As individuals, we are each better off when we make use of a common resource without making a contribution to the maintenance of that resource. However, if every individual acted in this manner, the common resource would be depleted and all individuals would be worse off. Social groups often display a high degree of coordinated behaviour that serves to regulate such conflicts of interest. When this behaviour emerges without the intervention of a central authority, we tend to attribute this behaviour to the existence of social norms (Axelrod 1986). A social norm is said to exist within a given social setting when individuals act in a certain way and are punished when seen not to be acting in accordance with the norm. Dunbar (1996, 2003) suggests that social structure and group size play important roles in the emergence of social norms and cooperative group behaviour.
All social dilemmas are marked by at
least one deficient equilibrium (Luce and Raiffa 1957). It is deficient
in that there is at least one other outcome in which everyone is better
off. It is equilibrium in that no one has an incentive to change their
behaviour. The Prisoners' Dilemma is the canonical example of such a
social dilemma. The Prisoners' Dilemma is a 2
2 non-zero sum, non-cooperative game, where
non-zero sum indicates that the benefits obtained
by a player are not necessarily the same as the penalties received by
another player, and non-cooperative indicates
that no per-play communication is permitted between players. In its most
basic form, each player has 2 choices: cooperate or defect. Based on the
adopted strategies, each player receives a payoff.
Figure 5.9 shows some typical values used to explore the behaviour of the Prisoners' Dilemma. The payoff matrix must satisfy the following conditions (Rapoport 1966): defection always pays more, mutual cooperation beats mutual defection, and alternating between strategies doesn’t pay. Figure 5.9 also shows the dynamics of this game, the vertical arrows signify the row player’s preferences and horizontal arrows the column player’s preferences. As can be seen from this figure, the arrows converge on the mutual defection state, which defines a stable equilibrium.
Figure 5.9. Prisoners' Dilemma
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The payoff structure of Prisoners' Dilemma: the game has an unstable equilibrium of mutual cooperation, and a stable equilibrium of mutual defection, this is shown by the arrows, moving away from mutual cooperation to mutual defection
While the 2-person Prisoner’s Dilemma has been applied to many real-world situations, there are a number of problems that cannot be modelled. The Tragedy of the Commons is the best known example of such a dilemma (Hardin 1968). While n-person games are commonly used to study such scenarios, they generally ignore social structure, as players are assumed to be in a well-mixed environment (Rapoport 1970). In real social systems, however, people interact with small tight cliques, with loose, long-distance connections to other groups. Also, traditional n-person games don’t allow players to punish individuals that do not conform to acceptable group behaviour, which is another common feature of many social systems. To overcome these limitations, I will introduce the Norms and Meta-Norms games, which are variations on the n-person Prisoners' Dilemma. They can easily be played out on a network and allow players to punish other players for not cooperating. Figure 5.10 illustrates the structure of the n-person Prisoners' Dilemma, Norms and Meta-Norms games.
Figure 5.10. The architecture of the Norms and Meta-norms games (After Axelrod 1986)
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Both games start with a variation on the n-person Prisoner’s Dilemma. The Norms Game allows players to punish those players caught defecting. The Meta-Norms Game allows players to punish those players who do not punish defectors.
The Norms game begins when an individual (i) has the opportunity to defect. This opportunity is accompanied by a known chance of being observed defecting (S) by one of i's nearest neighbours. If i defects, he/she gets a payoff T (temptation to defect) of 3, and each other player that is connected to i, receives a payoff H (hurt by the defection) of -1. If the player does not defect then each player receives a payoff of zero. To this point the game is equivalent to an n-person Prisoners' Dilemma played on a network (Rapoport 1970). However should i choose to defect, then one of his n neighbours may see the act (with probability 1-S) and may choose to punish i. If i is punished he receives a payoff of P=-9, however the individual who elects to punish i also incurs an expense associated with dealing out the punishment of E=-2. Therefore the enforcement of a social norm to cooperate requires an altruistic sacrifice.
From the above description it can be seen that each player’s strategy has 2 dimensions. The first dimension of player i’s strategy is boldness (Bi ), which determines when the player will defect. Defection occurs when S < Bi . The second dimension of i’s strategy is vengefulness (V), which is the probability that a player will punish another player if caught defecting. The greater the vengefulness the more likely they are to punish another player.
The Meta-Norms game is an extension of the Norms game. If player i chooses to defect, and player j elects not to punish i, and i and j have a common neighbour k, and k observes j not punishing i, then k has can punish j. Again, j receives the penalty P=-9, and, like the norm game, k receives an expenses E=-2.
Like the Prisoners' Dilemma, the Norms game and Meta-Norms game have unstable mutual cooperation equilibrium and a stable mutual defection equilibrium. The altruistic punishment is also an unstable strategy, as punishing an individual also requires a self-sacrifice. The stable strategy is mutual defection with no punishment for defectors. However, the global adoption of this strategy means that the population as a whole is worse off than if the unstable equilibrium strategy of mutual cooperation with punishment for defectors is adopted.
Let us imagine 2 variables (B and V) that make up a strategy are each allowed to take on a value between [0,1]. The variables represent the probability of defecting and punishing respectively. The variables are each encoded as a 16 bit binary number (as per Axelrod 1986). The evolution of players' strategies proceeds in the following fashion: (1) A small world network of 100 players with a degree of randomness p is created; (2) Each player is seeded with a random strategy; (3) The score or fitness of each play is determined from a given player’s strategy and the strategies of the players in their immediate neighbourhood; (4) When the scores of all the players are determined, a weighted roulette wheel selection scheme is used to select the strategies of the players in the next generation; (5) A mutation operator is then applied. Each bit has a 1 per cent chance of being flipped; (6) Steps 3–5 are repeated 500 times, and the final results are recorded; (7) Steps 2–6 are repeated 10,000 times. (8) Steps 1–7 are repeated for p values between 0 and 1 in increments of 0.01. The above experimental configuration was repeated for both the Norms game and the Meta-Norms game. Figure 5.11 shows the results of these simulations.
Figure 5.11. Simulation results
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(A) Average values for boldness and vengefulness over social networks with varying degrees of randomness for the Norms game. (B) Average values for boldness and vengefulness over social networks with varying degrees of randomness for the Meta-Norms game. (C) Comparison of the fitness values for the Norms game and Meta-Norms game. (D) Trade-off between Boldness and Vengefulness
From the simulation results we can see that, regardless of the social structure, the first order altruistic punishment isn’t enough to enforce the social norm of mutual cooperation. Figure 5.11(A) shows that, regardless of the social structure, the vengefulness decreases to zero, and boldness increases toward one. Essentially all players are attempting to exploit the shared resource, with no fear of being punished. However, for the Meta-Norms game, with second-order punishment, there is a distinct set of circumstances when the population as a whole will not exploit the common resource. Figure 5.11(B) shows that, when the social structure is regular and highly clustered, players boldness decreases, but as the social structure becomes more random (and clustering breaks down), the boldness of a given player increases, and each individual attempts to exploit the common resource. However, the level of exploitation is lower than that observed in the Norms game. These differences in system behaviour are also seen in the average payoff received by a player (Figure 5.11(C)). The average payoff per player in the Meta-Norms game is always higher that that received in the Norms game. The average payoff for the Meta-Norms game maximises just before the transition to a state of global exploitation. Statistical analysis of the network structure reveals that this maximum payoff point coincides with the breakdown of clustering within the network. Finally, Figure 5.11(D) depicts the trade-off between vengefulness and boldness. The Norms game (squares) converges to a strategy of low vengefulness and high boldness. While the Meta-Norms game produces a range of behaviours (circles), from the plot it can be seen that there is a trade-off between boldness and vengefulness. The Meta-Norms game produces a wide variety of strategies. These strategies are governed by the topology of the underlying social network. The trade-off surface can be thought of as the set of viable strategies, as nonviable strategies (such as high boldness and vengefulness) are selected against.
The results from the previous section provide a number of interesting insights into the emergence of social norms and group behaviour. Social structure and second order interactions seem to play an important role in the evolution of group behaviour. In the wider literature, there are many recorded instances where these 2 factors have been observed to influence group behaviour. Here I will explore 3 examples.
Japanese macaque were among the first primates observed by humans to display innovation and diffusion of new novel behaviours to other group members (Reader and Laland 2004). While many individual animals invent new behaviour patterns, most new behaviours (even if they are beneficial) are unlikely to become fixed within the community. Reader and Laland (2002) have shown that there is a link between the social structure of primates and the frequency with which new technologies are uptaken. Populations that tend to be more cliquish are more likely to adopt a new behaviour as member of the clique help to reinforce the novel behaviour.
Dunbar (1996) has shown that there is a correlation between neocortex size and the natural group size of primates. Also correlated with neocortex size is the cliquishness of the social structure. Dunbar (2003) conjectures that the increase in neocortex size may mean that individuals can manage and maintain more group relationships. The ability to maintain more complex relationships may allow individuals to locally enforce social behaviour. It has also been observed that, when primate groups grow too large, social order breaks down and the troop split into 2 or more smaller troops in which social order is re-established (Dunbar 2003).
The notion of Meta-Norms is widely used in denunciation in communist societies. When authorities accuse someone of doing something wrong, others are called upon to denounce the accused. Not participating in this form of punishment is itself taken as a defection against the group and offenders are punished.
In this section, I explored the emergence and enforcement of social norms through the use of 2 variations on the n-persons Prisoners' Dilemma. The simulation results suggest that a combination of second order interactions, altruistic punishment and social structure can produce coherent social behaviour. Such features have been observed to enforce norms in a number of social systems. The results from this study open a number of interesting future directions:
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As conjectured by Dunbar (2003), social order in primate troops breaks down when the troop becomes too large. This raises the question: What is the relationship between link density, number of nodes and other network statistics, and how do these statistics influence the behaviour of evolutionary games such as those described in this chapter?
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Coalitions and factions form and dissolve through time. How do the general results change if the underlying network is allowed to evolve?
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Several studies (Luce and Raiffa 1957) have shown that concepts such as the Nash equilibrium don’t hold when rational players are substituted for human players. Do the patterns and tradeoffs described previously hold when rational computer players are replaced by human decision makers?
All these questions require further experimentation but can be explored in the context of the framework proposed here.