How stock prices are set?

#1
I tried hard to get the answer of several questions regarding stock market while surfing web; but failed. I also read several tutorials explaining stock markets, but did not get answer to my questions. I would be very grateful if you please answer them; it would be of help for me.

In tutorials about stock markets, I found Stock exchanges are the place where buyers and sellers meet; they have no other function. Every person interested in buying shares of a stock has to post a price on which he is ready to buy the shares which is called bid price, and every person interested in selling shares of a stock has to post a price on which he is ready to sell the shares which is called ask price. The current price of shares of a stock is determined by bid and ask prices.. Are these statements true? Lets assume that its true. At several places in tutorials, I found The stock market specialists choose the lowest ask price and the highest bid price to entertain first.; at several other places, I found The stock market specialists work to ensure that every person who is trading (a seller or buyer) get the best possible price.. These two statements are contradictory. If we assume that the former statement is true then it implies that stock market specialists are working with a single motive of maximizing their commission (i.e. difference between bid and ask price) which gives worst possible price to traders (i.e. sellers and buyers), as the buyer is able to get it at the price of lower bids and the seller is able to sell it at the price of higher asks. The former statement provides maximum benefits to stock market specialists and minimum benefits to traders. While the latter statement provides minimum benefits to stock market specialists and maximum benefits to traders. Could you please tell me about which statement is true and how the current price of shares of a stock is determined (which are displayed through stock exchanges networks)? Is it the average of bid and ask prices? Can a trader tell the broker about the bid/ask prices he want to post or its only broker who takes decision over prices to post (by seeing previous trading transactions etc.)? If highest bid price < lowest ask price, then its not possible for the specialists to do transaction; otherwise they need to give money from their pocket. In such a situation, what stock market specialists do? Do they ask brokers/traders to revise the bid and ask prices? If a trader approaches a broker to buy or sell shares then there must be something that motivate broker to try to trade at best possible price (i.e. lowest price in buying case and highest price in selling case). What are the factors that motivate a broker to try for best possible prices? If a broker posts a bid/ask price and is unable to make a trade in the following trade transaction, then is he given a chance to revise his bid/ask price (by seeing the transaction details)?
I need to get more clarified through one example. Let 6 people want to trade shares of a stock (say stock X); 3 want to buy and 3 want to sell. Person 1 (say P1) wants to buy 1 share, P2 wants to buy 3 share, P3 wants to buy 7 share, P4 wants to sell 2 shares, P5 wants to sell 4 shares and P6 wants to sell 5 shares. P1s bid is $6, P2s bid is $4, P3s bid is $3, P4s ask is $1, P5s ask is $2 and P6s ask is $5. How (and in what order) stock market specialists make trades between these 6 people (assuming that any of the six persons do not revise their ask/bid prices and any other person do not trade in stock X until the completion of transactions between the 6 people)?
 

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