Generalized contest success functions

The key element of models of contest is the contest success function (CSF) which specifies the winning probabilities of agents. The existing axiomatizations of CSFs assume that contestants can make only one type of investment. This paper generalizes these axiomatizations to the case where each agent can have multiple types of investments. This allows us to provide a unified framework to extend and interpret the results of Skaperdas (Econ Theory 7:283–290, 1996) and Clark and Riis (Econ Theory 11:201–204, 1998), and rationalize some seemingly ad hoc CSFs used by applied researchers. This paper has benefitted from the comments of Aicke Hinrichs, Rene Levinsky, and two anonymous referees. We are especially thankful to one of the referees for detailed comments which have helped improve every aspect of this paper.     

Contests where there is variation in the marginal productivity of effort

Summary. We provide a characterization of participants' behavior in a contest or tournament where the marginal productivity of effort varies across contestants and individual productivity is private information. We then consider the optimal design of such a contest. We first analyze contestant behavior for the usual type of contest, where the highest output wins. Abilities need not be independently distributed. We demonstrate that there is a unique symmetric equilibrium output function, that output is increasing in ability, and that marginal effort is increasing in ability, while effort decreases when the cost of effort increases. Next we consider the case where the highest output need not win, with independently distributed abilities. We analyze the contest designer's decisions in choosing contest rules optimal from her perspective. We show that the output produced, probability of winning, and contest designer's expected revenue are generally increasing in contestants' ability. We examine the relationship between the marginal cost of producing output and marginal utility per dollar of the net award for winning. Received: July 30, 1998; revised version: August 7, 2000     

Doping and cheating in contest-like situations

Individuals who compete in a contest-like situation (for example, in sports, in promotion tournaments, or in an appointment contest) may have an incentive to illegally utilize resources in order to improve their relative positions. We analyze such doping or cheating within a tournament game between two heterogeneous players. Besides the direct and indirect costs of doping, three major effects are identified which determine a player's decision to deviate from a no-doping situation — a cost effect, a likelihood effect and a base-salary effect. Moreover, the influence of heterogeneity and exogenous performance risk on a no-doping outcome, and the influence of ex-ante and ex-post doping tests on the players' investments are discussed.     

Foundations for contest success functions

In the literature, the outcome of contests is either interpreted as win probabilities or as shares of the prize. With this in mind, we examine two approaches to contest success functions (CSFs). In the first, we analyze the implications of contestants’ incomplete information concerning the ‘type’ of the contest administrator. While in the case of two contestants this approach can rationalize prominent CSFs, we show that it runs into difficulties when there are more agents. Our second approach interprets CSFs as sharing rules and establishes a connection to bargaining and claims problems which is independent of the number of contestants. Both approaches provide foundations for popular CSFs and guidelines for the definition of new ones.

“The strategic approach also seeks to combine axiomatic cooperative solutions and non-cooperative solutions. Roger Myerson recently named this task the ‘Nash program’.”(Rubinstein 1985, p. 1151)

     

Rent-setting in multiple winner rent-seeking contests

In this paper, we analyze a multiple winner rent-seeking contest where the number of winners is set by a self-interested regulator. The winners receive a license to compete in a market. The structure of competition in the market influences the number of winners through the preferences of the regulator. The model indicates that Cournot competitors can be better off than firms that are able to collude on output determination.     

Subsidizing Technological Innovations in the Presence of R&D Spillovers

Abstract. We analyze a situation where a principal wants to induce two firms to produce an output, for example electricity from renewable energy sources. Firms can undertake non-contractible investments to reduce production cost of the output. Part of these investments spills over and also reduces production cost of the other firm. Comparing a general price subsidy and an innovation tournament, we find that the principal's expected cost of implementing a given expected output is always higher under the tournament, even though this scheme may lead to more innovation.     

Reciprocity in the shadow of threat

This paper considers a partial equilibrium model of conflict where two agents differently evaluate a contested stake. Differently from common contest models, agents have the option of choosing a second instrument to affect the outcome of the conflict. The second instrument is assumed to capture positive investments in ‘conflict management’—labeled as ‘talks’. The focus is on the asymmetry in the evaluation of the stake: whenever the asymmetry in the evaluation of the stake is large there is no room for cooperation and a conflict trap emerges; whenever the degree of asymmetry falls within a critical interval, cooperation seems to emerge only in the presence of a unilateral concession; as the evaluations of the stake converge, only reciprocal concessions can sustain cooperation. Finally the concept of entropy is applied to measure conflict and conflict management.     

The effect of power imbalances on incentives to make non-contractible investments

We use an experiment to study the effect of ex-post sharing rules on relationship-specific investments in an incomplete contracting context. We find that no power structure can induce first-best investments and that equally productive partners reach more efficient outcomes with a balanced power structure (i.e., equal sharing of returns) than with an asymmetric one. In addition, we find evidence for behavioural effects: partners make higher investments and reach higher efficiency levels than own-payoff maximisation would suggest. This behaviour is in line with a model where decision-makers care about social efficiency. It is not consistent with inequity-averse preferences.     

Rent-seeking with uncertain discriminatory power

We consider a two-player Tullock rent-seeking contest with uncertain discriminatory power in the contest success function. We examine the cases where both players are either informed or uninformed about the size of the discriminatory power, as well as the case where only one player has private information about it. We show that in all three cases the contest has a unique (Bayesian) Nash equilibrium. In each case we characterize key properties of the equilibrium.     

Information in conflicts

We consider two-player contests for a prize of common but uncertain value. For settings where one player knows the value of the prize, while the other only knows its prior distribution, we give conditions for when the uninformed agent is ex ante strictly more likely to win the prize than is the informed agent. In the special case of a lottery contest, equilibrium expenditures are lower under asymmetric information than if either both agents are informed or neither agent is informed.     

Heterogeneity,impatience, and dynamic private provision of a discrete public good

We analyze dynamic private provision of a discrete public good by heterogeneous agents, who differ in terms of their levels of impatience, in a differential game framework. In contrast to the strategic complementarity result for homogeneous individuals, we show that an asymmetric completion Markov perfect equilibrium exists, where the individual contributions and the strategic behaviors depend crucially on an impatience differential: the difference in rates of time preference across groups of individuals. When this differential is insignificant, contributions of both types of individuals are strategic complements. On the other hand, when this impatience differential exceeds a threshold, the contributions of impatient individuals become strategic substitutes, whereas the contributions of patient individuals remain strategic complements. We show that group size has an interesting role to play in the strategic behavior: increasing the number of patient individuals aggravates the incentives to free‐ride by the impatient agents. We also derive a condition under which all the socially beneficial projects get completed in the equilibrium.     

Contest design and heterogeneity

I show that the effect of heterogeneity on contest investments depends on the structure of the competition, which implies that heterogeneity matters for optimal contest design. This insight helps to explain empirical evidence on the comparison of different contest structures.     

An investment contest to influence environmental policy

In an investment contest for environmental policy, polluters and victims of pollution invest in an increase of their marginal benefits of pollution and environmental quality, respectively. These investments influence time-consistent environmental policy. Investments will exceed their optimal level. The more victims there are, the lower aggregate investment, the lower payoffs for the victims and the higher payoffs for the polluters. The more polluters there are, the higher aggregate investment and the lower payoffs for polluters and victims. Asymmetries between polluter and victim result in less overinvestment and bias environmental policy in favour of the less productive side.     

Financial markets,the structure of long-term investments and labour income risks

The paper develops a model to analyse the feedback between financial markets, long-term capital investments and the risk of labour incomes. We study a situation where firms are owned by entrepreneurs, who are able to share and diversify their income risk by trading on financial markets. Workers, in contrast, cannot short-sell the flows from future labour endowments and thus do not have the same opportunities. We derive two central results. Firstly, even if financial markets offer perfect risk-sharing opportunities for entrepreneurs, the participation restriction for labour incomes leads to a constrained inefficient market allocation. The constrained inefficiency arises because the effect of long-term investments on the risk of wages is not internalized by state prices. Secondly, we show that in general it is not true that workers indirectly benefit when we go from a situation with no financial markets to a situation with perfect financial markets for entrepreneurs but restricted participation for workers. The results suggest that a policy solution might require either to close some financial markets or to create new ones. We argue why there is a strong case for the creation of new markets rather than for closing existing ones.     

When less is more: Rationing and rent dissipation in stochastic contests

This paper shows how to maximize revenue when a contest is noisy. We consider a case where two or more contestants bid for a prize in a stochastic contest where all bidders value the prize equally. We show that, whenever a Tullock contest yields under-dissipation, the auctioneer?s revenue can be increased by optimally fixing the number of tickets. In particular, in a stochastic contest with proportional probabilities, it is possible to obtain (almost) full rent dissipation. We test this hypothesis with a laboratory experiment. The results indicate that, as predicted, revenue is significantly higher in a lottery with rationing than in a standard lottery. On the other hand, an alternative rationing mechanism that does not limit total expenditures fails to increase revenue relative to a standard lottery.     

Contests with size effects through costs

Eggert and Kolmar [Eggert, W., Kolmar, M., in press. Contests with size effects. European Journal of Political Economy.] examine the effects of group size on individual and aggregate payoffs in contests where prizes increase in the number of players. In a reconsideration of their paper, this note examines contests in which the unit cost of effort decreases in the number of players. In contests for fixed prizes, an increase in the number of players decreases individual and aggregate payoffs. In contests in which players face a trade-off between appropriative and productive investments, an increase in the number of players decreases individual payoffs while increasing aggregate payoffs.     

Inter-firm knowledge diffusion, market power, and welfare

We propose an infinite-horizon quantity-setting differential game with learning spillovers and organizational forgetting to analyze the optimal management decisions affecting the evolution of the stock of know-how, and, in turn, the dynamics of productive efficiency. Specifically, we study the long run impact of inter-firm knowledge diffusion on market power, i.e. the ability of a firm to raise the price above the marginal cost, and welfare. We consider two types of processes through which knowledge is acquired: (i) passive learning, or learning-by-doing, where managers do not actively invest in information and (ii) active learning, or learning-by-investing, where managers acquire new and additional information through specific investments in human capital. We show that: under (i), knowledge diffusion reduces market power; under (ii), knowledge diffusion reduces market power as long as learning spillovers are sufficiently important. From a welfare viewpoint, we also show that: under (i), knowledge diffusion is always welfare-enhancing; under (ii), weak spillovers are required in order for knowledge diffusion to be welfare-enhancing.     

Investment in the absence of property rights; the role of incumbency advantages

In many situations the individuals who can generate some output must enter a contest for appropriating this output. This paper analyses the investment incentives of such agents and the role of incumbency advantages in the contest. Depending on the advantages, an increase in the productivity of the investment can decrease or increase the amount of investment. The results are applied to autocrats’ investment behavior and job specific investment in organizations.     

Competition and innovation: an experimental investigation

The paper analyzes the effects of more intense competition on firms’ investments in process innovations. More intense competition corresponds to an increase in the number of firms or a switch from Cournot to Bertrand competition. We carry out experiments for two-stage games, where R&D investment choices are followed by product market competition. An increase in the number of firms from two to four reduces investments, whereas a switch from Cournot to Bertrand increases investments, even though theory predicts a negative effect in the four-player case. The results arise both in treatments in which both stages are implemented and in treatments in which only one stage is implemented. However, the positive effect of moving from Cournot to Bertrand competition is more pronounced in the former case.     

Anticipation effects of technological progress on capital accumulation: a vintage capital approach

Due to embodied technological progress new generations of capital goods are more productive. Therefore, in order to study the effects of technological progress, a model must be analyzed in which different generations of capital goods can be distinguished. We determine in what way the firm adjusts current investments to predictions of technological progress. In the presence of market power we show that a negative anticipation effect occurs, i.e. current investments in recent generations of capital goods decline when faster technological progress will take place in the future, because then it becomes more attractive to wait for new generations of capital goods. In case that only investments in new machines are possible, actually a whole wave of anticipation phases arises.