Handicaps to improve reputation

Abstract

An agent may be able to address a task at different times, with the state of nature more favorable to the task in some periods than in others. Success on a task will therefore more greatly improve the agent’s reputation if he is constrained in choosing when to address the task than if he enjoys flexibility in timing. These considerations can explain why presidents emphasize achievements in their first 100 days in office, and why performance of the economy in only some periods of a president’s term affect elections.

Keywords

Reputation principal-agent policy making

1. Introduction

Why should voters care about unemployment or economic growth in the election year, ignoring other years? This paper offers an explanation. Suppose that an incumbent aims to signal his ability in controlling the economy. Voters may attribute good performance of the economy in an arbitrary period to luck, or to a good state of nature. But if the incumbent says he will stimulate the economy in December of each year, or in an election year, or in some other constrained period, then he sets up a test which allows him to demonstrate his ability. A similar rationale can explain why new presidents are eager to accomplish much in their first 100 days in office.¹ Indeed, many journalists saw President Carter as a failure because he achieved little in his first 100 days compared with Roosevelt’s achievements (Rozell, 1989, 40). And some presidents see achievement in the first 100 days as important, as when President Clinton pledged to have the most productive 100-day period in modern history (Gergen, 1993).

The ideas discussed below build on the insight by the evolutionary biologists Zahavi and Zahavi (1997): weak individuals are more likely captured by predators than are strong individuals, so that ornaments or handicaps which hurt the weak more than they hurt the strong can increase the difference in predation rates. So the stronger ones will choose to produce larger ornaments that handicap them, signaling their strength to potential predators. For example, a gazelle which sights a wolf and jumps high into the air several times before fleeing signals that it is a swift runner, easily able to outrun the wolf, and so discourages the wolf from chasing it. The cost a peacock incurs in carrying its elaborate and weighty tail-feathers, which interfere with food gathering, signals to potential mates that it is especially fit to provide for its offspring. Similarly, a master chess player proves his ability by playing with some missing pieces, or even playing blind. A player who wins under such handicaps must be exceptionally good.

This paper makes two contributions. The first is to show that agents who care about their reputations may choose to handicap themselves by working on tasks in a pre-defined sequence instead of working on them at the times which would be most effective. The second contribution lies with examining what handicaps would maximize the welfare of a principal who can replace an agent who has been revealed to have low quality.

2. Literature

A phenomenon this paper addresses is that voters evaluate an incumbent’s performance not over his full term of office, but over some more limited period. The introduction already mentioned the importance of a president’s first 100 days. Evidence also suggests that votes for president were best predicted by per capita change in GNP in the second quarter of the election year (Fair, 1978). Conditions in the rest of a president’s term are mostly irrelevant, with economic conditions in quarter 13 of a president’s term having half the effect of economic conditions in quarter 14 on the presidential election, and economic conditions in quarters 9–12 having about a sixth of the effect (Bartels, 2012).

An incumbent may increase his chances of winning election by pandering to the public, taking actions the public may incorrectly believe are best (Maskin and Tirole, 2004; Smart and Sturm, 2013). If a project will likely fail even under a skilled leader, a leader (whether skilled or not) may prefer projects likely to fail over projects likely to succeed (Majumdar and Mukand, 2004). Indeed, an incumbent with a bad reputation may favor a highly risky policy: if the policy fails, he would have lost the next election anyway, but if the policy succeeds his reputation and so his chances of re-election improve. This idea is applied by Hess and Orphanides (1995) to claim that a president with a bad reputation who goes to war gets an opportunity to improve his reputation. Relatedly, if voters learn about a politician’s ability from the performance of a new project he undertakes, then an incumbent ignorant of his own ability will adopt too many projects if he is at risk of losing re-election; he will adopt too few projects if he is likely to win re-election (Biglaiser and Mezzetti, 1997).

Reputational concerns may lead a politician to terminate a policy that he, but not the voters, knows succeeded (Beniers and Dur, 2007). Reputational concerns can also give rise to political correctness: an adviser who wishes to avoid a reputation for bias may not truthfully reveal his information (Morris, 2001).

3. Assumptions

3.1. Agents

The agent can work on two tasks, or issues, A and B. The outcome on a task is either success or failure. The agent holds office for at most two terms, each with two periods. The initial focus is on the first term. In each period, the agent works on only one task, A or B. Success on a task in one period makes the benefit of working on the same task in a later period equal to zero.

An agent has either high-ability (an H-type) or low ability (an L-type). The prior probability that the agent has high ability is h. A high-ability agent succeeds on each task regardless of the state of nature. A low-ability agent succeeds on a task only if the state of nature is favorable to that task in the period in which he works on it. The state of nature is favorable with probability f, with independent draws of this probability in each period. A bad outcome in any period indicates that the agent has low ability.²

3.2. Principal

The principal observes the outcomes of the agent’s actions, and observes the task the agent addressed in each period. The principal observes neither the state of nature nor the agent’s ability.

The timeline for the model with one term follows.

In period 1:

nature determines the agent’s ability;

state of nature affecting task is determined;

agent works on task A or task B.

In period 2:

state of nature affecting task is determined;

agent works on the task not previously addressed, or on the task addressed with failure;

outcomes of tasks realized;

principal estimates ability of agent.

4. Reputation and output over two periods

In a model with one term and two periods, there are four possible patterns of handicaps. (a) Full flexibility (or no handicap) in any period. (b) Specification of which task to address in period 1, allowing the agent to choose the task he addresses in period 2. (c) Specification of which task to address in period 1 and which task to address in period 2. (d) Specification of which task to address in period 2, allowing the agent to choose the task he addresses in period 1. As will be claimed below, strategy (b) dominates strategies (c) and (d), so most of the analysis will consider only strategies (a) and (b).³

4.1. Flexible timing

Consider the agent’s performance when he can choose the task to address in each period, or when he enjoys flexible timing. In period 1 the agent observes one of the four possible states of nature: favorable to both A and B, favorable to A but not to B, and so on. With probability 1 − (1 − f)² the state of nature favors at least one task. Without loss of generality, call the task the state of nature favors, if it favors any, task A. With probability (1 − f)² the state of nature is unfavorable to both tasks.

The probability at the end of period 2 that the principal believes the agent has high ability is determined as follows. A necessary condition for believing the agent has high ability is that the agent succeeded in both periods, in which case the probability that the agent has high ability is

pr (H | GG) \equiv \frac{pr (GG | H) pr (H)}{pr (GG | H) pr (H) + pr (GG | L) pr (L)} .

(1)

The probability of GG|L is the probability that a low-ability agent will succeed in both periods. He will succeed in period 1 if the state of nature favors either task A or task B. That event occurs with probability 1 − (1 − f)². In period 2, the agent works on the task not addressed in period 1. The probability that the state of nature favors success in period 2 on that task is f. Thus, the probability that a low-ability agent will succeed on both tasks, given that in each period he chooses the task on which to work is (1 − (1 − f)²)f.

It follows that the probability that, under flexible timing, an agent who succeeded on both tasks has high ability is

\frac{h}{h + ((1 - {(1 - f)}^{2}) f) (1 - h)} .

(2)

Consider next output. Success on a task yields output G; failure yields output 0. The output of a high-ability agent over the two periods is 2G.

Expected output of a low-ability agent over the two periods is calculated by considering all possible states of nature in period 1. With probability (1 − f)² nature favors no task in period 1, and so in period 1 the agent fails. In period 2 the agent succeeds if nature favors at least one of the tasks, in which case the output is G. So with probability (1 − f)²(1 − (1 − f)²) output in period 1 is 0 and output in period 2 is G

With probability 2f(1 − f) nature in period 1 favors exactly one of the tasks. So in period 2 the agent works on the other task, and expected output over the two periods is G + fG

With probability f² in period 1 nature favors both tasks. Expected output over the two periods is then again G + fG. Combining these cases, expected output over the two periods by a low-ability agent who is unconstrained is

\begin{matrix} {(1 - f)}^{2} (1 - {(1 - f)}^{2}) G + 2 f (1 - f) (G + f G) + f^{2} (G + f G) = \\ f G (- f^{3} + 3 f^{2} - 4 f + 4) . \end{matrix}

(3)

4.2. Constrained timing for one period

Consider an agent who must work on task A in period 1. If he failed in period 1, then in period 2 he can work on whichever task the state of nature favors in period 2. If he succeeded in period 1, then in period 2 he works on task B. Note that giving the agent flexibility in period 2 dominates giving him flexibility in period 1, because the flexibility in period 2 allows the agent to work on the task on which he failed in period 1 if the state of nature favors that task in period 2.

And note that the reputation of the agent is the same when he is constrained only in period 1 as when he is constrained in both periods. The reason is that the agent takes advantage of the flexibility in period 2 only if he failed on task A in period 1. But if he failed in period 1, then he has revealed himself to have low ability. The flexibility in period 2 allows him to have greater expected output in period 2, but success in period 2 following failure in period 1 has no effect on his reputation.

The probability that, under constrained timing, an agent who succeeded on both tasks has high ability is

\frac{h}{h + f^{2} (1 - h)} .

(4)

Expected output over two periods of a low-ability agent who must work on task A in period 1 is

\begin{matrix} f (G + f G) + (1 - f) (1 - {(1 - f)}^{2}) G = f G (f^{2} - 2 f + 3) \end{matrix} .

(5)

4.3. Comparing outputs and reputations

The posterior probabilities that an agent who succeeded on both tasks has high ability is given by (4) for constrained timing, and by (2) for flexible timing. The numerators of the two expressions are the same, but the denominators differ. The difference in the denominators is h + f²(1 − h) − (h + ((1 − (1 − f)²)f)(1 − h)) = f²(1 − h)( f − 1), which is negative. That is, the second denominator is greater than the first, making the posterior probability under flexible timing smaller. In other words, a high-ability agent would gain a better reputation under constrained timing than under flexible timing.

For a given h, the difference in reputation for an agent who succeeded on both tasks has the rough shape of a mountain. For G = 1 and h = 1/2, the difference reaches a maximum at f = 1/2, and the difference is about 7 percentage points. When h = 0.1, the difference reaches a maximum at f = 0.25, and the difference is 13 percentage points. When h = 0.9, the difference reaches a maximum at f = 0.65, and the difference is 1.4 percentage points. So a handicap is most useful in improving a high-ability agent’s reputation for low values of h, but not for particularly low values of f.

Therefore, a high-ability agent would favor the handicap: he knows he will succeed on both tasks, which given constrained timing improves his reputation.⁴ Using the intuitive criterion strengthens the result. For suppose agents commit to working on task A in period 1. A high-ability agent has no incentive to deviate. But a low-ability agent can improve his performance by deviating. Therefore, a rational principal would infer that an agent who works on task B rather than A in period 1 has low ability, thereby inducing agents who care about their reputations to constrain their timing.

In other words, avoidance of handicapping is not a credible commitment by an incumbent: once in office a high-ability agent has an incentive to reveal his type, and so has an incentive to handicap himself. That in turn makes it rational for a low-ability incumbent to attempt to hide his type by also handicapping himself.

We now turn to comparing outputs. A high-ability agent always succeeds, regardless of the state of nature. Expected output of a low-ability agent under flexible timing minus expected output under constrained timing is

f G (- f^{3} + 3 f^{2} - 4 f + 4) - (f G (f^{2} - 2 f + 3)),

(6)

which (for G > 0) is positive for all 0 < f < 1. When f = 0 or f = 1, the difference is zero. The difference reaches its maximum at f = 1/2. Output is never higher under constrained timing than under flexible timing, and for a broad range of parameters is lower.

We thus see a trade off between output and reputation: flexibility increases the output of a low-ability agent, but reduces the reputation of a high-ability agent. If the principal cares most about discovering the agent’s ability, he will favor constrained timing. If he cares most about performance of the current agent, he will favor flexible timing. The principal, say voters, may then well prefer an incumbent who does not impose handicaps on himself, and so will perform well when in office. But the incumbent has an incentive to constrain himself.

5. Replacing an agent

Consider now two terms, each with two periods, and the possibility of replacing an agent at the end of the first term. Constrained timing allows the principal to estimate more accurately the ability of an agent in office in the first term; the principal can more likely replace a low-ability agent and so improve performance in the second term.⁵ But constrained timing has the disadvantage of reducing the average performance of low-ability agents. Because the second term is the terminal one, there is no benefit of constraining an agent in that term, and so I assume that the agent is flexible in the second term. That assumption allows a low-ability agent to perform better in the second term than if he is constrained, and therefore reduces the benefits of constrained timing in the first term.

5.1. Replacement with flexible timing

Let an agent have flexibility in all periods of both terms. In period 2 of each term he can work on a task on which he failed in period 1 of that term. But, for simplicity, suppose that in each term he must start afresh; the tasks addressed in the first term are no longer relevant in the second term.

The timeline follows.

In period 1 of the first term:

nature determines the agent’s ability;

state of nature affecting task is determined;

agent works on task A or task B.

In period 2 of the first term:

state of nature affecting task is determined;

agent works on task not previously addressed, or on the task previously addressed with failure;

outcomes of tasks realized;

principal replaces an agent revealed to have low ability.

In period 1 of the second term:

nature determines the ability of a new agent, if appointed;

state of nature affecting task is determined;

agent works on task A or task B.

In period 2 of the first term:

state of nature affecting task is determined;

agent works on task not previously addressed, or on the task previously addressed with failure;

payoffs realized.

Consider outcomes in the first term (consisting of two periods) when the agent is unconstrained. With probability h the agent in the first term has high ability, he performs well on both tasks, is re-appointed, and performs well on both tasks in the second term. A good outcome yields a gain of G; the gain from a bad outcome is assumed to be 0. Then output under a high-ability agent over the four periods in the two terms is 4G.

With probability 1 − h the agent in the first term has low ability. He will be reappointed if he succeeded on both tasks in the first term. That occurs with probability (1 − (1 − f)²)f, and the gain in the second term is again fG(2 − f)(f² − f + 2).

An agent in the first term who had failed on at least one task is revealed to have low ability, so is replaced. His replacement has high ability with probability h, and low ability with probability 1 − h. So when the agent in his first term fails at least once, expected output in the second term is h2G + (1 − h)(G(1 − (1 − f)²) + fG).

Combining all of these terms, expected output over the two terms when timing is flexible in the first term is

\begin{matrix} h 4 G + \\ (1 - h) \\ [ \\ f G (- f^{3} + 3 f^{2} - 4 f + 4) + \\ (1 - {(1 - f)}^{2}) f (f G (- f^{3} + 3 f^{2} - 4 f + 4)) + \\ (1 - ((1 - {(1 - f)}^{2}) f)) (h 2 G + (1 - h) (f G (- f^{3} + 3 f^{2} - 4 f + 4))) \\ ] . \end{matrix}

(7)

5.2. Replacement with constrained timing for one period

Now consider constrained timing in the first term, so that in period 1 of the first term the agent must work on task A. If he succeeded in period 1, then in period 2 he works on task B. If he failed in period 1, then in period 2 he works on the task (A or B), if either, which the state of nature favors.

A high-ability agent will succeed on all tasks, generating output of 4G over the four periods of the two terms.

Turn now to a low-ability agent in the first term. With probability f² a low-ability agent succeeds on both tasks in the first term and is reappointed. His expected output in the second term is G(1 − (1 − f)²) + fG.

An agent is replaced if he failed in period 1 or if he failed in period 2 given that he succeeded in period 1. The probability a low-ability agent fails in period 1 when he is constrained in period 1 is (1 − f). The probability a low-ability agent who succeeds in period 1 fails in period 2 is the probability that no state of nature favors a task in period 2; this probability is (1 − f)².

With probability 1 − f² a low-ability agent in the first term fails on at least one task, and so is replaced. Expected output by the new agent in the second term is then h2G + (1 − h)(fG(2 − f)(f² − f + 2)).

An agent who failed on his task in period 1 chooses in period 2 to work on the task that best matches the state of nature.⁶ With probability (1 − f)² he failed in period 1, and so in period 2 he can work on task A or B, yielding success with probability (1 −(1 − f)²). Expected output over the two periods of the first term for a low-ability agent who is constrained to work on task A in period 1 was given by (5).

Expected output over the two terms when timing is constrained in period 1 is thus

\begin{matrix} h 4 G + \\ (1 - h) \\ [ \\ f G (f^{2} - 2 f + 3) + \\ f^{2} (f G (- f^{3} + 3 f^{2} - 4 f + 4)) + \\ (1 - f^{2}) (h 2 G + (1 - h) (f G (- f^{3} + 3 f^{2} - 4 f + 4))) \\ ] . \end{matrix}

(8)

5.3. Comparing outputs

Comparing outputs under flexible and constrained timing, it is clear that for h = 0, flexibility is best: flexibility in timing improves the performance of a low-ability agent, and no high-ability agents are to be found. For h = 1, it matters not whether the agent is constrained or flexible: he will do well in either case. For f = 1, an agent of any type will always succeed on the task he addresses, so outcomes are identical under constrained and flexible timing. For f = 0, a low-ability agent always fails regardless of timing, so outcomes under flexible and constrained timing are again the same.

For less-extreme values of f and h, consider the difference between (7) and (8):

f G (1 - h) (f - 1) (f^{5} h - 3 f^{4} h + 4 f^{3} h - f^{2} (4 h + 1) + f (2 h + 1) - 1) .

(9)

Numerical solutions show that the value of (9) is positive: flexible timing yields greater expected output, and the benefits of flexible timing are greatest when h is small. Nevertheless, an incumbent in period 1 who cares about his reputation or about reappointment has an incentive to work on a particular task, say A, in period 1, before he knows whether the state of nature favors that task. For if the principal expects the agent to work on task A in period 1, and the agent does not, then the principal may infer that the agent has low ability, and will terminate him.

If the principal cares more about the second term than about the first, then the principal will want a high-ability agent in the second term, and so will prefer constrained timing. The motive can apply, for example, for promotion decisions. A handicapped governor who succeeds is a better candidate for president than an unconstrained governor who had the same success. Constrained timing is also more attractive if a low-ability agent’s performance declines with length of service. By assumption, a low-ability agent should choose a task that matches the state of nature. Such responsiveness can decline with length of service, making replacement of a low-quality agent more attractive, and therefore making constrained timing more beneficial.⁷

6. Conclusion

It is well known that reputational considerations can distort an agent’s behavior. The model in this paper is an application of this major idea. What is new is consideration of the timing of policy, and consideration of handicaps, wherein an agent purposely limits his freedom of action.

Handicaps can apply to additional issues of timing. They can explain the length of the window, why 100 days rather than 10? A short window may cause even a high-ability type to fail. A long window allows even a low-ability type to succeed.

And handicaps appear outside issues of timing. The idea can explain what instruments an incumbent allows himself. Suppose a high-ability agent can succeed on a task even if he uses a weak instrument, whereas a low-ability agent requires using a powerful instrument or using several instruments. Then a high-ability agent would want to signal his type by restricting his use of instruments, which would force a low-ability agent to do the same to avoid signaling his low ability. These considerations can explain how an incumbent can benefit from limits on his power. Such limits can resemble ideology. For example, a Republican may limit policy to use private enterprise whereas liberal may want government to do the activity directly.

Lastly, handicaps can apply to military strategy. For example, Romans disapproved of the Fabian strategy used in the second Punic War, which favored wearing down an opponent through a war of attrition and skirmishes to disrupt supply, rather than pitched battles and frontal assaults. Codes of honor, which handicap an army, can benefit a country by showing that it has military prowess, able to win despite the handicaps. These reputational effects will be especially important if the country faces several potential enemies, or expects to repeatedly face the same enemy.

7. Notation

f Probability state of nature favors the task.

G Principal’s gain from success on a task.

h Prior probability that agent has high ability.

Footnotes

Acknowledgements

I am grateful to the editor and the referees for suggestions which much improved the paper.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Notes

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