Lottery versus all-pay auction contests: A revenue dominance theorem

We allow a contest organizer to bias a contest in a discriminatory way; i.e., she can favor specific contestants by designing the contest rule in order to maximize total equilibrium effort (resp. revenue). The two predominant contest regimes are considered, all-pay auctions and lottery contests. For all-pay auctions the optimal bias is derived in closed form: It implies extreme competitive pressure among active contestants and low endogenous participation rates. Moreover, the exclusion principle advanced by Baye et al. (1993) becomes obsolete in this case. In contrast, the optimally biased lottery induces a higher number of actively participating contestants due to softer competition. Our main result regarding total revenue comparison under the optimal biases reveals that the all-pay auction revenue-dominates the lottery contest for all levels of heterogeneity among contestants. The incentive effect due to a strongly discriminating contest rule (all-pay auction) dominates the participation effect due to a weakly discriminating contest rule (lottery).     

Buyer groups as strategic commitments

Buyer cooperatives, buyer alliances, and horizontal mergers are often perceived as attempts to increase buyer power. In contrast to prior research emphasizing group size, I show that even small buyer groups composed of buyers with heterogeneous preferences can increase price competition among rival sellers by committing to purchase exclusively from one seller. Without transfer payments, at least one buyer group exists for each pair of sellers and buyer groups membership is chosen to achieve indifference between the two sellers. With transfer payments, and just two sellers, the grand coalition is a coalition-proof subgame perfect equilibrium (CP-SPNE), though equilibria with arbitrarily many buyer groups also exist. With three sellers (and with more sellers when the distribution of buyers is symmetric), a CP-SPNE always exists, all coalition-proof equilibria are payoff equivalent and have at least one buyer group for each pair of firms, so the grand coalition is not an equilibrium.     

Dynamic legislative decision making when interest groups control the agenda

We consider dynamic decision making in a legislature, in which in each period legislators vote between the status quo (previous period?s policy) and a new bill. However, the agenda formation process is captured by interest groups, that is, the new bill on the agenda is determined by an all-pay auction among these groups. We show that convergence to the median voter of the legislature arises if interest groups are patient enough but not necessarily otherwise. We characterize the bound on the speed of convergence in a family of stationary equilibria in which policy bounces between right-wing and left-wing policies. We also show that convergence may be faster if organized interest groups represent only one side of the policy space, e.g., when only business and not consumer interests are organized.     

Difference-Form Contests and the Robustness of All-Pay Auctions

In much of the existing literature on rent-seeking games, the outcome of the contest is either infinitely sensitive or relatively insensitive to contestants' efforts. The current paper presents a family of contest games that permit characterization of equilibrium for all levels of sensitivity of the outcome to contestants' efforts. Specifically, the outcome of the contest depends on the difference between efforts, which encompass the lottery and the all-pay auction as polar cases. The equilibrium converges to that of the all-pay auction as the probability of winning the prize grows infinitely sensitive to one's effort, and the main qualitative features of equilibrium persist over a large parameter region. Journal of Economic Literature Classification Numbers: D44, D72.     

The lifeboat problem

We study an all-pay contest with multiple identical prizes (“lifeboat seats”). Prizes are partitioned into subsets of prizes (“lifeboats”). Players play a two-stage game. First, each player chooses an element of the partition (“a lifeboat”). Then each player competes for a prize in the subset chosen (“a seat”). We characterize and compare the subgame perfect equilibria in which all players employ pure strategies or all players play identical mixed strategies in the first stage. The partitioning of prizes can lead to coordination failure when players employ nondegenerate mixed strategies. In these equilibria some rents are sheltered and rent dissipation is reduced.     

Expected revenue of all-pay and first-price sealed-bid auctions with affiliated signals

In the general symmetric auction framework of Milgrom and Weber (1982) it is shown—as a new manifestation of thelinkage principle—that the all-pay sealed-bid auction yieldshigher expected revenue than the standard first-price sealed-bid auction. This raises the question why sealed-bid auctions of the standard first-price variety are observed in practice whereas the all-pay variety is not.     

Strategic Pricing, Consumer Search and the Number of Firms

We examine an oligopoly model where some consumers engage in costly non-sequential search to discover prices. There are three distinct price-dispersed equilibria characterized by low, moderate and high search intensity. The effects of an increase in the number of firms on search behaviour, expected prices, price dispersion and welfare are sensitive (i) to the equilibrium consumers' search intensity, and (ii) to the status quo number of firms. For instance, when consumers search with low intensity, an increase in the number of firms reduces search, does not affect expected price, leads to greater price dispersion and reduces welfare. In contrast, when consumers search with high intensity, increased competition results in more search and lower prices when the number of competitors in the market is low to begin with, but in less search and higher prices when the number of competitors is large. Duopoly yields identical expected price and price dispersion but higher welfare than an infinite number of firms.     

Best-of-three all-pay auctions

We study a best-of-three all-pay auction. It is shown that with values of winning and without values of losing, this auction is less productive (the players’ total expected effort is smaller) than the one-stage all-pay auction. However, with different values of losing over the contest’s stages it may be more productive than the one-stage all-pay auction.     

Optimal seedings in elimination tournaments

We study an elimination tournament with heterogenous contestants whose ability is common-knowledge. Each pair-wise match is modeled as an all-pay auction. Equilibrium efforts are in mixed strategies, yielding complex dynamics: endogenous win probabilities in each match depend on other matches’ outcome through the identity of the expected opponent in the next round. The designer seeds competitors according to their ranks. For tournaments with four players we find optimal seedings for three different criteria: (1) maximization of total tournament effort; (2) maximization of the probability of a final among the two top ranked teams; (3) maximization of the win probability for the top player. We also find the seedings ensuring that higher ranked players have a higher winning probability. We compare our predictions with data from NCAA basketball tournaments.     

Cooperation vs. competition in R&D: The role of stability of equilibrium

We consider a model in which firms first choose process R&D expenditures and then compete in an output market. We show the symmetric equilibrium under R&D competition is sometimes unstable, in which case two asymmetric equilibria must also exist. For the latter, we find, in contrast to the literature that total profits are sometimes higher with R&D competition than with research joint venture cartelization (due to the cost asymmetry of the resulting duopoly in the noncooperative case). Furthermore, these equilibria provide another instance of R&D-induced firm heterogeneity.     

Beyond chopsticks: Symmetric equilibria in majority auction games

Majority auction games are simultaneous sealed-bid auctions of identical objects among identical bidders who each want to win a specified fraction (more than a half) of the objects. Each bidder receives no benefit from winning less than the specified fraction and no additional benefit from winning more than it. Symmetric equilibria having simple, intuitive forms are shown to exist in first-price, second-price and all-pay versions of such games when the number of bidders is sufficiently large. This contrasts with earlier results for the two-bidder “pure chopstick” majority auction games where the only known equilibria are more complicated.     

Strategic Jump Bidding in English Auctions

This paper solves for equilibria of sequential bid (or English) auctions with affiliated values when jump bidding strategies may be employed to intimidate one's opponents. In these equilibria, jump bids serve as correlating devices which select asymmetric bidding functions to be played subsequently. Each possibility of jump bidding provides a Pareto improvement for the bidders from the symmetric equilibrium of a sealed bid, second-price auction. The expanded set of equilibria can approximate either first- or second-price outcomes and produce exactly the set of expected prices between those two bounds. These results contrast with standard conclusions that equate English and second-price auctions.     

Identity and group conflict

We experimentally investigate the effects of real and minimal identities on group conflict. In turn we provide a direct empirical test of the hypotheses coined by Amartya Sen that the salience of a real identity escalates conflict but that of a mere classification would not do so. In a baseline treatment, two groups – East Asians and Caucasians – engage in a group contest, but information on the racial composition of the groups is not revealed. In the minimal identity treatment each group is arbitrarily given a different color code, whereas in the real identity treatment the race information is revealed. Supporting Sen׳s hypotheses, we find that compared to the baseline, free-riding declines and conflict effort increases in the real identity treatment but not in the minimal identity treatment. Moreover, this occurs due to an increase in efforts in the real identity treatment by females in both racial groups.     

Private Provisions of a Discrete Public Good with Voluntary Participation

This paper studies the mechanism that a profit-making principal should adopt to provide a discrete public good when the values of the consumers are their private information and their participation is voluntary. The free-riding issue is resolved through threatened nonprovision of the good by the provider. Every bidder is asked to announce his or her virtual value as defined in Myerson (1981) . The public good is provided if and only if the sum of the bidders' announced virtual values exceeds the provision cost. When a provision decision results, each bidder pays an amount that is determined by the announcement of other consumers. No one pays when a nonprovision decision results. We find that this mechanism is implementable through an all-pay auction. A restricted profit-maximizing mechanism that implements efficient allocation is also characterized. As in Gradstein (1994) , when provision is always efficient, that is, the sum of consumers' values always exceeds the provision cost, efficient allocation is achievable through a profit-maximizer. However, this is not the case when provision is not efficient.     

Information acquisition in conflicts

This paper considers incentives for information acquisition ahead of conflicts. First, we characterize the (unique) equilibrium of the all-pay auction between two players with one-sided asymmetric information where one player has private information about his valuation. Then, we use our results to study information acquisition prior to an all-pay auction. If the decision to acquire information is observable, but not the information received, one-sided asymmetric information can occur endogenously in equilibrium. Moreover, the cut-off values of the cost of information that determine equilibrium information acquisition are higher than those in the first best. Thus, information acquisition is excessive. In contrast, with open or covert information acquisition, the equilibrium cut-off values are as in the first best.     

Where are the rent seekers?

In a remarkably simple and yet one of the most original and insightful observations of 20th century economics, Gordon Tullock pointed out that there are efficiency losses when public policies and political behavior create contestable rents. Tullock also observed that social losses from contesting rents appeared smaller than might be expected, so raising the question ‘where are the rent seekers?’ Tullock proposed that political accountability and ‘free-riding’ incentives in interest groups limit social losses from rent seeking. We affirm Tullock’s explanations, which apply differently under different political institutions. We compare Tullock with Gary Becker, who focused on deadweight losses from redistribution and concluded, in contrast to Tullock, that political redistribution is efficient. The comparison with Becker highlights the significance of the recognition of Tullock’s concept of rent seeking. By excluding rent-seeking losses from the social costs of redistribution, Becker could arrive at a conclusion more favorable than Tullock to an ideology that sees merit in extensive redistribution. Tullock’s model, although more encompassing of actual social costs of redistribution, would have been less welcome in the social democratic welfare state.     

Tie-breaks and bid-caps in all-pay auctions

We revisit the two bidder complete information all-pay auction with bid-caps introduced by Che and Gale (1998), dropping their assumption that tie-breaking must be symmetric. Any choice of tie-breaking rule leads to a different set of Nash equilibria. Compared to the optimal bid-cap of Che and Gale we obtain that in order to maximize the sum of bids, the designer prefers to set a less restrictive bid-cap combined with a tie-breaking rule which slightly favors the weaker bidder. Moreover, the designer is better off breaking ties deterministically in favor of the weak bidder than symmetrically.     

The all-pay auction with complete information

Summary In a (first price) all-pay auction, bidders simultaneously submit bids for an item. All players forfeit their bids, and the high bidder receives the item. This auction is widely used in economics to model rent seeking, R&D races, political contests, and job promotion tournaments. We fully characterize equilibrium for this class of games, and show that the set of equilibria is much larger than has been recognized in the literature. When there are more than two players, for instance, we show that even when the auction is symmetric there exists a continuum of asymmetric equilibria. Moreover, for economically important configurations of valuations, there is no revenue equivalence across the equilibria; asymmetric equilibria imply higher expected revenues than the symmetric equilibrium.We are grateful to Jacques Crémer, Chuangyin Dang, Jürgen Dennert, Chaim Fershtman, Martin Hellwig, Arthur Robson, Heinrich Ursprung, Eric van Damme, Ton Vorst, and the referees for helpful comments. We benefitted from presentations at the World Congress of the Econometric Society in Barcelona, the European Meeting on the Economics of Information at Tilburg University, the Midwest Mathematical Economics Meetings at the University of Illinois, and seminars at Texas A&M University, The Pennsylvania State University, Tilburg University and the University of Montreal. Baye is grateful for support from the CentER for Economic Research at Tilburg University and the Tinbergen Institute where earlier versions of this paper were completed. Kovenock acknowledges support from Erasmus Universiteit Rotterdam, the Tinbergen Institute, the Center for Economic Studies at the University of Munich, the Institut d'Analisi Economica CSIC at the Universitat Autonoma de Barcelona, the Krannert School of Management, and the Jay N. Ross Young Faculty Scholar Award.     

Nonconvergent Electoral Equilibria under Scoring Rules: Beyond Plurality

We use Hotelling's spatial model of competition to investigate the position‐taking behavior of political candidates under a class of electoral systems known as scoring rules, though the model also has a natural interpretation in the firm location context. Candidates choose ideological positions so as to maximize their support in society. Convergent Nash equilibria in which all candidates adopt the same policy were characterized by Cox (1987). Here, we investigate nonconvergent equilibria, where candidates adopt divergent policies. We identify a number of classes of scoring rules exhibiting a range of different equilibrium properties. For some of these, nonconvergent equilibria do not exist. For others, nonconvergent equilibria in which candidates cluster at positions spread across the issue space are observed. In particular, we prove that the class of convex rules does not have Nash equilibria (convergent or nonconvergent) with the exception of some derivatives of Borda rule. We also look at “two‐party” equilibria. Implications for the firm location model are discussed.     

The evolution of sanctioning institutions: an experimental approach to the social contract

A vast amount of empirical and theoretical research on public good games indicates that the threat of punishment can curb free-riding in human groups engaged in joint enterprises. Since punishment is often costly, however, this raises an issue of second-order free-riding: indeed, the sanctioning system itself is a common good which can be exploited. Most investigations, so far, considered peer punishment: players could impose fines on those who exploited them, at a cost to themselves. Only a minority considered so-called pool punishment. In this scenario, players contribute to a punishment pool before engaging in the joint enterprise, and without knowing who the free-riders will be. Theoretical investigations (Sigmund et al., Nature 466:861–863, 2010) have shown that peer punishment is more efficient, but pool punishment more stable. Social learning, i.e., the preferential imitation of successful strategies, should lead to pool punishment if sanctions are also imposed on second-order free-riders, but to peer punishment if they are not. Here we describe an economic experiment (the Mutual Aid game) which tests this prediction. We find that pool punishment only emerges if second-order free riders are punished, but that peer punishment is more stable than expected. Basically, our experiment shows that social learning can lead to a spontaneously emerging social contract, based on a sanctioning institution to overcome the free rider problem.