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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
#1
Canal
Land
Canal
4,4
-1,6
Land
6,-1
0,0
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The prisoners dilemma as a
collective-action game
Farmer #2
Farmer
#1
Canal
Land
Canal
4,4
-1,6
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
#1
Canal
Land
Canal
2.3,2.3
-1,6
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
#1
Canal
Land
Canal
5,5
2,6
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
#1
Stag
Rabbit
Stag
10,10
0,8
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 payoffdominant while the other is risk-dominant.
Hunter #2
Hunter
#1
Stag
Rabbit
Stag
10,10
0,8
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
#1
Canal
Land
Canal
5,5
2,10
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.
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