Frank Riedel

On Irreversible Investment

This paper presents a new and general approach to the theory of irreversible investment. We show that the optimal policy is a base capacity policy and derive general monotone comparative statics results. When the operating profit function is supermodular, the base capacity increases monotonically with the exogenous shock; and firm size is decreasing in the user cost of capital. Last but not least, the paper provides a general existence theorem for optimal policies.

OPTIMAL STOPPING WITH MULTIPLE PRIORS

We develop a theory of optimal stopping under Knightian uncertainty. A suitable martingale theory for multiple priors is derived that extends the classical dynamic programming or Snell envelope approach to multiple priors. We relate the multiple prior theory to the classical setup via a minimax theorem. In a multiple prior version of the classical model of independent and identically distributed random variables, we discuss several examples from microeconomics, operation research, and finance. For monotone payoffs, the worst-case prior can be identified quite easily with the help of stochastic dominance arguments. For more complex payoff structures like barrier options, model ambiguity leads to stochastic changes in the worst-case beliefs.

Low Price Equilibrium in Multi-Unit Auctions: The GSM Spectrum Auction in Germany

The second-generation GSM spectrum auction in Germany is probably the most clear cut example of a low price outcome in a simultaneous ascending-bid auction. The present paper gives an account of the events, describes the auction rules and market conditions, and provides a theoretical explanation of low price equilibria in simultaneous, ascending-bid auctions. In particular it is shown that the low price equilibrium that implements the efficient allocation is the unique perfect equilibrium of that game.

Brown–von Neumann–Nash dynamics: The continuous strategy case

In John Nash’s proofs for the existence of (Nash) equilibria based on Brouwer’s theorem, an iteration mapping is used. A continuous- time analogue of the same mapping has been studied even earlier by Brown and von Neumann. This differential equation has recently been suggested as a plausible boundedly rational learning process in games. In the current paper we study this Brown-von Neumann-Nash dynamics for the case of continuous strategy spaces. We show that for continuous payoff functions, the set of rest points of the dynamics coincides with the set of Nash equilibria of the underlying game. We also study the asymptotic stability properties of rest points. While strict Nash equilibria may be unstable, we identify sufficient conditions for local and global asymptotic stability which use concepts developed in evolutionary game theory.

The Third Generation (Umts) Spectrum Auction in Germany

The third generation UMTS auction in Germany raised an enormous amount of revenue, and at the same time achieved a more competitive market structure than other UMTS auctions in Europe. The present paper explains the design of that auction, and presents a game theoretic explanation of observed events during the crucial phase of that auction, which have puzzled several observers. In addition, the paper evaluates the merit of the German UMTS auction design, relative to the English design, that was predominantly employed in Europe.

Voronoi languages: Equilibria in cheap-talk games with high-dimensional types and few signals

We study a communication game of common interest in which the sender observes one of infinite types and sends one of finite messages which is interpreted by the receiver. In equilibrium there is no full separation but types are clustered into convex categories. We give a full characterization of the strict Nash equilibria of this game by representing these categories by Voronoi languages. As the strategy set is infinite static stability concepts for finite games such as ESS are no longer sufficient for Lyapunov stability in the replicator dynamics. We give examples of unstable strict Nash equilibria and stable inefficient Voronoi languages. We derive efficient Voronoi languages with a large number of categories and numerically illustrate stability of some Voronoi languages with large message spaces and non-uniformly distributed types.

Sleep Apnea Syndrome in Patients with Cardiac Pacemaker

Background: Heart rhythm disturbances are cardiac side effects in patients with sleep-disordered breathing (SDB), which in itself is considered to be a risk factor for bradycardic rhythm disturbances. Objective: We analyzed the prevalence and degree of SDB in patients who received a cardiac pacemaker due to bradycardic rhythm disturbances and investigated the relationship between the severity of an underlying SDB and the type of heart rhythm disturbance. Methods and Results: 192 patients (100 males, 92 females, mean age 62.2 ± 12.2 years) were studied using the portable screening device MESAM IV. The respiratory disturbance index (RDI) was calculated visually. The mean RDI in all patients was 9.13 ± 11.09/h, 11.7 ± 13.15/h in males and 6.33 ± 7.42/h in females. The prevalence ratio of SDB between men and women was 1.7:1, with significant differences in the respective severity (p < 0.05). The screening showed a prevalence of SDB (RDI > 10/h) of 32.3%. The highest prevalence was found in the group of patients with atrial fibrillation and bradycardia. However, there were no significant differences compared to other types of rhythm disturbances. The RDI in the population studied depended on age and body mass index, but not on the existence or type of rhythm disturbance and not on concomitant diseases. Conclusion: The prevalence of SDB in cardiac pacemaker patients is similar to that in patients of comparable age without a pacemaker. A heart rhythm disturbance does not seem to be an independent risk factor for development of SDB. Nevertheless, the differential diagnosis of bradycardic rhythm disturbances in this age group should include a screening for sleep apnea.

Optimal consumption choice under uncertainty with intertemporal substitution

We extend the analysis of the intertemporal utility maximization problem for Hindy-Huang-Kreps utilities reported in Bank and Riedel (1998) to the stochastic case. Existence and uniqueness of optimal consumption plans are established under arbitrary convex portfolio constraints, including both complete and incomplete markets. For the complete market setting, Kuhn-Tuckerlike necessary and sufficient conditions for optimality are given. Using this characterization, we show that optimal consumption plans are obtained by re- flecting the associated level of satisfaction on a stochastic lower bound. When uncertainty is generated by a Levy process and agents exhibit constant relative risk aversion, closed-form solutions are derived. Depending on the structure of the underlying stochastics, optimal consumption occurs at rates, in gulps, or singular to Lebesgue measure.

Existence of Arrow–Radner Equilibrium with Endogenously Complete Markets under Incomplete Information☆

Abstract A stochastic pure exchange economy with incomplete information is studied where one risky asset and a locally riskless bond are traded. The underlying risks are described by a k -dimensional Brownian motion with k ⩾2. The Brownian motion is not directly observable. Agents observe only asset prices and dividends. Existence of an Arrow–Radner equilibrium is established. In such an equilibrium, asset prices result such that every informationally feasible consumption plan can be financed by trading in the risky asset and the bond. In this sense, the resulting asset market is complete. Journal of Economic Literature Classification Numbers: G12, G14, D51.

The Strategic Use of Ambiguity

Ambiguity can be used as a strategic device in some situations. To demonstrate this, we propose and study a framework for normal form games where players can use Knightian uncertainty strategically. In such Ellsberg games, players may use Ellsberg urns in addition to the standard objective mixed strategies. We assume that players are ambiguity-averse in the sense of Gilboa and Schmeidler. While classical Nash equilibria remain equilibria in the new game, there arise new Ellsberg equilibria that can be quite different from Nash equilibria. A negotiation game with three players illustrates this finding. Another class of examples shows the use of ambiguity in mediation. We also highlight some conceptually interesting properties of Ellsberg equilibria in two person games with conflicting interests.