It is often said that examining which socioeconomic system is superior, capitalism or socialism, is irrelevant since the collapse of the former Soviet Union. Some critics have declared, "Capitalism finally won!" Is it true that we can only now select a certain type of capitalism? Not necessarily. We understand that a dichotomy such as market economy vs. planned economy is inappropriate to the study of contemporary economies. Still, it is important to re-consider the working of the market as a whole and evaluate its potential with an open mind. Here we discuss a theoretically fundamental question: what should be the role of the market? The market can be considered a multi-dimensional (economic, political, cultural, ethical and so on) entity, which does not exist in vacuo. We focus on the compatibility of equity with efficiency. This paper is organized as follows. In section 1 ("What role of the market?") we will briefly review the problems concerning the working of the market, to which we must give answer. In section 2 ("Separation of Efficiency from Equity") we examine the theoretical meanings of the market that neo-classical economics suggests. As is well known, it is imagined in it that economic transactions constitute solved political problems. This apolitical characterization of the market will be analyzed focusing upon the relationship between economics and politics in general, and the problem of power in economics in particular. In section 3 ("Neo-classical vs. Post-Marxian economics") we will examine the critique of neo-classical economics and the positive role of the market provided by Bowles and Gintis from the perspective of creating "an egalitarian market economy". A central feature of their argument concerns the attractiveness of the market. Focusing upon egalitarianism mainly in terms of wealth, ownership and labour relations, they omit discussion of many other important macro economic considerations. In Section 4 ("Separation of Equity from Efficiency") we then consider the standing of market in the perspective of traditional Mandan economics. In Section 5 ("What can market do?") we first reconfirm the importance of the theoretical challenges: "Whether or not we can design an efficient socio-economic system where people can enjoy liberty, fairness and equality?" We can argue that so-called efficiency realized through market undermines feelings of solidarity with others. The market system inherently involves inequality, which cannot be efficient from the social point of view. So the market requires the public control. We conclude that the market would not be abolished, but reconstructed as an institution to be more accountable to all of society. The market must be viewed as the servant of humanity, not its master