Which means that your company just gambles its money at the stock market by betting on its own price.
One reason an investor might buy shares is to try to take over the company, i.e., end up with all of its shares. Well, it is well known that the people executing a takeover often depend on getting access to the target's cash and other easily-measured (and consequently easily-sold) assets to help pay for the takeover. Specifically, if the company has $100 in cash, the takeover artist can afford to pay approximately $100 more for the company than if it has $0 in cash. (Specifically, if the target has $100 in cash, the takeover artist can go to one of his friends and says, "Loan me $100 and I'll pay you back with the cash held by the company I am planning on buying.) So since market cap is approximately what it would cost to buy the whole company, it makes sense that a reduction in cash would cause a reduction in market cap of approximately the same amount.
Note that even investors who would never contemplate taking over a company can profit by selling their shares to someone who would and does, so they have the same incentive for valuing the company's cash as the takeover artist does: namely, the takeover artist can afford pay more for a company with cash, so the price paid by the takeover artist to the investor will be commensurately higher.
Now the fact that management decided to buy the company's shares rather than deploy the cash in some other way does send a signal to the market that management believes the company's stock is good value with the result that the price might end up at $103 instead of $100. We might call that a "second-order effect" of the buyback. Another second-order effect is that fact that if the market knows a buyback is in progess, they'll tend to hold out for a higher price. (In other words, a big purchase tends to drive up the price; the company will spread the buying out over time to try to limit that effect.) But these second-order effect are limited in magnitude; they are very unlikely to cause the price to end up at, e.g., $200.
Therefore, in your above example if the shares were actually being sold at $2 each, then the company is doing a disservice to the investors by buying the stock back.
However, if the stock is under pressure (below its “fair” price), say 50 cents in the above example, then by buying the stock back, you are creating true value for your investors.
Problem of course is that rarely do people running the company think this way.