Behavioral patterns in social networks.

Conte, Anna ; Di Cagno, Daniela T. ; Sciubba, Emanuela 等

I. INTRODUCTION

Individual strategies for network formation can be extremely complex. The main reason for this is that a network differs from a series of bilateral relationships because of the value of indirect connections: any two economic agents who have to decide whether to establish a social tie take into account not only their own characteristics and the characteristics of the prospective partner but also their (and the prospective partner's) position in the social network.

The theoretical literature on endogenous network formation stems from the seminal contributions by Myerson (1991), Jackson and Wolinsky (1996), and Bala and Goyal (2000). These papers take a game-theoretic approach to the formation of social ties where the main idea is that players earn benefits from being connected both directly and indirectly to other players and bear costs for maintaining direct links. Predicted outcomes are typically not unique. Even for those cases where the equilibrium architecture is unique (e.g., the star network in information communication models a la Bala and Goyal or Jackson and Wolinsky), the coordination problem of which agent occupies which position in the network still remains.

In the presence of multiplicity of equilibria and coordination problems, it is hardly surprising that most experimental contributions on this topic have highlighted the difficulty in obtaining convergence to an equilibrium network architecture as predicted by the theory. (1)

Because the observed network structures are ultimately the outcome of individual linking decisions, one possible approach to overcome this difficulty is to investigate the process of network formation in order to identify patterns of individual behavior that can be resumed in prevailing linking strategies.

With this aim, we use data of a computerized experiment of network formation, where all connections are beneficial and only direct links are costly. The network formation protocol that we adopt, unlike the one used by most of the experimental literature that has focused on convergence, requires that links are not unilateral, but have to be mutually agreed in order to form. In particular, players simultaneously submit link proposals, but a connection is made only when both the involved players agree. We collect data from nine groups of six participants in each and a minimum of 15 rounds of network formation.

In this paper, we estimate a system of equations that model each player's decision on the opportunity to propose a link to any of his or her prospective partners in each round of the game. This approach allows us to take into account the fact that from a player's perspective, the decision to propose a link to one of his or her opponents is not separate from the decision to propose or not a link to another opponent; therefore, decisions made by the player in each round of the game are the result of a joint valuation.

Relying on the results of such an analysis, we attempt to categorize players' systematic behavior into a set of possible strategies adopted by the experimental subjects in our network formation game.

An obvious departing point to interpret these results is to consider myopic best response behavior, where players propose links so to maximize current profits while taking other players' link proposals as given. This is a useful benchmark because most of the theoretical literature on strategic network formation makes the assumption that agents behave according to myopic best response and focuses on Nash Equilibrium (and refinements of Nash Equilibrium) as the appropriate solution concept for the network formation game.

Given the payoff structure of the network formation game, which we consider here, myopic best response requires subjects to propose (direct) links to all those that in the current network are not already reached through indirect connections. It is always advantageous, for example, to propose a link to an isolated node. On the other hand, redundant links (i.e., links to those nodes that are reached also through indirect connections) should be deleted. Also, given that links are only established if they are requested by both the involved players, proposing a link to a player from which a link proposal has not been received is always a matter of indifference.

When all agents play according to myopic best response, the emerging network architecture is a minimal network, where no redundant links are active. The set of minimal networks also contains trivial network structures such as the empty network. The empty network is a Nash network (i.e., a network structure induced by agents who play myopic best response) because links have to be mutually agreed in order to form, hence not proposing while not being proposed is always a best response. To obtain more interesting and more focused predictions, the theoretical literature on network formation has proposed pairwise equilibrium networks as a suitable refinement of Nash networks. A network is a pairwise equilibrium network when it is: (1) Nash and (2) such that all mutually profitable links have been activated. Pairwise equilibrium networks are both minimal and connected, in that all nodes are connected and through the smallest number of links.

By accepting pairwise equilibrium as the appropriate solution concept for the network formation game, most of the experimental literature has focused on whether convergence to a minimally connected network structure, such as the star, or the chain, can be obtained in the laboratory.

Minimally connected graphs are often reached in our experimental groups (21 out of 157, which correspond to a 13% of the total network configurations) but are typically unstable. Convergence to a minimally connected network is only observed in one out of the nine experimental groups (Group 7), where the same minimally connected graph is reached and then kept for four rounds until the end of the session. Mostly, we observed connected graphs that are not minimally connected (68 out of 157, which correspond to 43% of the total network configurations).

We went on to speculate which behavior, other than myopic best response, may concur in explaining the observed network architectures.

We start our analysis with a preliminary investigation of the determinants of individual linking decisions (see Section III). In accordance with myopic best response behavior, we find that subjects are less likely to propose links to those that can already be reached through indirect connections. Moreover, we find a tendency to reciprocate link proposals and a tendency to propose links to those who have the largest number of connections, which is not necessarily in accordance with best response behavior.

This observation motivates the two residual patterns of behavior we consider in this study. Along with the myopic best response strategy, we consider the "reciprocator" and the "opportunistic" strategies.

A subject who follows the "reciprocator" strategy makes link proposals to all those from whom link proposals have been received. Other than maximizing expected profits, a reciprocator aims to establish the largest number of direct links. As for best response behavior, reciprocators will not leave profitable linking opportunities unexploited; however, unlike best response behavior, they may keep redundant links. Reciprocators maximize revenues, rather than profits, and do not care about minimizing the cost through which a high connectivity is obtained.

A subject who follows the "opportunistic" strategy activates those links (among the ones which are feasible, in that proposals have been received by the other party involved) that are most profitable because by linking to them, the opportunist obtains the largest number of indirect connections. While this behavior may seem closer to profit maximization because both costs and revenues of link formation are taken into account, it differs from myopic best response in that profitable link opportunities may be neglected (and at the same time, there is no guarantee that redundant links will be avoided).

We then go on to verify from our data whether the identified strategies are well represented. Finally, in order to discriminate among these three types of systematic behavior, we estimate a mixture model to establish if these strategies are well identified and separated in our sample.

We find that it is safe to assume that each subject in our sample belongs to one type, with mixing proportions approximately equal to 45%, 30%, and 25% for best response, reciprocator, and opportunistic types, respectively.

We notice that the payoffs achieved by the three types are not too dissimilar, with opportunists earning marginally less than myopic best responders and reciprocators.

Finally, we note that the propensity to adopt a certain strategy is group-driven, with subjects being more likely to best respond, to reciprocate, and to behave opportunistically when others in the same group also do.

A large body of the experimental literature has focused on the presence of behavioral types. For example, reciprocators have been studied along with expected utility maximizers in the context of trust games (see Fehr, Kirchsteiger, and Riedl 1993; Berg, Dickhaut, and McCabe 1995; McCabe, Rassenti, and Smith 1996, among others), ultimatum games (see Camerer and Thaler 1995; Giith, Schmittberger, and Swarze 1982; Roth 1995, among others), and prisoners' dilemma games and public goods games (see Clark and Sefton 2001; Fehr and Gachter 2000; Fischbacher, Gachter, and Fehr 2001, among others). The opportunistic behavior, with a similar acceptation than the one used here, has been studied in the context of prisoners' dilemma games (see Boone and van Witteloostuijn 1999, among others), and public goods games (see Dugar 2013, among others). A vast strand of literature that partially overlaps with the one just cited aims to characterize aggregate outcomes on the basis of the distribution of types in the experimental sample (see Andreoni and Miller 2002; Charness and Rabin 2002; Costa-Gomes and Weizsacker 2008; Fischbacher and Gachter 2010, among others).

We apply a similar analysis to a network formation game. The fact that behavioral types (similar to, but) other than myopic best response behavior are significantly represented in the population may explain why the observed network architecture fails to converge to a minimally connected graph as predicted by the theory--the latter being based on expected utility maximization, hence on a single behavioral type.

In the context of the network formation game, a reciprocator is an agent who always proposes a link to those from whom a link proposal has been received in the previous round; an opportunist is someone who attempts to link to those with a large number of connections, thereby aiming to obtain the largest indirect benefit by free-riding on the costly direct links established by others (see Seabright 2004, 5). Both types can either be seen as stemming from alternative preferences, as it is often assumed in the literature reviewed above, or may be the result of bounded rationality (see Gale, Binmore, and Samuelson 1995; Roth and Erev 1995, among others). In fact, the complexity of the network formation game implies that subjects may devise rules of thumb that are close, but not identical, to expected profit maximization, and easier to implement. Irrespective of its origin, the observed distribution across types matters for network formation in that it determines the network architecture which one will observe.

Myopic best response agents always tend to include isolated nodes and delete redundant links, whereby pushing the network architecture to a minimally connected graph. If at any stage, two reciprocators link up, then that link will not be deleted even when it is redundant, which may result in stable network configurations that are not minimal. Finally, the presence of opportunists along with myopic best response agents and reciprocators may favor, when prevalent, the emergence of asymmetric network configuration such as the star, over alternative architectures. For example, when there is a single myopic best response agent and everyone else is an opportunist, the network converges to a star (where the myopic best response agent is the hub). Hence, the exact mix of strategies represented in the population can help us predict which network architecture will emerge in equilibrium.

The paper proceeds as follows. Section II describes the experimental design: the model and the experimental procedure. Section III presents and discusses the results of the model of link proposals described in Appendix A. Section IV shows the characteristics of the three behavioral types that emerged from the analysis in Section III. Section V analyses the data in the light of these three behavioral types. Section VI develops the mixture model, and Section VII concludes. The econometric model of link proposals is explained in Appendix A. The instructions (in their English translation) can be found in Appendix B. (2)

II. THE EXPERIMENTAL DESIGN

A. The Model

We model network formation as a noncooperative simultaneous move game. As in Myerson (1991, 448), we assume that players' strategies are vectors of intended links and that links are only formed when they are mutually agreed, that is, desired by both parties involved. There are positive network externalities in that both direct and indirect connections are beneficial; however, direct links are costly.

Consider a set N of n [greater than or equal to] 3 players, indexed by i = 1,2, ..., n. Each player i submits a vector of intended links:

(1) [a.sub.i] = ([a.sub.i1], [a.sub.i2,] ..., [a.sub.in]).

An intended link is [a.sub.ij] = {0, 1} where [a.sub.ij] = 1 means that player i intends to link to player j while [a.sub.ij] = 0 means that player i does not intend to link to player j. A link between i and j is formed if and only if [a.sub.ij] - [a.sub.ji] = 1. We denote the formed link by [h.sub.ij] - [h.sub.ji] = 1, while we represent the fact that there is no mutually agreed link between i and j by setting [a.sub.ij] - [a.sub.ji] = 0. By convention, [a.sub.ii] - [h.sub.ii] = 0. A strategy profile for all players

a = ([a.sub.1], [a.sub.2], ..., [a.sub.n]),

induces an (undirected) network of links h = [{[h.sub.ij]}.sub.i,j[member of]N], where players are nodes and links are the edges between them. We say that i and j are connected in the graph h if there exists a path of adjoining nodes [k.sub.1], [k.sub.2], ..., [k.sub.m] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Denote by [n.sup.d.sub.i] the number of direct neighbors of player i, and by [n.sub.i] the number of his or her direct and indirect connections. More in detail, denote by [n.sup.d.sub.i] the number of elements of the set [N.sup.d.sub.i] = {j | [h.sub.ij] = l} and by [n.sub.i] the number of elements of the set [N.sub.i], = {j| there is a path in h from i to j}. Notice that if i and j are directly linked, then there is a path between them (of length 1): hence necessarily [n.sub.i] [greater than or equal to] [n.sup.d.sub.i]. Player i's payoff, given his or her position in the network h, is assumed to be equal to:

[[pi].sub.i] (h) = b x [n.sub.i] - c x [n.sup.d.sub.i],

where b and c represent, respectively, the unitary benefit from (direct and indirect) connections and the unitary cost of direct links and are such that b > c > 0.

In this game, players simultaneously announce all the links they wish to form and the resulting network is formed by the mutually announced links. "This game is simple and intuitive. However, given that link creation requires the mutual consent of the two involved parties, a coordination problem arises. As such, the game displays a multiplicity of Nash equilibria, and very different network geometries can arise endogenously" (Calvo-Armengol and Ilkilic 2009, 2).

Examples of network architectures are shown in Figure 1. The complete network, where every node is directly connected to every other, is an example of a connected graph. The complete network is clearly not minimal as there are many redundant links. Examples of minimally connected graphs are the star and the chain.

B. The Experimental Procedure

The experimental sessions were conducted in spring 2006 and 2008 at CESARE, LUISS University in Rome with a total of 54 participants. (3) Subjects were first-year Economics students. Each subject participated in only one session and none had previously taken part in a similar experiment. Each experimental session was made of two or three groups of six participants in each, playing together in a network formation game, and lasted between 30 and 45 minutes. Subjects' total earnings were determined by the sum of the profits in each round and were paid using a conversion rate of 100 points per euro. They earned approximately 32 [euro] on average, on top of a 5 [euro] participation fee.

[FIGURE 1 OMITTED]

While in the sessions that were conducted in spring 2006, we implemented two alternative treatments, with different cost parameters, in the present paper, we only analyze data from one of the two treatments, for which detailed parameters are given in the table below: (4)

                             Initial
              Participants   Endowment    Cost    Benefit

Groups 1-9        6            500         90       100

[FIGURE 2 OMITTED]

All relevant parameters were equal across participants and displayed on the screen at all time throughout the experiment.

At the beginning of each session, subjects were told the rules of conduct and provided with detailed written instructions, which were read aloud by the experimenters.

Sessions consisted of a minimum of 15 rounds, with a random stopping rule determining the end of the experiment. (5) In each round, subjects were asked to submit (anonymously and independently) a vector of intended links. The initial screen for each participant is shown in Figure 2.

Participants were represented on the screen by different symbols, which we considered neutral in that they did not provide subjects with any particular clue when deciding to establish a link with another player in the group. (6) Subjects did not know their symbol (or the other participants' symbols) in advance and could identify themselves on the screen because their symbol was circled in red. In order to guarantee not only individual but also group anonymity, participants were invited to the laboratory in groups of 18, with three groups being matched at the same time. Participants were not told in which of the three groups of six they were playing, nor could they identify the group from their seating. (7)

[FIGURE 3 OMITTED]

The screen also displayed the relevant parameters for the session at play. After all subjects had confirmed their choice of network partners, the computer checked which links were mutually desired and activated them. At the end of each round, profits were computed and displayed on the screen. Great care was taken in making sure that all available information was provided to the experimental subjects in a user-friendly way. For this reason, the graphical interface was designed such that actual links were visualized on the screen as a graph rather than as a list of activated ties or as a matrix of 0/1 connections.

As an example. Figure 3 shows the players' screen at the end of round number 4. It displays the graph of all active links, total revenues, costs, and profits in the round. It also provides information on past unmatched proposals: at the end of a round, each subject was informed of those players who proposed a link to them but to whom they did not reciprocate. At any time during the experiment, subjects had access to a great deal of information on past history: by clicking on the bar corresponding to each round, they were able to visualize the graph of active links and the profits obtained in that round.

III. THE MODEL OF LINK PROPOSALS

In this section, we analyze and discuss the determinants of link proposals. In doing so, we make the assumption of players having static expectations; that is, we assume that each player expects his or her opponents to make exactly the same choices in round t as in round t - 1. This is in line with most of the theoretical literature on network formation; also, to a certain extent, such expectations are induced by the design of the experiment itself: the networks that result from choices in previous rounds are portrayed on the computer screen together with all the relevant information and made accessible throughout the game. (8)

Using the system of equations described in Appendix A, we estimated the probability of each subject i proposing a link to any of his or her prospective partners j, with j = 1, ..., 5, in group g, with g = 1, ..., 9, as a function of the position of i and j in the network reached in the previous round, which is represented graphically on the subject's screen. More in detail, we estimated the probability of subjects proposing a link as a function of a number of variables that can be classified into four categories: the characteristics of the proposer-recipient relationship, which, in particular, include the lagged dependent variable; the characteristics of the prospective partner; the characteristics of the proposer herself; and the characteristics of the network of links observed in the previous round. We also control for experience.

This exercise is meant as a preliminary analysis aiming to verify whether there is a systematic behavior in players' link proposals that might be ascribed to the application of certain strategies and, consequently, to identify and to study such strategies.

A. Estimation Results

The estimation results of the five-equation multivariate dynamic probit model derived in Appendix A are reported in Table 1. (9)

The relationship between i and his or her opponent, j, in t - 1 is described by the first five regressors. (10) Let us start by observing that the coefficient on the variable "i proposes a link to j" (that is, the lagged dependent variable) is positive and strongly significant, which conveys the idea that subjects tend to build on what they did in the previous round. Given the strong statistical significance of the coefficient on "j proposes a link to i," it seems more likely that a link is proposed if the recipient demanded a link to the proposer in the previous round. This could denote both a behavioral tendency to reciprocate and a rational response. In fact, under the assumption of static expectations (i.e., if players expect their opponents to make the same choices in round t as they did in round t = 1), given that links have to be mutually agreed, a link can only be established by proposing a link to proposers in the previous round. The fact of i and j being linked directly plays no role here, in that the variable "i and j directly linked," which is an interaction term between "i proposes a link to j" and "j proposes a link to i," shows not to be statistically significant in any specification. Anyhow, there is some evidence on the tendency to cut redundant links through the negative and statistically significant coefficient on the variable "i and j are linked both directly and indirectly." The attitude not to form redundant links is corroborated by the negativity and statistical significance of the coefficient on the variable "i and j only indirectly linked," even if not in terms of all the specifications of the model. Therefore, the probability of proposing a link seems to diminish if i and j were previously linked both directly and indirectly and if they were already linked but only indirectly.

This first set of findings essentially describes the behavior of a myopic best responder, as delineated in the Introduction, but there is something more. The coefficient on the variable "j proposes a link to i" showed to be positive and strongly significant in any specification. This makes us conclude that, other than a tendency to best respond to the previously formed network, there might be subjects who simply reciprocate demanded links.

In our opinion, another possible motive of link formation can be extrapolated from the results regarding the probability of i proposing a link to j as a function of j's characteristics, which seem to portray the figure of a player acting in a rather opportunistic way. In effect, the estimation results disclose that players tend to propose links to those who have the largest number of connections and that demanding a link is more likely, the larger the number of the opponent's redundant links--an indicator of high connectivity. (11) If a player is instead isolated--that is, he or she has no connection of any sort--the other players do not seem to be willing to include him or her.

Among the variables that describe i in the previous round, we found strong evidence of the fact that the propensity to demand a link increases only if the breakdown rate in the previous round--measured as the difference between the number of links proposed and the number of links activated--increases. (12)

We also estimated the propensity to propose a link as a function of the characteristics of the network of links that emerged in the previous round. Despite the large number of variables representing the network structure tested, none of them seem to play a significant role in subjects' decision. An example is reported in the third column of Table 1. It shows that neither the coefficient on the number of redundant links nor that on the number of isolated nodes in the group is statistically significant. We, therefore, conclude that players did not take into account the global structure of the network established in the previous round when expressing their willingness to demand a link and choosing the receiver of that proposal.

Table 1 also shows that the correlation coefficient p is precisely estimated to be about - 0.20. It is also significantly different from zero and negative, as expected. This indirectly supports our reasons for dealing with individual link proposals as being jointly determined. The considerable magnitude of the standard deviation of the individual-specific propensity to demand links, [[sigma].sub.[alpha]], puts into evidence the heterogeneity of the population. Finally, the coefficient 5 is estimated to be positive and significantly different from zero, so indicating that the noise diminishes, the higher the level of experience that players accumulate by playing the network game for several rounds.

Given these results, in what follows, we study the distribution of three basic patterns of behavior adopted by the experimental subjects in our sample:

* players who reciprocate to those who demanded a link in the previous round unless they can be reached otherwise through indirect connections (under the assumption of static expectations this behavior corresponds, in fact, to profit maximization);

* players who act by simply reciprocating link proposals from the previous round;

* players who try to reach the largest number of nodes by reciprocating to those who exhibit a high connectivity.

As stated earlier, this exercise was meant to search for the leading motives of individual linking decisions, which essentially correspond to the maximization of expected profits, direct links, and expected profits per link. In what follows, we will delineate the behavioral rules that define these types of player, and we will try to establish whether these patterns of behavior are deliberately and systematically adopted by the subjects in our sample and, if so, in which proportion of the observed sample the different types are represented.

IV. STRATEGIES OF LINK FORMATION

In each round of link formation, individuals have 32 available strategies. For each player, excluding the link to oneself, a strategy is given by a 5-dimensional vector of Os and Is. For example, a possible strategy of Player 1 is to propose a link to each of the other five players in the game:

(1, 1, 1, 1, 1)

Strategy (0,0,0,0,0) corresponds to the choice of not proposing a link to any of the other players, while (1,1,0,0,0) corresponds to the choice of proposing to the first two players (other than Player 1) and not the other ones, and so forth.

Under the assumption of static expectations, each player expects the other five players to play the same strategy in round t that they played in round t - 1. Hence, given these expectations on what the others will play, each player responds by selecting one of the strategies in the strategy set. In order to understand whether the behavioral patterns defined in the previous section are in fact represented in our sample, we have to define the specific characteristics required of a strategy such that it pertains to each of the behavioral types. The strategies eligible to be assigned to a type are the following:

* a strategy is of myopic best response type if it maximizes the player's expected profit;

* a strategy is of reciprocator type if it maximizes the player's expected number of direct links;

* a strategy is of opportunistic type if it only activates those links that provide the largest expected profit per link.

Notice that a player who adopts a myopic best response strategy proposes a link to all those that cannot be indirectly reached (and does not reciprocate links to those that can be indirectly reached). Such a strategy maximizes expected profits because, under our parametric assumptions, the benefit obtained by reaching a node is larger than the cost of a link. Hence, unless a node can be reached at zero cost through indirect connections, a proposal to connect directly should always be reciprocated. A myopic best response strategy activates all possible links, except the redundant ones.

A player adopting a reciprocator strategy reciprocates all link proposals that he or she has received in the previous round. Given that only links that are mutually agreed are activated, by reciprocating all link proposals a player can activate the largest possible number of direct links. The main difference between myopic best response and reciprocator strategies is that the latter activate all possible links, including the redundant ones.

A player following an opportunistic strategy does not reciprocate all link proposals, but those which bring the largest profit. An opportunist that receives more than one link proposal always favors the link proposals received by those who have the largest number of connections. Unlike the reciprocator, an opportunist recognizes the value of indirect connections. However, unlike expected profit maximizers, opportunists may miss out on a profit-generating connection when, for example, they do not reciprocate a link proposal from a player that does not have any direct links. On the other hand, the fact that opportunists target highly connected individuals does not prevent them from maintaining redundant links.

While a reciprocator attempts to activate all possible direct links, the opportunist seems to recognize that larger profits can be obtained by restricting the number of direct links and by exploiting indirect connections. However, opportunists target the "wrong" lot of links for deletion: rather than deleting redundant links (as an expected utility maximizer would do), they delete links to those with lower connectivity.

Given any network configuration, the strategies that fit our behavioral types are not unique. To start with, the fact that links have to be mutually agreed in order to be formed introduces some (trivial) multiplicity. For any of the three strategies outlined above, proposing links to any number of players from whom a link proposal has not been received in the previous round brings exactly the same result in terms of network configuration and profits as not proposing to them at all. (13)

Moreover, for myopic best response behavior, there are nontrivial ways in which expected profit-maximizing strategies are not unique. Consider, for example, the case of being linked to two agents who are also linked to one another. One of the two links is redundant; however, a player would be indifferent as to which link to maintain and which link to delete. In this case, the fact that more than one strategy can be identified as a myopic best response is not trivial because such multiple strategies will correspond to the same payoff but to different network configurations.

Finally, there is clearly some overlap among the three strategy types. It may occur that the same strategy, for a given network configuration, can be classified as belonging to more than one type.

Consider, for example, the initial configuration of empty network where nobody is proposing any link. Under static expectations, no link proposals will be expected for the next round as well, hence all types will be indifferent as of making any link proposals or not. Any strategy choice, in this case, can be classified as a myopic best response, or as a reciprocator strategy, or as an opportunistic strategy.

Less trivially, it may, for example, occur that expected profit maximization requires all link proposals to be reciprocated (imagine the initial network configuration is a minimal network), so that myopic best response strategies will coincide with reciprocator strategies. Similarly, it may occur that all agents who propose to a given player have the same number of connections, so that the strategy of reciprocating to only the most connected agents (opportunist) coincides with the strategy of reciprocating to all (reciprocator).

While it is easy to construct examples of overlap across strategies, in general, the three types are distinct. In our experimental sample, 39% of the strategy choices can be assigned to a single strategy type (see Section V for more details).

V. ANALYSIS OF EXPERIMENTAL DATA AND BEHAVIORAL TYPES

In this section, we analyze the experimentally generated data in light of the behavioral types defined in the previous sections in order to verify whether the three strategies are represented in our sample.

In our experimental sample, 360 out of 888 (40.54%) of the individual choices appear as if they were made by best responders. In order to assess whether this is a high percentage of choices or not, we compare it to the proportion of times a player who selects a strategy at random ends up selecting a best response strategy. We did this by determining, for each player in each round, the proportion of strategies that account for best response strategies, given the network of links arising in the previous round. This comparison is particularly useful in our framework where the set of strategies that a best responder may wish to choose contains more than one strategy. Assume, for example, that in a typical round, the experimental network that has been formed is such that for the next round, half of the available strategies are of the best response type. In that case, even someone choosing a strategy at random would have a very good chance of selecting a best response strategy.

The result of this exercise showed that the average proportion of best response strategies in our sample, given the network emerging in the previous round, was 0.3195 with standard error (henceforth s.e.) equal to 0.0067. Consequently, the proportion of best responses effectively played in the sample (0.4054) was significantly larger than the proportion of best responses our players would have selected, had they picked one of the 32 strategies at random in each round, which establishes that a significant share of choices in our experiment correspond to a "deliberate" desire to best respond. (14)

We repeated the exercise with the other two types. Both reciprocator and opportunistic strategies are well represented in our sample: 331 (0.3727) choices can be accounted for as being dictated by the reciprocator strategy; 357 (0.4020) choices can be accounted for as being dictated by the opportunistic strategy. By comparing these proportions with the probabilities players had to select a reciprocator strategy (an opportunistic strategy) by picking a strategy at random in each round, given the network arising in the previous round, we noticed that similar to what was observed in the case of best responders, reciprocators (opportunists) seemed to be selecting their strategies deliberately. More in detail, the average proportion of choices that account for random reciprocator choices is 0.2420 (s.e. 0.0071), compared to 0.3727 in our experimental sample; the average proportion of choices that account for random opportunistic choices is, similarly, 0.2426 (s.e. 0.0071), compared to 0.4020 in our experimental sample.

Many choices can be captured by more than one strategy at a time both in the real and the simulated samples: there are instances when the reciprocator strategy coincides with a best response, an opportunistic strategy or both; there are other instances when reciprocator and opportunistic strategies coincide, or do not coincide, with best response behavior, and so forth. Table 2 shows the overlap between the strategies arising from our experimental sample. (15) The table shows that almost 39% of all choices can be ascribed to only one behavioral type, the remainder being captured by none of the three types, two types at a time or three. It also reveals that 74% of all choices in our experimental sample can be explained in the light of one of our three behavioral types. (16) This is quite a high proportion considering that in many cases, playing a certain strategy in such a game might be rather difficult.

By comparing average profits obtained through each of the three strategies, we found that the average profits obtained by best response choices were not significantly different from those obtained by reciprocators, but both best responders and reciprocators earned, on average, a profit larger than that earned by opportunists: best response choices yielded our experimental subjects an average of 175.056 (s.e. 7.901) experimental units, while reciprocators earned 182.931 (s.e. 7.433), and opportunists 158.655 (s.e. 7.344) experimental units. Figure 4 shows that this pattern holds not just on average, but also for most sessions. The fact that opportunists earned on average less than myopic best responders and reciprocators should not be too surprising. Given our parametric assumptions, connections are always profitable: indirect connections are more profitable than direct links; however both increase profits. The opportunist, by only targeting those connections that provide the highest payoff, may often miss out on linking opportunities by not reciprocating link proposals to those who would bring in a more modest, but still positive, payoff.

VI. THE MIXTURE ASSUMPTION

As seen in the last section, patterns of behavior often overlap so that the choice of a particular strategy is compatible with more than one behavioral rule. For this reason, discriminating between subjects according to their behavioral type is rather difficult if one merely observes the strategies selected by them. In this section, we wanted to verify whether subjects systematically adopted one of the three patterns of behavior under investigation so that the former can be framed alternatively within our definitions of the reciprocator type (RC), the best response type (BR), and the opportunistic type (OP). In order to assign subjects to these types, we estimated a finite mixture model (see McLachlan and Peel 2000) that will allow us to verify if these strategies are well identified and separated in our sample.

[FIGURE 4 OMITTED]

We proceeded by assuming that a proportion %BR of the population from which the experimental sample is drawn behaves according to the best response type; a proportion [[pi].sub.RC] behaves according to the reciprocator type; and finally a proportion [[pi].sub.OP] = 1 - ([[pi].sub.BR] + [[pi].sub.RC]) behaves according to the opportunistic type. Our mixture assumption is that each subject belongs to one of these types and that he or she cannot switch type across rounds. The parameters ([[pi].sub.BR], [[pi].sub.RC], [[pi].sub.OP]) are known as the mixing proportions and are estimated along with the other parameters of the model.

The likelihood contribution of subject i in group g then is:

(2) [L.sub.ig] = [[pi].sub.BR] x [l.sup.BR.sub.ig] + [[pi].sub.RC] x [l.sup.RC.sub.ig] + [[pi].sub.OP] x [l.sup.OP.sub.ig]

where [l.sup.BR.sub.ig], [l.sup.RC.sub.ig], [l.sup.OP.sub.ig] are the likelihood contributions of individual i under the hypothesis of his or her belonging to the best response type, the reciprocator type, and the opportunistic type, respectively. These are derived as follows.

We model the individual propensities to behave according to type q [member of] (BR, RC, OP) in a very simple way, that is, by assuming that there is an average propensity, [[gamma].sup.q.sub.g] to choose one of the strategies that comply with that type's rule, which is common to all the subjects of that type. [[gamma].sup.q.sub.g] has a subscript g because we allow it to vary across groups in order to capture possible coordination effects (group-specific fixed effects). In other words, we tested whether players are more likely to adopt a strategy if there are other players in his or her group of the same type. Thus, individual i's propensity to choose one of the strategies that correspond to type q is:

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here, [[epsilon].sub.ig,t] is an error term, distributed as a standard normal and independent of anything else in the model. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a latent variable representing player i's attitude to act according to strategic type q. The available data are an unbalanced panel because the number of rounds played by each group ([T.sub.g]) depends on a random stopping rule that decides, after Round 15, whether or not to continue with another round of the game.

The observational rule is the following:

[y.sup.q.sub.ig,t] = 1

if [s.sub.ig,t] complies with type q's behavioral rules

[y.sup.q.sub.ig,t] = - 1 otherwise.

The likelihood contribution of subject i, conditional on being of type q, is

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [PHI][x] is the standard normal cumulative distribution function.

Results are displayed in Table 3. In Specification 1, where all [[gamma].sup.q.sub.g], with q [member of] (BR,RC,OP), are estimated as common constants, we find that the predominant type is the best response type, followed by the reciprocator and the opportunistic type. Adding group-fixed effects to the three types significantly increases the log-likelihood according to the likelihood-ratio test ([chi square].sub.24] = 80.010, (17) P value < .0001). This makes again the best response type the most popular with a mixing proportion [[pi].sub.BR] = 0.452, followed by the reciprocator type with a [[pi].sub.RC] = 0.296 and the opportunistic type with a [[pi].sub.OP] = 0.252. Compatible with these results, we observed that adopting a certain strategy seems group-driven (e.g., players are more likely to best respond if they are in a group where there are other players who do best response).

Given the estimation results of the mixture model, we can compute the posterior probabilities of each experimental subject being of each type. Using Bayes' rule, we have the following posterior probabilities:

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for q [member of] {BR, RC, OP}. Posterior probabilities are reported on the simplex displayed in Figure 5. The 54 subjects are represented by circles in the graph: small circles represent a single subject; larger circles cluster subjects concentrated in that area of the simplex (the larger the circle, the more numerous the cluster). The closer a subject is to a vertex of the simplex, the greater the posterior probability for that subject of being of the type represented on that vertex. (18) Subjects in the bottom left corner are of the best response type; subjects in the top comer are of the reciprocator type; and, finally, those in the bottom right corner are of the opportunistic type. The majority of subjects are located very close to the vertices of the simplex, a minority to the edges, and only three are in the middle. The gray simplex in the center represents a virtual area of "uncertainty over types" and is empty in the case under examination. This finding confirms that the mixture model clearly separates the three types of individuals, with most of them being assigned to a particular type with quite a high posterior probability. (19)

[FIGURE 5 OMITTED]

VII. CONCLUSIONS

In this paper, we use experimentally generated data to analyze individual linking strategies in a network formation game.

By a system of equations modeling players' link proposals in each round of the game, we are able to distinguish between strategies that we name of the myopic best response type, of the reciprocator type, and of the opportunistic type.

We find that approximately 40% of the network formation strategies adopted by the experimental subjects can be accounted for as myopic best response strategies, 37% as reciprocator strategies, and 40% as opportunistic strategies. Adding myopic best response, reciprocator, and opportunistic behavior, we are able to explain approximately 74% of the observed choices. We show that each of these types of behavior is "deliberate" in that we have obtained shares of each behavior that are significantly different from what we would have obtained if agents had selected links at random.

Given that there is overlap between strategies, we have tested econometrically if a mixture assumption can be validated for our sample. We find that it is safe to assume that each individual belongs to one type, with mixing proportions approximately equal to 45%, 30%, and 25% for best response, reciprocator, and opportunistic types, respectively.

We observe that the average profits obtained by subjects following each of the strategies are not too dissimilar, with opportunists earning marginally less. We argue that this is because, by targeting only the links that have the highest connectivity, opportunists may miss out on profitable connections.

Finally, we discover that the individual attitude to adopt a certain strategy is heavily group-driven, with agents being more likely to best respond, for example, when others in the same group also do so.

The latter finding has very interesting policy implications. By having more subjects who have an individual propensity toward a certain behavior, we increase the attitude of other members of the same group to adopt that kind of behavior. Hence, by controlling the group composition in behavioral types, one could favor some network outcomes as opposed to others.

In this paper, we present the reciprocator and the opportunist as behavioral strategies other than myopic best response behavior. If agents are myopic and have static expectations, anything other than myopic best response is "irrational." In a more complex model, where agents are farsighted and averse to strategic uncertainty, rational behavior may share features with the strategies that we have outlined here. A rational farsighted agent may attempt to establish his or her reputation as a reliable connection by always reciprocating link proposals. Equally, an agent who is averse to strategic uncertainty may choose to keep redundant links. We do not attempt such modeling here, but acknowledge the possibility that in a more general model of network formation, the outlined behavioral patterns may indeed stem from expected utility maximization. This is a topic for future research.

ABBREVIATIONS

BR: Best Response

GHK: Geweke-Hajivassiliou-Keane

OP: Opportunistic

RC: Reciprocator

s.e.: Standard Error

doi: 10.1111/ecin.12191

APPENDIX A

THE ECONOMETRIC MODEL OF LINK PROPOSALS

In each round of the game, each subject submits a vector of choices concerning the opportunity to propose or not propose a link to any of his or her opponents. From a player's perspective, the decision to propose a link to one of the opponents is not separate from the decision to propose or not propose a link to another opponent. For this reason, all decisions made by a player in a round are not independent but they are the result of a joint evaluation and need to be analyzed as such. (20)

Let us consider a set of six players, indexed by i = 1, ... ,6. Each player in round t submits a 5-dimensional vector of intended links:

(A1) [s.sub.ig,t] = ([s.sub.i1g,t], ..., [s.sub.ijg,t], ..., [s.sub.i5g,t]).

Here, j = 1, ..., 5 represents i's prospective players; groups of opponents are indexed by g, with g = 1, ...,9; t= 1, ..., [T.sub.g] indicates the round number. The final round number, [T.sub.g], may differ by group because of a random stopping rule that decides, after round t= 15, whether or not to continue with another round of the game. [s.sub.ijg,t], equals 1 if subject / expresses his or her willingness to be linked to j\ it equals -1 otherwise.

The 5-dimensional vector of intended links (6), neglecting the subscripts of group and time, corresponds to the 6dimensional vector of intended links (1) without the element au, that is excluding the connection to oneself (i.e., the connection from i to i).

The vector of intended links [s.sub.ig,t] is the result of a complex decision process. In making his or her decisions, i needs to jointly evaluate the opportunity to propose a link to each of his or her five prospective players. In other words, i needs to consider the following system of equations:

(A2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here, [W.sub.ijg,t-1] is a vector of explanatory variables describing the characteristics of the relationship between i and j in the previous round, including the lagged dependent variable, [s.sub.ijg,t-1]; [X.sub.jgt-1] is a vector of characteristics of j as shown by the network that emerged in the previous round; [Y.sub.ig,t-1] is a vector of explanatory variables related to i's position in the network in the previous round; the explanatory variables in [Z.sub.g,t-1] describe the main features of the network resulting from players' link proposals in the previous round. There are also two regression intercepts, [[alpha].sub.i] and [[lambda].sub.g]. Intercept [[alpha].sub.i] varies across individuals (individual-specific time-invariant random effect) and is assumed to be common to all Equations in (A2). We also assume that it does not depend on any observable. It represents the individual-specific propensity to demand links and is assumed to be normally distributed across the population: [[alpha].sub.i] ~ N (0, [[sigma].sup.2.sub.[alpha]]). In a network formation game, individual decisions within a group may be well correlated because of unobservable common shocks to all individuals in the same group--for example, because all individuals observe the same sequence of graphs occurring during a session. Our method of controlling for dependence on unobservables within a session is to model the intercepts [[lambda].sub.g] as random unobservables (group-specific fixed effects). The term (1 + [delta](t - 2)) is introduced in order to capture the effect of experience on players' decisions. A positive (negative) [delta] implies that subjects' choices eventually become less (more) noisy. (21) [s.sup.*.sub.ijt,t]--the latent dependent variable representing subject's i propensity to demand a link to j--and [s.sub.ijg,t], the observed binary outcome variable, are related by the following observational rule:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Because players in each round jointly evaluate the opportunity to propose a link to any of their opponents, we expect that the choice of proposing a link to one of the prospective partners reduces the probability of proposing a link to the others. In other words, we expect to observe a negative correlation across the equations in (A2). (22) i's decision in each round can be framed within the class of M-equation multivariate dynamic probit models. Anyhow, we need to place some restrictions on the variance-covariance matrix of the errors and the coefficients on the system's variables. In particular, the joint distribution of the error terms is assumed to take the form:

(A3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here, error variances on the leading diagonal of V have values of 1 and the off-diagonal elements are all equal to [rho]. This hypothesis of equi-correlation of the error terms of the system of behavioral equations (A2) follows from the fact that there is no reason to assume that a certain pair of equations in (A2) is more or less correlated than another pair. Further, we assume that the coefficients on the variables in system (A2) do not vary across equations.

Estimation of the dynamic system (A2) requires an assumption about the initial observations [s.sub.ijg,1]. Because players do not know anything about their opponents and the group of players as a whole before the graph of the network resulting from Round 1 link proposals is shown to them, we can safely assume that the initial condition [s.sub.ijg,1] is completely random.

Let us define player i's likelihood contribution as:

(A4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here, [[PHI].sub.5](.) is the 5-dimensional normal cumulative distribution function with arguments [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN

ASCII] with j = 1, ... ,5; [OMEGA] is a symmetric 5x5 matrix whose elements on the leading diagonal are equal to 1 ([[sigma].sub.jj] = 1 for j = 1, ..., 5) and are equal to [[sigma].sub.jk] = [rho](2[s.sub.ikg,t] - 1) (for j,k = 1, ..., 5 and j [not equal to] k) anywhere else; f ([alpha]; 0, [[sigma].sup.2.sub.[alpha]]) is the normal density function, evaluated at [alpha], with mean 0 and variance [[sigma].sup.2.sub.[alpha]].

[FIGURE A1 OMITTED]

[FIGURE A2 OMITTED]

[[PHI].sub.5](*) is evaluated by the Geweke-Hajivassiliou-Keane (GHK) algorithm. (23) The likelihood function is maximized using 20-point Gauss-Hermite quadrature. (24)

APPENDIX B

EXPERIMENTAL INSTRUCTIONS

Welcome

This is an experiment on the formation of links among different subjects. If you make good choices, you will be able to earn a sum of money that will be paid to you in cash immediately after the end of this session.

You are one of six participants in this experiment; at the very beginning, the computer will randomly assign to you an initial budget (equal across participants). The computer will also randomly assign to you an icon (Dropper, Radio. Cube. Floppy disk, Hand lens, Hour glass) that will identify you throughout the experiment and will assign you an initial budget (equal across participants). The icon identifying you is circled in red on your screen.

[FIGURE A3 OMITTED]

The experiment consists of a random number of rounds: there will be at least 15 rounds, after which a lottery administered by the computer will determine whether there will be a further round or the experiment is over.

Each participant in this experiment represents a node. At the beginning of the experiment, all nodes are isolated. In each round, the computer will ask you whether you want to propose any link and to whom. You may propose 0, 1, or more links. The computer will collect the proposals from all participants and activate only the links desired by both of the two subjects involved (reciprocated proposals).

Your screen will show the graph of active links. The box at the bottom right corner of your screen will show you who has proposed a link to you in the previous round and to whom you have not reciprocated.

Each link that you manage to activate has a cost (equal across participants) that is indicated on the screen. In each round, the computer may reject your link proposals if they entail an expenditure that is higher than your budget for that round. (25)

Your revenues in each round are automatically computed and are given by the product by the revenue per node (equal across subjects and indicated on your screen) and the number of nodes that you manage to reach both through your direct links and through the links activated by other participants.

Computing costs and revenues [see Figure A1]

Example: subject Radio is directly linked to Floppy disk and Dropper and indirectly, that is through Dropper, to Hand lens.

In each round, the computer calculates out your profit and displays it on your screen. The overall profit from the experiment is given by the sum of your revenues in all rounds. At the end of the experiment, you will be paid in cash an amount in euros equivalent to 10% of your total profit.

More in detail

At the beginning of the experiment, please wait for instructions from the experimenters before touching any key.

When the experimenter asks you to do so, please double-click only once on the "Network Client" icon on your desktop.

The following screen will appear [see Figure A2].

The screen gives you all the information regarding the round that you are about to play.

Be careful: each round has a maximum time duration given by the number of seconds indicated in red at the top right corner of your screen. If you have not managed to make your choice by then, the computer will immediately proceed to the next round.

Your screen shows all the relevant data useful for the current round (available budget, costs, and revenues) as well as the results that you have obtained from each of the previous rounds.

At the end of each round, the graph will show the links activated by you and the other participants (as shown above) [see Figure A3], Moreover, the table that summarizes your performance in the current round will be updated. You will have the possibility to review the situation of previous rounds by clicking on the corresponding bar in the same table. The table at the bottom right corner of your screen gives you additional information on proposals that you have received but not reciprocated in the previous rounds.

When the message "Round is now active" appears at the bottom of your screen, you can make your choice by ticking the boxes corresponding to the icons that you want to propose a link to. When you are done, press "Confirm." When all participants have confirmed their choices, the computer will show the results of the round on the screen.

You will be advised at the beginning of a new round by a "New Round" message. Be careful: after the 15th round, red and green lights will flash on the screen. If the lights stop on green, you will play another round; if they stop on red, the experiment is over.

It is very important that you make choices independently and that you do not communicate with other participants during the experimental session.

At the end of the last round, the experiment is over, and you will be paid a sum in cash corresponding to your profit during the course of the whole experiment.

For any problem, please contact the experimenters.

Enjoy.

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(1.) More in detail: while convergence may be more easily achieved in experimental settings where the stable network architecture is the wheel (for positive results see Callander and Plott 2005; Falk and Kosfeld 2012; for a negative result see Bernasconi and Galizzi 2005), convergence is always problematic in frameworks where the prediction for the stable network is the star (Berninghaus et al. 2007; Falk and Kosfeld 2012; Goeree et al. 2009). Falk and Kosfeld (2012) and Berninghaus et al. (2007) highlight the role of complexity and the lack of coordination in preventing convergence.

(2.) The software used for the experiment has been developed by Andrea Lombardo and is available from the authors on request.

(3.) Here, we reanalyze the data from Treatment 1 in Di Cagno and Sciubba (2008), plus some newly collected data. More in detail, of the nine groups considered here, seven coincide with those analyzed for Treatment 1 in Di Cagno and Sciubba (2008) and were collected in spring 2006. In spring 2008, we collected data for two additional groups under the same experimental protocol used for the 2006 data. More independent observations than we had in 2006 were required for the econometric analysis conducted in this paper.

(4.) Treatment 1 and Treatment 2 for which the data were originally collected are analyzed in Di Cagno and Sciubba (2008). The econometric analysis which we conduct for the present paper is more sophisticated than that in Di Cagno and Sciubba (2008) and requires more variations in the data than we had for Treatment 2. More in detail. Treatment 2 parameters are such that the cost of each individual link is higher than the benefit obtainable from each connection. This implies that the optimal strategies for network formation under Treatment 2 are very different from Treatment 1. Under Treatment 2, in particular, myopic best response no longer prescribes agents to propose a link to everyone unless they can already be reached indirectly, as for Treatment 1. The equilibrium network under Treatment 2 is not a minimally connected network as in Treatment 1 but always the empty network. Hence, it would have been misleading to analyze the data from Treatments 1 and 2 under the same model. The analysis of data from Treatment 2 in a separate model was made problematic by the fact that there is very little variation in our data set for Treatment 2 (most subjects were proposing very few links most of the time) and a robust econometric analysis of the Treatment 2 data would have required us to run a much larger number of groups.

(5.) At the end of Round 15 (and of each additional round after that), a lottery administered by the computer decided if an additional round had to be played. The probability of new rounds was fixed at 50%. The lottery was visualized on participants' screens by two flashing buttons, one red (with a NO sign) and one green (with a YES sign). In our sample, one group ended up playing 15 rounds; instead, 16, 17, 18, and 20 rounds were played by two groups each.

(6.) In this setting, we wanted to avoid any salient coordination device that induces coordination in a particular network. In the pilot study for this experiment (see Di Cagno and Sciubba 2008), we labeled participants with A,B,C,D,E,F and we found that the alphabetical ordering captured most of the linking decisions. See also Bernasconi and Galizzi (2005) and Falk and Kosfeld (2012).

(7.) While we always invited 18 subjects to the laboratory, on a few occasions, we could only collect data for two groups of six. This was because not everyone who had registered for the experimental session showed up on time and, on one occasion, because the software for one of the three groups crashed.

(8.) In contrast to what we assume here, see Carrillo and Gaduh (2012) and Kirchsteiger et al. (2011) for experimental evidence on farsighted behavior in network formation.

(9.) We have estimated several specifications of the model of link proposals, using many combinations of parameters and interaction terms as well as different proxies to represent subjects' and networks' characteristics. In Table 1, we report the selection of results that, in our opinion, gives the clearest picture of the main findings. All other results are available from the authors on request.

(10.) In order to ease readability, without loss of generality, we omit to mention that each i and j belongs to one of the nine groups indexed by g. Extensive notation can be found in Appendix A.

(11.) Here again, the evidence is not extremely robust, making us suspect that only part of the population may adopt that kind of behavior.

(12.) In our setting, reaching a node directly when it is already reached indirectly is always more beneficial than not reaching that node at all. Hence, we do not infer any particular behavior front this finding.

(13.) Note that all behavioral types may initiate links to those from whom they have not received proposals. Each of the types does this out of indifference, because given the myopic assumption and given that links have to be mutually agreed, there are no positive expected profits from proposing links to those who have not proposed in the previous round.

(14.) As each strategy has 1/32 probability of being selected, the proportion of strategies that are of a certain type, given what happened in the previous round, can be interpreted as the probability a player has to select that type of strategy by picking one of the 32 strategies at random.

(15.) In Table 2, the first row shows, for example, that 107 (12.05%) choices in the experimental sample can be accounted for as best response, but not as reciprocator and/or opportunistic behavior. Instead, the fourth row shows that 68 (7.66%) choices in the experimental sample can be accounted for as both best response and reciprocator behavior but not as opportunistic behavior, and so forth.

(16.) Had players selected one of the 32 strategies at random, given the strategies effectively played in the previous round, the average proportion of choices captured by at least one of the three strategies under investigation would have been 0.4399 (s.e. 0.0067).

(17.) Here, 24 is the number of free parameters estimated in Specification 2 with respect to Specification 1 in Table 3, which correspond to eight group dummies per type.

(18.) For producing the simplex, posterior probabilities have been rounded to the closest 0.05.

(19.) This technique has been previously used by Conte and Levati (2014) and Conte and Moffatt (2014).

(20.) Cf. Di Cagno and Sciubba (2008), who disregard this characteristic of players' decisions, deal with each player's link proposals to his or her prospective partners as independent choices.

(21.) A positive [delta] would eventually reduce the error variance (that is constrained to be equal to 1 in Round 2 for identification purposes), consequently making the role of the stochastic disturbance less and less relevant in players' decisions and, in this sense, highlighting the role of experience accumulated throughout the game.

(22.) Suppose player i's decisions are uncorrelated. Then, the probability that i proposes a link to both Player 1 and Player 2 is 25%. A correlation of - 0.25 reduces this probability to about 21%, a correlation of - 0.5 to about 17%, and so on. A positive correlation would obviously increase such probability.

(23.) This is implemented in Stata by the mvnp() function, see Cappellari and Jenkins (2006).

(24.) The program is available from the authors on request.

(25.) The budget constraint is only mentioned in the instructions (and not in the main text of the paper) because the budget constraint is only relevant for Treatment 2 in Di Cagno and Sciubba (2008), where the net profit from each link (revenue generated minus cost of connection) is negative. Given that the experimental instructions are the same here, we have reported them verbatim. However, there is no role for budget constraints in the present paper because they never become effective.

Conte: Senior Lecturer in Quantitative Methods,Max Planck Institute of Economics, 07745, Jena, Germany; EQM Department, University of Westminster. London, NW1 5LS, UK. Phone +44 20 7911 5000, ext. 66593, Fax +44 20 7911 5839, E-mail A.Conte@westminster.ac.uk

Di Cagno: Professor of Economics, Department of Economics and Finance, LUISS University, 00198, Rome, Italy. Phone +39 06 8522 5744, Fax +39 06 8522 5949, E-mail ddicagno@luiss.it

Sciubba: Senior Lecturer in Economics, Birkbeck, University of London, London, WC1E 7HX, UK. Phone +44 20 7631 6450. Fax +44 20 7361 6416, E-mail e.sciubba@bbk.ac.uk

TABLE 1
Estimation Results of Three Specifications of the Model of Link
Proposals Detailed in Appendix A

                                 (1)          (2)          (3)
i-j relationship in t-1

i proposes a link to j           0.560 ***    0.400 ***    0.400 ***
                                 (0.062)      (0.060)      (0.060)

j proposes a link to i           0.467 ***    0.460 ***    0.459 ***
                                 (0.059)      (0.058)      (0.058)

i and j directly linked          -0.089       0.130        0.127
                                 (0.079)      (0.091)      (0.091)

i and j are linked both          -0.238 ***   -0.191 **    -0.221 **
directly and indirectly          (0.074)      (0.086)      (0.089)

i and j only indirectly linked   -0.124 ***   -0.076       -0.075
                                 (0.045)      (0.056)      (0.056)

Characteristics of j in t - 1

number of j's redundant links    0.081 ***    0.069 **     0.107 **
                                 (0.031)      (0.032)      (0.046)

= 1 if y is isolated, = 0        -0.086       -0.041       -0.049
otherwise                        (0.066)      (0.068)      (0.085)

= 1 if j has the largest         0.126 **     0.112 *      0.115 **
number of connections, = 0       (0.056)      (0.058)      (0.059)
otherwise

Characteristics of i int-1

number of i's redundant links    --           0.011        0.045
                                              (0.032)      (0.043)

= 1 if i is isolated, = 0        --           -0.067       -0.078
otherwise                                     (0.054)      (0.065)

= 1 if i has the largest         --           -0.034       -0.039
number of direct links, = 0                   (0.041)      (0.041)
otherwise

number of links proposed minus   --           0.154 ***    0.154 ***
number of links activated                     (0.025)      (0.025)

Network in t--1

number of redundant links in     --           --           -0.039
the group                                                  (0.035)

number of isolated nodes in      --           --           0.004
the group                                                  (0.020)

Error components

[delta]                           0.029 ***    0.026 **     0.027 ***
                                 (0.011)      (0.010)      (0.010)

[rho]                            -0.196 ***   -0.205 ***   -0.205 ***
                                 (0.018)      (0.018)      (0.018)

[[sigma].sub.[alpha]             0.246 ***    0.211 ***    0.208 ***
                                 (0.037)      (0.034)      (0.034)

Log-likelihood                   -2587.8      -2560.7      -2560.0

Number of observations           4440         4440         4440

Number of subjects               54           54           54

Number of groups                 9            9            9

Note: The coefficients on the group-fixed effects, [[lambda].sub.g],
are omitted. ***, **, and * indicate p values < .01, < .05, and
< .10, respectively.

TABLE 2
The Frequency of Choices in Our Experimental
Sample Captured by Each of the Three
Strategies Alone and All Possible Overlaps

                                 Strategy
                      Best
Frequency      %      Response   Reciprocator    Opportunistic

107          12.05    [check]          X               X
126          14.19       X          [check]            X
112          12.61       X             X            [check]
68            7.66    [check]       [check]            X
108          12.16    [check]          X            [check]
60            6.76       X          [check]         [check]
77            8.67    [check]       [check]         [check]
230          25.90       X             X               X
tot. 888

Note: The tick ([check]) indicates when a strategy is represented;
the cross (X) when it is not.

TABLE 3
Estimation Results of the Mixture Model

                          (1)         (2)
[[gamma].sup.BR]          CC          GFE

[[gamma].sup.RC]          CC          GFE

[[gamma].sup.OP]          CC          GFE

[[pi].sup.BR]            0.417       0.452
                        (0.090)     (0.087)

[[pi].sup.RC]            0.367       0.296
                        (0.090)     (0.073)

[[pi].sub.OP]            0.216       0.252
                        (0.072)     (0.073)

Log-likelihood         -512.645    -472.640

Observations              888         888

Number of subjects        54          54

Number of groups           9           9

Notes: CC indicates that a common constant is estimated; GFE
indicates that group-fixed effects are estimated. The results are
omitted. All mixing proportions are statistically significant (p
values < .01).