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 ∈ R N + be the initial rights vector: P = 1 − � i ∈ N α i > 0 0 < α 1 < α 2 < · · · < α n Let δ ∈ (0 , 1) be the discount factor pure Cake sharing game: Γ α,δ = < N, { A i } i ∈ N , { π i } i ∈ N > A i = [0 , ∞ ) is the set of pure strategies of player i ∈ N π i is the payoff function of player i ∈ N , defined by: δ t i α i t i ≤ max j � = i t j Ties?? π i ( t 1 , . . . , t n ) = δ t i ( α i + P ) t i > max j � = i t j A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 5/28
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
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
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 [ · · · ∞ · · · · · · t 3 t 5 t 2 t 4 t 7 π 7 ( t ) = δ t 7 ( α 7 + P ) A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 6/28
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 ˆ t 7 0 [ · · · ∞ · · · · · · t 3 t 5 t 2 t 4 t 7 t 7 ) = δ ˆ π 7 ( t ) = δ t 7 ( α 7 + P ) < π 7 ( t − 7 , ˆ t 7 ( α 7 + P ) A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 6/28
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 ˆ t 7 0 [ · · · ∞ · · · · · · t 3 t 5 t 2 t 4 t 7 t 7 ) = δ ˆ π 7 ( t ) = δ t 7 ( α 7 + P ) < π 7 ( t − 7 , ˆ t 7 ( α 7 + P ) There are several last claimants · · · ∞ 0 [ t 3 t 5 t 2 · · · · · · t 4 t 7 = t 1 π 7 ( t ) = δ t 7 α 7 A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 6/28
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 ˆ t 7 0 [ · · · ∞ · · · · · · t 3 t 5 t 2 t 4 t 7 t 7 ) = δ ˆ π 7 ( t ) = δ t 7 ( α 7 + P ) < π 7 ( t − 7 , ˆ t 7 ( α 7 + P ) There are several last claimants ˆ t 7 · · · ∞ 0 [ t 3 t 5 t 2 · · · · · · t 4 t 7 = t 1 t 7 ) = δ ˆ π 7 ( t ) = δ t 7 α 7 < π 7 ( t − 7 , ˆ t 7 ( α 7 + P ) A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 6/28
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 ˆ t 7 0 [ · · · ∞ · · · · · · t 3 t 5 t 2 t 4 t 7 t 7 ) = δ ˆ π 7 ( t ) = δ t 7 ( α 7 + P ) < π 7 ( t − 7 , ˆ t 7 ( α 7 + P ) There are several last claimants ˆ t 7 · · · ∞ 0 [ t 3 t 5 t 2 · · · · · · t 4 t 7 = t 1 t 7 ) = δ ˆ π 7 ( t ) = δ t 7 α 7 < π 7 ( t − 7 , ˆ t 7 ( α 7 + P ) Options: A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 6/28
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 ˆ t 7 0 [ · · · ∞ · · · · · · t 3 t 5 t 2 t 4 t 7 t 7 ) = δ ˆ π 7 ( t ) = δ t 7 ( α 7 + P ) < π 7 ( t − 7 , ˆ t 7 ( α 7 + P ) There are several last claimants ˆ t 7 · · · ∞ 0 [ t 3 t 5 t 2 · · · · · · t 4 t 7 = t 1 t 7 ) = δ ˆ π 7 ( t ) = δ t 7 α 7 < π 7 ( t − 7 , ˆ t 7 ( α 7 + P ) Options: Discretizing?? A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 6/28
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 ˆ t 7 0 [ · · · ∞ · · · · · · t 3 t 5 t 2 t 4 t 7 t 7 ) = δ ˆ π 7 ( t ) = δ t 7 ( α 7 + P ) < π 7 ( t − 7 , ˆ t 7 ( α 7 + P ) There are several last claimants ˆ t 7 · · · ∞ 0 [ t 3 t 5 t 2 · · · · · · t 4 t 7 = t 1 t 7 ) = δ ˆ π 7 ( t ) = δ t 7 α 7 < π 7 ( t − 7 , ˆ t 7 ( α 7 + P ) Options: Discretizing?? Mixing?? A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 6/28
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
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
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 = ( G 1 , G 2 , . . . , G n ) , π i ( G − i , t ) = A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 7/28
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 = ( G 1 , G 2 , . . . , G n ) , π i ( G − i , t ) = t A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 7/28
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 = ( G 1 , G 2 , . . . , G n ) , G j ( t − ) π i ( G − i , t ) = A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 7/28
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 = ( G 1 , G 2 , . . . , G n ) , � G j ( t − ) π i ( G − i , t ) = j � = i A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 7/28
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 = ( G 1 , G 2 , . . . , G n ) , � G j ( t − ) π i ( G − i , t ) = P j � = i A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 7/28
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 = ( G 1 , G 2 , . . . , G n ) , � G j ( t − ) π i ( G − i , t ) = α i + P j � = i A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 7/28
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 = ( G 1 , G 2 , . . . , G n ) , δ t ( α i + P � G j ( t − )) π i ( G − i , t ) = j � = i A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 7/28
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 = ( G 1 , G 2 , . . . , G n ) , δ t ( α i + P � G j ( t − )) π i ( G − i , t ) = j � = i A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 7/28
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
Motivation Winner-takes-all Contests Various Models of Contests Results The Characterization Result Theorem Let Γ α,δ be an n -player cake sharing game. A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 8/28
Motivation Winner-takes-all Contests Various Models of Contests Results The Characterization Result Theorem Let Γ α,δ be an n -player cake sharing game. Then, Γ α,δ has a unique Nash equilibrium. Moreover. . . A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 8/28
Motivation Winner-takes-all Contests Various Models of Contests Results The Characterization Result Theorem Let Γ α,δ be an n -player cake sharing game. Then, Γ α,δ has a unique Nash equilibrium. Moreover. . . A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 8/28
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
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
Motivation Winner-takes-all Contests Various Models of Contests Results Second Example: Sharing a Market (Varian 1980) Firm 1 Consumers Firm 2 A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 9/28
Motivation Winner-takes-all Contests Various Models of Contests Results Second Example: Sharing a Market (Varian 1980) Firm 1: α 1 α 1 Consumers Firm 2 A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 9/28
Motivation Winner-takes-all Contests Various Models of Contests Results Second Example: Sharing a Market (Varian 1980) Firm 1: α 1 α 2 α 1 Consumers Firm 2: α 2 A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 9/28
Motivation Winner-takes-all Contests Various Models of Contests Results Second Example: Sharing a Market (Varian 1980) Firm 1: α 1 α 2 α 1 P Consumers Firm 2: α 2 A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 9/28
Motivation Winner-takes-all Contests Various Models of Contests Results Second Example: Sharing a Market (Varian 1980) p 1 ≤ ¯ p Firm 1: α 1 α 2 α 1 P Consumers p 2 ≤ ¯ p Firm 2: α 2 A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 9/28
Motivation Winner-takes-all Contests Various Models of Contests Results Second Example: Sharing a Market (Varian 1980) p 1 > p 2 − → p 1 α 1 p 1 ≤ ¯ p Firm 1: α 1 α 2 α 1 P Consumers p 1 > p 2 − → p 2 ( α 2 + P ) p 2 ≤ ¯ p Firm 2: α 2 A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 9/28
Motivation Winner-takes-all Contests Various Models of Contests Results Second Example: Sharing a Market (Varian 1980) p 1 > p 2 − → p 1 α 1 p 1 ≤ ¯ p p 1 < p 2 − → p 1 ( α 1 + P ) Firm 1: α 1 α 2 α 1 P Consumers p 1 > p 2 − → p 2 ( α 2 + P ) p 2 ≤ ¯ p p 1 < p 2 − → p 2 α 2 Firm 2: α 2 A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 9/28
Motivation Winner-takes-all Contests Various Models of Contests Results Second Example: Sharing a Market (Varian 1980) p 1 > p 2 − → p 1 α 1 p 1 ≤ ¯ p p 1 < p 2 − → p 1 ( α 1 + P ) → p 1 ( α 1 + P Firm 1: α 1 p 1 = p 2 − 2 ) α 2 α 1 P Consumers p 1 > p 2 − → p 2 ( α 2 + P ) p 2 ≤ ¯ p p 1 < p 2 − → p 2 α 2 → p 2 ( α 2 + P p 1 = p 2 − 2 ) Firm 2: α 2 A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 9/28
Motivation Winner-takes-all Contests Various Models of Contests Results Second Example: Sharing a Market (Varian 1980) p 1 > p 2 − → p 1 α 1 p 1 ≤ ¯ p p 1 < p 2 − → p 1 ( α 1 + P ) → p 1 ( α 1 + P Firm 1: α 1 p 1 = p 2 − 2 ) α 2 α 1 P Consumers p 1 > p 2 − → p 2 ( α 2 + P ) p 2 ≤ ¯ p p 1 < p 2 − → p 2 α 2 → p 2 ( α 2 + P p 1 = p 2 − 2 ) Firm 2: α 2 A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 9/28
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
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 α i loyal consumers Strategic consumers: P Higher admissible price: ¯ p A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 10/28
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 α i 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
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 α i loyal consumers Strategic consumers: P Higher admissible price: ¯ p The equilibrium of the pricing game Only the two firms with less loyal consumers “compete” A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 10/28
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 α i 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” A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 10/28
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 α i 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
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 α i 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
Motivation Winner-takes-all Contests Various Models of Contests Results Outline Motivation 1 Winner-takes-all Contests 2 Various Models of Contests 3 Results 4 A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 11/28
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
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
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
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
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 A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 12/28
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
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
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
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: b i ( e ) : [0 , M ] → R A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 12/28
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: b i ( e ) : [0 , M ] → R continuous and weakly decreasing A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 12/28
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: b i ( e ) : [0 , M ] → R continuous and weakly decreasing Prize Payoff Funtions: p i ( e ) : [0 , M ] → R A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 12/28
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: b i ( e ) : [0 , M ] → R continuous and weakly decreasing Prize Payoff Funtions: p i ( e ) : [0 , M ] → R continuous and weakly decreasing A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 12/28
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: b i ( e ) : [0 , M ] → R continuous and weakly decreasing Prize Payoff Funtions: p i ( e ) : [0 , M ] → R continuous and weakly decreasing p i (0) > 0 A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 12/28
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: b i ( e ) : [0 , M ] → R continuous and weakly decreasing Prize Payoff Funtions: p i ( e ) : [0 , M ] → R continuous and weakly decreasing p i (0) > 0 Tie Payoff Functions: A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 12/28
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
Motivation Winner-takes-all Contests Various Models of Contests Results Winner-takes-all Contests Tie Payoff Functions T i : [0 , M ] × 2 N \{∅} → R : A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 13/28
Motivation Winner-takes-all Contests Various Models of Contests Results Winner-takes-all Contests Tie Payoff Functions T i : [0 , M ] × 2 N \{∅} → R : T1) T i ( e, { i } ) = p i ( e ) A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 13/28
Motivation Winner-takes-all Contests Various Models of Contests Results Winner-takes-all Contests Tie Payoff Functions T i : [0 , M ] × 2 N \{∅} → R : T1) T i ( e, { i } ) = p i ( e ) T2) For each S such that i / ∈ S , T i ( e, S ) = 0 A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 13/28
Motivation Winner-takes-all Contests Various Models of Contests Results Winner-takes-all Contests Tie Payoff Functions T i : [0 , M ] × 2 N \{∅} → R : T1) T i ( e, { i } ) = p i ( e ) T2) For each S such that i / ∈ S , T i ( e, S ) = 0 T3) . . . A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 13/28
Motivation Winner-takes-all Contests Various Models of Contests Results Winner-takes-all Contests Tie Payoff Functions T i : [0 , M ] × 2 N \{∅} → R : T1) T i ( e, { i } ) = p i ( e ) T2) For each S such that i / ∈ S , T i ( e, S ) = 0 T3) . . . Examples A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 13/28
Motivation Winner-takes-all Contests Various Models of Contests Results Winner-takes-all Contests Tie Payoff Functions T i : [0 , M ] × 2 N \{∅} → R : T1) T i ( e, { i } ) = p i ( e ) T2) For each S such that i / ∈ S , T i ( e, S ) = 0 T3) . . . Examples � p i ( e ) i ∈ S | S | T i ( e, S ) = 0 otherwise A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 13/28
Motivation Winner-takes-all Contests Various Models of Contests Results Winner-takes-all Contests Tie Payoff Functions T i : [0 , M ] × 2 N \{∅} → R : T1) T i ( e, { i } ) = p i ( e ) T2) For each S such that i / ∈ S , T i ( e, S ) = 0 T3) . . . Examples � p i ( e ) i ∈ S | S | T i ( e, S ) = 0 otherwise � p i ( e ) { i } = S T i ( e, S ) = 0 otherwise A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 13/28
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
Motivation Winner-takes-all Contests Various Models of Contests Results Winner-takes-all Contests Contest form: f := ( { b i } i ∈ N , { p i } i ∈ N , { T i } i ∈ N ) A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 14/28
Motivation Winner-takes-all Contests Various Models of Contests Results Winner-takes-all Contests Contest form: f := ( { b i } i ∈ N , { p i } i ∈ N , { T i } i ∈ N ) For each σ = ( e 1 , . . . , e n ) ∈ [0 , M ] n , w σ := argmax i ∈ N { e i } A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 14/28
Motivation Winner-takes-all Contests Various Models of Contests Results Winner-takes-all Contests Contest form: f := ( { b i } i ∈ N , { p i } i ∈ N , { T i } i ∈ N ) For each σ = ( e 1 , . . . , e n ) ∈ [0 , M ] n , w σ := argmax i ∈ N { e i } Contest with pure strategies: A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 14/28
Motivation Winner-takes-all Contests Various Models of Contests Results Winner-takes-all Contests Contest form: f := ( { b i } i ∈ N , { p i } i ∈ N , { T i } i ∈ N ) For each σ = ( e 1 , . . . , e n ) ∈ [0 , M ] n , w σ := argmax i ∈ N { e i } Contest with pure strategies: C f pure := ( { E i } i ∈ N , { u i } i ∈ N ) , where A Unifying Model of Winner-takes-all Contests Julio Gonz´ alez-D´ ıaz 14/28
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