Contract Theory Conference

I organized the 4th Japan-Taiwan Contract Theory Conference on December 4, 2010 at my institute, National Graduate Institute for Policy Studies (GRIPS), which turned out to be extremely successful. We had 49 participants, including 18 guests from Taiwan; enjoyed listening to six intriguing talks and discussion. I am grateful for all who kindly attended this event and helped me setting it up.
The following is the conference program with my brief comments.

First session: 10:05--11:55

Chair: Chia-Hui Chen (Academia Sinica)

Speaker 1: Chien-Lung Chen (National Tsing-Hua University) 

• How Can Agents Be Better-Off with Pay Caps?  

Speaker 2: Mami Kobayashi (Kinki University)  

• Market Liquidity and Capital Structure of Financial Institution

Comments: The first talk is about optimal wage scheme under pay caps (in addition to minimum wage constraints), and the second is about incentive issues on corporate finance and governance. Theoretical robustness and the connection to actual labor and financial markets were discussed.                                              

Second session: 13:10--15:00

Chair: Jong-Rong Chen (National Central University)

Speaker 1: Chi-Hsiang Liu (National Central University)

• Promotion, Relative and Individual Performance Pay

Speaker 2: Kohei Kawamura （University of Edinburgh）

• Eliciting Information from a Large Population

Comments: Both talks showed frontier works on contract theory. The first one is considerable extension of promotion model and the second is to combine the ideas of cheap talk and (sequential) Bayesian learning. The interpretation and potential applications of their results were discussed.

Third session: 15:20―17:10

Chair: Pao-Chih Cheng (National Central University)

Speaker 1: Hsiang-Ling Shen (National Taiwan University)

• Sponsored Search Auctions under Two Search Engines

Speaker 2: Takanori Adachi (Nagoya University)

• Political Accountability, Electoral Control and Media Bias (with Yoichi Hizen)

Comments: Two speakers talked about their preliminary works which are both very new and strongly connected to real life interests: sponsored search auctions and media bias. Since some parts of their models were not well shaped yet, ideas for further improvement were discussed and proposed.

Lecture 4 (Dutta): Dynamic Programming 3

Original article (link) posted: 06/10/2005

In the last class, we checked the strong relationship between the dynamic optimization problem and the functional equation. In this class, we examined continuity, monotonicity and concavity of the value function.

Assumptions (for continuity)
(A1) State Space: Borel subset of a metric space
(A2) Action Space: Metric space
(A3) Reward Function: Bounded, measurable and continuous
(A4) Transition Function: Weakly continuous
(A5) Feasibility Correspondence: continuous

Theorem (Maitra 68)
Under (A1)-(A5), the value function V is bounded and continuous function. Furthermore, there is a stationary Markovian policy, h, that is optimal.
Proof
Step1 Bellman operator T maps bounded and continuous function (denoted by C(S)) back into the same space.
Step2 C(S) is a complete metric space.
Step3 T is a contraction.
Step4 There is a measurable selection h from the Bellman equation.
Note) To prove Step1, we use the Maximum Theorem. For the other part of the proof, we rely on the established results.

Assumptions (for monotonicity)
(B1) State Space: A subset of R^n
(B2)=(A2)
(B3)=(A3) + increasing on S (State Space)
(B4)=(A4) + First Order Stochastically Increases on S
(B5)=(A5) + A higher state has a larger feasible set

Theorem (Monotonicity)
Under (B1)-(B5), the value function is bounded, continuous and increasing on S.

Note) Step2-4 are almost same as before. So, we essentially need to check only Step1.

Theorem (Strict Monotonicity)
If we have a strictly increasing reward function, then V is also strictly increasing.
Proof
It is easy to see from the definition of the functional equation.
You should not try to restrict the space to the set of strictly increasing functions, because it is not complete. (the limit may not be a strictly increasing function)

Assumptions (Concavity)
(C1)=(B1) + S is convex
(C2) Action Space: A subset of R^m and convex
(C3)=(B3) + reward function is concave
(C4)=(A4) + transition function is "concave"
(concavity here is defined in terms of Second Order Stochastic Dominance)
(C5)=(B5) + graph is convex

Theorem (Concavity)
Under (C1)-(C5), the value function is bounded, continuous, and concave.
With strict concavity of the reward function, V is bounded, continuous, and strictly concave. In this case, h becomes a continuous function.

Theory Seminar (Sigurdsson)

Original article (link) posted: 05/10/2005

Sigurdsson (2005) "Auctions as Mechanism: An Application to Bankruptcy Reorganization" Job Market Paper

In the paper, he proposes the new mechanism of reorganization of firms in cases of bankruptcy. As a matter of fact, a firm that files for bankruptcy must either liquidate under Chapter 7 of the Bankruptcy Code or reorganize under Chapter 11. The distribution follows the absolute value priority rule (APR) which states that no creditor shall receive any value until all claims senior to his have been paid in full. Legal scholars have proposed several mechanisms as alternatives to the current system of judicially supervised bargaining, widely believed to be costly, lengthy, and to result in inefficient capital structures and violations of the APR. A major drawback of those proposals is their reliance on cash payments.
In the mechanism he proposes, the entire reorganized firm is sold in a cash auction and the proceeds are distributed to creditors according to the APR. He mentions 4 advantages of his mechanism. That is, the mechanism
1) implements the APR for far more general capital structures than the simple debt and equity structures assumed in previous mechanisms.
2) all but eliminates the need for cash payments and therefore works under tight financial constraints.
3) offers the advantage of familiarity over its more novel competitors in an auction.
4) allocates ownership efficiently when creditors do not agree on the firm's value.

It seemed that the participants of the seminar liked his paper. Hope he will get a good job!!
By the way, his main advisor is Eric Maskin. I heard a rumor that he did not take a student, but it should be wrong. I would like to talk to him about my research too.

Auction theory: Myerson and mechanism design approach

Continued from the previous two articles, we still focus on Introduction of the auction book:

After referring early literature on auction theory, the authors introduce mechanism design theory and explain Myerson's path-breaking contribution.

The early work of Vickrey, Wilson, and Milgrom was largely focused on an equilibrium analysis and comparison of standard auction formats. Myerson led the development of mechanism design theory, which enables the researcher to characterize equilibrium outcomes of all auction mechanisms, and identify optimal mechanisms - those mechanisms that maximize some objective, such as revenues. His first application was to auctions.

Myerson (1981) determined the revenue-maximizing auction with risk-neutral bidders and independent private information. He also proved a general revenue equivalence theorem that says that revenues depend fundamentally on how the items are assigned - any two auction formats that lead to the same assignment of the items yield the same revenues to the seller.

The trick in Myerson's analysis was recognizing that any auction can be represented as a direct mechanism in which bidders simultaneously report their private information and then the mechanism determines assignments and payments based on the vector of reports. For any equilibrium of any auction game, there is an equivalent direct mechanism in which bidders truthfully report types and agree to participate. Hence, without loss of generality we can look at incentive compatible and individually rational mechanisms to understand properties of all auction games. Incentive compatibility respects the fact that the bidders have private information about their values; individual rationality respects the bidders voluntary participation decision. This key idea is known as the revelation principle (Myerson, 1979).

The following is a brief summary of other related papers.

Myerson and Satterthwaite (1983)

• applied revelation principle
• proved the general impossibility of efficient bargaining

Related Papers

• Cramton, GIbbons, and Klemperer (1987) showed that efficiency becomes possible when the traders jointly own the items
• Wilson (1993) also showed efficient mechanisms when the roles of buyers and sellers are not fixed ex ante

References
Cramton, Gibbons, and Klemperer, "Dissolving a Partnership Efficiently," Econometrica, 1987.
Myerson, "Incentive Compatibility and the Bargaining Problem," Econometrica, 1979.
Myerson, "Optimal Auction Design," Mathmatics of Operations Research, 1981.
Myerson and Satterthwaite,  "Efficient Mechanism for Bilateral Trading," Journal of Economic Theory, 1983.
Wilson, "Design of Efficient Trading Procedures," in Friedman and Rust ed., The Double Auction Market: Institutions, Theories, and Evidence, Addison-Wesley Publishing Company, 1993.

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.

Combinatorial Auctions: Introduction

This book is a great collection of papers on a rapidly growing filed, "combinatorial auctions."


Let me quote a couple of useful sentences below taken from Introduction written by the editors, Peter Cramton, Yoav Shoham, and Richard Steinberg.

• The study of combinatorial auctions thus lies at the intersection of economics, operations research, and computer science.
• There are numerous examples of combinatorial auctions in practice. As is typical of many fields, practice precedes theory. Simple combinatorial auctions have been used for many decades in, for example, estate auctions.
• Recently, a variety of industries have employed combinatorial auctions. For example, they have been used for truckload transportation, bus routes, and industrial procurement, and have been proposed for airport arrival and departure slots, as well as for allocating radio spectrum for wireless communications services.
• Auction theory is among the most influential and widely studied topics in economics over the last forty years. Auctions ask and answer the most fundamental questions in economics: who should get the goods and at what prices? In answering these questions, auctions provide the micro-foundation of markets. Indeed, many modern markets are organized as auctions.

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 :)

Kandori (1991)

Original article (link) posted: 01/10/2005

Kandori (1991) "Correlated Demand Shocks and Price Wars During Booms" RES, 58

The paper extends the analysis of Rotemberg and Saloner (1986) to the case of serially correlated demand shocks and derives the same counter-cyclical movement as in their case (i.i.d. case), provided the discount factor and the number of the firms satisfy certain relationship.
The key observation in Rotemberg and Saloner (1986) was that, if the sum of future profits is unaffected by today’s demand, firms must set the price relatively low when demand is high. The premise is clearly satisfied when the demand shocks are i.i.d.. This paper shows introducing Markov demand shocks also create the same situation in the following two cases. The first case is when the discount factor delta exceeds, but is close to (N-1)/N, where N is the number of the firm. It is shown that firms maintain a constant profit (which equals to the monopoly profit in the worst state) under all demand conditions. Therefore, the extent of the correlation in demand is irrelevant.
The second case arises when delta tends to unity while (1-delta)N is held constant. In this case, firms are enormously forward-looking and total future profit is mostly determined by the stationary distribution, which is independent of today’s demand position.

The result itself is not that surprising (comparing to the other papers by Kandori at least). However, he is amazingly good at selling his work, especially in the following two points;
First, he stresses the importance of Rotemberg and Saloner (1986) and their drawbacks as well. The motivation of extension of their paper becomes very clear and the reader necessarily gets interested in HIS work.
Second, his way of illustrating results is quite lucid and rigorous. Although, the results can be more or less expected to hold, it always is difficult to prove them in rigorously.
Those techniques should be useful for us. Let’s learn them by the papers by Kandori!

Interesting Papers that cite Kandori (1991)

Bagwell (2004) "Countercyclical Pricing in Customer Markets" Economica, 71
Bo (2001) "Tacit Collusion under Interest Rate Fluctuations" Job Market Paper
Harrington (2004) "Cartel Pricing Dynamics in the Presence of an Antitrust Authority" Rand

Decision Theory 301

This is complementary to the previous post, "Decision Theory 101 (link)." In Appendix A: Optimal Choice, the author (Professor Gilboa) concisely explains the flexibility of rational choice framework. I think that his argument is really important especially when we evaluate the recent developments of behavioral economics and consider its relationship with the traditional (or rational) approach.

For our purposes, it is worthwhile highlighting what this model (the consumers' problem: by yyasuda) does not include. Choices are given as quantities of products. Various descriptions of the products, which may be part of their frames, are not part of the discussion. The utility function measures desirability on a scale. We did not mention any special point on this scale, such as a reference point. Further, choices are bundles of products to be consumed by the consumer in question at the time of the problem. They do not allow us to treat a certain bundle differently based on the consumer's history of consumption, or on the consumption of others around them. Hence, the very language of the model assumes that the consumer does not care what others have, they feel no envy, nor any disappointment in the case when their income drops as compared with last period, and so on.

It is important to emphasize that the general paradigm of rational choice does not necessitate these constraints. For instance, instead of the n products the consumer can consume today, we may have a model with 2n products, reflecting their consumption today and their consumption yesterday. This would allow us to specify a utility function u that takes into account considerations such as aspiration levels, disappointment, and so forth. Or, we can use more variables to indicate the average consumption in the consumer's social group, and then the utility function can capture social considerations such as the consumer's ranking in society and so forth. Indeed, such models have been suggested in the past and have become more popular with the rise of behavioral economics. These models show that the paradigm of rational choice is rather flexible. Yet, the specific theory restricts the relevant variables to be independent of history, others' experiences, emotions, and other factors which might be among the determinants of well-being.

Decision Theory 101

Let me continue to quote some basics of decision theory (or economics) from the Gilboa's recent book, "Making Better Decisions."
In Appendix A: Optimal Choice, the author nicely illustrates the framework of decision theory and its key concepts such as axioms and utility function. The following might be especially helpful for those who are against or suspicious about the fundamental tool of economics, utility maximization.

A fundamental of optimal choice theory is the distinction between feasibility and desirability. A choice is feasible if it is possible for the decision maker, that is, one of the things that she can do. An outcome is desirable if the decision maker wishes to bring it about. Typically, feasibility is considered to be a dichotomous concept, while desirability is continuous: a choice is either feasible or not, with no shades in between; by contrast, an outcome is desirable to a certain degree, and different outcomes can be ranked according to their desirability.

We typically assume that desirability is measured by a utility function u, such that the higher the utility of a choice, the better will the decision maker like it. This might appear odd, as many people do not know what functions are and almost no one can be observed walking around with a calculator and finding the alternative with the highest utility. But it turns out that very mild assumptions on choice are sufficient to determine that the decision maker behaves as if she had a utility function that she was attempting to maximize. If the number of choice is finite, the assumptions (often called axioms) are the following:

1. Completeness: for every two choices, the decision maker can say that she prefers the first to the second, the second to the first, or that she is indifferent between them.
2. Transitivity: for every three choices a, b, c, if a is at least as good as b, and b is at least good as c, then a is at least as good as c.

It turns out that these assumptions are equivalent to the claim that there exists a function u such that, for every two alternatives a and b, a is at least as good as b if and only if u(a) ≧ u(b). (...) Any other algorithm that guarantees adherence to these axioms has to be equivalent to maximization of a certain function, and therefore the decision maker might well specify the function explicitly.