A Combinatorial Multiple Winner Contest with Package Designer Preferences Subhasish M. Chowdhurya and Dan Kovenockb a

School of Economics, Centre for Behavioral and Experimental Social Science, and ESRC Centre for Competition Policy, University of East Anglia, Norwich NR4 7TJ, UK b

Department of Economics, The University of Iowa, W284 John Pappajohn Business Building, Iowa City, IA 52242-1994, U.S.A.

Abstract This article analyzes a multi-winner contest with players distributed around a ring network. Players expend costly effort and a fixed number of adjacent players are selected as the winning coalition by applying a lottery contest success function to the sum of the individual bids of all possible groups of adjacent players. Hence, the players view other players as both friends and foes, and coalitions are formed endogenously. With symmetric valuations, we characterize the unique symmetric equilibrium and compare our results with previous studies of multi-winner contests. We characterize equilibrium for two specific examples under asymmetric valuations. Implications for the role of economic network and contest design issues are explored.

JEL Classifications: C72, D72, D74, D85 Keywords: Collective rent-seeking, Multiple-winners, Combinatorial contests, Package preferences, Networks, Endogenous coalition formation, Asymmetric valuations Corresponding author: Subhasish M. Chowdhury; email: [email protected] We appreciate the helpful comments of Robert Kluin, Roman Sheremeta, seminar participants at Purdue University, the University of Birmingham, the University of Cincinnati, the University of East Anglia, and the participants at the 3rd Annual International Conference at Burdwan University. Any remaining errors are our own.

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1. Introduction We construct a multiple winner contest in which the contest designer exhibits a package preference and only a subset of contestants, related to each other through the network structure of designer preferences, can be winners. This type of environment is typical of research contests, political coalitions, and tendering for government projects, to name just a few examples. Among the individual research papers submitted to an academic conference, only a subset of papers with similar research areas or methodologies is selected for presentation during a particular session. The criterion comes from a package designer preference in which the conference program committee wishes to select the highest quality papers possible from an area. A similar structure is seen in the market for certain government tenders and in modeling consumer product preferences (Salop, 1979; Lancaster, 1979). For some type of government tenders, the companies producing products with features closer to government tastes win the tender. In some markets, such as the video games market, when many new products are launched, usually consumers prefer one certain type of product. The games having features similar to those preferred by consumers capture significant market share, but the other new products become obsolete. Another application comes from political economy, where political parties are considered to be arranged in a ‘political compass’ in terms of their ideology (Lester, 1996). In a Westminster system parties with similar ideologies can form a coalition to control the government. Contests are economic or social interactions in which two or more players expend scarce resources in order to win one or more prizes. The resources expended by the players, which determine their respective probabilities of winning a prize, are usually viewed as sunk regardless of the identities of the winners. The function that maps the profile of expenditures, ,

,…,

, into the profile of probabilities of winning,

,

,…,

, is called the

contest success function (CSF). Arguably, the most popular CSF applied in the literature is the lottery CSF (Tullock, 1980) in which player ’s probability of winning is

/∑

if

and 1/ otherwise. In many situations there can be multiple winners in a contest. A distinct area of the literature addresses the issue of contests with multiple winners. A survey by Sisak (2009) offers examples such as rent-seeking, allocation of quotas, tender of government projects, and sets of promotions in which multiple winner contests are relevant. The existing literature on multiple winner contests, however, does not explicitly consider any network effect originating from designer preferences. 2

In the current study we construct a multi-winner contest in which

players are

distributed around the circumference of a circle. Players expend costly effort to win a prize and only

adjacent players are selected as winners using a lottery CSF. This structure serves

as an abstract representation of the contest environments discussed above in which the designer has package preferences that translate into a ring network.1 In case of the contest for acceptance to an academic conference, the research areas of different authors are common knowledge. The relationship between the research areas can be interpreted as an economic network between the authors. Given the network, an author writes a paper taking into account the package preference of the designer in order to have a higher probability of having the paper accepted in a session along with other related papers. In our model, we position authors horizontally around the circumference of a circle according to their research area. We assume that the quality of a paper is directly related to the research effort expended on that paper. Given the designer’s package preference, only a subset of authors positioned adjacently in an arc of the circle, representing interconnected research areas, is selected as winning. An author can invest a large amount of effort to write a high quality paper. However, if his neighboring authors do not also invest sufficient effort, then the author’s probability of selection is lower. Similarly, if one’s neighboring authors write high quality papers, then writing a low quality paper in that area might result in being selected. Hence, in a sense, the players in this game view other players as both friends and foes, and coalitions are formed endogenously. A similar logic applies for the other applications as well. There is a sizable body of literature analyzing multiple-winner contests. In the current context, we focus only on the literature involving the lottery CSF. 2 This literature broadly distinguishes between two types of contests: one in which the set of winners consists of individual players, and one which the winners are members of a group. Berry (1993) was the first to analyze the case in which the set of winners consists of individual players. In his model players expend costly effort to win one of

prizes. The probability that a particular player

wins a prize is the sum of the efforts of all k-player combinations which involve that player, divided by the sum of efforts in all possible k-player combinations. In the symmetric equilibrium, 1

A ring network is a network topology in which each node is connected to exactly two other nodes, forming a single continuous loop. Adjacent pairs of nodes are directly and other pairs of nodes are indirectly connected. 2 For all-pay auction, Glazer and Hassin (1988), Barut and Kovenock (1998), and Clark and Riis (1998a) analyze the problem under complete information whereas Moldovanu and Sela (2001, 2006) consider incomplete information.

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rent dissipation is only a fraction of the total rent. Clark and Riis (1996) show that the result in Berry (1993) is directly obtained through a selection process underlying the lottery CSF for which only the very first prize is awarded based on the players' rent-seeking outlays; the probability that a player wins one of the remaining prizes is independent of these outlays. Clark and Riis (1996) instead consider a k-iteration nested contest in which the winners are chosen sequentially using a lottery CSF. In such a case, contrary to Berry’s result, total rent is dissipated for a large number of players.3 The other type of multi-winner contest is one in which the set of winners are members of a winning group. A function that transforms the efforts of individual group members into the group effort is called a group impact function. The literature on group contests with lottery CSFs is started by Katz et al. (1990). They assume symmetric players within each group and a perfectsubstitutes group impact function, that is, the group effort is the sum of individual member efforts. 4 In this study multiple equilibria exist, but the equilibrium total effort is uniquely determined. Baik (2008) generalizes this structure by considering within group asymmetric valuations. In the equilibrium, within each group only the group member with the highest valuation expends effort and the rest of the group-members free ride by expending zero effort. If multiple players within a group have the highest valuation, then multiple equilibria exist. To our knowledge, there is only one study (Fu and Lu, 2009) that considers the case in between, in which there is partial restriction on forming the winning group of players. Fu and Lu (2009) show that if the players are homogeneous and are divided into several sub-contests, then under the Clark and Riis (1996) nested method, a grand contest always generates more equilibrium rent dissipation than any set of sub-contests. The current study also explores the case in which there is only partial ex-ante restriction on forming the group of winning players. A unique feature of this model is that the groups are not pre-specified, but the players know they are connected through a ring network. In other words, the subset of players with whom each player is likely to win one of the prizes is a common knowledge. Table 1 summarizes how the current analysis fits in the context of the literature on multiple-winner contests 3

In follow up studies Clark and Riis (1998b) consider a non linear contest success function and asymmetric values of prizes, Yates and Heckelman (2000) apply the Clark and Riis (1996) structure to analyze regulatory incentives and Szymanski and Valetti (2005) consider the importance of second prize in the case of asymmetric players. 4 Most follow up studies use the perfect-substitutes impact function, except Lee (2011), Kolmar and Rommeswinkel (2010), and Chowdhury et al. (2011) who analyze the weakest-link, Constant Elasticity of Substitution, and Bestshot impact functions, respectively.

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Table 1. Specification of the set of winners in multiple-winner contests

Structure of the model (considering only lottery CSF)

Existing studies

No restriction on the specification of the set of winners

Partial restrictions on the specification of the set of winners

Full restriction on the specification of the set of winners

n players expend efforts and k (< n) are chosen to be the winners

n players are divided into k sub-contests and from each sub-contest only one winner is chosen

groups of size k expend efforts and only one group is chosen to be the winner

Berry (1993), Clark and Riis (1996), Yates and Heckelman (2001)

Fu and Lu (2009) (N.B. the current study uses partial restriction, albeit with a different model structure)

Katz et al (1990) Baik (2008)

The current analysis is also one of the few studies to use the concept of an economic network within a contest framework. In Goyal et al. (2006) economics researchers endogenously determine the network in terms of co-authorship, but a contest structure is not discussed.5 To our knowledge, Poemmerenke (2006), Franke and Ozturk (2009), Szymanski (2011), and Tellone and Vergotte (2011) are the only contest models that incorporate economic networks. However, each of the studies is distinct from our formulation. In Pommerenke (2006), there is no explicit cost of expending effort and the economic network is determined endogenously by the decisions of the players. In Franke and Ozturk (2009), the networks are exogenously given but they restrict the analysis to bilateral conflicts. Telone and Vergottee (2011) consider endogenous network formation as a barrier to entry to a contest. Szymanski (2011) considers both a contest in which players are located along a line and one in which they are located on a circle, but considers only contests in which there is a single winner. We consider a multiple winner contest under a ring network that arises in the market model of Salop (1979), the consumer preference model of Lancaster (1979), and the political compass model of Lester (1996). Given the ring network, the total package-efforts of all sets of

5

They show that there exist many stars in economics, i.e., there are “highly connected economists work with economists who have few or no other co-authors”. In the current study, however, we abstract away from coauthorship and the network is pre-specified in terms of the research areas.

5

k adjacent players determine the probability of winning. This is a mirror case of a combinatorial auction where bidders place bids on a combination of items, or ‘package’. 6 In the current structure, the package preferences of the designer determine the winners from the packages of the groups of individual efforts. Hence, we term this contest a combinatorial group contest. We can interpret a combination of adjacent players as a possible coalition formed in attempt to win a prize. Players have ex-ante common knowledge about the possible coalitions that can form, but the actual winning coalition formed depends upon individual efforts. As a result the coalitions are determined endogenously.

2. Model 2.1. A Contest with Package Designer Preferences Consider a contest in which n players (1 through n) are distributed around the circumference of a 1, 2 …

circle (Figure 1). Denote the set of players as each player and

and player

1 and

is adjacent to players

. As in the model of Salop (1979),

1 , with player 1 adjacent to players 2 1. The distance along the circumference

is adjacent to players 1 and

between any two neighboring players is considered a ‘unit’. There are

indivisible prizes

and a player can win at most one prize. Figure 1. A Circular Structure: n (n-1)

1 2

(t-1) t (t+1)

6

Combinatorial auctions were first proposed by Rassenti et al. (1982). Simple combinatorial auctions have been used in estate auctions, bus routes, transportation, industrial procurement, and in the allocation of radio spectrums for wireless communications. For a detailed discussion of this issue please see Cramton et al. (2006).

6

The designer preferences are defined over the group effort of all

groups of

adjacent

players. The preference generates a choice function that is represented by a lottery form. The players expend costly effort and a set of k adjacent players are selected as winners using a lottery contest success function. The probability that any particular set of k adjacent players wins is equal to the sum of the efforts of the players in that set divided by the aggregate sum of efforts in all possible k-adjacent sets of players. All k prizes are valued equally by a given player, although this value may differ across players. Let

0 denote the value of a prize to player

let the effort expended by player be denoted by

and

0. The positions and valuations of the

players are common knowledge. The probability that player t wins a prize is denoted by ,

where

,

is the vector of the efforts of players

.

Definition 1 (k-neighboring players): If the distance along the circumference of the circle between any pair of players ,

is strictly less than

units, then player and player are 1. If ,

called k-neighboring players. We denote this relationship as

are not k-

0.

neighboring players then

Definition 2 (k-neighboring group): A k-neighboring group is a set of k mutually k-neighboring players. i.e., if

is a k-neighboring group then the cardinality of

is

and

,

1.

,

For the sake of simplicity we always count the k-neighboring players or k-neighboring groups in a clockwise manner. We denote a k-neighboring group that starts with player and ends with player , as ,

,

,

,…,

,

,

. Note that there are a total of n distinct k-neighboring groups:

, ,

. Also, a particular player is a member of several k-neighboring

groups. For example, players 5 to 12 are common in 10-neighboring groups ,

,

,

,

, and

. In the continuation we will make use of the following observation.

Observation. Each player

is a member of k distinct k-neighboring groups.

We assume a perfect-substitutes group impact function. i.e., the total effort expended by a k-neighboring group ,

∑

,

,

is the sum of the efforts of all players in

,

. We denote this sum by

. Consequently, the probability that the k-neighboring group 7

,

wins is

,

, ,

,

/

,

,…,

,

,

,

if , otherwise

,

1/n

The probability that player

wins a prize,

neighboring group that includes player wins. So,

,

,

0

, equals the probability that any k-

is the ratio of the sum of the effort levels of

all k-neighboring groups that include player , and the sum of all possible k-neighboring groups’ efforts.7 Formally,

, the contest success function, is:

, where M

0 for some i N

if

∑

k/n ∑

k

Since there are

if

j x

x

(1)

i N

0 .

players and each player is included in

distinct groups, the

denominator in (1) is the sum of the efforts expended by all possible k-neighboring groups. The numerator is the total effort expended by the k-neighboring groups that include player . From the observation stated above, player is contained in effort appears

times. Players

1 and

1(

distinct k-neighboring groups, so his own ) are contained in

neighboring groups that contain player , so their efforts appear 1 and

until finally players

1 distinct k-

1 times. This continues

1 are contained in only one k-neighboring group that

contains player , so their efforts appear only once. Equation (1) can be rewritten as P

,

M ∑

∑

k/n

if x if x

0 for some i N 0

i N

(1’)

(1’) makes it clear that a player can free-ride on neighbors and win with positive probability without expending any effort. Given the CSF, the payoff function for player is: ,

‐ ‐

with probablity P with probablity 1

, P

,

(2)

The lottery CSF is continuous everywhere except when every player expends zero effort. But such a strategy profile is never an equilibrium because spending an infinitesimally small amount of effort strictly increases a player’s payoff. Consequently we ignore the discontinuity of the lottery CSF. Using (1) in (2), the expected payoff to player t is 7

For the mathematical construction of the CSF, see Appendix A.

8

M

, Player

∑

seeks to maximize

x

∑

with respect to

, subject to the non-negativity constraint

0. The corresponding Lagrangian equation and the first order (Kuhn-Tucker) conditions of maximization are M

x

∑

∑ ∑

‐

∑

0, λ

λx

M

0, λ

1

∑

(3) λ

0

(4)

0

(5) . A unique symmetric equilibrium

We first examine the case of symmetric values exists in this case.8 Substituting

in (4) we obtain ∑

Isolating

∑

to find the best response function we obtain Max 0,

/

∑

The expected payoff function is strictly concave in 2 ∑

for

∑

(6)

0. ∑

0

Hence, the first order conditions are necessary and sufficient for maximization. Solving (6) for symmetric equilibrium we obtain (7) Plugging this into the symmetric equilibrium probability of winning and the equilibrium payoff we obtain the following. The calculations for (7) and (8) are given in Appendix B. and

0

(8)

The equilibrium payoff is increasing in the number of prizes awarded and in the prize value and decreasing in the number of players. The equilibrium individual effort is increasing in 8

A general asymmetric valuation case or the possibilities of asymmetric equilibria in symmetric valuation are beyond the scope of this study. We discuss the asymmetry in valuation in section 2.3.

9

the prize value, decreasing in the number of prizes awarded, but the effect of the number of , so equilibrium effort increases

players is not straightforward. Note that

2 . Proposition 1 summarizes the symmetric

(decreases) in the number of players if equilibrium results.

Proposition 1 (Symmetric equilibrium of the combinatorial contest). In the combinatorial group contest defined by (1) and (2), there exists a unique symmetric pure strategy Nash equilibrium. and each player earns a strictly positive

The equilibrium effort of each player is equilibrium expected payoff

.

We label the equilibrium rent dissipation ( symmetric values as

) of the combinatorial group contest under

, which is monotonically increasing in the number of players. The total

surplus, defined as the difference between the total prize value and the total equilibrium effort, is 1

. The total surplus is also decreasing in the number of players but increasing in the

number of prizes and the prize value.

2.2. Comparison of Equilibrium Rent Dissipation In this section we compare the degree of symmetric equilibrium rent dissipation in the combinatorial group contest with the equilibrium rent dissipation in the models of Berry (1993), Clark & Riis (1996), Fu and Lu (2009), and Katz et al (1990). We label the equilibrium rent dissipation in each of these models, respectively, as RSB , RSCR , RSFL , and RSK . To be consistent with our analysis, we use a symmetric version of the Katz et al. (1990) model where there are / pre-specified groups in the contest (assume n/k to be an integer). Each group consists of members, and each member of the winning group earns one prize (a total of

prizes). We also

use a single-sequence version of the Fu and Lu (2009) model where the players are divided into subsets and there are / players in each subset. Only one player per subset is chosen to be winner; hence, there are total

winners.

It is interesting to note that even with fundamentally different network structures, the equilibrium rent dissipation in the current study is exactly the same as that in Berry (1993) and the symmetric version of Katz et al. (1990), but differs from that in the Clark and Riis (1996) 10

and single-sequence Fu and Lu (2009) models. Proposition 2 summarizes the results for equilibrium rent dissipation. A proof of this proposition is included in Appendix C. Table 2 provides a summary of the models and the corresponding total rent dissipation. Proposition 2 (Ranking of the equilibrium rent dissipation). With symmetric prize values, the Clark and Riis (1996) contest has greater equilibrium rent dissipation than the single-sequence Fu and Lu (2009) contest which, in turn, exhibits strictly greater rent dissipation than the combinatorial group contest. Moreover, the equilibrium rent dissipation of the combinatorial group contest is equal to that in the contest defined by Berry (1992), and the symmetric group size contest of Katz et al. (1990). That is, RSCR

RSFL

RSCG

RSB

RSK .

Table 2. Comparison of equilibrium rent dissipation (symmetric value case) Treatment type

Berry (1993)

Clark & Riis (1996) Symmetric group size Katz et al. (1990) Single-sequence Fu and Lu (2009)

Current Study

Symmetric equilibrium rent dissipation

Description n players expend effort to win one of the k prizes. Winners are chosen using a one-shot lottery CSF. n players expend effort to win one of the k prizes. Winners are chosen using a nested lottery CSF. n/k groups with k players per group expend effort to win one prize. The winner is chosen using a lottery CSF. Players distributed in k sets (n/k players in each set) expend effort to be the sole winner in the respective set. Winners are chosen using a lottery CSF. n players arranged in a circle expend effort to win one of the adjacent k prizes. Winners are chosen using a one-shot lottery CSF.

n

k V/n

1

k n

n

k V/n

k n

k V/n

n

k V/n

k n n

j V j

2.3. Asymmetric Values The effects of asymmetric values in single-winner contests or group contests are well documented in the literature. Stein (2002) shows that in a single winner Tullock contest with asymmetric values, players’ equilibrium efforts are ranked in the same order as their values. Moreover, some players with sufficiently low values may expend zero effort in equilibrium. In a 11

group contest in which players within the same group have asymmetric values (Baik, 2008), in each group players with high values expend positive effort in equilibrium and the players with relatively low valuations free ride by expending zero effort. This phenomenon is described as ‘the exploitation of the great by the small’ (Olson, 1965). However, the effect of asymmetric values in general multiple winner contest has not been thoroughly explored. Analogous to the case of asymmetric values in single-winner contests, one may observe players expend zero effort in multiple winner combinatorial contests. However, unlike single-winner contests, in the combinatorial contest, depending on a player’s value and position, there may be incentive to exert zero effort in order to free-ride on others. In this sense our results are comparable to group contests with within group asymmetric values A general treatment of the case of asymmetric values is beyond the scope of this paper. However, to capture the intuition behind how the presence of asymmetric values affects equilibrium behavior, we examine a simple 4x2 example. In this example four players (1, 2, 3 and 4) are located at the corners of a square. Two adjacent players are chosen to be winners. So the possible sets of winners are 1 & 2 , 2 & 3 , 3 & 4 , and 4 & 1 . Figure 2 describes the structure. We consider two cases in which two of the four players have the same high value for the prize and the other two players have the same low value. The cases are distinguished by the relative positions of the high-value players. Figure 2. A 4x2 Asymmetric case 1

2

4

3

; i.e., the two high value players are positioned

First, assume

on two diagonal corners of the square, as are the two low value players. We call this the ‘Diagonal’ case. Note that under a symmetric equilibrium, players 1 and 3 employ identical strategies, and players 2 and 4 employ identical strategies. In such an equilibrium, low value players completely free ride on the efforts of the high value players by expending zero effort.

12

High value players expend a quarter of their valuation of the prize. So, the total rent dissipation is even lower than the standard Tullock contest. . i.e., players with identical

For the second case assume

values are positioned in adjacent corners. We call this the ‘Adjacent’ case. Again, in a symmetric equilibrium, players with identical values employ identical strategies. Low value players expend zero and completely free-ride in equilibrium. However, this case differs from the previous one in the effort expended by the high value players. Unlike the diagonal case, with adjacent high value players, each high value player is able to partially free-ride on the effort of his high value neighbor. As a consequence, in equilibrium each expends only half of the equilibrium effort in the Diagonal case. Proofs of these results appear in Appendix D.

2.4. Comparison of Results with Asymmetric Values In Proposition 2, we showed under symmetric valuation that the equilibrium rent dissipation of the combinatorial group contest is equal to that in the contest defined by Berry (1992), and the symmetric group size group contest (Katz et al., 1990). Now, under asymmetric valuation, we compare the equilibrium rent dissipation of the combinatorial contest with the levels arising in the contest defined by Berry (1992) and the group contests (Katz et al., 1990; Baik, 2008). In the Berry (1993) model with two high value and two low value players, two types of equilibrium arise. If the high value players value the prize sufficiently more than the low value 0 . So the equilibrium rent

low value players free ride by expending zero effort dissipation is a

result,

in 2

/6 and

2 then in the equilibrium each high value player expends

players, i.e.,

/3. But when

2 , then the low value players do not completely free ride. As 2

equilibrium

2

/

; and the equilibrium rent dissipation is 2

/

/

and 2

.

Moving on to the models of group contest, when there are two high and two low value players, then a group contest can either be between the high value group and the low value group (H-H Vs. L-L) as in Katz et al. (1990) or between groups where there is one high and one low value player in each group (H-L Vs. H-L) as in Baik (2008). In the first case, one cannot uniquely determine the equilibrium individual effort levels, but the H-H group and L-L group expend total efforts of LH / L

H

and HL / L 13

H

respectively. Thus, equilibrium rent

dissipation is LH/ L

H . In the second case, both the low value players expend zero effort and

both the high value players expend H/4 making the equilibrium rent dissipation H/2. From section 2.3, in the diagonal case of the current study, the equilibrium rent dissipation is

/2; and in the adjacent case the equilibrium rent dissipation is

/4. Table 3

summarizes and compares the asymmetric equilibrium results. Table 3. Comparison of the equilibrium rent dissipation in asymmetric 4x2 cases Model setting

Parameter restrictions

Total equilibrium rent dissipation

Berry (1993)

H

2

H/3

Berry (1993)

H

2

2LH/ L

Katz (1990)

(H and H) Vs (L and L)

LH/ L

Baik (2008)

(H and L) Vs (H and L)

H/2

Current Study: Diagonal case Current Study: Adjacent case

2 H (and 2 L) are diagonally positioned 2 H (and 2 L) are adjacently positioned

H/2

H 2L

H

H

H/4

This result enables us to rank the models in terms of the equilibrium rent dissipation. For example, when 3L

2 , then RAB

RAD

RAB

RAK

RAA

. Hence, if

the objective of a contest designer is to maximize the total rent seeking expenditure, then for a contest between groups, the Baik (2008) treatment is preferred to the Katz et al. (1990) treatment; i.e., a contest between mixed valued groups will provide higher rent dissipation than a contest between a high valued group versus a low valued group. In case of individual multiple winner contests, the ‘diagonal case’ of the current study is preferred over the Berry (1993) treatment; which in turn, is preferred to the ‘adjacent case’ of the current study.

3. Discussion We construct a multi-winner contest with players distributed along the circumference of a circle. Players expend costly effort and a fixed number

of adjacent players are selected as the winning

coalition by applying a lottery contest success function to the sum of the individual bids of all 14

possible groups of

adjacent players. We fully characterize the unique symmetric equilibrium

under symmetric valuation of prizes. Even with fundamentally different network structures; the equilibrium rent dissipation of the current model turns out to be the same as that in Berry (1993) and Katz et al. (1990). We characterize equilibrium for two specific examples under asymmetric valuations and compare the rent dissipation with related models. Both in symmetric value and asymmetric value cases it is possible to rank different multiwinner settings in terms of equilibrium rent dissipation. This is useful for contest designing purposes. In case of the conference example, as described in the introduction, a conference committee may want to maximize the total effort expended on research. If it is known that the value for conference participation to a set of authors is sufficiently more than other authors, and that most of the high value authors work in a similar areas (‘adjacent case’), then the committee should set a Berry (1993) type selection process. But, if the high value authors work in much dispersed areas (‘diagonal case’), then the committee should employ a combinatorial group contest as coined in this study. In case the value for participation is symmetric, then any of Berry (1993) type or combinatorial group contest type of selection process would result in the same total effort expended on research.

15

References Baik, K. H. (2008). Contests with group-specific public-good prizes, Social choice and Welfare, 30(1), 103-117. Barut, Y., & Kovenock, D. (1998). The Symmetric Multiple Prize All-Pay Auction with Complete Information, European Journal of Political Economy, 14, 627-644. Berry, S. K. (1993). Rent-seeking with multiple winners, Public Choice, 77, 437-443. Chowdhury, S.M., Lee, D., & Sheremeta, R.M. (2011). Top Guns May Not Fire: Best-shot Group Contests with Group-Specific Public Good Prizes, UEA Working Paper #24 Clark, D. & Riis, C. (1996). A multi-winner nested rent-seeking contest, Public Choice, 87, 177184. Clark, D.J. and Riis, C. (1998a) Competition over more than one prize. American Economic Review, 88, 276–289. Clark, D.J. and Riis, C. (1998b) Influence and the discretionary allocation of several prizes. European Journal of Political Economy, 14, 605–625. Cramton, P., Shoham Y., & Steinberg, R. (2006). Combinatorial Auctions. MIT Press. Fu, Q. & Lu J. (2009). The Beauty of “Bigness”: on Optimal Design of Multi-Winner Contests. Games and Economic Behavior, 66 (1), 146-161. Franke, J., & Ozturk, T. (2009). Conflict Networks. Ruhr Economic Papers 116. Glazer, A., & Hasin, R.(1988). Optimal Contests , Economic Inquiry, 26, 133-143. Goyal, S., Leij, M., and Moraga-Gonzalez, J. (2006). Economics: an Emerging Small World. Journal of Political Economy, 114 (2), 403-412. Hotelling, H. (1929). Stability in Competition, Economic Journal, 39 (153), 41-57. Katz, E., Nitzan, S. and Rosenberg, J., (1990). Rent seeking for pure public goods. Public Choice, 65, 49–60. Kolmar, M., and Rommeswinkel, H. (2010). Group Contests with Complementarities in Efforts, U. of St. Gallen Law & Economics Working Paper No. 2010-12. Lancaster, K. (1979). Variety, Equity, and Efficiency: Product Variety in an Industrial Society, Blackwell, Oxford. Lee, D. (2011). Weakest-link contests with group-specific public good prizes. European Journal of Political Economy, forthcoming.

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Lester, J. C. (1996). The Political Compass (And Why Libertarianism Is Not Right-Wing), Journal of Social Philosophy, 27(2), 176-186. Moldovanu, B., & Sela, A. (2001). The Optimal allocation of Prizes in Contests, American Economic Review, 91(3), 542-558. Olson, M. (1965). The Logic of Collective Action: Public Goods and the Theory of Groups. Harvard University Press, Cambridge, MA. Pommerenke, K. (2006). Cooperation with Rivals, UCSC Working Paper # 617. Rassenti, S.J., Smith, V.L. & Bulfin, R.L. (1982). A Combinatorial Auction Mechanism for Airport Time Slot Allocation, Bell Journal of Economics, 13, 402-417. Salop, S.C., (1979). Monopolistic competition with outside goods. Bell Journal of Economics, 10, 141–156. Sisak, D. (2009). Multiple-prize contests: The Optimal allocation of Prizes. Journal of Economic Surveys , 23, 82–114. Skaperdas, S. (1996). Contest Success Functions, Economic Theory, 7, 283-290. Stein, W.S. (2002). Asymmetric rent-seeking with more than two contestants. Public Choice, 113, 325–336. Szymanski, S. (2011). Contests on a line and around a circle. Paper presented at the conference on Tournaments, Contests and Relative Performance Evaluation at North Carolina State University. Szymanski, S., & Valletti, T.M. (2005) Incentive effects of second prizes. European Journal of Political Economy, 21, 467–481. Tellone, T., & Vergotte, W. (2011).

Endogenous network formation in Tullock contests,

working paper. Tullock, G. (1980). Efficient Rent Seeking. In James M. Buchanan, Robert D. Tollison, Gordon Tullock, (Eds.), Toward a theory of the rent-seeking society. College Station, TX: Texas A&M University Press, 97-112. Yates, A. J. & Heckelman, J. C. (2001). Rent-Setting in Multiple Winner Rent-Seeking Contests, European Journal of Political Economy, 17 (4), 835 -852.

17

Appendix (A) Construction of the Winning Probability (CSF): P

,

∑

∑

∑

∑ M ∑

∑

(B) Derivation of the Symmetric Equilibrium: From (6) we have x

BRF

∑

x

/ M

∑

V

x ; when V

V t

N, then x

x

/

x

n

n x x Also: P And: E π

k

1 x n

1

k

1 x

2

1 2x

V

n

1 x

2 V

V M

k

∑

P x V

x

1

V

k

2

V

1 2x V.

(C) Proof of Proposition 2: Equivalence of Equilibrium Rent Dissipation Berry (1993): The total equilibrium rent dissipation in Berry (1993) is:

V

Current study: Individual equilibrium rent dissipation of any player is:

V Hence, the overall

equilibrium rent dissipation: nx

V

Katz et al. (1990): It is easy to show that everybody expending zero effort is not equilibrium so we will have an interior solution at the group level. Define

/ as the number of groups

and as the prizes are divided into the winning group members. In this fashion, at a group level the number of prize is 1. Under symmetric equilibrium individual expenditures cannot be identified, but because of symmetry, each group will collectively play like an individual player

18

with valuation V. Thus, equilibrium group expenditure is: /

dissipation is:

and the overall rent

.

/

(D) Characterization of the Asymmetric 4x2 Equilibrium Using (1), the probability of winning a prize by player t is

∑

As a result, expected payoff for player t is: E

(D.1)

∑

for t = 1, 2, 3, 4. Denote ‘-t’ as the player located diagonally to player t. Player t maximizes (D.1) subject to the non-negativity constraint x

0. Denoting λ as the

Lagrangian multiplier, the corresponding Lagrangian equation and the first order conditions are: V

∑

x

λx

V

1

∑

0, λ where x

0, λ

(D.2)

λ

0

(D.3)

0

(D.4)

is the effort of the player positioned diagonally to player t. H

(D.1) Diagonal case

L :

Recall that because of the positional and value-wise symmetries, in a semi-symmetric equilibrium x

x and x

x and also λ

λ and λ

λ . So analyzing player 1 and 2’s

strategies would suffice for our analysis. From (D.3) we can write V

1

λ

0

(D.1.1)

V

1

λ

0

(D.1.2)

∑

∑

There are 4 possible distinct combinations of Lagrangian multipliers: (i) λ λ

λ

0 ; (ii) λ

λ

0 and λ

λ

λ

λ

λ

0

Case (i): λ

λ

0 and λ

λ

λ

0 ; (iii) λ

0 : This means x

possible because expending any small ε

λ x

0 and λ x

x

λ λ

0 and 0 ; (iv)

0 ; which is not

0 strictly increases the payoff of any player. Hence

case (i) is ruled out. 19

Case (ii): λ

λ

0 and λ

λ

0: This means x

x

these conditions into (D.1.1) and (D.1.2) yields λ

1

0 and x

H L

x

0. Substituting

0: a contradiction. Hence case

(ii) is also ruled out. λ

Case (iii): λ

0 and λ

λ

0 : This means x

x

0 and x

1

L H

H

x

Substituting these conditions into (D.1.1) and (D.1.2) yields x

x

and λ

0 . λ

0 . Hence case (iii) is a possible equilibrium where the low value players completely

free ride on the high value players. Case (iv): λ

λ

λ

λ

0: This means x

conditions into (D.1.1) and (D.1.2) yields H

0 and x

x

0. Substituting these

L : a contradiction. Hence case (iv) is ruled out.

H

(D.2) Adjacent case

x

L :

Again, recall that because of the positional and value-wise symmetries, in a semix and x

symmetric equilibrium x

x and also λ

λ and λ

λ . So analyzing player

1 and 3’s strategies would suffice our analysis. From (D.3) we can write V

1

λ

0

(D.2.1)

V

1

λ

0

(D.2.2)

∑

∑

λ

Again, there are 4 distinct combinations of Lagrangian multipliers: (v) λ λ

0; (vi) λ

λ

λ

λ

0 and λ

λ

0; (vii) λ

λ

0 and λ

λ

0 and λ

0; (viii) λ

λ

0

Case (v): λ

λ

0 and λ

λ

0: Following the same logic as in case (i) this is not

possible. Hence case (v) is ruled out. Case (vi): λ

λ

0 and λ

λ

0 : This means x

Substituting these conditions into (D.2.1) and (D.2.2) yields λ

x 1

0 and x H L

x

0 .

0: a contradiction.

Hence case (vi) is also ruled out. Case (vii): λ

λ

0 and λ

λ

0 : This means x

Substituting these conditions into (D.2.1) and (D.2.2) yields x 1

L H

x

0 and x x

H

and λ

x

0. λ

0 . Hence case (vii) is a possible equilibrium where the low value players completely

free ride on the high value players. 20

λ

Case (viii): λ

λ

λ

0: Following the same logic as in case (iv) this is not possible.

Hence case (viii) is also ruled out. (E) Derivation of Asymmetric Equilibrium Berry (1993) Following Clark and Riss (1996), Berry (1993) CSF can be written as P 1

. When n

∑

4 and k

2 then the CSF becomes P

maximize his expected payoff subject to x ∑

V

x

∑

∑

. A player will

0. The corresponding Lagrangian equation is:

λx

where λ is the Lagrangian multiplier. The F.O.C.s are as follows: ∑

V

1

∑

λ

0

(E.1) H

Without loss of generality we assume equilibrium x

x and x

and (E.2) is same as (D.4)

x ; also λ

L. Thus, In a semi-symmetric

λ and λ

λ . Hence we analyze only the

strategies of player 1 and player 3. Imposing the conditions in (E.1): H

1

λ

0

(E.3)

L

1

λ

0

(E.4)

∑

∑

0, λ

There are 4 distinct combinations of Lagrangian multipliers: (a) λ 0; (c) λ

0, λ

Case (a): λ

0; and (d) λ 0, λ

λ

0 (b) λ

0, λ

0

0: Following the same logic as in case (i) under Appendix (D), this is not

possible. Hence case (a) is ruled out. Case (b): λ

0, λ

0: This means x 1

(E.3) and (E.4) yields λ Case (c): λ

0, λ

H

0 and x

0: a contradiction. Hence case (b) is also ruled out.

L

0: This means x H

(E.3) and (E.4) yields x

0; substituting these conditions into

and λ

1

0 and x

0 ; substituting these conditions into

L H

. Hence case (c) is equilibrium where the low value

players completely free ride on the high value players only if 1 Case (d): λ

λ

and (E.4) yields x

0: This means x H L HL L H

L H

and x

0 and x

21

0 i.e., H

2 .

0; substituting these conditions into (E.3) HL

L H

L H

L H

. Which is possible only if H

2 .