Is there any research, theory, or anything which shows how much of a market has to be in power of a few companies to be a force against free market? According to Free market


  a free market is a system in which the prices for goods and services
  are determined by the open market and by consumers. In a free market,
  the laws and forces of supply and demand are free from any
  intervention by a government or other authority and from all forms of
  economic privilege, monopolies and artificial scarcities


Being monopolies and oligopolies a force against free market, is there any study, theory or anything which shows how much of a market has to be in power of a few companies for them to be able to alter the free market laws/rules?
The quote in the question isn't really rigorous about what a free market is, but it talks about monopolies and artificial scarcities, so I am interpreting the efficient outcome with price equal to marginal cost as being one necessary feature of what they understand as a free market.



Let's look at the cournot model of competition. There are $n$ firms, each with idiosyncratic marginal cost $c_i$. Each firm simultaneously chooses its output (supply) quantity, $q_i$. Given consumers' demand, this results in some market price $P(Q)$, where $Q=\sum_{i=1}^n q_i$ is the total industry output. We would expect the demand curve to be downward sloping: $P'(Q)&lt;0$.

Thus, a firm's profits are
$$\pi_i(q_i,\mathbf{q}_{-i})=\left(P(Q)-c_i\right)q_i.$$
The first-order condition for maximisation of profit is
$$\frac{\partial\pi_i}{\partial q_i}=P(Q)-c_i+\frac{\partial P(Q)}{\partial q_i}=0.$$
Noting that $$\frac{\partial P(Q)}{\partial q_i}=\frac{\partial P(Q)}{\partial Q},$$
the first-order condition can be re-written as
$$\frac{P(Q)-c_i}{c_i}=-\frac{\partial P(Q)}{\partial Q}\frac{Q}{P(Q)}\frac{q_i}{Q},$$
$$\frac{P(Q)-c_i}{c_i}=-\frac{1}{\eta}m_i,$$
where $\eta$ is the price elasticity of demand and $m_i=q_i/Q$ is firm $i$'s market share.

The left-hand side measures a firm's price-cost margin, or its market power. We can compute the average market power in the industry by weighting each firm by its market share:

$$\text{avg market power}=\sum_i m_i \frac{P(Q)-c_i}{c_i}=-\frac{\sum_i m_i^2}{\eta}.$$

Note that $\sum_i m_i^2$ is the HHI index for the market. So this equation tells us that the average market power in an industry will be positively related to the HHI. For any positive HHI, there will be a positive mark-up and the market will deviate from the perfectly competitive outcome in which artificial scarcities do not exist. Put differently, only if there is an infinite number of firms, each controling an infinitely small share of the industry do we obtain the textbook perfect competition outcome.