Bilateral Trade with Correlated Buyer and Seller Values

Temel Kavramlar

The buyer-offering mechanism provides an e/(e-1) ≈ 1.582 approximation to the optimal social welfare in bilateral trade problems, even when the buyer's and seller's values are drawn from a joint distribution.

Özet

The paper studies the bilateral trade problem where a seller owns a single indivisible item and a potential buyer seeks to purchase it. Previous research on this problem has largely assumed that the values of the seller and the buyer are drawn independently. In contrast, this paper investigates the more realistic and technically challenging scenario where the values are drawn from a joint probability distribution.
The key findings are:

The buyer-offering mechanism, in which the buyer makes a take-it-or-leave-it offer to the seller, provides an e/(e-1) ≈ 1.582 approximation to the optimal social welfare, even when the values are drawn from a joint distribution. This approximation ratio is close to the best known ratio for the case of independent distributions.

The buyer-offering mechanism is optimal among all one-sided dominant strategy mechanisms, in the sense that no such mechanism can provide a better approximation ratio.

Mechanisms in which both the buyer and the seller have dominant strategies cannot provide any constant approximation to the optimal social welfare when the values are drawn from a joint distribution.

For deterministic Bayesian incentive-compatible mechanisms, there exists a joint distribution of the buyer and seller values such that no such mechanism can provide a better than 1 + ln(2)/2 ≈ 1.346 approximation to the optimal social welfare.

The paper also presents impossibility results for Bayesian incentive-compatible mechanisms in the case of independent distributions, improving the state-of-the-art.

İstatistikler

The optimal welfare of a joint distribution F is OPT_F = E(s,b)∼F[max{b, s}].
The welfare of a mechanism M = (x, p) is M_F = E(s,b)∼F[(x(s, b) · b + (1 - x(s, b)) · s].
The approximation ratio of a mechanism M = (x, p) to the optimal welfare is OPT_F / M_F.

Alıntılar

"We prove that the buyer-offering mechanism guarantees an e/(e-1) ≈ 1.582 approximation to the social welfare even if the values are drawn from a joint distribution."
"We prove that the buyer-offering mechanism is optimal in the sense that no Bayesian mechanism where one of the players has a dominant strategy can obtain an approximation ratio better than e/(e-1)."
"We also show that no mechanism in which both sides have a dominant strategy can provide any constant approximation to the social welfare when the values are drawn from a joint distribution."

Önemli Bilgiler Şuradan Elde Edildi

Bilateral Trade with Correlated Values

by Shahar Dobzi... : arxiv.org 04-17-2024

https://arxiv.org/pdf/2308.09964.pdf

Daha Derin Sorular

How can the approximation ratio of the buyer-offering mechanism be further improved for joint distributions of buyer and seller values

To improve the approximation ratio of the buyer-offering mechanism for joint distributions of buyer and seller values, several strategies can be considered. One approach could involve refining the pricing strategy based on the correlation between buyer and seller values. By incorporating more nuanced pricing mechanisms that take into account the joint distribution characteristics, such as adjusting prices based on the covariance or other statistical measures of the values, the mechanism could potentially achieve a better approximation ratio. Additionally, exploring more sophisticated algorithms or optimization techniques to determine the optimal pricing strategy under correlated values could lead to further improvements in the approximation ratio. Furthermore, conducting a detailed analysis of the specific properties of the joint distribution and tailoring the mechanism accordingly could also help enhance the performance of the buyer-offering mechanism in such scenarios.

What are the implications of the impossibility results for Bayesian incentive-compatible mechanisms on the design of practical bilateral trade mechanisms

The impossibility results for Bayesian incentive-compatible mechanisms have significant implications for the design of practical bilateral trade mechanisms. These results highlight the limitations of one-sided dominant strategy mechanisms in achieving high approximation ratios for social welfare in bilateral trade scenarios with correlated values. As a result, designers and researchers may need to explore alternative approaches or mechanisms that go beyond one-sided strategies to improve the efficiency and effectiveness of bilateral trade mechanisms. This could involve developing more complex mechanisms that consider the interplay between buyer and seller incentives more comprehensively, potentially incorporating elements of game theory, mechanism design, or algorithmic optimization to achieve better outcomes. By understanding the constraints imposed by the impossibility results, designers can focus on innovative solutions that address the challenges posed by correlated values in bilateral trade.

Can the techniques used in this paper be extended to analyze the power of randomized Bayesian incentive-compatible mechanisms in bilateral trade problems

The techniques used in the paper can be extended to analyze the power of randomized Bayesian incentive-compatible mechanisms in bilateral trade problems. By adapting the framework and analysis presented in the paper to incorporate randomness or probabilistic elements in the mechanisms, researchers can explore the effectiveness and limitations of randomized strategies in optimizing social welfare and gains from trade in bilateral trade scenarios. This extension could involve studying the impact of randomness on incentive compatibility, exploring the trade-offs between deterministic and randomized mechanisms, and investigating the potential benefits of incorporating probabilistic elements in mechanism design. By applying similar analytical methods and theoretical frameworks to randomized mechanisms, researchers can gain insights into the comparative advantages and challenges of using randomized strategies in bilateral trade settings.