 4/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

First Example: Sharing a Cake

Mommy&Daddy Mark Sara

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 4/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

First Example: Sharing a Cake

Mommy&Daddy Mark Sara

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 4/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

First Example: Sharing a Cake

Mommy&Daddy Mark: α1 Sara α1

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 4/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

First Example: Sharing a Cake

Mommy&Daddy Mark: α1 Sara: α2 α1 α2

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 4/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

First Example: Sharing a Cake

Mommy&Daddy Mark: α1 Sara: α2 α1 α2 P

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 4/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

First Example: Sharing a Cake

Mommy&Daddy Mark: α1 Sara: α2 α1 α2 P

t1 t2

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 4/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

First Example: Sharing a Cake

Mommy&Daddy Mark: α1 Sara: α2 α1 α2 P

t1 t2

t1 < t2 − → α1 t1 < t2 − → α2 + P

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 4/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

First Example: Sharing a Cake

Mommy&Daddy Mark: α1 Sara: α2 α1 α2 P

t1 t2

t1 < t2 − → α1 t1 < t2 − → α2 + P t1 > t2 − → α1 + P t1 > t2 − → α2

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 4/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

First Example: Sharing a Cake

Mommy&Daddy Mark: α1 Sara: α2 α1 α2 P

t1 t2

t1 < t2 − → α1 t1 < t2 − → α2 + P t1 > t2 − → α1 + P t1 > t2 − → α2 t1 = t2 − → α1 + P

2

t1 = t2 − → α2 + P

2

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 4/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

First Example: Sharing a Cake

Mommy&Daddy Mark: α1 Sara: α2 α1 α2 P

t1 t2

t1 < t2 − → δt1(α1) t1 < t2 − → δt2(α2 + P) t1 > t2 − → δt1(α1 + P) t1 > t2 − → δt2(α2) t1 = t2 − → δt1(α1 + P

2 )

t1 = t2 − → δt2(α2 + P

2 )

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 4/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

First Example: Sharing a Cake

Mommy&Daddy Mark: α1 Sara: α2 α1 α2 P

t1 t2

t1 < t2 − → δt1(α1) t1 < t2 − → δt2(α2 + P) t1 > t2 − → δt1(α1 + P) t1 > t2 − → δt2(α2) t1 = t2 − → δt1(α1 + P

2 )

t1 = t2 − → δt2(α2 + P

2 )

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 4/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

The Cake Sharing Game

The Model

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 5/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

The Cake Sharing Game

The Model

N = {1, . . . , n} is the set of players

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 5/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

The Cake Sharing Game

The Model

N = {1, . . . , n} is the set of players Let α ∈ RN

+ be the initial rights vector:

P = 1 −

i∈N αi > 0

0 < α1 < α2 < · · · < αn

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 5/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

The Cake Sharing Game

The Model

N = {1, . . . , n} is the set of players Let α ∈ RN

+ be the initial rights vector:

P = 1 −

i∈N αi > 0

0 < α1 < α2 < · · · < αn Let δ ∈ (0, 1) be the discount factor

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 5/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

The Cake Sharing Game

The Model

N = {1, . . . , n} is the set of players Let α ∈ RN

+ be the initial rights vector:

P = 1 −

i∈N αi > 0

0 < α1 < α2 < · · · < αn Let δ ∈ (0, 1) be the discount factor Cake sharing game: Γ

pure

α,δ =< N, {Ai}i∈N, {πi}i∈N >

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 5/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

The Cake Sharing Game

The Model

N = {1, . . . , n} is the set of players Let α ∈ RN

+ be the initial rights vector:

P = 1 −

i∈N αi > 0

0 < α1 < α2 < · · · < αn Let δ ∈ (0, 1) be the discount factor Cake sharing game: Γ

pure

α,δ =< N, {Ai}i∈N, {πi}i∈N >

Ai = [0, ∞) is the set of pure strategies of player i ∈ N

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 5/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

The Cake Sharing Game

The Model

N = {1, . . . , n} is the set of players Let α ∈ RN

+ be the initial rights vector:

P = 1 −

i∈N αi > 0

0 < α1 < α2 < · · · < αn Let δ ∈ (0, 1) be the discount factor Cake sharing game: Γ

pure

α,δ =< N, {Ai}i∈N, {πi}i∈N >

Ai = [0, ∞) is the set of pure strategies of player i ∈ N πi is the payoff function of player i ∈ N, defined by: πi(t1, . . . , tn) =    δtiαi ti ≤ max

j=i tj

δti(αi + P) ti > max

j=i tj

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 5/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

The Cake Sharing Game

The Model

N = {1, . . . , n} is the set of players Let α ∈ RN

+ be the initial rights vector:

P = 1 −

i∈N αi > 0

0 < α1 < α2 < · · · < αn Let δ ∈ (0, 1) be the discount factor Cake sharing game: Γ

pure

α,δ =< N, {Ai}i∈N, {πi}i∈N >

Ai = [0, ∞) is the set of pure strategies of player i ∈ N πi is the payoff function of player i ∈ N, defined by: πi(t1, . . . , tn) =    δtiαi ti ≤ max

j=i tj Ties??

δti(αi + P) ti > max

j=i tj

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 5/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

A Negative Result

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 6/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

A Negative Result

There is no Nash equilibrium in pure strategies

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 6/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

A Negative Result

There is no Nash equilibrium in pure strategies

There is a unique last claimant 0 [ · · · ∞ t3 t5 t2 t4 t7 · · · · · · π7(t) = δt7(α7 + P)

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 6/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

A Negative Result

There is no Nash equilibrium in pure strategies

There is a unique last claimant 0 [ · · · ∞ t3 t5 t2 t4 t7 · · · · · · ˆ t7 π7(t) = δt7(α7 + P) < π7(t−7, ˆ t7) = δˆ

t7(α7 + P)

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 6/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

A Negative Result

There is no Nash equilibrium in pure strategies

There is a unique last claimant 0 [ · · · ∞ t3 t5 t2 t4 t7 · · · · · · ˆ t7 π7(t) = δt7(α7 + P) < π7(t−7, ˆ t7) = δˆ

t7(α7 + P)

There are several last claimants 0 [ · · · ∞ t3 t5 t2 t4 t7 = t1 · · · · · · π7(t) = δt7α7

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 6/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

A Negative Result

There is no Nash equilibrium in pure strategies

There is a unique last claimant 0 [ · · · ∞ t3 t5 t2 t4 t7 · · · · · · ˆ t7 π7(t) = δt7(α7 + P) < π7(t−7, ˆ t7) = δˆ

t7(α7 + P)

There are several last claimants 0 [ · · · ∞ t3 t5 t2 t4 t7 = t1 · · · · · · ˆ t7 π7(t) = δt7α7 < π7(t−7, ˆ t7) = δˆ

t7(α7 + P)

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 6/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

A Negative Result

There is no Nash equilibrium in pure strategies

There is a unique last claimant 0 [ · · · ∞ t3 t5 t2 t4 t7 · · · · · · ˆ t7 π7(t) = δt7(α7 + P) < π7(t−7, ˆ t7) = δˆ

t7(α7 + P)

There are several last claimants 0 [ · · · ∞ t3 t5 t2 t4 t7 = t1 · · · · · · ˆ t7 π7(t) = δt7α7 < π7(t−7, ˆ t7) = δˆ

t7(α7 + P)

Options:

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 6/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

A Negative Result

There is no Nash equilibrium in pure strategies

There is a unique last claimant 0 [ · · · ∞ t3 t5 t2 t4 t7 · · · · · · ˆ t7 π7(t) = δt7(α7 + P) < π7(t−7, ˆ t7) = δˆ

t7(α7 + P)

There are several last claimants 0 [ · · · ∞ t3 t5 t2 t4 t7 = t1 · · · · · · ˆ t7 π7(t) = δt7α7 < π7(t−7, ˆ t7) = δˆ

t7(α7 + P)

Options: Discretizing??

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 6/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

A Negative Result

There is no Nash equilibrium in pure strategies

There is a unique last claimant 0 [ · · · ∞ t3 t5 t2 t4 t7 · · · · · · ˆ t7 π7(t) = δt7(α7 + P) < π7(t−7, ˆ t7) = δˆ

t7(α7 + P)

There are several last claimants 0 [ · · · ∞ t3 t5 t2 t4 t7 = t1 · · · · · · ˆ t7 π7(t) = δt7α7 < π7(t−7, ˆ t7) = δˆ

t7(α7 + P)

Options: Discretizing?? Mixing??

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 6/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

Mixed Strategies

The extended model

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 7/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

Mixed Strategies

The extended model

A mixed strategy is a distribution function G, defined on [0, ∞)

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 7/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

Mixed Strategies

The extended model

A mixed strategy is a distribution function G, defined on [0, ∞) Given a strategy profile G = (G1, G2, . . . , Gn), πi(G−i, t) =

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 7/28

SLIDE 44

Motivation Winner-takes-all Contests Various Models of Contests Results

Mixed Strategies

The extended model

A mixed strategy is a distribution function G, defined on [0, ∞) Given a strategy profile G = (G1, G2, . . . , Gn), πi(G−i, t) = t

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 7/28

SLIDE 45

Motivation Winner-takes-all Contests Various Models of Contests Results

Mixed Strategies

The extended model

A mixed strategy is a distribution function G, defined on [0, ∞) Given a strategy profile G = (G1, G2, . . . , Gn), πi(G−i, t) = Gj(t−)

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 7/28

SLIDE 46

Motivation Winner-takes-all Contests Various Models of Contests Results

Mixed Strategies

The extended model

A mixed strategy is a distribution function G, defined on [0, ∞) Given a strategy profile G = (G1, G2, . . . , Gn), πi(G−i, t) =

j=i

Gj(t−)

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 7/28

SLIDE 47

Motivation Winner-takes-all Contests Various Models of Contests Results

Mixed Strategies

The extended model

A mixed strategy is a distribution function G, defined on [0, ∞) Given a strategy profile G = (G1, G2, . . . , Gn), πi(G−i, t) = P

j=i

Gj(t−)

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 7/28

SLIDE 48

Motivation Winner-takes-all Contests Various Models of Contests Results

Mixed Strategies

The extended model

A mixed strategy is a distribution function G, defined on [0, ∞) Given a strategy profile G = (G1, G2, . . . , Gn), πi(G−i, t) = αi + P

j=i

Gj(t−)

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 7/28

SLIDE 49

Motivation Winner-takes-all Contests Various Models of Contests Results

Mixed Strategies

The extended model

A mixed strategy is a distribution function G, defined on [0, ∞) Given a strategy profile G = (G1, G2, . . . , Gn), πi(G−i, t) = δt(αi + P

j=i

Gj(t−))

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 7/28

SLIDE 50

Motivation Winner-takes-all Contests Various Models of Contests Results

Mixed Strategies

The extended model

A mixed strategy is a distribution function G, defined on [0, ∞) Given a strategy profile G = (G1, G2, . . . , Gn), πi(G−i, t) = δt(αi + P

j=i

Gj(t−))

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 7/28

SLIDE 51

Motivation Winner-takes-all Contests Various Models of Contests Results

The Characterization Result

Theorem

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 8/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

The Characterization Result

Theorem

Let Γα,δ be an n-player cake sharing game.

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 8/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

The Characterization Result

Theorem

Let Γα,δ be an n-player cake sharing game. Then, Γα,δ has a unique Nash equilibrium. Moreover. . .

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 8/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

The Characterization Result

Theorem

Let Γα,δ be an n-player cake sharing game. Then, Γα,δ has a unique Nash equilibrium. Moreover. . .

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 8/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

Second Example: Sharing a Market (Varian 1980)

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 9/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

Second Example: Sharing a Market (Varian 1980)

Firm 1 Firm 2

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 9/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

Second Example: Sharing a Market (Varian 1980)

Consumers Firm 1 Firm 2

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 9/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

Second Example: Sharing a Market (Varian 1980)

Consumers Firm 1: α1 Firm 2 α1

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 9/28

SLIDE 59

Motivation Winner-takes-all Contests Various Models of Contests Results

Second Example: Sharing a Market (Varian 1980)

Consumers Firm 1: α1 Firm 2: α2 α1 α2

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 9/28

SLIDE 60

Motivation Winner-takes-all Contests Various Models of Contests Results

Second Example: Sharing a Market (Varian 1980)

Consumers Firm 1: α1 Firm 2: α2 α1 α2 P

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 9/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

Second Example: Sharing a Market (Varian 1980)

Consumers Firm 1: α1 Firm 2: α2 α1 α2 P

p1 ≤ ¯ p p2 ≤ ¯ p

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 9/28

SLIDE 62

Motivation Winner-takes-all Contests Various Models of Contests Results

Second Example: Sharing a Market (Varian 1980)

Consumers Firm 1: α1 Firm 2: α2 α1 α2 P

p1 ≤ ¯ p p2 ≤ ¯ p

p1 > p2 − → p1α1 p1 > p2 − → p2(α2 + P)

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 9/28

SLIDE 63

Motivation Winner-takes-all Contests Various Models of Contests Results

Second Example: Sharing a Market (Varian 1980)

Consumers Firm 1: α1 Firm 2: α2 α1 α2 P

p1 ≤ ¯ p p2 ≤ ¯ p

p1 > p2 − → p1α1 p1 > p2 − → p2(α2 + P) p1 < p2 − → p1(α1 + P) p1 < p2 − → p2α2

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 9/28

SLIDE 64

Motivation Winner-takes-all Contests Various Models of Contests Results

Second Example: Sharing a Market (Varian 1980)

Consumers Firm 1: α1 Firm 2: α2 α1 α2 P

p1 ≤ ¯ p p2 ≤ ¯ p

p1 > p2 − → p1α1 p1 > p2 − → p2(α2 + P) p1 < p2 − → p1(α1 + P) p1 < p2 − → p2α2 p1 = p2 − → p1(α1 + P

2 )

p1 = p2 − → p2(α2 + P

2 )

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 9/28

SLIDE 65

Motivation Winner-takes-all Contests Various Models of Contests Results

Second Example: Sharing a Market (Varian 1980)

Consumers Firm 1: α1 Firm 2: α2 α1 α2 P

p1 ≤ ¯ p p2 ≤ ¯ p

p1 > p2 − → p1α1 p1 > p2 − → p2(α2 + P) p1 < p2 − → p1(α1 + P) p1 < p2 − → p2α2 p1 = p2 − → p1(α1 + P

2 )

p1 = p2 − → p2(α2 + P

2 )

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 9/28

SLIDE 66

Motivation Winner-takes-all Contests Various Models of Contests Results

The Characterization Result and the Pricing Game

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 10/28

SLIDE 67

Motivation Winner-takes-all Contests Various Models of Contests Results

The Characterization Result and the Pricing Game

The pricing game

N firms. Each one with αi loyal consumers Strategic consumers: P Higher admissible price: ¯ p

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 10/28

SLIDE 68

Motivation Winner-takes-all Contests Various Models of Contests Results

The Characterization Result and the Pricing Game

The pricing game

N firms. Each one with αi loyal consumers Strategic consumers: P Higher admissible price: ¯ p

The equilibrium of the pricing game

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 10/28

SLIDE 69

Motivation Winner-takes-all Contests Various Models of Contests Results

The Characterization Result and the Pricing Game

The pricing game

N firms. Each one with αi loyal consumers Strategic consumers: P Higher admissible price: ¯ p

The equilibrium of the pricing game

Only the two firms with less loyal consumers “compete”

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Motivation Winner-takes-all Contests Various Models of Contests Results

The Characterization Result and the Pricing Game

The pricing game

N firms. Each one with αi loyal consumers Strategic consumers: P Higher admissible price: ¯ p

The equilibrium of the pricing game

Only the two firms with less loyal consumers “compete” Only the firm with less loyal consumers gains by “competing”

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SLIDE 71

Motivation Winner-takes-all Contests Various Models of Contests Results

The Characterization Result and the Pricing Game

The pricing game

N firms. Each one with αi loyal consumers Strategic consumers: P Higher admissible price: ¯ p

The equilibrium of the pricing game

Only the two firms with less loyal consumers “compete” Only the firm with less loyal consumers gains by “competing” Strategic consumers pay no more than loyal consumers

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 10/28

SLIDE 72

Motivation Winner-takes-all Contests Various Models of Contests Results

The Characterization Result and the Pricing Game

The pricing game

N firms. Each one with αi loyal consumers Strategic consumers: P Higher admissible price: ¯ p

The equilibrium of the pricing game

Only the two firms with less loyal consumers “compete” Only the firm with less loyal consumers gains by “competing” Strategic consumers pay no more than loyal consumers

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 10/28

SLIDE 73

Motivation Winner-takes-all Contests Various Models of Contests Results

Outline

1

Motivation

2

Winner-takes-all Contests

3

Various Models of Contests

4

Results

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 11/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

The Game

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 12/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

The Game

The players want to get a prize

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 12/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

The Game

The players want to get a prize Players choose efforts

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 12/28

SLIDE 77

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

The Game

The players want to get a prize Players choose efforts The highest effort gets the prize

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 12/28

SLIDE 78

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

The Game

The players want to get a prize Players choose efforts The highest effort gets the prize

Primitives

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Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

The Game

The players want to get a prize Players choose efforts The highest effort gets the prize

Primitives

Efforts: e ∈ E = [m, M]

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 12/28

SLIDE 80

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

The Game

The players want to get a prize Players choose efforts The highest effort gets the prize

Primitives

Efforts: e ∈ E = [m, M] (M = +∞ → E = [m, +∞))

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 12/28

SLIDE 81

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

The Game

The players want to get a prize Players choose efforts The highest effort gets the prize

Primitives

Efforts: e ∈ E = [0, M] (M = +∞ → E = [m, +∞))

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 12/28

SLIDE 82

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

The Game

The players want to get a prize Players choose efforts The highest effort gets the prize

Primitives

Efforts: e ∈ E = [0, M] Base Payoff Funtions: bi(e) : [0, M] → R

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 12/28

SLIDE 83

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

The Game

The players want to get a prize Players choose efforts The highest effort gets the prize

Primitives

Efforts: e ∈ E = [0, M] Base Payoff Funtions: bi(e) : [0, M] → R

continuous and weakly decreasing

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 12/28

SLIDE 84

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

The Game

The players want to get a prize Players choose efforts The highest effort gets the prize

Primitives

Efforts: e ∈ E = [0, M] Base Payoff Funtions: bi(e) : [0, M] → R

continuous and weakly decreasing

Prize Payoff Funtions: pi(e) : [0, M] → R

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 12/28

SLIDE 85

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

The Game

The players want to get a prize Players choose efforts The highest effort gets the prize

Primitives

Efforts: e ∈ E = [0, M] Base Payoff Funtions: bi(e) : [0, M] → R

continuous and weakly decreasing

Prize Payoff Funtions: pi(e) : [0, M] → R

continuous and weakly decreasing

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 12/28

SLIDE 86

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

The Game

The players want to get a prize Players choose efforts The highest effort gets the prize

Primitives

Efforts: e ∈ E = [0, M] Base Payoff Funtions: bi(e) : [0, M] → R

continuous and weakly decreasing

Prize Payoff Funtions: pi(e) : [0, M] → R

continuous and weakly decreasing pi(0) > 0

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 12/28

SLIDE 87

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

The Game

The players want to get a prize Players choose efforts The highest effort gets the prize

Primitives

Efforts: e ∈ E = [0, M] Base Payoff Funtions: bi(e) : [0, M] → R

continuous and weakly decreasing

Prize Payoff Funtions: pi(e) : [0, M] → R

continuous and weakly decreasing pi(0) > 0

Tie Payoff Functions:

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 12/28

SLIDE 88

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

Tie Payoff Functions

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 13/28

SLIDE 89

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

Tie Payoff Functions

Ti : [0, M] × 2N\{∅} → R:

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 13/28

SLIDE 90

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

Tie Payoff Functions

Ti : [0, M] × 2N\{∅} → R: T1) Ti(e, {i}) = pi(e)

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 13/28

SLIDE 91

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

Tie Payoff Functions

Ti : [0, M] × 2N\{∅} → R: T1) Ti(e, {i}) = pi(e) T2) For each S such that i / ∈ S, Ti(e, S) = 0

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 13/28

SLIDE 92

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

Tie Payoff Functions

Ti : [0, M] × 2N\{∅} → R: T1) Ti(e, {i}) = pi(e) T2) For each S such that i / ∈ S, Ti(e, S) = 0 T3) . . .

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 13/28

SLIDE 93

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

Tie Payoff Functions

Ti : [0, M] × 2N\{∅} → R: T1) Ti(e, {i}) = pi(e) T2) For each S such that i / ∈ S, Ti(e, S) = 0 T3) . . .

Examples

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 13/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

Tie Payoff Functions

Ti : [0, M] × 2N\{∅} → R: T1) Ti(e, {i}) = pi(e) T2) For each S such that i / ∈ S, Ti(e, S) = 0 T3) . . .

Examples

Ti(e, S) =

pi(e)

|S|

i ∈ S

therwise

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 13/28

SLIDE 95

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

Tie Payoff Functions

Ti : [0, M] × 2N\{∅} → R: T1) Ti(e, {i}) = pi(e) T2) For each S such that i / ∈ S, Ti(e, S) = 0 T3) . . .

Examples

Ti(e, S) =

pi(e)

|S|

i ∈ S

therwise

Ti(e, S) = pi(e) {i} = S

therwise

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 13/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 14/28

SLIDE 97

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

Contest form: f := ({bi}i∈N, {pi}i∈N, {Ti}i∈N)

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 14/28

SLIDE 98

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

Contest form: f := ({bi}i∈N, {pi}i∈N, {Ti}i∈N) For each σ = (e1, . . . , en) ∈ [0, M]n, wσ := argmaxi∈N{ei}

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 14/28

SLIDE 99

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

Contest form: f := ({bi}i∈N, {pi}i∈N, {Ti}i∈N) For each σ = (e1, . . . , en) ∈ [0, M]n, wσ := argmaxi∈N{ei} Contest with pure strategies:

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 14/28

SLIDE 100

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

Contest form: f := ({bi}i∈N, {pi}i∈N, {Ti}i∈N) For each σ = (e1, . . . , en) ∈ [0, M]n, wσ := argmaxi∈N{ei} Contest with pure strategies: Cf

pure := ({Ei}i∈N, {ui}i∈N), where

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 14/28

SLIDE 101

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

Contest form: f := ({bi}i∈N, {pi}i∈N, {Ti}i∈N) For each σ = (e1, . . . , en) ∈ [0, M]n, wσ := argmaxi∈N{ei} Contest with pure strategies: Cf

pure := ({Ei}i∈N, {ui}i∈N), where

Ei := [0, M]

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 14/28

SLIDE 102

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

Contest form: f := ({bi}i∈N, {pi}i∈N, {Ti}i∈N) For each σ = (e1, . . . , en) ∈ [0, M]n, wσ := argmaxi∈N{ei} Contest with pure strategies: Cf

pure := ({Ei}i∈N, {ui}i∈N), where

Ei := [0, M] and ui(σ) := bi(ei) + Ti(ei, wσ)

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 14/28

SLIDE 103

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

Contest form: f := ({bi}i∈N, {pi}i∈N, {Ti}i∈N) For each σ = (e1, . . . , en) ∈ [0, M]n, wσ := argmaxi∈N{ei} Contest with pure strategies: Cf

pure := ({Ei}i∈N, {ui}i∈N), where

Ei := [0, M] and ui(σ) := bi(ei) + Ti(ei, wσ)

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 14/28

SLIDE 104

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

Contest form: f := ({bi}i∈N, {pi}i∈N, {Ti}i∈N) For each σ = (e1, . . . , en) ∈ [0, M]n, wσ := argmaxi∈N{ei} Contest with pure strategies: Cf

pure := ({Ei}i∈N, {ui}i∈N), where

Ei := [0, M] and ui(σ) := bi(ei) + Ti(ei, wσ) Productivity functions

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 14/28

SLIDE 105

Motivation Winner-takes-all Contests Various Models of Contests Results

Winner-takes-all Contests

Contest form: f := ({bi}i∈N, {pi}i∈N, {Ti}i∈N) For each σ = (e1, . . . , en) ∈ [0, M]n, wσ := argmaxi∈N{ei} Contest with pure strategies: Cf

pure := ({Ei}i∈N, {ui}i∈N), where

Ei := [0, M] and ui(σ) := bi(ei) + Ti(ei, wσ)

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 14/28

SLIDE 106

Motivation Winner-takes-all Contests Various Models of Contests Results

Assumptions

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 15/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

Assumptions

Assumption: All-pay

For each i ∈ N, bi(·) is strictly decreasing

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 15/28

SLIDE 108

Motivation Winner-takes-all Contests Various Models of Contests Results

Assumptions

Assumption: All-pay

For each i ∈ N, bi(·) is strictly decreasing

Assumption: Winner-pays

For each i ∈ N, pi(·) is strictly decreasing and bi(·) is constant

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 15/28

SLIDE 109

Motivation Winner-takes-all Contests Various Models of Contests Results

Assumptions

Assumption: All-pay

For each i ∈ N, bi(·) is strictly decreasing

Assumption: Winner-pays

For each i ∈ N, pi(·) is strictly decreasing and bi(·) is constant

¯ ei := supe∈[0,M]{bi(0) ≤ bi(e) + pi(e)}

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 15/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

Assumptions

Assumption: All-pay

For each i ∈ N, bi(·) is strictly decreasing

Assumption: Winner-pays

For each i ∈ N, pi(·) is strictly decreasing and bi(·) is constant

¯ ei := supe∈[0,M]{bi(0) ≤ bi(e) + pi(e)}

Assumption: M-bounding

For each i ∈ N, ¯ ei < M

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 15/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

Assumptions

Assumption: All-pay

For each i ∈ N, bi(·) is strictly decreasing

Assumption: Winner-pays

For each i ∈ N, pi(·) is strictly decreasing and bi(·) is constant

¯ ei := supe∈[0,M]{bi(0) ≤ bi(e) + pi(e)}

Assumption: M-bounding

For each i ∈ N, ¯ ei < M

Impact (trade-off) functions: for each i ∈ N, Ii(e) = bi(0)−bi(e)

pi(e) A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 15/28

SLIDE 112

Motivation Winner-takes-all Contests Various Models of Contests Results

Assumptions

Assumption: All-pay

For each i ∈ N, bi(·) is strictly decreasing

Assumption: Winner-pays

For each i ∈ N, pi(·) is strictly decreasing and bi(·) is constant

¯ ei := supe∈[0,M]{bi(0) ≤ bi(e) + pi(e)}

Assumption: M-bounding

For each i ∈ N, ¯ ei < M

Impact (trade-off) functions: for each i ∈ N, Ii(e) = bi(0)−bi(e)

pi(e)

Assumption: No-crossing

For each pair i, j ∈ N, if there is e∗ such that Ii(e∗) < Ij(e∗), then Ii(e) < Ij(e) for all e

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 15/28

SLIDE 113

Motivation Winner-takes-all Contests Various Models of Contests Results

A First Result

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 16/28

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Motivation Winner-takes-all Contests Various Models of Contests Results

A First Result

Proposition

If the contest Cf

pure satisfies All-pay and M-bounding, then it does

not have any Nash equilibrium.

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 16/28

SLIDE 115

Motivation Winner-takes-all Contests Various Models of Contests Results

A First Result

There is no Nash equilibrium in pure strategies

There is a unique winner 0 [ · · · ∞ e3 e5 e2 e4 e7 · · · · · · ˆ e7 π7(σ) = bi(e7) + pi(e7) < π7(σ−7, ˆ e7) = bi(ˆ e7) + pi(ˆ e7) There are several winners 0 [ · · · ∞ e3 e5 e2 e4 e7 = e1 · · · · · · ˆ e7 π7(σ) = bi(e7) + 0 < π7(σ−7, ˆ e7) = bi(ˆ e7) + pi(ˆ e7)

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 17/28

SLIDE 116

Motivation Winner-takes-all Contests Various Models of Contests Results

A First Result

Proposition

If the contest Cf

pure satisfies All-pay and M-bounding, then it does

not have any Nash equilibrium.

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 18/28

SLIDE 117

Motivation Winner-takes-all Contests Various Models of Contests Results

A First Result

Proposition

If the contest Cf

pure satisfies All-pay and M-bounding, then it does

not have any Nash equilibrium. We need mixed strategies

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 18/28

SLIDE 118

Motivation Winner-takes-all Contests Various Models of Contests Results

A First Result

Proposition

If the contest Cf

pure satisfies All-pay and M-bounding, then it does

not have any Nash equilibrium. We need mixed strategies No ties with positive probability in equilibrium

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 18/28

SLIDE 119

Motivation Winner-takes-all Contests Various Models of Contests Results

Outline

1

Motivation

2

Winner-takes-all Contests

3

Various Models of Contests

4

Results

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 19/28

SLIDE 120

Generalized Models

All-pay Winner-pays M-bounding No-crossing

SLIDE 121

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

SLIDE 122

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

SLIDE 123

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction

(Politically Contestable Rents)

SLIDE 124

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction
X
(Politically Contestable Rents)

SLIDE 125

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction
X
(Politically Contestable Rents)
3. Politically Contestable Transfers

SLIDE 126

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction
X
(Politically Contestable Rents)
3. Politically Contestable Transfers
X

SLIDE 127

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction
X
(Politically Contestable Rents)
3. Politically Contestable Transfers
X
4. Bertrand Competition

SLIDE 128

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction
X
(Politically Contestable Rents)
3. Politically Contestable Transfers
X
4. Bertrand Competition

X

SLIDE 129

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction
X
(Politically Contestable Rents)
3. Politically Contestable Transfers
X
4. Bertrand Competition

X

5. Varian’s Model of Sales

SLIDE 130

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction
X
(Politically Contestable Rents)
3. Politically Contestable Transfers
X
4. Bertrand Competition

X

5. Varian’s Model of Sales
X

SLIDE 131

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction
X
(Politically Contestable Rents)
3. Politically Contestable Transfers
X
4. Bertrand Competition

X

5. Varian’s Model of Sales
X
6. Federalism and Economic Growth

SLIDE 132

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction
X
(Politically Contestable Rents)
3. Politically Contestable Transfers
X
4. Bertrand Competition

X

5. Varian’s Model of Sales
X
6. Federalism and Economic Growth

X

X

SLIDE 133

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction
X
(Politically Contestable Rents)
3. Politically Contestable Transfers
X
4. Bertrand Competition

X

5. Varian’s Model of Sales
X
6. Federalism and Economic Growth

X

X
7. Market Makers

SLIDE 134

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction
X
(Politically Contestable Rents)
3. Politically Contestable Transfers
X
4. Bertrand Competition

X

5. Varian’s Model of Sales
X
6. Federalism and Economic Growth

X

X
7. Market Makers
X

SLIDE 135

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction
X
(Politically Contestable Rents)
3. Politically Contestable Transfers
X
4. Bertrand Competition

X

5. Varian’s Model of Sales
X
6. Federalism and Economic Growth

X

X
7. Market Makers
X
8. Litigation Systems

SLIDE 136

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction
X
(Politically Contestable Rents)
3. Politically Contestable Transfers
X
4. Bertrand Competition

X

5. Varian’s Model of Sales
X
6. Federalism and Economic Growth

X

X
7. Market Makers
X
8. Litigation Systems
X

SLIDE 137

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction
X
(Politically Contestable Rents)
3. Politically Contestable Transfers
X
4. Bertrand Competition

X

5. Varian’s Model of Sales
X
6. Federalism and Economic Growth

X

X
7. Market Makers
X
8. Litigation Systems
X
9. Timing Games

SLIDE 138

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction
X
(Politically Contestable Rents)
3. Politically Contestable Transfers
X
4. Bertrand Competition

X

5. Varian’s Model of Sales
X
6. Federalism and Economic Growth

X

X
7. Market Makers
X
8. Litigation Systems
X
9. Timing Games
X

SLIDE 139

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction
X
(Politically Contestable Rents)
3. Politically Contestable Transfers
X
4. Bertrand Competition

X

5. Varian’s Model of Sales
X
6. Federalism and Economic Growth

X

X
7. Market Makers
X
8. Litigation Systems
X
9. Timing Games
X

SLIDE 140

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction
X
(Politically Contestable Rents)
3. Politically Contestable Transfers
X
4. Bertrand Competition

X

5. Varian’s Model of Sales
X
6. Federalism and Economic Growth

X

X
7. Market Makers
X
8. Litigation Systems
X
9. Timing Games
X
Discretizing??

SLIDE 141

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction
X
(Politically Contestable Rents)
3. Politically Contestable Transfers
X
4. Bertrand Competition

X

5. Varian’s Model of Sales
X
6. Federalism and Economic Growth

X

X
7. Market Makers
X
8. Litigation Systems
X
9. Timing Games
X
Discretizing??

No Crossing??

SLIDE 142

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction
X
(Politically Contestable Rents)
3. Politically Contestable Transfers
X
4. Bertrand Competition

X

5. Varian’s Model of Sales
X
6. Federalism and Economic Growth

X

X
7. Market Makers
X
8. Litigation Systems
X
9. Timing Games
X
Discretizing??

No Crossing??

Other models

SLIDE 143

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction
X
(Politically Contestable Rents)
3. Politically Contestable Transfers
X
4. Bertrand Competition

X

5. Varian’s Model of Sales
X
6. Federalism and Economic Growth

X

X
7. Market Makers
X
8. Litigation Systems
X
9. Timing Games
X
Discretizing??

No Crossing??

Other models

Second Price Auction

SLIDE 144

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction
X
(Politically Contestable Rents)
3. Politically Contestable Transfers
X
4. Bertrand Competition

X

5. Varian’s Model of Sales
X
6. Federalism and Economic Growth

X

X
7. Market Makers
X
8. Litigation Systems
X
9. Timing Games
X
Discretizing??

No Crossing??

Other models

Second Price Auction Second Price All-Pay Auction

SLIDE 145

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction
X
(Politically Contestable Rents)
3. Politically Contestable Transfers
X
4. Bertrand Competition

X

5. Varian’s Model of Sales
X
6. Federalism and Economic Growth

X

X
7. Market Makers
X
8. Litigation Systems
X
9. Timing Games
X
Discretizing??

No Crossing??

Other models

Second Price Auction Second Price All-Pay Auction War of Attrition

SLIDE 146

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction
X
(Politically Contestable Rents)
3. Politically Contestable Transfers
X
4. Bertrand Competition

X

5. Varian’s Model of Sales
X
6. Federalism and Economic Growth

X

X
7. Market Makers
X
8. Litigation Systems
X
9. Timing Games
X
Discretizing??

No Crossing??

Other models

Second Price Auction

→ First Price Auction

Second Price All-Pay Auction

→ (First Price) All-pay Auction

War of Attrition

→ Timing Games

SLIDE 147

Generalized Models

All-pay Winner-pays M-bounding No-crossing

1. First Price Auction

X

2. All-Pay Auction
X
(Politically Contestable Rents)
3. Politically Contestable Transfers
X
4. Bertrand Competition

X

5. Varian’s Model of Sales
X
6. Federalism and Economic Growth

X

X
7. Market Makers
X
8. Litigation Systems
X
9. Timing Games
X
Discretizing??

No Crossing??

Other models

Second Price Auction Second Price All-Pay Auction War of Attrition

SLIDE 148

Motivation Winner-takes-all Contests Various Models of Contests Results

Classification

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 21/28

SLIDE 149

Motivation Winner-takes-all Contests Various Models of Contests Results

Classification

All-pay (bi functions strictly decreasing)

All-pay auction (Politically contestable rents) Politically contestable Transfers Varian’s model of sales Market makers Litigation systems (Silent) Timing games

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 21/28

SLIDE 150

Motivation Winner-takes-all Contests Various Models of Contests Results

Classification

All-pay (bi functions strictly decreasing)

All-pay auction (Politically contestable rents) Politically contestable Transfers Varian’s model of sales Market makers Litigation systems (Silent) Timing games

Winner-pays (pi functions strictly decreasing)

First price auction Bertrand competition Federalism and economic growth (No M-bounding)

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 21/28

SLIDE 151

Motivation Winner-takes-all Contests Various Models of Contests Results

Discussion

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 22/28

SLIDE 152

Motivation Winner-takes-all Contests Various Models of Contests Results

Discussion

Positive Features of the model Limitations of the model

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 22/28

SLIDE 153

Motivation Winner-takes-all Contests Various Models of Contests Results

Discussion

Positive Features of the model

Generality

Limitations of the model

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 22/28

SLIDE 154

Motivation Winner-takes-all Contests Various Models of Contests Results

Discussion

Positive Features of the model

Generality Powerful to model asymmetries

Limitations of the model

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 22/28

SLIDE 155

Motivation Winner-takes-all Contests Various Models of Contests Results

Discussion

Positive Features of the model

Generality Powerful to model asymmetries Accounts for non-linear functions

Limitations of the model

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 22/28

SLIDE 156

Motivation Winner-takes-all Contests Various Models of Contests Results

Discussion

Positive Features of the model

Generality Powerful to model asymmetries Accounts for non-linear functions

Limitations of the model

Complete information

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 22/28

SLIDE 157

Motivation Winner-takes-all Contests Various Models of Contests Results

Discussion

Positive Features of the model

Generality Powerful to model asymmetries Accounts for non-linear functions

Limitations of the model

Complete information Multiple prizes

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 22/28

SLIDE 158

Motivation Winner-takes-all Contests Various Models of Contests Results

Discussion

Positive Features of the model

Generality Powerful to model asymmetries Accounts for non-linear functions

Limitations of the model

Complete information Multiple prizes No-crossing assumption

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 22/28

SLIDE 159

Motivation Winner-takes-all Contests Various Models of Contests Results

Outline

1

Motivation

2

Winner-takes-all Contests

3

Various Models of Contests

4

Results

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 23/28

SLIDE 160

Motivation Winner-takes-all Contests Various Models of Contests Results

Characterization under All-pay and M-bounding

Theorem (Characterization under All-pay and M-bounding)

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 24/28

SLIDE 161

Motivation Winner-takes-all Contests Various Models of Contests Results

Characterization under All-pay and M-bounding

Theorem (Characterization under All-pay and M-bounding)

If either n = 2 or ¯ e1 > ¯ e2 > ¯ e3, then EP f has a unique Nash equibrium

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 24/28

SLIDE 162

Motivation Winner-takes-all Contests Various Models of Contests Results

Characterization under All-pay and M-bounding

Theorem (Characterization under All-pay and M-bounding)

If either n = 2 or ¯ e1 > ¯ e2 > ¯ e3, then EP f has a unique Nash equibrium

G∗

1(e) =

   e < 0 I2(e) 0 ≤ e ≤ ¯ e2 , 1 e > ¯ e2 G∗

2(e) =

   e < 0 I∗

1 (e)

0 ≤ e ≤ ¯ e2 , 1 e > ¯ e2 G∗

i (e) =

e < 0

1 e ≥ 0 A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 24/28

SLIDE 163

Motivation Winner-takes-all Contests Various Models of Contests Results

Characterization under All-pay and M-bounding

Theorem (Characterization under All-pay and M-bounding)

If either n = 2 or ¯ e1 > ¯ e2 > ¯ e3, then EP f has a unique Nash equibrium It takes two to tango!

G∗

1(e) =

   e < 0 I2(e) 0 ≤ e ≤ ¯ e2 , 1 e > ¯ e2 G∗

2(e) =

   e < 0 I∗

1 (e)

0 ≤ e ≤ ¯ e2 , 1 e > ¯ e2 G∗

i (e) =

e < 0

1 e ≥ 0 A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 24/28

SLIDE 164

Motivation Winner-takes-all Contests Various Models of Contests Results

Characterization under All-pay and M-bounding

Theorem (Characterization under All-pay and M-bounding)

If either n = 2 or ¯ e1 > ¯ e2 > ¯ e3, then EP f has a unique Nash equibrium

G∗

1(e) =

   e < 0 I2(e) 0 ≤ e ≤ ¯ e2 , 1 e > ¯ e2 G∗

2(e) =

   e < 0 I∗

1 (e)

0 ≤ e ≤ ¯ e2 , 1 e > ¯ e2 G∗

i (e) =

e < 0

1 e ≥ 0

Otherwise, EP f has a continuum of Nash equibria

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 24/28

SLIDE 165

Motivation Winner-takes-all Contests Various Models of Contests Results

Characterization under All-pay and M-bounding

Theorem (Characterization under All-pay and M-bounding)

If either n = 2 or ¯ e1 > ¯ e2 > ¯ e3, then EP f has a unique Nash equibrium

G∗

1(e) =

   e < 0 I2(e) 0 ≤ e ≤ ¯ e2 , 1 e > ¯ e2 G∗

2(e) =

   e < 0 I∗

1 (e)

0 ≤ e ≤ ¯ e2 , 1 e > ¯ e2 G∗

i (e) =

e < 0

1 e ≥ 0

Otherwise, EP f has a continuum of Nash equibria All the Nash equilibria give raise to the same payoffs:

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 24/28

SLIDE 166

Motivation Winner-takes-all Contests Various Models of Contests Results

Characterization under All-pay and M-bounding

Theorem (Characterization under All-pay and M-bounding)

If either n = 2 or ¯ e1 > ¯ e2 > ¯ e3, then EP f has a unique Nash equibrium

G∗

1(e) =

   e < 0 I2(e) 0 ≤ e ≤ ¯ e2 , 1 e > ¯ e2 G∗

2(e) =

   e < 0 I∗

1 (e)

0 ≤ e ≤ ¯ e2 , 1 e > ¯ e2 G∗

i (e) =

e < 0

1 e ≥ 0

Otherwise, EP f has a continuum of Nash equibria All the Nash equilibria give raise to the same payoffs: η1 = b1(¯ e2) + p1(¯ e2) and, for each i = 1, ηi = bi(0)

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 24/28

SLIDE 167

Motivation Winner-takes-all Contests Various Models of Contests Results

Characterization under All-pay and M-bounding

Theorem (Characterization under All-pay and M-bounding)

If either n = 2 or ¯ e1 > ¯ e2 > ¯ e3, then EP f has a unique Nash equibrium

G∗

1(e) =

   e < 0 I2(e) 0 ≤ e ≤ ¯ e2 , 1 e > ¯ e2 G∗

2(e) =

   e < 0 I∗

1 (e)

0 ≤ e ≤ ¯ e2 , 1 e > ¯ e2 G∗

i (e) =

e < 0

1 e ≥ 0

Otherwise, EP f has a continuum of Nash equibria All the Nash equilibria give raise to the same payoffs: η1 = b1(¯ e2) + p1(¯ e2) and, for each i = 1, ηi = bi(0) Implications of the result

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 24/28

SLIDE 168

Motivation Winner-takes-all Contests Various Models of Contests Results

Characterization under Winner-pays

Theorem (Characterization under Winner-pays and M-bounding)

Assume that, for each i ∈ N, bi(·) equals constant bi ∈ R.

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 25/28

SLIDE 169

Motivation Winner-takes-all Contests Various Models of Contests Results

Characterization under Winner-pays

Theorem (Characterization under Winner-pays and M-bounding)

Assume that, for each i ∈ N, bi(·) equals constant bi ∈ R. Let ¯ e1 > ¯

e2. Then, EP f has no Nash equilibrium in pure

strategies but it has a continuum of mixed Nash equilibria. The equilibrium payoffs are such that η1 ∈ (b1, b1 + p1(¯ e2)] and, for each i = 1, ηi = bi.

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 25/28

SLIDE 170

Motivation Winner-takes-all Contests Various Models of Contests Results

Characterization under Winner-pays

Theorem (Characterization under Winner-pays and M-bounding)

Assume that, for each i ∈ N, bi(·) equals constant bi ∈ R. Let ¯ e1 > ¯

e2. Then, EP f has no Nash equilibrium in pure

strategies but it has a continuum of mixed Nash equilibria. The equilibrium payoffs are such that η1 ∈ (b1, b1 + p1(¯ e2)] and, for each i = 1, ηi = bi. Let ¯ e1 = ¯

e2. Then, the set of Nash equilibria of EP f is

nonempty if and only if there is S ⊆ N, |S| > 1, such that, for each i ∈ S, Ti(¯ e2, S) = 0

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 25/28

SLIDE 171

Motivation Winner-takes-all Contests Various Models of Contests Results

Characterization under Winner-pays

Theorem (Characterization under Winner-pays and M-bounding)

Assume that, for each i ∈ N, bi(·) equals constant bi ∈ R. Let ¯ e1 > ¯

e2. Then, EP f has no Nash equilibrium in pure

strategies but it has a continuum of mixed Nash equilibria. The equilibrium payoffs are such that η1 ∈ (b1, b1 + p1(¯ e2)] and, for each i = 1, ηi = bi. Let ¯ e1 = ¯

e2. Then, the set of Nash equilibria of EP f is

nonempty if and only if there is S ⊆ N, |S| > 1, such that, for each i ∈ S, Ti(¯ e2, S) = 0 All the Nash equilibria give raise to the same payoffs:

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 25/28

SLIDE 172

Motivation Winner-takes-all Contests Various Models of Contests Results

Characterization under Winner-pays

Theorem (Characterization under Winner-pays and M-bounding)

Assume that, for each i ∈ N, bi(·) equals constant bi ∈ R. Let ¯ e1 > ¯

e2. Then, EP f has no Nash equilibrium in pure

strategies but it has a continuum of mixed Nash equilibria. The equilibrium payoffs are such that η1 ∈ (b1, b1 + p1(¯ e2)] and, for each i = 1, ηi = bi. Let ¯ e1 = ¯

e2. Then, the set of Nash equilibria of EP f is

nonempty if and only if there is S ⊆ N, |S| > 1, such that, for each i ∈ S, Ti(¯ e2, S) = 0 All the Nash equilibria give raise to the same payoffs: η1 = b1 + p1(¯ e2) and, for each i = 1, ηi = bi

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 25/28

SLIDE 173

Motivation Winner-takes-all Contests Various Models of Contests Results

Characterization under Winner-pays

Theorem (Characterization under Winner-pays and M-bounding)

Assume that, for each i ∈ N, bi(·) equals constant bi ∈ R. Let ¯ e1 > ¯

e2. Then, EP f has no Nash equilibrium in pure

strategies but it has a continuum of mixed Nash equilibria. The equilibrium payoffs are such that η1 ∈ (b1, b1 + p1(¯ e2)] and, for each i = 1, ηi = bi. Let ¯ e1 = ¯

e2. Then, the set of Nash equilibria of EP f is

nonempty if and only if there is S ⊆ N, |S| > 1, such that, for each i ∈ S, Ti(¯ e2, S) = 0 All the Nash equilibria give raise to the same payoffs: η1 = b1 + p1(¯ e2) and, for each i = 1, ηi = bi Implications of the result:

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 25/28

SLIDE 174

Motivation Winner-takes-all Contests Various Models of Contests Results

Characterization under Winner-pays

Theorem (Characterization under Winner-pays and M-bounding)

Assume that, for each i ∈ N, bi(·) equals constant bi ∈ R. Let ¯ e1 > ¯

e2. Then, EP f has no Nash equilibrium in pure

strategies but it has a continuum of mixed Nash equilibria. The equilibrium payoffs are such that η1 ∈ (b1, b1 + p1(¯ e2)] and, for each i = 1, ηi = bi. Let ¯ e1 = ¯

e2. Then, the set of Nash equilibria of EP f is

nonempty if and only if there is S ⊆ N, |S| > 1, such that, for each i ∈ S, Ti(¯ e2, S) = 0 All the Nash equilibria give raise to the same payoffs: η1 = b1 + p1(¯ e2) and, for each i = 1, ηi = bi Implications of the result: Auctions

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 25/28

SLIDE 175

Motivation Winner-takes-all Contests Various Models of Contests Results

Characterization under Winner-pays

Theorem (Characterization under Winner-pays and M-bounding)

Assume that, for each i ∈ N, bi(·) equals constant bi ∈ R. Let ¯ e1 > ¯

e2. Then, EP f has no Nash equilibrium in pure

strategies but it has a continuum of mixed Nash equilibria. The equilibrium payoffs are such that η1 ∈ (b1, b1 + p1(¯ e2)] and, for each i = 1, ηi = bi. Let ¯ e1 = ¯

e2. Then, the set of Nash equilibria of EP f is

nonempty if and only if there is S ⊆ N, |S| > 1, such that, for each i ∈ S, Ti(¯ e2, S) = 0 All the Nash equilibria give raise to the same payoffs: η1 = b1 + p1(¯ e2) and, for each i = 1, ηi = bi Implications of the result: Auctions and Bertrand competition

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 25/28

SLIDE 176

Motivation Winner-takes-all Contests Various Models of Contests Results

Characterization under Winner-pays

Corollary

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 26/28

SLIDE 177

Motivation Winner-takes-all Contests Various Models of Contests Results

Characterization under Winner-pays

Corollary

Take a general Bertrand competition model (BM) with n firms

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 26/28

SLIDE 178

Motivation Winner-takes-all Contests Various Models of Contests Results

Characterization under Winner-pays

Corollary

Take a general Bertrand competition model (BM) with n firms If the cost function is the same for all firms and exhibits strictly decreasing average costs,

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 26/28

SLIDE 179

Motivation Winner-takes-all Contests Various Models of Contests Results

Characterization under Winner-pays

Corollary

Take a general Bertrand competition model (BM) with n firms If the cost function is the same for all firms and exhibits strictly decreasing average costs, then there is no Nash equilibrium (neither pure, nor mixed)

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 26/28

SLIDE 180

Motivation Winner-takes-all Contests Various Models of Contests Results

Characterizations

Characterizations without M-bounding?

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 27/28

SLIDE 181

Motivation Winner-takes-all Contests Various Models of Contests Results

Characterizations

Characterizations without M-bounding? Ties

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 27/28

SLIDE 182

Motivation Winner-takes-all Contests Various Models of Contests Results

Conclusions

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 28/28

SLIDE 183

Motivation Winner-takes-all Contests Various Models of Contests Results

Conclusions

Generalization of the results included in the models satisfying All-pay assumption

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 28/28

SLIDE 184

Motivation Winner-takes-all Contests Various Models of Contests Results

Conclusions

Generalization of the results included in the models satisfying All-pay assumption Characterization result under Winner-pays assumption

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 28/28

SLIDE 185

Motivation Winner-takes-all Contests Various Models of Contests Results

Conclusions

Generalization of the results included in the models satisfying All-pay assumption Characterization result under Winner-pays assumption Further extensions:

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 28/28

SLIDE 186

Motivation Winner-takes-all Contests Various Models of Contests Results

Conclusions

Generalization of the results included in the models satisfying All-pay assumption Characterization result under Winner-pays assumption Further extensions:

1

Relax No-crossing

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 28/28

SLIDE 187

Motivation Winner-takes-all Contests Various Models of Contests Results

Conclusions

Generalization of the results included in the models satisfying All-pay assumption Characterization result under Winner-pays assumption Further extensions:

1

Relax No-crossing

2

Multiple prizes:

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 28/28

SLIDE 188

Motivation Winner-takes-all Contests Various Models of Contests Results

Conclusions

Generalization of the results included in the models satisfying All-pay assumption Characterization result under Winner-pays assumption Further extensions:

1

Relax No-crossing

2

Multiple prizes: K prizes

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 28/28

SLIDE 189

Motivation Winner-takes-all Contests Various Models of Contests Results

Conclusions

Generalization of the results included in the models satisfying All-pay assumption Characterization result under Winner-pays assumption Further extensions:

1

Relax No-crossing

2

Multiple prizes: K prizes → K + 1 compete

3

Incomplete information

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 28/28

SLIDE 190

Motivation Winner-takes-all Contests Various Models of Contests Results

Conclusions

Generalization of the results included in the models satisfying All-pay assumption Characterization result under Winner-pays assumption Further extensions:

1

Relax No-crossing

2

Multiple prizes: K prizes → K + 1 compete

3

Incomplete information

Other applications:

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 28/28

SLIDE 191

Motivation Winner-takes-all Contests Various Models of Contests Results

Conclusions

Generalization of the results included in the models satisfying All-pay assumption Characterization result under Winner-pays assumption Further extensions:

1

Relax No-crossing

2

Multiple prizes: K prizes → K + 1 compete

3

Incomplete information

Other applications:

1

Hybrid auctions

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 28/28

SLIDE 192

Motivation Winner-takes-all Contests Various Models of Contests Results

Conclusions

Generalization of the results included in the models satisfying All-pay assumption Characterization result under Winner-pays assumption Further extensions:

1

Relax No-crossing

2

Multiple prizes: K prizes → K + 1 compete

3

Incomplete information

Other applications:

1

Hybrid auctions

2

New tool to analyze Bertrand competition models

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 28/28

SLIDE 193

Motivation Winner-takes-all Contests Various Models of Contests Results

Conclusions

Generalization of the results included in the models satisfying All-pay assumption Characterization result under Winner-pays assumption Further extensions:

1

Relax No-crossing

2

Multiple prizes: K prizes → K + 1 compete

3

Incomplete information

Other applications:

1

Hybrid auctions

2

New tool to analyze Bertrand competition models

3

. . .

A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 28/28

SLIDE 194

A Unifying Model of Winner-takes-all Contests

Julio Gonz´ alez-D´ ıaz

Kellogg School of Management (CMS-EMS) Northwestern University and Research Group in Economic Analysis Universidad de Vigo