Contests with multiple alternative prizes: Public-good/bad prizes and externalities

We study contests in which there are multiple alternative public-good/bad prizes, and the players compete, by expending irreversible effort, over which prize to have awarded to them. Each prize may be a public good for some players and a public bad for the others, and the players expend their effort simultaneously and independently. We first prove the existence of a pure-strategy Nash equilibrium of the game, then establish when the total effort level expended for each prize is unique across the Nash equilibria, and then summarize and highlight other interesting and important properties of the equilibria. Finally, we discuss the effects of heterogeneity of valuations on the players’ equilibrium effort levels and a possible extension of the model.     

Endogenous Group Formation in Contests: Unobservable Sharing Rules

We study contests in which players compete by expending irreversible effort to win a prize, the prize is awarded to one of the players, the winner shares the prize with other players in his group, if any, and each group's sharing rule is unobservable to the other groups and the singletons, if any, when the players expend their effort. The number of groups, their sizes, and the number of singletons are exogenous in the first model, whereas they are endogenous in the second model. We show that group formation occurs if the number of players is four or smaller, but does not occur otherwise. We examine the effect of endogenous group formation on total effort level and the profitability of endogenous group formation. In each of the two models, comparing the outcomes of the case of unobservable sharing rules with those of the case of observable sharing rules, we show that the two cases yield quite different outcomes.     

Contests with endogenous deadlines

This paper analyzes the problem of a contest designer who chooses a starting time and a deadline of the contest to maximize discounted total effort by the contestants. Each contestant secretly decides how much effort to exert between the starting time and the deadline. At the deadline, the contestant who exerted most effort wins a prize, which consists of the endowment of the designer and collected interest. The contest has a unique Nash equilibrium. In the main model, the designer should announce the contest immediately with a short deadline to promote intense competition. I analyze how the optimal starting time and deadline change for a variable contest prize, different types of asymmetries, a Tullock lottery contest success function, and different goal functions of the designer.     

Information Management and Incentives

We ask how the incentives of an agent are affected by an information management system that lets the agent receive information about the performance of a colleague before (“transparent firm”) rather than after he provides effort (“nontransparent firm”). Transparency is detrimental for incentives if the performance of the colleague provides information on the relative impact of the agent’s effort on his success probability. The findings imply that firms in which comparisons between employees play a minor role for compensation are transparent. Firms in which they play a major role sometimes choose to be nontransparent despite the flexibility gains transparency provides.     

Contests over joint production on networks

We consider a network of heterogeneous agents where each edge represents a two‐player contest between the respective nodes. In these bilateral contests, agents compete over an endogenous prize jointly produced using their own contest efforts. We provide a necessary and sufficient condition for the existence of Nash equilibrium and characterize the equilibrium total effort for every agent. Our model has insightful results regarding the network type, that is, depending on whether the network is bipartite or nonbipartite. Finally, considering the sum of all expected utilities as an efficiency notion, we investigate the optimal network structure.     

Competitive prizes: when less scrutiny induces more effort

We consider a principal who is keen to induce his agents to work at their maximal effort levels. To this end, he samples n days at random out of the T days on which they work, and awards a prize of B dollars to the most productive agent. The principal’s policy (B, n) induces a strategic game Γ(B, n) between the agents. We show that to implement maximal effort levels weakly (or, strongly) as a strategic equilibrium (or, as dominant strategies) in Γ(B, n), at the least cost B to himself, the principal must choose a small sample size n. Thus less scrutiny by the principal induces more effort from the agents.The need for reduced scrutiny becomes more pronounced when agents have information of the history of past plays in the game. There is an inverse relation between information and optimal sample size. As agents acquire more information (about each other), the principal, so to speak, must “undo” this by reducing his information (about them) and choosing the sample size n even smaller.     

Contest success function with the possibility of a draw: Axiomatization

In imperfectly discriminating contests the contestants contribute effort to win a prize but the highest contributed effort does not necessarily secure a win. The contest success function (CSF) is the technology that translates an individual's effort into his or her probability of winning. This paper provides an axiomatization of CSF when there is the possibility of a draw (the sum of winning probabilities across all contestants does not add up to one).     

Do missing values exist? Incomplete data handling in cross-national longitudinal studies by means of continuous time modeling

In cross-national longitudinal studies it is often impossible to administer the same measurement instruments at the same occasions to all sample units in all participating countries. This quickly results in large quantities of missing data, due to (a) missing measurement instruments in some countries, (b) missing assessment waves within or across countries, (c) missing data for individual sample units. As compared to cross-sectional studies, the problem of missing values is further aggravated by the fact that missing values are always associated with different time intervals between repeated observations. In the past, this has often been dealt with by the use of phantom-variables, but this approach is limited to simple designs with few missing value patters. In the present paper we propose a new way to think of, and deal with, missing values in longitudinal studies. Instead of conceiving of a longitudinal study as a study with \(T\) discrete time points of which some are missing, we propose to conceive of a longitudinal study as a way to measure an underlying process that develops continuously over time, but is only observed at some selected discrete time points. This transforms the problem of missing values into a problem of unequal time intervals. After a quick introduction to the basic idea of continuous time modeling, we demonstrate how this approach provides a straightforward solution to missing measurement instruments in some countries, missing assessment waves within or across countries, and missing data for individual sample units.     

Allocation of Prizes in Contests with Participation Constraints

We study all‐pay contests with an exogenous minimal effort constraint where a player can participate in a contest only if his effort (output) is equal to or higher than the minimal effort constraint. Contestants are privately informed about a parameter (ability) that affects their cost of effort. The designer decides about the size and number of prizes. We analyze the optimal prize allocation for the contest designer who wishes to maximize either the total effort or the highest effort. It is shown that if the minimal effort constraint is relatively high, the winner‐take‐all contest in which the contestant with the highest effort wins the entire prize sum does not maximize the expected total effort or the expected highest effort. Rather, a random contest in which the entire prize sum is equally allocated to all the participants yields a higher expected total effort as well as a higher expected highest effort.     

Probability and quantile estimation from individually micro-aggregated data

Micro-aggregation is a frequently used strategy to anonymize data before they are released to the scientific public. A sample of a continuous random variable is individually micro-aggregated by first sorting and grouping the data into groups of equal size and then replacing the values of the variable in each group by their group mean. In a similar way, data with more than one variable can be anonymized by individual micro-aggregation. Data thus distorted may still be used for statistical analysis. We show that if probabilities and quantiles are estimated in the usual way by computing relative frequencies and sample quantiles, respectively, these estimates are consistent and asymptotically normal under mild conditions.     

Contingent payments in selection contests

The early literature on research contests stressed the advantages of a fixed prize in inspiring R&D effort. More recently the focus has moved towards endogenizing the rewards to research activity in these tournament settings, since this can induce extra effort or enhance the surplus of the buyer. We focus on a research contest as a means of selecting a partner for an R&D enterprise, in an informational setting in which the established providers of R&D services know more about each others’ relative capabilities than does the buyer/sponsor. An alternative use of our model is in choosing between prospective patentees where the Patent Trading Office has less information on the patents than the competitors. This asymmetry creates a source of inefficiency if a rank order contest is used as a selection device; we show how the contest can be modified to improve selection efficiency, while maintaining its simplicity (as only ordinal information is required). The modification that we suggest involves endogenizing the prizes that are awarded contingent upon whether a contestant wins or loses the contest. Furthermore, the payment system and the selection mechanism are detail-free. This paper is part of the project “The Knowledge-Based Society” sponsored by the Research Council of Norway (project 172603/V10).     

Principal‐agent and Peer Relationships in Tournaments

Effort provision in tournaments may depend on both social preferences towards the competitor and social preferences towards the principal who designed the tournament. In a laboratory experiment, we analyze agents' behavior in different tournament settings that vary the distribution of the prize between agents. The principal chooses between a low and a high guaranteed payment. Standard economic theory predicts the same effort provision in all situations. Instead, our results show that both the generosity of the principal and the strict separation between winner and loser increase the effort level in tournaments. Moreover, agents focus their preferences either on the principal or on the agent. Copyright © 2014 John Wiley & Sons, Ltd.     

Deadlines in stochastic contests

We consider a two-player contest model in which breakthroughs arrive according to privately observed Poisson processes. Each player’s process continues as long as she exerts costly effort. The player who collects the most breakthroughs until a predetermined deadline wins a prize.We derive Nash equilibria of the game depending on the deadline. For short deadlines, there is a unique equilibrium in which players use identical cutoff strategies, i.e., they continue until they have a certain number of successes. If the deadline is long enough, the symmetric equilibrium distribution of an all-pay auction is an equilibrium distribution over successes in the contest. Expected efforts may be maximal for a short or intermediate deadline.     

Efficient Contests

In their seminal contribution, Lazear and Rosen (1981) show that wages based upon rank induce the same efficient effort as incentive‐based reward schemes. They also show that this equivalence result is not robust toward heterogeneity in worker ability, as long as ability is private information because it is not possible to structure contests to simultaneously satisfy self‐selection constraints and first‐best incentives. This paper demonstrates that efficiency can be achieved by a simple modification of the prize scheme in a mixed (heterogenous) contest where contestants learn their type after entry. If contestants know their type before entering the contest, rent extraction becomes an issue. Implications for optimal contest design are also explored. Finally, the relationship between effort maximizing contests and profit maximizing contests are discussed.     

MULTIPLE-PRIZE CONTESTS – THE OPTIMAL ALLOCATION OF PRIZES

Abstract.  Multiple-prize contests are important in various fields of economics ranging from rent seeking over labour economics, patent and R&D races to tendering for (governmental) projects. Hence it is crucial to understand the incentive effects of multiple prizes on effort investment. This survey attempts to outline, compare and evaluate the results from the literature. While a first prize always results in a positive incentive to invest effort, second and later prizes lead to ambiguous effects. Depending on the objective function, the characteristics of the individuals and the type of contest a different prize allocation is optimal.     

Targeted campaign competition,loyal voters,and supermajorities

We consider campaign competition in which candidates compete for votes among a continuum of voters by engaging in persuasive efforts that are targetable. Each individual voter is persuaded by campaign effort and votes for the candidate who targets more persuasive effort to this voter. Each candidate chooses a level of total campaign effort and allocates his effort among the set of voters. We completely characterize equilibrium for the majoritarian objective game and compare that to the vote-share maximizing game. If the candidates are symmetric ex ante, both types of electoral competition dissipate the rents from office in expectation. However, the equilibria arising under the two electoral objectives qualitatively differ. In majoritarian elections, candidates randomize over their level of total campaign effort, which provides support for the puzzling phenomenon of the emergence of supermajorities in majoritarian systems. Vote-share maximization leads to an equilibrium in which both candidates make deterministic budget choices and reach a precise fifty–fifty split of vote shares. We also study how asymmetry between the candidates affects the equilibrium. If some share of the voters is loyal to one of the candidates, then both candidates expend the same expected efforts in equilibrium, but the advantaged candidate wins with higher probability for majoritarian voting or a higher share of voters for vote-share maximization.     

New methods for analyzing structural models of labor force dynamics

The economic theory of decision-making under uncertainty is used to produce three econometric models of dynamic discrete choice: (1) for a single spell of unemployment; (2) for an equilibrium two-state model of employment and non-employment; (3) for a general three-state model with a non-market sector. The paper provides a structural economic motivation for the continuous time Markov (or more generally ‘competing risks’) model widely used in longitudinal analysis in biostatistics and sociology, and it extends previous work on dynamic discrete choice to a continuous time setting. An important feature of identification analysis is separation of economic parameters that can only be identified by assuming arbitrary functional forms from economic parameters that can be identified by non-parametric procedures. The paper demonstrates that most econometric models for the analysis of truncated data are non-parametrically under-identified. It also demonstrates that structural estimators frequently violate standard regularity conditions. The standard asymptotic theory is modified to account for this essential feature of many structural models of labor force dynamics. Empirical estimates of an equilibrium two-state model of employment and non-employment are presented.     

Bayesian inference for the mover–stayer model in continuous time with an application to labour market transition data

This paper presents Bayesian inference procedures for the continuous time mover–stayer model applied to labour market transition data collected in discrete time. These methods allow us to derive the probability of embeddability of the discrete‐time modelling with the continuous‐time one. A special emphasis is put on two alternative procedures, namely the importance sampling algorithm and a new Gibbs sampling algorithm. Transition intensities, proportions of stayers and functions of these parameters are then estimated with the Gibbs sampling algorithm for individual transition data coming from the French Labour Force Surveys collected over the period 1986–2000. Copyright © 2003 John Wiley & Sons, Ltd.     

Managerial Incentives Based on Acquisition of Information

In a moral hazard setting, we model the fact that the agent may get private signals about the final outcome of his effort before the public realization of this outcome. Actions affect both the distribution of the outcome and the quality of the agent's private information. We compare simple contracts, based on output only, with revelation contracts, based on output and messages about signals. Revelation contracts give the agent some discretionary power during the course of the relationship; they are optimal if and only if lowering effort does not increase the quality of private information in the sense of Blackwell (1953). In the context of managerial compensation schemes, the revelation contracts we analyze can be viewed as allowing the agent to exercise an option on the final profits before the realization of these profits. The theory thus provides an alternative justification of the widespread use of stock options in managerial compensation schemes, as opposed to compensation schemes that rely only on salary, bonus, and (restricted) stock plans.     

Optimal lottery

This article proposes an equilibrium approach to lottery markets in which a firm designs an optimal lottery to rank-dependent expected utility (RDU) consumers. We show that a finite number of prizes cannot be optimal, unless implausible utility and probability weighting functions are assumed. We then investigate the conditions under which a probability density function can be optimal. With standard RDU preferences, this implies a discrete probability on the ticket price, and a continuous probability on prizes afterwards. Under some preferences consistent with experimental literature, the optimal lottery follows a power-law distribution, with a plausibly extremely high degree of prize skewness.