2.1. Repeated Game
Consider the following prisoner’s dilemma as the stage game with expected payoffs, denoted by G:
The time structure is borrowed from AMP. This prisoner’s dilemma game is played continuously over the time interval . The payoffs in the matrix above are flow rates of payoff. The interest rate is denoted by r, and so if the payoff at time t is , then the net present value of payoffs for the whole game is . Notice r represents patience of players, and the smaller the r is, the more patient the players are.
It is assumed that actions have to be held fixed for a certain time interval denoted by . In other words, players can choose their actions only at periods . In effect, the “discount factor” is interpreted as .
Like AMP, I assume that the signals arrive following a stochastic process which depends on the pair of actions. Unlike AMP, however, I assume that the signal is
private. In particular, it follows the following process. Let
represent arrival or absence of a signal to player
i, where
s (resp.
o) denotes arrival (resp. absence). Given an action pair
, the probability that each player gets a signal within a time interval
is given by the following figure:
where
. It is clear that each element is smaller than 1. Later I will present an assumption that guarantees each element of the figure is strictly positive when
. The time interval
is assumed to be small enough so that the probability that a signal is observed more than once within
is negligible. Finally,
Figure 1 is stated in terms of expected payoffs. One interpretation is that the game ends with probability
r and players get payoffs only after it ends.
where g, l > 0.
Notice that the marginal probability that a private signal arrives is given by
In other words, is the arrival rate of a signal to a player. From this, we can interpret as a measure of noisiness of the signal. The private signal is noisier if is smaller. An extreme case is . In this case, the signal is completely noise.
Hence, is an arrival rate of an event in which the arrival of a signal differs among players. If is small, then signals are highly correlated in the sense that if one player gets the signal, then the other players will also get the signal, most likely.
I will assume that (i) private signals are noisy relative to the patience of players, (ii) the degree of their correlation depends on actions and it is higher when both players cooperate than when one of them defects, and (iii) regardless of action pair, the correlation is high enough so that given a player gets a signal, it is more likely that the other player also observes a signal. Formally, first
This assumption states the relationship between the noisiness of the private signals and the patience of the players. Roughly, the assumption is easily satisfied when (i) the private signal is noisy and (ii) the players are impatient. Second,
This assumption implies that , or the disagreement over arrivals happens more frequently when a player defects than when both players cooperate. To see the implication of this assumption, consider a fictitious case in which signals are public in the sense that both players could commonly observe the pair of signals , instead of their own signal . Then the event constitutes the arrival of bad news in terms of AMP—an event which occurs more frequently when a player defects.
Finally, assume
This assumption means that given if a player gets a signal, it is likely that the other player also observes a signal. To see this, just observe that
Additionally, notice that this assumption guarantees that the first element of the figure takes a strictly positive value when .
Note that similar assumptions—such as that signals based on past actions are highly correlated, and correlation is higher when they cooperate—appear in other contexts. See, for example, Fleckinger [
13] and Awaya and Do [
14].
I will provide some parametric examples that satisfy all these assumptions.
Example 1. Let and . Additionally, let for any a and It is clear that Assumption 1 is satisfied, because right-hand side is 0. Assumption 3 is clearly satisfied as well. To see Assumption 2, Example 2. Again let , , and To see that these satisfy Assumption 1, notice and The remaining assumptions follow by the same reason as above.
Now I am ready to define histories and strategies. Let
be the history of actions that player
i took at periods
, and
be the history of signals that player
i has observed by period
. Recall that
is assumed to be small enough so that the probability so that the signal is observed twice within
is negligible.
A pure strategy
for player
i is a sequence of functions
where
and for
where
∅ is the
null history (throughout the paper, subscripts indicate players, superscripts indicate periods, and bold letters indicate histories). An implicit assumption here is that a player can condition his action on a signal he observes in a certain instance. This assumption is innocuous because this event happens with probability zero.
2.2. Repeated Game with Communication
Next consider an infinite horizon repeated game version of the stage game in which players can communicate. They can send messages in a finite set and let . Communication entails no cost so that it is “cheap talk”. All messages are observed publicly and without any error.
Assume players send messages only when they decide their actions, that is, at . The communication is done by an instance without taking any physical time, and when players decide their actions they can take the message sent at that moment into account.
Formally, let
be the history of messages that player
i send at periods
, and let
A pure strategy in this game is a pair of
where
is a sequence of
message strategies and
is a sequence of
action strategies. Then
and for
Again it is assumed that a player can condition a message sent, or an action taken, on a signal observed at the instance.