All-Pay Contests. (Ron Siegel; Econometrica, 2009) PhDBA 279B 13 Feb Hyo (Hyoseok) Kang First-year BPP

All-Pay Contests (Ron Siegel; Econometrica, 2009) PhDBA 279B 13 Feb 2014 Hyo (Hyoseok) Kang First-year BPP

Outline 1 Introduction All-Pay Contests An Example 2 Main Analysis The Model Generic Contests Non-generic Contests Participation 3 Concluding Remarks Summary & Conclusion 4 Appendix: Supplementary Examples (Non-identical, Non-generic) When The Power Condition Fails When The Cost Condition Fails

1 Introduction All-Pay Contests An Example 2 Main Analysis The Model Generic Contests Non-generic Contests Participation 3 Concluding Remarks Summary & Conclusion 4 Appendix: Supplementary Examples (Non-identical, Non-generic) When The Power Condition Fails When The Cost Condition Fails

All-Pay Contests We study a class of games, all-pay contests, which capture general asymmetries and sunk/irreversible investments inherent in many scenarios: Lobbying Competition for market power Labor-market tournaments R&D races We provide a closed-form formula for players equilibrium payoffs and analyze player participation.

Each player chooses a costly score and can be in one of two states: winning or losing. The primitives of the contest are commonly known: the equilibrium payoffs represent economic rents (not information rents). Our cost functions allow for differing production technologies, costs of capital, and prior investments. Nonordered or state-dependent cost functions are also allowed. Separable contests When all investments are unconditional, each player is characterized by her valuation for a prize (the payoff difference between the two states) and a weakly increasing, continuous cost function that determines her cost of choosing a score independently of the state. It nests many models of competition that assume a deterministic relation between effort and prize allocation. Single- and multi-prize all-pay auctions are separable contests with linear costs.

An Example Three risk-neutral firms compete for a monopoly position (v = 1). Each firm chooses how much to invest in lobbying (=score). Firms 1 and 2 have better lobbying technologies. Firm 3 has an initial advantage, yet her cost for high scores is high. 1 K (with score 1 + ɛ) is a lower bound on 1 s expected payoff. It can be better off: firm 1 must employ a mixed strategy in any equilibrium.

An Example - What Our Results Imply Firm 1 s equilibrium payoff is 1 K. Firms 2 and 3 can guarantee themselves (no more than) 0 (Theorem 1). Precisely one player receives a strictly positive expected payoff. For low and strictly positive values of γ, all three players must participate ( Theorem 2). This participation results from the nonordered nature of players cost functions. Lowering the prize s value can lead to a positive payoff for player 3, making her the only player who obtains a positive expected payoff. A VARIANT: What if firm 3 has 0 marginal cost? Firm 3 wins with certainty. It may be that no player invests even if a contest provides a valuable prize (pure-strategy equilibrium).

The Model n players compete for m homogenous prizes (0 < m < n). The set of players: N = {1,..., n}. Players compete by each choosing a score simultaneously and independently. Player i chooses a score si S i = [a i, ), where a i 0 is her initial score. Each of the m players with the highest scores wins one prize.

Player i has preferences over lotteries w/ an outcome pair (s i, W i ): W i v i (s i ) (1 W i )c i (s i ) where v i : S i R is i s valuation for winning (c i : cost of losing). Given a profile of scores s = (s 1,..., s n ), player i s payoff is u i (s) = P i (s)v i (s i ) (1 P i (s))c i (s i ) where P i : j N S j [0, 1], player i s probability of winning s.t. 0 if s j > s i for m or more players j i P i (s) 1 if s i > s j for N m or more players j i any value in [0, 1] otherwise s.t. n j=1 P j(s) = m. A player s probability of winning depends on all players scores. Her valuation for winning and cost of losing depend only on her chosen core.

Three Assumptions Assumption 1: v i and c i are continuous and non-increasing. Conditional on winning or losing, a lower score is weakly preferable. Assumption 2: v i (a i ) > 0 and lim si v i (s i ) < c i (a i ) = 0. With the initial score, winning is better than losing, so prizes are valuable. However, losing with the initial score is preferable to winning with sufficiently high scores. Assumption 3: c i (s i ) > 0 if v i (s i ) = 0. If winning with score s i is as good as losing with the initial score, then winning with score s i is strictly better than losing with score s i : v i (s i ) = c i (a i ) = 0 > c i (s i )

This formulation allows the difference between a player s valuation for winning and her cost of losing to depend on her chosen score. Consider a competition for promotions in which the value of the prize for player i is fixed at V i but some costs are only borne if the player wins (ci W when she wins and ci L when she loses). Then: ( ) u i (s) = P i (s) V i ci W (s i ) (1 P(s)) ci L (s i ) Contests can capture players risk attitudes as well. Let v i (s i ) = f (1 P(s)) and c i = f (c L i ) for some strictly increasing f s.t. f (0) = 0.

Separable Contests Every player i s preferences over lotteries with outcomes (s i, W i ) depend only on the marginal distributions of the lotteries. The effect of winning or losing on a player s Bernoulli utility is additively separable from that of the score: and for V i = v i (a i ) > 0. v i (s i ) = V i c i (s i ) u i (s) = P i (s)v i c i (s i ) The value ci (s i ) could be thought of as player i s cost of choosing score s i. V i could be thought of as player i s valuation for a prize. All expenditures are unconditional, and players are risk neutral.

Key Concepts 1 Player i s reach r i is the highest score at which her valuation for winning is 0: r i = max{s i S i v i (s i ) = 0} 2 Player m + 1 is the marginal player. 3 The threshold T of the contest is the reach of the marginal player: T = r m+1 4 Player i s power w i is her valuation for winning at the threshold: w i = v i (max{a i, T }) In a separable contest, a player s reach is the highest score she can choose by expending no more than her valuation for a prize.

Generic Contests Generic Conditions 1 Power Condition: The marginal player is the only player with power 0. 2 Cost Condition: The marginal player s valuation for winning is strictly decreasing at the threshold. For every x [a m+1, T ), v m+1 (x) > v m+1 (T ) = 0 In a separable contest, in particular, for every x [a m+1, T ): c m+1 (x) < c m+1 (T ) = V m+1 Players in N W = {1,..., m} have strictly positive powers, and players in N L = {m + 1,..., n} have strictly nonpositive powers. Contests that do not meet the Generic Conditions can be perturbed slightly to meet them.

Payoff Result Theorem 1 In any equilibrium of a generic contest, the expected payoff of every player equals the maximum of her power and 0. Players in N W have strictly positive expected payoffs (N L has 0). A generic contest has the same payoffs in all equilibria. Proof We invoke Least Lemma, Tie Lemma, Zero Lemma, and Threshold Lemma to prove Theorem 1. A mixed strategy G i of i is a cumulative probability distribution that assigns probability 1 to her set of pure strategies S i. When a strategy profile G = (G 1,..., G n ) is specified, P i (x) is shorthand for player i s probability of winning when she chooses x a i with certainty and all other players play according to G. For an equilibrium (G 1,..., G n ), denote by u i = u i (G i ) player i s equilibrium payoff.

Step 1: Least Lemma A player s expected payoff in G is at least the maximum of her power and 0. It suffices to consider players with strictly positive power (N W ). In equilibrium, no player chooses scores higher than her reach with a strictly positive probability ( A1, A3). By choosing max{a i, T + ɛ} for ɛ > 0, a player i in N W beats all N m players in N L with certainty. For every player i in N W, by continuity of v i, u i v i (max{a i, T + ɛ}) ɛ 0 v i (max{a i, T }) = w i

Step 2: Tie Lemma Suppose that in G two or more players have an atom at a score x. Then players who have an atom at x either all win with certainty or all lose with certainty when choosing x. N : the set of players who have an atom at x: N 2. E: strictly positive-probability event that all players in N choose x. D E: the event in which a relevant tie occurs at x; i.e. the event in which m prizes are divided among the N players in N, with 1 m < N. Conditional on D > 0, at least one player i in N can strictly increase her probability of winning to 1 by choosing x + ɛ. Since i chooses x with strictly positive probability, x r i and v i (x) > c i (x) ( A1 A3).

By continuity of v i and c i, player i would be strictly better off by choosing a score x + ɛ D has probability 0 and P(E) = P(E L ) + P(E W ) E L E is the event that at least m players in N \ N choose scores strictly higher than x. E W E is the event that at most m N players in N \ N choose scores strictly higher than x. Either E L or E W have probability 0 by independence of players strategies (otherwise, D would have strictly positive probability). 1. If P(E) = P(E L ), then, without conditioning on E, at least m players in N \ N choose scores strictly higher than x with probability 1, so P i (x) = 0 for every player i in N. 2. Similarly, if P(E) = P(E W ), then P i (x) = 1 for every player i in N.

Step 3: Zero Lemma In G, at least n m players have best responses with which they win with probability 0 or arbitrarily close to 0. These players have an expected payoff of at most 0. J: a set of some m + 1 players S: the union of the best-response sets of the players in J s inf : the infimum of S Case 1: two or more players in J have an atom at s inf : N J: the set of such players in J. It cannot be that P i (s inf ) = 1 for every player i in N, because any player in J \ N chooses scores strictly higher than s inf with probability 1, even if the players in N \ J choose scores strictly lower than s inf with probability 1, because only m (J \ N ) = m (m + 1 N ) = N 1 > 0 prizes are divided among the N players in N. Thus, by Tie Lemma, P i (s inf ) = 0 for every player i in N.

Case 2: One player (= i) in J has an atom at s inf : P i (s inf ) = 0, since all m players in J \ {i} chooses scores strictly higher than s inf with probability 1. In cases 1 and 2, P i (s inf ) = 0 for some player i in J who has an atom at s inf, so s inf is a best response for this player at which she wins with probability 0. Case 3: No player in J has an atom at s inf. By definition of s inf, there exists a player i in J with best responses {x n } n=1 s inf. Since 1 1 P i (x n ) j J\{i} (1 G j(x n )), no player in J has an atom at s inf, and G is right-continuous: P i (x n ) 0 as n. Note that J was a set of any m + 1 players. That is, even if J includes all the winning players, there are at least n (m + 1) + 1 = n m players in N who have best responses with which they win with probability 0.

Comments on Three Lemmas They hold regardless of the Generic Conditions. The Least Lemma + the Power Condition: The m players in NW have strictly positive expected payoffs. The Least Lemma + the Zero Lemma: Under the Power Condition, the n m players in NL obtain expected payoffs of 0.

Step 4: Threshold Lemma The players in N W have best responses that approach or exceed the threshold and, therefore, have an expected payoff of at most their power. Players in N L \ {m + 1} have strictly negative powers. Their reaches and the supremum of their best responses are strictly below the threshold. There is some s sup < T s.t. G i (x) = 1 for every player i in N L \ {m + 1} and every score x > s sup. This implies that every player i in N W chooses scores that approach or exceed the threshold (i.e. has G i (x) < 1 for every x < T ). Otherwise, for some s in (s sup, T ), G i (s) = 1 for all but at most m 1 players in N \ {m + 1}. But then the marginal player could win with certainty by choosing a score in (max{a m+1, s}, T ) (NOTE: a m+1 < T ). This would give her a strictly positive payoff, a contradiction.

Take a player i in N W. Because G i (x) < 1 for every x < T, there exists a sequence {x n } n=1 of best responses for player i that approach some z i T. Since x n is a best response for player i, who has a strictly positive payoff by the Least Lemma and the Power Condition, v i (x n ) > 0. By A1 and A2, v i (x n ) > c i (x n ). By continuity of v i, we have u i = u i (x n ) = P i (x n )v i (x n ) (1 P i (x n )) c i (x n ) v i (x n ) xn z i v i (z i ) v i (T ) = w i (Proof of Theorem 1) The Least Lemma + the Threshold Lemma show that players in N W have expected payoffs equal to their power. We checked that players in N L have expected payoffs of 0. Since N L N W = N, the expected payoff of every player equals the maximum of her power and 0.

Discussion of the Payoff Characterization Equilibrium payoffs in generic contests depend only on players valuations for winning at the threshold. Only the reach of each player and valuations for winning at the threshold need to be computed. Players costs of losing do not affect payoffs. The number of players who obtain positive expected payoffs equals the number of prizes. Simon and Zame s (1990) show that an equilibrium exists for some tie-breaking rule. The following corollary of their result and the Tie Lemma above show that an equilibrium exists for any tie-breaking rule. Corollary 1 Every contest has a Nash equilibrium.

Contests That Are Non Generic Corollary 2 Every contest (generic or not) has at least one equilibrium in which every player s payoff is the maximum of her power and 0. Proof Sketch This can be proved by considering a sequence of generic contests that approach the original contest and an equilibrium for each contest in the sequence. Every limit point of the resulting sequence of equilibria is an equilibrium of the original contest in which payoffs are given by the payoff result.

Corollary 3 In any equilibrium of a contest in which all players are identical, all players have a payoff of 0. Proof Sketch In any equilibrium of any contest, identical players have identical payoffs, and the Zero Lemma shows that at least one player has payoff 0. When players are not identical and the contest is not generic, the payoff of a player in some equilibrium may be very close to her valuation for winning at initial score, even if her power is very low. See Example 1 and Example 2.

Participation Theorem 2 In a generic contest, if the normalized costs of losing for the marginal player are strictly lower than that of player i > m + 1, that is, c m+1 (max{a m+1, x} v m+1 (a m+1 ) < c i(x) v i (a i ) for all x S i s.t. c i (x) > 0, and the normalized valuations for winning for the marginal player are weakly higher than that of player i > m + 1, that is, v m+1 (max{a m+1, x} v m+1 (a m+1 ) v i(x) v i (a i ) for all x S i, then player i does not participate in any equilibrium. In particular, if these conditions hold for all players in N L \ {m + 1}, then in any equilibrium only the m + 1 players in N W {m + 1} may participate.

Comment on Theorem 2 When players costs are strictly ordered, at most m + 1 players participate in any equilibrium. However, a player in N L \ {m + 1} may participate if she has a local advantage w.r.t. the marginal player. In our first example, suppose player 3 did not participate. Players 1 and 2 play strategies that make all scores in (0, T ) best responses for both of them. For low values of γ > 0, player 3 could then obtain a strictly positive payoff by choosing a low score: a contradiction. Thus, player 3 must participate in any equilibrium, even though her expected equilibrium payoff is 0.

Proof of Theorem 2 (in Appendix) It suffices to prove the result for contests in which v i (a i ) = 1 i. Choose an equilibrium G of such a contest. Suppose player i > m + 1 that meets the conditions of the proposition participated in G. Let t i = inf {x : G i (x) = 1} < T and t i = max{a m+1, t i }. Then, t i < T Pi (t i ) < 1 (Pf. of the Threshold Lemma) For every δ > 0, Pm+1 ( t i + δ) P i (t i ) since by choosing ( t i + δ), player m + 1 beats player i for sure. For every δ > 0 s.t. t i + δ < r m+1 = T, we have v m+1 ( t i + δ) > 0 c m+1 ( t i + δ)

Therefore, u m+1 P m+1 ( t i + δ)v m+1 ( t i + δ) (1 P m+1 ( t i + δ))c m+1 ( t i + δ) P i (t i )v m+1 ( t i + δ) (1 P i (t i ))c m+1 ( t i + δ) By definition of participation, c i (t i ) > 0, so c i (t i ) > c m+1 ( t i ). Since P i (t i ) < 1 and v m+1 (max{a m+1, x}) v i (x) for all x S i, by continuity of v m+1 and c m+1, player m + 1 can choose t i + δ for sufficiently small δ > 0 s.t. P i (t i )v m+1 ( t i + δ) (1 P i (t i ))c m+1 ( t i + δ) > P i (t i )v i (t i ) (1 P i (t i ))c i (t i ) = u i (t i ) 0 u m+1 > 0, which contradicts the payoff result (w m+1 = 0).

Summary & Conclusion All-pay contests capture general asymmetries among contestants and allow for both sunk and conditional investments. Thm 1: Expected Payoffs The expected payoff of every player equals the maximum of her power and 0; i.e. max {w i = v i (max{a i, T }), 0} Thm 2: Participation Players that are disadvantaged everywhere w.r.t. the marginal player do not participate in any equilibrium Reach and power are the right variables to focus. The addition of a player makes existing players weakly worse off. The addition of a prize makes player m + 2 the marginal player. This lowers the threshold and makes existing players better off. Making prizes more valuable raises the threshold.

Example 1: The Power Condition Fails The Payoff Result Does Not Hold One prize of common value 1 Players costs are: { (1 α)x, if 0 x h c 1 (x) = (1 α)h + (1 + αh 1 h )(x h), { if x > h (1 ɛ)x, if 0 x h c 2 (x) = (1 ɛ)h + (1 + ɛh 1 h )(x h), { if x > h γx, if 0 x h c 3 (x) = γh + L(x h), if x > h for some small α, ɛ in (0, 1), small γ 0, h in (0, 1), and L > 0. Regardless of the value of L, the threshold is 1 and the Power Condition is violated ( at least two players have power 0). The Cost Condition is met (costs are strictly increasing at 1).

For any h in (0, 1), there exist some β > 0 and M > 0 s.t. if α, ɛ, γ < β and L > M, then (G 1, G 2, G 3 ) is an equilibrium, for: 0, if x < 0 (1 ɛ)h + γ G 1 (x) = 1 α ( x h 1), if 0 x h (1 ɛ)h + (1 + ɛh 1 h )(x h), if h < x 1 1, if x > 1 0, if x < 0 (1 α)h, if 0 x h G 2 (x) = (1 α)h + (1 + αh 1 h )(x h), if h < x 1 1, if x > 1 { x/h, if x h G 3 (x) = 1, if x > h

Players costs Players atoms and densities in the equilibrium As L increases, player 3 s power, 1 γh L(1 h), becomes arbitrarily low. As h 1 and ɛ, α, γ 0, for any value of L > M, player 3 wins with near certainty. Her payoff approaches the value of the prize: (1 ɛ)(1 α)h 2 γh 0. A slight change in player 1 s or 2 s valuation for the prize leads to a generic contest and destroys the equilibrium.

Example 2: The Cost Condition Fails The Payoff Result Does Not Hold Competition may stop before the threshold is reached. Two-player separable contest for one prize of common value 1: for some d in (0, 1). c 1 (x) = bx, for some b < 1 x d, if 0 x < d c 2 (x) = 1, if d x 1 2x 1, if x > 1 Reach (r i ) Power (w i ) 1 Player 1 b > 1 1 b > 0 Player 2 1 0 The Power Condition holds The Cost Condition fails: c 1 2 (c 2(r 2 )) = [d, 1].

(G 1, G 2 ) is an equilibrium in which player 1 has a payoff of 1 bd > w 1 for As b 1, w 1 0 0, if x < 0 G 1 (x) = x d, if 0 x d 1, if x > d 0, if x < 0 G 2 (x) = 1 bd bx, if 0 x d 1, if x > d However, for any value of b, as d 0, player 1 s payoff approaches 1 (=the value of the prize).

Example 2: Another Equilibrium (G 1, G 2 ) is an equilibrium in which both players payoffs equal their powers, for 0, if x < 0 G 2 (x) = 1 b + bx, if 0 x 1 1, if x > 1