Optional Public Good Games

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Optional Public Good Games

• Satyaki Mahalanabis (Y3111042)
• and
• Arindam Chakrabarty (Y3111008)

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• What is a Public Good Game ?
• Extension of Prisoners Dilemma to a population of
  selfish players
• Each population member participates in a game, in
  which it can cooperate with others or defect
• Extensively studied in both classical and
  evolutionary game theory(2,3)
• Simple two choice Prisoners Dilemma type game
  models predict that all players will have a
  tendency to defect
• leads to a deadlock

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The Game Model

• Game is played in groups of N players
• Each Player has an amount c to play with in each
  game
• Each can cooperate ( contribute c ) or defect (
  not contribute anything )
• The total contribution is multiplied r times and
  shared equally by the participants
• Assume n cooperators out of N participants

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Game Model Continued..

• Payoff of the cooperators
• Pc r c n / N c
• Payoff of the defectors
• Pd r c n / N
• 1 lt r lt N
• No contributor means no payoff
• Pd is always greater than Pc
• Defectors get more payoff than Cooperators
  greater tendency to defect than to cooperate

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Significance of above equations

• The 1st inequality states that if all individuals
  in group were to either cooperate or defect, then
  theyll be better off cooperating than defecting.
  
• How ?
• If all defect, n 0 Pd r c 0 / N 0
• If all cooperate, n N Pc r c N / N c
• r c c c (r 1) gt 0 since r gt 1

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Existing models of altruism

• The two - choice model fails to explain the
  existence of cooperation in society
• Several alternative models have been suggested to
  explain this anomaly (4)
• - kin selection
• - reciprocal altruism
• - altruistic punishment

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Kin selection

• Altruistic behaviour among relatives
• Motivated by genetic stake that individuals have
  not only in their children, but also in their
  grandchildren in fact in all their relatives
• Consider individuals A B, gene G in A. Then G
  will increase in frequency if the relatedness
  between A B exceeds the ratio between fitness
  cost (to A) fitness benefit (to B). (W. D.
  Hamilton, 4)

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Reciprocal altruism

• Altruism among non-relatives
• Altruistic acts motivated by expectation of
  similar returns in the future, i.e. based on
  economic rather than genetic ties (as with kin
  selection)
• Simultaneous cooperation benefits both but is not
  guaranteed
• But repeated interaction gives the cooperator a
  chance to retaliate, thereby decreasing the
  tendency to defect.
• Pitfall altruistic behaviour among cooperators
  gives defectors opportunity to comeback.

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Altruistic punishment

• After every round of the public good games,
  players can fine specific co-players (defectors)
• Fines do not go to punishers, hence punishment is
  an unselfish act
• Is actually costly to the punisher
• Tendency to contribute increases
• Can also be alternately modelled by rewarding
  cooperators

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Motivation

• Earlier models attained cooperative behaviour
  through enabling individuals direct their
  altruistic acts only towards certain other
  individuals
• Were more complicated since they involved
  measures of altruism discrimination
• Optional public good games provides a simple
  alternative in that each player can now choose
  not to participate in the game

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The new model

• Here again, a player has an amount c to play
  with.
• But now the player has 3 choices
• - cooperate (as earlier)
• - defect (as earlier)
• - not participate i.e. be a loner
• The loners get a small but fixed payoff
• Pl s c with 0 lt s lt r 1, so that a group
  of all loners is worse off than a group of all
  cooperators but better off than a group of all
  defectors.

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Evolution of strategies by players

• Assumptions
• - strategies are pre-specified they do
  not depend upon group composition, eg.,
  number of cooperators
• - a strategys payoff determines its
  growth within the population
• One strategy formulation is that each player
  occasionally adopts the strategy of a randomly
  chosen model with probability proportional to the
  difference between the models payoff its own
  if it is positive otherwise with probability 0.

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Evolution of strategies by players (contd.)

• Considering a continuous time model, the
  evolution of frequencies xi of strategies i is
  given by
• dxi / dt S xi xj (Pi Pj), 1 lt i, j lt n,
  or
• j
• dxi / dt xi (Pi Pavg),
• where,
• Pavg S xj Pj, is the populations
  average payoff
• n number of available strategies

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Evolutionary Dynamics of Strategies

• Average payoff of the population is given by
• Pavg -xc ( Pc Pd) ( 1 xl ) ( Pd s )
  s c
• The system can be modelled by a Hamiltonian H,
  where
• df / dt - ?H / ?z
• dxl / dt ?H / ?f
• H G L
• and G and L are functions of xl and f
  respectively, f xc / ( xc xd ) (3).

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Dynamics Continued

• For r lt 2, the population exhibits brief bursts
  of cooperation
• Tendency, however, is to not cooperate, i.e.,
  loners dominate
• For r gt 2, population is stable at an unique
  point inside the simplex S3, which is surrounded
  by closed orbits along which the population
  revolves
• Cooperation, defection and non participation
  alternate in the population leading to a rock -
  paper - scissors kind of policy dynamics

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Dynamics Continued

• The payoff for cooperators, defectors and loners
  are equal,
• Pc Pd Pl
• and is equal to s c, the quantities being
  averaged over time
• Thus in the long run everybodys payoff is the
  same
• The option of non participation leads to
  cooperation, defection and non participation
  competing against each other rather than
  everybody defecting, as in prisoners dilemma
  type game

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Population Dynamics
The Simplex S3 represents the population
fraction ec, ed and el correspond to all
cooperators, all defectors and all loners
respectively
Taken from 3
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Population dynamics
F(z) Pd - Pc , z represents the fraction of
loners
Taken from 3
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Future Directions

• To study further the mathematical formulation of
  the optional public good games model (3)
• To simulate optional public good game and verify
  the predictions of the mathematical model
  (1,3)
• To test the population behaviour over wide ranges
  of parameter values

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References

• 1   Christoph Hauert, Silvia De Monte, Josef
  Hofbauer and Karl Sigmund. Volunteering as Red
  Queen Mechanism for Cooperation in Public Goods
  Games. Science, Vol. 296  10 May 2002 2   John
  Batali and Phillip Kitcher. Evolution of Altruism
  in Optional and Compulsory Public Goods Games.
  Journal of Theoretical Biology (1995) 175, 161 -
  1713   Christoph Hauert, Silvia De Monte,
  Josef Hofbauer and Karl Sigmund. Replicator
  Dynamics for Optional Public Good Games. Journal
  of Theoretical Biology ( 2002 ) 218, 187-194.
• 4 Karl Sigmund and Christoph Hauert.
  Primer Altruism Primer. Current Biology, 2002,
  128R270-R272

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