OTCBB and Pink Sheets have far less regulation and restrictions than do the major exhanges.
For a company to be listed on the OTCBB, the company must provide its own market maker.
Market makers are only required to provide an order on one side of the market, so they can only buy if they wish or sell.
The amount of initial liquidity provided in a situation you describe is almost exclusively left to the issuer, as a market maker can easily be found, but the market maker must sponsor the issuer to get on the exchange, so considering the illiquidity of these companies, fees from sponsorship is how the market maker makes a majority of its revenue. A company can provide no inventory on either side or arrange simultaneous buys and sells to give the market maker an even inventory to provide liquidity on both sides of the market.
To be able to drive the price in either direction where there are no other buyers & sellers would mean that the market maker is trading with itself, and that would be market manipulation. Considering how few trades are, this would be discovered almost immediately.
Market makers hope to profit off of the bid ask spread, but since there is rarely any volume, as stated before, the majority of revenue is the result of listing fees.
In terms of price movement under conditions of illiquidity and market maker obligations, market makers are only obligated to provide an order on one side of the market, bidding or asking, so the structure of their liquidity provision depends upon the initial inventory acquired from the issuer and then of course the subsequent market trading.
If an issuer doesn't want to provide any initial long or short inventory to the market maker, the market maker will only provide Benjamin Graham type valuation bids.
If an issuer provides long inventory, the market maker will set a high sell price.
If an issuer provides short inventory, the market maker will set a low buy price.
If an issuer provides both long and short inventory, the market maker can provide a spread, albeit wide, with a mid equal to the average price of the inventory.
The market maker is not obligated to trade, only to provide liquidity to at least one side. If there is a torrent of buyers, the market maker may provide a bid along side them. If the market maker doesn't believe the market, it could sell small amounts at ever higher prices, but this would be extremely risky, subjecting the inventory to directional bias, which is the bane of market makers' existence.
In short, a market maker on OTCBB and Pink Sheets are not obligated to trade, only to post orders on one side of the market, regardless of where the market is trying to push the issue.