Elster (1996) argues that social norms play a role in the generation of emotions. Indeed, emotions are regulated by social norms. Appraisal theory (Frijda, 1986; Lazarus, 1991) explains the role of social norms in the generation and regulation of emotions.

The theory argues that emotion arises from the appraisal process and from the subsequent coping process. During the appraisal process, a person assesses his/her relationship with the environment. Several so-called appraisal variables are individually calculated. The variables serve as an intermediate description of the person-environment relationship. Coping determines how people respond to the appraised significance of events. Problem-focused coping strategies attempt to change the environment, whereas emotion-focused coping strategies alter the mental stance.

The effect of both strategies is a change in a person’s interpretation of his/her relationship with the environment. Staller and Petta (2001) integrated an appraisal process into their Tabasco architecture. By using SOAR as a vehicle, Gratch et al. (2006) provided a model of the processes underlying cognitive appraisal. Both approaches are based on a rule-based paradigm. The developers admit that the largest deficiency of their model concerns the impoverished social reasoning.

Jealousy has not been modeled explicitly by other researchers. They have focused on other emotions so far (Picard, 1997; Castelfranchi, 2000).

We model jealousy as one single variable by using a game theoretical approach. We assume that this single variable has been generated either from an antecedent appraisal process or it is given a value which can continuously change in a simulation process. Our agents only comprise one learning function. They are not modeling mind on an individualistic micro-level as demanded by Castelfranchi (2006). However, in Chap. 4, we do present a generic layered incremental architecture, so that the impact of several parallel emerging emotions to collective behavior can be simulated. Game theory, as explained in one of the previous chapters, is well suited to model the intra-relationships between agents.

Emotions play a big role during cooperation. Jealousy seems to play a special role (Barachini, 2015; Brockner, 1988; Pelham & Swann, 1989; Rydell & Bringle, 2007). Just during the COVID-19 pandemic crisis where common goods, such as vaccines, are rare for some parties, we witness a struggle between different social groups and even nations. Jealousy is the main driver in such socioeconomic conflicts. We investigate the impact of jealousy on cooperation, and we simulate the deterministic evolutionary dynamics of populations which are arranged in a lattice.

We concentrate our effort on the question how jealous agents can invade co-operators and vice versa. In this context, we are interested about the impact of rewards to the behavior of agents. We show how complex emergent patterns are generated in certain parameter regions, when spatial cooperation in finite populations is applied.

1 The Notion of Jealousy

Cooperation through information sharing is substantially influenced by an individual’s level of jealousy. This concept so far has been discussed most often in the context of close relationships. This emotion also plays an important role in the work context. The loss or merely the perceived threat of a loss involves the perception of a rival’s intrusion. This rival has the potential to reduce one’s self-esteem or undermine a valued relationship. It is hence a characteristic of jealousy that it is triadic, involving the focal individual, the rival, and the valued target person.

Rydell and Bringle (2007) distinguish two types of jealousy. Reactive jealousy refers to the emotional components of jealousy. Suspicious jealousy contains cognitive and behavioral components. They argue that manifestations and antecedents differ for these two types. Specifically, reactive jealousy that occurs after a major jealousy-evoking event and is thus more closely related to exogenous factors appears to be positively related to trust and negatively related to chronic jealousy. However, suspicious jealousy that occurs before major jealousy-evoking events have occurred is more closely related to endogenous factors and is positively associated with insecurity and chronic jealousy and negatively associated with self-esteem, i.e., an individual’s self-evaluation of its own competencies (Rosenberg, 1965), including an affective component of liking itself (Pelham & Swann, 1989). It has been found to positively influence employee attitudes, such as job satisfaction, motivation, and performance (Brockner, 1988).

We assume that suspicious jealousy plays an important role in collaboration by mediating the relationship between self-esteem and the willingness to share knowledge with a person who is a potential rival such that the less self-esteem an individual has, the more they may show suspicious jealousy toward potential rivals, and the less they will be willing to share their knowledge with those. Reactive jealousy, in turn, may negatively influence cooperative behavior only after the occurrence of an event that has evoked an individual’s jealousy.

In our simulations, we investigate suspicious jealousy for two reasons. Firstly, it is easy to implement because we use simple matrices for payoff calculations, and secondly, reactive jealousy would impose much more complexity, since we would have to construct specific artificial social injustice examples. However, in real life, reactive jealousy is much better identifiable than suspicious jealousy. Reactive jealousy can easily be detected due to its exogenous nature. The detection of suspicious jealousy needs deeper psychological investigations.

2 Cooperation Methods and Modeling: State of the Art

Cooperation implies that a donor pays a cost and a recipient gets a benefit in the short term. In the long term, the relation should be balanced. However, cooperation is always vulnerable to exploitation by defectors. Researchers identified well-established cooperation rules, such as kin selection and related schemes (Hamilton, 1964a, 1964b, 1996, 2001), graph and group selection and related schemes (Ohtsuki & Iwasa, 2004), indirect reciprocity, and spatial evolutionary game theory. Among others, punishment and cooperation were investigated by Sigmund et al. (2001) with prisoner’s dilemma games.

In practice, most of the updating procedures in prisoner’s dilemma games (Rapoport, 1999) use short memory strategies that utilize only the recent history of previous interactions. Li and Kendall (2014) prove that longer memory strategies outperform shorter memory strategies statistically in the sense of evolutionary stability. In another recent publication, Lu et al. (2018) show that long memory effects can change the cooperation behavior in the spatial prisoner’s dilemma game. Chen et al. (2016) studied the impact of reputation on evolutionary cooperation. Both approaches apply Fermi rules for the updating procedures and belong to the class of long-term memory strategies.

However, Conte et al. (1999) proved that prisoner’s dilemma games (Axelrod, 1984) cannot account for a theory of cooperation, since bargaining is not possible. Therefore, in a previous research, we applied an inter-disciplinary approach based on the business transaction theory of Barachini (2007), which will be explained in Chap. 5 in detail. We underpin the theory in a broad industry survey (150 European companies, 3 countries) (Barachini, 2009). Similar to Akerlof (1970), Spence (1973), and Stiglitz (1975), the theory is based on the idea that information is asymmetric in its nature.

Although Conte et al. criticized the approach of using prisoner’s dilemma games because bargaining is not possible, we apply a game theoretical approach based on a short-term memory strategy. We compare this deterministic spatial approach with the stochastic one in Chap. 5.

3 Methodology

In the stochastic approach, members of a population meet each other randomly. In the spatial approach, members of a population are arranged in a two- or higher-dimensional array. In each round, every individual agent, represented by a cell, plays a game with its immediate neighbors. After the game, each cell is occupied by its original owner or by one of the neighbors, depending on who scored the highest payoff in that round.

On a spatial multi-dimensional grid, each agent occupies a position on the grid, the cell, and interacts with all of its neighbors. In normal life, an agent can select from an infinite number of strategies. In a deterministic spatial game, each agent adopts the strategy with the highest payoff in its neighborhood. The agents are updated in synchrony.

The fate of a cell depends on its own strategy, the strategy of its neighbors, and the strategy of their neighbors. If we consider a three-dimensional cube containing 5 × 5 mini cubes per surface, and we observe the inner cube, then the fate of this inner cube depends on its own strategy, the strategies of its 26 neighbors (3*3*3−1), and the strategies of their 98 neighbors (the outer layer of the 5*5*5 cube). Thus, 125 cells in total determine what will happen to the inner cell of a cube.

To make it easier for the reader, we show experiments with two-dimensional arrays only. In this case, each cell has eight direct neighbors because we apply the Moore neighborhood. Thus, 25 cells in total determine what will happen to a cell. To repeat, an agent represented by a cell will retain its current strategy, if it has a higher payoff than all of its neighbors. Otherwise, the agent will adopt the strategy of that neighbor with the highest payoff. We have to admit that in the simulated environment, we have perfect knowledge about the characteristics of an agent. In real life, that sort of information is difficult to discover. For the software implementation, we used an external module known as “VirtualLabs,” which is publicly available at the University of Vienna (https://www.univie.ac.at/virtuallabs/).

In our examples, we simulate agents with only three characteristics. That means an agent can choose between three strategies. We have defectors (D), co-operators (C), and conditional co-operators (CC). The latter represent jealous co-operators. The level of jealousy is determined by α respectively by γ of the payoff matrix in Fig. 6.1. Both values vary between 0 and 1. The higher both values are, the higher the probability of cooperation, and the lower the jealousy factor. Consequently, an agent can choose between the three strategies C, D, and CC. The numbers in the following payoff matrices preceding the colon represent the payoff given to the line player, and the numbers after the colon represent the payoff given to the column player.

The payoff matrix of Fig. 7.1 can be interpreted as follows: If two co-operators interact, then both receive one point. If a defector meets a co-operator, the defector gets a payoff a > 1, and the co-operator gets the payoff 0. The interaction between two defectors leads to a very small positive payoff ε, for both. If a co-operator meets a conditional co-operator, then the co-operator gets payoffs between 0 and 1, and the conditional co-operator gets payoffs between 1 and a, with a >1. If a conditional co-operator meets a defector, then the conditional co-operator gets payoffs between 0 and ε, and the defector gets payoffs between ε and a. When two conditional co-operators meet, then they both have a payoff range in between 0 and a, a > 1.

Fig. 7.1
figure 1

Payoff matrix

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The distribution of the rewards of the agents correlates to the prisoner’s dilemma in which two agents try to maximize their payoffs by cooperating with or betraying the other agent (Rapoport, 1999). As discovered by the biologists Maynard-Smith and Price (1973), such distributions are also a common phenomenon in the logic of animal conflicts.

In case of setting α to 1 or 0 and setting γ to 1 or 0, we get the simple matrix consisting only of co-operators (C) and defectors (D). Since we simulate jealousy by the parameters α and γ, more jealousy means less cooperation, we have to keep the matrix above, but for simplicity we choose to set ε → 0. In fact, this simplification of the matrix in Fig. 7.1 yields the matrix of Fig. 7.2, which represents a prisoner’s dilemma. To explore different evolutions of populations, we vary the parameters a, α, and γ. We perform our experiments by investigating suspicious jealousy only, because we would have to construct specific artificial social injustice examples, so that reactive jealousy could be investigated. This would make our simulations even more complex.

Fig. 7.2
figure 2

Simplified payoff matrix

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As explained, modeling of suspicious jealousy in the deterministic spatial approach is performed by varying the parameters α and γ. Since their values vary in between 0 and 1 with a > 1, this is a large mathematical space, and our simulation results fill more than thousand books. Therefore, we present an extract of all possible outcomes by discussing some interesting parameter regions, especially those which are represented by an iterated prisoner’s dilemma (IPD). In the IPD game, two agents choose their mutual strategy repeatedly. They also have memory of their previous behaviors and the behaviors of their opponents. The payoff matrix of Fig. 7.2 yields an IPD only, if certain conditions are fulfilled.

As explained by Rapoport (1999) and Chong et al. (2007), the reward of the co-operators, in our example the value 1, needs to be higher than half of the sum of the sucker’s payoff and of the temptation value. In other words, “1 > (a + 0)/2” and “1 > (α + (1−α) a + α)/2.” Thus, a < 2 and α < 1, respectively γ < 1. These additional conditions exclude that the alternate strategy combinations “CD, DC, CD, DC...” yield better results than continuous cooperation. These conditions are set to prevent any incentive to alternate between cooperation and defection. In the subsequent simulations, we take these conditions into consideration. Therefore, our examples are of type IPD.

Our update rules are based on deterministic dynamics. Each cell in the lattice is given to whoever has the highest payoff in the direct neighborhood, and all cells are updated in synchrony. This approach allows us to study the properties of deterministic spatial dynamics in discrete time. Although the fate of each cell is deterministic, the overall population dynamics can be richer than in stochastic approaches.

We did not investigate asynchronous updating procedures, where if one cell is chosen at random, its own payoff and the payoffs of all neighbors are determined. Then the individual cell is updated. Asynchronous updating means overlapping generations; synchronous updating means no overlapping generations. Asynchronous updating introduces random choice. The synchronous approach will exactly reproduce the population dynamics when repeated with the same initial conditions.

We are investigating the influence of jealousy on cooperation in a spatial IPD by using a short memory updating procedure without statistical means. Subsequently, we show under which conditions co-operators can invade defectors and vice versa. Individual agents of the population are placed in a two-dimensional lattice. Throughout the rest of part II of this book, we apply the matrix of Fig. 7.2 in order to calculate the fate of the cells.

4 Co-operators Invaded by Defectors

We investigate the conditions for a single defector to invade a population of co-operators. Note that a co-operator could be of type C (unconditional cooperation) or CC (conditional cooperation, based on the level of suspicious jealousy).

Figure 7.3 shows the conditions for a single defector to invade a population of jealous co-operators. We set the level of suspicious jealousy to 10%, which means the cooperation levels of α and γ are set to 0.9. Based on the payoff matrix from Fig. 7.2, we can evaluate the payoff of the single defector (red cell). The single defector is surrounded by eight jealous co-operators (blue cells). Therefore, the payoff of the defector is eight times the D-CC combination (8 × γ × a) plus one D-D combination. Since the D-D combination yields 0, we get 8 × 0.9 × a = 7.2 a. Therefore, the red cell has payoff 7.2 a.

Fig. 7.3
figure 3

Payoffs of a single defector cell and of the surrounded jealous co-operator cells

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The value of the blue cell above the red cell is calculated as follows: the blue cell above the red cell is a jealous co-operator, and it is surrounded by seven other blue cells and one red cell. Therefore, the payoff of the jealous co-operator is seven times the CC-CC combination [7 × (0.9 × 0.9 + 0.1 × 0.9 × a)] plus one CC-D combination. Since the CC-D combination yields 0, we get “5.67 + 0.63 a” for this blue cell. Similarly, all the other cells of Fig. 7.3 are filled in by using the payoff matrix of Fig. 7.2.

If “7.2 a > 6.48 + 0.72 a,” which means a > 1, then the defector will take over all its neighbor cells. This will then yield Fig. 7.4.

Fig. 7.4
figure 4

9D square

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In Fig. 7.4, the central defector has a payoff of zero, and the rest is evaluated using our payoff matrix from Fig. 7.2.

  • The 9D square defector cells will turn into a single defector again, if “4.86 + 0.54 a > 4.5 a.” Thus, if “1.22 > a,” then Fig. 7.4 will again turn into Fig. 7.3.

  • The 9D square stays stable, if “6.48 + 0.32 a > 4.5 a > 5.67 + 0.63 a.” Thus, if “1.71 > a > 1.46,” then the configuration of Fig. 7.4 will not change.

  • There will be further growth of the 9D square, if “4.5 a > 6.48 + 0.32 a.” Thus, if “a > 1.55,” we have further linear growth of the 9D square to a 25D square, and so on, until eternity.

  • Fig. 7.4 will turn into the cross of Fig. 7.5, if “5.67 + 0.63 a > 4.5 a > 4.86 + 0.54 a.” Thus, if “1.46 > a > 1.22,” we get a cross, which will turn again into a single defector of Fig. 7.3. In this case, we have a period two oscillator.

Fig. 7.5
figure 5

Cross

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The above example is only one of many million, and it shows one deterministic pattern. Many more deterministic patterns have been observed. We also simulated jealous co-operators invading defectors. It is strongly dependent on the value of a (the reward) and the values of γ and α, whether the world is dominated by defectors or by co-operators or whether there is a dynamic balance. In certain parameter regions, the abundance of jealous co-operators depends on the starting configuration of the grid. In other parameter regions, the frequency of co-operators is almost constant, especially in very large arrays.

The simulations are all based on finite automata. There is one very interesting constellation, where dynamic kaleidoscopes and fractals are generated. The initial conditions for these kaleidoscopes are shown in Fig. 7.6.

Fig. 7.6
figure 6

Conditions for fractals and kaleidoscopes

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  • If “4.5 a > 6.48 + 0.72 a,” then defectors win at corners. That means a > 1.71.

  • If “2.7 a < 4.05 + 0.45 a,” then jealous co-operators win along lines. That means a < 1.8.

  • Thus, if “1.8 > a > 1.71,” then we have a clash of extremes.

Figures 7.7 and 7.8 show the fractals after 124 and after 128 rounds for the parameter a, using a jealousy factor of 10% for each co-operator on a 128 × 128 spatial grid. We started with a single defector.

Fig. 7.7
figure 7

Kaleidoscope after 124 update rounds

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Fig. 7.8
figure 8

Kaleidoscope after 128 update rounds

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Only by coloring the grid, the pattern can be optically observed:

  • Yellow is a defector that was a jealous co-operator in the previous round.

  • Green is a jealous co-operator that was a defector in the previous round.

  • Red is a defector that was a defector in the previous round.

  • Blue is a jealous co-operator that was a jealous co-operator in the previous round.

In Figs. 7.7 and 7.8, the fractals repeat themselves at the power of 2. It can be observed that the number of defectors converges toward 2/3. This can be observed after many thousand rounds. The jealous co-operators cannot be totally invaded by defectors in this case.

5 Defectors Invaded by Co-operators

For “1.8 > a > 1.71,” we show in Fig. 7.9 a fractal after 260 rounds, using a jealousy factor of 50% for each co-operator. We used again a 128 × 128 spatial grid, representing a population size of 16,384 cells. We started with a 9D block of co-operators within defectors:

  • White is a defector that was a jealous co-operator in the previous round.

  • Blue is a jealous co-operator that was a defector in the previous round.

  • Red is a defector that was a defector in the previous round.

  • Dark is a jealous co-operator that was a jealous co-operator in the previous round.

Fig. 7.9
figure 9

Fractal after 256 update rounds

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In Fig. 7.9, the fractals repeat themselves at the power of 2. It can be observed that the number of defectors converges toward 2/3. The defectors cannot be invaded by jealous co-operators. With the given jealousy factor of 50% for each co-operator and a reward range between 1.71 and 1.80, defectors cannot be invaded. Thus, the strategy of the defectors is evolutionary stable. In terms of Maynard-Smith, both special cases of Figs. 7.8 and 7.9 represent a cyclic evolutionary stable strategy (ESS).

6 Findings

In this chapter, we investigated update procedures based on deterministic dynamics, in a spatial grid. If update rules are determined by deterministic dynamics, then we can even produce fractals and kaleidoscopes, which repeat themselves at the power of 2 and where all types of cells can find their place. Thus, jealous cells are not eliminated under specific parameter conditions. This is a special case of equilibrium.

We conclude that under special constellations of the cells, the presented spatial game theoretical approach has a high variety in dynamical behaviors. This variety mainly depends on the payoff function, which is determined by the intensities of the reward and of suspicious jealousy, and a very specific constellation of edge and line cells in the grid. Whether less or more reward is needed depends on the jealousy level. The parameter regions are shifted accordingly.

If the level of jealousy would be determinable in real life, then we could make estimations of the collective behavior of groups, and even then, it could be chaotic. But is a chaotic behavior dangerous for groups and their managers? Yes and no! Kaleidoscopes indeed make the impression of chaos, but in the long run, the numbers of co-operators and defectors stay balanced. In spite that jealous co-operators are switching continuously from one cell to the next, the equilibrium of the group might not be endangered. Therefore, the other parameter regions could be more relevant for real-life investigations, since they can tip populations completely, as it was explained in Fig. 7.4 for the 9D square, when the reward is higher than 1.55.

We simulated suspicious jealousy by simply modifying the two parameters α and γ. The level of jealousy and the amount of the reward influence social behavior and group structures, as it has been shown in the colored pictures.

We suspect that behavior of humans in groups is still more complex, since beside egoism, trust, and jealousy, additional emotions, such as joy, fear, surprise, anger, or anticipation, need to be considered. It is very hard to control emotions during experiments with humans and to identify root-cause effects triggered by these emotions. Moreover, in reality, it is probably impossible to identify people and measure their exact levels of suspicious jealousy. This type of jealousy can be regarded as a typical disease pattern. In order to establish unambiguous psychological experiments with humans, all these facts need to be considered.

In real life, a mixture of stochastic and spatial interactions occurs. In highly automated companies, with fixed defined processes and few customer contacts, spatial internal interaction in the production line is probably dominant. In random crowds, stochastic interaction is dominant. If, for spatial interaction, we decide to equip artificial agents with emotions such as jealousy, then the dynamic behavior of the crowd will correlate with the emergent presented patterns.

In the next chapter, we model agent-based stochastic interactions. We present a generic-layered architecture which is capable to model emotions such as jealousy, joy, fear, surprise, anger, anticipation, etc. in parallel.