Theorists have posited that pure tit-for-tat strategies in iterative prisoners dilemma games were invulnerable. Is this correct? The authors seek to answer this question by examining the ability of various prisoners dilemma gaming strategies to withstand invasion by other competing strategies.

Bender and Swistak examine a gaming strategy universe that includes the strategies:

• Tit for tat - a player will initially cooperate and then in future rounds mimic the behavior of their opponent.
• Tit for 2 Tats - a player will cooperate for the first two rounds and then defect in rounds where their opponent defected in the previous two.
• Suspicious Tit for Tat - a player will initially defect and then will mimic their opponent in future rounds.
• Always Defect
• Always Cooperate
• Grim Trigger - Begin by cooperating, if opponent defects then always defect afterward.

These strategies were examined in pure conditions where only one existed, and then competing strategies were introduced. If a given strategy could withstand incursions by competing strategies it was deemed "stable".

Stability proved to be a continuum. All strategies proved to have points of equilibrium. At this point, a strategy can withstand its maximum level of incursion. That point is that strategy's maximum stability.

Strogatz examines the underlying process of creating patterned behavior in situations where there is no obvious conscious control or even intention. These phenomena arise from “coupled oscillation”—that is, the tendency of phenomena at all levels of existence to synchronize their rhythmic features. The classic example: southeast Asian fireflies that flash in synchrony over miles of countryside.

Other natural examples are discussed:

• earthquakes
• forest fires
• mass extinctions
• synchronization of female menarche
• heart attacks

The Mathematics of Sync

The underlying requirement for coupled oscillation or sync to occur is for phenomena to operate in cycles and for the players in the phenomena to be able to influence each other mutually. In addition, a catalyst may sometimes be necessary. One of variables is pulsed communication vs. continuous interaction: continuous interaction creates more complex, subtle sync.

Some observed qualities/principles of sync:

• One-to-one coupling grows to many-to-many coupling.
• Catalysts tend to initially sync some players but desync others further.
• If one oscillator kicks another over a threshold, they will remain synchronized forever.
• Self-organized criticality and cascading behavior (described by Per Bak’s statistics of cascades).
• Frequency pulling (from Norbert Wiener): the tendency of oscillators to pull stray patterns into sync with the group is the universal mechanism of self-organization.
• Influence function (amplitude), sensitivity function (tendency to speed up or slow down), and level of connectivity shape the individual behaviors of players in the system and the time it takes them to self-organize (Winfree).
• Whenever the whole is different from the sum of the parts; whenever cooperation or competition is going on, the governing equations are nonlinear.
• A threshold of similarity is equivalent to a “phase transition.”
• Kuramoto’s rule: oscillators symmetrically adjust by making the minimum adjustment that allows them to communicate.

Disturbances to an equilibrium system grow as a function of the similarity of the individual players; if the players are nearly identical, the disturbances grow exponentially.

Link between biology and physics: “mutual syncronization is analogous to a phase transition, like the freezing of water into ice. The main difference is that when oscillators freeze into sync, they line up in time, not space.”

Frequency pulling tends to produce a pattern distribution that is unlike the familiar bell curve; instead it has a tall, narrow central peak and two weak peaks on either side—this is a possibly a description of a “standard” distribution to a synchronized or self-organized system.

“Virtually all major unsolved problems in science today have this intricate character…a complex, self-organizing system where everyone changes the state of everyone else.” Examples cited: biochemical cell reactions that lead to cancer; stock market booms and crashes; emergence of consciousness from firings of brain neurons; origin of life in the chemical reactions of the primordial soup.

Kuramoto’s rule in more detail: the amount of adjustment between pairs of oscillators is given by the sine function of the ange between them, multiplied by a number called the “coupling strength,” which determines the maximum possible adjustment. Breakthrough in this idea was the symmetrical relationship between oscillators, compared to Winfree’s concepts of frequency pull and sensitivity.

Kuramoto continued: all systems will migrate toward a state in which the order parameter and speed of the pack are constants. There are ultimately only two such states: an order parameter of 0, in which the system will never display synchrony; a “partially synchronized” state consisting of three groups: a synchronized pack of average speed, a slower desynchronized swarm of dawdlers, and a faster desynchronized swarm of sprinters. This latter case is possible only up to a certain threshold of diversity. You can predict how ordered the pack will be as a function of with width of the bell curve.

David Welsch & Steve Reppert (Mass Gen Hospital): “the brain contains a population of oscillators with distributed natural frequencies which pull one another into synchrony and make a more accurate oscillator en masse than individually. Wiener anticipated all that, but he missed an important detail: Instead of cycling 10 times per second, these cells cycle about a million times slower. These are the cells of the circadian pacemaker, the internal chronometer that keeps us in sync with the world around us.”

Strogatz’s breakthrough idea was to view oscillators as fluids.

Sync and cooperation:

“Reproductive sync has benefits for all if the females in the group are cooperative…It could be that women unconsciously strive to ovulate and conceive in step with their friends (to allow them to share child-rearing and breast-feeding duties) and to keep out of step with their enemies (to avoid competing with them for scarce resources)….Female rats in a synchronized group produce larger and healthier offspring than those reared by a solo mother.”

Cooperation in the context of oscillators means ability to sense one another’s rhythms and react to stay in step. [implications for growth of sensors?]

“When the system was self-synchronizing, Winfree found that no oscillator was indispensable. There was no boss. Any oscillator could be removed and the process would still work. Furthermore, the pack did not necessarily run at the speed of its fastest member. Depending on the choice of influence and sensitivity functions, the group could run at a pace nearer the average speed of those in the pack, or it could go faster or slower than any of its members. It was all wonderfully counterintuitive. Group synchronization was not hierarchical, but it wasn’t always purely democratic either.” (p. 52-53).

Human problems that sync can help explain, solve, interpret:

What causes fads, crowd behavior, and mob psychology?

While much of sync theory focuses on rhythmic phenomena, repeating the same cycles, human behavior is more complex. Thresholds are a focus here. Relevant research comes from:

• Thomas Schelling: discoverer of the “tipping point.”
• Mark Granovetter: rioting behavior (interesting that it is hard to distinguish a person who will readily join a riot from one who will join only after someone else starts rioting; i.e., normal sociological and psychological segmentations won’t catch the difference); also the threshold is quite low.
• Duncan Watts: innovation modeling by network analysis reveals two tipping points: 1) islands link together creating global cascades; 2) a dilution effect when cascading vanishes entirely because a node has too many neighbors. Also the concept of a “vulnerable cluster” which is equivalent to “early adopters.”

Traffic congestion

A basic chaos theory problem; key research comes from:

• Dirk Helbing and Bernardo Huberman: auto traffic synchronizes at a certain density, with large trucks setting the gating speed.
• Boris Kerner and Hubert Rehborn: traffic slows to crawl as on-ramp increases density, but stays in that sync for two hours after the heavy on-flow, “trapped” in a stable, but suboptimal sync; they have to be “defibrillated” in order to regain free flow.

Intentional collective action:

Examples of this are dance, singing, “waves” at football games, audience applause (in Europe); and on the dark side: totalitarianism: Nietszsche: “In individual, insanity is rare, but in groups, parties, nations, and epochs it is the rule.”

How the brain gives rise to the mind:

Acts of cognition are linked to brief surges of neural synchrony.

• Christoph von der Malsburg (USC) asked what physical process binds all the chaos of sensations to allow the perception of a single object. Hypothesis: separate banks of neurons throughout the brain all oscillate in sync for a fraction of second.
• Charles Gray and Wolf Singer: tests hypothesis with cats.
• Jurgen Fell (U. of Bonn) tests human subjects (epileptics with implanted electrodes), found link between sync activity on first viewing words to be memorized and ability to recall later.
• Francisco Varela: “mooney face” experiments reveal a flurry of “gamma oscillations” that mark the moment of perception, recognition occurs when discharges in multiple regions of the brain sync, followed by “active desynchronization.”
• Christof Koch and Francis Crick: Consciousness may be the subjective experience of these states of synchrony passing through our brains.

“The study of social dilemmas is the study of the tension between individual and collective rationality. In a social dilemma, individually reasonable behavior leads to a situation in which everyone is worse off. The first part of this review is a discussion of categories of social dilemmas and how they are modeled.” The Prisoner’s Dilemma, the problem of providing public goods, and Hardin’s Tragedy of the Commons are three powerful metaphors that facilitated and structured research but also served as blinders since their limitations are often not recognized.

Models:

Kollock’s analysis divides dilemmas into two-person and N-person dilemmas. The key two-person dilemmas are the Prisoner’s Dilemma, the Assurance Game, and the Chicken Game. Each of these models is defined by the ordering of four possible outcomes: mutual cooperation, mutual defection, and either first or second person’s unilateral defection. Each of these outcomes generates an individual benefit for each person and is ordered by the benefit for the first person.

The Prisoner’s Dilemma models unsecured transactions, e.g. buying and selling over the Internet. The best outcome of a Prisoner’s Dilemma is unilateral defection of the first person, followed by mutual cooperation, mutual defection, and the worst outcome is the first person’s unilateral cooperation. Since defection has the highest potential benefit and cooperation the highest potential risk, the equilibrium of the Prisoner’s Dilemma is mutual defection. This equilibrium is deficient because the best outcome for both players is mutual cooperation.

The Assurance Game is similar to the Prisoner’s Dilemma except it models situations where mutual cooperation is more benefical for each player than unilateral defection, e.g. a project that requires collaboration. This extra motivation to mutually cooperate creates two equilibria, one optimal, which is mutual cooperation, and one deficient, which is mutual defection. The optimal equilibrium requires trust between the two persons sufficient to assure each other that the other will cooperate. Insufficient trust leads to the deficient equilibrium.

The Chicken Game is again similar to the Prisoner's Dilemma except mutual defection is the worst outcome, worse than unilateral cooperation. This replaces the Prisoner’s Dilemma’s mutual defection equilibrium by two equilibria, unilateral defection and unilateral cooperation because of the strong motivation to not mutually defect. The Chicken Game is a model for situations that require volunteer effort to avoid the worst outcome but where duplicate effort is less desirable.

Kollock divides N-person dilemmas into two types based on cost and benefit for each individual. The first type is known as the social fence,s where an individual is presented with an immediate cost that generates a benefit shared by all. The individual wants to avoid the cost but if all do, everyone is worse off. A common metaphor of the social fence is the provisioning of public goods, which are (to a varying degree) non-excludable and nonrival. The key characteristic of a public good dilemma is the production function which defines the relationship between the level of resources contributed and the level of public good provided. Production functions are classified into decelarating, linear, accelerating, and step functions. Various production functions can produce N-person versions of any of the 2-person dilemmas.

The second type is know as social trap where the “individual is tempted by an immediate benefit that produces a cost to all. If all succumb to the temptation, the outcome is a collective disaster.” The usual metaphor of the social trap is the tragedy of the commons. A key feature of commons dilemmas is that the benefits are non-excludable (or difficult to make excludable) and subtractable. The key characteristic of commons dilemmas is the carrying capacity of the commons which depends on the replenishment rate of the subtractable joint resource.

Important (but not inevitable) features that affect N-person dilemma dynamics and contrast them to two-person dilemmas are anonymity, diffusion of defection cost, and little or no direct control on others. Some of these features are also found in two-person dilemmas, e.g. blaming defection on out-of-control circumstances is a form of anonymity in two-person games.

Solutions:

“The second part of [Kollock’s paper] is an extended treatment of possible solutions for social dilemmas. These solutions are organized into three broad categories based on whether the solutions assume egoistic actors and whether the structure of the situation can be changed: Motivational solutions assume actors are not completely egoistic and so give some weight to the outcomes of their partners. Strategic solutions assume egoistic actors, and neither of these categories of solutions involve changing the fundamental structure of the situation. Solutions that do involve changing the rules of the game are [called] structural solutions.”

The motivation of not completely egoistic actors to cooperate is influenced by social value orientation, communication, and group identity. The social value orientation of a person seems to be acquired from the person’s social environment and is some linear combination of a cooperator who tries to maximize joint outcome, a competitor who tries to maximize own outcome relative to partner, and an individualist who tries to maximize own outcome. Kollock does not find any conclusive results in how to influence social value orientation but does find evidence that it varies between different countries.

The presence of communication positively affects cooperation rates. Communication enables a person to find out about others’ choices, to make explicit commitments, to appeal to what is the moral thing to do, and most importantly, to create or reinforce a sense of group identity. The effect of group identity is in fact so strong that it can affect cooperation rates even in the absence of communication. In-group behavior of individuals frequently includes personal restraint and treating Prisoner’s Dilemma situations as Assurance Games. However, in-group behavior implies out-group behavior with the potential to cause severe social costs due to intergroup conflicts.

“[Strategic solutions] rely on the ability of [egoistic] actors to shape to shape the outcomes and hence behavior of other actors. For this reason, many of these strategic solutions are limited to repeated two-person dilemmas.” Axelrod (see The Evolution of Cooperation) identifies three requirements for strategic solutions: ongoing relationships between actors (i.e. all expect shared dilemmas in their future), ability to identify each other, and ability to keep track of the other’s past behavior.

The most successful strategy in iterative Prisoner’s Dilemma tournaments (everyone against everyone) that meet these requirements is Tit-for-Tat which starts out with cooperation and then matches the partner’s previous behavior. This strategy transforms a repeated Prisoner’s Dilemma into a repeated Assurance Game since the only long-term outcome of this strategy is either mutual cooperation or mutual defection (the two equilibria of the Assurance Game). Key aspects of successful strategies in repeated Prisoner’s Dilemma tournaments are (1) to realize that it is not a zero-sum game hence does not benefit from a competitive social orientation (“don’t be envious”), (2) to not defect first, (3) to reciprocate both cooperation and defection, and (4) to be predictable so that the partner clearly understands one's strategy. One important caveat is that repeated Prisoner’s Dilemma tournaments assume perfect communication. In real life where communication is often imperfect more generous or forgiving strategies can avoid accidental cycles of recrimination.

Recent evidence suggests that the strategy of choosing partners is more important than the strategy used within a dilemma. In a modified version of iterative Prisoner’s Dilemma tournament actors can exit current relationships and choose alternative partners. A very successful strategy in this environment is Out-for-Tat which exits a relationship as soon as the partner defects. A more forgiving version that gives a defecting partner a second chance is even more successful.

Strategies for N-person dilemmas involve grim triggers, social learning, and group reciprocity. In a “grim trigger” strategy an individual only cooperates if all other group members cooperate and defects as soon as one other group member defects. Social learning is the basis of a cognitively less taxing class of strategies that involves imitating other group members and look for thresholds in public good provisioning instead of calculating marginal rates of return or figuring out dominating strategies. Group identity increases cooperation rates because group members follow strategies that assume that all members share a strong expectation of group reciprocity (reciprocity within the group).

Structural solutions change the rules of the dilemma thereby changing or eliminating it. One approach is to reinforce prerequisites for strategic solutions by introducing long-term accountability (shadow of the future) that influences individual reputations. However, accountability and reputation are not sufficient to escape the Prisoner’s Dilemma’s equilibrium of mutual defection (in two- or N-person version) if the means to encourage cooperation are too weak (e.g. production function for public good too flat or too much effort required to reach provisioning point).

Many people seem to positively weigh others’ outcomes since cooperation increases significantly as the benefits to others from one’s cooperation increase. Cooperation levels are also higher if group members are asked to contribute to a non-divisible public good that only benefits the whole group, probably due to an increased sense of group identity (see group reciprocity).

Cooperation in N-person dilemmas increases if individual contributions have (or are perceived to have) a discernable effect, i.e. make an efficacious contribution. For public goods with step-level production function one can create a minimal subgroup that requires every member to contribute in order to reach the provisioning point or let two groups compete for contributions, turning an N-person Prisoner’s Dilemma into an N-person Chicken Game. Another example are "matching grants" or "adopting" an individual from a large group of benefactors.

Increasing group size makes defection more anonymous and increases the cost of organizing. However, research results on cooperation depending on group size alone are inconclusive. In the case of highly non-rival goods with a threshold production function a larger group is more likely to contain a "critical mass" of cooperating individuals. Diversity of group members' interests and resources encourages formation of critical mass.

A common structural strategy for N-person dilemmas is the creation of boundaries in an attempt to make public goods or commons more excludable. There are three main approaches: The first one is to institute an external authority or trusted leader to govern access to commons. This approach appears to be less preferable if other structural changes are possible. Establishing an external authority can raise severe problems of justice, enforcement, corruption, and scalability. The second approach is to break up commons into private parcels assuming that individuals will take better care of own property than common property. However, privatization does not work for non-divisible goods, raises the social question of who gets to own commons, does not prevent owners to routinely destroy their own property (“tragedy of enclosure”), and requires institutional support to enforce private property rights. A third approach is to locally regulate “access to and use of common property by those who actually use and have local knowledge of the resource.” One key characteristic of successful and long-lasting local regulations is clearly defined boundaries.

Sanctions are a structural method to encourage cooperation where the outcomes themselves of N-person dilemmas are too weak of a motivator. However, the implementation of sanctions can be very expensive. Local monitoring and sanctioning systems are more practical and less costly. Another way to reduce cost is to use a graduated system of sanctions with low-cost conflict resolution. A sanctioning system is itself a public good and therefore poses a second-order dilemma. Communities with a high level of trust readily cooperate in a first-order dilemma but cooperate less in a second-order dilemma hence are less willing to support a sanctioning system. The opposite is true for communities with a high level of distrust.

Six Degrees begins in the beginning. Stanley Milgram's initial small world studies are analyzed. His findings in seeing if a group of people in Nebraska can get a letter to someone in Massachusetts are scrutinized. Milgram left a puzzle. Mathematically, six degrees of separation can be shown and intuitively it is appealing. But do social networks actually work that way?

Initially, Watts steps into the world of pure mathematic theory. Graph theory and random graphs are employed to build potential worlds in which connections can be made. These tools are detailed and their histories explained.

Watts and his colleagues then take the science to new levels, by introducing sociology, epidemiology, economics, and business models into this new multi-disciplinary science. Immediately, each new field of study brings with it new insights into network dynamics.

This convergence of disciplines reveals the social, transportation and technological networks that make up our world. These networks are, ultimately, made up of individuals. Individuals in turn relate back to the networks and define how they operate.

Socially, people relate to their network by clustering. Clusters are logical organizations of network elements. In a social context, we might cluster in terms of a religion, a favorite author, a school we are attending or an affinity for a type of food. Some of these have very close physical distance, while others have a social distance with members spread out over a large area.

Networks of this type are, to various extents, “scale-free” networks. If graphed these networks roughly follow a classic power law trend where the level of connectivity between two nodes in a network increases dramatically as more nodes are connected. Real-world scale-free networks tend to have highly connected hubs which rapidly, purposely, and efficiently transmit pertinent or pervasive content from one location to another. In social circles, these are networkers. In the airline network these are hub airports. In traffic they would be freeway interchanges.

Due to this architecture, the Internet and modern air transport have combined to greatly decrease the role of proximity in our social networks. This has had great impacts on commerce, tourism, cultural sensitivity and other social factors. However, it has also led to great risks in the transmission of diseases, sensitivity to distant economic fluctuations, and rapid spread of misinformation.

These dynamics create a type of network that Duncan calls simultaneously robust and vulnerable. Their strength and weakness is that, with rapid transmission from cluster to cluster, anything can move quickly from one location or group to another. He uses the example of Toyota, whose network of suppliers was organized in such a way as to quickly compensate for and recover from a potential economic catastrophe.

Stable scale-free networks do not rely on a rigid hierarchy to provide direction in times of crisis. Rather, the structure of the network itself can rapidly respond to an unforeseen situation.

Their network was arranged in such a way as to foster and reward communication. This communication helped cope with ambiguous or unplanned situations. Rather than paralyzing Toyota while people waited for a decision from a rigid hierarchy, the contractors in the network were able to analyze the calamity and provide a rapid response to it.

As mentioned above, this robustness also rapidly transmits malicious content as well. The Melissa Virus, SARS and Ebola are analyzed to show why the network did or did not transmit them and, when it did, how they eventually died out.

Watts ends this book by summarizing that the multidimensional nature of social distance is sometimes counterintuitive and subjective. People can feel close in a network sense to people they are physically distant from and, conversely, socially distant from people physically nearby.

He continues by warning that social and physical distances have shrunk. People can quickly travel from place to place and economies are highly interdependent. The sheer number of dependencies in the modern world may yield surprising results from seemingly insignificant actions.

He finishes by showing the stability of our networks with the example of how New York adapted to the 9-11 attacks. The City bounced back to semi-normal operations within a week. During the disaster, the best laid plans of emergency operations staff were scuttled by the utter unavailability of facilities and services designed to copy with disasters. The network will provide.

The self-regarding and outcome oriented picture of human behavior presented in traditional economics does not explain why humans care so much about each other and about how social interaction is carried out, not just the end goals. The Ultimatum Game, designed by Werner Guth, is just one illustration of how real people will not always follow the dictates of self-interested rationality. Two subjects are given a sum of money, one is given the power to divide the sum, and the other can either accept or reject (in which case neither get any money). Research from conducting hundreds of trials of the game with thousands of students in Europe, Japan and the USA has shown that the responders frequently reject low offers and proposers frequently propose near equal divisions, even though it is to their monetary disadvantage. While early experiments on undergraduates seemed to suggest that there was a universal sense of fairness, extended research in different cultures (hunter-gatherers, slash-and-burn agriculturists, nomadic pastoralists) has exposed much cultural variation in responses, indicating that local cultural conditions play an important role in how people approach cooperation.

While mean proposals for university students from all over the world was usually between 42 and 48 percent, mean proposals from this cross-cultural study varied from 25 to 57 percent. Rejection rates, the action of the responders, also varied considerable between groups. Individual-level economic and demographic variables did not explain behavior as well as group-level behavior, and game play often could be connected to the people's common patterns of interaction. For example, the Orma recognized that one of the experiment's games was similar to the harambee, a local institution of giving to public goods like roads and schools. They began calling it 'the harambee game' and displayed highly prosocial behavior. In other groups, like the Au and Gnau, frequent rejection of generous offers can be explained by a cultural association with gift-giving: accumulating gifts, even if unsolicited, can imply a lowered status and force the receiver into future obligations or political alliance. The cross-cultural study showed that, in the case of groups at the extremes of behavior, "contrasting behaviors seem to reflect their differing patterns of everyday life, not any underlying logic of hunter-gatherer life ways."

The effect of market integration on cooperation to obtain a monetary reward can be explained easily: individuals from market-oriented societies when put in the context of one of the games are able to seek analogues in their daily activities of using and trading money with strangers. "Those who do not customarily deal with strangers in mutually advantageous ways may be more likely to treat anonymous interactions as hostile or threatening, or as occasions for the opportunistic pursuit of self-interest."

The seven games explored here, ultimatum, public goods, dictator, prisoner's dilemma, trust, gift-exchange, and third party punishment, can be used both as metaphor to describe prototypical situations in the social world and as a tool to predict the behavior of players in the context of other players' likely actions. Data on the responses of real players can help guide the formation of successful and sustainable institutions for collective action. In a public goods game, for instance, contributions to the public good declined over repeated periods as cooperative players eventually became frustrated with an instigating group of free-riders. Once the structure of the game is altered to allow for punishment of free riders, the average contribution rises steeply to over 95 percent of the endowment. The actual rate of punishment does not have to be that high to generate this increase either; "the mere threat of punishment, and the memory of its sting from past punishments, is enough to induce potential free riders to cooperate."

Another alteration that increases cooperation is permitting communication. "Communication allows the conditional cooperators to coordinate on the cooperative outcome and it may also create a sense of group identity." In the trust game, an investor gives an amount to a trustee, which is tripled and the trustee can give any amount from all to nothing back to the investor. Positive reciprocity, a sense of obligation to repay trusting investors that arises in the trust game, is an important key to harnessing cooperation. Implicit social contracts built on the basis of positive reciprocity are cheaper to implement and can be more successful than explicit contracts.

The environment of our evolutionary adaptation can theoretically explain the origin of these preferences in repeated game settings. Evolution equips people with the cognitive ability to learn social norms and resulting strategies rather than having them hard-wired into the brain. This accords with the game theory conclusion that the best strategy depends on the structure of social relations and potential for norms to take hold and be effective.