Invasion of Optimal Social Contracts
1. Introduction
Here, we investigate how the dynamics of social contracts are influenced by incentives promoting diversity. Specifically, we use a staghunt game to model the social contract and a snowdrift game to model the diversity incentive. The population is partitioned into groups, and each individual belongs to a subset of groups where the staghunt game is played. We assume that there is one group where individuals are encouraged to adopt a strategy that deviates from the majority. In this group, the individuals play a snowdrift game. The strategies evolve through imitation dynamics, where those yielding higher payoffs spread at higher rates. Our results show that moderate snowdrift incentives are sufficient to shift the equilibrium towards the optimal social norm, which becomes stable even after the diversity incentives are turned off. However, if individuals interact with several others in different groups, more significant snowdrift incentives may be necessary due to overall social pressure. We also demonstrate that similar outcomes can be observed in populations structured in square lattices, where the staghunt and snowdrift games are played within a range of neighbors. Our analysis is supported by computer simulations and analytical approximations that allow us to explain the results using simple game theory concepts.
2. Model
$$\begin{array}{c}\hfill \mathrm{Staghunt}=\left(\begin{array}{cc}1& \delta \\ 1r& 0\end{array}\right).\end{array}$$
The best solution is to adopt the same strategy as the partner. Because $r>0$, the norm A is the social optimum. However, if everyone is adopting B and a single individual moves to the social optimum A, this individual faces the risk of receiving $\delta $, which is worse than 0.
$$\begin{array}{c}\hfill \mathrm{Snowdrift}=\left(\begin{array}{cc}1& {\delta}^{\prime}\\ 1+{r}^{\prime}& 0\end{array}\right).\end{array}$$
The best solution is to adopt the opposite strategy to your partner. Notice that if strategy A interacts with strategy B, the individual adopting B receives $1+{r}^{\prime}$, which is larger than the payoff that A receives, which is ${\delta}^{\prime}$. That is, the diversity incentive is not benefiting the social contract A. As a last remark, the payoff obtained when both players adopt A is larger than that obtained when both players adopt B. However, this is not an issue because the diversity incentive is most important when strategy A is the minority.
$${\pi}_{f}={\pi}_{f}^{Sh}+\alpha {\pi}_{f}^{Sd}.$$
Notice that the parameter q determines the extent to which the staghunt payoff affects the cumulative payoff, which increases with the number of groups that an individual is connected to.
$${P}_{f\leftarrow z}=\frac{1}{1+exp\left(({\pi}_{z}{\pi}_{f})/K\right)},$$
where K is the selection intensity representing the population level of irrationality. We set $k=0.1$ to allow some level of irrationality but still maintain the effective selection of the most successful strategies. Our results are robust to variations of K. However, if K is too large, the payoff difference has a low impact on the outcome and the evolution become close to neutral evolution. Each generation consists of N repetitions of the imitation step.
3. Results
$$\frac{dx}{dt}=$$
where x is the fraction of individuals adopting A, while ${\pi}_{A}$ and ${\pi}_{B}$ are the average payoffs of individuals adopting A and B, respectively (see Appendix A). For $n=1$, there is a single wellmixed group where all individuals play the two games. Thus, the cumulative payoffs are effectively determined by the sum of the matrices of the two games:
$$M=\left(\begin{array}{cc}1+\alpha & \delta +\alpha {\delta}^{\prime}\\ 1r+\alpha (1+{r}^{\prime})& 0\end{array}\right).$$
, and is given by:
$${x}^{*}=\frac{\delta \alpha {\delta}^{\prime}}{\delta +r\alpha ({\delta}^{\prime}+{r}^{\prime})}.$$
Let us first recall the main results for the staghunt dynamics by setting $\alpha =0$. If the fraction of A at time t is such that $x<{x}^{*}$, because ${\pi}_{A}$
for x in that range, the population goes to the state $x=0$ (all B). If $x>{x}^{*}$, the population goes to state $x=1$ (all A). Thus, if a small amount of A (a fraction $\u03f5$) invades a population initially at $x=0$, as long as $\u03f5<{x}^{*}$, the invader has no chance. The invasion must be large enough to overcome the invasion barrier determined by the unstable equilibrium ${x}^{*}$.
$$\frac{\delta}{{\delta}^{\prime}}<\alpha <\frac{r}{{r}^{\prime}},$$
then, not only can A invade, but it will certainly dominate the population. Equation (8) is obtained by a simple Nash equilibrium analysis of the payoff matrix in Equation (6), which coincides with the fixed point analysis of the replicator Equation [48]. For $\alpha $ in this interval given by Equation (8), the payoff structure becomes equivalent to that of a Harmony game, where A is a global attractor of the dynamics. If the goal of the snowdrift incentive is to shift the population to the social optimum without ending at a coexistence equilibrium, then the moderate $\alpha $ solution is the best. Thus, if the incentive provided by the snowdrift is moderate, the optimal social contract can invade the population and persists even if the snowdrift incentive is turned off.
changes as the fraction x of A and the incentive $\alpha $ vary. If there is no snowdrift incentive ($\alpha =0$), only a massive conversion of A will drive the population to the norm B. However, if a moderate incentive is provided, any initial fraction of A will convert the population. As expected, if $\alpha $ is excessively large, the dynamics change and coexistence will be the final state, independently of the initial conditions.
$$M=\left(\begin{array}{cc}\tilde{q}+\alpha & \tilde{q}\delta +\alpha {\delta}^{\prime}\\ \tilde{q}(1r)+\alpha (1+{r}^{\prime})\end{array}\right)$$
where $\tilde{q}=1+q$
is the number of groups where the stag hunt is played. The dynamics are now determined by the impact of the staghunt payoff relative to the snowdrift payoff, which is controlled by the parameters $\alpha $ and q.
$$\tilde{q}\frac{\delta}{{\delta}^{\prime}}<\alpha <\tilde{q}\frac{r}{{r}^{\prime}}.$$
$$\alpha \frac{{r}^{\prime}}{r}<\tilde{q}<\alpha \frac{{\delta}^{\prime}}{\delta},$$
then the effective game is the harmony game and the strategy A can invade and dominate. In other words, only if the number of groups is moderate can the new social contract invade and dominate.
, while in Figure 3b, the fraction of A is shown as a function of the snowdrift incentive $\alpha $. The optimum influence of the incentive $\alpha $ over the success of A is felt for a moderate number of groups since x is at its maximum value for intermediate $\tilde{q}$ (see Figure 3a). If the system is highly connected, then stag hunt dominates, and the social norm A disappears when $x\left(0\right)$ is low. Notice that the incentive $\alpha $ has to be low to keep snowdrift from dominating the dynamics, as shown in Figure 3b.
To further investigate the robustness of our results, we also consider a square lattice version of our model. Each agent occupies a site in a square lattice and plays stag hunt and snowdrift with all the sites within a range of ${R}_{SH}$ for stag hunt and ${R}_{SD}$ for snowdrift, with ${R}_{SH}\ge {R}_{SD}$. The distance between two neighboring sites is set to be 1. The interaction ranges delimit groups and play a similar role to the parameter q in the previous version of the model.
. Let ${x}_{eq}$ be the fraction of A at equilibrium for a nonzero $\alpha $. If $\alpha $ is set to zero after the system reaches equilibrium, the population moves to $x=1$ if ${x}_{eq}>{x}_{SH}^{*}$ because, in this case, the population is in the basin of attraction of A. In the limit of infinite $\alpha $, the condition ${x}_{eq}>{x}_{SH}^{*}$ for $n=1$ is given by
$$\frac{\delta}{{\delta}^{\prime}}<\frac{\delta +r}{{\delta}^{\prime}+{r}^{\prime}}.$$
4. Discussion and Conclusions
The persistence of social contracts can be attributed to the fact that it is typically disadvantageous to deviate from established norms, thus making it difficult for a new state of equilibrium to emerge, even if it offers greater payoffs. However, our findings indicate that introducing an incentive for diversity facilitates establishing a new equilibrium.
We use the staghunt game to model the social contract and the snowdrift game to model the incentives. Our model assumes that individuals have a close group of friends and trusted contacts, where they are more likely to adopt new norms. Our findings demonstrate that if the snowdrift incentive is more moderate, then the population is driven to the optimum social contract, which is an equilibrium even without the incentive. However, if the incentive is large, the population is trapped in a mixed equilibrium where the old and new social contracts coexist. In this case, if the population successfully overcomes the staghunt invasion barrier, the incentive can be turned off, and the staghunt dynamics drives the population towards the best social contract.
Our model does not account for social contracts that are groupspecific. Instead, we assume that all individuals, regardless of their group affiliation, have equal options and obtain payoffs from the same staghunt game. Furthermore, as several societies regard diversity and inclusion as important moral values, we should stress that our model does not advocate against these values when we turn off the snowdrift incentive after achieving the new social optimum. We assume that moving to the new equilibrium is advantageous for all individuals involved. Consequently, the lesson is that policies promoting diversity can aid in coordinating actions toward new equilibria.
Changing social norms is a complicated matter, and we acknowledge that numerous factors likely influence the evolution of social contracts. Our research aims to illuminate one fundamental incentive structure: social norms do not encourage deviant behavior and policies that promote diversity and inclusion facilitate the emergence of different norms. When a block rests on an inclined plane held in place by static friction, gravity acts on it even if it remains stationary. Likewise, although there may be several other social forces, incentives for diversity and inclusion are forces pushing the population towards the desired social optimum.
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