公共財的穩定網路結構

行政院國家科學委員會專題研究計畫 成果報告

公共財的穩定網路結構

研究成果報告(精簡版)

計 畫 類 別 ： 個別型

計 畫 編 號 ： NSC 95-2415-H-002-022-

執 行 期 間 ： 95 年 08 月 01 日至 96 年 07 月 31 日

執 行 單 位 ： 國立臺灣大學經濟學系暨研究所

計 畫 主 持 人 ： 袁國芝

計畫參與人員： 博士班研究生-兼任助理：劉于鵬

碩士班研究生-兼任助理：陳俊廷

大學生-兼任助理：賴奕成、許喬茵

處 理 方 式 ： 本計畫可公開查詢

中 華 民 國 96 年 10 月 29 日

行政院國家科學委員會專題研究計畫 成果報告

公共財的穩定網路結構

研究成果報告(精簡版)

計畫類別： 個別型

計畫編號：NSC 95－2415－H－002－022

執行期間： 2006 年 08 月 01 日至 2007 年 07 月 31 日

執行單位： 國立臺灣大學經濟學系暨研究所

計畫主持人：

袁國芝

計畫參與人員： 兼任助理：碩士班研究生及大學部學生陳俊廷等四人

處理方式： 本計畫可公開查詢

中 華 民 國 96 年 10 月 29 日

行政院國家科學委員會補助專題研究計畫

□ 成 果 報 告

□期中進度報告

公共財的穩定網路結構

計畫類別：■ 個別型計畫 □ 整合型計畫

計畫編號：NSC 95－2415－H－002－022

期間： 2006 年 08 月 01 日至 2007 年 07 月 31 日

計畫主持人：袁國芝

共同主持人：

計畫參與人員：

兼任助理：碩士班研究生及大學部學生陳俊廷等四人

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本成果報告包括以下應繳交之附件：

□赴國外出差或研習心得報告一份

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處理方式：除產學合作研究計畫、提升產業技術及人才培育研究計畫、

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執行單位：國立臺灣大學經濟學系暨研究所

中 華 民 國 96 年 10 月 29 日

A Stable Network Structure of Public Goods

Kuo-chih Yuan

Department of Economics,

National Taiwan University

October 29, 2007

Abstract

We aim to analyze the relationship between public goods provision and network structures, and the stability of the networks. We determine the stable networks by examining the properties of pairwise stability. A proper form of utility function also is introduced in our model to replace the network value function, which often makes the equilibrium outcomes depended on the network structures. We claim that a weakly stable net-work always exists but some architectures of netnet-works may not be sup-ported by strong stability.

Keywords: Stable networks, public goods. JEL Classi…cation: C72, D62, H41.

1

Introduction

Network structures play a very important role in determining the outcome of economic situations. Also the science of social networks is a signi…cant …eld of sociological study and has been extensively documented. In recent years, theoretical models of network can be found in a vast literature and widely ex-amined in various applications, such like the relationship between social network structures and labour market outcomes, the trade and exchange of goods in non-centralized markets, research and collusive alliances among companies, election results in political party networks.

However, most of those studies emphasis that the network structure is the only key determinant of individuals’utility in the society. An individual’s value function only depends on a full or a reduced structure of the network. External-ities are generated by particular linkages and special structures, rather than the provision of goods o¤ered by connected components. I would like to argue that a piece of puzzle is missing under such assumptions. For instance, a music-loving student may connect her laptop to another user’s computer through a campus

I am grateful for …nancial support from the National Science Council, Taiwan (NSC95-2415-H002-022).

network, and gets permit to share music …les in other persons’computers. Thus we can say that her utility is not only depended on the number of computers connected to her laptop, but also the storage size of music …les in those comput-ers. No one will get bene…t by connecting to an empty storage computer. Once a certain network is formed, those students may start to consider whether they should go to buy a music CD or wait until other students buy and store their music CDs in their computers. This example shows us that a variant amount of public goods provision will a¤ect an individual’s value even when the network structure is …xed. I would like to consider a network formation model which can …t in the above situation and …nd what the equilibrium network structure is.

There is a well established and vast literature in network formation. One question often asked in literature is how we predict a network is likely to form when individuals have the discretion to decide their connections. One also would like to check how e¢ cient is a network and how does an equilibrium formation of network depend on the way that the value of a network is allocated among the individuals. Jackson (2003) surveys the literature on the formation of net-works in recent years and shows us a clear picture of the relationships between di¤erent formation models. Bala and Goyal (2000) present a model to explain how a network is formed under a non-cooperative game. Their basic notion of network formation is based on the situation that an individual trades o¤ the costs of forming and maintaining connections against the potential rewards from making so. They provide a characterization of the architecture of equilibrium networks, and study the dynamics of network formation. They …nd that indi-vidual e¤orts to access bene…ts o¤ered by others lead to the emergence of an equilibrium network. And the possible limiting networks are few simple archi-tectures, or generalizations of these simple architectures. Watts (2001) studies the process of network formation in a dynamic framework and determines which network structures the process converges to. She shows that if individuals are myopic and the bene…t from maintaining an indirect link of length two is greater than the net bene…t from maintaining a direct link, then the unique e¢ cient network is di¢ cult to form. By assuming that individuals’payo¤s are depended on the network of connections among them, Jackson and Watts (2002) analyze a dynamic formation and stochastic evolution of networks. In their model, indi-viduals can from and serve links over time based on the improvement between the current and new networks. They claim that predictions still can be made in spite of the likelihood that the stochastic process will lead to any given networks. Although most of studies are focusing on the process of network formation, there are still few researches to analyze how the payo¤ to one individual is a¤ected by another connected individual’s action. Sundararajan (2005) presents a model of local network e¤ects in which the payo¤ to a individual connected in a net-work is in‡uenced by the actions of her neighbors. He studies the adoption of a network good that displays local network e¤ects, and shows that the symmetric Bayes-Nash equilibria of an adoption game can be strictly Pareto-ranked based on a scalar neighbor-adoption probability value. Bramoulle and Kranton (2007) introduce a social learning model under a network. They show their structure

can foster specialization and specialization can have welfare bene…ts. They also claim that the extreme specialized equilibrium is the only outcome. Yuan (2005) demonstrates the possible multiplicity of Nash equilibria in a local public goods network. Unlike the assumption of homogenous players in BK’s model, I o¤er a heterogeneous setting of players’preferences. Thus my model will have a more general result in public goods network study.

In this paper, I will present a network formation model which involves two main issues. First, the social networks are formed by individual decisions. I suppose a link with another individual allows access to the bene…ts available to the latter. However, an indirect link won’t generate externalities. This assump-tion especially …ts many intellectual property cases. It is illegal to download video game software from another individual’s computer, but it is legal to play with the host through network connections. This means an individual only can receive bene…t from another person who she has a direct link with. Secondly, individuals also make decisions to choose the provision of goods under the social network structures.

2

Model

The outline assumption of the game of local public goods provision in networks comes from Yuan (2005). Let us consider a local public goods game with n players. We assume the circumstance in which the local public goods are pro-vided by means of private provision by players. Let I = f1; 2; ; ng, n 2 be the set of players. For any pair of players i; j 2 I, the pair-wise relationship be-tween the two players is represented by a variable rij 2 [0; 1]. rijis a parameter

which represents the degree of externalities of the local public goods provided by player j to player i. It shows how much service player i can receive from one unit of player j’s provision. In other words, one unit of the local public goods provided by player j will o¤er player i the same quantity of service as rij unit

does when the rij unit is provided by player i herself. When rij = 1, we say

that player i can receive a full service of player j’s local pubic good provision. Or we can say player j’s provision is just like a purely public good to player i. A player always receives the whole service of her own provision, so rii = 1.

When rij = 0, it refers to the case that player i receives no service from player

j’s provision, or we can say player j’s provision is like a private good to player i.

In this local public goods model, we assume that there is a boundary to limit the users of a speci…c local public good. If we suppose player j provides a local public good, then rij = 0 when player i is located outside the boundary.

Now suppose that each player only provides one speci…c local public good. Since two di¤erent kinds of local public goods may have two di¤erent e¤ective ranges of externalities, one player could bene…t by another player’s provision but the reverse situation is not true. It means rij= 1 but rji= 0 for some i; j 2 I.

Di¤erent local public goods may also o¤er di¤erent degrees of externalities, so rij 6= rji for some i; j 2 I. We even can assume the territory or the degree of

externalities varies with the level of a provision. The variable rij then should

be replaced by a function rij : [0; xj] ! [0; 1].

To avoid our analysis involving those complex situations of the asymmetric externalities between di¤erent pubic goods or units, we assume that the local public good are perfectly substitute. Although the provisions from di¤erent positions should be viewed as distinct local public goods, we think that the ter-ritory (or the degree) of the externality is exactly same for any unit of provision at any position. This assumption immediately implies rij = rji = 1 for any

i; j 2 I.

Let Libe a set of players and de…ned by Li fj j j 6= i, j 2 I and rij> 0g.

We can say that any player who belongs to the set Li is located in the certain

area where the residents receive the externality of player i’s provision. Since we have assumed that rij = rji for any i; j 2 I, then we can imply that j 2 Li if

and only if i 2 Lj.

Let G(I; r) be a simple graph in which at most one edge (i.e., either one edge or no edges) may connect any two vertices. A network r is the collection of the pair-wise relationships, r = frijgi;j2I. The players, on the integers 1; 2; ; n,

are located on the vertices of a simple graph. When rij = 1, we say that the two

players i, j are linked and one edge will connect the two corresponding vertices. And rij = 0 refers to the case of no link. To avoid confusion, we assume no

vertex is self-connected. Li, the set of players who are linked to player i, is

written by Li = fj j j 6= i, j 2 I and rij = 1g. Notice that we do not consider

player i belonging to the set Li. In other words, a player is not linked to herself.

For any given game G with a set of players I and a network r, there always is a corresponding graph G(I; r), and vice versa.

Let the local public good provided by a player be also indexed by the same set of integers. The player i’s provision of the local public good is denoted by xi, xi 0. The utility function of player i is:

ui = Vi(Xi) ci(xi), (1) where Xi= n X j=1 rijxj= xi+ X j2Li xj. (2)

Xi denotes the total service quantity which player i can receive from all

players. We also de…ne X i = Xi xi. X i is the total service quantity

which player i can receive from the other players’ provisions. Let Vi( ) be an

increasing, di¤erentiable and concave function. ci( ) is the cost function for

player i. Suppose

ci(xi) = cixi. (3)

The marginal cost of providing any unit of the local public good is constant, but each player has her own marginal cost. We also assume that V0

i(0) > cifor

all i 2 I. Let us de…ne xi by assuming that xi solves Vi0(xi ) = ci. Here xi can

be seen as the optional amount of consumption for player i if she pays for her own provision. It is clear that no player i will provide an amount of xi such

that xi> xi. Since Vi is an increasing, di¤erentiable and concave function and

V_i0(0) > ci, we know xi must exist and it is unique and positive.

2.1

Equilibrium Provision

In the local public goods model, each player will choose a provision xi as her

strategy. Any strategy xi > xi is strictly dominated for player i. Hence we

can let the strategy space of xi is Si = [0; xi_]. This …ts the …rst condition in

proposition. We assumed the utility function (or payo¤ function) ui= Vi(Xi)

cixi and Vi( ) : R+[ f0g ! R is an increasing, di¤erentiable and concave

function. cixi is a linear function of xi. So ui is a di¤erentiable and concave

function which …ts the second proposition condition. Therefore we can guarantee that there always exists a pure strategy Nash equilibrium in the local public goods model.

Although the above proposition tells us that a pure strategy Nash equilib-rium does exist in the local public goods game. We still need to …nd a certain method which can allow us to practically examine an equilibrium. In the fol-lowing lemma, we will give the necessary and su¢ cient conditions for a pure strategy Nash equilibrium.

Theorem 1 (x₁; x₂; ; x_n) is a pure strategy Nash equilibrium if and only if: (1) X_i = xi if xi > 0 and (2) Xi xi if xi = 0.

Proof. Necessity. Suppose (x₁; x₂; ; x_n) is a Nash equilibrium, then x_i should maximize player i’s payo¤: ui = Vi(xi+ X i) cixi for any i 2 I.

For any x_i, we know either x_i > 0 or x_i = 0. The …rst order condition V0

i(xi + X i) ci = 0 must hold when player i chooses xi = xi > 0. By

the de…nition of xi, we know xi + X i = Xi = xi. Thus we have proven the

necessary condition (1). If x_i = 0 and we suppose X_i < xi, then Vi0(Xi) =

V0

i(X i) > Vi0(xi) = ci (because Xi = xi + X i = X i and Vi is concave).

This result is in con‡ict with the maximum condition of Nash equilibrium. So we know X_i xi when xi = 0. We have proven the necessary condition (2).

Su¢ ciency. Suppose players have chosen a provision vector (x₁; x₂; ; x_n) which satis…es the conditions (1) and (2). For those x_i > 0, X_i = xi implies

x_i = xi X i. After giving the other players’choices (x1; ; xi 1; xi+1; ; xn), @ui

@xi jxi=xi= V

0

i(Xi) c = Vi0(xi ) c = 0 shows us xi is the best response for

player i. For any other x_i = 0, @ui

@xi jxi=0= V

0

i(Xi) c Vi0(xi) c = 0 (because

X_i xi and Vi is concave) also tells us the choice xi = 0 is the best response

for player i by given (x₁; ; x_{i 1}; x_i+1; ; x_n). Therefore any (x₁; x₂; ; x_n) which satis…es conditions (1) and (2) must be a Nash equilibrium.

De…nition 2 An equilibrium (x₁; x₂; ; x_n) is specialized if x_i > 0 then x_j = 0 for all j 2 Li; An equilibrium (x1; x2; ; xn) is distributed if 9i 2 I st. xi > 0

2.2

Stable Equilibrium

By examining the possible provision pro…les, we can easily …nd the multiplicity of equilibria in certain networks by using the theorem 1. For example, there three possible equilibria (2,0,2), (0,3,2) and (1,1,1) in a three-player star network case, when the network g = fr12; r23g and (x1; x2; x3) = (2; 3; 2). This makes

the task to judge the pairwise stability extremely di¢ cult because players may be uncertain about their payo¤s once a new link is formed or an existed link has been vanished.

To solve the uncertain problem, we wish to reduce the number of equilibria in our setting. We consider stable equilibria by using the notion of Nash taton-nement. De…ne bi(x i) as player i’s best response function to a pro…le x iand

de…ne b as the collection of these players’ best responses b = (b1; ; bn). An

equilibrium x is stable if and only if a positive number > 0 such that for any vector " satisfying 8 j"ij < and xi+ "i 0 the sequence x(t) de…ned by

x(0)_{= x + " and x}(t+1)_{= b(x}(t)_{) converges to x.}

this standard notion yields a result that all distributed equilibria are unsta-ble. However, multiple stable specialized equilibria may still exist.

Theorem 3 An equilibrium x = (x₁; x₂; ; x_n) is stable if and only if x is specialized and for any x_i = 0, X _i > xi .

Proof. If x is distributed, we know it is not stable by applying Bramoulle and Kranton’s proof. Suppose now x is specialized but 9i, st. xi = 0, X i = xi.

We can prove x is unstable still by applying B&K’s proof. Since an equilibrium must be either distributed or specialized, and a specialized equilibrium is either 9i, st. xi = 0, X i = xi or for any xi = 0, X i > xi. We have proven the

…rst two types of possible equilibria are not stable. If we can prove all equilibria in the third type are stable, then we can get both necessary and su¢ cient conditions. Consider that x is specialized and for any x_i = 0, X _i > xi. Let

= 1

2 mink2fijxi=0gfX k xkg . For 8 j"ij < and xi + "i 0, We know

x (1)_{= x . Thus x is stable.}

Now we know once the network g is given, we can …gure out all possible stable equilibria. However, a stable equilibrium does not exist in some given network.

3

Heterogeneity and Stability

Herterogenrity of players’preferences may cause multiplicity of stable equilibria in public goods provision. We know that the concept of stable networks must carefully be chosen and always connected to the sense of stability of provision. Thus we may need to make more assumptions on players’preferences in order to de…ne stable networks of public goods provision.

De…nition 4 A network of public goods g is strongly pairwise stable with respect to a provision pro…le x if (1) x is a stable equilibrium under g; (2) for any i 2 I

and j 2 Li, ui(x) ui(x0) for any stable equilibrium x0 under the network

g rij; and (3) for any i 2 I and j =2 Li, if ui(x) ui(x0) then uj(x) < ui(x0)

for any stable equilibrium x0 _{under the network g + r} ij.

De…nition 5 A network of public goods g is weakly pairwise stable with respect to a provision pro…le x if (1) x is a stable equilibrium under g; (2) for any i 2 I and j 2 Li, ui(x) ui(x0) for at least one stable equilibrium x0 under

the network g rij; and (3) for any i 2 I and j =2 Li, if ui(x) ui(x0) then

uj(x) < ui(x0) for at least one stable equilibrium x0 under the network g + rij.

We have de…ned two di¤erent concepts of network stability: strongly pair-wise stability and weakly pairpair-wise stability. The strongly pairpair-wise stability guarantees that no player can make herself possibly better o¤ by severing any single link, and any two players never make them both possibly better o¤ by forming a new link. We also can consider the strongly stability as a result of conservatively risk-aversion. Players take no risk when they play a network formation game under the concept of strong stability. On the other hand, the weakly pairwise stability presents players’ risk-loving attitude. Even there is only a small chance that the player possibly makes herself better o¤, she will prefer to form or severe a link to make the chance possible.

It is not too di¢ cult to understand that weakly pairwise stability is a neces-sary condition of strongly pairwise stability. We now need to prove that weakly pairwise stable networks always exist.

Theorem 6 A network of public goods g is always weakly pairwise stable with respect to some provision pro…le x.

Proof. We need to know how to …nd a pro…le x which is able to make g weakly pairwise stable. We develop a procedure to solve this problem. First we assign xi = xi if xi = maxfx1; ; xng and xj = 0 for all j 2 Li. Then consider a

new network g _frijg, which is the remaining network after exclude the player

i and i’s neighbour. Then we repeat the assigning process again until all xi has

been assigned a number. Then we can prove that the network g will be weak pairwise stable with respect to the pro…le x. We will leave this proof to the Appendix.

We can give an example to show how a network g can be weakly pairwise but failed to be strongly pairwise stable. Consider a network g = fr12; r13g

and (x1; x2; x3) = (2; 3; 2). We can understand that network g with respect

to (0; 3; 0) is weakly pairwise stable but not strongly pairwise stable. But the g = fr12; r13; r23g is a strongly pairwise stable network. To …nd the result, we

can check all the possible cases no matter there is newly forming a link or sever an existed link. All possible change have been described by the …gures among Fig. 1 and Fig. 4.

4

Conclusion

This study built a model of public goods provision with endogenous network formation under the concept of pairwise stability. We …rst generalize the mean-ing of specialized and distributed equilibria under the settmean-ing of heterogeneous preferences. In the notion of Nash tatonnement, we …nd that we may still have multiple stable equilibria. We also …nd that the a maximal independent set of order 2 of networks is no longer the necessary and su¢ cient condition for stable equilibria. Our …ndings di¤er from those obtained by Bramoulle and Kranton (2007) for a homogeneous preferences model.

In our analysis of pairwise stability we identi…ed the strong and weak stabil-ity between risk aversion dominance and risk loving dominance. When players’ link decisions become risk averse, an endogenous network is much easier to be form through the formation game, and we proven that a weakly stable network always exist in the public goods provision model. However, if players link deci-sions become risk dominated, the concept of strong stability leads to a result of possibly empty set of stable networks.

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