# Robust bilateral trade with discrete types

- 31 Downloads

## Abstract

Bilateral trade problem is the most common market interaction in which a seller and a buyer bargain over an indivisible object, and the valuation of each agent about the object is private information. We investigate the cases where mechanisms satisfying Dominant Strategy Incentive Compatibility (DIC) and Ex-post Individual Rationality (EIR) properties can exhibit robust performance in the face of imprecision in prior structure. We start with the general mathematical formulation for the bilateral trade problem with DIC, EIR properties. We derive necessary and sufficient conditions for DIC, EIR mechanisms to be Ex-post efficient at the same time. Then, we define a new property—Allocation Maximality—and prove that the Posted Price mechanisms are the only mechanisms that satisfy DIC, EIR and Allocation Maximal properties. We also show that Posted Price mechanism is not the only mechanism that satisfies DIC and EIR properties. The last part of the paper introduces different sets of priors for agents’ types and consequently allows ambiguity in the problem framework. We derive robust counterparts and solve them numerically for the proposed objective function under box and \(\phi \)-divergence ambiguity specifications. Results suggest that restricting the feasible set to Posted Price mechanisms can decrease the objective value to different extents depending on the uncertainty set.

## Keywords

Mechanism design Robustness Ambiguity \(\phi \)-Divergence## Mathematics Subject Classification

90C05 91B26## References

- Bayrak HI, Pınar MÇ (2016) Generalized second price auction is optimal for discrete types. Econ Lett 141:35–38CrossRefGoogle Scholar
- Bayrak HI, Güler K, Pınar MÇ (2017) Optimal allocation with costly inspection and discrete types under ambiguity. Optim Methods Softw 32(4):699–718CrossRefGoogle Scholar
- Bayraksan G, Love DK (2015) Data-driven stochastic programming using phi-divergences. In: The operations research revolution, INFORMS, pp 1–19Google Scholar
- Ben-Tal A, Bertsimas D, Brown DB (2010) A soft robust model for optimization under ambiguity. Oper Res 58(4–part–2):1220–1234CrossRefGoogle Scholar
- Ben-Tal A, Den Hertog D, De Waegenaere A, Melenberg B, Rennen G (2013) Robust solutions of optimization problems affected by uncertain probabilities. Manag Sci 59(2):341–357CrossRefGoogle Scholar
- Bose S, Ozdenoren E, Pape A (2006) Optimal auctions with ambiguity. Theor Econ 1(4):411–438Google Scholar
- Carroll G (2017) Information acquisition and robust trading mechanisms. Unpublished manuscript, Stanford University, StanfordGoogle Scholar
- De Castro LI, Yannelis NC (2010) Ambiguity aversion solves the conflict between efficiency and incentive compatibility, Technical report, Discussion Paper. Center for Mathematical Studies in Economics and Management ScienceGoogle Scholar
- Flesch J, Schröder M, Vermeulen AJ (2013) The bilateral trade model in a discrete setting. Department of Quantitative Economics, Maastricht University, MaastrichtGoogle Scholar
- Flesch J, Schröder M, Vermeulen D (2016) Implementable and ex-post IR rules in bilateral trading with discrete values. Math Soc Sci 84:68–75CrossRefGoogle Scholar
- Gilboa I, Schmeidler D (1989) Maxmin expected utility with non-unique prior. J Math Econ 18(2):141–153CrossRefGoogle Scholar
- Hagerty KM, Rogerson WP (1987) Robust trading mechanisms. J Econ Theory 42(1):94–107CrossRefGoogle Scholar
- Koçyiğit Ç, Bayrak HI, Pınar MÇ (2018) Robust auction design under multiple priors by linear and integer programming. Ann Oper Res 260(1–2):233–253CrossRefGoogle Scholar
- Kos N, Manea M (2009) Efficient trade mechanisms with discrete values, Technical report. Working paperGoogle Scholar
- Kouvelis P, Yu G (2013) Robust discrete optimization and its applications, vol 14. Springer, BerlinGoogle Scholar
- Matsuo T (1989) On incentive compatible, individually rational, and ex post efficient mechanisms for bilateral trading. J Econ Theory 49(1):189–194CrossRefGoogle Scholar
- Myerson RB, Satterthwaite MA (1983) Efficient mechanisms for bilateral trading. J Econ Theory 29(2):265–281CrossRefGoogle Scholar
- Othman A, Sandholm T (2009) How pervasive is the Myerson–Satterthwaite impossibility? In: IJCAI, pp 233–238Google Scholar
- Pardo L (2005) Statistical inference based on divergence measures, vol 185. Chapman & Hall/CRC, LondonGoogle Scholar
- Pınar MÇ (2018) Robust trade mechanisms over 0–1 polytopes. J Comb Optim 36(3):845–860CrossRefGoogle Scholar
- Pınar MÇ, Kızılkale C (2017) Robust screening under ambiguity. Math Program 163(1):273–299CrossRefGoogle Scholar
- Tawarmalani M, Sahinidis NV (2005) A polyhedral branch-and-cut approach to global optimization. Math Program 103:225–249CrossRefGoogle Scholar
- Vohra RV (2011) Mechanism design: a linear programming approach, vol 47. Cambridge University Press, CambridgeGoogle Scholar
- Vohra RV (2012) Optimization and mechanism design. Math Program 134(1):283–303CrossRefGoogle Scholar
- Wächter A, Biegler LT (2006) On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math Program 106(1):25–57CrossRefGoogle Scholar