Don't buy more stock in a company than necessary to retain a majority lead. Thirteen shares of stock in a company guarantees a majority.
If low on cash, create the least expensive companies possible. You can purchase more shares for the same amount of cash, thus increasing the probability of a majority or minority holder's bonus.
Have a diversified portfolio of stock. Don't put all your faith in one company.
More money is made through mergers than anywhere else. Merging a company in which you have majority is more profitable than making it unmergable at the beginning of the game.
Create as many companies as possible. You always receive a free share of stock (unless all 25 shares are held by other players).
One share of stock can often lead to a majority or minority holder's bonus.
Often retaining stock in a dead company is better than trading it for stock in the controlling company.
Remember, in the two player game the Department of Energy is also involved in all mergers. A greater number of shares is often required in the two player game than the others to receive a majority or minority.
When first learning to play Power Players, turn the "Stock holdings visible to other players" option on in the Preferences. This can help you better understand purchasing strategies by watching what the computer players purchase.