Delegation based on cheap talk

Abstract

We study a real-effort environment, where a delegator has to decide if and to whom to delegate a task. Applicants send cheap-talk messages about their past performance before the delegator decides. We experimentally test the theoretical prediction that information transmission does not occur in equilibrium. In our experiment, we vary the message space available to the applicants and compare the information transmitted as well as the level of efficiency achieved. Depending on the treatment, applicants can either submit a Number indicating past performance, an Interval in which past performance falls, or a free Text message. We observe that messages contain information in all treatments. Interestingly, information transmission occurs only in the treatments where messages are intervals or free text. Social welfare is higher if messages are intervals or free text than precise numbers. Gender and ethnicity stereotypes only influence delegation in the Number treatment, where no information transmission takes place.

A Appendix

1.1 A.1 Cheap-talk delegation game

We consider a sequential move game with one principal, two agents and incomplete information. There are one delegator and two senders, indexed by $i\in I=\left\{ D,1,2\right\}$. Types $\theta _{1},\theta _{2},\theta _{D}\in \varTheta$ are independently drawn from the same distribution f with support $[0,\theta ^{max}]$ and mean $\bar{\theta }$. In this context, a type refers to individual productivity. After observing their own type $\theta _{i}$, two senders simultaneously send a message $m_{i}\in M_{i}$ to the delegator. $M_{i}$ denotes the message space for sender i, which can be of smaller, equal or greater cardinality than the sender’s type space.

The delegator, after learning his own productivity $\theta _{D}$ and the two messages $m_{1},m_{2}$, decides if she wants to conduct a project herself or to delegate to one of the two senders, i.e. $d\in A=\{0,1,2\}$. If the principal decides to conduct her own project, i.e. $d=0$, then the delegator’s own productivity $\theta _{D}$ determines the project outcome. If sender i is chosen for delegation, i.e. $d=i$, then sender i receives a bonus, $B>0$, from the delegator and the outcome of the project is determined by the chosen player’s productivity $\theta _{i}$. To make the question interesting, we assume that the least productive delegator is at least indifferent to delegate under the prior distribution, i.e. $\bar{\theta }\ge B.$ The payoff functions, once delegation has occurred and the types are revealed are as follows:

$$\begin{aligned} U_{D}={\left\{ \begin{array}{ll} \theta _{d}-B &{} \text {if }d\ne 0\\ \theta _{D} &{} \text {if }d=0 \end{array}\right. }, \end{aligned}$$

for the delegator, and

$$\begin{aligned} U_{i}={\left\{ \begin{array}{ll} B &{} \text {if }d=i\\ 0 &{} \text {if }d\ne i \end{array}\right. }, \end{aligned}$$

for the senders $i=1,2.$

Together, the requirements of consistency of beliefs and sequential rationality in a Perfect Bayesian Nash Equilibrium imply that all messages sent in equilibrium necessarily need to lead to the same probability $p_{i}$ for a sender to be delegated to.

Proposition 1

If two messages $m'_{i}$ and $m''_{i}$ are both sent in equilibrium, then $p_{i}\left( m'_{i},\cdot \right) =p_{i}\left( m''_{i},\cdot \right)$ follows.

Proof

In what follows we will show that in a Perfect Bayesian Equilibrium information transmission that leads to message-dependent delegation decisions is impossible. In equilibrium, we require consistent beliefs. In our case this means that delegators form beliefs about the expected future project outcomes depending on to whom they delegate, i.e. $\mu _{1}\left( m_{1}\right)$ and $\mu _{2}\left( m_{2}\right)$. Note that rational beliefs about the productivity of one sender cannot depend on the message of the other sender, as the other sender has no other information than the prior. The beliefs have to be formed using Bayes’ Rule and in equilibrium have to be compatible with equilibrium messaging behaviour. Sequential rationality for the delegator implies:

$$\begin{aligned} d^{*}={\left\{ \begin{array}{ll} i &{} \text {if }\mu _{i}>\mu _{-i}\wedge \mu _{i}>\theta _{D}+B \\ D &{} \text {if }\max \left\{ \mu _{1},\mu _{2}\right\} <\theta _{D}+B \end{array}\right. } \end{aligned}$$

Note that a strategy for sender i is a mapping from her type space $\Theta _{i}$ into her message space $M_{i}$. Now suppose that a sender is assessing her probability $p_{i}$ of being delegated to. In equilibrium, a sender will maximize this probability assuming that the other sender and the delegator play their equilibrium strategies. Moreover, the sender in equilibrium will take into account how the delegator updates her beliefs. Therefore in equilibrium for the senders, the following will hold

$$\begin{aligned} m_{i}^{*}\left( \theta _{i}\right) =\arg \max _{m_{i}}p_{i}\left\{ m_{i},m_{-i}^{*}\left( \theta _{-i}\right) ;\mu _{i}\left( m_{i}\right) ,\mu _{-i}\left( m_{-i}\right) ;F\right\} \forall \theta _{i}\in \Theta _{i}. \end{aligned}$$

Observe that $p_{i}$ does not depend on $\theta _{i}$. This implies that all messages that are sent in equilibrium necessarily need to lead to the same delegation probability. Messages that lead to a lower probability cannot be sequentially rational and therefore cannot be part of an equilibrium. $\square$

This proposition either implies that all messages $m_{i}$ sent in equilibrium induce the same posterior expected ability $\mu _{i}$ or that induced differences in beliefs do not lead to differences in delegation behaviour for any possible delegator type. Our assumption that the abilities of the delegator and the senders are independently drawn from the same distribution rules out the latter, as long as the least productive delegator is prepared to delegate if she holds the prior beliefs.

Proposition 2

If $\bar{\theta }\ge B,$ then for any two messages $m'_{i}$ and $m''_{i}$ sent with positive probability in equilibrium induce assessments $\mu \left( m'_{i}\right) =\mu \left( m''_{i}\right) =\bar{\theta }$ holds.

Proof

To see this, first assume that $p_{i}\left( m_{i},\cdot \right)$ is positive but below unity for all equilibrium messages sent by i and $\mu _{i}\left( m''_{i}\right) >\mu _{i}\left( m'_{i}\right)$ for two messages sent with positive probability. The probability to be delegated to is given by

$$\begin{aligned} p_{i}=prob\left\{ \mu _{i}(m_{i})>\theta _{D}+B\right\} \cdot prob\left\{ \mu _{i}(m_{i})>\mu _{-i}|\mu _{i}(m_{i})>\theta _{D}+B\right\} . \end{aligned}$$

Note that the first probability is strictly increasing in $\mu _{i},$ while the second probability is positive and non-decreasing, which implies that $\mu _{i}\left( m''_{i}\right) >\mu _{i}\left( m'_{i}\right)$ and $p_{i}\left( m'_{i},\cdot \right) =p_{i}\left( m''_{i},\cdot \right)$ are not compatible for positive $p_{i}.$ It remains to be shown that different believed abilities together with delegation probabilities of zero or one are not possible in equilibrium. For delegation probabilities of one, all types of delegators would need to find it optimal to delegate. This is not possible since delegation is a dominated strategy for a delegator of type $\theta ^{max}$. Now turn to the case where equilibrium messages might imply different believed productivities but no delegation occurs ($p_{i}\left( m_{i},\cdot \right) =0$) for any equilibrium message. Note that an equilibrium delegation probability of zero for all messages sent requires that the least productive delegator type does not find it worthwhile to delegate regardless of the message. This implies that $\mu _{i}\left( m_{i}\right) \le B$ for all i and messages sent in equilibrium, which given our assumption on $\bar{\theta }\ge B$ implies $\mu _{i}\left( m_{i}\right) =B$ for all messages sent with positive probability in equilibrium and $\bar{\theta }=B$. This follows from the fact that the selection of a subset of a distribution with a conditional expected value below the distribution means necessarily implies that the non-selected part of the distribution has a conditional expectation above the distribution mean.

The analysis above has shown that in any equilibrium senders send messages that are not informative with respect to the expected productivity. In what follows, we refer to this result, when we speak of the impossibility of information transmission. As messages do not contain any information, senders in equilibrium will only use their own productivity and prior beliefs in order to decide on delegation. Whenever a delegator estimates her productivity below the mean productivity plus bonus, she will delegate. Delegating to either sender or any stochastic tie-breaking procedure is part of equilibrium as long as it does not condition on the messages. Since delegation in equilibrium only uses information that is available without messages, the equilibrium efficiency level is not influenced by the ability of senders sending messages or the type of messages they can send. $\square$

Corollary 1

Allowing agents to send messages in equilibrium does not improve social welfare.

This result immediately motivates our research. The natural question that arises is if there are behavioural mechanisms (such as lie-aversion) that allow humans to improve welfare over the non-message case. Moreover, one might ask if the kind of message senders can send has an impact on how much efficiency can be improved. Finally, we are interested in if stereotypes influence delegator decisions and efficiency.

1.2 A.2 Some additional analysis

1.2.1 A.2.1 Information contained in messages

See Fig. A.1 and Tables A.1, A.2.

Fig. A.1

Size of a lie in terms of bands. Sample size: 71 in the Number treatment, 70 in the Interval treatment

Table A.1 Regressions of messages across treatments

Table A.2 Probit regression on messages being untruthful

1.2.2 A.2.2 Delegation, performance and profitable messages

See Fig. A.2.

Fig. A.2

Delegation fractions by performance and messages

1.3 A.3 Sample instructions for the free-text treatment