Echenique (2003)

Echenique, F. (2003), The Equilibrium Set of a Two Player Game with Complementarities is a Sublattice, Economic Theory, 22: 903-905.  Link 

Summary  I prove that the equilibrium set in a two-player game with complementarities, and totally ordered strategy spaces, is a sublattice of the joint strategy space.

It is widely known that in games with strategic complementarities (GSC), i.e., best reply correspondings are monotone increasing for all players, the set of pure-strategy Nash equilibria forms a non-empty complete lattice. This result implies the existence of smallest and largest Nash equilibria.

The current paper looks further into the structure of the equilibrium set of GSC, and shows that under certain restrictions the equilibrium set becomes not just a complete lattice but also a sublattice, as is written down in the Summary above.

The practical importance of this result, according to the author, is:

If the equilibrium set of a game is a sublattice, then we can find new equilibria from knowing that two profiles are equilibria, and by taking the componentwise join and meet of players’ strategies.

Note that we cannot obtain such strong property by a complete lattice structure alone. In this sense, a sublattice is indeed critical. The proof is extremely simple, which makes use of the observation that (a) when there are only 2 players and (b) their strategy spaces are completely ordered, (c) the other player's strategy must be completely ordered. The property (c) does not hold either (a) or (b) is not satisfied. Once (c) is verified, the rest of the proof is just to follow the definition of GSC.

My random thought: A sublattice property also arises in one-to-one two-sided matching markets (but neither in one-to-many nor many-to-many markets). I'm wondering if the idea of this paper can be somehow connected to the sublattice property of the set of stable matchings.

A final remark: A nicely written but a bit uninformative note.

In Memory of Prof. Hayami

I found an interesting article (through this tweet by Prof. Sawada) about Prof. Yujiro HAYAMI, a leading agricultural/development economist who passed away last December:
"Death of a layman's economist" (by Prof. Abdul Bayes at Financial Express)

The article says:

Leaving aside for a moment his hundreds of technical articles published in reputed international journals, I shall take the privilege of citing from his famous book on development economics that I have mentioned before. Using the concept of Prisoner's Dilemma - when two persons convicted of a murder are kept in separate cells without one knowing what the other person is saying to the investigating officer - he illustrates how much loss the inability among people to establish cooperative relationship due to lack of communication and trust could generate for the society. This loss can happen in all economic transactions. "For example, in the transaction of a commodity, a buyer may try to reduce payment to a seller on the false charge of quality deficiency in delivered commodities. Then, the seller will deliver low-quality commodities thereafter. As their mutual distrust is heightened, they will stop transactions and thereby close off a mutually profitable business opportunity".

Prof. Hayami's famous book mentioned above is the following.

It is very interesting to know that Hayami has applied game theoretical ideas to his work on developmental studies. I should have tried to talk with him (Actually, Prof. Hayami was my colleague in GRIPS...)

Hayami cites another example regarding the costs of non-cooperation: "If employment is so insecure that employees may be discharged any moment, they would make little effort to acquire specific knowledge and skill for efficient work in his organisation. Their employer would then be inclined to discharge these employees for their lack of effort. In this way, cooperative relationship will not be established with the little accumulation of skill and knowledge needed for efficient functioning of this organisation". 

According to Hayami hypothesis, if mutual trust between particular individuals is thus elevated to a moral code in the society, huge savings would be made in transaction costs. If such cooperative negotiations could be guaranteed, business plans could be promoted ex-post much more flexibly and efficiently than by rigid ex-ante specification of contingencies, especially in long-term transactions subject to high risk and uncertainty. By and large, trust is a social capital that needs to be nurtured, if necessary, by structuring the cooperative relations into a hierarchical organisation. If mutual trust between workers and management ceases to work, the cost of monitoring and enforcing the contract would be large.

(made bold by yyasuda)

What is an Auction?

I found a nice introductory description of auction from the viewpoint of economics or game theory in the following textbook on auction theory:

The below is the (partial) quotation from the chapter 2.3.1, titled "What is an Auction?":

[A]uctions Have become an effective tool to implement public policy. Their use now ranges from the allocation of radio spectrum necessary for mobile communication, to spot markets trading electricity and pollution permits, as well as being widely used in government procurement. 

We can now define an auction by one of its central properties: as a market clearing mechanism, to equate demand and supply. Other market mechanisms include fixed price sales (as in a supermarket) or bargaining (as in the negotiated sale of a house or a used car). Within the class of market mechanisms which allocate scarce resources, one particular characteristic of the auction is that the price formation process is explicit. That is, the rules that determine the final price are usually well-understood by all parties involved. 

Auctions are often used in the sale of goods for which there is no established market. Auctions were instrumental in the mass privatization in Eastern Europe given the absence of a price system that could guide the valuation process for firms being privatized. Rare or unique objects are typically sold in auctions as the markets for these objects are likely to be very thin. However, auctions are also used to sell Treasury bills and the markets for these assets are very thick. The reason is that only governments can legally produce such bonds and therefore the sale in an auction is an exercise in revenue maximization. 

Auctions are more flexible than a fixed price sale and perhaps less time-consuming than negotiating a price. Auctions are used to sell hundreds of goods, such as bales of wool or used cars, in a few hours. One can imagine how many hours it would take to sell 100 used cars through negotiated sales.

Independence and Implementation

Continued from the previous post on bargaining theory, I am going to quote other parts from the Introduction. This time, my focus is on editor's (Prof. Thomson) view on a controversial axiom of Nash solution, contraction independence(*), as well as his brief survey on implementation of bargaining solutions.

(*) Contraction independence is often called the axiom of independence of irrelevant alternatives (IIA), which is defined as follows:

Contraction independence If, keeping the disagreement point constant, the feasible set contracts but the alternative chosen as solution outcome remains feasible, then it should remain the solution outcome.

On contraction independence
Contraction independence has been the object of the sharpest criticisms. Nash himself expressed some misgivings about it and Luce and Raiffa's (1957) objections are well known. In evaluating a bargaining situation, it is unavoidable and probably desirable that it be simplified and summarized, that it be reduced to its essential features. The issue is how much and what information should be discarded in the process, and one can make a convincing case that contraction independence ignores too much. Indeed, it covers contractions that affect the shape of the feasible in ways that seem relevant, for instance, the elimination of the only alternatives at which a particular agent's payoff is higher than at the initial solution outcome and the other agent's payoff lower than at the initial solution outcome; contraction independence prevents solutions from responding to such eliminations.
The main counterargument to this criticism was made by Nash himself: in practice, information is usually lacking about which alternatives are truly available, and a compromise under evaluation only competes against others that are not too different from itself. Modeling this lack or information explicitly is what an investigator should perhaps do, but there are also advantages to keeping the model simple; contraction independence is a formal way to express the idea.

On implementation
Whether a solution is implementable depends on the type of game forms that are used, and on the behavioral assumptions made about how agents confronted with such games behave. For implementation by normal form games and when agents calculate best responses taking as given the choices made by the other agents, a critical property for what is then called Nash-implementability is Maskin-monotonicity (actually an invariance property with respect to enlargements of the lower contour set at the chosen alternative). Most solutions are not Maskin-monotonic and therefore not Nash-implementable by normal form games. However, an implementation of the Kalai-Smorodinsky solution by a sequential game is possible (Moulin, 1984). Later contributions delivered the Nash solution (Howard, 1992), the egalitarian solution (Bossert and Tan, 1995)./// For subgame perfect implementation, a general theorem is offered by Miyagawa (2002). It covers all solutions obtained, after normalizing problems so that the ideal point has equal coordinates, by maximizing a monotone and quasi-concave function of the agents' payoffs. Implementation is by means of a stage game and the equilibrium notion is subgame perfection.

References
Bossert and Tan (1995) "An arbitration game and the egalitarian bargaining solution", Social Choice and Welfare, Volume 12, Number 1, 29-41.
Luce and Raiffa (1957), Games and Decisions: Introduction and Critical Survey, Wiley.
Miyagawa (2002) "Subgame-perfect implementation of bargaining solutions", Games and Economic Behavior, Volume 41, Issue 2, 292-308.
Moulin (1984) "Implementing the Kalai-Smorodinsky bargaining solution", Journal of Economic Theory, Volume 33, Issue 1, 32-45.

Bargaining Solutions: Nash, Egalitarian and Kalai-Smorodinsky

A leading researcher in bargaining theory, Professor William Thomson, recently edited the notable collection of papers in axiomatic bargaining:

Bargaining and the Theory of Cooperative Games: John Nash and Beyond


The following is quoted from the Introduction written by the editor. The parts I refer below focus on three representative solutions of the bargaining problem(*) in the literature, while his Introduction covers much more materials. I strongly recommend those who are interested in bargaining theory to read this insightful survey article.

Almost sixty years ago, Nash (1950) published his seminal paper on what is now known as the "axiomatic theory of bargaining"./// He formulated a list of properties, or "axioms", that he thought a solution should satisfy, and he established the existence and the uniqueness of a solution satisfying all of the axioms; this solution is now called the "Nash solution".

Nash's model has been one of the most successful paradigms of game theory. His paper is the founding stone of a literature that now comprises several hundred theoretical papers. The Nash solution is presented in all game theory textbooks./// Together with the Shapley value (Shapley, 1953) and the core (Gillies, 1959), it constitutes the obligatory background on cooperative games in most economics graduate programs.

In spite of the large number of reasonable solutions that one can easily define, only three solutions and variants have consistently emerged: in addition to the Nash solution, they are egalitarian(**) and Kalai-Smorodinsky solutions(***)./// The dominance of these three solutions and these variants is a central conclusion to be drawn from the literature.

The Nash solution has come out somewhat more often than the other two, and the claim can perhaps be made that it is indeed special./// But the argument is a little dangerous. Earlier, we talked about the theorist's need to simplify and summarize in order to analyze, and in axiomatic analysis simplification often takes the form of independence and invariance axioms. The Nash solution satisfies many invariance conditions, thus it is not much of a surprise that it should have dome out often. On the other hand, the monotonicity axioms that have generally led  to the Kalai-Smorodinsky and egalitarian solutions are readily understood and endorsed by the man on the street.

It is mainly on the basis of monotonicity properties that the Kalai-Smorodinsky solution should e seen as an important challenger to the Nash solution, the egalitarian solution presenting another appealing choice. This latter solution enjoys even stronger monotonicity requirements and like the Nash solution, it satisfies strong independence conditions. Unlike both the Nash and Kalai-Smorodinsky solutions, it requires interpersonal comparisons of utility however, which, depending upon the context, may be seen as a desirable feature or a limitation.

(*) A bargaining problem consists of a pair (S, d) where S, the feasible set, is the subset of alternatives, and d, the disagreement point, is a point of S./// A bargaining solution defined on a class of problems is a function that associates with each problem (S, d) in the class a unique point of S, the solution outcome of (S, d).

(**) The egalitarian solution (Kalai, 1977a) selects the maximal point of S at which utility gains from d are equal. More generally, by making utility gains proportional to a fixed vector of weights, we obtain the weighted egalitarian solution relative to these weights (Kalai, 1977b)

(***) The Kalai-Smorodinsky solution (Kalai and Smorodinsky, 1975) selects the maximal point of S that is proportional to the profile of maximal payoffs that agents can separately reach among the points of S that dominate d.

References
Gillies (1959) "Solutions to general non-zero-sum games" in Contributions to the Theory of Games IV, Princeton University Press, 47-85.
Kalai (1977a) "Nonsymmetric Nash solutions and replications of 2-person bargaining", International Journal of Game Theory, Volume 6, Number 3, 129-133.
Kalai (1977b) "Proportional Solutions to Bargaining Situations: Interpersonal Utility Comparisons", Econometrica, Vol. 45, No. 7 (Oct., 1977), 1623-1630.
Kalai and Smorodinsky (1975) "Other Solutions to Nash's Bargaining Problem", Econometrica, Vol. 43, No. 3 (May, 1975), 513-518.
Nash (1950) "The Bargaining Problem", Econometrica, Vol. 18, No. 2 (Apr., 1950), 155-162.
Shapley (1953) "A Value for n-Person Games" in Contributions to the Theory of Games II, Princeton University Press, 307-317.

Experimental Game Theory in GEB

I found an interesting website in Games and Economic Behavior (GEB), one of the leading academic journals in game theory. As titled "Two decades of experimental game theory in Games and Economic Behavior," this special online issue shows 17 articles on experimental game theory which have been published in GEB. It says:

Assembling this Virtual Special Issue on Experimental Game Theory has been an eye-opener. The first step was to go back through all the issues to get a bigger picture of the range of papers that we have published in this area. Games and Economic Behavior (GEB) was founded in 1989 at a time when there really wasn’t a subfield of experimental game theory as such. It wasn’t until a year later that this journal published its first article based on laboratory experiments, in the March 1990 issue – exactly twenty years ago.

See here for the detailed information.

Aumann's survey

Original article (link) posted: 20/10/2005

Aumann (1985) “What is Game Theory Trying to Accomplish?” in Frontiers of Economics

This is a survey article by Professor Aumann, a Nobel Laureate of this year. The paper consists of 18 sections. The first 8 sections concerned with generalities of game theory with particular attention of the concept of science, sections 9-17 illustrate four equilibrium concepts (Nash Equilibrium, Core, Von Neumann-Morgenstern Stable Set, and Shapley Value) with applications, and section 18 concludes. In the first half of the paper, he put his point of view about game theory (or economics) as a science, which is very deep and insightful. It is highly recommended to read first 8 chapters for those who are interested in such questions as “What is (social) science?” and “What is the definition of science or truth?”.
His main claim is stated in Introduction as follows;

A solution concept (could be replaced with “a scientific theory”) should be judged more by what it does than by what it is; more by its success in establishing relationships and providing insights into the workings of the social processes to which it is applied than by considerations of a priori plausibility based on its definition alone.

Auction theory: Vickrey and early literature

Continued from the previous post, let me quote interesting parts from the editors' introductory summary.The following nicely illustrates the contribution of the pioneer of auction theory, William Vickrey.

Vickrey's seminal paper (Vickrey, 1961), mentioned in his 1996 Nobel Prize in economics, introduced the independent private value model, demonstrated equilibrium bidding behavior in a first-price auction, and then showed that truthful bidding could be induced as a dominant strategy by modifying the pricing rule: let each bidder pay the social opportunity cost of his winnings, rather than his bid. Finally, he showed in an example what would later be proven generally as the revenue equivalence theorem: different auction mechanisms that result in the same allocation of goods yield the same revenue to the seller.

Then, the authors explain a few important papers in the early literature of auction theory since Vickrey. The followings are my summary.

Wilson (1969)

• (pure) common value
• first analysis of equilibrium bidding with common values
• demonstrated the importance to avoid (what would be later called) the winner's curse

Milgrom (1981)

• common + private values
• discovered the importance of monotone likelihood ratio property (MLRP)
• showed that MLRP + conditional independence implies that

1. bidders use monotonic bidding strategies
2. a monotonic strategy satisfying the first-order condition constitutes an equilibrium

Milgrom and Weber (1982)

• affiliated values: if one bidder has a high signal of value, it is more likely that the signals of the other bidders are high
• showed that under affiliated values

1. Vickrey's revenue equivalence result no longer holds when we introduce a common value element
2. ascending auctions yield higher revenues than sealed-bid auctions

References
Milgrom, "Rational Expectations, Information Acquisition, and Competitive Bidding," Econometrica, 1981.
Milgrom and Weber, "A Theory of Auctions and Competitive Bidding," Econometrica, 1982.
Vickrey, "Counterspeculation, Auctions, and Competitive Sealed Tenders," Journal of Finance, 1961.
Wilson, "Competitive Bidding with Disparate Information," Management Science, 1969.

Frontiers of Science

I have been to Potsdam in Germany on Nov. 11 - 14 to attend 7th Japanese-German Frontiers of Science Symposium 2010 (link). It's a really interdisciplinary conference jointly organized by Alexander von Humboldt Foundation and Japan Society for the Promotion of  Science.

I was a invited speaker of the social science session titled "New Methods in Decision Making" (session list), and talked about "Recent Developments in Market Design and its Applications to School Choice" (slide). It was quite exciting to give a presentation to researchers from completely different fields, mainly from natural science. Although I didn't have enough time to cover the details of my own studies, many of them seem to get surprised to see how powerful and useful game theoretical tools are.

I also enjoyed the talks and discussions in other sessions very much. Most of topics were unfamiliar to me of course, but their frontier works looked truly exciting. This was a wonderful opportunity indeed! Many thanks to the organizers and participants :)

Shapiro (1983)

Original article (link) posted: 29/09/2005

Shapiro (1983) "Premiums for High Quality Products as Returns to Reputations" QJE 98

Think about the market where producers can change product quality over time and consumers cannot observe quality prior to purchase. Then, what will happen? To answer this question, Shapiro (1983) develops a model that explores the implications of firm-specific reputations in a perfectly competitive environment. The one of the most interesting results is that in the equilibrium, firms produce higher quality products earn larger premiums. The premiums are needed for the following two reasons;
First, there is a cost to establish reputation and to offset the cost, positive return (=premium) is needed. Without a premium, no firm chooses high quality.
Second, in this market, sellers can always increase profits in the short-run by reducing the quality of their products (="fly-by-night strategy"). To prevent this deviation, positive return on the faithful path (which dominates the short-run return induced by fly-by-night strategy) is needed.
Although, the above point has already been recognized (Klein and Leffler (1981) explored this idea informally), this is the first paper which models reputation under competitive markets.

Tirole (1988) (2.6.2) provides a simplified version of the Shapiro model; one firm, and two types of qualities. He also points out two problems of Shapiro's model, which are the reliance of infinite-horizon time and bootstrap aspects of the equilibria. As is easily seen, only the lowest quality product is provided in each period with finite horizon model (by backward induction). Bootstrap aspects mean that reputation matters only because consumers believe it matters. Indeed, if, for example, consumers believe the firms produce the low quality no matter what the past history, then their expectation would again be fulfilled. In other words, the analysis suggests only that repetition may offer incentives to supply quality, not that it necessarily will.

Note) In the paper, this possibility is excluded since the author poses strong assumption about reputation formation; the expected quality of the firm's product at t is simply the product quality he chooses at t-1, i.e., R(t)=q(t-1). This simple adjustment expectation turns out to be a rational expectation. However, as Tirole mentions, there are other rational expectation equilibria and Shapiro (1983) does not mention them.

Finally, notice that Kreps and Wilson (1982) and Milgrom and Roberts (1982) showed that reputation effects can be obtained even with a finite horizon by introducing asymmetric information about firm's type. Their models also pin down the equilibrium strategy and high quality is necessarily observed.

References

Klein and Leffler (1981) "The Role of Market Forces in Assuring Contractual Performance" JPE, 81
Kreps and Wilson (1982) "Reputation and Imperfect Information" JET, 27
Milgrom and Roberts (1982) "Predation, Reputation, and Entry Deterrence" JET, 27
Tirole (1988) "The Theory of Industrial Organization" MIT Press

Game Theory in Finance

What is going on in the up-front academic research in finance? I found a concise description of the field of finance from the great survey article:
"Finance Applications of Game Theory"by Franklin Allen and Stephen Morris (1998, link)

In Introduction, they say the following:

1. Introduction
Finance is concerned with how the savings of investors are allocated through financial markets and intermediaries to firms, which use them to fund their activities. Finance can be broadly divided into two fields. The first is asset pricing, which is concerned with the decisions of investors. The second is corporate finance, which is concerned with the decisions of firms. Traditional neoclassical economics did not attach much importance to either kind of finance. It was more concerned with the production, pricing and allocation of inputs and outputs and the operation of the markets for these. Models assumed certainty and in this context financial decisions are relatively straightforward. However, even with this simple methodology important concepts such as the time value of money and discounting were developed.
Finance developed as a field in its own right with the introduction of uncertainty into asset pricing and the recognition that classical analysis failed to explain many aspects of corporate finance.

Although the paper was written more than 10 years ago, game theoretical perspectives in finance has still not been widespread. If you are interested in these materials, you should definitely check this. Here is the abstract of the paper:

Abstract
Traditional finance theory based on the assumptions of symmetric information and perfect and competitive markets has provided many important insights. These include the Modigliani and Miller Theorems, the CAPM, the Efficient Markets Hypothesis and continuous time finance.
However, many empirical phenomena are difficult to reconcile with this traditional framework. Game theoretic techniques have allowed insights into a number of these. Many puzzles remain. This paper argues that recent advances in game theory concerned with higher order beliefs, informational cascades and heterogeneous prior beliefs have the potential to provide insights into some of these remaining puzzles.

Complementarity and supermodularity

I found a nice summary of key concepts in game theory, complementarity and supermodularity, which are especially important for auction and matching theory.


"Supermodularity and supermodular games" byXavier Vives
in the new palgrave dictionary of economics:


The below is quoted from Xavier's survey.

The basic idea of complementarity is that the marginal value of an action increases with the level of other actions available. The mathematical concept of supermodularity formalizes the idea of complementarity. The theory of monotone comparative statics and supermodular games provides the toolbox to deal with complementarities.

This theory, in contrast to classical convex analysis, is based on order and monotonicity properties on lattices. Monotone comparative statics analysis provides conditions under which optimal solutions to optimization problems change monotonically with a parameter.

The theory of supermodular games exploits order properties to ensure that the best response of a player to the actions of rivals increases with their level. The power of the approach is that it clarifies the drivers of comparative statics results and the need of regularity conditions; it allows very general strategy spaces, including indivisibilities and functional spaces such as those arising in dynamic or Bayesian games; it establishes the existence of equilibrium in pure strategies; it allows a global analysis of the equilibrium set when there are multiple equilibria, which has an order structure with largest and smallest elements; and finally, it finds that those extremal equilibria have strong stability properties and there is an algorithm to compute them.

Aumann's View on Science and Game Theory

I was impressed by Robert Aumann, when I met him at the game theory conference in Brazil (link). And, after coming back to Japan, I got impressed again to see what he has written on the preface of the volume of his "Collected Papers." His view on science and game theory is very close to mine (perhaps, I have been unconsciously affected by his view or similar idea spread among theorists).

[A]ll the papers in the collection concern game theory, its applications and its tools. Beyond the subject matter, they also share a common methodological theme: they deal with relationships. Science is often characterized as a quest for truth, where truth is something absolute, which exists outside of the observer. But I view science more as a quest for understanding, where the understanding is that of the observer, the scientist. Such understanding is best gained by studying relations - relations between different ideas, relations between different phenomena, relations between ideas and phenomena.

(...)

Indeed, the idea of relationship is fundamental to game theory. Disciplines like economics or political science use disparate models to analyze monopoly, oligopoly, perfect competition, public goods, elections, coalition formation, and so on. In contrast, game theory uses the same tools in all these applications. (...) Perhaps the most exciting advance in game theory in recent years has been the connection with evolution: The realization that when properly interpreted, the fundamental notion of Nash equilibrium, which a priori reflects the behavior of consciously maximizing agents, is the same as an equilibrium of populations that reproduce blindly without regard to maximizing anything.

Aumann's message forward to Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis by Roth and Sotomayor (1990), which he describes as a book chronicles one of the outstanding success stories of the theory of games,  is also insightful. I again share his view on evaluating the good "matching" of theory and practice.

The theoretical part of the story begins in 1962, with the publication of the famous Gale-Shapley paper, "College Admissions and the Stability of Marriage." Since then, a large theoretical literature has grown from this paper, which is thoroughly covered in this book. But the most dramatic development came in 1984, when Roth published his discovery that the Gale-Shapley algorithm had in fact been in practical use already since 1951 for the assignment of interns to hospitals in the United States; it had evolved by a tirial-and-error process that spanned more than half a century.

Historical drama on the creation of game theory

I just came back to Tokyo after attending the SAET conference (link) in Singapore and the world congress of the Econometric Society (link) in China (Shanghai). It was a great experience to visit the two Asian countries that I have never been before. I really enjoyed the conferences, meeting lots of people such as my friends, co-authors, teachers, and big names.

Now it's time to go back to my work! Many thanks to everyone I met there. Hope all of you would have a productive academic year starting from September :)

The short article below is what I had prepared before I left Japan:

A new book on the early history of game theory came out recently. It focuses on the two founding fathers, Von Neumann and Morgenstern.

"Von Neumann, Morgenstern, and the Creation of Game Theory: From Chess to Social Science, 1900–1960" by Robert Leonard, Cambridge University Press (link)

As a reviewer's comment, Harold W. Kuhn, a professor emeritus of mathematics at Princeton university describes the book as follows:

The publication of The Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern in 1944 was hailed by one reviewer as 'one of the major scientific achievements of the first half of the twentieth century.' Another reviewer signaled that 'the techniques applied by the authors in tackling economic problems are of sufficient generality to be valid in political science, sociology, or even military strategy.' In this exemplary study in the history of economics, Robert Leonard has given us a masterful account of the gestation of this work, starting with the importance of chess in European intellectual life at the beginning of the twentieth century and ending with the military applications of game theory at the RAND Corporation during the middle of the century.

The predecessor of the work is the author's 1995 article in the Journal of Economic Literature, which won the Best Article Award of the History of Economics Society.

Robert Leonard (1995) "Von Neumann, Morgenstern, and the Creation of Game Theory From Chess to Social Science, 1900–1960" (link)

Bayesian Games

Original article (link) posted: 22/09/2005

The following is the memo about Bayesian Games. All the sentences are quoted from Myerson (1991) "Game Theory" (Chapter 2.8 and 2.9).

Background
A game with incomplete information is a game in which, at the first point in time when the players can begin to plan their moves in the game, some players already have private information about the game that other players do not know.
The initial private information that a player has at this point in time is called the type of the player.
Harsanyi (1967-68) argued that a generalization of the strategic form, called the Bayesian form, is needed to represent games with incomplete information.

Consistent model
Most of the Bayesian games that have been studied in applied game theory have beliefs that are consistent with a common prior. One reason for this tendency to use consistent models is that consistency simplifies the definition of the model. Furthermore, inconsistency often seems like a strikingly unnatural feature of a model. In a consistent model, differences in beliefs among players can be explained by differences in information, whereas inconsistent beliefs involve differences of opinion that cannot be derived from any differences in observations and must be simply assumed a priori.

Agreeing to disagree
In a sports match, suppose it is common knowledge among the coaches of two teams that each believes that his own team has a 2/3 probability of winning its next game against the other team, then the coaches' beliefs cannot be consistent with a common prior. In a consistent model, it can happen that each coach believes that his team has a 2/3 probability of winning, but this difference of beliefs cannot be common knowledge among the coaches. (see Aumann, 1976)

Bayesian Games are general enough?
To describe a situation in which many individuals have substantial uncertainty about one another's information and beliefs, we may have to develop a very complicated Bayesian-game model with large type sets and assume that this model is common knowledge among the players. This result begs the question; is it possible to construct a situation for which there are no sets of types large enough to contain all the private information that players are supposed to have, so that no Bayesian game could represent this situation?
Mertens and Zamir (1985) showed under some technical assumptions, that no such counterexample to the generality of the Bayesian game model can be constructed, because a universal belief space can be constructed that is always big enough to serve as the set of types for each player.
Although constructing an accurate model for any given situation may be extremely difficult, we can at least be confident that no one will ever be able to prove that some specific conflict situation cannot be described by any sufficient complicated Bayesian game.

References

Aumann (1976) "Agreeing to Disagree" Annals of Statistics, 4
Harsanyi (1967-68) "Games with Incomplete Information Played by 'Bayesian' Players" Management Science, 14
Mertens and Zamir (1985) "Formulation of Bayesian Analysis for Games with Incomplete Information" IJGT, 14

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.

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

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.

Refinements and Selection

Original article (link) posted: 25/07/2005

Myerson (1991) mentions the distinction between equilibrium refinements and criteria for selection among the set of equilibria.

A refinement of Nash equilibria is a solution concept that is intended to offer a more accurate characterization of rational intelligent behavior in games. However, the ultimate refinement that exactly characterizes rational behavior can still include multiple equilibria for many games. (e.g., the Battle of the Sexes)
A selection criterion is then any objective standard, defined in terms of the given structure of the mathematical game, that can be used to determine the focal equilibrium that everyone expects. Equity and efficiency criteria of cooperative game theory can be understood as such selection criteria.
(p. 241)

According to him, the concept of "cheap-talk" should be considered as an equilibrium selection criterion (NOT as a refinement), because it is based on the assumption that players share a rich natural language for communication, and hence rational and intelligent players may not reach that equilibrium without some cultural environment.