solves the condition in Lemma 1 for the parameterization given in Table 2B. 10
Proposition 3. In our experimental economies with private monitoring, the efficient
outcome can be sustained as an equilibrium without using personal punishment. Instead,
this outcome cannot be sustained as an equilibrium by relying exclusively on personal
punishment.
Recall that with personal punishment an agent has the option, at a cost, to lower the
current earnings of his opponent only after observing the outcome of the prisoners’
dilemma. In a one-shot interaction, choosing personal punishment is a dominated action
because it is costly for the punisher. In our design the interaction is indefinitely repeated,
but personal punishment is still individually suboptimal for the same reason it is in the
one-shot game.
Personal punishment is dominated for two reasons. First, it does not trigger a faster
contagion to the state of economy-wide defection. In our design agents are anonymous,
randomly matched in each period, and can only observe actions and outcomes in their
pair. Hence, to someone outside the match, a choice of personal punishment is no more
visible than a choice of defection. Because of private monitoring, personal punishment is
no more efficient than a “grim trigger” defection strategy, and in addition, it is costly.
Second, the sole use of personal punishment cannot sustain cooperation, even with
public monitoring. The reason is that personal punishment is not a credible threat because
after observing a defection, it is never individually optimal to pay the cost for personal
punishment.11 For instance, a strategy where agents always cooperate and respond to a
defection only with personal punishment for the period cannot sustain cooperation. After
the opponent defects, an agent has no incentive to inflict personal punishment because it
simply adds a further loss. Additionally, the incentive to defect in following periods
Contagion equilibria as in Kandori (1992) are not robust to adding a small amount of noise in the
observation of individual behavior. With noise, equilibria arise similar to those in the continuum limit
where individual behavior is unobservable (e.g., see Al-Najar and Smorodinsky, 2001, Fudenberg, Levine,
and Pesendorfer, 1998, Levine and Pesendorfer, 1995). One can suppose that the larger the population, the
greater the instances of noise in observability. To lessen such instances in our experimental economies, we
work with four-agent economies, the smallest possible number that allow pairwise anonymous matching.
11 On the contrary, defecting after having observed a defection is an optimal strategy.
14