The theory and empirics behind the social ties model

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The theory and empirics behind the

social ties model.

Thesis by: V.M. de Graaff

10683976

Master Economics: Behavioural Economics and Game Theory

ECTS: 45

ABSTRACT

The theoretical social ties model of van Dijk and van Winden (1997) is adapted to allow for forward-looking behavior, group ties formation and to test its performance. The model is estimated on 4 experimental datasets of repeated linear and nonlinear public good games involving 2 and 4 players. Performance of this model is compared with the performance of an inequality aversion and a simple fixed social preference model. The estimation findings provide evidence that the history of social interaction between individuals is a key determinant of preferences. Findings on both within-sample and out-of-sample tests suggests that the predictive performance of the social ties model in tracking the dynamic contribution patterns is quite good for all 4 experimental datasets. Moreover, these findings also indicate that the performance of the model is slightly better for nonlinear and small groups public good games compared to linear and bigger group public good games. Additionally, empirical tests indicate that the performance of the model is as good or better as an inequity aversion model and a fixed preferences model. In conclusion, the dynamic, emotional driven social ties model can account well for the dynamic contributions patterns in repeated nonlinear/linear public good games.

Key words: Social preferences, social interaction, social ties, public goods games, experimental economics

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NTRODUCTION

Almost all economic models assume that individuals are only pursuing their material self-interest and do not concern about social goals. The fundamental assumption is that individuals only care about themselves and therefore act in their own self-interest. This means that agents have purely selfish preferences and infinite computing abilities, which permits them to effortlessly maximize their own utility given their opinions about the environment. However, predictions made by these rationality models are in conflict with numerous data on observed behavior. Apparently, for a diversity of (non-)strategic reasons, individuals may show behavior that appears to be in conflict with the standard rationality model.

Therefore, many researchers have constructed alternative models in order to explain non-selfish behavior of individuals. Most of these newly proposed models disregard the greed assumption. However, they preserve the rationality assumption and incorporate the development of new types of preferences. Grouped under the wide label of “social preferences”, these models assume interdependent utility function of various forms, implying that one’s utility is also depending on the utility of the individuals one is interacting with (see for a survey, Sobel 2005). In other words, economic models of interdependent utilities formalize the care for others in their own choice by constructing the others utility into the individuals own utility.

A few of these interdependent utility models are based purely on distributional preferences, like altruism or inequity aversion. This implies that the chosen action is based on the different distributions of income that are available to the decision maker. These models imply that an individual’s utility function is elaborated with a parameter representing differences in payoffs with other individuals (See for example: Kirchsteiger, 1994; Levine, 1998; Fehr and Schmidt, 1999; Bolton and Ockenfels, 2000; Charness and Rabin, 2002). For example, a main insight of the model of Fehr and Schmidt (1999) is that there is an important interaction between the distribution of preferences in a given population and the strategic environment and therefore they state that the results they found can explain a variety of non-selfish behavior.

Another class of models tries to explain non-selfish behavior by constructing reciprocity into the agents’ preferences; that is answering an (un-) kind action by a costly

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(un-)kind action. Sobel (2005) argues that these preferences contain “intrinsic reciprocity”. This means that individuals are willing to sacrifice their own material consumption to increase the consumption of others in response to kind behaviour while they are willing to sacrifice their material consumption to decrease other individuals’ material consumption in response to unkind behaviour. For example, in cox et al. (2007) the authors propose a parametric model that incorporates an individual’s desire to harm or help others depending on the emotional states that arises from status considerations and reciprocity motives. In addition, some other examples of papers that incorporates beliefs about the intentions or about the kindness of the type of the other individual into the person’s own utility function are Rabin (1993), Falk and Fischbacher (2006), Freeman (2004), and Dufwenberg and Kirchsteiger (2004).

In conclusion, multiple theories and models are developed by economist that try to explain the non-selfish behavior often observed in experiments (see for survey: Fehr and Schmidt 2006; Sobel, 2005). These other-regarding preferences relate to the distribution of

outcomes (like inequity aversion models) or to reciprocity constructed either on the (un-)kindness of other individuals’ behavioral intentions or the type of other individuals (for

example, selfish, spiteful or altruistic).

Though many different interdependent utility models brought valuable insights into the workings of (economic) individual behavior, they also raised several problems. The first problem is that evidence for these models appears to be mixed and results depend a lot on the type of game or environment being analysed. Secondly, the reliance of psychological game theory and the existence of multiple equilibria makes reciprocity models unnecessarily complex. Another more basic problem for interdependent utility models is the specification of α, the weight put on the other’s payoff. For example, the models of Rabin (1993), Dufwenberg and Kirchsteiger (2004), Falk and Fischbacher (2006) and Charness and Rabin (2002) all construct explicit functional forms for α. They give plausible arguments to selected experimental evidence but none of these papers describes observable behavioral assumptions on α that can describe behavior (Problem is identified by Sobel 2005). As we see later on, an important goal of this thesis is to try to tackle this problem. Finally, models that formalize purely distributional preferences assume that these preferences are stable and in most cases, preferences are seen as a stable personal trait. For example, in the Levine’s signalling model (1998), the weight one attaches to his counterparts utility is influenced by one’s own belief

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about the weight the other individual is attaching to one’s own utility, and where the real weight is assumed to be a fixed personality trait. Hence, mostly no time or history dependence is included in such models. Therefore, empirical tests are performed rather on static distributional choices, on one-shot games or on the last period if the game is a repeated one. Either there is no interaction between players included or learning and reputation-building issues are ignored. In the paper of Fehr and Schmidt (1999), they discuss this “problem”. They acknowledge that their model cannot explain the evolution of the interaction over time. In its place, they focus on the explanation of the stable behavioral patterns that emerge after several periods. Though it is interesting to examine the behavioural patterns that could emerge during an interaction, it is not certain that such patterns indeed will emerge. Later in this thesis, we indeed see that reaching stable behavioural patterns may not always happen. Moreover, it is fascinating to study how players reach certain equilibria, so it is too easy to disregard the evolution of play. Moreover, not only the history of play is absent in most models, but also the affective processes are mostly neglected. There are only a few exceptions where economists took the effect of emotions on individual decision making into account. For example, the model on loss aversion (Tversky and Kahneman, 1991) regret (Loomes and Sugden, 1982; Bell, 1982) and disappointment (Loomes and Sugden, 1986). However, until recent times, affective processes have been lacking in most behavioral models of (social) decision-making.

As discussed in the previous paragraph, the newly proposed social preferences models have three essential characteristics. Namely, they do not itemize any emotional mechanism that are, at least indirectly, referred to; they focus on static equilibria; and, they assume stable other-regarding preferences. The social tie model introduced by van Dijk and van Winden (1997) differs in these respects. This model imply that affective processes and the history of play are both a driving force in explaining individual preferences. In their paper, the dynamics

of social ties and local public good provision, they proposed a formalization of the

development of social ties between players facing the problem of private provision of public goods. In this model, social ties are modelled as the weight attached to another individuals well-being in one’s own utility function. This weight is allowed to be dynamic and is assumed to be determined by two driving forces: the interaction history and affective impulses generated by the behavior of the counterpart. While the model still assumes the utility

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maximization hypothesis, it makes social preferences endogenous and specific to each interaction player. In other words, the social ties model assumes that the accumulated affective experience is important for the sign and size of the weight attached to the specific counterpart’s interest. This gives the development of social ties an intrinsically dynamic nature and allow for changes in the strength of the tie. This process allows for individual differences in personality and these differences are important for the speed with which positive or negative ties develop.

Since the introduction of this model, there is some behavioural and neurological evidence that the model can capture behavior remarkably well in two and four-player nonlinear public good games. In a behavioural paper written by van Dijk et al. (2002), the writers conclude that there is strong evidence of the development of social ties in a pairwise nonlinear public good experiment. In line with the finding of van Dijk et al. (2002), the experiment conducted by Sonnemans et al. (2006) shows that prolonged interactions in an nonlinear public good game leads to the development of social ties, where the strength of the tie depends on the success of the interaction. Moreover, findings on this experiment also suggest that individuals discriminate between group members based on their relative contributions. Though both papers provide evidence for the existence of affective tie formation, parameter estimation on the model was not carried out. At the neural level, both the experiments of Fahrenfort et al. (2012) and Bault et al. (2014) have shown that humans indeed form interpersonal ties depending on the success of the interaction. Both papers credible argue that there is a correlation between the activity in the posterior superior temporal sulcus (pSTS) and both the interaction success as the liking ratings of an interacting partner.

Though the existence of the formation of social ties received indirect support through multiple behavioural and neurological experiments, direct behavioural evidence of the model is scarce. Only in the papers of Pelloux et al. (2015) and Bault et al. (2014), the performance of the model is empirically tested. Pelloux et al. (2015) gives a direct behavioural assessment of the social ties model by scrutinizing the performance of the model through both within-sample as out-of-within-sample tests. In order to do this, they adapt the theoretical social ties model (van Dijk and van Winden, 1997) to permit for forward looking behavior and stochasticity. This thesis is highly inspired by the paper of Pelloux et al. (2015) therefore, I will follow the same

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general outline. In their paper, they empirically test the model on the data for three nonlinear public good game experiments, while I will empirically test the model on data for both linear as nonlinear public good games. Testing the performance of the model in different frameworks could enhance the robustness of the behavioural findings and can help us identifying and explaining potential differences in the outcomes.

In this thesis, I will behaviourally evaluate the social ties model using 4 different datasets. The theoretical social ties model (van Dijk and van Winden, 1997) will be adapted by implementing a probabilistic choice mechanism. Furthermore, the model is extended to allow for both forward-looking behavior and the formation of “group” ties. This provides an explicit dynamic model. This explicit model will then be used to scrutinize the performance through both out-of-sample as within-sample tests for linear and nonlinear public good game datasets concerning groups of 2 and 4 players. This is the first attempt that the performance of the model in explaining individual behavior is investigated for a linear public good game. Furthermore, the model is also used for the first time to examine whether individuals could form an affective tie towards a group of individuals. After finishing all the empirical tests for the social ties model, I will also investigate the performance of a simple fixed preferences model and an inequality aversion model (Fehr and Schmidt, 1999; Bolton and Ockenfels, 2000). Comparing the estimation results for the social ties model versus the fixed preferences and inequality aversion model will give us a more complete picture of the relative performance of the social ties model.

An important question that I would like to answer in this thesis is whether the social ties model could explain individual behavior better or as well than other-regarding preferences models. Since, at least theoretically, the social ties model has the potential to overcome the deficiencies that characterize most behavioural models, answering this question could help behavioural economics in understanding human motivation in individual economic decision making. The organization of this thesis is as follows. Section 2 starts with the theoretical underpinnings of the formation of affective social ties followed by the introduction of the formal model. In section 3, the model is empirically tested by conducting several tests on 3 experimental datasets. Besides giving an extended analysis on the social ties model in section 4, an inequality aversion model and a fixed social preferences model are empirically tested. In the last part of section 4, the fourth dataset will be used to test whether

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individuals could form ties with a group of individuals. Moreover, the experimental design of the fourth experiment will allow me to test a forward-looking version of the social ties model. Finally, concluding remarks are given in section 5.

2 S

OCIAL TIES

This section deals with the formulation of the social ties model (van Dijk and van Winden, 1997). In section 2.1, the theoretical underpinnings of the formation of affective social ties will be given. Then, section 2.2 will implement and extend the theoretical social tie model to allow for stochasticity and forward-looking behavior. At last, chapter 2 will finish with some concluding remarks.

2.1 S

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Many findings from psychology suggest that interpersonal attachment is one of the most important fundaments of human motivation. The psychologist Ribal (1963) stated that in human interaction, motivation is determined by the need for nurturance, which represent the motivational need to give, and the need for succorance, that denotes the motivational need to receive. Later, in the survey written by Baumeister and Leary (1995), the writers hypothesize that humans form social attachments eagerly and resist the termination of existing bonds. Extensive psychological evidence (Baumeister and Leary, 1995) underline the hypothesis that the longing for affective interpersonal relationships is an essential human motivation. From an evolutionary viewpoint, this eagerness to form bonds between humans makes sense. For example, as a behavioral mechanism to bond if it does good to you and flee or fight what is bad. Furthermore, more and more studies indicate that mental and physical problems are more common among individuals who lack close personal relationships (S. Cohen, 2004). Individuals that have no bonds marked by positive concern and caring, tends to be more unhappy, depressed, anxious and have a higher morbidity and mortality rate (see for review; B.N. Uchino, 2006). Not only psychologist argue that interpersonal attachment is a driving force in human motivation. Sociologists also have persuasively argued that affective interpersonal relationships has a profound effect on the economic interaction in markets and organizations (See Coleman,1984; Granovetter, 1985; Uzzi, 1996). For example, Laywer and Thye (1999) review psychological and sociological research on emotions. They conclude that the solidarity and cohesion of relationships based on reciprocal exchange may be sensitive to

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emotional expressions in the exchange process. Nonetheless, this prosperity of evidence, processes of interpersonal attachment have mostly been ignored in formal models of social interaction. This is also the case for recent models in behavioral economics and game theory that allow for social preferences, like the distributional and reciprocity preferences models. The incorporation of affective processes (sentiments and emotions) into these models is rare, meaning that the social glue for attachment is missing. Van Winden (2008) argues that incorporating affective processes would give relations between individuals (or groups of individuals) individualized, historical, and context dependent content. Granovetter (1985) argues that this would help to bond the gap between an undersocialized and an oversocialized interpretation of human behavior that characterizes numerous models. The undersocialized models are characteristic for neoclassical models often used in economics while the oversocialized models in economics and sociology assuming individuals with fully internalized social norms. Granovetter (1985) state that “under- and oversocialized accounts are paradoxically similar in their neglect of ongoing structures of social relations, and a sophisticated account of economic action must consider its embeddedness in such structures” (see page 481).

Recently, there is a rapidly growing literature that suggests that affection (emotions) in interpersonal relationships play an important role in economic decision making (Elster, 1998; Loewenstein 2000, Fehr and Smidt 2006, Camerer et al. 2005 and Bault et al. (2008)). Alfred Marshall (1961 [1890]) argues that labor market rigidity is related to the reluctance of workers to migrate due to affective relationships. Later on, economists where especially interested in the impact of affective relations within the family (e.g., Barro, 1974; Becker, 1981; Ott, 2012). Sometimes, interpersonal relations outside the family domain also received some attention. For example, the importance of affective interpersonal relations for the functioning of organizations (Rotemberg, 1994), industrial customer-supplier relationships (Lövblad et al. 2012) and foreign direct investment (Laar, 2006)). Furthermore, such relations, though implicitly, plays a role in a wider range of topics. For example, in efficiency-wage theory (Akerlof, 1982) or private provision of insurance (Arnott and Stiglitz, 1991). This tendency, the increasing importance of affect in interpersonal relationships, is also supported by recent neuroscientific findings (Fehr and Camerer, 2007; Sanfey, 2007; Bault et al., 2014 and Fahrenfort et al. (2012)). Other neural studies of human social behavior have also begun to

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investigate affective attachment such as sympathy (Decety and Chaminade, 2003), friendship (Fareri et al., 2012) and romantic attachment (Zeki, 2007). Despite the wealth of evidence presented in this paragraph, processes of affective interpersonal attachment in formal models of social interaction have mostly been neglected. There are only a handful of models in which the formalization of interpersonal behavior is the outcome of an interacting affective system (for example, the models of Battigalli and Dufwenberg (2007), Loewenstein and O’Donoghue (2007), van Dijk and van Winden (1997) and van Winden (2001). This thesis will give a direct behavioural evaluation of the social ties model introduced by van Dijk and van Winden (1997).

Before the introduction of the formal social tie model in the next section, the question what exactly a social tie is have to be addressed. In simple words, a social tie denotes a caring about the well-being of a specific other individual, based on sentiments experienced while interacting with that other individual. These sentiments are the affective components of interpersonal attachment and are the key element of a social tie. Hence, sentiments that individuals have for one another is closely related to the extend they care about each other’s well-being, which suggests a formalization of utility interdependence. In other words, the more individuals care for someone in a positive (negative) way, the stronger (weaker) is the affective bond. However, a social tie is not merely determined by the weight that someone attaches to the well-being of another individual. This is only when people are neutral towards individuals they do not know. If individuals have positive or negative sentiments towards strangers, a social tie with a particular individual only exists if the weight attached to the other’s welfare differs from a generalized other (a stranger’s welfare). Furthermore, it has to be mentioned that in most sociological literature a social tie is usually seen as a mutual two-sided relationship. A tie between individuals i and j would contain i’s sentiments about j and vice versa. However, in this paper, I will speak of i’s social tie with j without directly mentioning

j’s sentiment about i. This distinction is important, as sentiments between people are not

necessarily symmetric (Wellman, 1988). Besides the fact that sentiments do not have to be symmetric during an interaction, there is a growing consensus among experimentalists that different types of individuals exists (See Ledyard 1995).

Now that a general notion of a social tie is provided, it is interesting to address how social ties develop. Existing literature generally acknowledged that ties develop through prolonged interaction and erode when interaction ends (Coleman 1990, Levine and Moreland

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1998). The interaction between people is either valued positively or negatively. Based on this, positive or negative sentiments are formed (Feld 1981, Fararo 1989). Hence, it may be expected that social ties could develop during a repeated public good game and that these ties will depend on the payoff that is generated in the interaction process. In conclusion, the social ties is dynamic, it is nourished by emotional experiences and it is individualized (related to a specific other individual while not assuming trait-like other regarding preferences).

2.2 S

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Without loss of generality, this subsection presents a logarithmic transformation of the multiplicative Cobb-Douglas specification of the social ties model introduced by van Dijk and van Winden (1997). Transforming the model to a logarithmic version is introduced in the paper of van Winden et al. (2008). While the paper of Pelloux et al. (2015) uses a linear version of the social ties model, a logarithmic transformation would not affect the model’s estimation results in a qualitative sense. Adapting the model to a logarithmic, discrete time version of the theoretical model will allow us to empirically test the model on data for repeated public good games. A myopic version of the model is introduced in the first part of this subsection (2.2.1). Similar to the work of Pelloux et al. (2015), a social ties model which allow for forward-looking behavior is presented in part 2.2.2.

2.2.1 Myopic individuals

The behavioral model implemented in this thesis is based on the theoretical social ties model introduced by van Dijk and van Winden (1997). Based on multiple psychological findings (see for example, Frijda, 1986; or Coleman 1990), this model assumes that social ties develop as an unconscious by-product of extended interaction that produces positive (negative) emotions if valued positively (negatively), and that these social ties deteriorate over time. The development of positive and negative bonds during interaction are formalized via the concept of an interdependent utility function by letting the weight attached to another individual’s utility to express the bond developed during an interaction with another individual. The crucial difference compared to most other interdependent utility functions is that the weight of the bond is dynamic and evolves over time depending on the positive or negative experiences that individuals encounter during the interaction. In theory, the social ties model is interesting because it can account for various kinds of behavior often observed in economic literature,

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such as selfish behavior, altruism, spite, inequity aversion and reciprocity (see van Winden (2012) for a more elaborate account).

The mathematical model introduced here is a logarithmic discrete time implementation of the theoretical social ties model. It consists of multiple equations where dyads of individual i and j are considered. The first equation is defined as:

𝑈𝑖𝑡 = 𝑙𝑛𝑃𝑖𝑡 + 𝛼𝑖𝑗𝑡∙ 𝑙𝑛𝑃𝑗𝑡 (Eq.1)

The social tie at time t for individual i with individual j is formalized by attaching a weight (𝛼𝑖𝑗𝑡)

to j’s payoff (denoted as 𝑃𝑗𝑡) in the utility function of individual i (denoted as 𝑈𝑖𝑡).

Furthermore, individual i’s utility also consists of his own payoff (𝑃𝑖𝑡). Hence, in this model

representation, the social tie is the weight in one’s own utility for the payoff received by the individual who he is interacting with. The weight can be negative, positive or null. In this thesis, it is assumed that the absolute value of the social tie is one (which is the weight of individual

i’s own utility). This is a reasonable assumption as it is unlikely that individuals are valuing

other individuals’ well-being more than their own (See for evidence; Goeree et al., 2002). Furthermore, it should be noticed that when the weight of the social tie is null, the utility function is a traditional utility maximization function often encountered in mainstream economics.

As said before, the sentiments that an individual experiences is not a fixed trait, therefore, the weight that an individual puts on the counterparts well-being is allowed to fluctuate throughout the interaction. This is a departure of the traditional assumption of stable preferences and shifts to a deeper level by anchoring the parameters of the model to psychological processes. Therefore, the parameters of the model consist of history retention, in which the number of periods of play influences the social tie, and sentiments, the extent to which an emotional impulse will influence the social tie:

𝛼_𝑖𝑗𝑡 = 𝛿1∙ 𝛼_{𝑖𝑗𝑡−1}+ 𝛿2∙ 𝐼_{𝑖𝑗𝑡−1} (Eq.2)

The parameter 𝛿1 specifies the tie persistence (is related to memory), which is inversely

related to the decay of the social tie, and 𝛿2 can been seen as the tie proneness (related to

sentiments) of individual i. Furthermore, 𝐼𝑖𝑗𝑡−1 is the emotional impulse for player i and is

initiated by player j in period t-1. The memory related parameter 𝛿1 > 0 presents the history

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number of periods of play that influences player’s i behavior significantly. The second

parameter, 𝛿2 > 0, is related to the sentimental impulse that player i experiences during the

interaction with player j. This impulse, 𝐼𝑖𝑗𝑡−1, is generated by the last action taken by the other

player compared to a reference point. In case of a public good game, the impulse is the

difference between 𝑗′𝑠 most recent contribution to the public account and a reference

contribution. Hence, the impulse function is defined as:

𝐼𝑖𝑡 = 𝑔𝑗𝑡− 𝑔𝑖𝑡

𝑟𝑒𝑓

(Eq. 3)

In this function, 𝑔𝑗𝑡 stands for the contribution in the public account by player j in period t.

𝑔_𝑖𝑡𝑟𝑒𝑓 is the reference level about player 𝑗′𝑠 contribution to the public account. For now, the

type of reference is not constrained. Many possibilities, like the standard Nash Equilibrium or one’s own behavior should been considered. Hence, either the reference level could be endogenous or exogenous determined.

The social tie model is based on the emotional assessment of a situation. As said before, this is in stark contrast with most social preferences models. These models hinges on beliefs about the type or the intentions of the opponents, and put more emphasize on the cognitive side of decision making (e.g. Levine, 1998). The psychological and neurological work of Zajonc (1984) and Ledoux (1997) also emphasize this; they argue in favour of the primacy of affect. Furthermore, psychological studies showed that there is a difference between experiences versus description based choices (see for a survey, Rakow and Newell, 2010). For example, Denrell (2005) show that the sequential sampling of experiences could affect social impression formation greatly. Hence, preferences formation in the social tie model are emotionally formed and is out of the cognitive control by what is experienced, cognition will then take effect to make the best possible decision according to these preferences. The next step is to move to the cognitive part. This is done by implementing a probabilistic choice mechanism using the traditional logit form. Therefore, the choice probability for each action is determined by:

𝜋_𝑖𝑘𝑡 = 𝑒𝜃𝑖𝑈𝑖𝑘𝑡

∑𝐾_𝑘=0𝑒𝜃𝑖𝑈𝑖𝑘𝑡 (Eq.4)

Where 𝜋𝑖𝑘𝑡 is the probability that individual i chooses contribution k (𝑘 ∈ [0; 𝐾]) in round t

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individual’s behavior and perfect utility maximization. When 𝜃𝑖 → 0 , then choices are

randomly taken and have an uniform distribution. On the other hand, if 𝜃𝑖 → ∞, then the

probabilistic choice mechanism reduces to perfect utility maximization where individual i choices with certainty the action maximizing his utility. For a given individual, the log-likelihood of choosing a particular stream of decisions over the whole game can be expressed as:

𝐿𝑜𝑔𝐿𝑖 = ∑𝑇𝑡=1∑𝐾𝑘=0𝑑𝑖𝑘𝑡∙ ln (𝜋𝑖𝑘𝑡) (Eq.5)

where 𝑑𝑖𝑘𝑡 denotes a dummy variable. If this variable is one, then action 𝑘 is chosen in period

𝑡.

Before continuing to the next section of this thesis, it is worthwhile to illustrate how the social ties model works. An individual will consider an action as a reference. Then, he will witness the action chosen by the other individual and will compare it to his/her reference. This evaluation is called the emotional impulse. An action that will be perceived as kinder than the individual’s reference will create a positive impulse and subsequently will increase the social tie. On the other hand, when an action is perceived as less kind than the individual’s reference, then it triggers negative emotions and will therefore lower the social tie. Hence, the social tie that an individual will develop towards another individual could be seen as a stock variable of all the emotional impulses activated by the interaction. Two parameters will

determine the strength of the impact of an emotional impulse on the tie, namely, 𝛿1 (the

memory or tie-persistence parameter) and 𝛿2 (the emotionality or tie-proneness parameter).

Reciprocity is introduced in the model through the tie-proneness parameter; when 𝛿2 = 0,

the behavior of the counterpart does not enter the utility function and the tie. If the parameter is strictly positive then this will increase the tendency of the individual to reciprocate. In conclusion, actions from the other individual that is perceived as unkind (kind) will trigger negative ties that will induce the individual to try to lower (increase) the other’s payoff. Equation 2 describes the tie mechanism and embodies the affective part of this dual-process model. Direct cognitive control can be applied neither on the social tie nor on the emotional impulse. The individual cannot achieve a certain goal merely by choosing these variables. Individual preferences are determined through emotional responses generated in an automatic way by the interaction. The cognitive part of the model is introduced by equation 4, where the probability of each action is determined.

14 2.2.2 Forward looking individuals

Thus far, the social tie model assumes that individuals are myopic and do not look into the future. Some small adaptions transforms the model to a forward-looking model (see Pelloux et al., 2015). The intertemporal utility of an individual with limited foresight is defined by the next equation:

𝑉𝑖𝑡 = 𝑈𝑖𝑡+ 𝜆𝑖∙ 𝑈𝑖𝑡+1 (Eq.6)

Where 𝑉𝑖𝑡 is the individual’s i expected intertemporal utility in round t, which both depends

on the utility of this period and expected utility of the next period. The parameter 𝜆𝑖 ∈ [0; 1]

signifies the discount factor applied to the utility expected by individual i in the next period. Since the individuals will maximize their utility only over the next two periods, they have

limited foresight. Additionally, it should be noticed that if 𝜆𝑖 = 0, then individuals are myopic

and the social tie model is described by the first five equations. Furthermore, it is required to model the way expectations concerning the next period contribution of the other individual are formed by forward-looking agents. This is assumed to follow a simple adaptive expectation formation process described in the next equation:

𝑔_{𝑖𝑗𝑡+1}𝑒𝑥𝑝 = 𝛽 ∙ 𝑔_𝑖𝑡+ (1 − 𝛽) ∙ 𝑔_𝑖𝑗𝑡𝑒𝑥𝑝 (Eq.7)

Thus, the expected amount that will be contributed by individual j in the next period is assumed to be a convex combination of the expectation for the current period and the individual’s own contribution for the current period. The parameter 𝛽 is therefore assessing the expected reciprocity from the interaction individual.

2.3 S

In the previous subsections, a logarithmic version of a myopic and a forward-looking social tie model is given. In contrast to most other-regarding preferences models, three distinguishing characteristics have been constructed into the same model. Firstly, the social tie model is dynamic, due to its evolving nature during the interaction. Secondly, it is fed by emotional experiences (impulses). At last, the model is individualized, it is related to a specific other person and does not assume trait like other-regarding preferences. Trait-like other-regarding preferences, like spitefulness or altruism are permitted by the model by leaving open what

the initial weight (𝛼𝑖𝑗𝑡) is. More specifically, the extended social ties model allow for several

forward-15

looking model reduces to the myopic model implying that individuals does not take the future into account and act as if they are myopic. Furthermore, the limited foresight implied by the forward-looking model gives a strategic component to the model. A forward-looking individual would be able to contribute more in a public good game for purely strategic reasons as he expect his own contribution to have a positive impact on the other individual’s next period’s contribution. This feature seems very important as it can potentially explain the decay of contribution levels towards the end of a finitely repeated game (the often-encountered end effect). Second, the standard selfish preference model is embedded in this social ties model,

merely by setting the value of the initial tie (𝛼_𝑖𝑗0) and the value of the tie-proneness parameter

(𝛿2) equal to zero. The model, then reduces to the standard selfish model (𝑈𝑖𝑡 = 𝑙𝑛𝑃𝑖𝑡).

Thirdly, when 𝜃𝑖 → ∞, then standard perfect utility maximization with no errors is obtained.

Fourthly, when the initial tie is not null (𝛼𝑖𝑗0≠ 0), there is not a deteriorating trend in the tie

between rounds (𝛿1 = 1) and there is no updating process for the initial tie (𝛿2 = 0), then

“stable” standard social preferences is obtained.The direction of these stable preferences will

depend on the value of the initial tie. If the initial tie is lower then zero (𝛼𝑖𝑗0 < 0), then

unkindness will occur. Individuals “derive” utility for decreasing their counterpart’s utility. On

the other hand, when the value of the initial tie is positive (𝛼_𝑖𝑗0 > 0), then the individual have

altruistic preferences. The individual “derive” utility by increasing the counterpart’s utility. Concerning the application of the social ties model to a public good environment, the value of the tie could give insights on the behavioural outcomes. If the value of tie tends to

one (𝛼𝑖𝑗𝑡 → 1), then fully cooperative behavior is reached. In contrast, when the value of tie

tends to minus one ( 𝛼𝑖𝑗𝑡 → −1 ), then maximally uncooperative behavior is induced.

Furthermore, the social ties model can also describe complex distributional social preferences, like inequity aversion. For example, in the case of a symmetric public good game where the contributions of the agents are costly and assume that both agents take their own

contribution as a reference (𝑔_𝑖𝑡𝑟𝑒𝑓= 𝑔_𝑖𝑡). Then, a higher contribution of the agent in a period

compared to the other agent not only suggests disadvantageous inequality, but also creates a negative emotional impulse in the social tie model. Hence, the agent will decrease his contribution in the next period. The same line of reasoning implies that a lower contribution than the other will create an advantageous inequality and a positive impulse and this will push the agent towards a higher contribution in the next period. Furthermore, the paper of van

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Winden (2012) argues that the model can also explain various kinds of behavior that are important in explaining cooperation in prisoners dilemma environments. These are mimicking behavior, direct/indirect reciprocity, in-groups and out—groups distinction and emotional and behavioral contagion. It is clear, at least theoretically, that this social ties model turns out to be remarkably flexible. However, this flexibility is empirically restricted because the parameters can be estimated (as will be shown in the next section). From a wider viewpoint, by incorporating affective bonds in the standard model, adaptive interpersonal relationships generated through interaction are to be expected. This means that social ties are changing over time dependent on personal traits, past experiences and different contexts. An important feature of the model is that adaptions show an affective component that are not founded on changes in beliefs.

3 T

ESTING THE MODEL TO VARIOUS GAMES

An important objective of this paper is to test the degree to which the social ties model could explicate observed behavior in several experimental datasets. I will limit myself to three repeated (non-)linear public good games experiments with fixed partner matching. The first two datasets involves a four-player public good game (Sonnemans et al., 2006; Rand et al., 2009) whereas the third one contains a two-player game (Van Dijk et al., 2002). The datasets of Sonnemans et al., (2006) and Van Dijk et al. (2002) are also used in the paper of Pelloux et al. (2015) to estimate the parameters of the social ties model. In sections 3.1 and 3.2, I will elaborate on the experimental design and the aggregate behavioral results of these datasets. Then, I will estimate the model at the group-level for each dataset (section 3.3). In the last section (section 3.4), the group level estimates will be used to produce both dynamic within-sample as out-of-within-sample predictions about the development of contributions throughout the public good game.

3.1 E

In the three experiments, the subjects play a repeated public good game where they have to allocate monetary units between a private and a public account. This section merely discusses the general design of the public good game experiments. For a more elaborate account of the all the techniques used during the experiments, see the papers of Sonnemans et al. (2006), Rand et al. (2009) and van Dijk et al. (2002).

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3.1.1 Sonnemans et al. (2006)

In Sonnemans et al. (2006), subjects are randomly assigned to groups of four and stay in the same group throughout the experiment. The group anonymously play a nonlinear public good game for 32 periods. Each subject have an endowment of 10 monetary units (from now on MU) in each period that they could distribute between a public- and a private account. Every MU invested in the public account yields seven MU to each of the group members. The private

account yields a nonlinear payoff of 21 ∗ 𝑥 − 𝑥2, where 𝑥 denotes the MU in the private

account. Furthermore, subjects pay a fixed cost of 60 MU in each round. Hence, the equation that represents the total payoff for each subject in each period is:

𝑝𝑎𝑦𝑜𝑓𝑓 = 21 ∗ 𝑥 − 𝑥2 _{+ 7 ∗ 𝑦 − 60}

where 𝑦 stands for all MU invested in the public account. Due to the nonlinearity of the payoff function, the Nash equilibrium lies in the interior of the contribution space. This one shot Nash equilibrium is to contribute 3 MU to the public account. The Pareto optimum is to contribute the whole endowment of 10 MU. At the end of each period, subjects received feedback on both the individual contributions as their own payoff for that round.

3.1.2 Rand et al. (2009)

Just as in the experiment of Sonnemans et al. (2006), subjects in Rand et al. (2009) are anonymously assigned to fixed groups of 4. Some other features of the experimental design have important differences. The first difference is that subjects play a linear public good game for 50 rounds. Furthermore, the monetary allocation task between the public and private account consists of 20 instead of 10 MU. Hence, the action space on how much to contribute to the public account is twice as large in this experimental setup. The payoff to the private account (x) is the amount of MU that the subjects invest to this account. The payoff to the public account (y) is 1.6 times the total amount in the public account divided by four. The payoff function for each period is described by the equation:

𝑝𝑎𝑦𝑜𝑓𝑓 = 𝑥 + (1.6 ∗ 𝑦)/4

Both the one shot Nash equilibrium as the Pareto optimum are on the boundary of the action space. The Nash equilibrium is not to contribute to the public account whereas the Pareto optimum is to invest every MU to the public account. This crucial difference with a non-linear public good game has some implications for the forecasts made by the social ties model. Hence, in a linear public good game, we suspect that the model makes much more all or

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nothing predictions conditional on the relative values of the social ties and of the return of MU to the public account. If the social ties with all of the three group members are negative, then it will certainly lead to a null contribution while in a non-linear game, individuals with egocentric and anti-social preferences would not necessarily choice the same contribution level. In other words, forecasts about the development of contributions across periods are probably less accurate for a linear game then a nonlinear game.

3.1.3 Van Dijk et al. (2002)

The experimental design of van Dijk et al. (2002) closely resembles that of Sonnemans et al (2006). In this experiment, subject plays a nonlinear public good game in which subjects are anonymously matched in pairs. The pairs play the game for 25 periods and after each period, each subject receive feedback on their counterpart’s contribution to the public account. Each subject have to allocate an endowment of 10 MU into a private (x) and a public account (y).

The private account yields a nonlinear payoff of 28 ∗ 𝑥 − 𝑥2 and both subjects of the pair

receives 14 MU of every MU invested in the public account. In each period, the subjects have to pay a fixed cost of 110 MU. Hence, the payoff function for each period is given by:

𝑝𝑎𝑦𝑜𝑓𝑓 = 28 ∗ 𝑥 − 𝑥2+ 14 ∗ 𝑦 − 110

The one shot game Nash equilibrium is to contribute 3 MU to the public account. Choosing the full contribution of 10 MU is the Pareto optimum.

The three experiments discussed above, have some variations in their design. Sonnemans et al. (2006) and van Dijk et al. (2002) both have a nonlinear payoff function and a similar Nash equilibrium/Pareto optimum. The main difference is the amount of subjects playing the game against each other. The experiment of Rand et al. (2009) has the most dissimilarities in their experimental design; subjects play a linear public good game with the Nash equilibrium and the Pareto optimum at the edges of the action space. These three datasets offer a decent opportunity to analyse the robustness of the behavioral outcomes in different contexts.

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3.2 A

Figure 1: average contribution behavior

In this section, I will shortly present the aggregated behavioral outcomes of the three experiments. In Sonnemans et al. (2006), the average contribution to the public account is 6.52 MU, which is 65% of the total endowment (see appendix A for descriptive statistics). Figure 1 shows the development of the average contribution to the public good; we can see an increase towards stabilization at 7 MU. Clearly, the figure shows that the end effect kicks inn for the Sonnemans (2006) experiment. The average contribution in the last round (3.39 MU) has dropped by almost a halve compared to period 29 (6.57 MU). Interestingly, both the relative frequencies of the Nash equilibrium as the full contribution Pareto optimum are the highest in the action space (see appendix B). Subjects choose 18 percent of the time the Nash equilibrium while 26 percent of choices was the Pareto optimum.

The data of Rand et al. (2009) display less cooperation because the average contribution is at 45 percent of the total endowment (20 MU available; average is 9.08 MU). Starting from 13.30 MU in the first period, cooperation is rapidly decreasing to stabilize nearby 8 MU. Fascinatingly, the data does not appear to have an end effect. It even shows a slight increase towards the end. This is due to the experimental design; Rand et al (2009) did not announce the number of periods that the subjects had to play the game. Noticeably, the relative frequencies of the Nash equilibrium (32 percent) and the Pareto optimum (27 percent) are totalling 60 percent of all the contributions. Almost none of the relative

0,0000 2,0000 4,0000 6,0000 8,0000 10,0000 12,0000 14,0000 0 5 10 15 20 25 30 35 40 45 50 co n tr ib u tion to p u b lic goo d Period Sonnemans et al. (2006) Rand et al. (2009) van Dijk et al. (2002)

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frequencies of the contributions in the interior of the action space surpasses the 2 percent. This indicate that a substantial part of the subjects either fully cooperate or defect.

Data on van Dijk et al. (2002), concerning pairs, reveal a little less cooperation than Sonnemans et al. (2006) with an average contribution of 6.01 MU (60 percent of the total endowment). Starting from an average of 5 MU in the first period, cooperation gradually grows to stabilize a little above 6.50 MU. Again, in the last few periods cooperation falls because of the often-observed end-effect, though the effect is slightly less than in Sonnemans et al. (2002). The average contribution drops by 1.95 MU between period 24 and period 25. Again, appendix B shows that the Nash equilibrium and the Pareto optimum are among the highest relative frequencies of contributions to the public account (both frequencies are close to 20 percent).

3.3 E

One of the main objectives of this paper is to test to what degree this model fits the data in public good experiments. In order to do this, parameter estimation is done using the maximum likelihood method with. In this sub-section, the estimation on the three experimental dataset will be run at the group level. Before we continue to discuss the estimation results, I would like to note that group level estimation on the data of Sonnemans et al. (2006) and van Dijk et al. (2002) is also done in the paper of Pelloux et al. (2015). In this thesis, the model that is estimated is slightly different because it is a logarithmic version of the social ties model instead of linear version. This will lead to slightly different estimation results but the results does not differ in a qualitative sense.

Previous work unveils that for Sonnemans et al. (2006) and van Dijk et al. (2002) the model performce was the highest when the reference level was set to the Nash equilibrium

and the starting value for the tie was set to zero (𝛼𝑖𝑗0= 0) compared to other static and

dynamic specifications of the reference point (see Pelloux et al., 2015). For Rand et al. (2009), multiple static and dynamic specifications of the reference point for the impulse function are

considered. Moreover, both the specification where an initial tie is set to zero (𝛼𝑖𝑗0 = 0) and

one (𝛼𝑖𝑗0 = 1) are considered. Appendix C show the group estimates of these different model

specifications. Findings show that a reference level of 10 has the lowest log-likelihood and Akaike Information Criteria (AIC) implying that this model best fits the data. However, the

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specification with a reference level of zero (Nash Equilibrium) appear to perform better in forecasting contribution levels (lowest root mean square error). Since, both the data on Sonnemans et al (2006) and van Dijk et al. (2002) indicate that the reference level set equal to the Nash equilibrium shows the best performance, I will also use the Nash equilibrium as a reference level for Rand et al. (2002). Furthermore, I assume that the subjects starts with a

null tie (𝛼𝑖𝑗0= 0) indicating that subjects start with self-centred preferences that might evolve

during the interaction. Appendix C shows that when the initial tie is set to one (𝛼𝑖𝑗0 = 1) only

slightly change the estimates since this value only affects the first few rounds of the game.

Table 1: Group level estimation

Parameters Sonnemans et al. (2006) (Std Err)

Rand et al. (2009) (Std Err)

Van Dijk et al. (2002) (Std Err) 𝜃 4.7059* 7.9429* 8.7484* (0.3532) (0.1481) (0.4139) 𝛿1 0.2494* 0.1599* 0.5852* (0.0473) (0.0337) (0.0365) 𝛿2 0.1363* 0.0405* 0.0863* (0.0121) (0.0017) (0.0074) Log-Likelihood AIC Wald 𝝌𝟐 -3716.590 4.1513 9278.110* -8641.243 5.4027 17106* -1835.170 3.8698 12501.27* * p<0.01

Table 1 present the results of group level estimations. For all three datasets, the parameters are significant at the 1 percent level; the model significantly describe subject’s choices. Results on the log-likelihood and AIC indicate that the quality of fit of our three samples is the highest for van Dijk (2002) and the lowest for Rand et al. (2009). Concerning the two nonlinear public good games (Sonnemans et al, 2006 and van Dijk et al. 2002), we can

notice that the tie persistence parameter (𝛿1) of Sonnemans et al. (2006) takes a lower value.

This indicate that the effect of history of play on the decision is shorter. Meanwhile, the tie

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the emotional impulse have the greatest instant influence on tie formation. A relative low tie persistence parameter and a relative high tie proneness parameter could be because subjects experience a more powerful immediate reciprocity during the game. An obvious reason could be that it is harder to keep track of all the subjects’ streams of contributions in groups compared to pairs. This causes that subjects puts more emphasize on the immediate history of play. Only, the findings of Rand et al. (2009) contradicts that subjects in groups base their behavior more on the immediate history of play. Both the tie proneness as the tie persistence parameters have the lowest values for our three samples. This may be because subjects played a linear public good game where both the Nash equilibrium as the Pareto optimum are at the edges of the action space. As seen in appendix B, 60 percent of the choices made are on these edges, which could induce that subjects put more emphasis both on the self-centered as the full cooperation strategy.

Some important inferences of the estimation results should be addressed before we move to the behavioral forecasts section. First, for all of our three samples, the parameter

𝛿1 is significantly different from zero implying that players consider the history of play in

making their contribution decisions. Counterpart’s behavior significantly influences contribution behavior for multiple rounds contradicting most reciprocity models that assume that individuals would only adapt their behavior based on the last round of play. Additionally,

the parameter 𝛿2 is also significantly different from zero implying that the emotional

component of the model influence contribution decisions. Moreover, because the tie

proneness parameter (𝛿2) is significantly different from zero implies that the model of selfish

preferences is rejected.

3.4 B

Constructing forecasts on subjects’ contributions to the public good is the main task of this section. In order to scrutinize the quality of fit, I start with using the estimated parameters of each sample to make individual forecasts on the same dataset. In other words, the estimated parameters are used to check whether the model is able to forecast the period-to-period trajectory of contributions. Then, I will present the outcomes of out-of-sample forecasts where I use the parameter estimates of one of the samples to forecast behavior in the other two.

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Some assumptions are required in order to make forecasts on individual behavior. In

the first round, the social ties is assumed to be zero (𝑎𝑖𝑗1 = 0) and thus all subjects start with

selfish preferences indicating that they always contribute the standard Nash equilibrium. Therefore, the social tie formation starts from period two onwards; thus forecasts on individual behavior also starts from period two. Another assumption is that the behavior of the counterparts is taken as given. Generating forecasts are founded on this pre-determined behavior. Forecasted contributions of each subject, thus, are the one that is most probable according to the model (see equation 4).

Figure 2: Behavioral forecasts (selected interactions)

Sonnemans et al. (2006)

Rand et al. (2009)

van Dijk et al. (2002)

Figure 2 shows forecasts, using group level estimates, on some illustrative interactions while appendix D presents predications on all of the interactions for each of the three samples. In each of these figures, G (P) denotes groups (pairs) numbers followed by the players (P) number within the group. The data of Sonnemans et al. (2006) shows that for a majority of groups, the model appears to forecast behavior quite well. In groups with stable behavior over multiple rounds, the quality of the forecasted contribution is high (for example G2, G6, G8, and G12). For groups displaying more complex contribution dynamics, the model either could forecast period-to-period changes quite well (like G6 depicted above) or it just forecast the general trend of behavior (G5, G7 and G11).

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For Rand et al. (2009) the model appears to do a relative poor job in forecasting contribution levels. The model seems to fail in making qualitative forecasts on most groups that exhibited chaotic behavior and where coordination fails (for example G1, G6, G7, G11 and G16). For some other chaotic groups, period-to-period changes on contribution are captured well by the model (see for example G2 in figure 2).

The figure on van Dijk et al. (2002) shows us that forecasts on a huge majority of pairs are nearly identical to real contribution levels, especially when there is coordination between players (see for example; P5, P9, P10, P11 and P16). When contribution dynamics are more complex, the model seems to captures period-to-period variations better compared to the other two datasets. Even though for some pairs the model is just been able to track the overall trend of behavior (see P8 and P15 in figure 2).

Testing the quality of fit for forecasts on contributions using figures is a good place to start but a more formal approach is required. In order to delve deeper into testing the forecasting power of the model, I will display both within as out-of-sample forecasts. In other words, the parameter estimates (see table 1) of each of our three samples is used to make forecasts within their own sample and in the two remaining ones. To test the quality of fits, I computed the root mean squared error (RMSE) and the mean absolute error (MAE), which reflects the difference between the actual contribution and the contributions forecasted by the model. The lower the values of the RMSE’s and the MAE’s are, the higher is the forecasting power of the model.

Table 2 (see next page) reports the RMSE and the MAE for all possible combinations. It is immediately clear that, as already argued earlier, the model makes the best within sample behavioral forecasts for the dataset of van Dijk et al. (2002). The value of the RMSE is just below 2, which indicates that the model performs slightly better than Sonnemans et al. (2006). The result of Rand et al. (2009) suggests that the model has the worst quality of fit (RMSE of 7.81). Moving from within- to out-of-sample forecasts, the decrease in forecasting performance is relatively low. The highest increase in the RMSE is when forecasting out-of-sample involves the Rand et al. (2009) data. The RMSE statistic rises by 30 percent using the estimates of Sonnemans et al. (2006) and 35 percent using the estimates of van Dijk et al. (2002).

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Altogether, two interesting implications should be noticed. Firstly, it appears that model performance is better for non-linear public good games compared to linear public good games. Secondly, the model could better forecast contribution behavior when the group composition is relatively small; a two-player game performs better than groups consisting of 4 players. This may be because keeping track of all the players’ streams of contributions is more challenging in bigger groups.

Table 2: RMSE and MAE based on group-level estimates

Predicted dataset Sonnemans et al. (2006)

Rand et al. (2009)

Van Dijk et al. (2002)

Within sample predictions

2.8929 7.8071 1.9984

(1.9400) (4.7969) (1.3070)

Out of sample predictions Set of parameters estimates used

for prediction:

Sonnemans et al. (2006) - 10.1431 2.0809

(6.8841) (1.3125)

Rand et al. (2009) 2.7864 - 2.5465

(2.3986) (2.0724)

Van Dijk et al. (2002) 2.9972 10.4809 -

(2.0219) (7.2566)

Note: Mean average errors are in parentheses

4 A

DDITIONAL ANALYSIS

This section will delve deeper into the examination on the data in order to provide a more thorough understanding of the model’s performance. Firstly, the performance of the model at the individual level will be examined. Then, the performance of the social ties model will be compared to an inequity aversion model and a fixed social preference model. The last

26

subsection is devoted to the analysis of a modified social ties model, which allow for “group” ties. This dataset also allow to examine the forward-looking model.

4.1 E

In section 3.3 and 3.4, the performance of the model was analysed using group-level estimates. Now it is time to estimate the model for each subject separately. This gives us the opportunity to check the heterogeneity of the estimated parameters

Model estimation on a sample with only 25 to 50 observations might be unfavourable to the estimation accuracy but it could still be informative. In contrast to group estimation, individual estimation could provide a valuable insight into the distributions of each parameter. The distributions of the relative frequencies for all three parameters are displayed in appendix E. From these tables, I had to eliminate findings on 48 players since the estimation did not converge or the parameters exhibit extreme values due to an absence of variability of contribution behavior. Nevertheless, data on each parameter is available for 109 subjects. For

the tie persistence parameter (𝛿1), the distribution seems to be uniform, except for values

close to zero. Apparently, a significant amount of individuals (around 35 percent had a tie persistence below 0.02) does not consider the history of play in making contribution choices;

they appear to display only instantaneous reciprocity. The mean of the 𝛿1 estimates is 0.33,

which is more or less the same than the average of the three group estimates. The tie

proneness (𝛿2) parameter distribution reveals us that around 85 percent of the values are

between 0 and 0.2 which seem quite low. Nonetheless, individual estimation suggests that for

only 10 players the tie proneness parameter is zero (𝛿2 = 0) meaning that they do not develop

a social tie. At last, I could utilize the individual parameters to examine if there are specific interactions among them. To accomplish this, I derived the Pearson product-moment correlation coefficient between individual parameters. A negative and significant correlation

( 𝑟 = −0.2447𝑎𝑛𝑑 𝑝 𝑣𝑎𝑙𝑢𝑒 = 0.0103 ) between 𝛿1 and 𝛿2 is computed indicating that a

greater impact on the social tie goes hand in hand with a shorter lasting impact of the history of play.

4.2 S

In this subsection, I am going to compare the performance of the social ties model to several behavioral models. As already argued in section 3.3, behavioral models like the standard

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model of selfish preferences and most reciprocity models are already rejected. This leaves us with two other behavioral models, namely, an inequity aversion model and a fixed social preferences model.

In the papers of Fehr and Schmidt (1999) and Bolton and Ockenfelds (2000), the writers argued that individuals tends to have inequity aversion preferences meaning that they don’t “like” income differences. The purpose of the model was to explain behavior in one-shot games, though, it seems captivating to scrutinize how this model will perform in a dynamic setting. Interestingly, as already argued in section 2, the social ties model is able to describe inequity aversion preferences, that is when the own contribution will be chosen as the reference contribution in the impulse function (see equation 3). The inequity aversion model that I will estimate is slightly different and have the following expected utility function:

𝑈𝑖𝑘𝑡 = 𝑙𝑛(𝑃𝑖𝑘𝑡) − 𝛼𝑖∙ ∑ max(ln (𝑃𝑗𝑘𝑡) − 𝑙𝑛(𝑃𝑖𝑘𝑡); 0) 𝑛 𝑗 − 𝛽𝑖∙ ∑ max(ln (𝑃𝑖𝑘𝑡) − 𝑙𝑛(𝑃𝑗𝑘𝑡); 0) 𝑛 𝑗

where 𝑃𝑖𝑘𝑡(𝑃𝑗𝑘𝑡) is the expected payoff of subject 𝑖 (𝑗) in period 𝑡 for a contribution level 𝑘,

given the expectation 𝑖 has on 𝑗′𝑠 contribution. The parameter 𝑎𝑖 signifies disadvantageous

inequity aversion while the parameter 𝛽𝑖 denotes advantageous inequity aversion. Compared

to the myopic social ties model, the expected contribution does affect the ranking of the utility function. Since there is no data available on the expectations that players form on their counterpart’s contribution, I will use the counterpart’s contribution of the previous period as the expected contribution in period 𝑡.

Table 3 (see next page) reports the parameter estimates, the log-likelihood, the Akaike information criteria and the root mean squared error of the model. Clearly, the estimated parameters for the samples of Sonnemans et al. (2006) and van Dijk et al (2002) are insignificant while for van Dijk et al. (2002) two parameters are significant at the 5 percent level. Furthermore, the table shows us that for each of our three samples the value of β (advantageous inequality) is higher than the value of α (disadvantageous inequality). This result is rather strange because it means that for a given payoff, an individual would preferably earn less, compared to more, than his counterpart. Additionally, it seems that individuals are willing to remunerate more than one monetary unit to reduce advantageous inequality by one monetary unit (because β is larger than one), which is a very odd result. Apparently, the

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empirical findings reflects that the lack of grasps of the model on the data is noteworthy. Moreover, the lack of grasp is also highlighted by the relative low value of θ. The closer these values tends to zero, the more random are the forecasted choices. Overall, we can straightforwardly conclude that inequity aversion cannot explain the interaction dynamics in linear/nonlinear public good games.

Table 3: Inequality aversion model estimation

Parameters Sonnemans et al. (2006) (Std Err)

Rand et all. (2009) (Std Err)

Van Dijk et al. (2002) (Std Err) 𝜃 0.8185 (0.8040) 0.5818 (1.9337) 1.7509* (0.8066) α 2.2067 (2.1463) 2.8229 (9.1831) 1.5678 (0.8296) 𝛽 4.0491 (3.5676) 3.0327 (8.5634) 2.7267* (1.0715) Log-Likelihood AIC RMSE (MAE) -3309.694 3.8165 2.6302 (1.6717) -8423.988 5.3744 7.0858 (4.2149) -1688.942 3.7104 2.2002 (1.2971) Note: *p<0.05

Now, the performance of a simple fixed social preferences model will be compared to the social ties model. The utility function of this fixed social preferences model has the following form:

𝑈_𝑖𝑡 = ln(𝑃_𝑖𝑘𝑡) + 𝛼𝑖 ∙ ∑ ln (𝑃𝑗𝑘𝑡)

𝑁 𝑗=1

where 𝑃𝑖𝑘𝑡 (𝑃𝑗𝑘𝑡) is the expected payoff of subject 𝑖 (𝑗) for a contribution level 𝑘 in period 𝑡.

Again, I assume adaptive expectation formation on contribution; the expectation of player 𝑖 about his counterparts’ contribution is the contribution of the previous round. This model has two alternations compared to the social ties model (see equation 1), namely, the parameter 𝑎 does not depend on 𝑗 and 𝑡. Each player has generalized social and time invariant feelings concerning all other players.

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Table 4: Fixed social preference model estimation

Parameters Sonnemans et al. (2006) (Std Err)

Rand et al. (2009) (Std Err)

Van dijk et al. (2002) (Std Err) 𝜃 2.4841* (0.4106) 10.1995* (0.3388) 5.3765* (0.4741) α 0.6534* 0.4385* 0.5571* (0.0720) (0.0055) (0.0295) Log-Likelihood AIC RMSE (MAE) -3991.672 4.6010 3.5214 (2.5179) -9164.665 5.8461 6.7169 (5.0198) -2015.819 4.4250 2.4778 (2.0779) Note: *p<0.01

Table 4 present the estimation results for the fixed social ties model. All the parameter estimates are significant at the one percent level. Furthermore, the estimated values of the parameter α lies in the credible range of 0 ≤ α ≤ 1. It seems that the fixed social preferences model is able to explain behavior for our three samples though for Sonnemans et al. (2006) and van Dijk et al. (2002) both the AIC and the RMSE statistics are higher than the social ties model (see table 1 and 2). This indicate that the quality of fit on the data for the dynamic social ties model is higher. Remarkably, the relative performance of the fixed social preference model compared to the social ties model seems to be mixed for the Rand et al. (2009) data. The computed AIC indicate that the quality of fit is better for the social ties model while the RMSE suggest that the forecasting performance is better for the fixed social preferences model. After careful contemplation, it is reasonable to conclude that the performance of the social ties model for nonlinear public good games is slightly better than the fixed social preferences model.

4.3 T

“

”

Until now, social ties towards specific other individuals were explored but it is not unreasonable to believe that individuals could form an universal affective social tie towards a