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We analyze a symmetric Bayesian game in which two players individually contribute to fund a discrete public good; contributions are refunded if they do not reach a threshold set by the seller of the good. We characterize the distributions of players' private values that can support a symmetric equilibrium in continuous piecewise-linear strategies, and we calculate these strategies. Allowing the seller to charge an entry fee before players make their private contributions, we show these piecewise-linear equilibrium strategies maximize the seller's expected profit over all incentive compatible selling mechanisms.

In the European Union Emissions Trading System, the supply of carbon permits is determined in a decentralized manner by Member States. Alternatively, the supply could be determined by an EU central authority. We analyze whether decentralization leads to lower total abatement costs under various assumptions about the behavior of Member States and the privacy of information about their abatement costs. If Member States do not behave strategically, then decentralization is preferred, regardless of whether abatement costs are private information. If the Member States behave strategically, then decentralization may or may not be preferred, depending on the degree of uncertainty about abatement cost parameters, the variation in emission endowments, and the number of Member States.

We study a rent-seeking contest in which the players' valuations of the prize are private information. We determine a Bayesian equilibrium & give conditions under which the equilibrium exists. Although players are ex ante symmetric, increased possibilities for ex post lopsidedness lead to less aggressive bidding. (Lopsidedness increases as players' values become less positively correlated or as the variation in possible values increases.) We also compare the private-information contest to a related public-information contest in which the realizations of values are common knowledge. The contests are equally efficient & players are indifferent between the two, but risk-averse sellers of the prize are not. 1 Table, 3 Figures, 1 Appendix, 5 References. Adapted from the source document.