Abstract
This paper examines different Brownian information structures over varying time intervals. We focus on the non-limit case, and on the trade-offs between information quality and quantity when making a decision whether to cooperate or defect in a prisoners’ dilemma game. In the best-case scenario, the information quality gains are strong enough so that agents can substitute information quantity with information quality. In the second best-case scenario, the information quality gains are weak and must be compensated for with additional information quantity. In this case, information quality improves but not quickly enough to dispense with the use of information quantity. For sufficiently large time intervals, information degrades and monitoring becomes mostly based on information quantity. The results depend crucially on the particular information structure and on the rate at which information quality improves or decays with respect to the discounting incentives.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
A notable variation of the original model is Fudenberg and Olszewski (2011), who study repeated games with stochastic asynchronous monitoring. Outside the limit case, Fudenberg et al. (2014) show that if players wait long enough, then it is likely that everybody will have observed the same signal and a folk theorem may be possible. Kamada and Kominers (2010) also consider a time-varying information structure. However, their argument and information notions are different; see also Kandori (1992).
With this information structure Fudenberg and Levine (2007) have shown that full efficiency is possible at the limit.
With this information structure Fudenberg and Levine (2007) have shown that cooperation is possible at the limit, but not full efficient payoffs.
We have restricted our analysis to the simplest setting. The results obtained in the following sections generalize to other discount factors, payoff structures and games.
A strategy is public if it depends only on the public history (of signals) and not on the private history (of signals and of individual efforts). Given a public history, a profile of public strategies that induces a Nash equilibrium on the continuation game from that time on is called a PPE.
Sannikov and Skrzypacz (2007) and Fudenberg and Levine (2007) have shown that the best equilibrium is obtained with a “grim” strategy that prescribes mutual effort as long as the public signal falls within the region bounded by some critical threshold/s. Once these thresholds are crossed, i.e. the public signal falls in the region defined by the set \(S_{\tau },\) the punishment stage is initiated.
The results obtained are particularly robust. The predictions of the model do not depend significantly on the value of the parameters. Nonetheless, clearly, the greater the difference between these parameters (i.e. greater the impact of deviations in the drift or in the noise components of the process), the easier it is to see the reported effects. Figures 1a, b, 2a, b and 3a, b are sufficiently representative of the information structures considered in the present paper.
In the case where a deviation increases the drift of the process, the common one-sided threshold \(\overline{s}_{\tau },\) which is employed to distinguish the observations that suggest mutual effort \(\left\{ s_{\tau }<\overline{s}_{\tau }\right\} \) from the observations that suggest deviation \(S_{\tau }=\left\{ s_{\tau }\ge \overline{s}_{\tau }\right\} \), has a symmetric intuition.
In technical terms, the difference in results between Sect. 4.1 and this section is related to the fact that the noise component converges to zero slower (at rate \(\tau ^{1/2}\)) than the drift component, which converges to zero faster (at a rate \(\tau \)). This aspect is crucial at the limit.
References
Abreu, D., Milgrom, P., & Pearce, D. (1991). Information and timing in repeated partnerships. Econometrica, 59, 1713–1733.
Abreu, D., Pearce, D., & Stacchetti, E. (1986). Optimal cartel equilibria with imperfect monitoring. Journal of Economic Theory, 39, 251–269.
Abreu, D., Pearce, D., & Stacchetti, E. (1990). Toward a theory of discounted repeated games with imperfect monitoring. Econometrica, 58, 1041–1063.
Faingold, E., & Sannikov, Y. (2011). Reputation in continuous-time games. Econometrica, 79(3), 773–876.
Fudenberg, D., Ishii, Y., & Kominers, S. (2014). Delayed-response strategies in repeated games with observation lags. Journal of Economic Theory, 150, 487–514.
Fudenberg, D., & Levine, D. (2007). Continuous time limits of repeated games with imperfect public monitoring. Review of Economic Dynamics, 10(2), 173–192.
Fudenberg, D., & Levine, D. (2009). Repeated games with frequent signals. Quarterly Journal of Economics, 124, 233–265.
Fudenberg, D., & Olszewski, W. (2011). Repeated games with asynchronous monitoring of an imperfect signal. Games and Economic Behavior, 72, 86–99.
Kamada, Y., & Kominers, S. (2010). Information can wreck cooperation: A counterpoint to Kandori (1992). Economics Letters, 107, 112–114.
Kandori, M. (1992). The use of information in repeated games with imperfect monitoring. Review of Economic Studies, 59, 581–594.
Mirrlees, J. (1974). Notes on welfare economics, information and uncertainty. In M. Balch, D. McFadden & S. Wu (Eds.), Essays on economic behavior under uncertainty (pp. 243–258). Amsterdam, North-Holland: American Elsevier Pub. Co.
Osório, A. (2012). A folk theorem for games when frequent monitoring decreases noise. The B.E. Journal of Theoretical Economics, 12(1), 1–27. (Article 11).
Osório, A. (2015). Some notes and comments on the efficient use of information in repeated games with poisson signals. Operations Research Letters, 43(3), 288–292.
Porter, R. (1983). Optimal cartel trigger price strategies. Journal of Economic Theory, 29, 313–338.
Rao, B. P. (1999). Statistical inference for diffusion type processes. Arnold Publishers.
Sannikov, Y. (2007). Games with Imperfectly observable actions in continuous time. Econometrica, 75, 1285–1329.
Sannikov, Y., & Skrzypacz, A. (2007). Impossibility of collusion under imperfect monitoring with flexible production. American Economic Review, 97, 1794–1823.
Sannikov, Y., & Skrzypacz, A. (2010). The role of information in repeated games with frequent actions. Econometrica, 78, 847–882.
Author information
Authors and Affiliations
Corresponding author
Additional information
I would like to thank David Rahman, Juan Pablo Rincón-Zapatero and Ricardo Ribeiro, as well as several seminars and congresses participants for helpful comments and discussions. I am also grateful to the editor and two anonymous referees for their constructive comments. Financial support from the GRODE and the Spanish Ministerio of Ciencia y Innovación project ECO2016-75410-P is gratefully acknowledged. All remaining errors are mine.
Rights and permissions
About this article
Cite this article
Osório, A. Brownian Signals: Information Quality, Quantity and Timing in Repeated Games. Comput Econ 52, 387–404 (2018). https://doi.org/10.1007/s10614-017-9685-5
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10614-017-9685-5