Quantum game theory

My friends working at University of Tokyo recently published an article about quantum game theory on Journal of Physics.

Yohei Sekiguchi, Kiri Sakahara, and Takashi Sato (2010), "Uniqueness of Nash equilibria in a quantum Cournot duopoly game," Journal of Physics A: Mathematical and Theoretical, Volume 43, Number 14

Here is a link to the article. Congratulations!

Unfortunately, I don't know anything about quantum game theory. According to wikipedia (link), it is explained as follows:

Quantum game theory is an extension of classical game theory to the quantum domain. It differs from classical game theory in three primary ways:

1. Superposed initial states,
2. Quantum entanglement of initial states,
3. Superposition of strategies to be used on the initial states.

Anyways, it is surprising that game theorists publish their papers on physics journals. Hum, quantum game theory might be worth trying to study...


During the conference in Brazil (link), I had a chance to attend one of the presentations by physicists, whose topic is not about quantum game theory though:

"Distinguishing the Opponents: Mutual Cooperation is Never Destroyed"
by Lucas Lages Wardil (Universidade Federal de Minas Gerais)

The paper investigates the evolution in network structures. Unlike previous works, he considers that each agent can take a contingent action, i.e., strategy, rather than a unconditional action which has been commonly assumed in the literature. That is, depending on whom to play with, each agent will take different actions; with a certain updating process each agent changes her (contingent) action against a specific opponent. In this framework with extended agents' types, he shows that cooperation (in prisoner's dilemma) becomes easy to sustain under certain networks and imitation dynamics.

In evolutionary biology, where this kind of research is widely investigated, it is unrealistic to regard contingent actions as a agents' type, because a type is considered to be genetic. In Economics, we usually examine contingent action plans in rational frameworks but it is uncommon in bounded rational frameworks such as evolutionary game. The reason (I guess) is that it is difficult to argue why and how an agent with such a complicated action plan follows a irrational/heuristic adjustment process to update her behaviors.

Anyways, I found this paper by a physicist very interesting. In some sense, his research connects biology and economics (although further justification/interpretation seems to be necessary to apply his models in these fields). There might be many things that we economists can learn from physicists.

Reny's new exsistence theorem

In the plenary session on the third day of BWGT conference (link), Professor Philip Reny talked about his new research on monotone pure strategy equilibria:

Title: On the Existence of Monotone Pure Strategy Equilibria in Bayesian Games (link to pdf)

Abstract: We generalize Athey's (2001) and McAdams' (2003) results on the existence of monotone pure strategy equilibria in Bayesian games. We allow action spaces to be compact locally-complete metrizable semilattices and type spaces to be partially ordered probability spaces. Our proof is based upon contractibility rather than convexity of best reply sets. Several examples illustrate the scope of the result, including new applications to multi-unit auctions with risk-averse bidders.

According to Prof. Reny, while the topic of the paper is related to many fields such as mathematical economics, mechanism design, and auctions, there are two seminal papers that strongly motivated his research. Athey (2001) first establishes the sufficient conditions to guarantee the existence of monotone pure strategy equilibria in Bayesian games with one-dimensional and totally ordered type and action spaces. The key condition is a Spence-Mirlees single-crossing property. McAdams (2003) extends Athey's analysis to multi-dimensional and partially ordered spaces.

Prof. Reny succeeded to derive weaker conditions than McAdams in Bayesian games with multi-dimensional strategy spaces, and also extend to the infinite type and action spaces. The key insight is to use a fixed point theorem derived by Eilenberg and Montgomery (1946) instead of Kakutani's (used by Athey) or Glicksberg's (used by McAdams) ones. The latter two theorems require best reply sets to be convex while the former requires only contractibility, which turns out to be (almost) automatically satisfied in Bayesian games.
His main result says the following:

Theorem: (Under some conditions) If, whenever the other players employ monotone pure strategies, each player's set  of monotone pure-strategy best replies is nonempty and join-closed, then a monotone pure strategy equilibrium exists.

Note that a subset of strategies is join-closed if the pointwise supremum of any pair of strategies in the set is also in the set.

The idea of join-closedness (in the different context, though) recently showed up when I discussed my jointwork on the structure of stable matchings with co-authors. It may have some connection...

References
Susan Athey (2001), "Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information,"  Econometrica, Vol. 69: 861-889.
Samuel Eilenberg and Deane Montgomery (1946), "Fixed Point Theorems for Multi-Valued Transformations," American Journal of Mathematics, Vol. 68: 214-222.
David McAdams (2003), "Isotone Equilibrium in Games of Incomplete Information," Econometrica, Vol. 71:1191-1214.

AER (June 2010)

The latest volume of the American Economic Review (link to the table of contents) just arrived by mail. One may get interested in the lead-off articles written by the Nobel laureates 2009 in Economic Sciences:

"Beyond Markets and States: Polycentric Governance of Complex Economic Systems" by Elinor Ostrom
"Transaction Cost Economics: The Natural Progression" by Oliver E. Williamson

Other articles that attracted my interest are the following:

• "A Study of the Internal Organization of a Bidding Cartel" by John Asker
• "Bargaining with Arrival of New Traders" by William Fuchs and Andrzej Skrzypacz
• "The Burden of the Nondiversifiable Risk of Entrepreneurship" by Robert E. Hall and Susan E. Woodward
• "What Causes Industry Agglomeration? Evidence from Coagglomeration Patterns" by Glenn Ellison, Edward L. Glaeser and William R. Kerr

I downloaded the pdf files of the above articles, trying to read them on the way to Brazil tomorrow, where I will attend a great game theory conference: The Second Brazilian Workshop of the Game Theory Society (BWGT2010, link).

Great statistician passed away

I heard a sad news: Eminent statistician David Blackwell has died at 91

Professor Blackwell, though a mathematician or statistician, is quite famous to us economists for his significant contribution in mathematical statistics and dynamic programming.
Here is a link to the wikipedia article about him.

He has written many influential papers such as:

• "Conditional Expectation and Unbiased Sequential Estimation," Annals of Mathematical Statistics, Vol. 18, No. 1 (Mar., 1947), pp. 105-110
• "Equivalent Comparisons of Experiments," Annals of Mathematical Statistics, Vol. 24, No. 2 (Jun., 1953), pp. 265-272  
• "Discounted Dynamic Programming," Annals of Mathematical Statistics, Vol. 36, No. 1 (Feb., 1965), pp. 226-235

Interestingly, despite his deep thoughts and ingenious works on each research topic, he described himself as follows:

I've worked in so many areas—I'm sort of a dilettante. Basically, I'm not interested in doing research and I never have been.

May he rest in peace.

IO Seminar (Daughety and Reinganum)

Original article (link) posted: 21/09/2005

Daughety and Reinganum "Imperfect Competition and Quality Signaling"

The paper investigates the one-shot oligopoly model where firms produce substitute products with associated vertical quality measure. Each firm has private information about its quality (="type") and signaling effects by pricing are captured. Their main focus is comparison between separating equilibria of incomplete information (about vertical quality) and that of complete information.
As main results, they show the following;

1) incomplete information (signaled via prices) softens price competition, and imperfect competition can reduce the degree to which firms distort their prices to signal their types
2) low-quality firms always prefer playing the incomplete information game to the full-information analog
3) if the proportion of high-quality firms is great enough, they also prefer incomplete information to full-information

It is very difficult for me to judge their contribution in this field because there are so many papers in the literature and some of them look quite similar to this paper at least for those who are not familiar with this line of research. The model in the paper tries to capture the both effects of incomplete information and imperfect competition, which makes it very complicated. Although the main results mentioned above sound interesting, similar kind of qualitative results can be derived in simpler model I guess. For example, the comparison between a separating equilibrium and a pooling one in a simple Spence type signaling model has the implications quite similar to (2) and (3). Of curse, except for the results (1)-(3), they show a bunch of comparative static and some of them are interesting and not obvious. However, I would like to say it should be needed for them to say why such a complicated model is used to explain the results most of which were already known and hence not surprising.
The literature review (Section 2) is comprehensive. So, if you are interested in the paper, it might be better to read some key references before deeply tackle this paper.

Interesting papers in References

Bagwell and Riordan (1991) "High and Declining Prices Signal Product Quality" AER, 81
Mailath (1989) "Simultaneous Signaling in an Oligopoly Model" QJE, 104
Martin (1995) "Oligopoly Limit Pricing: Strategic Substitutes, Strategic Complements" IJIO, 13

Grossman (1989)

Original article (link) posted: 20/09/2005

I was taking a look at Sanford Grossman's book "The Informational Role of Prices". This is a collection of his articles focusing on informational role of prices or that of contracts. All the articles except for the first one were reprinted from journals. Here, I take a memo for the chapter one which is an introduction of the book.

The main theme of the book is to propose models which incorporate two aspects of the role of prices at the same time. He says;

I have elaborated a model of economic equilibrium that is based upon the idea that prices have a dual role: They constrain behavior by affecting the costs or benefits of acts, but they also convey information about what will be the costs and benefits of the acts.

In the framework of Marshall or Walras, people are merely constrained by prices, and so, no one learns anything from prices. That is, their models cannot capture the second role of prices. To be more precise, in the Walrasian model, the demand function specifies a desired level of holdings of the security at each particular price, irrespective of whether that price is a market clearing price. However, a trader might be induced to adjust his "demand" function to reflect the fact that the price at which market clears conveys information. To take this informational role into account, the author assumes that the consumer faces a price that is a real offer of another person, or the outcome of a market process.
In short, his idea is to redefine the "demand" as an expression of desired holdings at prices that are "market clearing", i.e., each consumer chooses his demand at "p" to maximize his expected utility conditioned on both his private information and on the information contained in the event that "p" is a market clearing price. As a consequence, there is no desire to recontract after observing that a particular price is the market clearing price because each person has already incorporated it.
In the rest of the chapter 1, the author introduces the concept of "uninformed traders" or "noise traders" who demand a security for noninformational reasons. Incorporating them into his models, he tries to explain the stock market crush in 1987. Interestingly, he claims this event supports his models with rational agents rather than irrational behavior of traders, which is described as follows.

Some have suggested that this is strong evidence against investor rationality, and point to the October 1987 episode as another example of irrational behavior. In contrast, I think that these events and the excessive volatility of stock prices relative to the volatility of expected payoffs are evidence in favor of the type of "rationality" embodied in the R.E. approach outlined above, rather than evidence for irrationality. As I argue below, once the Walrasian notion of demand is eliminated, the volatility phenomena can be seen as an expression of a sophisticated trading strategy rather than irrationality.

Interesting papers in References

Grossman and Stiglitz (1976) "Information and Competitive Price Systems" AER, 66-2
Kreps (1988) "In Honor of Sandy Grossman, Winner of the John Bates Clark Medal" J. of Economic Perspectives, 2-2
Kyle (1985) "Continuous Auctions and Insider Trading" Econometrica, 53
Roll (1984) "Orange Juice and Weather" AER, 74

The following book might be useful for those who are interested in the line of research mentioned above.

Brunnermeier (2001) "Asset Pricing under Asymmetric Information" Oxford University Press

IO Seminar (Rob and Fishman)

Original article (link) posted: 14/09/2005

Rob and Fishman "Is Bigger Better? Customer base expansion through word of mouth reputation" forthcoming in JPE

The paper develops a modeling framework in which a firm regards its reputation as a capital assets whose value is maintained through a process of active and continuous investment. Firms are required to investment for each period to produce high quality products. The quality of the product is only known to a consumer who buys it from the firm (experience good assumption), and she will tell this information to a new consumer with some probability. This information spread captures "word of mouth reputation".

Their main finding is that investment in quality is positively related to the size of customer base which is defined as the number of consumers who are aware of the firm's reputation. This is because reputation is costly to acquire and takes a long time to regain once it has been lost, and hence, a good reputation is more valuable to a firm the larger its customer base is. The model predicts that the larger is a firm, the more it invests in quality, and the higher is the average quality it delivers.

Interesting papers in references

Horner (2002) "Reputation and Competition" AER, 92
Mailath and Samuelson (2001) "Who wants a Good Reputation" RES, 68
Shapiro (1983) "Premiums for High Quality Products as Returns to Reputation" QJE, 98
Tadelis (2002) "The Market for Reputation as an Incentive Mechanism" JPE, 92

Their contribution in the literature is stated as follows.

What differentiates our approach from all these papers is that reputation in our model spreads in the market through word of mouth, or referrals - consumers tell other consumers about their experience, causing some firms to grow and other firms to decline. As a consequence, a firm starts out small, grows gradually, and changes its investment as its reputation is established. These interrelated processes of firm growth, reputation formation, and the links between age, size, and investment in quality represent our main contribution to the literature.

Lecture 1 (Dutta): Math Preliminaries

Original article (link) posted: 13/09/2005

The class by professor Dutta started with mathematical preliminaries. It might be good to study those concepts and some theorem again because I left myself unclear of some of them during taking a math class in my first year.
Well, I am thinking to write a brief summary for each week (The class is once every week on Monday). Here is the first one.

Topics in the class

1) Correspondings and Maximum Theorem
2) Contraction Mapping Theorem

What we covered in the class

In (1):
(a) The definitions of correspondings and several versions of continuities: upper semi-continuity (USC), lower semi-continuity (LSC) (and continuity).
(b) Maximum Theorem and its proof. Note) Maximum Theorem says that the maximum value is continuous and the maximizer is USC in parameters under some conditions.

In (2):
(c) The def. of contraction, Cauchy sequences and complete metric space.
(d) Contraction Mapping Theorem and its proof. Note) The theorem says that if there is a contraction corresponding and its domain is a complete metric space, then there exists a unique fixed point.

Comments

(a) I've often mixed up USC and LSC, but finally the difference seems to be clear for me.
(b) I need to reconsider the proof. It's not so complicated but not that easy either.
(c) I realized that I had forgotten the def. of complete metric space...
(d) The proof is much easier than (b). Uniqueness is almost straight forward.

Recommended readings

SLP Chapter 3
Sundaram (1995) "A Course in Optimization Theory"

Microeconomic Analysis - G6218

Original article (link) posted: 13/09/2005

Today, I attended the first class of G6218 which is an advanced microeconomics for upper class students by Professor Dutta. The course covers basic issues in dynamic economics and consists of the following three parts.

1) Dynamic Programming: The theory of single-agent models
2) Repeated/Dynamic Games: The theory of multi-agent models
3) Application: A strategic analysis of climate change

Although there is no required textbook, the two books listed below seems to be useful.

Stokey, Lucas and Prescott (1989) "Recursive Methods in DYnamic Economics" for (1)
Fudenberg and Tirole (1991) "Game Theory" for (2)

Since professor Dutta is very good at teaching and the topics are interesting for me and deep in mathematics, I am thinking to take this course. It should be better to bring SLP from Princeton... (I asked my friends in Princeton to keep my books.)

JEP for FREE

Here comes a great news!

Online issues of the Journal of Economic Perspectives published since 1999 are now publicly accessible at no charge, compliments of the American Economic Association.

You can access to the Journal of Economic Perspectives from here. The symposium in the latest volume (Spring 2010), Con out of Economics, contains interesting debate among leading econometricians that may attract much attention.