The following is a short version of An essay on Kandori and Matsushima (1998) "Private Observation, Communication and Collusion" (Econometrica, 66), the term paper of the class by Professor Dutta.

Summary
The present paper analyzes the role of communication and the possibility of cooperation in a long term relationship, when the actions of the players are imperfectly observed and each player receives only private signal.
The analysis of such a situation, a repeated game with (imperfect) private monitoring, is known to be a hard problem in game theory due to its fairly complex mathematical structure, particularly due to the lack of common information shared by players. Under private monitoring, the distribution of the private histories is no longer common knowledge after a deviation (off the equilibrium path), because only the deviator takes her deviation into account for up-dating her belief while other players cannot. This means the continuation play off the equilibrium path is not even an equilibrium of the original game. Therefore, the recursive structure found in the public monitoring case, i.e., the property that the continuation payoff after any history is chosen from the identical set of equilibrium payoffs, is destroyed under private monitoring (hence, we cannot apply dynamic programming techniques provided by Abreu, Pearce and Stacchetti (1990)).
Indeed, in sharp contrast to the well-explored case of repeated games under public information (with the celebrated Folk Theorems by Fudenberg, Levine and Maskin (1994)), little had been known about the private monitoring case until recently. This is unfortunate because this class of games admits a wide range of applications such as collusion under secret price-cutting, exchange of goods with uncertain quality, and observation errors.
In this paper, the authors introduce communication in the model with private monitoring to overcome the basic difficulty of this subject. Namely, they assume that at the end of each period players can communicate what they privately observed. The announced messages generate publicly observable history, and the players can play different equilibria depending on the history of communication. Facilitating communication as a coordination device, the authors construct equilibria in which players reveal their private information truthfully, and show that the folk theorem obtains under a set of mild assumptions.
Finally, it should be noticed that their results provide a theoretical support for the conventional wisdom that communication plays an important role in sustaining collusion.

Introducing Communication
To overcome the difficulties associated with private monitoring, the authors introduce communication which generates publicly observable history, and enables players to play different equilibria depending on the history of communication. Thanks to this communication, the recursive structure is recovered, and one can apply the results in previous literature, e.g., the characterization of equilibrium payoff sets provided by Abreu, Pearce and Stacchetti (1990) and Fudenberg and Levine (1994), or Folk Theorems given by Fudenberg, Levine and Maskin (1994).

Main Result
Folk theorem can obtain given Condition 1-3 listed below.

Condition 1
If player j has a perfectly undetectable deviation at the minimax point for player i, j has no incentive to take it.
Condition 2
If either player i or j (but not both) deviates with certain probabilities from a pure action profile wich generales an extremal point, the other players can statistically detect it.
Condition 3
The players other than i and j can statistically discriminate player i's (possibly mixed) deviations from player j's at any pure action profile wich generales an extremal point.

Remark 1
These conditions guarantee that every Pareto-efficient profile and each minimax strategy profile are enforceable, which is sufficient to establish the following Folk Theorem. Roughly speaking, Condition 2 and 3 correspond to the pair-wise identifiability condition and Condition 1 is replaced with the individually full-rank condition in Fudenberg, Levine and Maskin (1994). The former is sufficient for Nash-threat version of the Folk Theorem, while the latter, in addition to the former, is sufficient for minimax version of the Folk Theorem.

Folk Theorem
Suppose that there are more than two players (n>2) and the information structure satisfies Condition 1-3. Also suppose that the dimension of V is equal to the number of players. Then, any interior point in the set of feasible and individually rational payoffs can be achieved as a sequential equilibrium average payoff profile of the repeated game with communication, if the discount factor is close enough to 1.

Basic Idea
In their communication model, one must induce each player to reveal her signal truthfully. To do so, they consider the equilibria in which each player's future payoff is independent of what she communicates. If this is the case, she is just indifferent as to what she says, and truthful revelation becomes a (weak) best response.
As one might expect, this can be done if there are at least three players and the information structure can distinguish different players' deviations. A player's private information can be used to determine when and how to transfer payoffs among other players. (In two player case, this transfer is no longer available and hence the punishment of the other player necessarily invites welfare loss.)

Remark 2
Roughly speaking, efficiency under publicly observable signals can be achieved if players can be punished by "transfers". If the information structure allows us to tell which player is suspect, we can transfer the suspect player's future payoff to the other players. This can provide the right incentives without causing welfare loss, compared to the case where all players are punished simultaneously.
Remark 3
The authors also examine the possibility of providing strict incentives to tell the truth. It is shown that when private signals are correlated, there is a way to check if each player is telling the truth and we can construct the equilibria in which the players have strict incentives for truth telling.
Remark 4
If there are two players, or if the information structure fails to distinguish different players' deviations, the above idea cannot be utilized. However, even in such cases, the Folk Theorem can be obtained by infrequent communication. This is based on the idea of Abreu, Milgrom and Pearce (1991) that delaying the release of information helps to achieve efficiency.
Remark 5
To analyze the equilibria, they employ the method developed by Fudenberg and Levine (1994). Instead of directly solving the repeated game, this method first considers simple contract problems associated with the stage game. Then, the solutions to those contract problems are utilized to construct the set of equilibrium payoffs of the repeated game.

Conclusion
As I explained in Summary, the characterization of equilibria in repeated games with private monitoring have been an open question, because the games lack recursive structure and are hard to analyze. The present paper shows that communication is an important means to resolve possible confusion among players in the course of collusion during repeated play. Namely, as they introduce communication to generate publicly observable history, the authors recover the recursive structure and show a Folk Theorem.
However, it should be noticed that they did not show the necessity of communication for a Folk Theorem. In principle, there is a possibility that a Folk Theorem holds even without communication. Indeed, the analyses of private monitoring have been rapidly developed (not yet achieve the complete characterization of equilibrium, though) since Kandori and Matushima (1998). Therefore, I would like to mention the recent literature on repeated games with private monitoring, which concludes this essay (I relied on the excellent survey by Kandori (2002) for the remain part).

Recent Literature
Sekiguchi (1997) is the first paper to construct an equilibrium which is apart from the repetition of the stage game equilibrium under private monitoring. He shows that efficiency can be approximately achieved (without communication) in the prisoner's dilemma model, if the information is almost perfect.
Bhaskar and Obara (2002) extend Sekiguchi's construction to support any point Pareto dominating (d,d) (in the prisoner's dilemma), when monitoring is private but almost perfect. Sekiguchi-Bhaskar-Obara type of equilibrium is called "belief-based" equilibrium because they facilitate the coordinated punishment idea.
Piccione (2002) and Ely and Valimaki (2002) introduce a completely different, useful technique to support essentially the same area under almost perfect monitoring. In contrast to "belief-based" equilibrium by S-B-O, their equilibrium utilizes the uncoordinated punishment idea, and hence is named "belief-free" equilibrium.
Matsushima (2004) extends Ely and Valimaki's construction and show that their Folk Theorem continues to hold even if monitoring is far from perfect, as long as private signals are distributed independently.
Ely, Horner and Olszewski (2005) give the most general results in two-player repeated games with private monitoring. Using "belief-free" strategies, they provide a simple and sharp characterization of equilibrium payoffs. While such strategies support a large set of payoffs, they are not rich enough to generate a Folk Theorem in most games besides the prisoner's dilemma, even when information is almost perfect.

References
Abreu, Milgrom and Pearce (1991) "Information and timing in repeated partnerships" Econometrica, 59
Abreu, Pearce and Stacchetti (1990) "Toward a Theory of Discounted Repeated Games with Imperfect Monitoring" Econometrica, 58
Bhaskar and Obara (2002) "Belif-based Equilibria in the Repeated Prisoners' Dilemma with Private Monitoring" Journal of Economic Theory, 102
Ely, Horner and Olszewski (2005) "Belief-free Equilibria in Repeated Games" Econometrica, 73
Ely and Valimaki (2002) "A Robust Folk Theorem for the Prisoner's Dilemma" Journal of Economic Theory, 102
Fudenberg and Levine (1994) "Efficiency and Observability with Long-Run and Short-run Players" Journal of Economic Theory, 62
Fudenberg, Levine and Maskin (1994) "The folk theorem with imperfect public information" Econometrica, 62
Kandori (2002) "Introduction to Repeated Games with Private Monitoring" Journal of Economic Theory, 102
Matsushima (2004) "Repeated Games with Private Monitoring: Two Players" Econometrica, 72
Piccione (2002) "The Repeated Prisoner's Dilemma with Imperfect Private Monitoring" Journal of Economic Theory, 102
Sekiguchi (1997) "Efficiency in the Prisoner's Dilemma with Private Monitoring" Journal of Economic Theory, 76