A Game Theory version of Hardin’s “Tragedy of the Commons”

September 2006

Based on the set-up in Hardin’s article, one can illustrate the ‘tragedy of the commons’ as a ‘non-cooperative game with complete information.’

Assumptions

Two herders

At the carrying capacity of a commons, each herder earns zero economic profits.

Each head of cattle sells for $1

Each herder incurs one-half of the marginal cost of each additional head of cattle

Marginal costs (MC) are increasing

MC = -1 if the commons gets one more head of cattle

MC = -1.5 if the commons gets a second head of cattle

Solution to the game

The dominant strategy for each herder is to defect:

If Herder 2 cooperates, ‘defect’ is the dominant strategy for Herder 1, since 0.5 is preferred to 0

If Herder 2 defects, ‘defect’ is the still dominant strategy for Herder 1, since -0.25 is preferred to –0.5.

If Herder 1 cooperates, ‘defect’ is the dominant strategy for Herder 2, since 0.5 is preferred to 0

If Herder 1 defects, ‘defect’ is the still dominant strategy for Herder 2, since -0.25 is preferred to –0.5.

So while the social optimum is mutual cooperation (0,0), the herders will end up at the sub-optimal Nash equilibrium (-0.25,-0.25)

Herder 2 à Herder 1	Cooperate	Defect
Cooperate	(0,0)	0-0.5=-0.5, 1 – 0.5 = 0.5)
Defect	(1 – 0.5 = 0.5,0-0.5=-0.5)	(1-0.5 – 0.75 = -0.25, 1-0.5 – 0.75 = -0.25)