Gaming and strategic ambiguity in incentive provision

Multi-task principal-agent models e. When efforts on different tasks are technological substitutes for the agent, incentives on one task crowd out incentives on others, and as a result, even on easily measured tasks, optimal incentives may be low-powered.

Models of self-enforcing contracts e. In our model, the ambiguous incentive scheme that we term ex post discretion can be beneficial even when the performance measures are contractible. None of the models above studies the potential benefits of exogenous randomization, as is used in the scheme we term ex ante randomization. The potential benefits of exogenous randomization have also been explored in hidden-information models, especially those studying the design of optimal tax schedules.

Stiglitz and Pestieau, Possen, and Slutsky , among others, have shown that randomization can facilitate the screening of privately-informed individuals and is especially effective when private information is multi-dimensional.

In our hidden-action cum hidden-information setting, in contrast, ex ante randomization in fact eliminates the need for screening, as we show in Section 7. The costs and benefits of transparency in incentive design are also explored in Jehiel , Rahman , and Lazear Jehiel shows in an abstract moral hazard setup that a principal may gain by keeping agents uninformed about some aspects of the environment e. Rahman examines how a principal can provide incentives for an individual tasked with monitoring and reporting on the behavior of an agent.

He shows that randomization by the principal over what he asks the agent to do allows the principal to incentivize the monitor effectively. Finally, Lazear , in a model in which agents have no hidden information, explores high-stakes testing in education and the deterrence of speeding and terrorism, identifying conditions under which a lack of transparency can have beneficial incentive effects.

In the testing context, a crucial role is played by the exogenous restriction on the number of topics that can be tested, whereas in our model, even when all tasks can be measured and rewarded, deliberate ambiguity about the weights in the incentive scheme can be desirable.

The paper perhaps most closely related to ours is MacDonald and Marx Since task outcomes are binary, contracts are automatically linear in each outcome and specify at most four distinct payments. In this simple environment, it is possible to solve for the optimal contract, and they show that the more complementary the tasks are for the principal, the more the optimal reward scheme makes successes on the tasks complementary for the agent. In fact, their optimal outcome could be implemented using the scheme we term ex post discretion, which requires less commitment power and uses ex post payment schemes that are simpler, because they are separable in the task outcomes.

Section 2 outlines the model, and Section 3 analyzes the optimal deterministic incentive scheme. Section 4 studies ex ante randomization and ex post discretion. Section 5 identifies settings in which ambiguous contracts are dominated by deterministic schemes. Section 6, which is the heart of the paper, identifies environments in which optimally weighted ambiguous contracts dominate the best deterministic scheme.

Section 7 discusses extensions and robustness of our results, and Section 8 concludes. Proofs not provided in the text are in the appendix. The efforts chosen by the agent are not observable by the principal. In addition, at the time of contracting, the agent is privately informed about his cost of exerting efforts.

The two types of agent are assumed to have the same level of reservation utility, which we normalize to zero in certainty-equivalent terms. Incentive schemes will be compared according to the expected payoff generated for the principal. Below we consider a variety of incentive schemes. Thus the agent is uncertain at the time he chooses his efforts about which performance measure will be more highly rewarded, and by varying the level of k, the principal can affect how much this uncertainty matters to the agent.

In this case, the optimal deterministic linear contract can take one of two possible forms. The second form of contract rewards performance on only one task which we take to be task 1 below and so induces the agent to exert effort on only one task, and it uses performance on the other task to provide insurance for the agent, by exploiting the correlation between the shocks to the performance measures.

The SD contract induces the agent to exert effort on both tasks, while the OT contract elicits effort on only one task. This benchmark is important because, as we will see, there are environments in which optimally designed ambiguous contracts allow the principal, even in the presence of hidden information, to achieve a payoff arbitrarily close to that achievable in this benchmark.

In choosing between C1bal , C2bal and C1OT , C2OT in the NHI setting, the principal faces a trade-off between the more balanced efforts induced by the former and the lower risk cost imposed by the latter. In principle, three possible patterns of effort could emerge: a both types of agent exert balanced efforts, b one type exerts balanced efforts and the other type focused effort, or c both types exert focused effort.

Relative to an SDM, an ADM has the benefit of inducing one type of agent to choose balanced efforts, but it imposes more risk on that agent type and also necessitates leaving rents to the other type. The reason is the rents that hidden information forces the principal to leave to one agent type when he uses the ADM. This discontinuous drop reflects the impossibility, for even a small degree of privately-known preference across tasks, of inducing balanced efforts from both types with a deterministic scheme.

Hence in this case, both types of agent would optimally exert effort only on their preferred task. Finally, the smaller is the parameter k, the more different are the two possible compensation schedules and the more costly is the risk imposed by the randomization, so the smaller is the optimal effort gap. Thus EAR is more robust to the introduction of private information on the part of the agent than is the best deterministic scheme.

The performance of ex ante randomization does not display this extreme sensitivity. We can prove that under interim randomization, the unique Bayes-Nash equilibrium is the same as the outcome described in Proposition 2. This results in the most balanced profile of effort choices, assessed ex ante, and also avoids leaving any rent to either type of agent. The effort-balancing incentives generated by EAR do, however, come at a cost in terms of the risk imposed on the risk-averse agent.

Equations 5 and 6 show that increasing k has three effects. Second, a larger k, because it induces the agent to choose less balanced efforts, raises the cost of compensating the agent for the risk imposed by the randomization per se.

Finally, a larger k reduces the cost per unit of aggregate effort induced of the risk imposed on the agent from the shocks to measured performance. In general, the optimal design of a contract with EAR involves a trade-off between these three different effects. Weighting the different performance measures more equally in the two possible compensation schedules is costly in terms of effort balance and thereby in terms of the risk imposed by the randomization, but is helpful in allowing better diversification of the measurement errors.

The next proposition describes how the optimal value of k varies with several parameters of the contracting environment and with the level of aggregate effort to be induced. Furthermore, the optimal level of k is smaller, the smaller is B i. In Section 6, where we identify environments where ambiguous schemes outperform deterministic ones, we will build on these results.

In Section 7. For now, though, we turn to a second class of ambiguous contracts. With EPD, as with EAR, the closer k is to 1, the more similar are the two possible compensation schedules, and if k were equal to 1, EPD would involve no discretion at all and would collapse to the SD contract. Comparing Propositions 4 and 2 reveals important similarities, as well as important differences, between the incentives and payoffs generated by EPD and EAR. Part i holds in each case for exactly the same reason.

Just as for EAR, the first-order conditions for interior optimal efforts then imply equation 4. Under EPD, this insurance motive is still present, because at the time the agent chooses efforts, he is uncertain about which compensation schedule the principal will select.

These results parallel those for EAR. What is the cost of the risk imposed by ex post discretion on the agent, and how does it compare to that imposed by ex ante randomization? Section 4. As with EAR, increasing k has three distinct effects. On the other hand, a larger k improves the diversification of the measurement errors.

This section identifies three environments in which both types of ambiguous incentive scheme are strictly dominated by a deterministic contract. In general, therefore, the principal faces a trade-off in choosing between ambiguous and deterministic incentive schemes.

Ambiguous schemes are typically better at inducing balanced efforts, while deterministic schemes have the advantage of imposing lower risk costs on the agent per unit of aggregate effort induced. The three conditions identified in Proposition 6 are ones under which this trade-off does not in fact arise.

Optimal primaries more. There are two candidates, one of whom is a higher quality candidate. Voters reside in m different states and receive noisy private information about the Voters reside in m different states and receive noisy private information about the identity of the superior candidate. States vote in some order, and this order is chosen by a social planner. We provide conditions under which the ordering of the states that maximizes the probability that the higher quality candidate is elected is for states to vote in order from smallest to largest populations and most accurate private information to least accurate private information.

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The Nature of Tournaments more. This paper characterizes the optimal way for a principal to structure a rank-order tournament in a moral hazard setting as in Lazear and Rosen With considerable generality, it is optimal to give rewards to top performers that With considerable generality, it is optimal to give rewards to top performers that are smaller in magnitude than corresponding punishments to poor performers.

The paper also identifies four reasons why the principal might prefer, instead, to give larger rewards than punishments: i risk aversion R is small relative to absolute prudence P ; ii the distribution of shocks to output is asymmetric and the asymmetry takes a particular form; iii the principal faces a limited liability constraint; and iv there is agent heterogeneity of a particular form.

Intuition is given as to why these factors affect the optimal prize schedule. This paper uses the theory developed by Green and Stokey to relate the results about tournaments to the structure of the optimal individual contract.

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We show that random or ambiguous incentive schemes induce more balanced efforts from an agent who performs multiple tasks and who is better informed about the environment than the principal is. On the other hand, such random schemes impose more risk on the agent per unit of effort induced.

By identifying settings in which random schemes are especially effective in inducing balanced efforts, we show that, if tasks are sufficiently complementary for the principal, random incentive schemes can dominate the best deterministic scheme.