Hybrid All-Pay and Winner-Pay Contests Online Seminar at SSE in Stockholm, June 10, 2020 Johan N. M. Lagerl¨ of Dept. of Economics, U. of Copenhagen Email: johan.lagerlof@econ.ku.dk Website: www.johanlagerlof.com June 9, 2020 J. Lagerl¨ of (U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 10, 2020 1 / 26
Introduction: Contests (1/3) Contests are common in economic, social and political life: sports, military combat, war; political compet’n, rent-seeking for rents allocated by regulator; marketing, advertising, patent races, relative reward schemes in firms, beauty contests between firms, litigation. A common modeling approach: Contestant i chooses x i ≥ 0 to max π i = v i p i ( x 1 , x 2 , . . . x n ) − x i x r where p i is a differentiable contest success funct. ( p i = j ). i � n j =1 x r Gordon Tullock’s motivation for studying the dissipation rent: Empirical studies in the 1950s: DWL appears to be tiny. Tullock: Maybe a part of profits adds to the cost of monopoly. p m DWL MC quantity J. Lagerl¨ of (U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 10, 2020 2 / 26
Introduction: Hybrid contest (2/3) A hybrid contest : In some contests, each contestant can make both all-pay investments and winner-pay investments . Example: The competitive bidding to host the Olympic games. All-pay investments : Candidate cities spend money upfront, with the goal of persuading members of the IOC. Winner-pay investments : A city commits to build new stadia and invest in safety arrangements if being awarded the Games. To fix ideas, consider the following formalization: Contestant i chooses x i ≥ 0 and y i ≥ 0 to maximize π i = ( v i − y i ) p i ( s 1 , s 2 , . . . s n ) − x i , subject to s i = f ( x i , y i ). J. Lagerl¨ of (U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 10, 2020 3 / 26
Introduction: Other examples (3/3) Further examples Competition for a government contract or grant: All-pay investments : Time/effort spent on preparing proposal. Winner-pay investments : Commit to ambitious customer service. A political election: All-pay investments : Campaign expenditures. Winner-pay investments : Electoral promises (costly if they deviate from the politician’s own ideal policy). Rent seeking to win monopoly rights of a regulated market: All-pay investments : Ex ante bribes (how Tullock modeled it). Winner-pay investments : Conditional bribes. J. Lagerl¨ of (U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 10, 2020 4 / 26
Literature review (1/1) Two earlier papers that model a hybrid contest: Haan and Schonbeek (2003) . They assume Cobb-Douglas—which here is quite restrictive. Melkonyan (2013) . CES but with σ ≥ 1. Symmetric model. Hard to check SOC. My analysis: (i) other approach which yields easy-to-check existence condition; (ii) assumes general production function and CSF; (iii) studies both symmetric and asymmetric models. Other contest models with more than one influence channel: Sabotage in contests (improve own performance and sabotage the others’ performance): Konrad (2000), Chen (2003). War and conflict (choice of production and appropriation): Hirschleifer (1991) and Skaperdas and Syroploulos (1997). Multiple all-pay “arms” (maybe with different costs): Arbatskaya and Mialon (2010). J. Lagerl¨ of (U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 10, 2020 5 / 26
A model of a hybrid contest (1/3) n ≥ 2 contestants try to win an indivisible prize. Contestant i chooses x i ≥ 0 and y i ≥ 0 to maximize the following payoff: π i = ( v i − y i ) p i ( s ) − x i , subject to s i = f ( x i , y i ) , where s = ( s 1 , s 2 , . . . , s n ) and s i ≥ 0 is contestant i ’s score . v i > 0 is i ’s valuation of the prize. p i ( s ) is i ’s prob. of winning (or contest success function, CSF). x i is the all-pay investment : paid whether i wins or not. y i is the winner-pay investment : paid i.f.f. i wins. It is a one-shot game where the contestants choose their investments ( x i , y i ) simultaneously with each other. J. Lagerl¨ of (U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 10, 2020 6 / 26
A model of a hybrid contest (2/3) Assumptions about the production function f ( x i , y i ) Thrice continuously differentiable in its arguments. Strictly increasing in each of its arguments. Strictly quasiconcave. Homogeneous of degree t > 0: ∀ k > 0 f ( kx i , ky i ) = k t f ( x i , y i ). Satisfies f (0 , 0) = 0. Inada conditions to rule out x i = 0 or y i = 0. Example (CES): � � t σ σ − 1 , σ − 1 σ − 1 f ( x i , y i ) = α x + (1 − α ) y α ∈ (0 , 1) , σ > 0 σ σ i i J. Lagerl¨ of (U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 10, 2020 7 / 26
A model of a hybrid contest (3/3) Assumptions about the contest success function p i ( s ) n n � � p i ( s ) ∈ [0 , 1] , with p i ( 0 ) ≤ 1 and p i ( s ) = 1 for all s � = 0 , i =1 i =1 Twice continuously differentiable for all s ∈ ℜ n + \ { 0 } . Strictly increasing and strictly concave in s i . Strictly decreasing in s j for all j � = i . If s i = 0 and s j > 0 for some j � = i , then p i ( s ) = 0. Any values of p i ( 0 ) ≤ 1 allowed, although p i ( 0 ) < 1 for all i . Later I assume that p i ( s ) is homogeneous in s . Example (extended Tullock): w i s r i p i ( s ) = � n , w i , r > 0 . j =1 w j s r j J. Lagerl¨ of (U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 10, 2020 8 / 26
Analysis (1/7) One possible approach: Plug the production function into the CSF. Take FOCs w.r.t. x i and y i . Used by Haan and Schoonbeek (2003) and Melkonyan (2013), assuming Cobb-Douglas and CES, respectively. My approach: Solve for contestant i ’s best reply in two steps: 1 Compute the conditional factor demands. That is, derive optimal x i and y i , given s (so also given s i ). 2 Plug the factor demands into the payoff and then characterize contestant i ’s optimal score s i (given s − i ). Important advantage: a single choice variable at 2, so easier to determine what conditions are required for equilibrium existence. J. Lagerl¨ of (U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 10, 2020 9 / 26
Contestant i solves (for fixed p i ): min x i , y i p i y i + x i , subject to f ( x i , y i ) = s i . The first-order conditions ( λ i is the Lagrange multiplier): ∂ L i ∂ L i = 1 − λ i f 1 ( x i , y i ) = 0 , = p i − λ i f 2 ( x i , y i ) = 0 . ∂ x i ∂ y i So, by combining the FOCs: � 1 � x i � � 1 = f 1 ( x i , y i ) def = g ⇒ x i = y i h , p i f 2 ( x i , y i ) y i p i def = g − 1 ). where h is the inverse of g (i.e., h By plugging back into s i = f ( x i , y i ) and rewriting, we obtain: � � 1 � 1 � s i t Y i ( s i , p i ) = , X i ( s i , p i ) = Y i ( s i , p i ) h . f ( h (1 / p i ) , 1) p i Contestant i ’s payoff: π i ( s ) = p i ( s ) v i − C i [ s i , p i ( s )], where def C i [ s i , p i ( s )] = p i ( s ) Y i [ s i , p i ( s )] + X i [ s i , p i ( s )] . A Nash equilibrium of the hybrid contest: � � A profile s ∗ such that π i ( s ∗ ) ≥ π i s i , s ∗ , all i and all s i ≥ 0. − i J. Lagerl¨ of (U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 10, 2020 10 / 26
Analysis (3/7) The cost-minimization problem and the h function � � y i x i m x i slope = − g y i y i 45 ◦ h ( m ) 45 ◦ s i = f ( x i , y i ) Y � � x i g slope = − 1 y i p i x i x i m y i X (a) Cost (b) Graph of the g (c) Graph of the h minimization. function. function. J. Lagerl¨ of (U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 10, 2020 11 / 26
Analysis (4/7) Equilibrium existence Define the following elasticities: � � � � � � � � 1 1 f 1 h , 1 h def 1 pi pi The elasticity of output w.r.t. x i : η = . � � � � p i 1 f h , 1 pi � � h ′ � � 1 1 1 def � . pi pi The elasticity of substitution: σ = − p i � 1 h pi = ∂ p i def s i The elasticity of the win probability w.r.t. s i : ε i ( s ) p i . ∂ s i We have that η ∈ (0 , t ), σ > 0, and ε i ∈ (0 , 1). Assumption 1. The production function and the CSF satisfy: � � � � 1 1 t ≤ 1 and ε i ( s ) η σ ≤ 2 (for all p i and s ). p i p i Proposition 1. Suppose Assumption 1 is satisfied. Then there exists a pure strategy Nash equilibrium of the hybrid contest. J. Lagerl¨ of (U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 10, 2020 12 / 26
Assume a CES production function, t = 1, r ≤ 1, and w i s r w i i p i ( s ) = � n and p i (0 , ∙ ∙ ∙ , 0) = � n . j =1 w j s r j =1 w j j α 1 1 3 ( r σ − 2 ) 2 σ def 4 Θ( σ, r ) = 1 1+ ( r σ − 2 ) 2 σ α ∗ 1 Assumption 1 satisfied 4 σ 0 σ ∗ 0 2 4 15 20 r r r r J. Lagerl¨ of (U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 10, 2020 13 / 26
Recommend
More recommend
