z + 3( h - z )-[l + 3( y - l )] ≥ δ 3(V3 - V4)
⇒ δ ≤ 3×2( z -1 )+( h - y )
h-z
since from (1) we have V3 - V4 = 3(h - z). It is easy to see that for our parameterization this
is satisfied by all δ∈(0,1) since -2(z—l)+ (h—y) = 34.
h-z
Finally, consider V2 ≥ V~ 2 . This inequality is immediately rewritten as:
2 22
z + Ξ( h - z )-[ l + 2 ( y - l )] ≥ δ 3 (V3-V4 )
3 z - l + 2(h - y)
⇒ δ ≤ ××--------—
2 h-z
which always holds for all parameters since z l+2(h y) = 34.
h-z
The intuition is simple. Cooperating instead of sanctioning after observing a defection
may be helpful to the player, since it delays the spread of the sanction. However, doing so
generates a current loss to the player since he earns y (instead of h ) if he meets a
cooperator, and l (instead of z ) if he meets a deviator. Therefore, the player must be
sufficiently impatient to prefer play of Z to Y—clearly, the smaller l and y , the greater
the incentive to follow with the sanction. Our parameterization insures that this incentive
exists for all δ∈(0,1) .
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