J Econ Manage Strat. 2020;29:377–400. wileyonlinelibrary.com/journal/jems © 2020 Wiley Periodicals, Inc.
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377 DOI: 10.1111/jems.12343O R I G I N A L A R T I C L E
Contests over joint production on networks
Serhat Doğan
1| Kerim Keskin
2| Çağrı Sağlam
11Department of Economics, Bilkent University, Ankara, Turkey
2School of Business, ADA University, Baku, Azerbaijan
Correspondence
Kerim Keskin, School of Business, ADA University, AZ1008, Baku, Azerbaijan.
Email:kkeskin@ada.edu.az
Abstract
We consider a network of heterogeneous agents where each edge represents a two‐player contest between the respective nodes. In these bilateral contests, agents compete over an endogenous prize jointly produced using their own contest efforts. We provide a necessary and sufficient condition for the existence of Nash equilibrium and characterize the equilibrium total effort for every agent. Our model has insightful results regarding the network type, that is, depending on whether the network is bipartite or nonbipartite. Finally, considering the sum of all expected utilities as an efficiency notion, we investigate the optimal network structure.
K E Y W O R D S
efficiency, endogenous prize, joint production, networks, optimal network, Tullock contests
J E L C L A S S I F I C A T I O N C72; D74
1 | INTRODUCTION
Since the seminal works of Tullock (1980) and Lazear and Rosen (1981), contest‐like situations have been extensively investigated in the literature. In any such conflict, the contending parties expend nonrefundable resources to increase their chances of winning some prizes. Real‐life applications range from sports and warfare to firm competition and political campaigns.
In the contest theory literature, the existing studies mostly concentrate on a single conflict with two or more contenders while overlooking possible interdependencies between different conflicts. These models are insufficient to explain complex real‐life situations such as wars between countries in which a country is fighting against several others at the same time (because of different issues, over different territories, etc.) or market competitions between multi- product firms in which a firm is simultaneously competing with several others at the same time (in different markets, for different products, etc.). Arguably, the former situation is a pure conflict, whereas the latter situation includes some sense of collaboration as there may be positive externalities attached to the firms' contest efforts (e.g., increased market awareness or increased production efficiency). In this paper we are interested in the latter situation.
For a more concrete example of market competition, consider a set of firms deciding on their advertising expenditures to promote their own products. Each firm produces different types of goods; and for each type of good a firm produces, there is another firm producing a close substitute. Advertising a specific good increases the consumers' willingness to pay for that good as well as for its close substitute; and whichever firm is more successful than its
competitor ends up with a higher market share and a higher expected revenue. Along similar lines, consider also a set of political parties deciding on the content of their political campaigns for an upcoming election. Each political party has certain opinions on different topics; and for each view explicitly supported by a political party, there is another political party challenging that view. Expressing an opinion on a particular topic leads to a debate, which in turn attracts the swing voters' attentions to that topic, so that whichever political party wins the debate ends up with an increased number of supporters. From a game‐theoretic perspective, two firms competing in the same market or two political parties debating over an issue are engaged in a contest game where they exert costly efforts to increase their expected revenues or expected number of votes. Considering the set of all firms or political parties, the overall interaction can be described as a network including a bilateral contest game on each edge.
In the sense that our interest is mostly in strategic conflicts between the agents of a network, our model is most closely related to the model proposed by Franke and Öztürk (2015).1 These authors model the structure of local conflicts with strategic interdependencies as a network where nodes represent the conflicting parties and edges represent the bilateral conflicts in which the winner collects a rent. Following an implicit characterization of Nash equilibrium, they concentrate on conflict intensity in certain types of networks and investigate cases of peaceful conflict resolution as a notion of efficiency. In the current paper, we analyze conflicts over joint production instead of studying pure conflicts (e.g., war) as in Franke and Öztürk (2015). Accordingly, we capture a sense of collaboration between the agents of a network. This calls for a new modeling, which comes with important analytical and intuitive differences.
Analytically, we study bilateral contests with endogenous prizes (increasing in both agents' effort levels) rather than rent‐seeking contests.2Intuitively, the nature of competition is different since exerting more effort not only increases one's winning probability, but also one's winning prize, as well as the winning prize for the other party. Attached is a different efficiency implication: in contrast to rent‐seeking contests, in which efforts are usually considered social waste, exerting more effort might increase social welfare in our model since efforts are now productive.
It is worth noting that there are other studies in the“crossroads” of contests and networks, focusing on the strategic interactions between several conflicting parties. For example, Huremovic (2015) investigates network stability in a model of strategic network formation, after generalizing the results by Franke and Öztürk (2015). In more recent papers, Hiller (2017) analyzes a network with friendship and enmity links such that an agent tries to extract payoffs from his/her enemies in different bilateral contests not by choosing efforts but only through the help of his/her friends;
and Grandjean, Tellone, and Vergote (2017) study an n‐player contest on a network that influences the winning prizes directly through the number of neighbors. These papers explore different conflict situations that lack the sense of collaboration our model captures. Furthermore, they are rather interested in a network formation analysis, hence are missing a detailed characterization of equilibrium strategies for a fixed network.
Throughout this paper, we analyze Nash equilibrium. We first provide a necessary and sufficient condition for the existence of equilibrium and then characterize the equilibrium total effort for every agent. Our results indicate that the network type significantly influences the equilibrium characteristics. In particular, if the network is nonbipartite, then the agents in each bilateral conflict exert the same amount of effort in an equilibrium; whereas if the network is bipartite, then the agents in the stronger group are equally more advantaged in their contests as they exert a common multiple of the equilibrium efforts exerted by their respective rivals. Afterward, we proceed to an optimal network analysis in which we consider the sum of all expected utilities as an efficiency notion. When the agents are symmetric, we prove that any nonbipartite network is optimal. For the asymmetric case, we numerically show that there are cases where any non- bipartite network is optimal as well as cases in which there exists a certain type of bipartite network dominating all nonbipartite networks. For the latter, our numerical analysis indicates that the optimal network is generally such that the number of agents in the partitioned sets are close to each other, although strong asymmetries are also possible.
Our main contribution is twofold: (a) We contribute to the analysis of strategic conflicts (see Anderton &
Carter,2009; Dechenaux, Kovenock, & Sheremeta,2015; Konrad,2009, among others). Unlike the conventional models focusing on a single conflict between two or more parties, we analyze multiple simultaneous conflicts between a group of heterogeneous agents. On top of that, as we investigate contests over joint production, our analysis is closely related to the literature on contests with endogenous prizes (see Chowdhury & Sheremeta, 2011; Chung, 1996; Keskin &
Sağlam,2018; Lee & Kang,1998, among others). (b) We contribute to the analysis of games played on networks (see Abreu & Manea, 2012; Bramoullé & Kranton, 2007, 2014; Galeotti, Goyal, Jackson, Vega‐Redondo, & Yariv, 2010, among others);3 and in particular, to the emerging literature on the analysis of contests on networks. Our model investigates potentially very complex conflict situations between heterogeneous agents of a given network. Interest- ingly, with such interdependent conflicts embedded in a network, our model is missing a typical assumption in network games: strategic complementarity or substitutability.
The rest of the paper is organized as follows. In Section2, we formulate our model. In Section 3, we analyze the existence and characteristics of its Nash equilibria. After investigating the optimal network structure in Section4, we conclude in Section5.
2 | THE MODEL
Let A be a set of n agents. A network G on A is represented by the pair (A, E), where A is the set of nodes and E⊂ A × A denotes the set of edges. Here, we consider simple networks for which no edge has a weight or direction, so that if (i, j)∈ E, then (j, i) ∈ E as well. We are particularly interested in connected networks for which there is a path between any pair of nodes. Denoting the neighbors of agent i by N(i) = {j|(i, j)∈ E}, this implies that N(i) ≠ ∅ for every i ∈ A.
We consider a two‐player contest on every edge (i, j) ∈ E that is to be played by the respective nodes i, j ∈ A. Let eij∈ [0, ∞) denote the contest effort exerted by agent i in his/her contest against j on the edge (i, j). We assume the following Tullock‐type contest success function:
P e e e
e e
( , ) =
+ ,
i ij ji
ij
ij ji
where α ∈ (0, 1) can be labeled as an effectiveness parameter. In this context, depending on the model interpretation, Pi(eij, eji) can either represent agent i's probability of winning the respective contest or agent i's share from the respective winning prize. Furthermore, each agent i∈ A has a productivity parameter i
( )
12, 1 ; and for each contest he/she participates in, agent i's winning prize is determined by a Cobb‐Douglas joint production function:V e eij( ,ij ji) =e eij ji .
i 1 i
This assumption can be motivated by referring to the “advertising” and “politics” interpretations provided in the Introduction. For the former, Vij(eij, eji) represents the total revenue for firm i in case it has a hundred percent market share in the respective market. Advertising efforts by firms i and j jointly increase the consumers' willingness to pay, which in turn increases the total revenue generated in this market. As for the latter, Vij(eij, eji) represents the total number of swing voters who would start supporting party i in case it wins the debate on the respective topic. Debating efforts by political parties i and j jointly attract the swing voters' attention to that topic, which in turn increases the number of voters influenced by the debate outcome. Notice that agent i has a particular ability to convert its own contest effort into a winning prize (denoted by βi), which is assumed to be higher than its ability to convert agent j's effort.4Notice also that if the agents were symmetric, then they would have the same production function with powers β and 1 − β; whereas when the agents are heterogeneous in their productivity parameters, a constant returns to scale production function would imply the assumed structure.5
In Figure1, we present an illustrative example for n = 9 agents. In this example we see that agent 1 participates in two different two‐player contests. One of these contests is against agent 5 and the other is against agent 6. In a similar manner, agent 2 participates in three different contests on the edges (2, 5), (2, 7), and (2, 8). Throughout the paper, we are interested in such conflict situations and investigate the set of Nash equilibria of the model.
At this point, the case of zero effort requires further attention. Notice that a Cobb‐Douglas joint production function implies that if agent j chooses not to exert any effort in the contest against agent i, then agent i cannot obtain Vij(eij, ej, i) > 0 no matter how much effort he/she exerts. In fact, since contest efforts are costly, agent i's best response would be to exert zero effort as well. Here we make an additional assumption to avoid this undesired outcome, so that we can concentrate
F I G U R E 1 An illustrative example
only on the cases in which all agents actively participate in all of their contests in the equilibrium. In particular, we assume that if eji= 0, then agent i has a different production function that returns a positive amount for every eij> 0. Assuming that the function is increasing and concave in eij, we prefer not to specify a functional form, aiming to keep the reader's attention on the function Vijabove. Be that as it may, we must emphasize that the case of zero effort would not be realized in any Nash equilibrium, because now, choosing not to exert any effort will never be a best response for some agent.6,7 We further assume that for every i∈ A: α + βi< 1. This indicates that we concentrate on cases in which the expected revenue of agent i∈ A in his/her contest against j ∈ N(i), which is denoted by Pi(eij, eji)Vij(eij, eji), exhibits diminishing marginal returns in agent i's own contest effort. Otherwise, a solution to an agent's optimization problem might not exist under certain values of model parameters, and such existence conditions would turn out to be quite intractable considering the heterogeneity embedded in the network's edge structure.
Finally, let ei=∑j∈N(i)eijdenote the total contest effort exerted by agent i∈ A. We assume a quadratic cost function8 for every agent i∈ A:
ci( ) =ei kei2,
where k > 0 is some constant andei denotes the collection of all efforts exerted by agent i on all of his/her edges.
Similarly,e = (e1,…, en) denotes the collection of all efforts exerted by all agents on all edges. Accordingly, the total expected utility of agent i∈ A can be written as
U e
e e e e ke
e ( ) =
+ .
i
j N i ij
ij ji
ij ji i
( )
1 2
i i
It is agent i's objective to maximize this total expected utility by choosingeigiven the effort levelejfor any j∈ N(i).
3 | EQUILIBRIUM ANALYSIS
In this section, we analyze the set of Nash equilibria of our model. Consider agent i∈ A and one of his/her competitors j∈ N(i). Given the competitor's effort eji> 0, we solve the utility maximization problem for agent i. The first‐order condition with respect to eijcan be written as
U e
e e e e e
e e ke
= ( + ( + ))
( + ) 2 = 0.
i ij
ij ji ji i ij ji
ij ji
i
+ 1 1
2
i i
(1)
Defining the ratio rij= eij/eji, we can rewrite the first‐order condition above as
U e
r e e e r
r e ke
r r
r ke
= ( + ( + 1))
( + 1) 2 = 0
= ( + ( + 1))
( + 1) 2 = 0.
i ij
ij ji ji i ji ij
ij ji
i
ij i ij
ij
i
+ 1
2 2
+ 1
2
i
i
Our first result highlights that at any equilibrium, every agent i∈ A chooses the same ratio against any of his/her competitors.
Lemma 1. At any Nash equilibrium, every agent i A has a unique ratio ri such that for every i j( , ) E: ri=e eij/ ji. This implies that for every i A and j N i( ): ri= 1/rj.
Proof. See the appendix.
Given the Tullock‐type contest success function, Lemma 1 implies that an agent's equilibrium prize share or winning probability is the same across all of his/her contests. Accordingly, the result indicates that an agent prefers to distribute his/her total equilibrium effort in such a way that there is no discrimination against any contest. To put it differently, it is not worthwhile for an agent to pay more attention to a certain subset of edges.
Now, we divide the following analysis into two cases specifying the network structure: (a) nonbipartite networks and (b) bipartite networks. A network G = (A, E) is said to have an odd cycle if for some odd k , there exists a sequence of agents i1, i2,…, ik, i1such that each pair of consequent nodes is adjacent. A network is bipartite, if it does not contain any odd cycle; that is if there exists a partition of A into A1and A2such that A1∩ A2=∅, A1∪ A2= A, and if (i, j)∈ E then either i∈ A1and j∈ A2or vice versa. If not, the network is nonbipartite.
Lemma 2. Consider any Nash equilibrium for a given network. If the network is nonbipartite, then for every i A r: i=r= 1. On the other hand, if it is a bipartite network for a partition A A{ 1, 2}, then there exists a ratiorsuch that for every i A1and j A2: ri=r and rj= 1/r.
Proof. See the appendix.
For any nonbipartite network, we see that the equilibrium ratio r* = 1. This implies that if there exists an odd cycle in the network, then for any contest between any two agents i, j∈ A, we would have eij*=e*ji at the equilibrium. This makes the rest of the equilibrium analysis straightforward. Using the first‐order condition (1), we can establish that the total effort exerted by agent i∈ A is
e e
( ) = + 2k
8 .
i* i
i (2)
The comparative statics analysis is fairly obvious. Any distribution of these total efforts over the agents' respective edges would constitute a Nash equilibrium as long as eij*=e*jiis satisfied on every edge.
As for a bipartite network, we still need to prove that there exists a unique equilibrium ratio r* and to determine which ratio is being used at an equilibrium. From this point onward, when a bipartite network for a partition {A1, A2} is considered, the ratio r always refers torA1, which is the ratio used by any member of A1. Furthermore, unless otherwise stated, we assume that rA1 rA2. This means that rA1=r 1.
Proposition 1. For any given network G= ( , ), there exists a unique equilibrium ratio r*.A E
Proof. From earlier arguments, we already know that the equilibrium ratio is unique, for instance, r* = 1, for a nonbipartite network.
Now assume that the network is bipartite. Recall that the first‐order condition (1) can be written as
U e
r r
r ke
= ( + ( + 1))
( + 1) 2 = 0.
i ij
i i i
i
i
+ 1
2
i
Solving forei, we have
e r r
= k r( + ( + 1)) 2 ( + 1)
i i i i
i
+ 1
2
i
for every i A1. Notice that this equation uniquely determines agenti's total efforteiin terms of ri, i, and some exogenously given model parameters. Furthermore, for every j A2, a symmetric equation can be written. By Lemma2, for every i A1 and j A r2: i=r and rj= 1/r. Accordingly, we can write that
e r e r r
( , ) = k r( + ( + 1))
2 ( + 1) ,
i i* + 1 i
2
i (3)
e r e r r
k r
r r r
k r
(1/ , ) =(1/ ) ( + ((1/ ) + 1)) 2 ((1/ ) + 1)
= ( + ( + 1))
2 ( + 1) .
j *j
j
j
+ 1
2 1
2
j
j
(4)
Within the proof of Lemma1, we have shown that
r r
r
r r
r r ( + ( + 1))
( + 1) =
( + 1) +
( + 1)
i i
+ 1
2
+ 1 + 1
2
i i i
is monotonically decreasing inr.9Thus, we can safely deduce that e r( , )i is decreasing inr. Conversely, e(1/ , )r j turns out to be increasing inr.
Our proof will be complete, if we show that the solution to the following equation exists and is unique:
e r( , ) = r e(1/ , ).r
i A i
j A
j
1 2
(5)
This equation utilizes the facts that the ratio of the total effort exerted by all agents in A1on all of their edges to the total effort exerted by all agents in A2 on all of their edges should be equal tor. Equation (5) can also be written as
r
r r r r r
r r r r
( + 1) + =
( + 1) +
i A i A
i
j A j A
j
1 1 +1
1 1
i i j j
1 1 2 2
Then, the following are true.
• As r , we have r +i 1 0and r + 1 , so that the LHS 0; as well as we have r +1/(r + 1) and r1 i , so that the RHS .
• As r 0, we have r +i 1 and r + 1 1, so that the LHS ; as well as we have r +1/(r + 1) 0 and r1 i 0, so that the RHS 0.
The observations above jointly imply the existence of r* solving Equation (5). In addition to that, since each e *i
is decreasing inr, it must be that the LHS is decreasing inr; and similarly, since eache*j is increasing inr, it must be that the RHS is increasing inr. Accordingly, there exists a unique r* solving Equation (5), which is revealed to be the equilibrium ratio of our model.
The equilibrium ratio r* can be interpreted as the unbalance of contest efforts over the whole network. When r* = 1, there is a balance such that the two contenders exert equal efforts in a contest and end up with equal prize shares or winning probabilities. When r* > 1, the contest efforts are unbalanced as one of the contenders starts to play more aggressively. Moreover, the level of unbalance increases as r* increases. From Lemma1, we know that an agent does not prefer paying more attention to a certain subset of his/her edges. As it turns out, the fact that every agent follows such a no‐discrimination policy creates a perfect balance of contest efforts over the whole network for nonbipartite networks. On the other hand, how agents are partitioned into two groups would be important for bipartite networks.
There is one stronger group, employing the ratio r* > 1, and each member of that group has a relatively higher equilibrium prize share or winning probability in his/her contests with the members of the opposing group. Inter- estingly, no group member ends up with a greater advantage, because given the unique ratio r*, the equilibrium prize shares or winning probabilities are the same across all bilateral contests. That is, it is as if each partitioned group acts as a single agent at the equilibrium.
As for the comparative statics results, within the proof of Lemma1, we have already mentioned how the equili- brium total efforts respond to an increase in r. Now, recalling that the equilibrium total efforts e(r, βi) and e(1/r, βj) are given by Equations (3) and (4), we concentrate on the respective effects of increases in βior βj.
Remark 1. The following are true for the equilibrium total efforts:
(i) The function e r( , )i is increasing in i.
(ii) The function e(1/ , )r j is increasing in j if and only if
r r
< 1 r
log 1 + .
j (6)
Proof. See the appendix.
Notice that e(r, βi) and e(1/r, βj) are not symmetric in terms of their respective responses to increases in βiand βj. This asymmetry is a result of the fact that agents employ different ratios: r and 1/r. To be more precise, if an agent is a member of the stronger group with a relatively higher ratio (in Remark1, it is agent i), then his/her equilibrium total effort always increases in own productivity parameter. If not, there are cases in which having a higher productivity parameter leads to a lower equilibrium total effort. For instance, in the weaker group, there is a critical value char- acterized by the RHS of inequality (6) such that if a group member has a productivity parameter higher than that critical value, he/she would be discouraged and exert lower total effort. The critical value decreases in r, so that if agent j's group gets weaker, he/she becomes more likely to be discouraged.
Our analysis above does not consider any edge structure neither for nonbipartite networks nor for bipartite net- works. This issue is highlighted by the next remark.
Remark 2. Given any nonbipartite network, the equilibrium ratio r* is always 1. Given a bipartite network for a partition A A{ 1, 2}, the equilibrium ratio r* is independent of the edge structure.
The equilibrium ratio r* is the unique solution to Equation (5). It also uniquely characterizes the equilibrium total efforts e *i and e*j as given by Equations (3) and (4) above. Any distribution of these total efforts over the agents' respective edges would constitute a Nash equilibrium as long as the equilibrium ratio r* is preserved on each edge.
Because of this further requirement that the equilibrium ratio should be preserved on every edge (i, j)∈ E,10showing that there exists a unique solution to Equation (5) does not suffice for the existence of a Nash equilibrium. This is the reason why the geometry of the network might prevent the existence of an equilibrium, so that we need a condition over the network structure to obtain an existence result. In the following we formalize these arguments.
Definition 1. Consider any network G= ( , ) such that each node iA E A is assigned a value ci +. Let
c c
c = ( , …,1 n), and let an allocationa = ( , …,a1 aE )overEbe such that for every i j( , ) E: a( , )i j =a( , )j i 0. An allocation is said to constitute a solution forc if for every i A:
ci= a .
j N i i j ( )
( , )
Lemma 3. Consider any network G= ( , ) such that each node iA E A is assigned a value ci +. Let
c c
c = ( , …,1 n). There exists an allocationa = ( , …,a1 a| |E )constituting a solution forcif and only if for any S A:
c c .
i S i
j N S j ( )
Moreover, there exists a positive allocation constituting a solution forc if and only if, in addition to the condition above, the inequality is strict when S N N S( ( )).
Proof. See the appendix.
Lemma 3 implicitly states that the existence of a solution depends on whether the network is sufficiently well‐
connected or not, because a given set of nodes S⊂ A should be connected to sufficiently many number of nodes for the respective inequality to be satisfied. Notice that when c is fixed, as the network becomes more connected, ∑i∈Sci
remains constant, while∑j∈N(S)cjincreases. Accordingly, we can claim that once the existence of a solution is acquired, forming new edges would not violate the inequality; hence, there exist some thresholds on E such that a solution exists once those thresholds are exceeded.
Below, Theorem1 restates the lemma by expressing the equilibrium efforts in terms of r* and then assigning an appropriate value forc. Proposition1already confirms the existence of a unique equilibrium ratio r* under all possible
cases, and now the theorem characterizes for which networks this ratio can be utilized to construct a Nash equilibrium.
We can see that as in Lemma3, the respective inequality persists as the number of edges increases. For instance, we would not have any nonexistence problem on a complete network. However, if the network is not complete, then an equilibrium might not exist depending on the network structure and the model parameters. To be more precise, the theorem implicitly characterizes a minimum collective strength condition for the neighbors of any set of agents S⊂ A, such that a weighted sum of the total equilibrium efforts exerted by the members of N(S) should not be less than the same for the members of S.11An extreme example would be a star network, and even then, a Nash equilibrium would exist provided that the center agent is strong enough to compete with all of the other agents.12
Theorem 1. Given any network G= ( , ), there exists a Nash equilibrium if and only if for the equilibrium ratioA E r* and for any S A:
r
r e r r
r e r
+ 1 ( , ) + 1
( , )
*
* * *
* *
i S i
i
i i
j N S j
j
j j
( )
(7)
such that the inequality is strict when S N N S( ( )).
Proof. Assume that there exists a Nash equilibrium e*= ( , …,e*1 e*n). Note that for any i j E
( , ) : e eij* */ ji=ri*= 1/r*j. Moreover, the total effort exerted by both agents in the respective contest is equal to
e e e e
r r
r e r
r e
+ = + = + 1
= + 1
* * * * .
*
*
* * *
* *
ij ji ij
ij i
i i
ij j
j ji
Consider an allocation a= ( , …,a1 a| |E ) over E such that ai j = r e*
r ij ( , )
*+ 1
*
i
i . Let ci= r e r( , )*
r i i
*+ 1
*
i
i . Then, for
any S A:
c = a = a a = c,
i S i
i S j N i i j
i S j N S i j
j N S i N N S i j
j N S j ( )
( , )
( ) ( , )
( ) ( ( )) ( , )
( )
so that
r
r e r r
r e r
+ 1 ( , ) + 1
( , ).
*
* * *
* *
i S i
i
i i
j N S j
j
j j
( )
Conversely, note thatr e r( , )*
r i i
*+ 1
*
i
i is equal to the total effort exerted by both agents in all of the contests agenti participates in. Set
c r
r e r
= + 1
( , )
*
* *
i i
i
i i
and c= ( , …,c1 cn). Utilizing Lemma 3, there exists a positive allocation a= ( , …,a1 a| |E ) over E constituting a solution forc, so that for every i A:
ci= a .
j N i i j ( )
( , )
Now, let eij*= r a
r + 1 ( , )i j
*
*
i
i . Then,
e r
r a r
r c e r
= + 1 =
+ 1 = ( , ).
* *
*
*
* *
j N i ij
j N i i i
i j i
i
i i i
( ) ( )
( , )
The collection of theseeij*values constitutes a Nash equilibrium, with no player exerting zero effort in some of his/her contests.
It is worth noting that our analysis indicates that for any given network there exists a unique equilibrium ratio that characterizes the equilibrium total effort for every agent. However, by definition, a strategy profile includes information about each agent's contest efforts on all of his/her edges. In that sense, for any given network, there might exist multiple Nash equilibria yielding the same levels of total efforts. More precisely, for example, if agent i∈ A has two edges (i, j), (i, j′) ∈ E and is supposed to exert an equilibrium total effort of e*i in these contests, then there might exist two Nash equilibria with eij*+eij*=ei*and eij +eij = *ei .
In Figure2, we return to the illustrative example provided at the beginning of this section. Noting that the example is a bipartite network for a partition {A1, A2} with A1= {1, 2, 3, 4} and A2= {5, 6, 7, 8, 9}, we calculate the unique equilibrium ratio r* as 0.888. Accordingly, the equilibrium total efforts are uniquely characterized by e = 0.3631* , e = 0.3872* , e = 0.4563* , e = 0.4334* , e = 0.3125* , e = 0.3906* , e = 0.3377* , e = 0.3908* , and e = 0.4179* . These total efforts can be distributed over the respective edges, as illustrated in Figure 2. Since the equilibrium ratio r* is preserved in every two‐player contest on the network, these values of contest efforts constitute a Nash equilibrium. However, it is possible to find other values of contest efforts satisfying the same ratio and adding up to the same equilibrium total efforts. In that sense, there is a multiplicity of equilibria.
3.1 | Further results
Thus far, we have characterized the unique equilibrium ratio r* as well as the equilibrium total effort e *i for every i∈ A and for all types of networks: (a) nonbipartite networks and (b) bipartite networks. For (a), the results appear to be fairly straightforward. Especially for (b), we now extend our equilibrium analysis to have a better understanding of equilibria. We start with Proposition 2 below, which will lead to three additional results.
Proposition 2. Given a bipartite network for a partition A A{ 1, 2}: (i) if i Ae r( , ) =i r j A e(1/ , )r j
1 2 , then r= *;r
(ii) if i Ae r( , ) <i r j Ae(1/ , )r j
1 2 , then r> *; andr
(iii) if i Ae r( , ) >i r j Ae(1/ , )r j
1 2 , then r< *.r
Proof. See the appendix.
Following this proposition, our first observation is that given a bipartite network for a partition {A1, A2}, if the equilibrium ratio r* = 1, we would have
e(1, ) = + 2k
i 8
i
F I G U R E 2 A numerical example of an equilibrium [Color figure can be viewed at wileyonlinelibrary.com]
for every i∈ A.13At this equilibrium, Equation (5) is still satisfied:
A A
| | + 2 = | | + 2 .
i A i
j A
1 2 j
1 2
This observation leads to the following remark.
Remark 3. Given a bipartite network for a partition A A{ 1, 2},
A A
| | | |
2 > j A j i A i
1 2 2 1
if and only if r* > 1. The result is true also with converse inequalities.
Proof. Follows from Proposition2and the observation for r = 1 above.
In Remark 3, we characterize the cases in which the equilibrium ratio is relatively greater for a group of agents. The characterization is provided in terms of the group size and the sum of productivity parameters for the group members. Accordingly, the crowded group is stronger than the other group unless their sum of pro- ductivity parameters is sufficiently low. Furthermore, the result is more refined in a symmetric case. In parti- cular, if the agents are symmetric (i.e., for any i, j∈ A: βi= βj), given a bipartite network for a partition {A1, A2}, we would have∣A1∣ > ∣A2∣ if and only if r* > 1. Considering the converse inequalities, it is also implied that r* = 1 occurs only when∣A1∣ = ∣A2∣; so that if ∣A∣ is odd, then r* ≠ 1.
Returning to the asymmetric case, below we present two additional observations regarding agent transfers. We first consider a single agent transfer. As it turns out, a transfer from a group to the other always makes the latter group stronger.
Remark 4. Given a bipartite network for a partition A A{ 1, 2}, if we transfer an agent k from A1into A2, such that we end up with another bipartite network for the partition A{ 1\ { },k A2 { }}k , then the equilibrium ratio r*
decreases.14On the other hand, if there is such a transfer from A2 into A1, then the equilibrium ratio increases.
Proof. See the appendix.
We then analyze how the equilibrium ratio changes if two agents from different sets are substituted. In such a case, the outcome is unclear, since there are two transfers at once, one from A1into A2and the other from A2into A1. Yet, referring to the agents' productivity parameters, we can identify which group becomes advantaged.
Remark 5. Given a bipartite network for a partition A A{ 1, 2}, if we substitute an agent k A1 with some agent k A2, meaning that we end up with another bipartite network for the partition A{( 1\ { })k { }, (k A2 { }) \ { }}k k , then the equilibrium ratio decreases if and only if e r( ,* k) +r e* (1/ ,r* k) < ( ,e r* k) +r e* (1/ ,r* k).
Proof. See the appendix.
As mentioned earlier, in a bipartite network, the set of agents is partitioned into two groups. As it turns out, each group acts as a single agent at the equilibrium, which indicates a form of collaboration between the group members, perhaps not a strong collaboration as one would observe in a team contest, but still a prominent one with the idea of“an enemy of my enemy is my friend.” The remarks above rely on the fact that the equilibrium ratio is independent of the network's edge structure and deal with alternative partitions in which there are marginal changes in both groups. Remark4 emphasizes that even the least productive agent is important such that adding such an agent into a group motivates the group members, so that they end up with higher prize shares or winning probabilities. Remark 5 further illustrates that if it is a substitution of two agents, then it would be those agents' productivity levels that jointly determine which group is favored. These remarks are especially relevant in examples where the network structure is formed endogenously, either by the network
agents or by the third parties. For instance, they can be useful in answering the question: what cost should one pay to add a particular agent to a certain group or not to lose a particular agent from that group.15
Given any network G = (A, E) and any edge (i, j)∈ E, it is established that the ratio of agent i's equilibrium contest effort to agent j's equilibrium contest effort is some constant r > 0. Clearly, in the respective two‐player contest, agent i has a winning probability of rα/(1 + rα). This is true for each edge including agent i∈ A. In addition, if agent i exerts a contest effort of eijon the edge (i, j)∈ E, then agent j would exert eij/r. Accordingly, the winning prize for agent i∈ A from that contest would be
e e
r .
ij ij 1
i i
⎜ ⎟
⎛⎝ ⎞
⎠
This equation is a linear function of eij, hence the expected utility for agent i∈ A from the overall interaction on this network can be written as
U r
r e e
r ke
r
r r e ke
r
r e ke
= + 1
= + 1
= + 1 ,
i j N i
ij ij
i
j N i
ij i
i i
( )
1 2
( )
1 2
+ 1
2
i i
i
i
⎜ ⎟
⎛⎝ ⎞
⎠
which turns out to be a function of the equilibrium ratio, the total contest effort exerted by agent i, and some exogenously given model parameters. Notice that the first term of this expression,
r
r e
+ 1 i,
+i 1
can be labeled as the expected production by agent i, since the second term describes the agent's cost of effort.
Utilizing these arguments, we now present a new proposition stating the aggregate of all efforts, the expected total production, and the sum of all expected utilities.
Proposition 3. Consider a bipartite network for a partition A A{ 1, 2}; and for notational simplicity, let r* denote the corresponding equilibrium ratio. Then the aggregate of all efforts can be written as
e r r
k r
r r r
= ( + ( + 1)) k r
2 ( + 1) + ( + ( + 1))
2 ( + 1) .
* * *
*
* * *
*
i A
i
j A + 1 j
2
1
2
i j
1 2
Furthermore, the expected total production is
r r
k r
r r r
k r ( + ( + 1))
2 ( + 1) + ( + ( + 1))
2 ( + 1)
* *
*
* * *
*
i A
i
j A
j 2( + 1)
3
2(1 )
3
i j
1 2
and the sum of all expected utilities is
U*= U r( , ) +* U(1/ , ),r*
i A
i j A
j
1 2
where
( )( )
( )( )
( ) ( )
( )
( ) ( )
( )
U r
r r r
k r
U r
r r r r r
k r ( , ) =
+ + 1 (2 ) + 1
4 + 1
and
(1 , ) =
+ + 1 (2 ) + 1
4 + 1
.
*
* * *
*
*
* * * * *
*
i A
i i A
i i
j A
j j A
j j
2( + 1)
4
2(1 )
4
i
j
1 1
2 2
On the other hand, if the network is nonbipartite, then the aggregate of all efforts can be written as
e e n
= (1, ) = + 2k
8 .
*
i A i
i A i
Furthermore, the expected total production is n
k + 2
16
i A i
and the sum of all expected utilities is
U U n
k k
= (1, ) = + 2
16
( + 2 )
64 .
*
i A i
i A i i A i 2
Proof. See the appendix.
The calculation of the sum of all expected utilities sets the stage for the analysis of the optimal network structure, presented in the following section.
4 | OPTIMAL NETWORK ANALYSIS
In this section, we aim to understand which types of networks are optimal. The optimality condition is represented by the maximization of social welfare defined as the sum of all expected utilities (as calculated in Proposition3).16
Throughout this section, we conduct our analysis under the assumption that an equilibrium exists for the network under consideration. We start with the case in which agents are symmetric; that is for any i∈ A: βi= β. Proposition4 below analytically characterizes the set of optimal networks.
Proposition 4. Assume that the agents are symmetric. Then any nonbipartite network is optimal. Moreover, a bipartite network for the partition A A{ 1, 2}is optimal if and only if A| 1| = |A2|.
Proof. See the appendix.
In contrast, if there is heterogeneity in the agents' productivity parameters, the optimal networks may take different forms. To analyze the case of heterogeneity, we concentrate on the comparison between bipartite networks and nonbipartite networks. More precisely, we investigate the question whether one can find a bipartite network better than any nonbipartite network in a given situation. Even this turns out to be a challenging task. Accordingly, we restrict our analysis into the cases in which there are only two types of productivity parameters: β and β′. As we report below, such an analysis is sufficient to capture important insights out of the current model regarding the optimal network structure.
Furthermore, as it will be argued later in more detail, our results appear to be robust in the number of the types of productivity parameters.
(a)
(b)
(c)
(d)
F I G U R E 3 Black‐labeled (β, β′) pairs represent the cases in which there exists a bipartite network better than any nonbipartite network.
For all cases on the left, there is only one β′‐type agent; and from left to right, the number of β′‐type agents increases by one. (a) n = 8, α = .05, (b) n = 8, α = .2, (c) n = 9, α = .05, (d) n = 9, α = .2
For the sake of illustration, we consider certain numerical values for the number of agents n and the effectiveness parameter α. Let n ∈ {8, 9} and α ∈ {0.05, 0.2}. Let β and β′ vary in the range of 0.5 and 1 − α. Our findings are reported in Figure3. The parameters β and β′ are represented on the horizontal and vertical axes, respectively. Each graph on the LHS represents the case in which there is only one β′‐type agent; and from left to right, the number of β′‐type agents increases by one. The remaining cases in which there are relatively more β′‐type agents are omitted, as they would be symmetric around the diagonal to the respective cases in the figure. Any given (β, β′) pair is labeled black, if there exists a bipartite network that yields a sum of all expected utilities greater than any nonbipartite network does; if otherwise, the pair is labeled white.
For n = 8, we see that only in symmetric cases, that is only when β = β′, one cannot find a bipartite network better than any nonbipartite network, implying that the optimal network is nonbipartite. In all of the remaining cases, it appears that there always exists some bipartite network dominating all nonbipartite networks, so that the optimal network is bipartite. For n = 9, the result turns out to be the same in symmetric cases. Furthermore, in some asym- metric cases, there exists a region of β and β′ values for which the optimal network is still nonbipartite. Such a pair of productivity parameters appears to be close to the diagonal, that is the difference between β and β′ is sufficiently small.
In the remaining cases, the optimal network turns out to be bipartite.
Since all of our observations above are independent of the network's edge structure, we can further state that any nonbipartite network is optimal in white regions as long as the necessary and sufficient condition presented in Theorem1is satisfied. In contrast, being a bipartite network does not suffice to be optimal in black regions, since how agents are partitioned influences the sum of all expected utilities. Figure 3 does not distinguish between bipartite networks; but for each case, it simply considers the best bipartite network, whichever that is. Therefore, an optimal network analysis calls for a more detailed investigation in black regions.
For those black regions, we extend our optimality analysis referring to some numerical examples. We find that the type of optimal network may vary quite interestingly. For example, fixing n = 8, α = 0.2, β = 0.5, and β = 0.75, we make the following observations (Table 1). When there are four agents of β′‐type, the optimal network is for a partition {A1, A2}, with A1 including four β‐type nodes and A2 including four β′‐type nodes. Considering the cases in which there are five or six β′‐type nodes, the optimal network continues to separate the nodes into a group of β‐types against a group of β′‐types. These groups correspond to five against three nodes for the former and six against two nodes for the latter. Finally, if we assume that there are three or seven nodes of β′‐type, the optimal network occurs when
∣A1∣ = ∣A2∣ = 4, with one of these sets being mixed, including one β‐type node and three β′‐type nodes. Carrying out a similar analysis when n = 9, we obtain similar results (Table2).
As seen above, when there are four agents of β′‐type, the optimal network is for a partition {A1, A2}, with A1
including four β‐type nodes and A2including four β′‐type nodes. Considering the cases in which there are five or six
T A B L E 1 The partition {A1, A2} for the optimal network for n = 8 and α = .2
The types of agents A1 A2
5 β nodes, 3 β′ nodes {0.5, 0.75, 0.75, 0.75} {0.5, 0.5, 0.5, 0.5}
4 β nodes, 4 β′ nodes {0.75, 0.75, 0.75, 0.75} {0.5, 0.5, 0.5, 0.5}
3 β nodes, 5 β′ nodes {0.75, 0.75, 0.75, 0.75, 0.75} {0.5, 0.5, 0.5}
2 β nodes, 6 β′ nodes {0.75, 0.75, 0.75, 0.75, 0.75, 0.75} {0.5, 0.5}
1 β nodes, 7 β′ nodes {0.75, 0.75, 0.75, 0.75} {0.5, 0.75, 0.75, 0.75}
T A B L E 2 The partition {A1, A2} for the optimal network for n = 9 and α = .2
The types of agents A1 A2
5 β nodes, 4 β′ nodes {0.5, 0.75, 0.75, 0.75, 0.75} {0.5, 0.5, 0.5, 0.5}
4 β nodes, 5 β′ nodes {0.75, 0.75, 0.75, 0.75, 0.75} {0.5, 0.5, 0.5, 0.5}
3 β nodes, 6 β′ nodes {0.75, 0.75, 0.75, 0.75, 0.75, 0.75} {0.5, 0.5, 0.5}
2 β nodes, 7 β′ nodes {0.75, 0.75, 0.75, 0.75, 0.75, 0.75} {0.5, 0.5, 0.75}
1 β nodes, 8 β′ nodes {0.75, 0.75, 0.75, 0.75, 0.75} {0.5, 0.75, 0.75, 0.75}
β′‐type nodes, the optimal network continues to separate the nodes into a group of β‐types against a group of β′‐types.
These groups correspond to five against three nodes for the former and six against two nodes for the latter. Finally, if we assume that there are three or seven nodes of β′‐type, the optimal network occurs when ∣A1∣ = ∣A2∣ = 4, with one of these sets being mixed, including one β‐type node and three β′‐type nodes. Carrying out a similar analysis when n = 9, we obtain similar results (Table2).
These numerical observations also indicate why finding an analytical way of identifying the optimal network might be a challenging task. We must emphasize, however, that given any numerical values for n, α, and all βi's, the respective sum of all expected utilities can directly be calculated under each possible network structure by referring to Proposition3. And these calculations would reveal the optimal network.
Interestingly, among these optimal networks, the most efficient network is not necessarily the one for which the number of β′‐type nodes is the highest. This implies that a ceteris paribus increase in one agent's productivity parameter does not guarantee an increase in efficiency. For instance, when n = 8, the sums of all expected utilities for the respective optimal networks are 0.9208, 0.9588, 1.0044, 0.9879, and 0.9632, respectively; whereas when n = 9, those sums are 1.0668, 1.1090, 1.1326, 1.0972, and 1.0882, respectively. Accordingly, having five β′‐type nodes when n = 8 and having six β′‐type nodes when n = 9 turn out to be the most efficient. In such cases, if the productivity parameter for one of the β‐type nodes increases to β = 0.75, we would observe a decrease in the sum of all expected utilities.
In the existing literature on network games, special types of networks are commonly considered. Here we con- centrate on such network structures, depending on whether they are bipartite or not:
(i) Complete networks are nonbipartite.17
(ii) Star networks are bipartite for a partition {A1, A2}, with either∣A1∣ = 1 or ∣A1∣ = n − 1.
(iii) Ring networks are
(a) bipartite for a partition {A1, A2}, with∣A1∣ = ∣A2∣, if n is even;
(b) nonbipartite if n is odd.
(iv) Line networks are bipartite for a partition {A1, A2}, (a) with∣A1∣ − ∣A2∣ = 0, if n is even;
(b) with∣A1∣ − ∣A2∣ = 1, if n is odd.
Given that the network's edge structure has no influence on the results reported in Proposition3: (a) for odd values of n, any ring network and the complete network yield the same outcomes; and (b) for even values of n, any ring network for a partition {A1, A2} and any line network for the same partition yield the same outcomes.
Following this categorization, we refer back to the optimality analysis. For each white‐labeled pair of productivity parameters in Figure3, the complete network is optimal as well as any other nonbipartite network. This means that when n is odd, any ring network would also be optimal in white regions. On the other hand, given a black‐labeled pair, optimality is related to the network's partitional structure. In particular, we see that a star network in which one of the partitioned sets has a cardinality of one is rarely optimal. An exception might be when n = 3. For larger values of n, optimal bipartite networks are usually such that the cardinalities of the partitioned sets are close to each other. In that regard, there appears to be a large set of parameter values for which the optimal network is a line network. Notice also that if n is even, then whenever a line network is optimal, the respective ring network would be optimal as well.
Finally, if we consider three or more types of productivity parameters, then performing a similar analysis would be much less straightforward. Nevertheless, trying some other values for the number of types, we can safely report that our observations provided under two types of productivity parameters are mostly robust. For instance, if those different productivity parameters are sufficiently close to each other, meaning that (β1,…, βn) is close enough to the diagonal of the respective polytope, then it would not be possible to find a bipartite network that yields a higher sum of all expected utilities than any nonbipartite network does. By contrast, if those productivity parameters are sufficiently far from each other, then one can find at least one bipartite network dominating all nonbipartite networks.
5 | CONCLUDING REMARKS
In this paper, we have analyzed multiple simultaneous conflicts between numerous heterogeneous agents located on a network. Studying bilateral contests over joint production increasing in contest efforts, we have provided a necessary and sufficient condition for the existence of Nash equilibrium and characterized the equilibrium total effort for every
