On the Convergence of Tâtonnement for Linear Fisher Markets
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arXiv
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| Format: | Preprint |
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2024
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| author | Nan, Tianlong Gao, Yuan Kroer, Christian |
| author_facet | Nan, Tianlong Gao, Yuan Kroer, Christian |
| contents | Tâtonnement is a simple, intuitive market process where prices are iteratively adjusted based on the difference between demand and supply. Many variants under different market assumptions have been studied and shown to converge to a market equilibrium, in some cases at a fast rate. However, the classical case of linear Fisher markets have long eluded the analyses, and it remains unclear whether tâtonnement converges in this case. We show that, for a sufficiently small step size, the prices given by the tâtonnement process are guaranteed to converge to equilibrium prices, up to a small approximation radius that depends on the stepsize. To achieve this, we consider the dual Eisenberg-Gale convex program in the price space, view tâtonnement as subgradient descent on this convex program, and utilize last-iterate convergence results for subgradient descent under error bound conditions. In doing so, we show that the convex program satisfies a particular error bound condition, the quadratic growth condition, and that the price sequence generated by tâtonnement is bounded above and away from zero. We also show that a similar convergence result holds for tâtonnement in quasi-linear Fisher markets. Numerical experiments are conducted to demonstrate that the theoretical linear convergence aligns with empirical observations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_12526 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Convergence of Tâtonnement for Linear Fisher Markets Nan, Tianlong Gao, Yuan Kroer, Christian Computer Science and Game Theory Multiagent Systems Optimization and Control Tâtonnement is a simple, intuitive market process where prices are iteratively adjusted based on the difference between demand and supply. Many variants under different market assumptions have been studied and shown to converge to a market equilibrium, in some cases at a fast rate. However, the classical case of linear Fisher markets have long eluded the analyses, and it remains unclear whether tâtonnement converges in this case. We show that, for a sufficiently small step size, the prices given by the tâtonnement process are guaranteed to converge to equilibrium prices, up to a small approximation radius that depends on the stepsize. To achieve this, we consider the dual Eisenberg-Gale convex program in the price space, view tâtonnement as subgradient descent on this convex program, and utilize last-iterate convergence results for subgradient descent under error bound conditions. In doing so, we show that the convex program satisfies a particular error bound condition, the quadratic growth condition, and that the price sequence generated by tâtonnement is bounded above and away from zero. We also show that a similar convergence result holds for tâtonnement in quasi-linear Fisher markets. Numerical experiments are conducted to demonstrate that the theoretical linear convergence aligns with empirical observations. |
| title | On the Convergence of Tâtonnement for Linear Fisher Markets |
| topic | Computer Science and Game Theory Multiagent Systems Optimization and Control |
| url | https://arxiv.org/abs/2406.12526 |