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1
The Evolution of Cooperation within the Iterated
Prisoners dilemmaon a Social Network
2
Introduction

• The evolution of cooperation in the iterated
  prisoners 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.

3
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

4
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

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Belief Structures
Impacted by Socialization and Interaction

• 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 prisoners 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.

Shifts Behavioral Patterns
Characterized By

• 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.

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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
  Prisoners Dilemma implies many interesting
  things.
• Self-organized criticality (SOC) is introduced
  into the evolution of cooperation in the iterated
  prisoners dilemma.
• The distribution of the attractors over the
  games 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.