All-Pay Auctions with Affiliated Values

Abstract

This paper analyzes all-pay auctions where the bidders have affiliated values for the object for sale and where the signals take binary values. Since signals are correlated, high signals indicate a high degree of competition in the auction and since even losing bidders must pay their bid, non-monotonic equilibria arise.We show that the game has a unique symmetric equilibrium, and that whenever the equilibrium is non-monotonic the contestants earn no rents. All-pay auctions result in low expected rents to the bidders, but also induce inefficient allocations in models with affiliated private values. With two bidders, the effect on rent extraction dominates, and all-pay auction outperforms standard auctions in terms of expected revenue. With many bidders, this revenue ranking is reversed for some parameter values and the inefficient allocations persist even in large auctions.JEL CLASSIFICATION: D44, D82

War of Attrition with Affiliated Values

“…The linear dotted curve in Figure 1 shows the equilibrium path of the all-pay auction in the phase plane with the same parameters as in Case (iii) of the war of attrition. This illustrates why the case of partially overlapping supports does not appear in the all-pay auction (see Chi et al [2017] for detailed analysis).…”

“…This is an easier form to analyze, because the bidders always pay their own bid irrespective of the outcome. We show in Chi et al [2017] that the unique symmetric equilibrium of the all-pay auction, regardless of whether it is monotonic or not, retains at least following two properties: the support of the low type starts at zero and the union of the two bid supports is a connected interval. With this result in hand, suppose both bidders employ a monotone bidding strategy and consider the highest bid 11 When the bidders' types are independent, we have p…”

“…When contrasting to the all-pay auction, either of the two effects may dominate and hence the revenue ranking is ambiguous. 4 In Chi, Murto, and Välimäki [2017], we show that the possibility of inefficient allocations may not vanish even in a large (first-price) all-pay auction. 5 The equal rent property is a special feature of our binary signal structure.…”

“…This leads to the equilibrium where the two supports are disjoint. 12 In Chi et al [2017], we analyze the closely related model of (first-price) all-pay auction in the same binary setting, and show that φ H (L) ≥ φ L (L) is the sufficient and necessary condition for its unique symmetric equilibrium to be in monotone strategies. Hence, the result established here suggests that the war of attrition calls for a stronger condition for the existence of a monotone strategy equilibrium.…”

“…6 We have computed the allocation and information rents for that auction format in Chi et al [2017]. When comparing the war of attrition and the all-pay auction, we find that the revenue ranking depends on the primitives of the model that we have in mind.…”