An odd-lot buyback occurs when a company offers to purchase shares of its stock back from people who hold less than 100 shares. Investors can wind up with odd-lot shares in a number of ways, often through dividend reinvestment plans or a reverse-split. Odd lots shouldn't be confused with round lots, which are any lots that can be evenly divided by 100, or mixed lots, which are lots of more than 100 shares that cannot be evenly divided by 100.
A popular method that companies use to buy back stocks is called a Dutch auction. Shareholders who are interested in participating in the auction indicate a price range within which they would be willing to sell their stocks back. The company will buy back the shares from the lowest tendered offers, all at the same price. The price is the highest of the accepted offers.
This type of offer makes it less expensive both for the company (due to the reduced cost of servicing these small shareholder accounts) and for the shareholders (because they do not have to pay brokerage fees to sell their shares). A buyback also can increase a stock's price-to-earnings ratio by decreasing the number of outstanding shares.
Farewell, Odd-Lot Theory
Odd-lot buybacks are far less common than they used to be, primarily due to the extraordinary growth in online trading platforms. The competition among these platforms brought down the standard commission so that it is no longer prohibitively expensive for an investor to sell off odd numbers of shares. In the past, investors had to use brokerage houses to dispose of their shares, and commissions were higher for these odd-lot, smaller trades.
Smaller investors, who tend to buy in odd lots, are the source of "odd-lot theory," which basically assumes that small investors are less experienced and driven by emotion, as opposed to technical theory and logic. The theory assumes that these small investors, therefore, are always wrong, so traders should do the opposite. Odd-lot theory was popular in the days before electronic trading platforms became the norm; it has since fallen out of favor, more or less, and hasn't been seriously used as a way to gauge the market since the 1980s. (See also: A Breakdown of Stock Buybacks.)
This question was answered by Katie Adams.