In a duopoly with lotteries, a Stackelberg leader offering one lottery guarantees 1/4 of monopoly revenue for regular distributions, and in the worst case no leader can guarantee more than 1/e.
Approximating gains from trade in two-sided markets via simple mechanisms
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Approximately Optimal Mechanism Design for Competing Sellers
In a duopoly with lotteries, a Stackelberg leader offering one lottery guarantees 1/4 of monopoly revenue for regular distributions, and in the worst case no leader can guarantee more than 1/e.