options requires an increase in the expected utility at the start of the punishment regime,
so that m is increased. Since
δz
δx
< 0, increasing z implies a decreased normal-play level of
compliance. This reduced compliance lowers the union’s productivity, lowering utility. At
W = V
p
(ˆ
x), players cannot increase their punishment play expected utility any more, and
use their exit option. In so doing, their utility drops discontinuously from V
n
(ˆ
x) to V
p
(ˆ
x).
The shaded region represents utility loss from the exit option.
As long as the value of the exit option is greater than the expected utility at the start
of the punishment regime, the addition of the exit option affects utility and often, the
sustainability of the union. Without any option to quit the game, the optimal contribution
in equilibrium, while not at full compliance, is as close as possible. Contribution is sustained
by making the punishment regime severe enough that the players want to avoid it. Armed
with an option to exit the union, players first compare W to V
n
(s
∗
), the maximal expected
utility of participation in normal periods. If the exit option exceeds this amount, naturally
they will quit the union, as Case 4 describes. However, we see that even if the value of the
exit option is less than the expected value of the union, the union might not be sustainable.
As the exit option utility just surpasses the expected utility at the start of the punishment
regime (Case 2), players might have some room to adjust the punishment regime utility
upward by lessening the severity of the punishment. Players have many options for adjusting
the expected utility at the start of the punishment period: they can reduce the severity of
the single-period punishment (by complying more), they can reduce the duration of the
punishment regime, or they can lower the threshold, thereby increasing the tolerance for
non-compliance. While any one of these modifications raises the expected utility at the start
of the punishment regime, they all will lower the maximal utility obtainable during normal
periods.
11
While these adjustments may rescue the union, they come at a cost: a less severe
punishment induces less cooperative behavior from all members, reducing the utility from
the union, and therefore reducing player utility. As Case 3 describes, players are limited in
11
Only an increase in the level of patience could make the wait worthwhile, and in general, we assume
that the discount parameter is exogenously determined.
19
Convention