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.

Descriptive and Normative Theories

A great introductory book on decision theory written by a leading authority, professor Itzhak GIlboa, came out recently:



Professor Ehud Kalai at Northwestern University describes this book as follows:

"This book is extremely effective for anyone who wants to acquire quick, basic understanding of old and new concepts of decision theory, with a minimum level of technical details."

In the first chapter, the author indeed effectively explains two essential notions in economics, "descriptive" (sometime called "positive") and "normative" theories:

A descriptive theory is meant to describe reality. For instance, the claim that demand curves slope down attempts to tell us something about the world. Importantly, it does not make a value judgement and takes no stand on whether this feature of the world is good or bad.

A normative theory is a recommendation to decision makers, a suggestion regarding how they should make decisions. For instance, the claim that we should reduce income inequality is a normative claim. Note that the word "normative" does not mean here "the norm in a given society" as it does in other social sciences. The term only says something about the type of interaction between the theorist and the decision maker, namely, that this is an instance in which the former is trying to convince the latter to behave in a certain way.

The author continues to document the role of each theory, which is also very intuitive:

In decision theory it is often the case that a principle can be interpreted either descriptively or normatively. Consider the theory that each economic agent maximizes a utility function. I may propose it as descriptive, namely, arguing that this is a good description of real economic agents. And I may promote it as normative, in which case my claim will be that you would be wise to become such an agent. As a descriptive theory, the principle is tested for its correspondence to reality. The better it fits the data, the more successful it is. As a normative one, the principle should not fit reality. In fact, there is no point in giving decision makers recommendations that they anyway follow. Rather, the test is whether the decision makers would like to follow the principle.

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.

IO Seminar (Kadyrzhanova)

Original article (link) posted: 28/09/2005

Kadyrzhanova "The Leader-Bias Hypothesis: Monopolization and Industry Structure under Imperfect Corporate Control" Job Market Paper

The paper examines the effect of the imperfect corporate control in a dynamic oligopoly market with cost reducing R&D investments. The managers are assumed to have an over-producing incentive ("empire-building" hypothesis), and hence, they do not maximize firms' short-run profit without intervention of the shareholders. Corporate control serves to shift managers' preference from "empire-building" to "profit maximization".
The key observation is that shareholders may want to choose imperfect control because of the commitment benefit of the over-producing derived by "empire-building" preference of the manager. Indeed, she shows that even if there is no cost of corporate control, shareholders do not choose full control. Moreover, it is shown that shareholders are more willing to leave discretionary authority to managers when ahead of rivals, which results in lower turnover, higher concentration, persistently monopolized markets, and significantly lower consumer surplus.

Comments
I found the paper quite interesting. However, there were so many things she put in her presentation and the relationship among those are not clear enough for me. I am afraid that audiences also got little because her focus of the talk is vague. I think her presentation could have become much better if she had tried the followings:
1) Stress and make clear the contribution to the literature
2) Put more intuitive explanation of the main results
3) Be more confident on mathematical parts
4) Mention some actual story in markets or empirical facts as a motivation
5) Explain which element of the model is a key to derive the corresponding result

Interesting Papers in Reference

Athey and Schmutzler (2001) "Investment and Market Dominance" RAND 32 (1): 1-26
Bagwell, Ramey and Spulber (1997) "Dynamic Retail Price and Investment
Competition" Rand, 28(2), 207-227
Bolton, Brodley, and Riordan (2000) "Predatory Pricing: Strategic Theory and Legal Policy" Georgetown Law Journal, 88, pp. 2239-2330
Bolton and Scharfstein (1990) "A Theory of Predation Based on Agency Problems in Financial Contracting" American Economic Review 80(1): 93-106
Cabral and Riordan (1994) "The Learning Curve, Market Dominance and Predatory Pricing" Econometrica, 62, pp. 1115-1140
Fershtman and Judd (1987) "Equilibrium Incentives in Oligopoly" American Economic Review, 77(5), 927-940