Bounded Rationality, Group Formation and the Emergence of Trust: An Agent-Based Economic Model

Abstract

It is possible to model trust as an investment game, where a player in order to receive a reward or a better outcome, accepts a certain risk of defection by another player. Despite having achieved interesting insights and conclusions, traditional game theory does not predict the existence of trust between players who are selfish and exhibit maximizing behavior. However, experiments with these games reveal the presence of trust in player decision making. The purpose of this paper is twofold. First, it aims to build an agent-based economic model to show that trust revealed in these experiments can emerge from a simple set of dynamics. Using the generative methodology proposed by Epstein, we introduce natural selection, learning and group formation to the model to verify their impact on the emergence of trust between agents. Second, since the experiments reveal that the participants present bounded rational behavior, the paper aims to show that in an agent-based model, bounded rationality can be modelled through an artificial intelligence algorithm, the learning classifier system (LCS). As a result, we have observed that natural selection favors more selfish behavior. In addition, learning and the forming of groups increased trust in our simulations and they were able to reverse selfish behavior when introduced along with natural selection. The level of trust that emerged from the model with these three dynamics was similar to that observed in these experiments. Finally, it is possible to verify that the LCS was able to model bounded rational behavior in agents.

Appendix: Description of the Simulation Model

1.1 Initializing the Model

The environment of the model consists of a 7 $\times$ 7 dimension board linked by the edges. In this environment, there are two types of food: fruit and boars. At the simulation’s beginning, each cell is randomly assigned a capacity for each type of fruit drawn from an uniform distribution from 0 to 20. The initial amount of food for each cell is set at its maximum capacity. A total of 60 individuals is created and randomly assigned a position on the board. This position is defined as their home cell. They can search for food on the whole board, but all individuals return to their home cell after that. Each individual receives 200 calories in their creation, and their metabolism is set to 3 calories; that is, each round every individual spends 3 calories.

Each individual has a memory space to store information about the number of fruits and boars in the environment. However, the space is not large enough to store information about every cell in the simulation. For each type of food, each individual has three spaces of information resulting in a total of six spaces. They can “remember” three locations for each type of food and the amount of food available. As a premise, each agent can only “see” the cells that they are in and the cells surrounding it. Since the agents move around the board to gather fruits or hunt boars, they can collect information about other places, but as they return to their home cell, those cells may not be in sight anymore. They can also set a flag on the information in their memory as current—they can actually see it, or remembered—they can no longer see that information. Their memory is initialized with their home cell’s information and other two random locations around them for each type of food.

Next, the rules of the Learning Classifier Systems (LCS) are initialized: perception and memory of fruits (LCS5), perception and memory of boards (LCS1), Hunting Hour decision (LCS2), Hunting Place Decision (LCS3), Division of hunting spoils decision (LCS4) and Gathering place decision (LCS6)⁶. It is important to assert that each individual has his or her own set of LCS rules. At the moment of the agent’s initialization, these rules are assigned random values for each position and action. The initial fitness value is a parameter of the simulation and is set to 0.5. This parameter is the same for all the six LCSs.

1.2 LCS Rules

Two different individuals may use different decision making process modeled as two distinct set of LCSs. There is a total of six different LCS rules for each agent. Two of them (LCS1 and LCS5) are perception / editing rules used to interpret the environment and make it manageable to facilitate the decision making process. These rules are used to store information in the memory, and they were modeled as LCSs so that the decision making process is more adaptable and less restricted to the judgment of what information is really important. The genetic algorithm behind the LCS is responsible for judging what information is relevant to the decision making process. The remaining four rules are the decision ones. The six LCS will be detailed below.

1.2.1 LCSs About Perception and Memory of Fruits (LCS5) and Boars (LCS1)

The first perception rule is about the memory of fruits (LCS5) used to decide whether or not to store specific information in the memory. Each round, each agent looks around to recognize the environment. For each cell surrounding them, including the cell they are in, they run the LCS5. Each LCS has six rules that can be used to associate a pattern found in the environment and an action. Each rule of these six rules has a total of five positions representing information of the environment. Here, the environment can also include the state of that agent’s memory. These five positions are: (i) the position in memory with the smallest amount of fruits—assumes values from 1 to 3 according to the memory position, (ii) flag for the information with the smallest amount of fruits—assumes 1 for current and 2 for remembered, (iii) the position in memory with the second smallest amount of fruits—1 to 3 as in i, iv) flag for the information with the second smallest amount of fruit—1 or 2 as in ii, and (v) comparison of the amount in the cell being analyzed and the three information in the memory—assumes values from 1 to 4 being: 1 if the quantity of fruits in the cell is less than or equal to the smallest amount of fruits in memory, 2 if the quantity is more than the smallest and less than or equal to the second smallest, 3 if it is more than the second smallest and less than or equal the largest amount and 4 if it is more than the largest amount. Also, each position may assume the value 0 indicating a wild card. This value may correspond to any of the possible values and represent more general patterns .

For example, a classifier represented by the following numbers: 1 0 2 1 4 (figura1) maps a situation where the smallest number in the memory is in the first position, and it does not matter if this information is current or remembered. The second smallest number in the memory is in the second position and it is current information, and the amount of fruit in the cell is larger than all the information in the agent’s memory. An example of a classifier of the LCS for memory of fruits can be seen in Fig. 7.

Fig. 7

Example of a classifier for memory of fruits LCS

Each classifier is associated with an action: 0—not to consider the cell being analyzed, 1—substitute the smallest value in memory for the new information, 2—substitute the second smallest value in memory or 3—substitute the largest value. The function is to select the best rule to be used in case of more than one rule map the environment, and also to decide what rule should be discarded in case of a new rule is created. Each classifier has a fitness value associated with that. So, the same classifier of the example above associated with, for example, an action 3 means that whenever the agent finds the situation represented in the example above, he or she will store the information of the cell being analyzed in the position where the largest value is.

The second perception rule is related to the memory of boars (LCS1) and follows the same reasoning as the memory of fruits rule. The agents run the LCS in order to recognize the environment around them. There are five positions in each classifier and they are the same as in the perception rules for the fruits: (i) position in memory of the smallest amount of boars—1 to 3, (ii) flag—1 for current and 2 for remembered, (iii) second smallest amount of boars—1 to 3, (iv) flag—1 or 2 and (v) comparison of the amount of boars in the cell being analyzed and the information in memory—1 to 4 as in the fruit case. The actions are also the same: 0—discard the new information, 1 to 3—substitute the smallest, second smallest or largest value in memory respectively. And also, all classifiers have a fitness value. At the moment of initialization, all classifiers are assigned random values for the rules and actions and receive the initial value of 0.5 as fitness value.

1.2.2 LCSs About Hours (LCS2) and Place (LCS3) to Hunt

The other four LCS rules are the ones that model the decision making processes. The first of them is used to decide how many hours the pair of agents will hunt (LCS2) in each round. To make that decision, agents will use the history of hunt division decided by the Killer agent the Leader was paired with. The value is calculated as a percentage, so if the pair of agents hunted 4 boars resulting in 12 calories and the Killer shared 6 calories, the value stored will be 50%. The memory has six positions and registers the last six hunts in which the agent had participated as Leader. Each time there is a new division in a hunt, the oldest information is discarded, and the new information is appended. At the initialization, six random values are created in the memory for each agent.

Each rule has four positions representing the environment:

• 1. proportion between 3 times the max of boars and max of fruits in memory calculated by the formula $round(ln(\frac{3 \cdot max\,boars}{max\,fruits}))+1$. The number of boars is multiplied by 3 because of the calories a boar has compared to a fruit. The use of the logarithm is justified due to the dispersion of the information. If there are 20 boars and only 1 fruit, the relation is 60 and the logarithm is 4.1 . In the case of equal numbers of boars and fruits, the relation is 3 and the logarithm is 1.1. Finally, in the case of only 1 boar and 20 fruits, the relation is 0.15 and the logarithm is -1.9. Values are then restricted to the interval [1, 4] after being added 1. The value 0 will correspond to the wild card. In the case of the max of fruits being 0 the value is set to 4.

• 2. Proportion between 3 times the max of boars and max of fruits in memory as in the formula above but only for current values, that is, remembered values in memory are discarded. This allows different actions when there is certainty about the values.

• 3. The largest number of boars in memory divided by 5. Since only integer numbers are considered in the LCS rules, the program gets the ceiling of this calculation. The ceiling is used in order to consider the amount of 0 boars as a particular case. This value is added 1 and varies between 1—corresponding to 0 boars and 5—corresponding to 16 to 20 boars. 0 is then the wild card.

• 4. Mean of past hunt division multiplied by 4. Here the program also gets the ceiling in order to consider a 0 mean as a special case. This value is added 1 and varies between 1—corresponding to 0 division of hunting spoils and 4—corresponding to 75% to 100% division of hunting spoils. 0 is the wild card.

There are five possible actions number 1 to 5 corresponding to the total of hours the Leader will decide to hunt whenever he or she finds the situation represented by the classifier rule: (1) 0 h, (2) 2 h, (3) 5 h, (4) 8 h and (5) 10 h. The decision to restrict the number of hours to only five options is to keep the algorithm simple. Otherwise, there would be o total of 11 possible decisions, and it would be more difficult for a program to set a pattern. Also, each rule has a fitness value. At the initialization time, each agent receives one set of six rules, and all rules and actions are set to random values. The fitness value is set to 0.5.

The second decision LCS is the hunting place decision (LCS3). Each rule has five positions representing the environment: (i) the position in memory with the largest amount of boars—values from 1 to 3, (ii) flag for the largest amount of boars—1 current and 2 remembered, (iii) position in memory with the second largest amount of boars—values from 1 to 3, (iv) flag for the second largest amount of boars—1 or 2 and (v) flag for the smallest amount of boars—1 or 2. All these positions can receive the value 0 corresponding to the wild card. Actions vary from 1 to 3, corresponding to the place where the agent will decide to hunt. Similar to the other LCS, each rule has a fitness value. At initialization, all rules and actions receive random values and the fitness value is set to 0.5.

1.2.3 LCS Division of the Hunting Spoils (LCS4)

The third decision LCS is the division of the hunting spoils (LCS4). The Killer agent performs this action after the hunt had happened. To make this decision, the Killer agent will have to consider the history of hours of past hunts. Each agent has a memory of the last six hunts in which he was a Killer agent. At the initialization, all six positions are set to random values, considering only the possible values of hunting hours in the simulations: 0, 2, 5, 8 and 10.

Each rule has three positions representing the environment: (i) proportion between 3 times the max of boars and max of fruits in memory calculated in the same way as the first position of the LCS for the hunting hour decision—values from 1 to 4, (ii) hours spent hunting in the current hunt—values from 1–0 h to 5–10 h and (iii) mean of the last hunting hours divided by 2.5—values from 1–0 h to 5–7.5 to 10 h (the program also gets the ceiling in order to consider 0 h as a special case). All positions may have 0 as a wild card. Actions are: 1—0% division, 2—25%, 3—50%, 4—75% and 5—100%. Each rule has a fitness value. At initialization, all rules and actions receive random values, and the fitness value is set to 0.5.

1.2.4 LCS About Place to Gather Fruit (LCS6)

The last decision is the one that decides the place to gather fruits (LCS6), and it is very similar to the second decision that decides the hunting place (LCS3). Each rule has five positions: (i) the position in memory with the largest amount of fruits—values from 1 to 3, (ii) flag for the largest amount of fruits—1 current and 2 remembered, (iii) position in memory with the second largest amount of fruits—values from 1 to 3, (iv) flag for the second largest amount of fruits—1 or 2 and (v) flag for the smallest amount of fruits—1 or 2. 0 corresponds to the wild card. Actions vary from 1 to 3 corresponding to the place where the agent will decide to gather fruits. Moreover, each rule has a fitness value. At initialization, all rules and actions receive random values, and the fitness value is set to 0.5.

1.3 Execution Procedures

Each round begins with the growing back of the resources in each cell. Each cell increases its amount of each food by one unit, up to its maximum capacity in every round. So, since all the cells were initialized at their full capacity in the first round, nothing happens. Only in subsequent rounds that this growing back has any effect.

Next, the agents are paired. There are two types of pairing, depending on the simulation’s settings. In the group formation setting, agents are paired within their cells. In this procedure, agents are shuffled so that randomness is guaranteed. Agents are picked in pairs; the first agent is assigned the role of Leader, and the other one becomes the Killer. In case of an odd number of agents, the last one will be only a Gatherer.

Things change a little when there is no group formation . It is necessary to address the number of Gatherers in each situation to keep the numbers comparable. The number of cells with an odd number of agents is the same number of agents that become Gatherer in the group formation setting. So, in the case of no group formation, it is necessary to have the same number of Gatherers. So, in this case, agents are paired only after the Gatherers were assigned. Agents are shuffled in order to preserve randomness before role assignment.

The first decision in each round is for the Leaders to decide how many hours they will hunt (LCS2). However, before that, each Leader runs the perception / editing LCS for boars (LCS1). This procedure is run for every cell surrounding the agent, including the one the agent is in. These cells are shuffled to preserve randomness. In case one or more of these cells are already in the agents’ memory, the quantity is just updated before the LCS process starts. The flag indicating whether the information is current or remembered is also set to current and this cell is removed from the list .

For the remaining cells, the LCS procedure is run. First, there is a translation from the information in the memory and the cell being analyzed to the five codes listed in the previous subsection . Next, this condition is compared to the list of LCS rules to find any correspondence. A rule will be selected with a chance proportional to its fitness score among the corresponding rules in case of more than one correspondence. In case of no match, a new rule will be created, and it will be based on the rule that is more similar to the actual environment condition. The similarity is based on how many conditions of the rules are satisfied. Again, if more than one similar rule, the selection is made with a chance proportional to the fitness score among those with the most conditions satisfied. To create a new rule, all the satisfied conditions remain the same and the remaining conditions are randomly assigned to the environmental condition or a wild card. This new rule will replace an existing one selected with a chance proportional to the inverse of the fitness rule. Based on the selected or created rule, the memory is updated, replacing one of the existing information or discarded according to that rule’s action. The procedure continues until all the cells are analyzed. The rules used are stored in a list for future use in updating their fitness scores. This updating will depend on the hunting results and will be executed later.

After having his memory updated, the Leader will decide how many hours to spend hunting (LCS2). The procedure is similar in all LCS executions. First, there is a translation from the environmental conditions to the four codes described in the previous subsection. Next, the condition is compared to find correspondence and select the rule to be used based on fitness score or to create a new one in case of no correspondence based on similarity and fitness score. The rule used is stored for further evaluation and updating of its fitness score. This evaluation depends not only on the hunting results but also on the gathering activity and will be executed later.

Next, the Leader will run the LCS3 to decide where to hunt. The procedure is the same: translating the environmental conditions, select a rule or create a new one in case of no match all based on fitness scores, store the rule used for further evaluation and updating of fitness scores.

With the number of hours and the hunting place decided (LCS2 and LCS3), the agents can hunt. Both Leader and Killer move to the hunting place. The number of boars hunted is the minimum of the boars existing in the place and the hunting hours. The number of boars hunted is deducted from the quantity available in the cell.

With the number of boars hunted the Leaders can evaluate their LCS3 rule for the decision of hunting place. This evaluation only occurs in the learning mode. In case learning is disabled in the simulation, all fitness score remains with their initial value of 0.5. If learning is abled, the fitness score will be multiplied by a decreasing factor of 0.5 of older information’s importance. Then, the result is added with the total of calories obtained with the hunt, that is, the number of boars times 3.

After that, mutation or crossover can take place. There is a chance of 15% that they occur, only in learning mode, with half of that chance directed to mutation and a half to crossover. The procedure for mutation is to select a strong rule, with a chance proportional to its fitness score and a weak rule, proportional to the inverse of its fitness score. If by chance they are the same, a new weak rule is selected until they are different. A position in the strong rule is selected for mutation and a new value is selected for that position. This new rule is compared to the existing rules to verify that it is different from any of them. If there is an equal rule, the procedure continues until a different rule is obtained. The new rule replaces the weak rule in the LCS.

The crossover procedure is to select two strong rules and one weak rule based on fitness scores, all different from each other. A new rule is formed, taking information randomly from both strong rules. This new rule is compared to the existing ones, and if it is equal to any of them, a new rule is formed following the same procedure until it is different from all existing rules. This new rule then replaces the weak rule.

Now, it is time for the Killer to run the LCS4 to decide how to divide the hunting spoils. It follows the same procedures: environmental conditions, selection or creation of a rule, storage of the rule used. The decision will result in a percentage of division from 0% to 100% with increments of 25%. This percentage is multiplied by the number of calories or 3 times the number of boars hunted, and the result is rounded to have an integer number. The Killer will add the total of the calories obtained from the hunting subtracted by the total of calories shared to his or her calorie stock, and the Leader will add the total of calories shared to his or her calorie stock. After that, the Killer will return home.

The Leader will then evaluate the hunting results to update the perception LCS1 for the memory of boars. Since these rules were used as many times as needed to evaluate all cells surrounding the Leader, the update process of the fitness scores will consider all the times the rule was used. The criterion to evaluate these rules is the hunting results. But it is also necessary to consider the hours dedicated to hunt. For example, hunting one boar in a 2 h hunt is different from hunting the same one boar in 10 h hunting. So, the result is normalized by multiplying the number of boars by ten – the total possible hunting hours and dividing by the number of hunting hours, so that, hunting one boar in a 2 h hunting is equivalent to hunting five boars in a 2 h hunting. This fitness value is then divided by the number of uses of the LCS. If a rule was used more than once, it would be considered all the times it was used. So, we get a measure of the average fitness value per use of a rule. The procedure then selects all the rules that have been used and update its fitness value. For each rule, the current value of the fitness score decreases by the factor $0.5 ^ {times\, the\, rule\, was\, used/times\, all\, rules\, were\, used}$, that is, it considers the proportion this rule was used among all uses. If a rule was used half of the times, the decreasing factor would be $0.5 ^ {(1/2)} ~ 0.71$. If a rule was used all the time, the decreasing factor would be 0.5. Then, the average fitness value is added as many times as the rule was used. So, if a rule was used half of the time, it will receive half of the fitness value. If it was used all the time, it would receive all the fitness value. After updating the fitness scores, there is a possibility for mutation or crossover. The procedure is the same for all LCSs and it was described above, and it only occurs in the learning mode.

Before returning home, the Leader will run the perception / editing LCS1 for boars to prepare for the next round. Since he or she is in a different position from home, it is worthwhile to see the environment and gather information for the next decision.

After the hunt, it is now time to gather fruit for the remaining hours of the day. All agents perform that task unless they have spent all 10 h hunting. The first step is to run the perception / editing LCS5 for fruits. The procedure is the same as in the boars’ case but using the LCS and memory for fruits. With the memory updated, agents run the LCS6 for the decision of where to gather fruits. Again, the procedure is the same as the boars’. After deciding where to gather fruits, the agent will move to the gathering position. The number of fruits gathered will be the minimum between the total of fruits available in the gathering location and the remaining hours of the day. This amount is deducted from the number of fruits available in the location. Each fruit will provide one calorie that will be added to the agent’s calorie stock.

The next step is to update the fitness value of the LCS6 rule used to decide where to gather fruit, which will only happen in the learning mode. With learning, the fitness value is multiplied by the decreasing factor of 0.5 and the total of calories obtained with the gathering of fruits is added. After that, there is a chance of 15% for mutation or crossover. The procedure is the same as in other LCSs.

Having updated the fitness value and the LCS6 rule of gathering place , it is time to update the fitness value of the perception / editing LCS5 for fruits. The procedure resembles that used for the perception / editing LCS1 for boars. The result is also normalized, multiplying the number of fruits by ten and dividing by the number of gathering hours. The average fitness per use is calculated by dividing the fitness value by the number of uses of the LCS, considering every rule’s use. Each rule is multiplied by the factor $0.5 ^ {times\, the\, rule\, was\, used/times\, all\, rules\, were\, used}$, and the average fitness value is added as many times as the rule was used. After updating the fitness value, there is a possibility for mutation or crossover with the same probability as in other LCSs only in the learning condition.

Before the agents return home, they run the perception / editing LCS5 for fruits again to prepare for the next round.

At this moment, the agents in the simulation have performed all possible feeding activities, and it is possible for the Leader agents to update the fitness value of the hunting hours (LCS2) and for the Killer agents to update the fitness value of the division (LCS4). To update the fitness value of the hunting hours LCS2, the fitness value is multiplied by 0.5 and added to the value of calories obtained in that round, which is the total of calories sent by the Killer plus the calories obtained from fruits.

To update the fitness value of the division LCS, it is necessary to consider the possibility of indirect reciprocity. If the Killer only considers the result of the last round, then the short term results may direct the rules’evolution, resulting in more selfish behavior. So, to allow indirect reciprocity, the fitness value of the division LCS will consider the long term results. It will be considered the average of the next six rounds of the Killer’s total calories (boars + fruits). This opens the possibility that good faith in dividing the hunting spoils may generate more trust in the future, resulting in more calories in the long term. So, only after six rounds it will be possible to update this LCS rule’s fitness value. Currently, the rule used six rounds ago is the one to have its fitness value updated. This value is decreased, and the average of total calories obtained in the previous six rounds is added. Also, there is a possibility for mutation and crossover. All this only occurs in the learning mode.

After all, rules have been used and their fitness values have been updated, all agents have their calories stock deducted by their metabolism. If the simulation is in the selection condition, all agents with a negative or zero number on their calorie stock die and are removed from the simulation. Also, if the selection is enabled, there is a probability that each agent reproduces. This probability depends on the agent’s calorie stock and is zero for low stocks, around 5% for calories stocks of 600 (three times their initial calorie stock), and 10% for high stocks. The offspring inherit the minimum between half their parent’s stock and 200 calories and receive a copy of all their parents’ LCSs rules. The parent has their calorie stock deducted from the amount passed to his or her offspring. The offspring is assigned to a random position on the board.

Finally, after reproduction, there is a possibility of 0.1% for each agent to migrate to a random position in the board. After that, the round ends and a new one starts. A total of 4000 rounds are run in each simulation.