##### Auctions with Entry Costs

In Donna and Espín-Sánchez (2018) we study the sequential water auctions in Mula (Spain). When goods are substitutes bidders face decreasing marginal returns. In this case, after the first auction, the distribution of valuations for the winner and for the losers are asymmetric, even if they were symmetric for the first auction. Due to decreasing marginal returns on water, the winner now has a lower distribution of valuations than the losers.

Inspired by this event, with Álvaro Parra and Yuzhou Wang we decided to study second-price auctions with participation costs when bidders are asymmetric. We expanded the framework to allow for asymmetric costs, arbitrary distribution of valuations, and to allow for more than two groups of players.

We characterize the equilibrium in general second-price auctions with entry costs by introducing a notion of the firm’s *strength*, that uses the publicly known characteristics of bidders and ranks them by their ability to endure competition. *Strength* is the hypothetical participation cutoff value that would make a bidder indifferent to participate, conditional on all other bidders using the same strategy as the bidder. Although this strategy is generally not an equilibrium, it neatly captures a bidder’s ability to endure competition. A stronger bidder indicates that the bidder is willing to participate in the auction at a lower valuation even while facing competitors who are also participating at lower valuations.

We show that an equilibrium where players’ strategies are ranked by *strength*, or *herculean* equilibrium, exists in general environments. Moreover, when the distributions of valuations are concave, the *herculean* equilibrium is the unique equilibrium.

- “Entry Games under Private Information” (Online Appendix) with Álvaro Parra and Yuzhou Wang,
*The Rand Journal of Economics*, 2023, Vol. 54, No. 3, 512-540.

##### Entry Games

Our tools apply more generally than second price auctions with participation costs. We explore how they can be applied to entry games under private information. The tools could also be applied to other games where the actions taken by the players are strategic substitutes.

In Espín-Sánchez et al (2022) we study firm entry decisions when firms have private information about their profitability. We generalize current entry models by allowing general forms of market competition and heterogeneity among firms. *Post-entry* profits depend on market structure, firms’ identities, and entering firms’ private information. We characterize the equilibrium in this class of games by introducing a notion of the firm’s *strength* and show that an equilibrium where players’ strategies are ranked by *strength*, or *herculean* equilibrium, always exists. Moreover, when profits are elastic enough with respect to the firm’s private information, the *herculean* equilibrium is the unique equilibrium of the entry game.

We study firm entry decisions when firms have private information about characteristics that affect their profitability and that of their competitors. Here are some examples on how to solve for the equilibria in such games. Monopoly profits are π_{i}(v_{i})=v_{i} – K_{i}; and duopoly profits are π_{i}(v_{i},v_{j}) = γv_{i} – ρv_{i} – δ – K_{i}, where K_{i} is the entry cost. The distributions of types are v_{1}~U[0,1] and v_{2}~U[0,α].

*The orange dot allows you to change the values of the parameters (α horizontally and δ, γ or ρ vertically). The orange line is the response function for firm 1 and the purple line is the response function for player 2.*

- Example 1: Type-independent Extensive Margin. π
_{i}(v_{i})=v_{i}-0.5, π_{1}(v_{1},v_{2}) = v_{1}– 0.5 – 0.5 and π_{2}(v_{2},v_{1}) = v_{2 }– 0.5δ – 0.5

- Example 2: Extensive Margin. π
_{i}(v_{i})=v_{i}-0.5, π_{1}(v_{1},v_{2}) = 0.5v_{1}– 0.5 and π_{2}(v_{2},v_{1}) = 0.5γv_{2}– 0.5

- Example 3: Intensive Margin. π
_{i}(v_{i})=v_{i}-0.5, π_{1}(v_{1},v_{2}) = v_{1}– v_{2}– 0.5 and π_{2}(v_{2},v_{1}) = v_{2}–*ρ*v_{1}– 0.5