Original article (link) posted: 04/09/2005
Professor Davis at Columbia University put the note "Ph.D. Thesis Research: Where do I Start?" on his web site.
There are many interesting advices. Especially the following two are helpful for me.
Don’t Take Courses!
By the third year of a PhD program, your job is research, not more courses! You can take more courses (of course), but you should have a very good reason for doing so. Acceptable reasons include (a) It is a course that takes you to the frontier of research in an area in which you plan to do research or (b) It develops mathematical or econometric techniques that you plan to use in short order. The reason that I advise not taking courses is that it is a convenient, comforting, and seemingly rationalizable way of avoiding the harder, more frustrating, but necessary conversion from being a consumer of research to being a producer of research. Focus on your primary task – developing your own research program.
Don’t teach!
. . . more than you have to. For many, teaching is attached to a stipend or is otherwise economically unavoidable. In this case, do what you must! Moreover, there are some real intellectual and practical advantages from doing a couple of terms of TA work. Explaining the concepts to others is very useful in consolidating them in yourself. But beyond this, the returns become strongly negative. Your job is research – and anything that distracts you from this is a heavy cost. The first cost, which may seem remote at the time that you are deciding on the teaching, is that it could delay completion of the thesis by a year or more. An even larger cost is if it crowds out time to write a really great thesis. As a PhD student, your time is very valuable; treat it that way.
Hum, as Professor Obara recommended to me, I shouldn't teach this year even like a small group seminar... Also, instead of attending classes, it must be better for me to use time for focusing on my own research or going to seminars.
2010-06-19
2010-06-15
Information in Repeated Games
Original article (link) posted: 31/08/2005
Let's think about Green and Porter model.
What happens if demand becomes less stochastic so that the firms have better obserbability on their output??
Kandori (1992) "The Use of Information in Repeated Games with Imperfect Monitoring"(RES 59) answers this question in a general framework.
In the paper, using the concept called "quasi-garbling" which is Blackwell's definition of informativeness, Kandori elegantly shows that
in the general model of discounted repeated games with imperfect monitoring, the set of payoffs attainable via pure-strategy sequential equilibria becomes larger (in the sense of set inclusion) as the observability of the past actions increases. The intuition behind the assertion is that the more accurately "cheating" is detected, the easier it is to enforce coordination.
Thus, the answer to the first question is
in an oligopoly model of the Green-Porter type, the best symmetric equilibrium becomes better and the most severe symmetric punishment (the worst equilibrium) becomes more severe, as the demand becomes less noisy.
The papers written by Kandori are all clear and elegant! How come he could write so many great papers... Every time I read his paper, I feel losing my confidence and realize how far where he was from where I am standing now :(
P.S.
Professor Kandori was my senior thesis advisor at University of Tokyo.
Let's think about Green and Porter model.
What happens if demand becomes less stochastic so that the firms have better obserbability on their output??
Kandori (1992) "The Use of Information in Repeated Games with Imperfect Monitoring"(RES 59) answers this question in a general framework.
In the paper, using the concept called "quasi-garbling" which is Blackwell's definition of informativeness, Kandori elegantly shows that
in the general model of discounted repeated games with imperfect monitoring, the set of payoffs attainable via pure-strategy sequential equilibria becomes larger (in the sense of set inclusion) as the observability of the past actions increases. The intuition behind the assertion is that the more accurately "cheating" is detected, the easier it is to enforce coordination.
Thus, the answer to the first question is
in an oligopoly model of the Green-Porter type, the best symmetric equilibrium becomes better and the most severe symmetric punishment (the worst equilibrium) becomes more severe, as the demand becomes less noisy.
The papers written by Kandori are all clear and elegant! How come he could write so many great papers... Every time I read his paper, I feel losing my confidence and realize how far where he was from where I am standing now :(
P.S.
Professor Kandori was my senior thesis advisor at University of Tokyo.
2010-06-12
A note on Abreu (1986)
Original article (link) posted: 20/08/2005
There has been no systematic attempt to study the maximal degree of collusion sustainable by credible threats for arbitrary values of the discount factor. In view of the motivation for moving from static to repeated models, this has some claims to being the essential question at issue.
As argued by Abreu (1983, Ph.D thesis), the fundamental determinant of the limits of collusion is the severity of punishments with which potential deviants from cooperative behavior can credibly be threatened. Accordingly, this paper concentrates principally on characterizing strategy profiles which yield optimal (in the sense of most severe) punishments.
A particular class of paths called two-phase punishments plays a central role. A two-phase punishment is symmetric; in addition it is stationary after the first period, i.e., in the second phase. I show that the optimal two-phase punishment is:
(1) Globally optimal for a certain range of parameter values.
(2) An optimal symmetric punishment.
(3) More severe than Cournot-Nash reversion.
(4) Easily calculated. (It is completely characterized by a pair of simultaneous equations.)
(5) The second phase of the optimal two-phase punishment is the most collusive symmetric output level which can be sustained by the optimal two-phase punishment itself.
...
(5) says that the optimal two-phase punishment consists of a stick and carrot; furthermore, the carrot phase is the most attractive collusive regime which can credibly be offered when optimal two-phase punishments are used to deter defections. Thus when we solve for optimal two-phase punishments, we simultaneously determine the maximal degree of collusion sustainable by optimal symmetric punishments. It is worth remarking that the stick-and-carrot property is not a curiosum, but arises naturally from the structure of the problem.
Optimal asymmetric punishments have a rather complicated structure and thus far elude description as complete as that provided for optimal symmetric punishments in the earlier section.
(All quoted from Abreu (1986))
I would like to say something about asymmetric cases below.
For delta large enough, optimal symmetric punishments yield 0 payoff hence they are the most severe punishment (notice that a firm's minmax payoff in the component game is zero). And in this case, all firms simultaneously minmax one another in the first phase of the punishment.
However, if an optimal symmetric punishment yields firms positive payoffs (more than their minmax payoffs), then it is not globally optimal.
Abreu gives characterizations of optimal asymmetric punishments. But they are complicated and much less sharper than those with symmetric cases.
There has been no systematic attempt to study the maximal degree of collusion sustainable by credible threats for arbitrary values of the discount factor. In view of the motivation for moving from static to repeated models, this has some claims to being the essential question at issue.
As argued by Abreu (1983, Ph.D thesis), the fundamental determinant of the limits of collusion is the severity of punishments with which potential deviants from cooperative behavior can credibly be threatened. Accordingly, this paper concentrates principally on characterizing strategy profiles which yield optimal (in the sense of most severe) punishments.
A particular class of paths called two-phase punishments plays a central role. A two-phase punishment is symmetric; in addition it is stationary after the first period, i.e., in the second phase. I show that the optimal two-phase punishment is:
(1) Globally optimal for a certain range of parameter values.
(2) An optimal symmetric punishment.
(3) More severe than Cournot-Nash reversion.
(4) Easily calculated. (It is completely characterized by a pair of simultaneous equations.)
(5) The second phase of the optimal two-phase punishment is the most collusive symmetric output level which can be sustained by the optimal two-phase punishment itself.
...
(5) says that the optimal two-phase punishment consists of a stick and carrot; furthermore, the carrot phase is the most attractive collusive regime which can credibly be offered when optimal two-phase punishments are used to deter defections. Thus when we solve for optimal two-phase punishments, we simultaneously determine the maximal degree of collusion sustainable by optimal symmetric punishments. It is worth remarking that the stick-and-carrot property is not a curiosum, but arises naturally from the structure of the problem.
Optimal asymmetric punishments have a rather complicated structure and thus far elude description as complete as that provided for optimal symmetric punishments in the earlier section.
(All quoted from Abreu (1986))
I would like to say something about asymmetric cases below.
For delta large enough, optimal symmetric punishments yield 0 payoff hence they are the most severe punishment (notice that a firm's minmax payoff in the component game is zero). And in this case, all firms simultaneously minmax one another in the first phase of the punishment.
However, if an optimal symmetric punishment yields firms positive payoffs (more than their minmax payoffs), then it is not globally optimal.
Abreu gives characterizations of optimal asymmetric punishments. But they are complicated and much less sharper than those with symmetric cases.
2010-06-09
Two essential papers by APS
Original article (link) posted: 20/08/2005
What's the difference between APS (1986) and APS (1990)? The distinction is similar to that of Abreu (1986) and Abreu (1988). The earlier papers (both published in JET), APS (1986) and Abreu (1988), analyze the optimal strategies in actual oligopoly models by using powerful properties in repeated games established by them. The later papers (both in Econometrica) extend the results in generalized situations. Although these two Econometrica papers are more general and sophisticated than those of JET, they may have some drawbacks such as, lack of motivation, too concise explanation (to understand), and more seriously, showing no actual optimal strategies. Two papers in JET are strongly motivated by the actual collusion problems in IO, and hence you can see how useful the properties they found are.
Two APS papers heavily rely on the technique used in dynamic programming. Using those technique, they reduce the repeated game to a static structure from which can be extracted the optimal equilibria in question. That is,they construct a new game by truncating the discounted supergame as follows:
after each first-period history, replace the sequential equilibrium successor by the payoffs associated with that successor.
APS (1990) summarize the distinction of the two papers.
First, it relaxes the restriction of symmetry, showing the theory capable of embracing both asymmetric equilibria of symmetric games and arbitrary asymmetric games. Secondly, the sufficiency of using bang-bang reward functions in efficiently collusive equilibria is strengthened to a necessity theorem. Finally, we provide an algorithm useful in computing the sequential equilibrium value set.
(APS (1990), p.1044)
Finally, I would like to remind you of the difference between Abreu and APS. These papers are essentially different in the sense that the former analyze perfect monitoring cases whereas the latter consider imperfect monitoring cases. Moreover, even though all these papers are inspired by dynamic programming technique, their focus is different more or less. The argument in Abreu is about equilibrium paths, but that in APS is about equilibrium payoff sets.
References
Abreu (1986) "Extremal Equilibria of Oligopolistic Supergames" JET, 39
Abreu (1988) "On the Theory of Infinitely Repeated Games with Discounting" Econometrica, 56
Abreu, Pearce and Stacchetti (1986) "Optimal Cartel Equilibria with Imperfect Monitoring" JET, 39
Abreu, Pearce and Stacchetti (1990) "Toward a Theory of Discounted Repeated Games with Imperfect Monitoring" Econometrica, 58
What's the difference between APS (1986) and APS (1990)? The distinction is similar to that of Abreu (1986) and Abreu (1988). The earlier papers (both published in JET), APS (1986) and Abreu (1988), analyze the optimal strategies in actual oligopoly models by using powerful properties in repeated games established by them. The later papers (both in Econometrica) extend the results in generalized situations. Although these two Econometrica papers are more general and sophisticated than those of JET, they may have some drawbacks such as, lack of motivation, too concise explanation (to understand), and more seriously, showing no actual optimal strategies. Two papers in JET are strongly motivated by the actual collusion problems in IO, and hence you can see how useful the properties they found are.
Two APS papers heavily rely on the technique used in dynamic programming. Using those technique, they reduce the repeated game to a static structure from which can be extracted the optimal equilibria in question. That is,they construct a new game by truncating the discounted supergame as follows:
after each first-period history, replace the sequential equilibrium successor by the payoffs associated with that successor.
APS (1990) summarize the distinction of the two papers.
First, it relaxes the restriction of symmetry, showing the theory capable of embracing both asymmetric equilibria of symmetric games and arbitrary asymmetric games. Secondly, the sufficiency of using bang-bang reward functions in efficiently collusive equilibria is strengthened to a necessity theorem. Finally, we provide an algorithm useful in computing the sequential equilibrium value set.
(APS (1990), p.1044)
Finally, I would like to remind you of the difference between Abreu and APS. These papers are essentially different in the sense that the former analyze perfect monitoring cases whereas the latter consider imperfect monitoring cases. Moreover, even though all these papers are inspired by dynamic programming technique, their focus is different more or less. The argument in Abreu is about equilibrium paths, but that in APS is about equilibrium payoff sets.
References
Abreu (1986) "Extremal Equilibria of Oligopolistic Supergames" JET, 39
Abreu (1988) "On the Theory of Infinitely Repeated Games with Discounting" Econometrica, 56
Abreu, Pearce and Stacchetti (1986) "Optimal Cartel Equilibria with Imperfect Monitoring" JET, 39
Abreu, Pearce and Stacchetti (1990) "Toward a Theory of Discounted Repeated Games with Imperfect Monitoring" Econometrica, 58
2010-06-07
Price Rigidities
Original article (link) posted: 16/08/2005
In reality, prices cannot be adjusted continuously.
...
On the demand side, past prices may affect the firms' current goodwill through consumers' learning about the good or switching costs. On the supply side, past prices affect current inventories.
...
The presence of price rigidities raises the possibility that price reactions are not bootstrap reactions but are simply attempts to regain or consolidate market share.
(Tirole (1988), p.253-4)
In Maskin and Tirole (1988), they assume two firms choose their prices asynchoronously, at odd (even) periods, firm 1 (2) chooses its price. They consider Markov strategies, the simple pricing strategies that depend only on the "payoff-relevant information", and look for a perfect equilibrium in which the firms use Markov strategies. (They call it "Markov perfect equilibrium")
Despite the restriction to simple (Markov) strategies, multiple equilibria exist (indeed, there also exist several kinked-demand-curve equilibria). However, it can be shown that in any Markov perfect equilibrium, profits are always bounded away from the competitive profit (which is 0).
...
The intuition here is that if firms were stuck in the competitive price region, with the prospects of small profits in the future, a firm could raise its price dramatically and lure its rival to charge a high price for at least some time (the rival would not hurry back to nearly competitive prices). Thus tacit collusion is not only possible (as in the supergame approach) but necessary. Furthermore, it can be shown that there exists only one pair of equilibrium strategies that sustain industry profits close to the monopoly profit. These strategies from a symmetric kinked-demand-curve equilibrium at the monopoly price, and they are the only symmetric "renegotiation-proof" equilibrium strategies (whatever the current price, the firms cannot find an alternative Markov perfect equilibrium that they both prefer).
(Tirole (1988), p.256)
In reality, prices cannot be adjusted continuously.
...
On the demand side, past prices may affect the firms' current goodwill through consumers' learning about the good or switching costs. On the supply side, past prices affect current inventories.
...
The presence of price rigidities raises the possibility that price reactions are not bootstrap reactions but are simply attempts to regain or consolidate market share.
(Tirole (1988), p.253-4)
In Maskin and Tirole (1988), they assume two firms choose their prices asynchoronously, at odd (even) periods, firm 1 (2) chooses its price. They consider Markov strategies, the simple pricing strategies that depend only on the "payoff-relevant information", and look for a perfect equilibrium in which the firms use Markov strategies. (They call it "Markov perfect equilibrium")
Despite the restriction to simple (Markov) strategies, multiple equilibria exist (indeed, there also exist several kinked-demand-curve equilibria). However, it can be shown that in any Markov perfect equilibrium, profits are always bounded away from the competitive profit (which is 0).
...
The intuition here is that if firms were stuck in the competitive price region, with the prospects of small profits in the future, a firm could raise its price dramatically and lure its rival to charge a high price for at least some time (the rival would not hurry back to nearly competitive prices). Thus tacit collusion is not only possible (as in the supergame approach) but necessary. Furthermore, it can be shown that there exists only one pair of equilibrium strategies that sustain industry profits close to the monopoly profit. These strategies from a symmetric kinked-demand-curve equilibrium at the monopoly price, and they are the only symmetric "renegotiation-proof" equilibrium strategies (whatever the current price, the firms cannot find an alternative Markov perfect equilibrium that they both prefer).
(Tirole (1988), p.256)
2010-06-02
Remarks on APS by Tirole (1988)
Original article (link) posted: 04/08/2005
Tirole (1988) mentions APS in supplementary section. His description about optimal collusions is very clear, so I will quote it here.
Abreu, Pearce and Stacchetti (1986, 1990) show that one can indeed restrict attention to a collusive phase and a punishment phase, characterized by payoffs V+ and V-, where V+ and V- are now the best and worst elements in the set of symmetric perfect equilibrium payoffs. Furthermore, the collusive phase and the punishment phase take simple forms. In the collusive phase, the firms produce output q+. The punishment phase is triggered by a tail test, i.e., it starts if the market price falls under some threshold level p+. Thus, the collusive phase is qualitatively similar to that presumed in Porter (1983) and Green and Porter (1984). The punishment phase, however, does not have a fixed length; rather, it resembles the collusive phase. The two firms produce (presumably high) output q- each. If the market price exceeds a threshold price p-, the game remains in the punishment phase; if it lies below p-, the game goes back to the collusive phase. Thus, the evolution between the two phases follows a Markovian process. The reader may be surprised by the "inverse tail test" in the punishment phase. The idea is that a harsh punishment requires a high output (higher than is even privately desirable); to ensure that the firms produce a high output, it is specified that in the case of a high price (which signals a low output) the game remains in the punishment phase. (Notice that if one restricted punishments to be of the Cournot type, the optimal length of punishment would be T = "infinity", from the APS result on the harshest possible punishment V-.)
(Tirole (1988), p.265)
Tirole (1988) mentions APS in supplementary section. His description about optimal collusions is very clear, so I will quote it here.
Abreu, Pearce and Stacchetti (1986, 1990) show that one can indeed restrict attention to a collusive phase and a punishment phase, characterized by payoffs V+ and V-, where V+ and V- are now the best and worst elements in the set of symmetric perfect equilibrium payoffs. Furthermore, the collusive phase and the punishment phase take simple forms. In the collusive phase, the firms produce output q+. The punishment phase is triggered by a tail test, i.e., it starts if the market price falls under some threshold level p+. Thus, the collusive phase is qualitatively similar to that presumed in Porter (1983) and Green and Porter (1984). The punishment phase, however, does not have a fixed length; rather, it resembles the collusive phase. The two firms produce (presumably high) output q- each. If the market price exceeds a threshold price p-, the game remains in the punishment phase; if it lies below p-, the game goes back to the collusive phase. Thus, the evolution between the two phases follows a Markovian process. The reader may be surprised by the "inverse tail test" in the punishment phase. The idea is that a harsh punishment requires a high output (higher than is even privately desirable); to ensure that the firms produce a high output, it is specified that in the case of a high price (which signals a low output) the game remains in the punishment phase. (Notice that if one restricted punishments to be of the Cournot type, the optimal length of punishment would be T = "infinity", from the APS result on the harshest possible punishment V-.)
(Tirole (1988), p.265)
2010-05-31
Great professor Matsushima
Original article (link) posted: 03/08/2005
At the last part of the renegotiation section in Pearce (1992), I found the interesting description about Matsushima (1990) "Structure of Renegotiation in Infinitely Repeated Games" mimeo, Stanford University.
I cannot end this catalogue without mentioning an intriguing paper by Matsushima (1990). His idea is that just as an equilibrium specifies what will happen if its "instructions" are not obeyed, societies have metacodes indicating what happens when a social convention (equilibrium) is breached. The ensuing analysis is highly ingenious; to my astonishment, Matsushima emerges from a jungle of infinite sequences of social conventions and breaching rules, with an existence result. I will not try to explain the motivation for the solution concept; on that score, despite some enjoyable discussions with the author, I remain mystified.
(Pearce (1992), p.166)
By the way, I could not find Matsuhima's paper; he does not seem to have published the paper, and Stanford does not have the working paper either. If you know something about this paper, please let me know. (maybe I should ask him directly)
At the last part of the renegotiation section in Pearce (1992), I found the interesting description about Matsushima (1990) "Structure of Renegotiation in Infinitely Repeated Games" mimeo, Stanford University.
I cannot end this catalogue without mentioning an intriguing paper by Matsushima (1990). His idea is that just as an equilibrium specifies what will happen if its "instructions" are not obeyed, societies have metacodes indicating what happens when a social convention (equilibrium) is breached. The ensuing analysis is highly ingenious; to my astonishment, Matsushima emerges from a jungle of infinite sequences of social conventions and breaching rules, with an existence result. I will not try to explain the motivation for the solution concept; on that score, despite some enjoyable discussions with the author, I remain mystified.
(Pearce (1992), p.166)
By the way, I could not find Matsuhima's paper; he does not seem to have published the paper, and Stanford does not have the working paper either. If you know something about this paper, please let me know. (maybe I should ask him directly)
2010-05-28
Regional Caps on JRMP
I attended a lunch seminar on microeconomics at University of Tokyo today (May 27th); Fuhito Kojima, one of the most productive game theorists in my ages, talked about his recent matching paper with Yuichiro Kamada (a rising star at Harvard):
"Improving Efficiency in Matching Markets with Regional Caps: The Case of the Japan Residency Matching Program" (joint with Yuichiro Kamada)
Their work is strongly motivated by the actual centralized matching system used in the Japan residency matching program (JRMP). JRMP, as it follows NRMP in the U.S., employs the Gale-Shapley algorithm to assign doctors to hospitals. However, due to the vacancy problem at (mainly) rural hospitals, JRMP recently introduced "regional caps" to each prefecture in order to control the numbers of doctors in popular area.
The current paper theoretically investigates the effects of imposing such an exogenous caps to the GS algorithm. Based on the new stability concept they define, which incorporates regional cap constraints in a natural way and coincides with the usual stability if there were no caps, they show the followings:
Here is a technical remark. To define new algorithm, we have to decide two things (which are absent from the usual GS algorithm), (1) target caps and (2) an order of hospitals. As I remember correctly, the resulting outcome is independent of the choice of (1) (whenever target caps satisfy weak feasibility conditions), but the speaker didn't mention whether the outcome is also invariant to (2) or not. I may better ask him later...
I found the paper really interesting: motivation is clear, results look nice, contains important policy implication, can be applied to different matching problems, and so on. It might be desirable if they could provide some policy recommendation with respect to the choice of regional caps. Although actual caps are typically determined by politics or something other than economic theory I suppose, some benchmark analysis should be helpful. Anyways, I like this paper very much :) (We may perhaps find its title on the front page of some top journal in the near future!)
"Improving Efficiency in Matching Markets with Regional Caps: The Case of the Japan Residency Matching Program" (joint with Yuichiro Kamada)
Their work is strongly motivated by the actual centralized matching system used in the Japan residency matching program (JRMP). JRMP, as it follows NRMP in the U.S., employs the Gale-Shapley algorithm to assign doctors to hospitals. However, due to the vacancy problem at (mainly) rural hospitals, JRMP recently introduced "regional caps" to each prefecture in order to control the numbers of doctors in popular area.
The current paper theoretically investigates the effects of imposing such an exogenous caps to the GS algorithm. Based on the new stability concept they define, which incorporates regional cap constraints in a natural way and coincides with the usual stability if there were no caps, they show the followings:
- The current Japanese mechanism, i.e., exogenous regional caps + GS, is not a stable matching mechanism.
- There exists a new algorithm (natural extension of the GS) that always a stable and constrained efficient matching.
- This new mechanism is group strategy-proof for doctors, that is, any group of doctors does not have an incentive to manipulate their preferences.
Here is a technical remark. To define new algorithm, we have to decide two things (which are absent from the usual GS algorithm), (1) target caps and (2) an order of hospitals. As I remember correctly, the resulting outcome is independent of the choice of (1) (whenever target caps satisfy weak feasibility conditions), but the speaker didn't mention whether the outcome is also invariant to (2) or not. I may better ask him later...
I found the paper really interesting: motivation is clear, results look nice, contains important policy implication, can be applied to different matching problems, and so on. It might be desirable if they could provide some policy recommendation with respect to the choice of regional caps. Although actual caps are typically determined by politics or something other than economic theory I suppose, some benchmark analysis should be helpful. Anyways, I like this paper very much :) (We may perhaps find its title on the front page of some top journal in the near future!)
2010-05-26
Folk theorems by FLM (1994)
Original article (link) posted: 03/08/2005
According to Pearce (1992), Fudenberg, Levine and Maskin (1994) "The Folk Theorem in Repeated Games with Imperfect Public Information" Econometrica, 62 represent the state of art in discounted folk theorems for a broad range of information structure, and "anyone interested in repeated games should read it closely".
It is difficult to state their folk theorems without relying on mathematical symbols. Here, I try to summarize the key insight as I quote the relevant sentences (in italic) from Pearce (1992).
To derive folk theorems, they consider two conditions, the individual full rank condition and the pairwise full rank condition. The former is needed to ensure that a player's different possible actions can be distinguished, and hence encouraged or discouraged. The latter is needed to acquire information which discriminates statistically between deviations by some player and others.
Without the individual full rank condition, it may not be possible to induce players to play some strategy, no matter what rewards are attached to signal realizations. In contrast,
this (="the individual full rank condition") guarantees that any behavior can be enforced if arbitrary continuation payoffs can be used.
The failure of the pairwise full rank condition explains the inefficient results by Radner, Myerson and Maskin (1986).
The problem there was that the only way to enforce good behavior was to punish both players in the event that output is low. Efficient (or nearly efficient) cooperation in a model where no player's actions are observed, generally requires that, when one player's continuation payoff is reduced, another's must be increased; surplus should be passed back and for the amongst players, not thrown away.
With the additional restrictions on the information structure guaranteed by the full rank conditions, FLM prove a folk theorem of virtually the same degree of generality as for perfect monitoring.
As a final remark, it should be noticed that the pairwise full rank condition does not necessarily be satisfied at equilibrium strategy profiles.
It would have been reasonable to guess that, to prove that a desired profile "gamma" can be enforced (almost) efficiently, it would be necessary to impose pairwise full rank relative to deviations form "gamma". By contrast, all that is actually assumed is that, for each "i" and "j", there is some distinguishing "alpha" that allows i's and j's deviations to be distinguished, and not necessarily the same "alpha" for each pair of players! FLM demonstrates that a profile as close as desired to "gamma" can be found that puts a little weight on the strategies used in the "distinguishing profiles," and discriminates as required between deviations of different players.
I should read FLM again...
According to Pearce (1992), Fudenberg, Levine and Maskin (1994) "The Folk Theorem in Repeated Games with Imperfect Public Information" Econometrica, 62 represent the state of art in discounted folk theorems for a broad range of information structure, and "anyone interested in repeated games should read it closely".
It is difficult to state their folk theorems without relying on mathematical symbols. Here, I try to summarize the key insight as I quote the relevant sentences (in italic) from Pearce (1992).
To derive folk theorems, they consider two conditions, the individual full rank condition and the pairwise full rank condition. The former is needed to ensure that a player's different possible actions can be distinguished, and hence encouraged or discouraged. The latter is needed to acquire information which discriminates statistically between deviations by some player and others.
Without the individual full rank condition, it may not be possible to induce players to play some strategy, no matter what rewards are attached to signal realizations. In contrast,
this (="the individual full rank condition") guarantees that any behavior can be enforced if arbitrary continuation payoffs can be used.
The failure of the pairwise full rank condition explains the inefficient results by Radner, Myerson and Maskin (1986).
The problem there was that the only way to enforce good behavior was to punish both players in the event that output is low. Efficient (or nearly efficient) cooperation in a model where no player's actions are observed, generally requires that, when one player's continuation payoff is reduced, another's must be increased; surplus should be passed back and for the amongst players, not thrown away.
With the additional restrictions on the information structure guaranteed by the full rank conditions, FLM prove a folk theorem of virtually the same degree of generality as for perfect monitoring.
As a final remark, it should be noticed that the pairwise full rank condition does not necessarily be satisfied at equilibrium strategy profiles.
It would have been reasonable to guess that, to prove that a desired profile "gamma" can be enforced (almost) efficiently, it would be necessary to impose pairwise full rank relative to deviations form "gamma". By contrast, all that is actually assumed is that, for each "i" and "j", there is some distinguishing "alpha" that allows i's and j's deviations to be distinguished, and not necessarily the same "alpha" for each pair of players! FLM demonstrates that a profile as close as desired to "gamma" can be found that puts a little weight on the strategies used in the "distinguishing profiles," and discriminates as required between deviations of different players.
I should read FLM again...
2010-05-25
Information and timing
Original article (link) posted: 03/08/2005
With imperfect monitoring, shrinking the period length implies less discounting from one period to the next, but also leaves less time for players to observe signals relevant to behavior.
...
So there are two effects of reducing the period length: an effective increase in patience, which we know from the monotonicity result tends to increase the average value set, and a worsening of information, which Kandori (1992) elegantly shown to decrease the set of equilibrium values. Either of these two effects can dominate in a particular case.
...
If the period of fixed action is a year, under plausible parameter values cooperation could be sustained very profitably. But suppose that instead auctions can be changed daily. The only way to encourage cooperation is to punish the event that there is no goods news, which has probability near 1 whether anyone shirks or not.
...
Ironically, in this case the player's ability to respond quickly to information destroys all possibility of cooperation. This suggests that delaying the release of information might actually be valuable in partnerships; Abreu, Milgrom and Pearce (1991) show that for high "delta", information delays can virtually eliminate the inefficiency that Radner, Myerson and Maskin (1986) identified.
(Pearce (1992), p.156)
Abreu, Milgrom and Pearce (1991) "Information and Timing in Repeated Partnerships" Econometrica, 59
Kandori (1992) "The Use of Information in Repeated Games with Imperfect Monitoring" RES, 59-3
Radner, Myerson and Maskin (1986) "An example of a Repeated Partnership Games with Discounting and with Uniformly Inefficient Equilibria" RES, 53
With imperfect monitoring, shrinking the period length implies less discounting from one period to the next, but also leaves less time for players to observe signals relevant to behavior.
...
So there are two effects of reducing the period length: an effective increase in patience, which we know from the monotonicity result tends to increase the average value set, and a worsening of information, which Kandori (1992) elegantly shown to decrease the set of equilibrium values. Either of these two effects can dominate in a particular case.
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If the period of fixed action is a year, under plausible parameter values cooperation could be sustained very profitably. But suppose that instead auctions can be changed daily. The only way to encourage cooperation is to punish the event that there is no goods news, which has probability near 1 whether anyone shirks or not.
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Ironically, in this case the player's ability to respond quickly to information destroys all possibility of cooperation. This suggests that delaying the release of information might actually be valuable in partnerships; Abreu, Milgrom and Pearce (1991) show that for high "delta", information delays can virtually eliminate the inefficiency that Radner, Myerson and Maskin (1986) identified.
(Pearce (1992), p.156)
Abreu, Milgrom and Pearce (1991) "Information and Timing in Repeated Partnerships" Econometrica, 59
Kandori (1992) "The Use of Information in Repeated Games with Imperfect Monitoring" RES, 59-3
Radner, Myerson and Maskin (1986) "An example of a Repeated Partnership Games with Discounting and with Uniformly Inefficient Equilibria" RES, 53
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