Anecdotal evidence reveals that an import quota is not always filled when the quota is specified in terms of a market-share limit instead of a quantity limit. In a simple Cournot duopoly, we provide a theoretical rationale for this outcome. Imposing a market-share quota eliminates pure strategy equilibria. Instead, a mixed-strategy equilibrium arises where only the domestic firm mixes choices. The quota is binding under one of the two equilibrium domestic strategies, but it is not binding under the other. Compared to a tariff that restrains the foreign market share to an equivalent level, domestic profits are always higher and consumer surplus is always lower under the market-share quota. Social welfare is lower under the market-share quota when the domestic firm uses its "constrained" strategy, but this outcome may be reversed when the domestic firm uses its "unconstrained" strategy. These results may continue to hold when the foreign firm can move output costlessly between markets.