In this chapter, we model jealousy of agent-based stochastic interactions from a very different perspective. We study the impact of jealousy to cooperation models by using dynamic artificial agents and agent populations following combined characteristics. We simulate a dynamic multi-player public goods game, and we use the Business Transaction Theory (Barachini, 2007) as a basic layer for information valuation.

Our approach enables us to simulate small to large populations in a short time, and it enables flexibility in changing agent characteristics. The technical framework is based on the Eclipse development environment. The artificial agents are programmed in Java.

Since we simulate human properties, we need to understand the basic characteristics of cooperation. In knowledge management, exchange and sharing are the activities that characterize knowledge-driven collectives. More precisely, when we talk about exchange and sharing, we talk about information transfer between humans.

Since repeated prisoner’s dilemma games fall short for modeling truly cooperative behavior (Conte et al., 1999) because the possibility of bargaining is inexistent, we use, in addition to evolutionary game theory, artificial agents for social simulation. The approach combines the advantages of several disciplines, namely, artificial intelligence, psychology, and economy.

1 The Economic Perspective of Communication

In cooperation, a donor pays a cost and a recipient gets a benefit. However, cooperation is always vulnerable to exploitation by defectors. Researchers identified well-established cooperation rules, such as kin selection and related schemas (Hamilton 1964a, 1964b, 1996, 2001), graph and group selection and related schemas (Ohtsuki & Iwasa, 2004), indirect reciprocity, and spatial evolutionary game theory. Among others, punishment and cooperation were investigated by Sigmund et al. (2001). As explained above, Conte et al. (1999) proved that prisoner’s dilemma games cannot account for a theory of cooperation since bargaining is not possible.

The research results gathered from the different disciplines encourage the application of an inter-disciplinary approach. To investigate cooperation methods from an information economics perspective, we developed the Business Transaction Theory, which relates information exchange to modern portfolio theory. We underpin the theory in a broad industry survey (150 European companies, 3 countries) published in JKM (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.

Akerlof (1970) experimented with car markets and the asymmetric information situation between the seller, who knows everything about the car, and the buyer, who knows only what the seller has told him. The fact that the potential buyer is aware of his lack of information leads to the default assumption of the buyer that the used car is a “lemon” (a car of low quality). Therefore, the potential buyer will always bargain for a lower price, which drives high-quality cars out of the market so that finally only “lemons” will be offered. This process is called “adverse selection.” Akerlof observed that this effect can be weakened by factors such as “repeated sales” and “reputation,” which, however, is less true for the insurance, labor, and credit markets. Spence’s job market studies (Spence, 1973) yielded similar results.

Stiglitz took the work of Spence as a basis and focused especially on the insurance market. He introduced the concept of “screening,” which is a self-selection mechanism, where agents are offered a variety of contracts. The selections done by the agents lead to a revelation of their risk level, as risk averse agents tend to choose contracts which charge lower premiums, but higher deductibles. According to Stiglitz, the Pareto-optimal outcome would be a full screening that identifies each agent’s true capabilities, but it cannot be seen as market equilibrium due to missing sustainability.

The theories of Akerlof, Spence, and Stiglitz focus on specific markets. Market participants exploit asymmetric information to gain profits. The market of the Business Transaction Theory is characterized by information itself. No specific market is necessary. Thus, it is the basic underlying information exchange mechanism between humans that is characterized by the Business Transaction Theory, independent from any specified markets. We apply this theory as the basic communication mechanism also for agents.

2 Business Transaction Theory

The Business Transaction Theory defines two types of information exchange: Type-1 is the immediate exchange of information in both directions. Thus, donor and recipient both provide information. This type of duplex information provision can be mapped to over-the-counter (OTC) businesses of banks. The difference to the well-known OTC business is that there are no intermediaries involved such as dealers and brokers because the so-called information market is generated implicitly in the minds of the parties involved.

Type-2 of information exchange is more complicated because information flow is unidirectional at first. This is the case when we offer information to individuals in the hope to get even more valuable information back some day in the future. Type-2 of information exchange can be mapped to options.

In the investment world (Sharpe et al., 1995), an option is a type of contract between two people where one person, the writer, grants the other person, the buyer, the right to buy a specific asset at a specific price within a specific time. Alternatively, the contract may grant the other person the right to sell a specific asset. The variety of contracts containing an option feature is enormous.

Type-2 of information exchange can be mapped to the call option for stocks. It gives the buyer the right to buy a specific number of shares of a specific company from the option writer at a specific purchase price at any timeFootnote 1 up to a specific date. Figures 8.1 and 8.2 show the profit and loss (P&L) graphs of a buyer and a seller. The buyer of a call option will have to pay the writer a premium in order to get the writer to sign the contract. The fair value of an option can be evaluated using the binomial option pricing model or the more recent method by Black-Scholes (in Sharpe et al., 1995):

Fig. 8.1
figure 1

P&L graph for “buy a call”

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Fig. 8.2
figure 2

P&L graph for “write a call”

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$$ \mathrm{Fair}\ \mathrm{value}=\mathrm{N}\left(\mathrm{d}1\right)\times \mathrm{Ps}-\mathrm{E}\times \mathrm{N}\left(\mathrm{d}2\right)/{\mathrm{e}}^{\mathrm{RT}} $$

where:

  • d1: (ln(Ps/E) + (R + 0.5 σ2)T)/σ × sqrt(T)

  • d2: d1 − σ × sqrt(T)

  • Ps: Current market price of underlying stock

  • E: Exercise price of option

  • N: Normal

  • R: Compound risk-free rate of return

  • σ: Risk of the underlying stock

  • sqrt: Square root

  • T: Time remaining until expiration

Figure 8.1 relates the value of a call option with an exercise price of 200 to the price of the underlying stock of expiration. If the stock price is below 200, the option will be worthless when it expires. In this case, the writer will gain (see Fig. 8.2) the premium. If the price is above 200, the option can be exercised for 200 to obtain a security with a greater value, resulting in a net gain to the option buyer that will equal the difference between the securities market price and the 200-exercise price. However, margin requirements, commission payments, and other market-making activities make the procedure more complicated in practice.

In the Business Transaction Theory, Type-2 of information exchange means that one person (the buyer) provides information and hopes to get even more valuable information in the future (see Figs. 8.1 and 8.2). The information offered to the writer has some value—the premium. The buyer hopes to get back another type of information that is at least as valuable as the premium. Thus, the underlying traded goods are not stocks, but information.

In the case of the classical stock market, the Black-Scholes formula is based on statistics, the exercise price is known, the risk of the underlying common stock can be evaluated, and the option has a well-defined expiration date.Footnote 2

In the case of information brokerage, it is more difficult to evaluate a fair price for a piece of information since we do not even know the value of the underlying good because it is an unknown piece of information which might, or might not, be offered by the writer sometime in the future. In the Black-Scholes formula, the current market price of the underlying stock can be evaluated. In contrast, no objective evaluation can be performed for information generated by humans since information is always evaluated subjectively. The evaluation function might even be similar, but due to differing context knowledge, the same piece of information may still be evaluated differently on an individual basis. Therefore, statistics as in the Black-Scholes formula cannot be applied immediately in the Business Transaction Theory, since Ps, E, R, and σ now represent subjectively assessed values.

The parameter T is indeterminable since it is not known when or even whether at all valuable information will be received in the future. As a consequence, all parameters have to be simulated in a scientific environment so that the value of the information transferred between agents could be evaluated. Additionally, the social behavior of agents reacting to information needs to be modeled.

3 Modeling Approach

We combine cognitive models of social agents with evolutionary game theory. As the basic information exchange mechanism between agents, we use business transactions. In this section, we describe the implementation of the business transaction layer, the research design for simulating jealousy, and the structure of the framework.

3.1 Implementation of the Business Transaction Layer

We simulate cooperative behaviors of finite populations by simulating the Black-Scholes parameters and by using payoff matrices as they are used in evolutionary game theory. The latter is used to evaluate certain states, i.e., goals of individual agents and crowds; the former is used to simulate the dynamical changing utility of information that is exchanged between agents. For calculating the Black-Scholes parameters, we propose the following procedures:

As explained in the previous section, the utility of information must be calculated or estimated. For communities of practice (CoPs) which are established on the web, Schmidt (2000) proposes two practical approaches in his experiments. He introduces the term “Knowledge Euro” for his approaches. One approach allows the information provider to simply fix a price before the information is delivered. The other leaves it up to the receiver to judge the value of the information. In the latter case, it is up to the receiver to pay a price on a donation basis. In this way, we also consider bargaining, which cannot be modeled in repeated prisoner’s dilemma games (Conte et al., 1999).

We propose to calculate the Black-Scholes parameters in the following way:

  • Ps: The utility of the current market price of underlying information, a future value, can be fixed by the receiver (donation basis).

  • E: The utility of the exercise price can be fixed by the provider (writer).

  • T: The time whether and when both agents meet in the future can be simulated through different models, i.e., by simulating a Poisson distribution for meetings. If no meeting takes place at time T, then the call is not exercised (information provision is not reciprocated).

  • R: The compound risk-free rate can be set to a constant parameter.

  • σ: The risk of the underlying information can be simulated in different ways.

We can simulate closed marketplaces, where prices behave according to certain functions (logarithmic, linear, scattered, etc.), or we memorize prices fixed by information producers and receivers on donation basis and calculate the variance of these historical data.

Another possibility is the direct calculation of the so-called implicit volatility. This can be achieved in the following way: We put the current market price of the information (either determined by buyer or writer) on the left-hand side of the Black-Scholes formula. Next, all the other factors, except σ, are entered on the right-hand side, and a value for σ, the only unknown variable, is found. This value denotes the implicit volatility of information.

Implicit volatility may be explained in a simple example on a monetary basis: If we assume that the risk-free rate is 6% and that a 6-month call option with an exercise price of 40 € (the writer’s estimated utility of future information) sells for 4 € (utility of buyer’s information), then the price of the underlying good is 36 € (the writer’s real utility of future information). Different estimates of σ can be plugged into the right-hand side of the Black-Scholes formula, until a value of 4 € is achieved. In this example, an estimated value of .4 (40%) for σ will result in a number for the right-hand side of the Black-Scholes formula that is equal to 4 €, the current market price of the call option (the utility of the information provided by the option buyer) that is on the left-hand side. In this example, σ can be estimated for a 6-month option not only for an exercise price of 40 € but also for higher or lower exercise prices. Following Sharpe et al. (1995), after sampling σ values over a sufficiently large range of exercise prices, we can calculate a best estimate for σ as average over these values.

By using payoff matrices similar as described in Chap. 4 and the above-described parameter calculations, the value of an individual’s payoff is dependent on the achievement of its goals which in turn can be interpreted as an indicator of its fitness. The same holds for groups.

We simulate the parameters of the Black and Scholes formula as explained above, and we use the Business Transaction Theory (BTT) as the basic evaluation mechanism for rating the information that is exchanged in between artificial agents constituting finite populations of different sizes.

One of the major goals of economic theory has been to explain how cooperation among individuals can happen in a decentralized setting. It was thought that contracts between individuals could suffice to steer their behavior. But the assumption that a contract could completely specify all relevant aspects of a relationship and could be enforced at zero costs to the exchanging parties is not applicable to many forms of cooperation.

Explicit contracts, economic institutions, and also individuals depend on incentive mechanisms involving strategic interaction. However, some research results show that incentives sometimes undermine moral sentiments (Bowles, 2008) and need to be introduced very carefully. Therefore, we concentrate our research toward incentive mechanisms and their use with respect to jealousy. Our agents are able to learn using reinforcement and evolutionary adaptation and therefore can change their behavior so that they can achieve individual or collective goals. We play several rounds with different group size and agent characteristics of a specific multi-player public goods game “establish homes” and evaluate the results with payoff matrices as described in Chap. 4.

An individual agent will exert a significant influence to the other agent’s cooperative behavior, in particular on their knowledge-sharing behavior. The more agent characteristics we have, the better we can simulate reality. However, we need to reduce complexity in order to keep our experiments tractable. Therefore, we use jealousy only in a first step. The intensity of jealousy is dynamically changed during the game playing process. The dynamic adaption depends on the success or failure to reach group goals and on incentives such as praise and blame which are optionally injected into the public goods game “establish homes” described subsequently.

3.2 Research Design for Jealousy

We model a dynamic multi-player public goods game with a finite population. From one round to the next, we monitor the agent characteristics. The agent characteristics are modified from one round to the next according to specific rules, such as haploid reproduction,Footnote 3 or according to reinforced coordination strategies.Footnote 4

Similar to the simulations in Chap. 4, we assume a population of unconditional co-operators, conditional (jealous) co-operators, and egoist agents.Footnote 5 Consider a population in which agents can live and work alone or in groups. By the fitness of an agent, we mean the expected accumulated knowledge calculated with the business transaction function (Black-Scholes) during one round minus the probability that the agent leaves the group.

Agents can also cooperate on a conditional or unconditional basis in a group, each producing an amount b at cost c. All benefits and costs are expressed in fitness units, as calculated by the business transaction function. We assume that the output of a group is shared equally by all members, so if all agents cooperate, each has a net group fitness benefit b − c > 0.

Groups consist of three types of actors. The first type, whom we call jealous co-operators, works conditionally and cooperates until a certain level of fitness is reached in their counterparts. The second type, whom we call egoists, maximizes fitness. They work and cooperate only to the extent that the expected fitness cost exceeds the expected cost of defecting. The third type, whom we call co-operators, works unconditionally, and they always cooperate (high agreeability). The agent types can change dynamically from round to round.

Agents change their type with probability 1 − ß from one round to the next. With probability ß/2, an agent takes on each of the other two types. We call ß the rate of change.Footnote 6 Also, with probability 1 − ß, egoistic agents inherit the estimate of s > 0 (the cost of being eliminated or isolated in the group because of excessive egoism) from the previous round. With probability ß, an agent in the next round is a mutant whose s is drawn from a uniform distribution on [0, 1]. Thus, s is an endogenous variable.

Jealous co-operators mutate to egoists as soon as they meet agents with sufficiently high accumulated fitness. Suppose an egoist agent defects (i.e., does not work or only gathers information) a fraction Ss of the time, so the average rate of defecting is given by S = (1 – P*fj − fc)*Ss, where fj is the fraction of the group, which consists of jealous co-operators, fc is the fraction consisting of co-operators, and P is the probability that a jealous agent cooperates. The fitness value of the group output is n*(1 − S)*b, where n is the size of the group.

Since output is shared equally, each member receives (1 − S)*b. The loss to the group from an egoist defecting is b*Ss. The fitness cost of the working function, which can be written as D(Os), where Os = 1 − Ss, is increasing and convex in its argument. Expending effort always benefits the group more than it costs the workers, so Os*b > D(Os) for Ss (0, 1]. Thus, at every level of effort, Os, working and cooperating helps the group more than it hurts the worker and co-operator.

Further, we assume that the cost of effort function is such that in the absence of jealous co-operators, members face a public goods problem (i.e., an n-player prisoner’s dilemma), in which the dominant strategy is to contribute very little or nothing. We model jealousy as follows. The fitness cost of a conditional co-operator to detect egoists is Cp > 0. A member defecting at rate Ss will not be selected for cooperation with probability fj*Ss. Egoist agents, given their individual assessment s of the cost of being ignored, and with the knowledge that there is a fraction fj of jealous agents in their group, choose a level of defecting, Ss, to maximize expected fitness. Writing down the expected fitness cost of working, FCW (Ss), as the cost of effort plus the expected cost of being ignored, plus the agent’s share in the loss of output associated with one’s own defection, we get FCW(Ss) = D(1 − Ss) + s*fj*Ss + Ss*b/n. Then, egoist agents select Ss, the value which minimizes the fitness cost of working (FCW).

The expected contribution of each group member to the group’s population in the next round is equal to the member’s fitness minus (for the egoist agent) the likelihood of defection. Different kinds of incentives may be imposed per round on individual agents or on complete groups if more than one group is simulated. However, research results show that incentives may undermine moral sentiments and therefore are producing worse group results as if they would not have been imposed (Falk & Kosfeld, 2006). Both argue that “control aversion” may be a reason that incentives degrade performance. Fehr and Rockenbach (2003) argued that even if incentives reduce the total gains of a group, their use may give the principal a sufficient large slice of the smaller pie to motivate the principal to use them.

By positive incentives, we simply mean the donation of additional scores to the agent in a multi-player public goods game, in reality praise for cooperation. An agent’s goal might be to establish a house for himself. He gets a certain amount of scores for reaching the goal. On the other hand, he is getting scores when he helps establish houses for others. The more houses established in the group, the fewer scores the agent might get for his established house, but the more scores will be distributed to the group. Negative incentivesFootnote 7 will reduce the distribution of scores to individual agents or groups.

As a variation of the experiment “establish homes,” we would even be able to impose “signals,” as defined by Spence (1973), so that agents can be distinguished by their education level.Footnote 8 According to our framework, we hope to simulate various combinations of agent characteristics.

3.3 Framework

We can simulate jealousy in isolation, subsequently with signaling and incentive injection. Inside the communication layer in between agents, we use implicit volatility. Later, we can use linear functions. Utility of information is fixed by receiver and provider. This approach shows a clear order of investigation from simple to medium complexityFootnote 9 inside the communication layer as well as in the higher level of the experimental setting. Simulations of more complex agent characteristics might be intractable.

On the highest level (Layer-3), we implement the agent characteristics agreeableness, jealousy, power, and combinations thereof. Optionally, we can impose signaling. On the second level (Layer-2), we implement knowledge payoff (fitness), cooperation readiness (conditional and unconditional) with payoff matrices, and agent-type modification functions (primitive learning). On the lowest level (Layer-1), we implement BTT (see Fig. 8.3).

Fig. 8.3
figure 3

Layer model

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In its highest complexity, six parameters (jealousy, power, agreeableness, one signaling type, positive incentive, negative incentive) can be simulated with three different risk calculations of the underlying.

4 Results and Findings

We describe results for jealous and egoistic agents for the game “establishing homes.” Three types of experimental settings which are exemplary for our investigations are presented.

We started with a finite population of size 100, 1000, and 10,000. We limited our experiments up to 10,000 rounds. From one round to the next, we monitor the different agent populations incorporating the agent characteristics described above. Agent characteristics were modified from one round to the next according to haploid reproduction and simple learning mechanisms by modifying fitness as explained in the previous chapters. In the communication layer of BTT, we apply implicit volatility. We observed in all three settings that after 10,000 rounds and in case of small population sizes even earlier, the population stabilized.

First experiment: we assumed 90% unconditional co-operators, 9% jealous co-operators, and 1% egoists as an initial setting. The group fitness at the end was proportional to the group size. The distribution of egoists, jealous co-operators, and unconditional co-operators stayed stable. This is a typical setting for an evolutionary stable strategy (ESS).

Second experiment: we left the number of egoists with 1% but raised the jealous co-operators up to 50%. We observed that the co-operators were extinguished first and after that the jealous co-operators, and at the end, we ended up with egoists only. Finally, the group fitness was much lower than in the first experiment.

Third experiment: same as the second experiment, but we added positive incentives for cooperation in the sense that an agent got additional scores when helping its neighbor building its house. We observed that jealous co-operators migrated to unconditional co-operators. We ended up with slightly less than 1% egoists, and the rest turned into unconditional co-operators. The jealous co-operators were completely extinguished (see Fig. 8.4). The group fitness was proportional to the group size and better than in the first experiment.

Fig. 8.4
figure 4

Third experiment

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Analyzing Fig. 8.4, we observe that there is a convergence even in very large populations. In this case, the system needs more rounds so that a stable state can be reached. Note that stability is not always achievable. However, for the third type of experiment, this was always the case. Therefore, we conclude that not only jealousy but also egoism might be influenced to a certain extent when proper incentives are applied. However, egoists seem to be much more robust toward incentives.

Incentives might have a big influence on jealousy and even on egoists. The results are encouraging since they indicate that jealousy might be suppressed by applying appropriate incentives. The results though seem to contradict Falk and Kosfeld’s investigations (Falk & Kosfeld, 2006) who found that the morale of chiefs can be undermined through incentives. In our examples, we experimented with groups without hierarchies. Therefore, both experiments are not directly comparable.

The applicability of this result to human beings remains questionable since jealousy is a complex property of human nature. It might easily happen that a jealous person suppresses its behavior as long as gains from incentives outperform other disadvantages. In the case of absence of incentives, this person might quickly switch back to its old behavior pattern. This might especially be true for the suspicious jealousy type of person.

In this chapter, we have simulated agent behavior based on stochastic interactions. We found similar high variety in the dynamic behavior of agents as in the spatial deterministic approach. The variety depends upon the different specific rules, such as haploid reproduction, or different reinforced coordination strategies and on payoff functions. In any case, it indicates clearly that emotional modeling is dependent upon utility functions and implementation methods. We also did not expect such a high variety in the dynamic behavior of agents in the spatial deterministic approach. Originally, our intention to simulate human emotional behavior with agents was driven by practical limits with human settings. In those settings, humans often cheat, and therefore, the results are questionable. In any case, it can be concluded that the outcome of the simulation of socio-cognitive properties including emotion depends on the applied methodology.

Note that our simulations are based on exogenous parameters and we use game theoretical approaches to analyze social behavior of agents. We are not able to simulate emotions on intra-agent basis. In human brains, emotions and social behavior are driven by bottom-up regulation systems. These systems are driven by neurotransmitters on chemical basis such as dopamine, noradrenalin, or peptides such as endorphins. Some of these transmitters are spread to a couple of neurons in parallel in a diffuse way which is not yet completely understood.

Moreover, even if we could encode all secrets of the human brain, intelligent digital agents simulating all known human properties and emotions simultaneously (digital selves) might cause some intractable problems. Nevertheless, we try to show in the next chapter under which preconditions digital selves might manage digitized service chains.