9: Collective-Action Games

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9 Collective-Action Games

• In collective-action games the interests of
  society and the individuals are in conflict

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Collective-Action Games

• Come in three familiar forms
• Prisoners dilemma games
• Chicken games
• Assurance games
• Typically they involve many players, although we
  shall simplify them to the case of two for
  clarity.

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Collective-Action Games

• The Prisoners dilemma as a collective-action game

Consider the standard public goods problem. Two
farmers share a supply of water from the
mountains. Each can work on maintaining the canal
that supplies them both with water, or they can
work on their own crops. This leads to payoffs of
the following form.
Farmer 2 Farmer 2
Canal Land
Farmer 1 Canal 4,4 -1,6
Farmer 1 Land 6,-1 0,0
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The prisoners dilemma as a collective-action game
Farmer 2 Farmer 2
Canal Land
Farmer 1 Canal 4,4 -1,6
Farmer 1 Land 6,-1 0,0

• This is the standard prisoners dilemma.
• Each farmer tends his own land and we get the
  socially sub-optimal Nash equilibrium 0,0.
• Not the socially optimal 4,4.

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The prisoners dilemma as a collective-action game
II
Farmer 2 Farmer 2
Canal Land
Farmer 1 Canal 2.3,2.3 -1,6
Farmer 1 Land 6,-1 0,0

• Again we see that there is a divergence between
  the socially optimal -1,6 or 6,-1 and the
  Nash 0,0.
• Here there is a particularly difficult problem to
  solve as both farmers prefer to be the one
  receiving 6.

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Chicken as a collective action game
Now our trusty yeomen must coordinate to avoid
the pitfalls of the chicken game.
Farmer 2 Farmer 2
Canal Land
Farmer 1 Canal 5,5 2,6
Farmer 1 Land 6,2 0,0
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Collective-Action Games

• The Stag Hunt (An assurance game)
• The French philosopher, Jean Jacques Rousseau,
  presented the following situation. Two hunters
  can either jointly hunt a stag (an adult deer and
  rather large meal) or individually hunt a rabbit
  (tasty, but substantially less filling). Hunting
  stags is quite challenging and requires mutual
  cooperation. If either hunts a stag alone, the
  chance of success is minimal. Hunting stags is
  most beneficial for society but requires a lot of
  trust among its members.

Hunter 2 Hunter 2
Stag Rabbit
Hunter 1 Stag 10,10 0,8
Hunter 1 Rabbit 8,0 7,7
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Collective-Action Games

• The Stag Hunt
• There are two pure strategy equilibria. Both
  players prefer one equilibrium to the other.
  However, the inefficient equilibrium is less
  risky as the payoff variance over the other
  player's strategies is lower. Specifically, one
  equilibrium is payoff-dominant while the other is
  risk-dominant.

Hunter 2 Hunter 2
Stag Rabbit
Hunter 1 Stag 10,10 0,8
Hunter 1 Rabbit 8,0 7,7
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Collective-Action Games

• A Possible Solution Correlated Equilibria
• Consider the following version of the Chicken Game

Farmer 2 Farmer 2
Canal Land
Farmer 1 Canal 5,5 2,10
Farmer 1 Land 10,2 0,0
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Collective-Action Games

• Correlated Equilibria
• Each farmer prefers a different equilibrium,
  making sure that they are coordinated on the same
  one is good for society.
• Suppose that each could give their strategy
  choice to a trusted third party.
• The trusted third party tells them that they will
  allocate a role to one farmer according to some
  random mechanism
• Farmer one is told that a coin will be flipper
  and if it comes out heads he gets to play Land
  otherwise he must play Canal.
• Both farmers observe the coin toss.
• Each farmer gets an expected value of 6 from
  this.
• Deviating from the allocated role hurts either
  farmer and will not occur.
• This is a correlated equilibrium.