In a recent paper, Reny (2011) generalized the results of Athey (2001) and McAdams (2003) on the existence of monotone strategy equilibrium in Bayesian games. Though the generalization is subtle, Reny introduces far-reaching new techniques applying the fixed point theorem of Eilenberg and Montgomery (1946, Theorem 5). This is done by showing that with atomless type spaces the set of monotone functions is an absolute retract and when the values of the best response correspondence are non-empty sub-semilattices of monotone functions, they too are absolute retracts. In this paper, we provide an extensive generalization of Reny (2011), McAdams (2003), and Athey (2001). We study the problem of existence of Bayesian equilibrium in pure strategies for a given partially ordered compact subset of strategies. The ordering need not be a semilattice and these strategies need not be monotone. The main innovation is the interplay between the homotopy structures of the order complexes that are the subject of the celebrated work of Quillen (1978), and the hulling of partially ordered sets, an innovation that extends the properties of Reny’s semilattices to the non-lattice setting. We then describe some auctions that illustrate how this framework can be applied to generalize the existing results and extend the class of models for which we can establish existence of equilibrium. As with Reny (2011), our proof utilizes the fixed point theorem in Eilenberg and Montgomery (1946).
A single seller faces a sequence of buyers with unit demand. The buyers are forward-looking and long-lived but vanish (and are replaced) at a constant rate. The arrival time and the valuation is private information of each buyer and unobservable to the seller. Any incentive-compatible mechanism has to induce truth-telling about the arrival time and the evolution of the valuation.
We derive the optimal stationary mechanism, characterize its qualitative structure and derive a closed-form solution. As the arrival time is private information, the agent can choose the time at which he reports his arrival. The truth-telling constraint regarding the arrival time can be represented as an optimal stopping problem. The stopping time determines the time at which the agent decides to participate in the mechanism. The resulting value function of each agent can not be too convex and has to be continuously differentiable everywhere, reflecting the option value of delaying participation. The optimal mechanism thus induces progressive participation by each agent: he participates either immediately or at a future random time.
A single seller faces a sequence of buyers with unit demand. The buyers are forward-looking and long-lived. Each buyer has private information about his arrival time and valuation where the latter evolves according to a geometric Brownian motion. Any incentive-compatible mechanism has to induce truth-telling about the arrival time and the evolution of the valuation.
We establish that the optimal stationary allocation policy can be implemented by a simple posted price. The truth-telling constraint regarding the arrival time can be represented as an optimal stopping problem which determines the first time at which the buyer participates in the mechanism. The optimal mechanism thus induces progressive participation by each buyer: he either participates immediately or at a future random time.
This paper presents new results on the existence of pure-strategy Bayesian equilibria in specified functional forms. These results broaden the scope of methods developed by Reny (2011) well beyond monotone pure strategies. Applications include natural models of first-price and all-pay auctions not covered by previous existence results. To illustrate the scope of our results, we provide an analysis of three auctions: (i) a first-price auction of objects that are heterogeneous and imperfect substitutes; (ii) a first-price auction in which bidders’ payoffs have a very general interdependence structure; and (iii) an all-pay auction with non-monotone equilibrium.
Injury rates in thirteen U.S. women’s college sports and four U.S. girls’ high school sports are examined in this paper. The sports are categorized as high injury (H) or low injury (L) and differences in injury rates between the two are examined. Estimates are presented of the injury savings that would result if the H sports were changed to have injury rates similar to those in the L sports. The estimated college savings are 13,610 fewer injuries per year and 2,020 fewer healthy years lost-to-injury per year. The estimated high school savings are 143,900 fewer injuries per year and 24,300 fewer healthy years lost-to-injury per year. For concussions the savings are 2,750 per year for college and 49,390 per year for high school. The estimated dollar value (in 2015 dollars) of the total injury savings is between $122 million and $505 million per year for college and between $1.3 billion and $4.9 billion per year for high school.
This paper examines the history of U.S. infrastructure since 1929 and in the process reports an interesting fact about the U.S. economy. Infrastructure as a percent of GDP began a steady decline around 1970, and the government budget deficit became positive and large at roughly the same time. The infrastructure pattern in other countries does not mirror that in the United States, so the United States appears to be a special case. The overall results suggest that the United States became less future oriented beginning around 1970. This change has persisted. This is the interesting fact. Whether it can be explained is doubtful.