Optimal Collusion with Internal Contracting

In this paper, we develop a model of collusion in which two firms play an infinitelyrepeated Bertrand game when each firm has a privately-informed agent. The colluding firms, fixing prices, allocate market shares based on the agents information as to cost types. We emphasize that the presence of privately-informed agents may provide firms with a strategic opportunity to exploit an interaction between internal contracting and market-sharing arrangement : the contracts with agents may be used to induce firms truthful communication in their collusion, and collusive market-share allocation may act to reduce the agents information rents.Optimal collusion, internal contract, privately-informed agents, price-fixing

Collusion with Internal Contracting

In this paper, an infinitely-repeated Bertrand game is considered. The model has a two-tier relationship; two firms make a self-enforced collusive agreement and each firm writes a law-enforced contract to its privately-informed agent. The main finding is that in optimal collusion, interaction between intra-firm (internal) contracting and inter-firm collusion may be exploited; inter-firm collusion may enhance the efficiency of internal contract, and conversely, internal contracting may facilitate collusioncollusion, internal contract, repeated games, market allocation

Impossibility of large phase shifts via the "giant Kerr effect" with single-photon wavepackets

An approximate analytical solution is presented, along with numerical calculations, for a system of two single-photon wavepackets interacting via an ideal, localized Kerr medium. It is shown that, because of spontaneous emission into the initially unoccupied temporal modes, the cross-phase modulation in the Schrodinger picture is very small as long as the spectral width of the single-photon pulses is well within the medium's bandwidth. In this limit, the Hamiltonian used can be derived from the "giant Kerr effect" for a four-level atom, under conditions of electromagnetically-induced transparency; it is shown explicitly that the linear absorption in this system increases as the pulse's spectral width approaches the medium's transparency bandwidth, and hence, as long as the absorption probability remains small, the maximum cross-phase modulation is limited to essentially useless values. These results are in agreement with the general, causality- and unitarity-based arguments of Shapiro and co-workers.Comment: 8 pages, 2 figures, to be submitted to Physical Review

Advertising and Collusion in Retails Markets

We consider non-price advertising by retail firms that are privately informed as to their respective production costs. We first analyze a static model. We construct an advertising equilibrium, in which informed consumers use an advertising search rule whereby they buy from the highest-advertising firm. Consumers are rational in using the advertising search rule, since the lowest-cost firm advertises the most and also selects the lowest price. Even though the advertising equilibrium facilitates productive efficiency, we establish conditions under which firms enjoy higher expected profit when advertising is banned. Consumer welfare falls in this case, however. We next analyze a dynamic model in which privately informed firms interact repeatedly. In this setting, firms may achieve a collusive equilibrium in which they limit the use of advertising, and we establish conditions under which optimal collusion entails pooling at zero advertising. More generally, full or partial pooling is observed in optimal collusion. In summary, non-price advertising can promote product efficiency and raise consumer welfare; however, firms often have incentive to diminish advertising competition, whether through regulatory restrictions or collusion.non-price advertising, retail firms, advertising equilibrium

Reply to "Comment on "Some implications of the quantum nature of laser fields for quantum computations''''

In this revised reply to quant-ph/0211165, I address the question of the validity of my results in greater detail, by comparing my predictions to those of the Silberfarb-Deutsch model, and I deal at greater length with the beam area paradox. As before, I conclude that my previous results are an (order-of-magnitude) accurate estimate of the error probability introduced in quantum logical operations by the quantum nature of the laser field. While this error will typically (for a paraxial beam) be smaller than the total error due to spontaneous emission, a unified treatment of both effects reveals that they lead to formally similar constraints on the minimum number of photons per pulse required to perform an operation with a given accuracy; these constraints agree with those I have derived elsewhere.Comment: A reply to quant-ph/0211165. Added more calculations and discussion, removed some flippanc

"Modes of the universe" study of two-photon deterministic, passive quantum logical gates

We use the "modes of the universe" approach to study a cavity-mediated two-photon logical gate recently proposed by Koshino, Ishizaka and Nakamura. We clarify the relationship between the more commonly used input-output formalism, and that of Koshino et al., and show that some elements of this gate had been anticipated by other authors. We conclude that their proposed gate can work both in the good and bad cavity limits, provided only that the pulses are long enough. Our formalism allows us to estimate analytically the size of the various error terms, and to follow the spectral evolution of the field + cavity system in the course of the interaction.Comment: 9 pages, 8 figure

Gate fidelity of arbitrary single-qubit gates constrained by conservation laws

Recent investigations show that conservation laws limit the accuracy of gate operations in quantum computing. The inevitable error under the angular momentum conservation law has been evaluated so far for the CNOT, Hadamard, and NOT gates for spin 1/2 qubits, while the SWAP gate has no constraint. Here, we extend the above results to general single-qubit gates. We obtain an upper bound of the gate fidelity of arbitrary single-qubit gates implemented under arbitrary conservation laws, determined by the geometry of the conservation law and the gate operation on the Bloch sphere as well as the size of the ancilla.Comment: Title changed; to appear in J. Phys. A: Math. Theor.; 19 pages, 2 figure

Approximate programmable quantum processors

A quantum processor is a programmable quantum circuit in which both the data and the program, which specifies the operation that is carried out on the data, are quantum states. We study the situation in which we want to use such a processor to approximate a set of unitary operators to a specified level of precision. We measure how well an operation is performed by the process fidelity between the desired operation and the operation produced by the processor. We show how to find the program for a given processor that produces the best approximation of a particular unitary operation. We also place bounds on the dimension of the program space that is necessary to approximate a set of unitary operators to a specified level of precision.Comment: 8 page