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     

On Elimination in Contests: A Perspective on Output Maximization

This paper investigates the desirability of adding a preliminary elimination stage for output maximization in a winner‐take‐all contest framework in which the contestant who achieves the highest (random) output wins. We find that, generally, the desirability of an elimination stage does not monotonically depend on the productivity of the effort; adding a preliminary stage can improve output for both concave and convex production functions. This result contrasts sharply with current insight from effort maximization, which argues that adding a preliminary stage can increase effort supply only if the production function is concave.     

Incomplete information in rent-seeking contests

We consider a variant of the Tullock lottery contest. Each player’s constant marginal cost of effort is drawn from a potentially different continuous distribution. In order to study the impact of incomplete information, we compare three informational settings to each other; players are either completely informed, privately informed about their own costs, or ignorant of all cost realizations. For the first and the third setting, we determine the unique pure-strategy Nash equilibrium. Under private information, we prove existence of a pure-strategy Bayesian Nash equilibrium and identify a sufficient condition for uniqueness. Assuming that unit cost distributions all have the same mean, we show that under ignorance of all cost realizations ex ante expected aggregate effort is lower than under both private and complete information. Ex ante expected rent dissipation, however, is higher than in the latter settings if we focus on the standard lottery contest and assume costs are all drawn from the same distribution. Between complete and private information, there is neither a general ranking in terms of effort nor in terms of rent dissipation.     

Uncertain contest success function

In this article, contestants play with a certain probability in Contest A and with the complementary probability in Contest B. This situation is called contest uncertainty. In both contests, effort is additively distorted by a contest noise parameter which affects the sensitivity of the contest success function (CSF). In Contest A (B), this parameter is linearly added to (subtracted from) effort. We analyze the interaction of contest uncertainty and contest noise on contestant behavior and profit. For symmetric contestants, contest noise has an ambiguous effect on effort and profit. We show that more contest uncertainty can imply greater effort. Furthermore, an introduction of an infinitesimal degree of contest uncertainty can have a large impact on effort and profit. Based on the analysis, this article presents the contest organizer's incentive to manipulate the degree of uncertainty in the contest. For profit or effort maximization, the contest organizer should always eliminate any uncertainty. If contestants are asymmetric, more contest noise increases effort as well as competitive balance if both Contests A and B have the same probability of occurrence.     

Overbidding and overspreading in rent-seeking experiments: Cost structure and prize allocation rules

We study experimentally the effects of cost structure and prize allocation rules on the performance of rent-seeking contests. Most previous studies use a lottery prize rule and linear cost, and find both overbidding relative to the Nash equilibrium prediction and significant variation of efforts, which we term ‘overspreading.’ We investigate the effects of allocating the prize by a lottery versus sharing it proportionally, and of convex versus linear costs of effort, while holding fixed the Nash equilibrium prediction for effort. We find the share rule results in average effort closer to the Nash prediction, and lower variation of effort. Combining the share rule with a convex cost function further enhances these results. We can explain a significant amount of non-equilibrium behavior by features of the experimental design. These results contribute towards design guidelines for contests based on behavioral principles that take into account implementation features of a contest.     

Liquidity, Quality, Production Cost, and Welfare in a Search Model of Money

This paper examines a search model of money with divisible commodities of high and low quality, while keeping the assumptions of indivisible money and unit-inventory constraint. With no direct barter and a higher fixed cost of producing high relative to low quality, an increase in the money stock encourages the production of high-quality output by trading off the larger trading opportunities against the significance of higher fixed cost. As long as the fixed-cost differential between high and low quality is sufficiently small relative to the utility gain from high-quality consumption, the quality improvement outweighs the negative effect of higher money stocks on aggregate production, and hence implies higher welfare.     

The optimal multi-stage contest

This paper investigates the optimal (effort-maximizing) structure of multi-stage sequential-elimination contests. We allow the contest organizer to design the contest structure using two instruments: contest sequence (the number of stages, and the number of contestants remaining after each stage), and prize allocation. When the contest technology is sufficiently noisy, we find that multi-stage contests elicit more effort than single-stage contests. For concave and moderately convex impact functions, the contest organizer should allocate the entire prize purse to a single final prize, regardless of the contest sequence. Additional stages always increase total effort. Therefore, the optimal contest eliminates one contestant at each stage until the finale when a single winner obtains the entire prize purse. Our results thus rationalize various forms of multi-stage contests that are conducted in the real world.     

Contest architecture

A contest architecture specifies how the contestants are split among several sub-contests whose winners compete against each other (while other players are eliminated). We compare the performance of such dynamic schemes to that of static winner-take-all contests from the point of view of a designer who maximizes either the expected total effort or the expected highest effort. For the case of a linear cost of effort, our main results are: (1) If the designer maximizes expected total effort, the optimal architecture is a single grand static contest. (2) If the designer maximizes the expected highest effort, and if there are sufficiently many competitors, it is optimal to split the competitors in two divisions, and to have a final among the two divisional winners. Finally, if the effort cost functions are convex, the designer may benefit by splitting the contestants into several sub-contests, or by awarding prizes to all finalists.     

Asymmetric technologies in contests

I consider a contest between two risk-neutral players over a common-value prize, in which one player has a linear cost-of-effort function and the other a strictly convex cost-of-effort function f. I show that if the value of the prize is above (below) a certain threshold level, then the equilibrium aggregate effort in this contest is larger (smaller) than in a contest in which both players are characterized by the strictly convex cost-of-effort function f, and smaller (larger) than in a contest in which both players are characterized by a linear cost-of-effort function. Therefore, in contrast to the general result in the literature, asymmetry in contests can increase competition.     

Cost Signals under Uncertain R&D Outcomes

When it is difficult for firms to differentiate their products from those of their competitors, research and development (R&D) spending on process innovation to lower the cost of production is crucial for profitability. However, the information asymmetry in production costs that results from innovation reduces the efficiency of all firms in a market for a homogeneous good. We employ a signalling game to discuss the feasibility of utilising R&D spending and output levels as cost signals in an environment of quantity competition. The results show that a firm does not spend its money on R&D solely to signal the type of cost. Rather, R&D spending may be chosen as a cost signal over the output level only if expenditures on R&D can lead to a sufficiently high probability of reducing production costs.     

Entrepreneurship and productivity: The slow growth of the planned economies

Trends in gross domestic product (GDP) and total factor productivity (TFP) growth in the former socialist economies seem to indicate that these economies were converging to unusually low long-run growth rates in the late 1980s. In this paper we develop an endogenous growth model of entrepreneurship that is able to account for the difference in long-run performance between centrally planned economies and market-oriented ones. Long-run growth rates of output and productivity are determined by the growth of the stock of managerial knowledge, which in turn depends on the share of the population involved in entrepreneurial activities and on the time that spent on those activities. We analyze the effect of two characteristics of centrally planned economies on their growth performance. First, in centrally planned economies factors of production are distributed by the central planner to the firms' managers through a contest that uses up some of the managers' productive effort. Second, the leadership is “egalitarian,” in the sense that it treats individuals with different abilities equally. We show that these two features reduce the fraction of people becoming entrepreneurs/managers, as well as their managerial effort, which in turn reduces long-run output and TFP growth. Furthermore, we find that centrally planned economies have lower income inequality and slightly higher capital–output ratios, which is consistent with these countries' experiences.     

The Welfare Effects of Incentive Schemes

This paper computes the change in welfare associated with the introduction of incentives. We calculate by how much the welfare gains of increased output due to incentives outweigh workers' disutility from increased effort. We accomplish this by studying the use of incentives by a firm in the check-clearing industry. Using this firm's production records, we model and estimate the worker's dynamic effort decision problem. We find that the firm's incentive scheme has a large effect on productivity, raising it by 12% over the sample period for the average worker. Using our parameter estimates, we show that the cost of increased effort due to incentives is equal to the dollar value of a 5% rise in productivity. Welfare is measured as the output produced minus the cost of effort; hence, the net increase in the average worker's welfare due to the introduction of the firm's bonus plan is 7%. Under a first-best scheme, we find that the net increase in welfare is 9%.     

Multi-activity contests

In many contests, players can influence their chances of winning through multiple activities or “arms”. We develop a model of multi-armed contests and axiomatize its contest success function. We then analyze the outcomes of the multi-armed contest and the effects of allowing or restricting arms. Restricting an arm increases total effort directed to other arms if and only if restricting the arm balances the contest. Restricting an arm tends to reduce rent dissipation because it reduces the discriminatory power of the contest. But it also tends to increase rent dissipation if it balances the contest. Less rent is dissipated if an arm is restricted as long as no player is excessively stronger than the other with that arm. If players are sufficiently symmetric in an arm, both players are better off if that arm is restricted. Nevertheless, players cannot agree to restrict the arm if their costs of using the arm are sufficiently low.     

Loss aversion and the quantity–quality tradeoff

Firms face an optimization problem that requires a maximal quantity output given a quality constraint. But how do firms incentivize quantity and quality to meet these dual goals, and what role do behavioral factors, such as loss aversion, play in the tradeoffs workers face? We address these questions with a theoretical model and an experiment in which participants are paid for both quantity and quality of a real effort task. Consistent with basic economic theory, higher quality incentives encourage participants to shift their attention from quantity to quality. However, we also find that loss averse participants shift their attention from quality to quantity to a greater degree when quality is weakly incentivized. These results can inform managers of appropriate ways to structure contracts, and suggest benefits to personalizing contracts based on individual behavioral characteristics.     

In-house competition, organizational slack, and the business cycle

Multiplant firms pit their facilities against each other for production assignments. The present paper studies the consequences of this practice in a model where production is limited by capacity constraints and asymmetric information allows facilities to accumulate slack. It shows the amount of slack per unit of output to be pro-cyclical. Indeed, as capacity constraints become more acute in economic booms, the power of in-house competition for quota assignments is reduced and slack per unit of output increases, while the opposite is true in downturns. Moreover, in downturns firms may use higher cost facilities even when lower cost plants are not running at capacity since this boosts X-efficiency in low-cost plants.     

Experimental tests of Tullock’s contest with and without winner refunds

We examined experimentally the two-agent, complete-information Tullock’s contest, with and without refund for the winner. We find that the average bids in the refund group are higher than the average bids in the group without a refund, consistent with the theory. However, the auctioneer does not increase his profit if he changes the design of the contest by reimbursing the winner’s cost of effort. We also find underbidding for the low-valuation players and overbidding for the high-valuation player in a contest with a refund. Some players chose the corner solution of staying out of the game by biding zero.     

Repeated Contests with Asymmetric Information

The same contestants often meet repeatedly in contests. Behavior in a contest potentially provides information with regard to one's type and can therefore influence the behavior of the opponents in later contests. This paper shows that if effort is observable, this can induce a ratchet effect in contests: high ability contestants sometimes put in little effort in an early round in order to make the opponents believe that they are of little ability. The effect reduces overall effort and increases equilibrium utility of the contestants when compared with two unrelated one-shot contests. It does, however, also introduce an allocative inefficiency since sometimes a contestant with a low valuation wins. The model assumes an imperfectly discriminating contest. In an extension I show that, qualitatively, results are similar in a perfectly discriminating contest (all pay auction).     

THE DETERMINANTS OF THE QUANTITY‐QUALITY BALANCE IN MONOPOLY*

This paper describes how a monopolist manipulates the balance of quantity and quality in order to increase revenue when its customers treat quantity and quality as substitutes. This ‘skewing’ of quality depends on the characteristics of customer's demand for quality. Customers differ in demand for quality, because they differ in either (i) their preferences and/or (ii) their time cost per unit. The monopolist is constrained to supply the same quality of good to all customers. The price and quality per unit are described under the assumption the monopolist (i) profit maximises; (ii) maximises social welfare subject to a profit constraint. The determinants of the skewing of quantity and quality are found under third‐degree price discrimination and uniform pricing.     

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.     

A collaborating inventory model in a supply chain

The present article investigates an economic order quantity/ economic production quantity model in three-layer (manufacturer, vendor and retailer) supply chain management. In each stage, the products may undergo non-conforming quality items which have less value in the market. This model maximizes a collaborating expected profit function while production rate, order quantity, number of shipments with equal sizes are decision variables and unit production cost is a function of production rate. Numerical example is illustrated to test the model.