Sequential Two-Prize Contests

Transcription

1 Sequential Two-Prize Contests Aner Sela September, 2009 Abstract We study two-stage all-pay auctions with two identical prizes. In each stage, the players compete for one prize. Each player may win either one or two prizes. We analyze the unique subgame-perfect equilibrium of our model with two players where each player s marginal values for the prizes are decreasing, constant or increasing. Jel Classification: D44, D82, J31, J41. Keywords: Multi-stage contests, All-pay auctions. 1 Introduction In winner-take-all contests, all contestants including those who do not win the prize, incur costs as a result of their e orts. However, only the winner receives the prize. Winner-take-all contests have been applied to numerous settings including rent-seeking and lobbying in organizations, R&D races, political contests, promotions in labor markets, trade wars, military con icts and biological wars of attrition. The most common winner take-all contest is the all-pay auction. In the standard (one-stage) all-pay auction each player submits a bid (e ort) for the prize and the player who submits the highest bid receives the prize, but, independently of success, all players bear the cost of their bids. In the economic literature, all-pay auctions are usually studied under complete information where each player s value for the prize is common Department of Economics, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel. anersela@bgu.ac.il 1

2 knowledge, 1 or under incomplete information where each player s value for the prize is private information to that player and only the distribution of the players values is common knowledge. 2 Most of this literature has focused on one-stage all-pay auctions with a single prize which is awarded to the player with the highest bid 3 or, alternatively, on one-stage all-pay auctions with several prizes which are awarded to the players with the highest bids. 4 In the real world, however, we can nd numerous multi-stage contests with several prizes where in each stage a single prize is awarded or several prizes are awarded simultaneously. For example, the situation in which employees seek to climb the organizational hierarchy often consists of several types of well-de ned positions, and political contests where any candidate, regardless of whether he wins or loses, may compete again in the next round of the contest for the same job. In this paper, we analyze the equilibrium behavior in sequential two-prize all-pay auctions under complete information where the prizes are identical. Each player may win more than one prize, and each player s marginal values for the rst and the second prize are either decreasing, constant or increasing. The di erence between the players marginal values in both stages accords well with environments in which there are synergies between prizes. A synergy between prizes does not necessarily have to be the same for both players. In fact, it could even be positive for one player and negative for the other. In other words, our two-stage all-pay auction encompasses any variation between the players marginal values for prizes, and therefore the model suits any dynamic environment. Because it is possible to win more than one prize, our model is more complex than the standard one-stage all-pay auction with a single prize. In one-prize contests, the players strategies are quite simple since the players have only two options, either to win the prize or to lose it, while in sequential multi-prize contests more intricate strategies are involved since each player may win more than one prize and therefore players may face many options that depend on the identity of the winner in each stage. Furthermore, each of these options may have a di erent e ect on the chance of each 1 See, for example, Hillman and Riley (1989), Baye, Kovenock and de Vries (1993), Che and Gale (1998)), Kaplan, Luski and Wettstein (2003) and Konrad (2006). 2 See, for example, Amman and Leininger (1996), Krishna and Morgan (1998), Moldovanu and Sela (2001) and Cohen, Kaplan and Sela (2008). 3 Baye, Kovenock and de Vries (1996) provided a complete characterization of equilibrium behavior in the complete information all-pay auction with one prize. 4 See, for example, Barut and Kovenock (1998) and Moldovanu and Sela (2001, 2006). 2

3 player to win the other prizes in the later stages. In particular, in sequential multi-prize contests, each player has to decide in which stages he will compete to win and in which stages he will quit and conserve his e ort (resources) for the other rounds. In our sequential two-prize all pay auction with two players, we nd the unique sub-game perfect equilibrium where no player quits in the rst stage such that both players compete to win both of the prizes. The players use mixed strategies in both stages and therefore each has a chance to win both prizes as well as none of them. The players behavior in the second stage is similar to that of the standard one-stage all-pay auction, but their behavior in the rst stage is completely unique. For example, in contrast to the standard one-stage all-pay auction, both players may have positive expected payo s. Moreover, even if a player is weaker than the other player (his marginal values are lower than those of his opponent) he may have a positive expected payo. On the other hand, if both players marginal values are increasing, then one of the players necessarily has an expected payo of zero (similar to the standard one-stage all-pay auction). We also show that, in contrast to the standard one-stage all-pay auction, both players may have expected payo s of zero even if they have di erent marginal values for the prizes awarded in the contest. In our sequential two-prize all-pay auction with more than two players the equilibrium is not unique, and a complete characterization of the equilibrium behavior is quite complicated. However, the extension of the equilibrium of our model with two players to the case with more than two players is quite simple. We illustrate this extension where the players marginal values are decreasing. A similar extension holds for any relation between the players marginal values. There is an extensive literature on multi-stage contests, particularly multi-stage all-pay auctions. For example, Moldovanu and Sela (2006) studied two-stage all-pay auctions under incomplete information where the winners of the sub-contests at the rst stage compete among themselves in a second stage. Groh et al. (2008) studied the same model of a two-stage all-pay auction under complete information for the purpose of nding the optimal seeding from the designer s point of view. Konrad and Kovenock (2005) analyzed a multi-stage contest in which two players compete against each other in a sequence of all-pay auctions, and the player who is the rst to accumulate a su ciently larger number of wins than his rival is awarded the single prize. Konrad and Kovenock (2009) studied best-of-k contests with two players where players are 3

4 matched in the all-pay auction and the rst to win k+1 2 games is the winner of the contest. 5 In contrast to our multi-prize model, in all of the above papers the players compete for one prize that is allocated to the winner in the nal stage. The paper most related to our work is Kovenock and Roberson (2009) who studied a two-stage game in which players compete in a complete information all-pay auction in each stage, where winning (or losing) in the rst stage a ects the winner s utility function and his opponent s utility in the second stage. One of the main di erences between their model (as well as those discussed above) and ours is that while in their model winning in a stage provides an advantage for the winner in the next stage, in our model winning in the rst stage may yield either an advantage or disadvantage for the winner in the second stage depending on the relation between the players marginal values for the prizes in both stages. The paper is organized as follows: In Section 2 we introduce our sequential two-prize all-pay auction, and in Section 3 we analyze the unique equilibrium behavior in two-prize all-pay auctions with two players. In Section 4 we show several conclusions about the players expected payo s in the two-prize all-pay auction. In Section 5 we illustrate an extension of our results to the case with more than two players. Section 6 concludes. 2 The model We consider a sequential all-pay auction with two players which we denote by i = a; b; and two stages which we denote by t = 1; 2. In each of the stages a single (identical) prize is awarded. Let v i j denote player i0 s marginal value for winning his j-th prize. That is, if player i wins only one prize his value is v1 i and if he wins two prizes his value is v1 i + v2: i We assume that all players marginal values are common knowledge. Each player i submits a bid (e ort) x 2 [0; 1) in the rst stage. The player with the highest bid wins the rst prize and all the players pay their bids. The players know the identity of the winner in the rst stage before the beginning of the second stage, such that the players values in the second stage are common knowledge. The player with the highest bid in the second stage wins the second prize and all the players pay their bids. 5 Klumpp and Polborn (2006) studied best-of-k contests where players are matched in the Tullock contest. 4

5 3 Equilibrium In order to analyze the subgame-perfect equilibrium of a sequential two-prize all-pay auction we begin by analyzing the second stage and go backwards to the rst stage. 3.1 The second stage Assume that player i s value in the second stage is v i ; i = a; b, and, without loss of generality, assume that the players values satisfy v a v b : According to Baye, Kovenock and de Vries (1996), there is always a unique mixed-strategy equilibrium in which players a and b randomize on the interval [0; v b ] according to their e ort cumulative distribution functions, which are given by v a F b (x) x = v a v b v b F a (x) x = 0 Thus, player a s equilibrium e ort is uniformly distributed, that is F a (x) = x v b while player b s equilibrium e ort is distributed according to the cumulative distribution function F b (x) = va v b + x v a The respective expected payo s in the second stage are u a 2 = v a v b and u b 2 = 0: Now, given the equilibrium strategies in the second stage we can analyze the equilibrium strategies in the rst stage. 3.2 The rst stage Assume that player i s value in stage t = 1; 2 is v i t: If player a wins in the rst stage his payo in this stage is v a 1 and then, if v a 2 > v b 1; the winning in the rst stage guarantees to player a an expected payo of v a 2 v b 1 in the second stage, otherwise, if v a 2 v b 1, his expected payo in the second stage is zero. Thus, if player a wins in the rst stage, his expected payo is v a 1 + maxfv a 2 v b 1; 0g: If player a doesn t win in the rst stage then, if v a 1 > v b 2; his expected payo in the second stage is v a 1 v b 2, otherwise, if v a 1 v b 2, his expected payo in the second stage is zero. Thus, if player a does not win 5

6 in the rst stage his expected payo is maxfv a 1 v b 2; 0g: A similar argument holds for player b: The induced marginal value of each player in the rst stage is the di erence between his expected payo in the contest if he wins or not in the rst stage. Thus, the induced marginal values of the players in the rst stage are: bv a 1 = v a 1 + maxfv a 2 v b 1; 0g maxfv a 1 v b 2; 0g (1) bv b 1 = v b 1 + maxfv b 2 v a 1; 0g maxfv b 1 v a 2; 0g Note that since the players marginal values are positive, the induced marginal values given by (1) are positive as well. Then, using the induced marginal values, the players equilibrium strategies in the rst stage of the sequential two-prize all-pay auction are stated in the following result. Theorem 1 In the unique subgame-perfect equilibrium of the sequential two-prize all-pay auction, the players strategies in the rst stage are as follows: 1. If bv a 1 bv b 1; player a s equilibrium e ort is distributed according to F a 1 (x) = x bv b 1 while player b s equilibrium e ort is distributed according to F b 1 (x) = x + bva 1 bv b 1 bv a 1 The respective expected payo s in the rst stage are u a 1 = bv a 1 bv b 1 and u b 1 = 0: Then, the players total expected payo s in the contest are a = bv a 1 bv b 1 + maxfv a 1 v b 2; 0g (2) b = maxfv b 1 v a 2; 0g 2. If bv a 1 < bv b 1; player a s equilibrium e ort is distributed according to F a 1 (x) = x + bvb 1 bv a 1 bv b 1 while player b s equilibrium e ort is distributed according to F b 1 (x) = x bv a 1 6

7 The respective expected payo s in the rst stage are u a 1 = 0 and u b 1 = bv b 1 bv a 1. Then, the players total expected payo s in the contest are a = maxfv a 1 v b 2; 0g (3) b = bv b 1 bv a 1 + maxfv b 1 v a 2; 0g Proof. Assume rst that bv a 1 > bv b 1. Then, the players randomize on the interval [0; bv b 1] according to their e ort cumulative distribution functions, F a 1 (x) and F b 1 (x); which are given by the indi erence conditions: (v a 1 + maxfv a 2 v b 1; 0g)F b 1 (x) + (maxfv a 1 v b 2; 0g)(1 F b 1 (x)) x = bv a 1 bv b 1 + maxfv a 1 v b 2; 0g (v b 1 + maxfv b 2 v a 1; 0g)F a 1 (x) + (maxfv b 1 v a 2; 0g)(1 F a 1 (x)) x = maxfv b 1 v a 2; 0g Using the induced values bv a 1; bv b 1; the above indi erence conditions can be written as bv a 1F b 1 (x) x = bv a 1 bv b 1 (4) bv b 1F a 1 (x) x = 0 The system of equations (4) describing the players mixed strategies F1 a (x); F1 b (x) in the rst stage of the sequential two-prize all-pay auction is identical to the system of equations describing the equilibrium strategies of players in the standard one-stage all-pay auction where the players values are the induced marginal values bv 1; a bv 1. b Hence, according to Baye, Kovenock and de Vries (1996), there is a unique mixed strategy equilibrium in the rst stage of the sequential two-prize all-pay auction which is given by: F a 1 (x) = x bv b 1 (5) F b 1 (x) = x + bva 1 bv b 1 bv a 1 where the players expected payo s are u a 1 = bv a 1 v b 1 and u b 1 = 0: Similarly, if bv a 1 < bv b 1; the players randomize on the interval [0; bv a 1] according to their e ort cumulative distribution functions, F a 1 (x) and F b 1 (x); which are given by the indi erence conditions: (v a 1 + maxfv a 2 v b 1; 0g)F b 1 (x) + (maxfv a 1 v b 2; 0g)(1 F b 1 (x)) x = maxfv a 1 v b 2; 0g (v b 1 + maxfv b 2 v a 1; 0g)F a 1 (x) + (maxfv b 1 v a 2; 0g)(1 F a 1 (x)) x = bv b 1 bv a 1 + maxfv b 1 v a 2; 0g 7

8 or alternatively by: bv b 1F a 1 (x) x = bv b 1 bv a 1 (6) v a 1F b 1 (x) x = 0 Thus, the explicit distribution functions are: F a 1 (x) = x + bvb 1 bv a 1 bv b 1 (7) F b 1 (x) = x bv a 1 where the players expected payo s are u a 1 = 0 and u b 1 = bv b 1 bv a 1: 4 Results According to Theorem 1, the players expected payo s in the rst stage of the contest are determined by their induced values. Even if player a; for example, has the larger marginal value in the rst stage (v1 a > v1) b he may stay out of the contest with a positive probability. Moreover, he has an expected payo of zero in the rst stage if he does not have the higher induced value (bv 1 a < bv 1). b By (1), the di erence between the players induced values is given by bv 1 a bv 1 b = (v1 a v1) b + v a 2 v1 b v a 1 v2 b (8) Using equation (8) we can derive several results about the players total expected payo s in the contest. Proposition 1 In the sequential two-prize all-pay auction, if each player s marginal values are decreasing, the player with the highest marginal value has the highest total expected payo. Proof. We assume that v a 1 v b 1: 6 Suppose rst that bv a 1 bv b 1 0: Then, since the marginal values are decreasing, by our assumption we obtain that v a 1 v b 1 v b 2: Then, by (2) and (8), player a 0 s total expected payo is 6 A similar argument holds if v a 1 < vb 1 : a = (v1 a v1) b + v a 2 v1 b 8

9 and player b 0 s total expected payo is b = maxfv b 1 v a 2; 0g Thus, the di erence between the players expected payo s satis es a b (v a 1 v b 1) 0 Suppose now that bv a 1 bv b 1 < 0: Then, by (3) player a s total expected payo is a = v a 1 v b 2 and player b s total expected payo is b = (v b 1 v a 1) + (v a 1 v b 2) v b 1 v2 a + maxfv1 b v2; a 0g: Thus, the di erence between the players expected payo s satis es a b (v a 1 v b 1) 0 In contrast to the standard one-stage all-pay auction with two players in which only the player with the higher value has a positive expected payo, we have Proposition 2 In the sequential two-prize all-pay auction, both players may have positive expected payo s. Furthermore, even if both marginal values of a player are lower than both marginal values of his opponent, he still might have a positive expected payo. Proof. By (2) and (3), if v b 1 > v a 2 and v a 1 > v b 2 no player in the sequential two-prize all-pay auction has an expected payo of zero. Now, suppose that minfv a 1; v a 2g > maxfv b 1; v b 2g: Then, by (8) we obtain that bv b 1 bv a 1 = 2v b 1 v b 2 v a 2 Thus, by (3 ), if 2v1 b v2 b v2 a > 0; player b has a positive expected payo although both of his marginal values are lower than player a s marginal values. Similar to the standard all-pay auction with two players in which at least one of the players has an expected payo of zero, we obtain 9

10 Proposition 3 In the sequential two-prize all-pay auction, if both players marginal values are increasing, then one of the players necessarily has an expected payo of zero. Proof. By (2) and (3) we need to show that bv 1 a bv 1 b 0 implies that v1 b v2 a 0, and bv 1 a bv 1 b 0 implies that v1 a v2 b 0: Below we show that bv 1 a bv 1 b 0 implies that v1 b v2 a 0: 7 Assume that bv 1 a bv 1 b 0: Suppose now that v1 b v2 a > 0: Then, since player a s marginal values are increasing, by (8) we obtain, bv a 1 bv b 1 = (v a 1 v b 1) + (v b 1 v a 2) v a 1 v2 b = (v a 1 v2) a v a 1 v2 b < 0 Thus we obtain a contradiction to our assumption bv 1 a bv 1 b 0 and therefore the di erence v1 b v2 a is not positive. In contrast to the standard all-pay auction with two players in which both players have expected payo s of zero i they have identical values, we obtain Proposition 4 In the sequential two-prize all-pay auction, both players may have expected payo s of zero even if they have di erent marginal values. Proof. By (2) and (3), no player has a positive expected payo in the contest i the following three conditions are satis ed: v a 2 > v b 1 (9) v b 2 > v a 1 and given (9) and (8) bv a 1 bv b 1 = 2(v a 1 v b 1) + (v a 2 v b 2) = 0 (10) For example, if v b 2 = 2v a 1 and v a 2 = 2v b 1; the conditions (9) and (10) are satis ed. 5 Extensions Up to this point we considered sequential contests with only two players. It is important to note that similar to the standard one-stage all-pay auction with more than two players (n > 2), in our model with more 7 In a similar way it can be shown that bv a 1 bv b 1 0 implies that va 1 v b

11 than two players, the equilibrium strategies are not necessarily unique. Below we show the extension of the equilibrium of our model with two players to the case with more than two players. The players strategies in the second stage are similar to the standard all-pay auction (see Baye, Kovenock and de Vries, 1996) and therefore we focus only on the equilibrium strategies in the rst stage where the players marginal values are decreasing. 8 In this equilibrium, the two players with the highest induced marginal values are the only active players in the rst stage. First, we denote by v max h the h-highest marginal value in the set fv1g i i2n [ fv2g i i2n where N is the set of players. That is, v max 1 is the highest marginal value and v max 2n is the lowest marginal value. Denote the two active players by a and b. Now, we have four possible cases to deal with as follows: 1) v1 a = v max 1 and v1 b = v max 2 ; v2 b = v max 3 : In this case the induced marginal values of players a and b in the rst stage are: bv a 1 = v max 1 (v max 1 v max 3 ) = v max 3 bv b 1 = v max 2 (v max 2 v max 4 ) = v max 4 Since for every player i 2 N fa; bg; bv i 1 v max 4 ; and bv a 1 bv b 1; by Theorem 1 we obtain that in the rst stage players a and b randomize on the interval [0; bv 1] b according to the e ort cumulative distribution functions which are given by (5). Thus, the respective expected payo s in the rst stage are u a 1 = v max 3 -v max 4, and u i 1 = 0 for all i 6= a: 2)v1 a = v max 1 ; v2 a = v max 3 and v1 b = v max 2 : In this case the induced marginal value of players a and b in the rst stage are: bv a 1 = v max 1 (v max 1 v max 4 ) = v max 4 bv b 1 = v max 2 (v max 2 v max 3 ) = v max 3 Since for every player i 2 N fa; bg; bv i 1 v max 4 ; and bv b 1 bv a 1; by Theorem 1 we obtain that in the rst stage players a and b randomize on the interval [0; bv 1] a according to the e ort cumulative distribution functions which are given by (7). Thus, the respective expected payo s in the rst stage are u b 1 = v max 3 -v max 4 ; and u i 1 = 0 for all i 6= b: 8 Similar arguments hold for other relations among the players marginal values. 11

12 3) v a 1 = v max 1 ; v a 2 = v max 2 and v b 1 = v max 3 : In this case the induced marginal value of players a and b in the rst stage are: bv a 1 = (v max 1 + v max 2 v max 3 ) (v max 1 v max 4 ) = v max 2 + v max 4 v max 3 bv b 1 = v max 3 Now, if v max 2 + v max 4 2v max 3 ; then bv 1 a bv 1: b Thus by Theorem 1 we obtain that in the rst stage players a and b randomize on the interval [0; bv 1] b according to the e ort cumulative distribution functions which are given by (5). Thus, the respective expected payo s in the rst stage are u a 1 = v max 2 + v max 4 2v max 3, and u i 1 = 0 for all i 6= a. Otherwise, if v max 2 + v max 4 < 2v max 3, then bv 1 a < bv 1, b and we obtain that in the rst stage players a and b randomize on the interval [0; bv 1] a according to the e ort cumulative distribution functions which are given by (7). Thus, the respective expected payo s in the rst stage are u b 1 = 2v max 3 v max 2 v max 4 ; and u i 1 = 0 for all i 6= b: 4) v1 a = v max 1 ; v1 b = v max 2 and v1 c = v max 3. 9 In this case the induced marginal values of players a and b in the rst stage are: bv a 1 = v max 1 (v max 1 v max 3 ) = v max 3 bv b 1 = v max 2 (v max 2 v max 3 ) = v max 3 Since for every player i 2 N fa; bg; bv i 1 v max 3 ; and bv a 1 = bv b 1; by Theorem 1 we obtain that in the rst stage players a and b randomize on the interval [0; bv a 1] according to the e ort cumulative distribution functions which are given by (5). Thus, the respective expected payo s in the rst stage are u i 1 = 0 for all i: 6 Concluding remarks In the sequential two-prize all-pay auction we presented, if players have diminishing, constant or increasing marginal returns with respect to winning additional auction, the players implicit valuations in the rst stage depend non-linearly on both of the marginal valuations of both of the players. Hence, the players behavior in the rst stage is unexpected which suggest that we are dealing with an interesting and challenging multi-stage model. 9 Player c is not an active player. 12

13 Indeed, our results showed that the characteristics of the equilibria in the standard one-stage and sequential two-prize all-pay auctions are not the same. For example, in a one-stage all-pay auction the player with the lowest value necessarily has an expected payo of zero for any equilibrium, while in a sequential two-prize all pay auction with two players, even if a player has two marginal values which are both smaller than those of his opponent, he may have a positive expected payo in the rst stage. Furthermore, in contrast to the standard one-stage all-pay auction, on the one hand, both players may have positive expected payo s, and on the other hand, both players may have expected payo s of zero even if they have di erent marginal values in the sequential two-prize all-pay auction. References [1] Amman, E., Leininger, W.: Asymmetric All-Pay Auctions with Incomplete Information: The Two- Player Case. Games and Economic Behavior 14, 1-18 (1996) [2] Barut, Y., Kovenock, D.: The Symmetric Multiple Prize All-Pay Auction with Complete Information. European Journal of Political Economy 14, (1998) [3] Baye, M., Kovenock, D., de Vries, C.: Rigging the Lobbying Process. American Economic Review 83, (1993) [4] Baye, M., Kovenock, D., de Vries, C.: The All-Pay Auction with Complete Information. Economic Theory 8, (1996) [5] Che, Y-K., Gale, I.: Caps on Political Lobbying. American Economic Review 88, (1998) [6] Cohen, C., Kaplan, T.R., Sela, A.: Optimal Rewards in Contests. RAND, Journal of Economics 39(2), (2008) [7] Groh, C., Moldovanu, B., Sela, A., Sunde, U.: Optimal Seedings in Elimination Tournaments. Economic Theory, forthcoming [8] Hillman, A., Riley, J.: Politically Contestable Rents and Transfers. Economics and Politics 1, (1989) 13

14 [9] Kaplan, T.R., Luski, I., Wettstein, D.: Innovative Activity and Sunk Costs. International Journal of Industrial Organization 21, (2003) [10] Klumpp, T., Polborn, M.: Primaries and the New Hampshire E ect. Journal of Public Economics, 90, (2006) [11] Konrad, K.A.: Silent Interests and All-Pay Auctions. International Journal of Industrial Organization 24, (2006) [12] Konrad, K.A, Kovenock, D.: Equilibrium and E ciency in the Tug-of-War. CEPR Discussion Paper, (2005) [13] Konrad, K., Kovenock, D.: Multi-Battle Contests. Games and Economic Behavior 66, (2009) [14] Kovenock, D., Roberson, B.: Is the 50-State Strategy Optimal? Journal of Theoretical Politics 21(2), (2009) [15] Krishna, V., Morgan, J.: An Analysis of the War of Attrition and the All-Pay Auction. Journal of Economic Theory 72(2), (1997) [16] Moldovanu, B., Sela, A.: The Optimal Allocation of Prizes in Contests. American Economic Review 91, (2001) [17] Moldovanu, B., Sela, A.: Contest Architecture. Journal of Economic Theory 126(1), (2006) 14