Social structure formation in a network of agents playing a hybrid of ultimatum and dictator games

04/11/2019 ∙ by Jan E. Snellman, et al. ∙ 0

Here we present an agent based model involving a hybrid of dictator and ultimatum games being played in a co-evolving social network. The basic assumption about the behaviour of the agents is that they try to attain superior economic positions relative to other agents. As the model parameters we have chosen the relative proportions of the dictator and ultimatum game strategies being played between a pair of agents in a single social transaction and a parameter depicting the living costs of the agents. The motivation of the study is to examine how different types of social interactions affect the formation of social structures and communities, when the agents have the tendency to maximize their social standing. We find that such social networks of the agents invariably undergo a phase change from simple chain structure to more complex networks as the living cost parameter is increased. The point where this occurs, depends also on the relative proportion of the dictator and ultimatum games being played. We find that it is harder for complex social structures to form when the dictator game strategy in social transactions of agents becomes more dominant over that of the ultimatum game.



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I Introduction

Studies of human social behaviour suggests that striving for superior social positions is a fundamental characteristic of the human nature AHH2015 , and this feature has even measurable effects on human brain activity ZTCB2008 ; ISS2008 ; KMD2012 ; UP2014 . That being the case, however, one could ask why the resultant competition between individuals does apparently not preclude humans from forming structured societies? This is an intriguing question since humans are inherently sociable and show clear tendency to form complex social and societal structures.

An obvious answer to this question is that humans are dependent on each other for survival, which in turn implies that without either environmental or social pressure humans would not develop complex communities at all (see, e.g. ASGB2014 and references within). In order to model the societal effects of the tendency of superiority maximisation we introduced the better-than-hypothesis, or BTH, in SGGBK2017 . As the name implies, the working assumption of BTH is that humans are motivated primarily by the aim to maximize their status in society, an assumption which is shared by Alfred Adler’s school of individual psychology adler . In SGKBK2018 we discovered an abrupt behavioural change in a model derived from BTH, which could be interpreted as formation of social structures due to outside pressure.

Human communities are by nature connected and perpetually changing, so they can be considered as dynamic networks of individuals. Consequently there have been a number of attempts at the theoretical level to model the community formation process using the network theory. There are naturally many different aspects related to the subject in addition to the influence of outside pressures, and thus most studies have concentrated so far on such aspects as the roles of opinion formation HN2006 ; IKKB2009 ; SGGBK2017 ; LGAGI2005 ; JA2005 ; GK2006 ; SM2013 , deception IGDKB2014 ; BGDIK2015 or information exchange EZ2000 ; W2015 .

The model we studied in SGKBK2018

concerned a social network with simulated agents playing dictator game with their neighbours. The agents were allowed to change their own connections, and the model was governed by two parameters, one of which, the memory parameter, measures how fast the agents forget the way they were treated, and the other, the cost parameter, measures the proportion of money spent on living costs. We found that when one varies either one of these parameters the social networks produced by the model consist of disconnected chains that lengthen until the parameters reach certain values, after which the networks become much more connected and complex. This phase transition was clearly visible from the susceptibility and shortest path length plotted as functions of these parameters: they exhibited a sharp peak at certain points.

Phase transitions in models of social behaviour have been observed and studied before. They are of particular interest not only because of the insights they provide into various social phenomena, but also because they can be handled and interpreted in terms of analogous phase transitions of the physical systems. Many models of community, opinion and hierarchy formation have been found to involve phase changes. In HN2006 , for instance, the combined effect of co-evolving opinions and social network structure on community and opinion formation was studied, with a phase change taking place as the relative influence of the two processes was varied. Similar, but more general, approaches were used in EMV2006 and MJTKK2019 to study the role of homophily in the dynamics of the social networks and in the segregation of network structures, respectively. In CCV2016 hierarchy formation was presented as a result of a phase change when social and cognitive constraints were involved.

The original dictator game mentioned above is a problem in decision theory first proposed in KKT1986 , where one of the players, the dictator, is tasked with dividing a fixed reward between the players at will. It is different from the closely related ultimatum game GT1990 , in which the other player, called accepter in the context of the game, is allowed to either reject or accept the offer. In case of rejection, neither player gets anything, while in the case of acceptance the division is realised as it stands. Both games were constructed to demonstrate the limitations of rational economic behaviour.

The ultimatum and dictator games allow the testing of human economic behaviour, especially the assumption of rationality (see e.g.S1955 and HS ). Providing that humans rationally seek to maximize their own profits, in the dictator game the dictators should reserve everything for themselves, while in the ultimatum game one could expect that the accepter should accept any nonzero amount offered, and that the proposers should offer the minimum possible amounts. However, in the actual experiments with real humans the proposers in the ultimatum game and dictators of the dictator game have a tendency to offer substantial amounts to the other player, and unbalanced propositions in the ultimatum game (especially those in favour of the proposer) tend to be rejected by the accepters (see, for example, E2011 ; metalyysi ; CC2008 ; FLK2015 ). These results immediately raise the question on the rationality assumption of human behaviour. As we argued in SGGBK2017 , assuming humans to rationally seek superior position over the others means in the case of the ultimatum game that accepters should accept offers of over of the full prize, and that the proposers should realise this and adjust their offers accordingly.

In this study we extend our earlier model presented in SGKBK2018 by letting the simulated agents play ultimatum game as well as dictator game, and take a look at the effects this has on the behavioural phase change. The motivation for this study is to test, how different modes of social interactions change the structures and dynamics of social networks of humans following the BTH.

This paper is organised such that in the next Section II we define the network model of individuals playing the game with a strategy that is a hybrid of the ultimatum and dictator game strategies. This is followed by the simulation results presented in Section III. Finally we draw conclusions in Section IV.

Ii Network model of agents playing ultimatum and dictator games

Let us consider a set of linked agents forming a social network and playing a hybrid of the ultimatum and dictator game with all their network neighbours. For both of these games each agent has its individual offering rate , while in the case of the ultimatum game each agent has also a threshold for accepting the offer (see X2010 ).

The agents have a choice between two different social strategies, playing either the dictator game or ultimatum game with their neighbors, by making the choice between them at random. The simulation proceeds in cycles, in each of which the agents play either the dictator game with probability

or the ultimatum game with probability , with all the agents they are connected to. The game being played is chosen independently for each transaction between the agents, so the agent could play ultimatum game with agent and dictator game with agent in a single cycle. The nth cycle is denoted by . The agents will always play in the same order.

This double strategy model is similar to the one in Ref. SGKBK2018 , with the difference that with probability the accepting agent gets to either accept or reject the division offered by the proposing agent. Thus, if the agent makes an offer to agent the transaction occurs if either the dictator game is played or if . In the case that the division is realised the accumulated wealth and of the agents change as follows:


where is the time before the transaction and is the time immediately after it.

In each cycle, the amount as the ”living costs” is deducted from the accumulated wealth of the agents, such that the wealth of an agent can only be reduces to zero in the worst case. Here the cost parameter turns out to be very important in shaping the structure of the network when the game is played. This parameter was also one of the two model parameters in the model presented in SGKBK2018 , the other being a memory parameter, which for the sake of simplicity we do not employ in this study. Of these two parameters it could be said that different values of the relative proportions and of the dictator and ultimatum games played by the agents correspond to different rules of social interaction and that the parameter can be interpreted as an external pressure or need forcing the agents to cooperate.

Now, the wealth acquired by agent in a cycle can be written as follows,


where is the set of those neighbours of the agent that will accept, or be forced to accept (depending on whether the agent gets to play either ultimatum or dictator game with agent ), the offer , and, conversely, is the set of the neighbouring agents’ offers the agent is either willing or forced to accept. If the full set of agents is denoted by and the set of neighbours of denoted by , can be written as follows


where is the set of all agents that either accept or are forced to accept the offer , i.e.



is a random variable that determines, whether the agent

gets to play dictator or ultimatum game with agent .

The agents compare themselves to other agents on the basis of their accumulated wealth. According to the BTH, the change in the utility of the agents in a cycle can be written in the following form


Then using Eg. (6) we can determine the contribution by the agent to the utility of agent , as was done in Ref. SGKBK2018 . The cumulative form of this contribution can be written as follows


where is an element of the adjacency matrix:




where is the number of agents in .

The agents choose their offering rate and the acceptance threshold according to Eq. (6) using a mix of simple hill-climbing and random walk methods. The agents have the values of and assigned randomly to them at the start of the simulations, and they seek to find better values by choosing a random direction on the -plane and changing their and values to that direction as long as , defined by Eq. (6), remains positive. When becomes negative, the agents choose randomly a new direction to proceed. The whole process can then be written in the following form:


where is the length of the step the agents take into their chosen direction . As stated, the direction is the angle that is reassigned a different value every time is negative, i.e.


where is chosen randomly. As for the step length , it is linearly reduced from to in the course of the first time steps using simulated annealing technique.

The agents redefine their social relations after each cycle according to such that the agent will form a social link with the agent if both and are non-negative, provided that the link between agents and does not already exist. Similarly, agent will cut an existing link with agent if .

Next we perform comprehensive computer simulations of this model of networked agents interacting with their linked neighbors by playing a hybrid of ultimatum and dictator games. In this we focus on getting insight into the social structure formation and phase change behavior therein.

Figure 1: The social network of agents with (panel (a)), and (panel (b)) and (panel(c)).
Figure 2: The susceptibility and the path lengths in the ultimatum (, panels (a) and (b)) and dictator modes (, panels (c) and (d)). The curves are for four different network sizes with total agent number shown in the legend.

Iii Results

In this study the simulations were run for time steps, at which point they were halted. To obtain reliable statistics we performed the simulations times and also took averages over the second half of the time-series due to the strongly fluctuating nature of the system. Since our main interest is in studying possible phase changes present in the model, we are mostly interested in the shortest path length and the susceptibility

of the social networks, defined as the second moment of

-sized clusters, excluding the largest connected component of the network:


Our model experiences a phase change behavior as the cost parameter is increased, that is the social networks of the simulated agents change from collections of simple pairs to longer chains to more connected complex networks, as depicted in Fig. 1 for the case. However, similar transition occurs for all the values of probability . In essence what happens is that the short chains of agents lengthen until they start joining to other chains and become more and more entangled and networked, which resembles phase transition due to crystallization process in freezing liquids.

The average susceptibility and the shortest path length as the functions of are shown in Fig. 2 for several network sizes. The cases shown are for the pure dictator and ultimatum game models, i.e. and , respectively. For the case we observe a strong peak in both and at , which only becomes stronger with increasing number agents. The fact that the height of the peak increases with the size of the network suggests that the phase transition is genuine. The situation is quite similar with the case, although with some differences, the most important of which is that the point of transition is now less well defined and broader than in the previous case, and it occurs at instead of . A smaller difference is that, while has virtually no influence on the shortest path length and susceptibility below the transition point, its influence is noticeably greater past the transition point.

Figure 3: The susceptibility (panels (a),(c), (e), (g) and (i)) and the path lengths (panels (b),(d), (f), (h) and (j)) as functions of the probability of dictator game being played, . For panels (a) and (b) the cost parameter has a constant value of , for panels (c) and (d) , for panels (e) and (f) , for panels (g) and (h) and for panels (i) and (j) .

Since the transition points are at different spots for the pure ultimatum or dictator strategy modes of the model, what happens between these modes, i.e. for a hybrid strategy, is an interesting question. In Fig. 3 the shortest path length and susceptibility are shown as a function of the probability for four different values of cost parameter : , , , . In this figure we observe very tall but broad peaks in both quantities for all values. As observed in the case of variable in Fig. 2, the peak rises strongly as a function of the number of agents in the network. Also a significant feature of the peak is that it seems to occur at higher values of as is increased, such that for the peak occurs at and for at .

Figure 4: The average degree (panel (a)), shortest path length (panel (b)) and susceptibility (panel (c)) near the phase transition points for , and . The vertical lines show either maximal values of the shortest path length and susceptibility, or the points at which the average degree exceeds the value of .

In Fig. 4 we take a closer look at how the average degree, the shortest path length and the susceptibility near the phase transition points behave for different game strategy proportions, i.e. (pure ultimatum strategy), (equal mix of ultimatum and dictator strategies) and (pure dictator strategy) near the points where the latter quantities attain their maximum values according to Fig. 2. In Fig. 4 the maximal values of the susceptibility and shortest path length are marked with vertical lines, along with the approximate points at which the average degree rises above the value of . For the average degree value is significant, because the social networks consisting of chains have average degrees below that value. Thus the phase change exhibited by this model is indicated by the average degrees rising above this value. We see that for the (pure ultimatum strategy) case the maxima of the shortest path length and susceptibility occur relatively close to the value of at which the average degree rises above , but that these points diverge for other cases of shown. In the (equal mix of ultimatum and dictator strategy) case it would seem that while the maximal value of the shortest path length occurs at a clearly lower value of than where the average degree rises above , the point at which the shortest path length clearly starts to decline occurs roughly at the same point as where the average degree passes . The (pure dictator strategy) case behaves in very different manner than the other cases: While the average degree tends to be monotonically rising in the other cases, in the case there is a slight declining phase between and . Also, while the shortest path length and susceptibility for the and cases tend to be rising until they reach their maxima and then decline, this pattern is only roughly reproduced by the susceptibility in the case. The shortest path length in the case exhibits a slight rising trend in the declining phase.

Figure 5: Phase diagram in the

-plane for N = 100, calculated by using two different methods. The solid line shows the estimate from linear interpolation, while the dashed line with the error bars shows the results of the Monte Carlo method.

The results presented in Figs. 2, 3 and 4 allow us to draw a rough phase diagram for the system, as depicted in Fig. 5, for the pair of model parameters, i.e. the living cost parameter and relative proportions, and of the dictator and ultimatum games, respectively. The phases depicted in the figure concern concern the changes in the network structure such that in the upper part of the curve one obtains mostly chains, while in the lower part there are networks connected with more than two neighbors. The point at which the average degree of agents surpasses the value in the simulations is taken to represent the onset of the phase transition. Since the exact parameter values at which this happens are essentially never captured in the numerical data, linear interpolation is used to estimate where these crossings take place. The phase transition curve, depicted as a solid line in Fig. 5, runs approximately from point to , in a monotonically rising but sub-linear manner.

Due to the very rough nature of the linear interpolation used to construct the (solid) phase transition line in Fig. 5, we were not able to make any reasonable error estimates in this case. To gain some idea on the size of the errors, we used a Monte Carlo method to obtain an alternative phase change curve and its error estimates. We did this by selecting five values of , , , , and , and then randomly choosing values for in close vicinity of the phase curve indicated by the linear interpolation method. Running the simulations for times we got a value for the average degree for thus chosen pair of parameters, and if this exceeded the value of by small enough margin (less than to be exact), the parameter pair was stored. Repeating these steps we obtained for each of the chosen values of an ensemble of values of

, for which the mean and standard deviation could be calculated. If the ensemble seemed to terminate at the limits of the search area, the search area was expanded until the ensemble fitted within it. The mean values for

acquired in this way are shown in Fig. 5 as asterisks connected by a dashed line, and the standard deviations provide the error bar estimates. Finally, the number of points from which the means and standard deviations were calculated ranged from () to ().

As seen in Fig. 5, the results of the Monte Carlo method agree rather closely with those obtained by linear interpolation, though the difference between the methods, small as it is, increases as increases. The errors, as given by the Monte Carlo method turn out to be relatively small: in most cases they barely exceed the size of the asterisk markers. The convergence of the results of the two methods and the small errors mean that the phase diagram shown in Fig. 5 can be considered reasonably accurate.

The reason why the point of phase change is different for the ultimatum and dictator game models is that in the former case, the agents need to take into account the possibility of rejection when making proposals, while in the latter case they do not need to. For any values of the relative proportions and of the dictator and ultimatum games being played, respectively, the phase change occurs when the number of neighbours needed to make profit is more than , and so the chains change into complex social networks. In the case of the dictator game the agents need at least neighbours using similar division strategies to make any profit (as argued in SGKBK2018 ), consequently it is there where the phase change occurs, as seen in Fig. 5. When , the agents need more than relations because some of them might decide to turn down their offers and consequently the phase change occurs at a lower value of the living cost parameter .

Iv Conclusions

In this study we have created an agent-based network model combining ultimatum and dictator games, where the modelled agents play dictator game with linked agents with the probability and ultimatum game with the probability . The agents can also redefine their social relations according to the conduct of other agents in the game. This is a generalisation of our earlier dictator game model introduced in SGKBK2018 , and as such the model presented here has the same parameter describing the living costs, but not the parameter describing forgiveness.

Our main focus in this study has been the apparent phase change seen in SGKBK2018 , in which the social networks produced by the models would change in form abruptly from chains of agents to more complex structure at certain parameter values. This change naturally manifests itself most clearly as a strong peak in the susceptibility and shortest path length of the social network, when these quantities are plotted as functions of parameter . We find that for the extreme cases of and , which correspond to the pure ultimatum game strategy and dictator game strategy, respectively, this transition occurs for different values of : and , respectively. Setting the parameter between these values and plotting the susceptibility and shortest path length as a function of the relative proportion of the dictator game strategy revealed a broad peak that indicates a similar phase transition taking place in this case as well.

The discovery of shifting phase transition point as a function of has implications for the applicability of BTH-style models to real human societies, the most profound of which relates to the role of the ground rules of social interactions in formation of social structures. The fact that this phase transition occurs at a lower value of when the ultimatum game becomes more prevalent means that in that case the agents are generally more inclined towards forming complex societies than when the dictator game is predominant. As the ultimatum game could be characterised to be more reciprocal than the dictator game, given that both players get to make a decision regarding the division, one could draw the conclusion that reciprocity in social interactions makes it easier for the agents to form more complex communities. This is a promising result for BTH, since one would intuitively expect this to be the case in the real human societies. Finally judging from our numerical results, the simulated societies based on purely dictatorial social interactions need more than thrice the amount of external pressure to maintain complex social structures than those based on ultimatums.

In order to further study the effect of reciprocity on social structure formation under BTH one would need to define a measure for the reciprocity of social interaction types, such as the dictator and ultimatum games, and then study the propensity of social structure formation as a function of this measure. However, defining such a measure is beyond the scope of this study, as is the devising of new social interactions that would have different ratings on that measure, as would be necessary for the aforementioned future research.

JES acknowledges the financial support of Niilo Helander’s foundation. The computational resources provided by the Aalto Science-IT project have been utilised in this work. KK acknowledges support from the Rutherford Foundation Visiting Fellowship at The Alan Turing Institute, UK, and from the European Community’s H2020 Program under the scheme INFRAIA-1-2014-2015: Research Infrastructures”, Grant agreement No. 654024 SoBigData: Social Mining and Big Data Ecosystem” (http://www. RAB wants to acknowledge financial support from Conacyt (Mexico) through project 283279.


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