Ilan Kremer

On randomized one-round communication complexity

Abstract. We present several results regarding randomized one-round communication complexity. Our results include a connection to the VC-dimension, a study of the problem of computing the inner product of two real valued vectors, and a relation between “simultaneous” protocols and one-round protocols.

Bidding with Securities: Auctions and Security Design

We study security-bid auctions in which bidders compete by bidding with securities whose payments are contingent on the realized value of the asset being sold. Such auctions are commonly used, both formally and informally. In formal auctions, the seller restricts bids to an ordered set, such as an equity share or royalty rate, and commits to a format, such as first or second-price. In informal settings with competing buyers, the seller does not commit to a mechanism upfront. Rather, bidders offer securities and the seller chooses the most attractive bid, based on his beliefs, ex-post. We characterize equilibrium payoffs and bidding strategies for formal and informal auctions. For formal auctions, we examine the impact of both the security design and the auction format. We define a notion of the steepness of a set of securities, and show that steeper securities lead to higher revenues. We also show that the revenue equivalence principle holds for equity and cash auctions, but that it fails for debt (second-price auctions are superior) and for options (a first-price auction yields higher revenues). We then show that an informal auction yields the lowest possible revenues across all possible formal mechanisms. Finally, we extend our analysis to consider the effects of liquidity constraints, different information assumptions, and aspects of moral hazard.

Divisible-Good Auctions: The Role of Allocation Rules

We examine the role of allocation rules in determining the set of equilibrium prices in uniform-price auctions. Beginning with Wilson (1979), the theoretical literature has argued that these auctions are subject to possible low equilibrium prices. We show that this is due to the way the asset is being divided. We focus on allocation rules that specify the way the asset is divided in cases of excess demand. This may have a dramatic effect on the set of equilibrium prices. In particular, we show that a simple allocation rule (pro rata) eliminates underpricing, while the allocation rule used in practice has a negative effect on equilibrium prices.

Information Aggregation in Common Value Auctions

A key question in information economics and finance is whether prices aggregate information in a competitive environment. That is, when agents are endowed with private information, does competition lead in the limit to prices which would occur if all information were public?. This paper examines the asymptotic properties of prices in common value auctions when the number of bidders becomes large. The main idea in this paper is to separate the issue of information aggregation from the issue of revenue efficiency. We also distinguish between the amount of information prices contain, and the degree to which prices approximate the asset’s value. Using this approach we get a simple characterization of the limiting distribution of prices. We are not only able to answer whether prices converge to the asset’s value, but also get an explicit form for the limiting distribution. Prices converge to the expected value of the asset conditional on the information possessed by what we call the “pivotal bidder.” This provides insights on how the auction format and the information structure influence the limiting behavior of prices. It shows that the limiting distribution of prices can be derived from the statistical properties of certain order statistics. This lets us abstract from the specific equilibrium bidding strategies. For example, we show that while the first and second price auction have different equilibrium strategies they share a similar limiting distribution of prices. These auctions yield different limiting distributions from auctions such as the English auction or kn auction. Our

Dynamic signaling and market breakdown

We consider the effects a public revelation of information (e.g. rating, grade) has on trading in a dynamic signaling model. Competing buyers offer prices to a privately informed seller who can reject them and delay trade. Delay is costly and the seller has no commitment to its duration. The external public information allows for signaling in equilibrium. More interestingly, we characterize the dynamics of trade and prices. If signals are noisy, no trade takes place just before the revelation of external information. If signals are fully revealing, then trade occurs even close to revelation, however, transaction prices are discontinuous.

Online trading algorithms and robust option pricing

In this work we show how to use efficient online trading algorithms to price the current value of financial instruments, such as an option. We derive both upper and lower bounds for pricing an option, using online trading algorithms.Our bounds depend on very minimal assumptions and are mainly derived assuming that there are no arbitrage opportunities.

On the informational inefficiency of discriminatory price auctions

We analyze bidding behavior in large discriminatory price auctions where the number of objects is a non-trivial proportion of the number of bidders. Bidders observe private signals that are affiliated with the common value. We show that the average price in the auction is biased downward from the expected value of the objects, even in the competitive limit. In particular, we show that conditional on relatively low signals, bidders bid the expected value of the objects conditional n their information and winning; while bids at higher signals flatten out and are below the expected value conditional on winning.

Ordering, revenue and anchoring in art auctions

We estimate the effect of ordering by value on revenues in sequential art auctions held by Sothebys and Christies. We exploit a pre determined rotation of which of these two houses holds their auction first during auction week in New York City. When the house that goes first has relatively expensive paintings compared to the other house, we find that the sale premium for the week is around 21% higher relative to the mean sale premium, and the fraction of paintings sold during the week is around 11% higher. We provide evidence that this is due to an anchoring effect.

Errata for: on randomized one-round communication complexity

Abstract. ((No abstract.))

Robust Option Pricing: Hannan and Blackwell Meet Black and Scholes

We apply methods developed in the literature initiated by Hannan and Blackwell on robust optimization, approachability and calibration, to price financial securities. Rather than focus on asymptotic performance, we show how gradient strategies developed to minimize asymptotic regret imply financial trading strategies that yield arbitrage-based bounds for option prices. These bounds are new and robust in that they do not depend on the continuity of the stock price process, complete markets, or an assumed pricing kernel. They depend only on the realized quadratic variation of the price process, which can be measured and, importantly, hedged in financial markets using existing securities. Our results also apply directly to a new class of options called timer options. Finally, we argue that the Hannan–Blackwell strategy is path dependent and therefore suboptimal with a finite horizon. We improve it by solving for the optimal path-independent strategy, and compare the resulting bounds with Black–Scholes.