The Evolution of Cooperation within the Iterated Prisoner’s dilemma on a Social Network

Introduction The evolution of cooperation in the iterated prisoner’s dilemma is a well researched and documented phenomenon. However, by changing a few of the underlying assumptions a more realistic and intriguing result regarding the emergence of competitive and cooperative behaviors is found. The first and foremost change posited is reevaluating the notion of utility. In this paper, utility is developed in terms of a mathematical belief structure.

Changes to the Game Structure Classical Evolutionary Game Theory –Use of Replicator Equations –Birth/Death Process Population Demographics Change –Random Draw to determine player pairings –Infinite Population Sizes –Strategies Constant Belief Based Utility Approach –Use of difference equations –Populations are constant (no birth/death process) –Player pairings retain memory (i.e. a network structure evolves) –Initial pairings based on nearest neighbor search –Strategies change as beliefs are changed

Belief Based Utility In the absence of external constraints, people tend to act in accordance with their beliefs. Needs and external driving factors may force people to act against their beliefs. In this model of the iterated prisoners dilemma two beliefs are modeled: hostility and retribution. Changes between belief and need can be modeled using a catastrophe geometry

In the case of the iterated prisoners dilemma we are interested in networked patterns of action that arise through competition and collaboration among actors. These interactions may impact the degree of cognitive dissonance among actors and (through loss) give rise to a need function altering their normal modes of behavior. The primary factors that drives this in the iterated prisoner’s dilemma is the accumulated wealth of a player. The manner in which wealth is generated is impacted by several factors that determine how a social network is constructed by actors. Belief StructuresImpacted by Socialization and Interaction Shifts Behavioral Patterns The shadow zone. Where patterns of behavior converges to need driven behavior (belief structures overridden). The illuminated zone. Patterns of behavior correspond to beliefs (cognitive dissonance is measured by amplitude.) The caustic: The dividing line between belief driven response and need driven response. Characterized By:

Results Initially, cooperative players P1 and P2 are overwhelmed by hostile players P3 and P4. However, shortly into the game P1 and P2 players begin to form coalitions and through favorable outcomes, arising from the tit-for-tat and cooperative P1 strategies, continue to grow relationships and accumulate wealth. As P1 players have a higher reputation among P2 players they are more readily accepted into the P2 coalition. Hostile players P1 and P2 attempt to invade these cooperative networks, but are not allowed due to their low reputation scores. Over time, hostile players go against their beliefs, first the P4 and later the P3 players, and begin to mimic the behavior of cooperative players. When this happens the social network is in equilibrium. The presence of critical points, seen in chart 6, that are chaotic attractors in the iterated Prisoner’s Dilemma implies many interesting things. Self-organized criticality (SOC) is introduced into the evolution of cooperation in the iterated prisoner’s dilemma. The distribution of the attractors over the game’s evolution implies a sense of temporal scale- invariance. That is the characteristic of adaptive behavior arising the temporal distribution of attractors came into being from a series of phase transitions that occurred during the evolution of the game itself. This implies self-adaptation on the part of the players.