  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 seller's
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.
