Hybrid All-Pay and Winner-Pay Contests

Transcription

1 Hybrid All-Pay and Winner-Pay Contests Seminar at DICE in Düsseldorf, June 5, 208 Johan N. M. Lagerlöf Dept. of Economics, U. of Copenhagen Website: June 2, 208 J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, 208 / 24

2 Introduction: What is a hybrid contest? /2) A hybrid contest: In some economic, social, or political situation, each one of a number of economic agents try to win an indivisible prize. To increase her probability of winning, each contestant makes 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 subject to s i = f x i, y i ). π i = v i y i ) p i s, s 2,... s n ) x i, J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

3 Introduction: What is a hybrid contest? /2) A hybrid contest: In some economic, social, or political situation, each one of a number of economic agents try to win an indivisible prize. To increase her probability of winning, each contestant makes 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 subject to s i = f x i, y i ). π i = v i y i ) p i s, s 2,... s n ) x i, J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

4 Introduction: What is a hybrid contest? /2) A hybrid contest: In some economic, social, or political situation, each one of a number of economic agents try to win an indivisible prize. To increase her probability of winning, each contestant makes 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 subject to s i = f x i, y i ). π i = v i y i ) p i s, s 2,... s n ) x i, J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

5 Introduction: Other examples 2/2) 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. Tullock s motivation: Empirical studies in the 950s: DWL appears to be tiny. Tullock: Maybe a part of profits adds to the cost of monopoly. J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

6 Literature Review /2) Two earlier papers that model a hybrid contest: Haan and Schonbeek 2003). They assume Cobb-Douglas which here is quite restrictive. Melkoyan 203). CES but with σ. 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 99) and Skaperdas and Syroploulos 997). Multiple all-pay arms maybe with different costs): Arbatskaya and Mialon 200). J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

7 Literature Review /2) Two earlier papers that model a hybrid contest: Haan and Schonbeek 2003). They assume Cobb-Douglas which here is quite restrictive. Melkoyan 203). CES but with σ. 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 99) and Skaperdas and Syroploulos 997). Multiple all-pay arms maybe with different costs): Arbatskaya and Mialon 200). J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

8 Literature Review 2/2) Multidimensional procurement) auctions: Che 2003), Branck 997), Asker and Cantillon 2008). Firms bid on both price and many dimensions of) quality. The components of each bid jointly determine a score. Auctioneer chooses bidder with highest score. Differences: In their models, not both all-pay and winner-pay ingredients. Not a probabilistic CSF. Optimal design of a research contest: Che and Gale 2003). A principal wants to procure an innovation. Fimrs choose both quality of innovation and the prize if winning. Thus, effectively, both all-pay and winner-pay ingredients. Differences: Not a probabilistic CSF so mixed strategy eq.), linear production function, mechanism design approach. J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

9 Literature Review 2/2) Multidimensional procurement) auctions: Che 2003), Branck 997), Asker and Cantillon 2008). Firms bid on both price and many dimensions of) quality. The components of each bid jointly determine a score. Auctioneer chooses bidder with highest score. Differences: In their models, not both all-pay and winner-pay ingredients. Not a probabilistic CSF. Optimal design of a research contest: Che and Gale 2003). A principal wants to procure an innovation. Fimrs choose both quality of innovation and the prize if winning. Thus, effectively, both all-pay and winner-pay ingredients. Differences: Not a probabilistic CSF so mixed strategy eq.), linear production function, mechanism design approach. J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

10 A model of a hybrid contest /2) n 2 contestants try to win an indivisible prize. Contestant i chooses x i 0 and y i 0 to maximize the following expected payoff: π i = v i y i ) p i s) x i, subject to s i = f x i, y i ), where s = s, 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. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

11 A model of a hybrid contest 2/2) Assumptions about p i s): Twice continuously differentiable in its arguments. Strictly increasing and strictly concave in s i. Strictly decreasing in s j for all j i. The contest is won by someone: n j= p js) =. Later I assume that p i s) is homogeneous in s. Assumptions about 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 ). Inada conditions to rule out x i = 0 or y i = 0. Examples: p i s) = w i si r n j= w, f x jsj r i, y i ) = [ αx σ σ ] + α)y σ tσ σ σ J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

12 A model of a hybrid contest 2/2) Assumptions about p i s): Twice continuously differentiable in its arguments. Strictly increasing and strictly concave in s i. Strictly decreasing in s j for all j i. The contest is won by someone: n j= p js) =. Later I assume that p i s) is homogeneous in s. Assumptions about 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 ). Inada conditions to rule out x i = 0 or y i = 0. Examples: p i s) = w i si r n j= w, f x jsj r i, y i ) = [ αx σ σ ] + α)y σ tσ σ σ J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

13 A model of a hybrid contest 2/2) Assumptions about p i s): Twice continuously differentiable in its arguments. Strictly increasing and strictly concave in s i. Strictly decreasing in s j for all j i. The contest is won by someone: n j= p js) =. Later I assume that p i s) is homogeneous in s. Assumptions about 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 ). Inada conditions to rule out x i = 0 or y i = 0. Examples: p i s) = w i si r n j= w, f x jsj r i, y i ) = [ αx σ σ ] + α)y σ tσ σ σ J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

14 Analysis /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 Melkoyan 203), assuming Cobb-Douglas and CES, respectively. My approach: Solve for contestant i s best reply in two steps: 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. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

15 Analysis /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 Melkoyan 203), assuming Cobb-Douglas and CES, respectively. My approach: Solve for contestant i s best reply in two steps: 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. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

16 Contestant i solves for fixed p i ): min xi,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 x i = λ i f x i, y i ) = 0, L i y i = p i λ i f 2 x i, y i ) = 0. So, by combining the FOCs: = f x i, y i ) def = g p i f 2 x i, y i ) xi where h is the inverse of g i.e., h def = g ). y i ) ) x i = y i h, p i By plugging back into s i = f x i, y i ) and rewriting, we obtain: [ ] t s i Y i s i, p i ) =, X i s i, p i ) = Y i s i, p i ) h f h /p i ), ) Contestant i s payoff: π i s) = p i s) v i C i [s i, p i s)], where C i [s i, p i s)] def = 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 si, s i), all i and all si 0. J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24 p i ).

17 Contestant i solves for fixed p i ): min xi,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 x i = λ i f x i, y i ) = 0, L i y i = p i λ i f 2 x i, y i ) = 0. So, by combining the FOCs: = f x i, y i ) def = g p i f 2 x i, y i ) xi where h is the inverse of g i.e., h def = g ). y i ) ) x i = y i h, p i By plugging back into s i = f x i, y i ) and rewriting, we obtain: [ ] t s i Y i s i, p i ) =, X i s i, p i ) = Y i s i, p i ) h f h /p i ), ) Contestant i s payoff: π i s) = p i s) v i C i [s i, p i s)], where C i [s i, p i s)] def = 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 si, s i), all i and all si 0. J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24 p i ).

18 Contestant i solves for fixed p i ): min xi,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 x i = λ i f x i, y i ) = 0, L i y i = p i λ i f 2 x i, y i ) = 0. So, by combining the FOCs: = f x i, y i ) def = g p i f 2 x i, y i ) xi where h is the inverse of g i.e., h def = g ). y i ) ) x i = y i h, p i By plugging back into s i = f x i, y i ) and rewriting, we obtain: [ ] t s i Y i s i, p i ) =, X i s i, p i ) = Y i s i, p i ) h f h /p i ), ) Contestant i s payoff: π i s) = p i s) v i C i [s i, p i s)], where C i [s i, p i s)] def = 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 si, s i), all i and all si 0. J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24 p i ).

19 Contestant i solves for fixed p i ): min xi,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 x i = λ i f x i, y i ) = 0, L i y i = p i λ i f 2 x i, y i ) = 0. So, by combining the FOCs: = f x i, y i ) def = g p i f 2 x i, y i ) xi where h is the inverse of g i.e., h def = g ). y i ) ) x i = y i h, p i By plugging back into s i = f x i, y i ) and rewriting, we obtain: [ ] t s i Y i s i, p i ) =, X i s i, p i ) = Y i s i, p i ) h f h /p i ), ) Contestant i s payoff: π i s) = p i s) v i C i [s i, p i s)], where C i [s i, p i s)] def = 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 si, s i), all i and all si 0. J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24 p i ).

20 Contestant i solves for fixed p i ): min xi,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 x i = λ i f x i, y i ) = 0, L i y i = p i λ i f 2 x i, y i ) = 0. So, by combining the FOCs: = f x i, y i ) def = g p i f 2 x i, y i ) xi where h is the inverse of g i.e., h def = g ). y i ) ) x i = y i h, p i By plugging back into s i = f x i, y i ) and rewriting, we obtain: [ ] t s i Y i s i, p i ) =, X i s i, p i ) = Y i s i, p i ) h f h /p i ), ) Contestant i s payoff: π i s) = p i s) v i C i [s i, p i s)], where C i [s i, p i s)] def = 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 si, s i), all i and all si 0. J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24 p i ).

21 Contestant i solves for fixed p i ): min xi,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 x i = λ i f x i, y i ) = 0, L i y i = p i λ i f 2 x i, y i ) = 0. So, by combining the FOCs: = f x i, y i ) def = g p i f 2 x i, y i ) xi where h is the inverse of g i.e., h def = g ). y i ) ) x i = y i h, p i By plugging back into s i = f x i, y i ) and rewriting, we obtain: [ ] t s i Y i s i, p i ) =, X i s i, p i ) = Y i s i, p i ) h f h /p i ), ) Contestant i s payoff: π i s) = p i s) v i C i [s i, p i s)], where C i [s i, p i s)] def = 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 si, s i), all i and all si 0. J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24 p i ).

22 Analysis 3/7) The cost-minimization problem and the h function y i Y slope = g xi slope = p i X a) Cost minimization. y i ) s i = f x i, y i ) x i m 45 g xi y i ) b) Graph of the g function. x i y i x i y i h m) 45 c) Graph of the h function. m J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

23 Analysis 4/7) Equilibrium existence Define the following elasticities: The elasticity of output w.r.t. x i : η The elasticity of substitution: σ ) p i ) p i [ ) ] ) h, h pi pi [ ) ] f h., pi ) pi pi ) h. pi def = f def = h The elasticity of the win probability w.r.t. s i : ε i s) = def p i s i s i p i. We have that η 0, t), σ > 0, and ε i 0, ). Assumption. The production ) ) function and the CSF satisfy: i) t and ε i s) η σ 2 for all p pi pi i and s); Proposition. Suppose Assumption is satisfied. Then there exists a pure strategy Nash equilibrium of the hybrid contest. J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, 208 / 24

24 Analysis 4/7) Equilibrium existence Define the following elasticities: The elasticity of output w.r.t. x i : η The elasticity of substitution: σ ) p i ) p i [ ) ] ) h, h pi pi [ ) ] f h., pi ) pi pi ) h. pi def = f def = h The elasticity of the win probability w.r.t. s i : ε i s) = def p i s i s i p i. We have that η 0, t), σ > 0, and ε i 0, ). Assumption. The production ) ) function and the CSF satisfy: i) t and ε i s) η σ 2 for all p pi pi i and s); Proposition. Suppose Assumption is satisfied. Then there exists a pure strategy Nash equilibrium of the hybrid contest. J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, 208 / 24

25 Analysis 4/7) Equilibrium existence Define the following elasticities: The elasticity of output w.r.t. x i : η The elasticity of substitution: σ ) p i ) p i [ ) ] ) h, h pi pi [ ) ] f h., pi ) pi pi ) h. pi def = f def = h The elasticity of the win probability w.r.t. s i : ε i s) = def p i s i s i p i. We have that η 0, t), σ > 0, and ε i 0, ). Assumption. The production ) ) function and the CSF satisfy: i) t and ε i s) η σ 2 for all p pi pi i and s); Proposition. Suppose Assumption is satisfied. Then there exists a pure strategy Nash equilibrium of the hybrid contest. J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, 208 / 24

26 Analysis 4/7) Equilibrium existence Define the following elasticities: The elasticity of output w.r.t. x i : η The elasticity of substitution: σ ) p i ) p i [ ) ] ) h, h pi pi [ ) ] f h., pi ) pi pi ) h. pi def = f def = h The elasticity of the win probability w.r.t. s i : ε i s) = def p i s i s i p i. We have that η 0, t), σ > 0, and ε i 0, ). Assumption. The production ) ) function and the CSF satisfy: i) t and ε i s) η σ 2 for all p pi pi i and s); Proposition. Suppose Assumption is satisfied. Then there exists a pure strategy Nash equilibrium of the hybrid contest. J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, 208 / 24

27 Assume a CES production function, t =, r, and p i s) = α w i s r i n j= w js r j and p i 0,, 0) = w i n j= w. j 3 4 σ rσ 2) + rσ 2) 2 σ Θσ, r) = def 2 α 4 Assumption satisfied r 4 r σ 5 r 20 r σ J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

28 Analysis 6/7) To check the SOC with Melkoyan s analytical approach is cumbersome and in the end he relies on numerical simulations: [... ] one can demonstrate, after a series of tedious algebraic manipulations, that a player s payoff function is locally concave at the symmetric equilibrium candidate in 7) if and only if [large mathematical expression]. [... ] Numerical simulations indicate that this inequality is violated only for extreme values of the parameters [... ]. In addition to verifying the local second-order conditions, I have used numerical simulations to verify that the global second-order conditions are satisfied under a wide range of scenarios. J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

29 Characterization of equilibrium Recall: π i s) = p i s) v i C i [s i, p i s)]. The FOC with an equality if s i > 0): π i s) s i = p i s) v i C s i, p i ) C 2 s i, p i ) p i s) 0. s i s i Use Shephard s lemma, C 2 s i, p i ) = Y i [s i, p i s)]: [v i Y i s i, p i s))] p i s) s i C s i, p i ), ) with an equality if s i > 0. Proposition 2. Suppose Assumption is satisfied. Then s = s,..., s n) is a pure strategy Nash equilibrium of the hybrid contest if and only if condition ) holds, with equality if s i > 0, for each contestant i. J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

30 Characterization of equilibrium Recall: π i s) = p i s) v i C i [s i, p i s)]. The FOC with an equality if s i > 0): π i s) s i = p i s) v i C s i, p i ) C 2 s i, p i ) p i s) 0. s i s i Use Shephard s lemma, C 2 s i, p i ) = Y i [s i, p i s)]: [v i Y i s i, p i s))] p i s) s i C s i, p i ), ) with an equality if s i > 0. Proposition 2. Suppose Assumption is satisfied. Then s = s,..., s n) is a pure strategy Nash equilibrium of the hybrid contest if and only if condition ) holds, with equality if s i > 0, for each contestant i. J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

31 Characterization of equilibrium Recall: π i s) = p i s) v i C i [s i, p i s)]. The FOC with an equality if s i > 0): π i s) s i = p i s) v i C s i, p i ) C 2 s i, p i ) p i s) 0. s i s i Use Shephard s lemma, C 2 s i, p i ) = Y i [s i, p i s)]: [v i Y i s i, p i s))] p i s) s i C s i, p i ), ) with an equality if s i > 0. Proposition 2. Suppose Assumption is satisfied. Then s = s,..., s n) is a pure strategy Nash equilibrium of the hybrid contest if and only if condition ) holds, with equality if s i > 0, for each contestant i. J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

32 Characterization of equilibrium Recall: π i s) = p i s) v i C i [s i, p i s)]. The FOC with an equality if s i > 0): π i s) s i = p i s) v i C s i, p i ) C 2 s i, p i ) p i s) 0. s i s i Use Shephard s lemma, C 2 s i, p i ) = Y i [s i, p i s)]: [v i Y i s i, p i s))] p i s) s i C s i, p i ), ) with an equality if s i > 0. Proposition 2. Suppose Assumption is satisfied. Then s = s,..., s n) is a pure strategy Nash equilibrium of the hybrid contest if and only if condition ) holds, with equality if s i > 0, for each contestant i. J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

33 Characterization of equilibrium Recall: π i s) = p i s) v i C i [s i, p i s)]. The FOC with an equality if s i > 0): π i s) s i = p i s) v i C s i, p i ) C 2 s i, p i ) p i s) 0. s i s i Use Shephard s lemma, C 2 s i, p i ) = Y i [s i, p i s)]: [v i Y i s i, p i s))] p i s) s i C s i, p i ), ) with an equality if s i > 0. Proposition 2. Suppose Assumption is satisfied. Then s = s,..., s n) is a pure strategy Nash equilibrium of the hybrid contest if and only if condition ) holds, with equality if s i > 0, for each contestant i. J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

34 A Symmetric Hybrid Contest /4) Assumption 2. The CSF is symmetric and homogeneous of degree 0. Note that, thanks to Assumption 2: p i s, s,..., s) = εn) s i ns Use this in the FOC and impose symmetry: v y ) εn) [ ns = C s, ] = n ts C def, where εn) = ε i,,..., ). [ s, n ] = [ ] y ts n + x v y ) t εn) = y + nx. And from before, x = hn)y. The last equalities are linear in x and y, so easy to solve. Proposition 3. Within the family of sym. eq., there is a unique pure strategy equilibrium: s = f [hn), ] y ) t, x = hn)y, and y = t εn)v + nhn) + t εn). J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

35 A Symmetric Hybrid Contest /4) Assumption 2. The CSF is symmetric and homogeneous of degree 0. Note that, thanks to Assumption 2: p i s, s,..., s) = εn) s i ns Use this in the FOC and impose symmetry: v y ) εn) [ ns = C s, ] = n ts C def, where εn) = ε i,,..., ). [ s, n ] = [ ] y ts n + x v y ) t εn) = y + nx. And from before, x = hn)y. The last equalities are linear in x and y, so easy to solve. Proposition 3. Within the family of sym. eq., there is a unique pure strategy equilibrium: s = f [hn), ] y ) t, x = hn)y, and y = t εn)v + nhn) + t εn). J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

36 A Symmetric Hybrid Contest /4) Assumption 2. The CSF is symmetric and homogeneous of degree 0. Note that, thanks to Assumption 2: p i s, s,..., s) = εn) s i ns Use this in the FOC and impose symmetry: v y ) εn) [ ns = C s, ] = n ts C def, where εn) = ε i,,..., ). [ s, n ] = [ ] y ts n + x v y ) t εn) = y + nx. And from before, x = hn)y. The last equalities are linear in x and y, so easy to solve. Proposition 3. Within the family of sym. eq., there is a unique pure strategy equilibrium: s = f [hn), ] y ) t, x = hn)y, and y = t εn)v + nhn) + t εn). J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

37 A Symmetric Hybrid Contest /4) Assumption 2. The CSF is symmetric and homogeneous of degree 0. Note that, thanks to Assumption 2: p i s, s,..., s) = εn) s i ns Use this in the FOC and impose symmetry: v y ) εn) [ ns = C s, ] = n ts C def, where εn) = ε i,,..., ). [ s, n ] = [ ] y ts n + x v y ) t εn) = y + nx. And from before, x = hn)y. The last equalities are linear in x and y, so easy to solve. Proposition 3. Within the family of sym. eq., there is a unique pure strategy equilibrium: s = f [hn), ] y ) t, x = hn)y, and y = t εn)v + nhn) + t εn). J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

38 A Symmetric Hybrid Contest /4) Assumption 2. The CSF is symmetric and homogeneous of degree 0. Note that, thanks to Assumption 2: p i s, s,..., s) = εn) s i ns Use this in the FOC and impose symmetry: v y ) εn) [ ns = C s, ] = n ts C def, where εn) = ε i,,..., ). [ s, n ] = [ ] y ts n + x v y ) t εn) = y + nx. And from before, x = hn)y. The last equalities are linear in x and y, so easy to solve. Proposition 3. Within the family of sym. eq., there is a unique pure strategy equilibrium: s = f [hn), ] y ) t, x = hn)y, and y = t εn)v + nhn) + t εn). J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

39 A Symmetric Hybrid Contest /4) Assumption 2. The CSF is symmetric and homogeneous of degree 0. Note that, thanks to Assumption 2: p i s, s,..., s) = εn) s i ns Use this in the FOC and impose symmetry: v y ) εn) [ ns = C s, ] = n ts C def, where εn) = ε i,,..., ). [ s, n ] = [ ] y ts n + x v y ) t εn) = y + nx. And from before, x = hn)y. The last equalities are linear in x and y, so easy to solve. Proposition 3. Within the family of sym. eq., there is a unique pure strategy equilibrium: s = f [hn), ] y ) t, x = hn)y, and y = t εn)v + nhn) + t εn). J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

40 A Symmetric Hybrid Contest /4) Assumption 2. The CSF is symmetric and homogeneous of degree 0. Note that, thanks to Assumption 2: p i s, s,..., s) = εn) s i ns Use this in the FOC and impose symmetry: v y ) εn) [ ns = C s, ] = n ts C def, where εn) = ε i,,..., ). [ s, n ] = [ ] y ts n + x v y ) t εn) = y + nx. And from before, x = hn)y. The last equalities are linear in x and y, so easy to solve. Proposition 3. Within the family of sym. eq., there is a unique pure strategy equilibrium: s = f [hn), ] y ) t, x = hn)y, and y = t εn)v + nhn) + t εn). J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

41 A Symmetric Hybrid Contest /4) Assumption 2. The CSF is symmetric and homogeneous of degree 0. Note that, thanks to Assumption 2: p i s, s,..., s) = εn) s i ns Use this in the FOC and impose symmetry: v y ) εn) [ ns = C s, ] = n ts C def, where εn) = ε i,,..., ). [ s, n ] = [ ] y ts n + x v y ) t εn) = y + nx. And from before, x = hn)y. The last equalities are linear in x and y, so easy to solve. Proposition 3. Within the family of sym. eq., there is a unique pure strategy equilibrium: s = f [hn), ] y ) t, x = hn)y, and y = t εn)v + nhn) + t εn). J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

42 A Symmetric Hybrid Contest /4) Assumption 2. The CSF is symmetric and homogeneous of degree 0. Note that, thanks to Assumption 2: p i s, s,..., s) = εn) s i ns Use this in the FOC and impose symmetry: v y ) εn) [ ns = C s, ] = n ts C def, where εn) = ε i,,..., ). [ s, n ] = [ ] y ts n + x v y ) t εn) = y + nx. And from before, x = hn)y. The last equalities are linear in x and y, so easy to solve. Proposition 3. Within the family of sym. eq., there is a unique pure strategy equilibrium: s = f [hn), ] y ) t, x = hn)y, and y = t εn)v + nhn) + t εn). J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

43 Proposition 4. Effect of more contestants on x and y : x n < 0 σn) > n n 2) hn) n ) [ + t εn)], y n nn 2)hn) > 0 σn) > ; n )nhn) and if σn), then necessarily x n y < 0 and > 0. n In order to understand the above: More contestants means a lower probability of winning. This lowers the relative cost of investing in y i. So whenever σn) is sufficiently large, y x n > 0 and n < 0. But if σn) small, the derivatives must have the same sign. For: y n n x = σn) + y n n x follows from x = hn)y ). As σn) 0, the production function requires x i and y i to be used in fixed proportions a Leontief production technology). J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

44 The total amount of equilibrium expenditures in the symmetric hybrid model is defined as R H = def nc [ s, n]. The corresponding amount in the all-pay contest: R A = t εn)v. Proposition 5, part a). In the symmetric model: R H = y [ v )R A = v [ + nhn)] + ]. R A In particular, for any finite n, we have R H < R A. The payoff suggests the intuition: π i = v i y i ) p i s) x i. Proposition 5, part b). In the symmetric model, suppose p i s) = φs i )/ n j= φs j), where φ is a strictly increasing and concave function satisfying φ0) = 0. Then R H is weakly increasing in n if and only if: i) σ n) + 4n tr n ) 2 ; 2) or ii) inequality 2) is violated and h n) / Ξ L, Ξ H ). See figure! J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

45 The total amount of equilibrium expenditures in the symmetric hybrid model is defined as R H = def nc [ s, n]. The corresponding amount in the all-pay contest: R A = t εn)v. Proposition 5, part a). In the symmetric model: R H = y [ v )R A = v [ + nhn)] + ]. R A In particular, for any finite n, we have R H < R A. The payoff suggests the intuition: π i = v i y i ) p i s) x i. Proposition 5, part b). In the symmetric model, suppose p i s) = φs i )/ n j= φs j), where φ is a strictly increasing and concave function satisfying φ0) = 0. Then R H is weakly increasing in n if and only if: i) σ n) + 4n tr n ) 2 ; 2) or ii) inequality 2) is violated and h n) / Ξ L, Ξ H ). See figure! J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

46 A Symmetric Hybrid Contest 4/4) Illustration of result b) Assume CES, t =, and n = 0. α 3 4 R H decreasing in n at n = Assumption satisfied σ J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

47 Asymmetric Hybrid Contests /2) I assume n = 2 and I study three models: The CSF is biased in favor of one contestant. One contestant has a higher valuation than the other. I also endogenize the degree of bias. Assumption 3. The CSF is given by p i s) = w i si r w s r + w. 2s2 r The following three equations define equilibrium values of p, y, and y 2 : y i = Υp ) def = rtpi p i )v i rtpi p i ) + p i + h w 2 v rt 2 p i [ ) h p w v rt p f [ rtp p ) + p + h ), for i =, 2, and Υp ) = 0, where ] r, p )] rt ) ) The equilibrium is unique if rη σ. pi pi [ ) h p p )f [ rtp p ) + p + h ] r, )] rt. p J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

48 Asymmetric Hybrid Contests /2) I assume n = 2 and I study three models: The CSF is biased in favor of one contestant. One contestant has a higher valuation than the other. I also endogenize the degree of bias. Assumption 3. The CSF is given by p i s) = w i si r w s r + w. 2s2 r The following three equations define equilibrium values of p, y, and y 2 : y i = Υp ) def = rtpi p i )v i rtpi p i ) + p i + h w 2 v rt 2 p i [ ) h p w v rt p f [ rtp p ) + p + h ), for i =, 2, and Υp ) = 0, where ] r, p )] rt ) ) The equilibrium is unique if rη σ. pi pi [ ) h p p )f [ rtp p ) + p + h ] r, )] rt. p J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

49 Asymmetric Hybrid Contests /2) I assume n = 2 and I study three models: The CSF is biased in favor of one contestant. One contestant has a higher valuation than the other. I also endogenize the degree of bias. Assumption 3. The CSF is given by p i s) = w i si r w s r + w. 2s2 r The following three equations define equilibrium values of p, y, and y 2 : y i = Υp ) def = rtpi p i )v i rtpi p i ) + p i + h w 2 v rt 2 p i [ ) h p w v rt p f [ rtp p ) + p + h ), for i =, 2, and Υp ) = 0, where ] r, p )] rt ) ) The equilibrium is unique if rη σ. pi pi [ ) h p p )f [ rtp p ) + p + h ] r, )] rt. p J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

50 Asymmetric Hybrid Contests 2/4) A Biased decision process w w 2 but v = v 2 ) Among the results: a) p > p 2 y < y 2 Cs, p ) > Cs 2, p 2 ). b) Evaluated at symmetry w = w 2 ): p w > 0, y w < 0, y 2 w > 0, x w > 0 x 2 w < 0 σ2) > rt. Different valuations v v 2 but w = w 2 ) Among the results: a) p > p 2 y v < y 2 v2. b) v y > v 2 y 2 Cs, p ) > Cs 2, p 2 ). J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

51 Asymmetric Hybrid Contests 2/4) A Biased decision process w w 2 but v = v 2 ) Among the results: a) p > p 2 y < y 2 Cs, p ) > Cs 2, p 2 ). b) Evaluated at symmetry w = w 2 ): p w > 0, y w < 0, y 2 w > 0, x w > 0 x 2 w < 0 σ2) > rt. Different valuations v v 2 but w = w 2 ) Among the results: a) p > p 2 y v < y 2 v2. b) v y > v 2 y 2 Cs, p ) > Cs 2, p 2 ). J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

52 An Endogenous Bias w chosen, but v v 2 and w 2 fixed) Timing of events in the game: A principal chooses w to maximize R H = Cs, p ) + Cs 2, p 2 ). 2 w becomes common knowledge and the contestants interact as in the previous analysis. Assumption 3. The production function is of Cobb-Douglas form: f x i, y i ) = xi α y β i, for α > 0 and β > 0. Results: The equilibrium values of p and w satisfy: If v = v 2, then p = 2 and ŵ = w 2. If v > v 2, then p > 2. If v > v 2, then ŵ < w 2 at least if v v 2 is very small or big. My intuition for results: Contestant is more valuable as a contributor as v > v 2 ). Hence, she should be encouraged to use x, as all-pay investments are more conducive to large expenditures. This is achieved by making winner-pay inv. costly: p > 2. To generate p > 2, v > v 2 is more than enough, so bias can be in favor of Contestant 2. Might not be robust. J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

53 An Endogenous Bias w chosen, but v v 2 and w 2 fixed) Timing of events in the game: A principal chooses w to maximize R H = Cs, p ) + Cs 2, p 2 ). 2 w becomes common knowledge and the contestants interact as in the previous analysis. Assumption 3. The production function is of Cobb-Douglas form: f x i, y i ) = xi α y β i, for α > 0 and β > 0. Results: The equilibrium values of p and w satisfy: If v = v 2, then p = 2 and ŵ = w 2. If v > v 2, then p > 2. If v > v 2, then ŵ < w 2 at least if v v 2 is very small or big. My intuition for results: Contestant is more valuable as a contributor as v > v 2 ). Hence, she should be encouraged to use x, as all-pay investments are more conducive to large expenditures. This is achieved by making winner-pay inv. costly: p > 2. To generate p > 2, v > v 2 is more than enough, so bias can be in favor of Contestant 2. Might not be robust. J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

54 Numerical example t = r = v 2 = w 2 = ) Plot of plot p and ŵ against v for three different values of α: 0.9 the blue, dotted curve), 0.5 the green, dashed curve), and 0. the red, solid curve) p v a) The high-valuation contestant s probability of winning ŵ v b) The weight in the CSF that is assigned to the high-valuation contestant s score. J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

55 Main results and contributions: /) The analytical approach borrowing from producer theory): Generality, tractability, and an existence condition. 2 A larger n leads to substitution away from all-pay investments. But only if the elasticity of substitution is large enough. 3 Total expenditures always lower in hybrid contest than in all-pay. 4 Total exp tures can be decreasing in n also shown by Melkoyan). 5 Asym. contests in terms of valuations and bias): Sharp predictions about relative size of investm s and of expenditures. 6 Endogenous bias: High-valuation contestant more likely to win but the bias is against her the latter might not be robust). J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24

56 Possible avenues for future work /) Sequential moves: first x, y ), then x 2, y 2 ). 2 Applying the producer theory approach to other contest models with multiple influence channels. 3 Experimental testing. Relatively sharp predictions. But risk neutrality might be an issue? 4 Further work on asymmetric contests. More than two contestants. Can a contestant be hurt by a bias in favor of her? Can a contestant benefit from an increase in rival s valuation? 5 Contest design in broader settings. J. Lagerlöf U of Copenhagen) Hybrid All-Pay and Winner-Pay Contests June 5, / 24