# Cooperation under incomplete information on the discount factors

- 297 Downloads
- 4 Citations

## Abstract

In repeated games, cooperation is possible in equilibrium only if players are sufficiently patient, and long-term gains from cooperation outweigh short-term gains from deviation. What happens if the players have incomplete information regarding each other’s discount factors? In this paper we look at repeated games in which each player has incomplete information regarding the other player’s discount factor, and ask when full cooperation can arise in equilibrium. We provide necessary and sufficient conditions that allow full cooperation in equilibrium that is composed of grim trigger strategies, and characterize the states of the world in which full cooperation occurs. We then ask whether these “cooperation events” are close to those in the complete information case, when the information on the other player’s discount factor is “almost” complete.

## Keywords

Repeated games Incomplete information Cooperation Common belief Rationalizability Prisoner’s dilemma## References

- Blonsky M, Probst D (2008) Communicating Trustworthiness. Mimeo,Google Scholar
- Chassang S, Takahashi S (2011) Robustness to incomplete information in repeated games. Theoretical Economics 6:49–93CrossRefGoogle Scholar
- Dekel E, Fudenberg D, Morris S (2007) Interim Correlated Rationalizabiliy. Theoretical Economics 2:15–40Google Scholar
- Einy E, Haimanko O, Moreno D, Shitovitz B (2008) Uniform Continuity of the Value in Zero-Sum Games with Differential Information. Mathematics of Operations Research 33:552–560CrossRefGoogle Scholar
- Kajii A, Morris S (1997) Common p-Belief: The General Case. Games and Economic Behavior 18:73–82CrossRefGoogle Scholar
- Lehrer E, Pauzner A (1999) Repeated Games with Differential Time Preferences. Econometrica 67:393–412CrossRefGoogle Scholar
- Lehrer E, Yariv L (1999) Repeated Games with Incomplete Information on One Side: The case of different discount factors. Mathematics of Operations Research 24:204–218CrossRefGoogle Scholar
- Maor C (2010) Cooperation under Incomplete Information on the Discount Factors, M.Sc. Thesis Tel Aviv University. http://arxiv.org/abs/1012.3117
- Maschler M, Solan E, Zamir S (2013) Game Theory. Cambridge University Press,Google Scholar
- Monderer D, Samet D (1989) Approximating Common Knowledge with Common Beliefs. Games and Economic Behavior 1:170–190CrossRefGoogle Scholar
- Monderer D, Samet D (1996) Procimity of Information in Games with Incomplete Information. Mathematics of Operations Research 21:707–725CrossRefGoogle Scholar
- Morris S (1999) Approximate Common Knowledge Revisited. International Journal Game Theory 28:385–408CrossRefGoogle Scholar
- Morris S, Kajii A (1997) The Robustness of Equilibria to Incompelte Information. Econometrica 65:1283–1309CrossRefGoogle Scholar
- Watson J (1999) Starting small and renegotiation. Journal of Economic Theory 85:52–90CrossRefGoogle Scholar
- Watson J (2002) Starting small and Commitment. Games and Economic Behavior 38:176199CrossRefGoogle Scholar
- Weinstein J, Yildiz M (2007) A Structure Theorem for Rationalizability with Application to Robust Predictions of Refinements. Econometrica 75(2):365–400CrossRefGoogle Scholar