Probabilistic fair behaviors spark its boost in the Ultimatum Game: the strength of good Samaritans

Fairness is one of cornerstones that underpin human civilization, and plays a vital role for solutions to those great challenges we are facing, like climate change, aggravated economic inequality, and pandemics. A canonical model for understanding fairness is the Ultimatum Game (UG) Güth et al. (1982): two players are asked to split a certain amount of money, one (the proposer) proposes a division between herself/himself and the other player (the responder), who either accepts or rejects. If the agreement is reached, they share the money accordingly; otherwise, they earn nothing. According to the paradigm of Homo economicus Solow and Simon (1957); Samuelson and Nordhaus (2005), people are self-interested and act in a fully rational manner, the proposers should make the smallest possible offer, because the responder is expected to accept it still because “something is better than nothing”. The rational solution for the UG thus ends with a very unfair split.

A large number of experiments conducted in different countries and with varied incentives, however, reveal that this is not really the scenario how human act when playing the game. The majority of the offers are between $40\%$ and $50\%$ of the total amount, and the offers below $20\%$ are only $3\%$ and with more than half chance are rejected by the responders Güth et al. (1982); Thaler (1988); Bolton and Zwick (1995); Kagel and Roth (1995); Guth and Kocher (2014). It turns out that we human are remarkably fond of fairness even at the cost of the potential monetary loss. Similar observations are also made for non-human species, such as monkeys Brosnan and de Waal (2003) and chimpanzees Proctor et al. (2013), though arousing some debates for the latter Keith Jensen and Tomasello (2007).

Many efforts have been made to understand the above discrepancies. For example, one psychological explanation is that the decision-making for human in the game may be not simply based upon the expected payoff, but also the payoff difference for the two evolved players Kirchsteiger (1994); Bethwaite and Tompkinson (1996); Fehr and Schmidt (1999). Also, the rejection of a low offer can also be taken as punishment on the proposers, because they lose much more than the responders. In fact, a great variety of theories are proposed in the past thirty years to understand the emergence of fairness Debove et al. (2016). These theoretical studies indicate that reputation Nowak et al. (2000); Andre and Baumard (2011), noise Binmore and Samuelson (1994); Gale et al. (1995), empathy Sanchez and Cuesta (2005); Page and Nowak (2002), population structures Page and Sigmund (2000); Kuperman and Risau-Gusman (2008); Sinatra et al. (2009), and role-alternating Iranzo et al. (2011) each could facilitate the evolution of fairness. Note that, all these models are developed within the paradigm of Homo economicus, players are supposed to be full self-interested and fully rational, all their actions are strictly economic-driven.

This assumption, however, has been challenged by behavioral economists, who have identified the facts that psychological and emotional are important factors in decision-making Colin F. Camerer and Rabin (2003); Loewenstein and Lerner (2003), and the related neural basis in the brain are also reported Sanfey et al. (2003); Rilling and Sanfey (2011). In our daily lives, there are some occasions where individuals make their decisions up to some other incentives, such as intrinsic moral incentives; people may occasionally want to a good samaritan to do a good deed without an economic return being expected at all. When they act as such, they are not purely self-interested, and can be regarded as irrational to some extent in the sense that they are now not only economic-driven but also partially other incentive-driven, e.g. morality-driven Alexander (2007); Frey (2000). It has been argued that our moral sense has developed a sophisticated defense mechanism that enables individuals to survive and the groups they belong to to thrive over millions of years evolution, the morality-driven acts have their place in social activities  Boehm (2012). Thus, it’s reasonable to assume the presence of good samaritans, and revealing how they impact the evolution of fair acts would be helpful for understanding the emergence of fairness in many realistic circumstances.

In this work, we fill this gap from the perspective of evolutionary game theory  Kuperman and Risau-Gusman (2008). By relaxing the assumption of Homo economicus, players are allowed to probabilistically act as a fair player from time to time, we investigate the impact of these behaviors on the economic-driven behaviors and the fairness level for the whole population. By tuning the probability, we show that a small such probability is sufficient to drive the whole population to be fair players, an even lower probability is able to sustain this state once it is reached. These form a first-order phase transition of the fairness level and a hysteresis structure is seen. Further analytical treatment and numerical simulations show that these observations are robust to whether the population is unstructured or homogeneously structured, or with different updating rules. Heterogeneous networked population, is further found to enhance the leverage effect, especially when the hub nodes act as fair players; the transitions, however, become continuous.

The paper is organized as follows: we introduce our model in Sec. 2II. In Sec. 3III, we show results for the population with 1d lattice and well-mixed structures. In Sec. 4IV, A mean-field theory is developed for explanation and insights. Erdős-Rényi random networks and Barabási-Albert scale-free networks are also examined to study the impact of underlying population structures in Sec. 5V. Finally, we conclude our work together with some discussions in Sec. 6VI.

Let’s consider a population of $N$ individuals playing the Ultimatum Game, where they could either be well-mixed or structured (square lattice, random networks, and scale-free networks) in our studies. Following the common practice Güth et al. (1982); Debove et al. (2016), the total amount of money to be divided by the two players in each game is equal to the unity. The strategy for a player $i$ is characterized by two parameters $p_{i},q_{i}\in[0,1]$, where $p_{i}$ is the offer that the player $i$ is willing to give to the responder when acting as a proposer, while $q_{i}$ denotes the acceptance threshold when in the role of responder. Their initial values are chosen randomly and independently from the uniform distribution $[0,1]$.

Suppose now two players $i,j$ are now engaged in an interaction, with their strategies $S_{i}\!=\!(p_{i},q_{i})$ and $S_{j}\!=\!(p_{j},q_{j})$, respectively. When they are in the role of proposer and responder with equal chance, the expected payoff for player $i$ versus player $j$ is

$$E_{ij}=\left\{\begin{array}[]{lr}1-p_{i}+p_{j},\quad\quad\mbox{if}\ p_{i}\geqslant q_{j}\ \mbox{and}\ p_{j}\geqslant q_{i}\\ 1-p_{i},\quad\quad\quad\quad\,\mbox{if}\ p_{i}\geqslant q_{j}\ \mbox{and}\ p_{j}<q_{i}\\ p_{j},\quad\quad\quad\quad\quad\;\;\>\mbox{if}\ p_{i}<q_{j}\>\mbox{and}\ p_{j}\geqslant q_{i}\\ 0,\quad\quad\quad\quad\quad\;\;\>\>\,\mbox{if}\ p_{i}<q_{j}\ \mbox{and}\ p_{j}<q_{i}.\\ \end{array}\right.$$

(1)

In our numerical experiments, the evolution is as following synchronous updating procedure. At the beginning of each round, with the spontaneous fair probability $\rho$, each player chooses to behave as a good Samaritan with the fair strategy $S_{h}\!\equiv\!(p_{h},q_{h})$, with $p_{h}\!=\!q_{h}\!=\!C_{h}$ and $C_{h}=0.5$ here, the half-half split; otherwise it uses the strategy obtained in the last round. Next, every player interacts with all their nearest-neighbors up to the underlying networks, their payoffs are collected according to the rule given by Eq.(1) and are then added up, denoted as $\pi_{i}$ for player $i$. To this end, all players update their strategies proportional to their payoffs, including those adopting the strategy $S_{h}$, i.e., the player $i$ imitates the strategies in its neighborhood, including itself, according to the Moran rule with the probability Roca et al. (2009)

$$w(S_{j}\rightarrow S_{i})=\frac{\pi_{j}}{\sum_{j\in\Omega^{\prime}_{i}}\pi_{j}},$$

(2)

where $\Omega^{\prime}_{i}=\Omega_{i}\cup\{i\}$, and $\Omega_{i}$ denotes the neighborhood of player $i$. The strategy imitation is subject to small mutations ($\delta p,\delta q)\in[-\varepsilon,\varepsilon]$. Notice that, according to the above rules, the choice of being a good Samaritan is only temporal, no memory is considered; the subsequent action taken is based upon Eq.(2), which can be considered as economic-driven and rational.

Obviously, when the spontaneous fair probability $\rho=0$, our model is recovered to the classic UG within the strict economic-driven paradigm, while the other extreme case of $\rho=1$ is completely mortality-driven, fully fair scenario is expected but is rarely seen. Realistic scenarios are supposed to occur in between the two extremes $0\!<\!\rho\!<\!1$, where both incentives work together. In order to measure the fairness level of the population, we calculate the time averages of offer $\langle p\rangle$ and acceptance threshold $\langle q\rangle$ as the order parameters. A ideal fairness scenario corresponds to $\langle p\rangle\!=\!\langle q\rangle\!=\!0.5$, the half-half split; whereas the solution for the rational economic man approaches zero if no any other mechanism works. Note that, after the transient evolution, the value of $q_{i}$ is almost always slightly smaller than $p_{i}$ for each individual to maximize its payoff, therefore $\langle q\rangle\approx\langle p\rangle$, the so called empathetic scenario Sanchez and Cuesta (2005); Page and Nowak (2002), and we thus only focus on the evolution of the offer $\langle p\rangle$ to avoid redundancy. Finally, if not stated otherwise, the population size $N=1000$, and the mutation strength $\varepsilon=0.001$.

To have a better understanding of the above results, we developed a mean-field theory based on the replicator equation (RE) Taylor and Jonker (1978); Roca et al. (2009). Consider an infinitely large well-mixed population, each individual spontaneously chooses the strategy $S_{h}=(C_{h},C_{h})$ with rate $\hat{\rho}$ as above. According to the above numerical results, the players typically are either driven to the high fairness state $S_{h}$, or to a low fairness state $S_{l}=(p_{l},q_{l})$ normally with $p_{l}\!\approx\!q_{l}\!<\!0.2$.

Next, let’s denote the fraction for the high and low fairness subgroups are respectively $f_{h}$ and $f_{l}$. According to the spirit of RE, the evolution of two fractions

$$\dot{f}_{h}=\hat{\rho}_{{}_{0}}f_{h}(\Pi_{h}-\bar{\Pi})+\hat{\rho}f_{l},$$

(3)

$$\dot{f}_{l}=\hat{\rho}_{{}_{0}}f_{l}(\Pi_{l}-\bar{\Pi})-\hat{\rho}f_{l},$$

(4)

where $\Pi_{h,l}$ are their payoffs for the two subgroups and $\bar{\Pi}$ is the average payoffs for the whole population. The first term on the right hand side thus describes the selection process due to their relative fitness respective to the whole population with the rate $\hat{\rho}_{{}_{0}}$, corresponding to the economic-driven selection. The second term captures the spontaneous transition from the low fairness subgroup $S_{l}$ to the high fairness subgroup $S_{h}$ with the rate $\hat{\rho}$, corresponding to the morality-driven process. Note that, what really matters here for the competition of the two processes is their relative frequencies, therefore we set $\hat{\rho}_{{}_{0}}=1$ without loss of generality and vary the spontaneous fair rate $\hat{\rho}$. Since $f_{h}+f_{l}=1$ has to be satisfied, we shall only focus on Eq. (3). With some algebra (see Appendix), we obtain the evolutionary equation for $f_{h}$,

$$\dot{f}_{h}=(1-f_{h})[(1-\hat{\rho})f^{2}_{h}+(\hat{\rho}-C_{h})f_{h}+\hat{\rho}].$$

(5)

There are three fixed points

$$\dot{f}^{*(1,2,3)}_{h}=1,\frac{C_{h}-\hat{\rho}\pm\sqrt{(\hat{\rho}-C_{h})^{2}-4\hat{\rho}(1-\hat{\rho})}}{2(1-\hat{\rho})}.$$

(6)

The bifurcation diagram for $C_{h}=0.5$ is shown in Fig. 4(a), where $f^{*(1)}_{h}=1$ is alway a stable solution; one of the other two solutions $f^{*(2)}_{h}$, however, is unstable. A bistable region is seen when the spontaneous rate $\hat{\rho}<\hat{\rho}_{c}$, the region terminates via the saddle-node bifurcation Kuznetsov (2004); Strogatz (2015) when the two branches collide, i.e. $5\hat{\rho}^{2}-5\hat{\rho}+\frac{1}{4}=0$, leading to $\hat{\rho}_{c}=\frac{1}{2}-\frac{1}{\sqrt{5}}\approx 0.0528$. Therefore, the predicted bistable region is within $\hat{\rho}\in[0,\hat{\rho}_{c})$.

Importantly, the revealed bifurcation diagram shows that even in the classic UG (i.e. the case of $\hat{\rho}=0$), the dynamics already holds bistable properties, which was pointed out in a previous mini-UG game Nowak et al. (2000). The presence of spontaneous transition process is only to enhance the likelihood for the evolution to the $f^{*(1)}_{h}$ branch. But when $\hat{\rho}\rightarrow 0$, the basin of attraction of the solution $f^{*(1)}_{h}$ is small that random initial conditions almost always fall into the competing basin for the other stable solution $f^{*(3)}_{h}$, unless peculiar initial condition being prepared. Furthermore, even starting from $f^{*(1)}_{h}$, the mutation stochasticity could kick the system out of its basin and finally the system evolves into the solution $f^{*(3)}_{h}$. That’s why we see the jump from $S_{h}$ to $S_{l}$ when $\hat{\rho}$ is decreased before $\hat{\rho}=0$ being reached in the hysteresis loop. These facts may explain why the bistability nature of UG has long being neglected because the two basins of attraction are not comparable, and thus no related observation can be made. As the spontaneous transition rate $\hat{\rho}$ is increased, the basin of attraction of the solution $f^{*(1)}_{h}$ increases and instead the one of $f^{*(3)}_{h}$ shrinks. When the two are comparable, bistable properties are easy to see. Finally, when $\hat{\rho}>\hat{\rho}_{c}$ the basin of $f^{*(3)}_{h}$ disappears, and $f^{*(1)}_{h}$ becomes the only stable solution.

The bifurcation diagrams with different $C_{h}$ are also shown in Fig. 4(b), they show that the bistability nature is robust. The region increases as a high level of fairness is desired, while it disappears as $C_{h}\rightarrow 0$. This is reasonable because a higher level of fairness requires more effort to spark, e.g. the extreme case of $C_{h}=1$ requires nearly $20\%$ spontaneous fair acts per round to make the whole population to follow, while only $1.4\%$ is sufficient to guarantee its evolution into the target state $S_{h}=(0.25,0.25)$.

A full bifurcation diagram (see Appendix) indicates that the bistable region is actually much larger, the fair society ($f^{*(1)}_{h}=1$) theoretically can even survive in the case of $\hat{\rho}<0$, which can be interpreted as spontaneous transition from fair state $S_{h}$ to $S_{l}$. This prediction is, however, presumptively difficult to observe either in numerical simulations or in real societies, because the basin of attraction of $f^{*(1)}_{h}=1$ now becomes very small and thus extremely sensitive to any perturbation.

Note that, our mean-field treatment is self-consistent because we assume the existence of the two types of players from very beginning. Its shortcoming is also obvious that it fails to describe the transient dynamics, such as the random initial conditions used in our work.

In the real world, the population is structured neither in any regular fashion nor unstructured as studied above, the interpersonal connectivities are often depicted as complex networks Watts and Strogatz (1998). Here, we investigate Erdős-Rényi (ER) random networks Bollobás (2001) and Barabási-Albert (BA) scale-free networks Barabási and Albert (1999), which represent typical cases of homogeneous and heterogeneous complex networks. We examine their impact on the outcome of our model. For comparison, all network sizes are also $N$=1000, and the average degree $\langle k\rangle=4$.

Fig. 5(a) reports the case of ER random networks, where similar observations are made. The bistable region within the first-order fairness transition $0.01<\rho<0.05$ is almost the same compared to the above two cases. The hysteresis in the inset, indicates the $\rho_{c2}$ is also actually larger, around 0.075, due to the same reason given in the well-mixed population, but with a lesser extent. This then provides evidence that the boost of the overall fairness level by probabilistic fair behaviors is robust for homogenous structured populations.

For BA scale-free networks, however, the transition from low to high level of fairness is no longer in a discontinuous manner, see Fig. 5(b). In the absence of spontaneous fair act $\rho=0$, the average strategy $(\langle p\rangle,\langle q\rangle)\approx(0.12,0.1)$. As the probabilistic fair acts set in, the value of $\langle p\rangle$ continuously improves as $\rho$ is increased, and full fairness state $S_{h}$ is reached when $\rho>0.02$, smaller than the typical value of $\rho_{c2}\approx 0.05$ in the ER network case. This means that the network heterogeneity further facilitates the boost of fairness sparked by the probabilistic fair behaviors, but in a smooth way.

Intuitively, on heterogeneous networks, the hub nodes are expected to outperform the peripheral nodes in boosting the fairness level because they can affect much more neighbors. To validate, only one hub with the largest degree is chosen to spontaneously act within $S_{h}$ with probability $\rho$, the rest are purely economic-driven, evolving according to Eq.(2). Surprisingly, when $\rho>0.2$, the desired fairness for the whole population is reached, see the inset in Fig. 5(b). This means, when the hub players spontaneously act in the fair manner, their impact will disproportionally spreads to the whole population that could be unexpectedly to overturn the fairness level.

Heuristically, one might guess the hubs are more easily to form fair clusters that entrain the rest of the network to be in the state $S_{h}$, just like the synchronization process Gómez-Gardeñes et al. (2007) or the cooperation evolution Santos and Pacheco (2005) on the scale-free networks in previous studies. This guess however is not true, the fact is much more complex; the hubs are found that not more likely to form fair clusters, in fact the evolution of fairness is strongly fluctuating when $\rho$ is small, long intermittency of low fair state are found with nontrivial statistical features, which will be presented elsewhere.

In summary, in this work we relax the assumption of Homo economicus and allow probabilistic spontaneous fair acts, and focus on their impact on the evolution of the fairness level. We find that as this probability increases, a first-order PT is identified for all cases except for the heterogeneously structured populations. The level of fairness for the population could abruptly jumps from a low fairness state into a high fairness state. Once the population is heterogeneously structured, such as in a scale-free network manner, the transition becomes continuous, and the network heterogeneity is found to further enhance this leverage effect. Further simulations show that these findings are robust against model details, e.g., when the synchronous updating is replaced with asynchronous version Roca et al. (2009), or the Moran rule is replaced with the Fermi rule Szabó and Tőke (1998). Our mean-field treatment reveals that there is a bistable fairness structure for the evolution in UG model, and explains why the spontaneous acts make the jump to the high fairness state easier. The implication of our findings are that once occasional spontaneous fair acts are present, the emergence of a high level of fairness is possible. Since probabilistic fair behaviors are indeed often seen in the real life, the observations made here is expected to work as a plausible mechanism for the emergence of fairness.

Interestingly, part of these observations were confirmed in a previous experiment work by Brenner et al Brenner and Vriend (2006), where the human play together with the artificial players with assigned strategies, they found that a lower acceptance rate or higher $q$ leads to higher offers, i.e. a higher level of fairness. This observation was previously modeled by integrating both fairness motive and adaptive learning hypotheses, and found that the presence of a small fraction of “tough” responders who reject unfair offers indeed results in a leverage effect for the whole population Xiong et al. (2014). Though, our work points out the possibility of abrupt jump and the underlying of bifurcation structure, besides the fairness motive alone is sufficient to account for the leverage effect.

From the control point of view, our work may also provide a strategy to promote social fairness in some contexts where the fairness level is not satisfied. Instead of probabilistic fair acts, a fraction of individuals are pinned to be fair players, who are supported financially to keep that strategy, and $\rho$ is interpreted as the fraction of pinned players. Preliminary simulations show that this pinning control is very efficient to achieve a high fairness state, with relatively low control costs, which will be presented elsewhere.

We note that our model is slightly deviated from the paradigm of Homo economicus that spontaneous fair acts are allowed and we focus on its consequence by evolution. The assumption per se, however, is made a prior, motivated by the daily experience. Our work cannot explain why occasional fair behaviors are present, which has been an important subject in psychology Boehm (2012).

We are supported by the Natural Science Foundation of China under Grants Nos. 61703257 and 11747309, and by the Fundamental Research Funds for the Central Universities GK201903012. ZJQ is supported by the Natural Science Foundation of China under Grants No. 12165014.