11 research outputs found
Nash equilibria in fisher market
Much work has been done on the computation of market equilibria. However due to strategic play by buyers, it is not clear whether these are actually observed in the market. Motivated by the observation that a buyer may derive a better payoff by feigning a different utility function and thereby manipulating the Fisher market equilibrium, we formulate the Fisher market game in which buyers strategize by posing different utility functions. We show that existence of a conflict-free allocation is a necessary condition for the Nash equilibria (NE) and also sufficient for the symmetric NE in this game. There are many NE with very different payoffs, and the Fisher equilibrium payoff is captured at a symmetric NE. We provide a complete polyhedral characterization of all the NE for the two-buyer market game. Surprisingly, all the NE of this game turn out to be symmetric and the corresponding payoffs constitute a piecewise linear concave curve. We also study the correlated equilibria of this game and show that third-party mediation does not help to achieve a better payoff than NE payoffs
On the Efficiency of the Walrasian Mechanism
Central results in economics guarantee the existence of efficient equilibria
for various classes of markets. An underlying assumption in early work is that
agents are price-takers, i.e., agents honestly report their true demand in
response to prices. A line of research in economics, initiated by Hurwicz
(1972), is devoted to understanding how such markets perform when agents are
strategic about their demands. This is captured by the \emph{Walrasian
Mechanism} that proceeds by collecting reported demands, finding clearing
prices in the \emph{reported} market via an ascending price t\^{a}tonnement
procedure, and returns the resulting allocation. Similar mechanisms are used,
for example, in the daily opening of the New York Stock Exchange and the call
market for copper and gold in London.
In practice, it is commonly observed that agents in such markets reduce their
demand leading to behaviors resembling bargaining and to inefficient outcomes.
We ask how inefficient the equilibria can be. Our main result is that the
welfare of every pure Nash equilibrium of the Walrasian mechanism is at least
one quarter of the optimal welfare, when players have gross substitute
valuations and do not overbid. Previous analysis of the Walrasian mechanism
have resorted to large market assumptions to show convergence to efficiency in
the limit. Our result shows that approximate efficiency is guaranteed
regardless of the size of the market
Nash Social Welfare Approximation for Strategic Agents
The fair division of resources is an important age-old problem that has led
to a rich body of literature. At the center of this literature lies the
question of whether there exist fair mechanisms despite strategic behavior of
the agents. A fundamental objective function used for measuring fair outcomes
is the Nash social welfare, defined as the geometric mean of the agent
utilities. This objective function is maximized by widely known solution
concepts such as Nash bargaining and the competitive equilibrium with equal
incomes. In this work we focus on the question of (approximately) implementing
the Nash social welfare. The starting point of our analysis is the Fisher
market, a fundamental model of an economy, whose benchmark is precisely the
(weighted) Nash social welfare. We begin by studying two extreme classes of
valuations functions, namely perfect substitutes and perfect complements, and
find that for perfect substitutes, the Fisher market mechanism has a constant
approximation: at most 2 and at least e1e. However, for perfect complements,
the Fisher market does not work well, its bound degrading linearly with the
number of players.
Strikingly, the Trading Post mechanism---an indirect market mechanism also
known as the Shapley-Shubik game---has significantly better performance than
the Fisher market on its own benchmark. Not only does Trading Post achieve an
approximation of 2 for perfect substitutes, but this bound holds for all
concave utilities and becomes arbitrarily close to optimal for Leontief
utilities (perfect complements), where it reaches for every
. Moreover, all the Nash equilibria of the Trading Post mechanism
are pure for all concave utilities and satisfy an important notion of fairness
known as proportionality
Achieving Diverse Objectives with AI-driven Prices in Deep Reinforcement Learning Multi-agent Markets
We propose a practical approach to computing market prices and allocations
via a deep reinforcement learning policymaker agent, operating in an
environment of other learning agents. Compared to the idealized market
equilibrium outcome -- which we use as a benchmark -- our policymaker is much
more flexible, allowing us to tune the prices with regard to diverse objectives
such as sustainability and resource wastefulness, fairness, buyers' and
sellers' welfare, etc. To evaluate our approach, we design a realistic market
with multiple and diverse buyers and sellers. Additionally, the sellers, which
are deep learning agents themselves, compete for resources in a common-pool
appropriation environment based on bio-economic models of commercial fisheries.
We demonstrate that: (a) The introduced policymaker is able to achieve
comparable performance to the market equilibrium, showcasing the potential of
such approaches in markets where the equilibrium prices can not be efficiently
computed. (b) Our policymaker can notably outperform the equilibrium solution
on certain metrics, while at the same time maintaining comparable performance
for the remaining ones. (c) As a highlight of our findings, our policymaker is
significantly more successful in maintaining resource sustainability, compared
to the market outcome, in scarce resource environments
Wages and Utilities in a Closed Economy
The broad objective of this paper is to propose a mathematical model for the
study of causes of wage inequality and relate it to choices of consumption, the
technologies of production, and the composition of labor in an economy. The
paper constructs a Simple Closed Model, or an SCM, for short, for closed
economies, in which the consumption and the production parts are clearly
separated and yet coupled. The model is established as a specialization of the
Arrow-Debreu model and its equilibria correspond directly with those of the
general Arrow-Debreu model. The formulation allows us to identify the
combinatorial data which link parameters of the economic system with its
equilibria, in particular, the impact of consumer preferences on wages. The SCM
model also allows the formulation and explicit construction of the consumer
choice game, where expressed utilities of various labor classes serve as
strategies with total or relative wages as the pay-offs. We illustrate, through
examples, the mathematical details of the consumer choice game. We show that
consumer preferences, expressed through modified utility functions, do indeed
percolate through the economy, and influence not only prices but also
production and wages. Thus, consumer choice may serve as an effective tool for
wage redistribution
Nash equilibria in fisher market
Much work has been done on the computation of market equilibria. However due to strategic play by buyers, it is not clear whether these are actually observed in the market. Motivated by the observation that a buyer may derive a better payoff by feigning a different utility function and thereby manipulating the Fisher market equilibrium, we formulate the Fisher market game in which buyers strategize by posing different utility functions. We show that existence of a conflict-free allocation is a necessary condition for the Nash equilibria (NE) and also sufficient for the symmetric NE in this game. There are many NE with very different payoffs, and the Fisher equilibrium payoff is captured at a symmetric NE. We provide a complete polyhedral characterization of all the NE for the two-buyer market game. Surprisingly, all the NE of this game turn out to be symmetric and the corresponding payoffs constitute a piecewise linear concave curve. We also study the correlated equilibria of this game and show that third-party mediation does not help to achieve a better payoff than NE payoffs