A standard result from auction theory is that bidding truthfully in a second price auction is a weakly dominant strategy. The result, however, does not apply in the presence of Cost Per Action (CPA) constraints. Such constraints exist, for instance, in digital advertising, as some buyer may try to maximize the total number of clicks while keeping the empirical Cost Per Click (CPC) below a threshold. More generally the CPA constraint implies that the buyer has a maximal average cost per unit of value in mind. We discuss how such constraints change some traditional results from auction theory. Following the usual textbook narrative on auction theory, we focus specifically on the symmetric setting, We formalize the notion of CPA constrained auctions and derive a Nash equilibrium for second price auctions. We then extend this result to combinations of first and second price auctions. Further, we expose a revenue equivalence property and show that the sellers revenue-maximizing reserve price is zero. In practice, CPA-constrained buyers may target an empirical CPA on a given time horizon, as the auction is repeated many times. Thus his bidding behavior depends on past realization. We show that the resulting buyer dynamic optimization problem can be formalized with stochastic control tools and solved numerically with available solvers.
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