Nonverifiability: Nash Implementation

In section 6.2, we have just seen how the principal and the agent can achieve ex post efficiency through an ex ante contract when they are both risk neutral. This contract uses only the agent’s message but fails to achieve efficiency when the agent is risk-averse or when nonresponsiveness occurs. We now propose a slightly more complicated implementation of the ex post efficient allocation that also works in these cases. The new feature of this implementation comes from the fact that both the principal and the agent must now send a report on the state of nature at date t = 2. Requesting both the principal and the agent to report the state of nature moves us somewhat beyond the technics that have been the focus of this volume. This extension is only a small detour into a multiagent setting. It is needed to assess the true importance of the nonverifiability constraint in principal-agent models.

The principal now offers to play a game with the agent, the outcome of which is enforced by the uniformed court of law. In this game, the principal is, like the agent, an active player, so that the mechanism to be played is a two-player game. However, as this game is played ex post under complete information, the characterization of the optimal two-player mechanism is relatively straightforward.

In this context, a general mechanism should involve two message spaces, one for the principal, say Mp , and one for the agent, Ma. Still denoting by the set of feasible allocations, we have the following definition:

Definition 6.1: A mechanism is a pair of message spaces Ma and Mp and a  mapping  g˜(·) from  M = Ma  × Mp   into ,  which  writes  as  g˜(ma, mp ) = (q˜(ma, mp ), t˜(ma, mp)) for  all  pairs  (ma, mp ) belonging  to  M.

To fix ideas, let us assume that the principal and the agent have respective utility functions V = S(q, θ) − t and U = t C(q, θ). In this context, the first-best allocation rule a(θ) = (t(θ), q(θ)) is such that


We consider that those traders play a Nash equilibrium6 of the mechanism (M, g˜(·)).

A Nash equilibrium of the mechanism (M, g˜(·)) is a pair of message functions (ma(·), mp*(·)),  which  satisfy  the  following  incentive  conditions:  for  the  principal,

and for the agent,

According to (6.7), when the principal conjectures that the agent’s strategy is given by ma(θ) in state θ, his best response is mp*(θ). Similarly, (6.8) states that the agent’s strategy ma(θ) is  a  best  response  to  the  principal’s  behavior.

We can now state the definition that follows:

Definition 6.2: An allocation rule a(θ) from Θ to  is implementable in Nash equilibrium by a mechanism (M, g˜(·)) if there exists a Nash equilib- rium (ma(·), mp*(·)) such  that  for all θ in Θ.

When the message spaces Ma and Mp are reduced to the set of possible types Θ, we have the following definitions:

Definition 6.3: A direct revelation mechanism is a mapping g(·) from Θ2 to ,  which  writes  as  is the agent’s (resp. principal’s) report in Θ.

Definition 6.4: A direct revelation mechanism g(·) is truthful if it is a Nash equilibrium for the agent and the principal to report truthfully the state of nature.

Denoting the set of Nash equilibria of the direct revelation mechanism g(·) in state θ by Ng(θ), we have the following definition:

Definition 6.5: The allocation a(θ) is implementable in Nash equilib- rium by the direct revelation mechanism g(·) if the pair of truthful report-ing strategies of the principal and the agent forms a Nash equilibrium of g(·) ((θ, θ) in Ng(θ) for all θ in Θ) such that a(θ) = g(θ, θ) for all θ in Θ.

Truthful direct revelation mechanisms must thus satisfy the following Nash incentive constraints:


We can now prove a new version of the revelation principle in this complete information environment.

Proposition 6.1: Any allocation rule a(θ) that is implemented in Nash equilibrium  by  a  mechanism  (M, g˜(·)) can  also  be  implemented  in  Nash equilibrium by a truthful direct revelation mechanism.

Proof:     The  mechanism  (M, g˜(·))  induces  an  allocation  rule     are  the  messages  of  the  agent  and the  principal  in  state  θ at  the  Nash  equilibrium  of  (M, g˜(·)) that  we  consider.  Let us define a direct revelation mechanism g(·) from Θ2  into , such that g(θ, θ) = g˜ ◦ m(θ) where  m(θ) = (ma(θ), mp*(θ)).  For  all  states  of  nature  θ,  we  have  thus .  We check that it is a Nash equilibrium for the players to report the truth when they face the direct revelation mechanism g(·). For the principal, we have

Taking for  any  θ’ in  Θ,  we  obtain

Finally, we get

Hence, the principal’s best response to a truthful reporting strategy by the agent is also to report truthfully.

Proceeding similarly for the agent, we prove that the agent’s best response is also to report his type truthfully. Hence, truthful reporting is a Nash equilibrium.

The important question at this point is to determine which restrictions are really put on allocations by the incentive compatibility constraints (6.9) and (6.10). In particular, we would like to know under which conditions the first-best allo- cation rule a(θ) = (t(θ), q(θ)) is implementable as a Nash equilibrium of the direct revelation mechanism played by the principal and the agent. It turns out that incentive compatibility in this multiagent framework imposes very few restrictions on the set of implementable allocations.

To see that, let us first consider the simple case where the principal’s utility function does not depend directly on θ, i.e., his utility is given by V = S(q) − t.

Figure 6.4: Nash Implementation of the First-Best with the No-Trade Option as Punishment

The agent also has the standard linear cost function of chapter 2, U = t − θq. We know that the first-best allocation entails producing outputs q(θ), such that S’(q(θ)) = θ. Using transfers t(θ) = θq(θ) allows then the principal to extract all of the agent’s rent.

A truthful direct revelation mechanism g(·) that implements in Nash equilib- rium the first-best allocation rule a(θ) = (t(θ), q(θ)) can be summarized by a

matrix (Figure 6.4), where the lines (resp. columns) represent the agent’s (resp. principal’s) possible reports in Θ = {θ, θ¯}. In each box of the matrix, we have rep-resented the transfer-output pair corresponding to the reports made by the principal and the agent.

For instance, when both the principal and the agent report to the court that θ has realized, the contract (t, q) is enforced. The principal gets a net surplus S(q) − t = S(q) − θq, and the agent gets tθq = 0 if the true state of nature is θ. If they disagree, the no-trade option is enforced, with no output being produced and no transfer being made.

The important point to note is that the same game form must be played by the agent and the principal, whatever the true state of nature θ. Indeed, the state of nature being nonverifiable, the transfers and outputs in each box of the matrix cannot be made contingent upon it. The goal of this mechanism is to ensure that there exists a truthful Nash equilibrium in each state θ that implements the first-best allocation a(θ) = (t(θ), q(θ)).

Proposition 6.2: Assume that preferences are given by V = S(q) − t and U = t − θq, then the first-best allocation rule is Nash-implementable.

Proof: Let us check that truthtelling is a Nash equilibrium of the direct revelation mechanism g(·) in each state of nature. Consider first state θ. Given that the agent reports θ, the principal gets S(q) − t = S(q) − θq by reporting the truth and zero otherwise. By assumption, trade is valuable when θ realizes (S(q) − θq > 0), and telling the truth is a best response for the principal. The agent is indifferent between telling the truth or not when the principal reports θ, because tθq = 0. Hence, he weakly prefers to tell the truth as a best response. Consider now state θ¯. Given that the agent reports θ¯, the principal gets  by reporting the truth and zero otherwise. By assumption, trade is also valuable when  θ¯  realizes .  Telling  the  truth  is  a  best  response  for  the principal. Similarly, the agent is indifferent between telling the truth or not when the principal reports truthfully, because . He weakly prefers to tell the truth. This ends the proof that truthtelling is a Nash equilibrium of g(·).

It is important to note that, when θ realizes, the pair of truthful strategies is  not  the  unique  Nash  equilibrium  of  the  direct  mechanism  g(·).  Indeed,  (θ¯, θ¯) is another Nash equilibrium in this state of nature. The agent strictly gains from misreporting  if  the  principal  does  so,  because .  Also,  the principal prefers to report θ¯ if the agent does so, because he obtains .

There are two possible attitudes vis-à vis this multiplicity problem. First, one may forget about it and argue that truth-telling should be a focal equilibrium. This attitude rests on a relatively shaky argument in the absence of a definitive theory of equilibrium selection. Moreover, some authors have shown in related models that the nontruthful equilibrium may sometimes Pareto dominate the truthful one from the players’ point of view.7 In this case, the focus on the truthful equilibrium is less attractive. This argument is less compelling in our context since the two equilibria cannot be Pareto-ranked: the agent does better in the nontruthful equilibrium, than in the truthful one, but the principal does worse.

The second possible attitude towards the multiplicity of equilibria is to take it seriously and to look for mechanisms that ensure that the first-best allocation is uniquely implementable. This second attitude is the route we are going to take now.

Definition 6.6: The first-best allocation rule a(θ) is uniquely imple- mentable  in  Nash  equilibrium  by  the  mechanism  (M, g˜(·)) if  the  mecha-nism has a unique Nash equilibrium for each θ in Θ and it induces the allocation a(θ).

In  the  definition  above,  we  do  not  restrict  a  priori  the  mechanism  g˜(·) to be a direct revelation mechanism. It could well be that the cost of obtaining unique implementation is the expansion of the space of messages that the agent and the principal use to communicate with the court. Such extensions are often used in multiagent (three or more) frameworks. In our principal-agent model, those extensions are not needed, provided that one conveniently defines the out- of-equilibrium-path punishments.

For the time being, let us consider a truthful direct revelation mechanism like that shown in figure 6.5.

Figure 6.5: Nash Implementation of the First-Best with General Punishments

The  outcomes  may  be  different  from  the  no-trade  option used above, in order to give more flexibility to the court in designing off-the- equilibrium punishments, ensuring both the truthful revelation and the uniqueness of the equilibrium. Let us now see how it is possible to do so.

The conditions for having a truthful Nash equilibrium in state θ are, for the principal,

and for the agent

Similarly, the conditions for having a truthful Nash equilibrium in state θ¯ are, for the principal,

and for the agent,

Let us now turn to the conditions ensuring that there is no nontruthful, pure- strategy Nash equilibrium in either state of nature. Consider a possible nontruthful equilibrium  (θ¯, θ¯) when  state  θ  realizes.  Given  that  (6.13)  is  needed  to  satisfy the  principal’s  incentive  constraint  in  state  θ¯,  the  only  way  to  break  the  possible equilibrium is to induce a deviation by the agent. Therefore, we must have

Now consider a possible nontruthful pure-strategy Nash equilibrium (θ, θ) when state θ¯ realizes. Given that (6.11) is needed to ensure the principal’s incen- tive constraint in state θ, the only way to break the possible equilibrium is again to induce a deviation by the agent:

Figure 6.6: Off-the-Equilibrium Path Punishments

A truthful direct revelation mechanism g(·), which uniquely implements the first-best as a Nash equilibrium, exists when the conditions (6.11) through (6.16) are all satisfied by a pair of punishment contracts . We now have the following proposition:

Proposition 6.3: Assume preferences are given by V = S(q) − t and U =  t − θq. A truthful direct revelation mechanism g(·) exists that uniquely implements in Nash equilibrium the first-best allocation rule a(θ).

Proof: The clearest way of doing this proof is to draw a picture. In Figure 6.6, we have represented the first-best allocation a(θ) and the possible punishments .

Because the first-best allocation requires that the agent produce efficiently and gets zero rent, the indifference curves of the principal are tangent to the zero-profit lines of the agent in each state of nature at points A and B. First, the θ-agent incentive compatibility constraint (6.12) and the condition (6.16) define a subset C where (tˆ1, qˆ1) may lie (crossed area in figure 6.6). Within this subset, the principal’s incentive constraint (6.13) further reduces the set of possible punishments (tˆ1, qˆ1) to the area E close to the origin (shaded area in figure 6.6). This set is nonempty, because the principal’s indifference curve does not go through the origin  when  trade  is  valuable  in  state  θ¯ (namely  when ).

Similarly, the agent’s incentive constraints (6.14) and (6.15) define a subset D of possible values for the punishment (tˆ2, qˆ2) (crossed area in figure 6.6). In figure 6.6, this full set satisfies the principal’s incentive compatibility constraint (6.11). More generally, by strict concavity of the principal’s indifference curve  going  through  B,  there  exists  a  nonempty  subset  of  D  that  lies strictly above this indifference curve. All those points obviously lie above the prin- cipal’s indifference curve W = S(q) − t going through A.

Proposition 6.3 yields a very striking result. It says that direct revelation mech- anisms are enough to ensure efficiency always if the court can design punishments in a clever way. There is no need to use more complex mechanisms in this simple and rather structured principal-agent model.

More generally, one may wonder if the requirement of unique Nash imple- mentation imposes some structure on the set of allocation rules a(θ) = (t(θ), q(θ)) that can be implemented this way. Indeed, this structure exists. Before describing it, we need another definition, which we cast in the general case where the prin- cipal’s and the agent’s preferences are respectively given by V = S(q, θ) − t and U = t C(q, θ).

Definition 6.7: An allocation rule a(θ) = (t(θ), q(θ)) is monotonic if and only if for any θ in Θ, such that a(θ) # a(θ’) for some θ in Θ, there exists an allocation (tˆ, qˆ) such that one of the two conditions below is true:

These inequalities have a simple meaning.8 The allocation rule a(·) selects the pair a(θ) = (t(θ), q(θ)) in state θ and not in state θ, if there exists another allocation (tˆ, qˆ) such that either the principal or the agent prefers this allocation to a(θ) when the state of nature is θ.

Under the assumptions of proposition 6.3, the first-best allocation rule a(·) is monotonic. Indeed, first we note that a(θ) # a(θ¯). Second, the principal’s utility function being independent of θ, there does not exist any allocation (tˆ, qˆ) such that condition (P ) holds. Lastly, there exists (tˆ, qˆ) such that condition (A) holds. In state θ, the set of such pairs is the set C in figure 6.6. In state θ¯, it is the set D. The monotonicity of allocation rules is an important property that follows immediately from unique implementation in Nash equilibrium, as is shown in proposition 6.4.

Proposition 6.4: Consider an allocation rule a(·), which is uniquely implemented  in  Nash  equilibrium  by  a  mechanism  (M, g˜(·));  then  the allocation rule a(·) is monotonic.

Proof:     The  mechanism  (M, g˜(·))  uses  the  message  spaces  Ma and  Mp .  If  the allocation  rule  a(·) is  uniquely  implementable  in  Nash  equilibrium  by  g˜(·),  we know  that,  in  state  θ,  there  exists  a  pair  of  strategies  (ma(θ), mp(θ))  such  that , and these strategies form a Nash equilibrium

Moreover, a(θ) being different from a) for a θ different from θ, a(θ) is not a Nash equilibrium in state θ. This means that either the principal (or the agent) finds  it  strictly  better  to  send  a  message  m˜p  rather  than  mp(θ) (or  m˜a  rather  than ma(θ) in  the  case  of  the  agent).  For  the  principal,  this  means  that

For the agent, this means that

In each case, it is easy to show that the allocation rule a(·) is monotonic. Take  in  the  first  case  (the  principal’s  devi-ation)  and  in  the  second  case  (the  agent’s deviation).

The intuitive meaning of proposition 6.4 is rather clear. In order to prevent an allocation implemented in one state of nature θ to be also chosen in another state θ, either the principal or the agent must deviate and choose another message in  state  θ.  Hence,  the  mechanism  g˜(·) that  uniquely  implements  the  allocation rule a(·) must include an allocation (tˆ, qˆ), which is worse than (t(θ), q(θ)) for both agents in state θ but better for at least one in state θ. In this case, the latter player’s preferences are reversed between states θ and θ, breaking a possible equilibrium that would also implement a(θ) in state θ.

The monotonicity property is a necessary condition satisfied by an allocation rule that is uniquely implementable in Nash equilibrium.

The remaining question is to know how far away this property is from suf- ficiency. With more than two agents (n ≥ 3), Maskin (1999) shows that monotonicity plus another property, no veto power, is also sufficient for unique Nash implementation. With two agents only, Dutta and Sen (1991) and Moore and Repullo (1990) have provided necessary and sufficient con- ditions for unique Nash implementation in more general environments than the principal-agent relationship analyzed in this chapter.

Source: Laffont Jean-Jacques, Martimort David (2002), The Theory of Incentives: The Principal-Agent Model, Princeton University Press.

Leave a Reply

Your email address will not be published. Required fields are marked *