A principal who offers a contract may renege when her default option is sufficiently attractive. The size of this temptation, which measures her commitment power, is often her private information. This paper asks how contracting outcomes change under this information asymmetry. Disciplining off-path beliefs with the Intuitive Criterion, I find that every type of principal behaves and earns payoffs exactly as if she were commonly known to have the least commitment power. Hidden commitment power is therefore powerless. The result delivers an unambiguous policy lesson on how to mitigate this information asymmetry prior to contracting: only measures that improve the worst case have value. Applied to credit rating, it rationalizes the monotone-partitional structure widely used in practice.
Keywords: commitment, signaling, intuitive criterion, credit rating.
In many principal-agent relationships, a principal incentivizes an agent by promising, through a contract, to respond to the agent’s action in a specified way. This promise is not automatically credible. Once the agent has acted, the principal chooses between honoring the contract and taking a default option that always remains open to her. She reneges whenever the default payoff exceeds the honoring payoff. The contract binds her only when the temptation to default is small. The ex-post enforcement failure driven by this temptation gives rise to the principal’s limited commitment,3 whose severity varies with the size of the default temptation. Limited commitment bites because default typically harms the agent’s incentives. Anticipating that a sufficiently tempted principal will renege, the agent curtails the very action the contract was meant to elicit, and contracting deteriorates.
Two applications make the structure concrete, and the first recurs as the paper’s running example. In debt issuance, an issuer (principal) raises funds from an investor (agent) by offering a debt contract with an interest rate. After the investor sinks an investment, the issuer either repays the loan with interest or reneges. If she reneges, a court liquidates her endowment and transfers it to the investor. Because the investor’s recovery upon default is the fixed endowment rather than a return that scales with the investment, default depresses his incentive to invest. An issuer whose endowment is small relative to the promised repayment therefore cannot credibly support a high interest rate. In government procurement, an agency (principal) procures a service from a supplier (agent) under a payment rule. After the supplier delivers, the agency either pays as promised or reneges. If it reneges, it is held liable for breach and bears a statutory penalty. Because the supplier is left uncompensated when the agency reneges, default depresses his incentive to provide quality. An agency facing only mild penalties therefore cannot credibly promise a generous payment.
The size of the default temptation, which inversely measures the principal’s commitment power, varies across principals. For example, an issuer with a more valuable endowment, or an agency facing harsher penalties, reneges less often. Moreover, these default consequences are typically the principal’s private information: the issuer knows the market value of her assets better than the investor does, and the agency knows its own legal exposure better than the supplier does. Commitment power is therefore hidden.
This paper studies the contracting consequences of hidden commitment power. Facing a contract offer, the agent must solve two problems at once. The first is what action to choose, anticipating that the principal may strategically default. The second is what to infer about the principal’s type from the offer itself. The two are intertwined, since the inference governs which types the agent expects to renege, and hence the action he is willing to take. The principal therefore faces a signaling tension: a less tempted principal prefers to separate and exploit her greater credibility, while a more tempted one imitates her to borrow it. In fact, a more-tempted principal can secure at least the payoff of any less-tempted one by imitating her offer, and consequently, full separation never arises in equilibrium.
Hidden commitment power is powerless. Under a mild and standard refinement of off-path beliefs, the Intuitive Criterion of chkr1987?, every type of principal behaves and earns payoffs exactly as if she were commonly known to have the weakest commitment power, that is, the largest temptation to default. Theorem 2 establishes that in every perfect Bayesian equilibrium (PBE) surviving the Intuitive Criterion, every type earns the symmetric-information payoff of the weakest type and offers a contract that the weakest type would offer under common knowledge about her type. The contract offer conveys no information about the principal’s type, no type is rewarded for privately facing a less attractive default option, and the outcome is invariant to the prior. The applied bite is stark. In debt issuance, issuers with any additional collateral that stays hidden do no better than the most financially constrained one. In government procurement, every agency contracts on the terms the least-deterred agency could honor, so stronger legal exposure goes unrewarded if it is hidden from the supplier.
The intuition unfolds in two parts.
First, additional commitment power is valuable in equilibrium precisely when default is entailed on the equilibrium path. If an equilibrium has all principal types honoring, they must offer contracts even the weakest type will not renege on, which caps the incentives any of them can provide at the weakest type’s symmetric-information level. The only equilibrium pattern that lifts any type above this worst-case benchmark is default-separation: a pooled contract that some higher types honor while some lower types renege on. Every equilibrium therefore takes one of two forms. Under full pooling at the worst case, all types honor a contract that the weakest type can sustain on her own. Under default-separation, a shared contract splits the pooling types into an honoring group and a reneging group. In debt issuance, default-separation is one interest rate that high-collateral issuers repay and low-collateral issuers renege on. It is the only route by which the issuer can beat the worst-case pool.
Second, intuitive beliefs break down every default-separation pattern. The deviation that does the work exploits two features of any default-separation outcome. Motive separation: a reneging type does not pay the contract’s promised terms, the interest in debt or the payment in procurement, so the contract does not bind her payoff; an honoring type, by contrast, fulfills the promise and thus earns payoff sensitive to the contract’s terms. Partial credibility: under default-separation, honoring occurs with intermediate probability; as a result, were the agent to conjecture a higher honoring probability, an honoring type could cut her contract’s cost while the agent’s action is maintained. These two features imply the existence of an off-path contract with worse incentives: it can be attractive to honoring types, because it lowers what they owe, but always unattractive to reneging types, whose payoff depends only on the agent’s action, which falls due to the worse incentives. The Intuitive Criterion then forces the agent to rule out the reneging types when the principal deviates to this off-path contract. His belief thus concentrates on the honoring types, induces the action they want, and validates the deviation. In debt issuance, the deviation offers a lower interest rate that maintains the investment once the investor believes he faces a high-collateral issuer for sure. This is a trade only an issuer who repays on the equilibrium path would take. Default-separation thus cannot survive refinement.
Putting the two parts of the intuition together yields the powerlessness conclusion: an equilibrium delivers payoffs above the worst-case level only through default-separation, but no such equilibrium is supported by intuitive beliefs. The force is asymmetric: it rules out default-separation yet leaves intact full pooling on a contract the weakest type can honor. In every surviving outcome, all types then earn the weakest type’s symmetric-information payoff. The high-collateral issuer ends up funded as though her collateral were the market’s lowest, and the well-protected agency contracts as though it were the least deterred.
The contracting result has an immediate implication for information design before contracting. Suppose a designer can publicly disclose information about the principal’s type, inducing a common posterior belief before the contracting game is played. Because Theorem 2 applies to the post-disclosure game, the contracting outcome under any posterior depends on it only through the posterior’s weakest type. Disclosure therefore improves welfare only by raising the worst case. Corollary 1 makes this precise. A signal that leaves the weakest type in every posterior is welfare-equivalent to no disclosure, and improving on it requires inducing some posterior that excludes the weakest type. When disclosure carries a complexity cost, weakly increasing in the number of distinct signal realizations,4 Proposition 3 shows that every optimal signal is monotone-partitional. Such a signal sorts types into intervals and reveals which interval contains the type, with each interval’s lower threshold its only payoff-relevant feature. Applied to debt issuance, the information designer chooses precisely a credit rating system. A monotone-partitional information structure thus bundles issuers into ordered bands, and each issuer borrows on the terms warranted by the floor of her band. This result rationalizes the letter-grade systems common in credit-rating practices (e.g., AAA, AA+, \(\ldots\), D at S&P and Aaa, Aa1, \(\ldots\), C at Moody’s).
The argument proceeds from a concrete instance to the general result and its uses. Section 2 develops a tractable debt-issuance example, where the primitives are easy to state and the refinement argument can be displayed in full. Section 3 sets up the general model. Section 4 states the assumptions and builds the symmetric-information benchmark. Section 5 introduces the Intuitive Criterion and states the main result, Theorem 2. Section 6 draws out the implication for credit rating, and Section 7 discusses extensions.
The paper sits at the intersection of three literatures: hidden commitment in contracting, signaling, and information design.
Hidden commitment in contracting. The closest predecessors are hala2012?, lima2013?, kart2018?, and fakl2019?, who, like this paper, study contracting in which a privately informed principal faces limited enforceability of her promises. Working in repeated relational settings, they select equilibria by Pareto or principal-optimality. This paper instead casts the problem as a signaling game and disciplines it with the Intuitive Criterion, the weakest standard refinement. The selection then runs the opposite way: even this minimal discipline singles out the Pareto-dominated, principal-least-preferred outcome, in sharp contrast with their selection. The friction itself is close to the imperfect commitment studied by best2001?. There, as here, the principal can commit to part of an allocation but retains residual ex-post discretion. In this paper, that discretion is the choice between honoring the promise and taking the default option, governed by the size of the default temptation. One essential departure is that the privately informed party is the principal rather than the agent, the feature that introduces the signaling constraint.
Complementary to the contract enforceability at issue here is a second commitment concept, dynamic commitment. It is studied in self-enforcing relational agreements (thwo1988?), control rights and incomplete financial contracting (agbo1992?), and debt under default and renegotiation (hamo1998?). Relatedly, dosk2022? study limited commitment in dynamic mechanism design. More broadly, the point that a friction can annihilate the value of commitment is also articulated by bagwell?, who shows that, in a leader-follower game, even a slight noise in the follower’s observation of the committed action collapses the first-mover advantage to the simultaneous-move outcome. The present paper joins this classic view in a contracting setting, where a friction again collapses the outcome to the least-commitment benchmark. None of these papers, however, considers asymmetric information about commitment.
Signaling. The analysis is a signaling problem in the tradition of spen1978? and chkr1987?, but does not rely on global single-crossing. Rather, within a default-separation outcome, honoring and reneging types differ in their marginal utility for higher actions at the deviation contracts, and the Intuitive Criterion exploits this endogenous, local difference to overturn the pattern. Signaling without global single-crossing also appears in feht2002?, argm2007?, frka2019?, and chis2022?, who study distinct primitive preferences such as two-dimensional types or double-crossing indifference curves. Pooling also arises in kart2009? and bibo2018?, through forces unrelated to the one here.
Information design. The monotone-partitional optimal signal of Proposition 3 resembles results in mens2021?, gole2018?, and dwma2019?, but arises for a different reason. The partition is driven by Theorem 2: designer welfare depends on each posterior only through its lowest type. lirs22? also study information design, with a sender whose limited credibility is common knowledge.
This section develops a debt-issuance example that contains every economic force of the general model. The example shows how limited and hidden commitment power shape contracting, and how the Intuitive Criterion selects among the resulting equilibria.
A debt issuer (principal) raises funds from an investor (agent) by offering an interest rate \(r\in[0,1]\). After observing \(r\), the investor chooses an investment size \(i\geq0\) at investment cost \(i+\frac{1}{2}i^2\). The investment yields the issuer a non-verifiable gain of \(2i\), capturing the long-run benefits the issuer obtains by financing her project. At maturity, the issuer owes the investor a repayment of \(i+ri\), the borrowed funds \(i\) plus the interest \(ri\).
The contracting friction lies in whether the issuer honors this repayment. She owns an endowment whose value \(\theta>0\) is verifiable at court. Faced with the repayment \(i+ri\), she may choose to honor it. Alternatively, she may renege, in which case the court verifies the endowment, liquidates it, and transfers \(\theta\) to the investor. The two players’ payoffs upon honoring (\(H\)) and default (\(D\)) are \[\label{example-payoffs} \begin{align} u_{\text{issuer}}^H(i,r)&=2i-(i+ri)=i-ri,\qquad &u_{\text{investor}}^H(i,r)&=(i+ri)-\left(i+\tfrac{1}{2}i^2\right)=ri-\tfrac{1}{2}i^2, \\ u_{\text{issuer}}^D(i,\theta)&=2i-\theta, &u_{\text{investor}}^D(i,\theta)&=\theta-\left(i+\tfrac{1}{2}i^2\right). \end{align}\tag{1}\] Two structural features of 1 drive the analysis. First, the issuer’s payoffs are strictly increasing in the investment \(i\) under both honoring and reneging, so she always benefits from a larger investment. Second, the investor’s default payoff depends only on the endowment \(\theta\), not on the interest rate \(r\). Default thus deprives the investor of the contractual return, while leaving in place the costly investment. Aggressive investment is what the issuer wants, but it is precisely the condition that tempts her to renege, which in turn weakens the investor’s incentive to invest in the first place.
The issuer privately observes her commitment-power type \(\theta\in\Theta=\{\theta_L,\theta_H\}\) with \(\theta_H>\theta_L>0\). The investor holds a full-support prior belief \(\mu^0\in\Delta\Theta\), with \(\mu^0_H:=\mu^0(\theta_H)\in(0,1)\). The timing is
The issuer privately observes \(\theta\).
The issuer publicly announces an interest rate \(r\in[0,1]\).
The investor updates his belief about \(\theta\) and chooses an investment \(i\geq0\).
The issuer chooses whether to repay \(i+ri\) in full or renege, in which case the court enforces transfer of \(\theta\).
The relevant solution concept is Perfect Bayesian Equilibrium (PBE). Each PBE specifies the issuer’s type-contingent interest-rate offer, the investor’s belief inferred from each contract, his investment strategy, and the issuer’s honoring decision in each contingency. All components are consistent with Bayes’ rule on the equilibrium path and payoff maximization at every information set.
I first record what happens when \(\theta\) is publicly known. The investor’s honoring payoff \(ri-\tfrac{1}{2}i^2\) has best response \(i=r\). If the issuer honors, her payoff is \(i-ri=r-r^2\), maximized at \(r=\frac{1}{2}\). Hence, when \(\theta\) is sufficiently high, the issuer has full commitment and offers \(\frac{1}{2}\).
When \(\theta\) is low, however, \(r=\frac{1}{2}\) may not be feasible: the issuer honors \(r\) if and only if \(\theta\geq i+ri\). To raise \(r\) above what the endowment supports invites default. The issuer does not want this either, because reneging dissipates the endowment without restoring the investor’s investment incentive. So under limited commitment the issuer offers the highest \(r\) she can credibly honor, which, using \(i=r\), means \(\theta=i+ri=r(1+r)\). Solving yields \[\label{example-benchmark-rate} r^{\text{SI}}(\theta)=i^{\text{SI}}(\theta)=\frac{\sqrt{4\theta+1}-1}{2},\tag{2}\] together with issuer payoff \[\label{example-benchmark-payoff} U^{\text{SI}}(\theta)=r^{\text{SI}}(\theta)-r^{\text{SI}}(\theta)^2.\tag{3}\] The full-commitment threshold is \(\theta^*=\frac{3}{4}\). When \(\theta\geq\theta^*\), the issuer can credibly offer \(r=\frac{1}{2}\). When \(\theta<\theta^*\), she is constrained by her endowment and offers \(r^{\text{SI}}(\theta)\). For \(\theta<\theta^*\), the rate \(r^{\text{SI}}(\theta)\), the investment \(i^{\text{SI}}(\theta)\), and the issuer payoff \(U^{\text{SI}}(\theta)\) are all strictly increasing in \(\theta\). Higher commitment power thus monotonically improves contracting.
When \(\theta\) is private, the equilibrium analysis is more subtle. To see why, suppose for contradiction that the two types play their symmetric-information offers and let \(\theta_L<\theta^*<\theta_H\), so \(r_L=r^{\text{SI}}(\theta_L)<\frac{1}{2}=r_H\). The low type would gain by deviating to \(r_H\). Doing so induces the higher investment \(i_H=\frac{1}{2}\), because the investor would believe she faces the high type. The low type would then renege, paying her endowment \(\theta_L\) rather than the interest \(r_Hi_H\). Default payoff \(2i_H-\theta_L>2i_L-\theta_L\geq U^{\text{SI}}(\theta_L)\) exceeds the on-path payoff, giving the low type a higher payoff than the high type. Hence, full separation is not sustainable. More broadly, any separating outcome in which the high type’s offer induces a higher investment invites low-type imitation, because the low type faces a larger room to renege after such imitation.
Pooling is therefore ubiquitous in equilibrium. The pooling equilibria may take two structures.
Full pooling (FP) Both types offer the same \(r\), and both honor it. Honoring is feasible only if \(\theta_L\geq i+ri\). The investor anticipates honoring with probability one, so his best response is \(i=r\). Combining, this requires \(\theta_L\geq r(1+r)\), namely \(r\leq r^{\text{SI}}(\theta_L)\). The largest such interest rate is \(r^{\text{FP}}=r^{\text{SI}}(\theta_L)\). Both types’ payoffs in this FP outcome equal \(U^{\text{SI}}(\theta_L)\), the lowest-type benchmark.
Default-separation (DS) Both types offer the same \(r\), but only the high type honors it, while the low type reneges. Under default-separation, the investor expects honoring with probability \(\mu^0_H\) and reneging with probability \(\mu^0_L\), so his best response is \(i=\mu^0_H r-\mu^0_L\) (the first-order condition on his expected payoff in default-separation).5 Default-separation requires \(\theta_L<i+ri\leq\theta_H\) at the equilibrium investment, so the low type indeed reneges and the high type indeed honors. Many parameter values support such a default-separation PBE. The illustrative example used in what follows takes \(\theta_L=\tfrac{1}{10}\) (well below the full-commitment threshold \(\theta^*=\tfrac{3}{4}\), so the low type is severely constrained), \(\theta_H>\tfrac{3}{10}\), and \(\mu^0_H=0.8\) (large enough to make the investor’s optimism plausible). Under these values, the strategy profile \((r^{\text{DS}},i^{\text{DS}})=(0.5,0.2)\) is a PBE. Indeed, \(i^{\text{DS}}=0.2=0.8\times0.5-0.2=\mu^0_H r^{\text{DS}}-\mu^0_L\) is the investor’s first-order condition. The repayment \(i^{\text{DS}}+r^{\text{DS}}i^{\text{DS}}=0.3\) lies between \(\theta_L=0.1\) and \(\theta_H\), and on-path deviations are deterred by suitably pessimistic off-path beliefs. In fact, a continuum of default-separation PBEs exists, varying both the offered rate and the implied investment.
Appendix 8 gives one example of a richer pattern combining pooling and partial separation, beyond the two structures above. Its purpose is to demonstrate that the PBE space is not exhausted by full pooling and default-separation. The general PBE outcomes split into two categories. Category 1 consists of full pooling, which has an essentially unique outcome at \(r^{\text{FP}}=r^{\text{SI}}(\theta_L)\) and is Pareto-dominated. Category 2 contains all other PBEs. Crucially, every Category 2 PBE must involve default-separation: several types offer the same contract, with the higher types among them honoring it while the rest renege. The Intuitive Criterion, established next, eliminates Category 2 in its entirety, leaving Category 1 as the unique surviving outcome.
A standard tool for disciplining off-path beliefs in signaling games is the Intuitive Criterion of chkr1987?. Adapted to this setting, it states the following. Fix a PBE in which the type-\(\theta\) issuer earns equilibrium payoff \(U^*(\theta)\), and consider any off-path interest rate \(r'\). After observing \(r'\), the investor evaluates each type under the most favorable belief, that the issuer is of the highest type, together with his best response to that belief. He should not assign positive probability to a type \(\theta'\) whose payoff from \(r'\) is then strictly below \(U^*(\theta')\), provided some other type \(\theta''\) would weakly benefit from \(r'\) under the same belief. A PBE survives the Intuitive Criterion if, at every off-path \(r'\), the investor’s belief assigns zero probability to all types ruled out in this manner. I then call it an intuitive PBE.
The Intuitive Criterion eliminates default-separation. To see this in the example, revisit the default-separation PBE \((r^{\text{DS}},i^{\text{DS}})=(0.5,0.2)\) with \(\theta_L=\tfrac{1}{10}\) and \(\mu^0_H=0.8\). Consider the off-path rate \(r'=0.2-\epsilon\) for small \(\epsilon>0\). The best possible belief is \(\theta_H\). Under it the investor expects full commitment, so his best response is \(i'=r'=0.2-\epsilon\).
The low type is guaranteed worse off after deviating. On path, the low type reneges and earns \(2i^{\text{DS}}-\theta_L=0.4-0.1=0.3\). After deviating to \(r'\), the repayment \(i'(1+r')\approx0.24\) still exceeds \(\theta_L=0.1\), so she reneges again and earns \(2i'-\theta_L=0.3-2\epsilon\). Default payoff depends only on the action \(i\), and a lower action means strictly lower payoff. The low type thus cannot benefit from offering \(r'\) no matter the investor’s response.
The high type can benefit. On path, the high type honors and earns \(i^{\text{DS}}-r^{\text{DS}}i^{\text{DS}}=0.2(1-0.5)=0.1\). After deviating to \(r'\), with the best belief \(\theta_H\) inducing \(i'=0.2-\epsilon\), she still honors (since \(\theta_H>i'(1+r')\)), and her payoff is \(i'-r'i'=i'(1-r')\approx0.2\cdot0.8=0.16>0.1\). The high type substitutes credibility for incentives: by securing the belief that she honors for sure, she sustains nearly the same investment while charging a much lower rate.
The Intuitive Criterion therefore requires that, after observing \(r'\), the investor assigns probability one to the high type. This makes \(i'=r'\) the investor’s best response, and the high type strictly prefers the deviation. The default-separation PBE is broken. The same logic, formalized in the next sections, rules out every PBE involving default-separation. The only surviving PBE outcome is full pooling at the lowest-type benchmark.
Theorem 1 (Debt-issuance example). In every intuitive PBE, every type of issuer offers \(r^{\text{SI}}(\theta_L)\), the investor invests \(i^{\text{SI}}(\theta_L)\), and the issuer earns \(U^{\text{SI}}(\theta_L)\).
Two features of default-separation jointly produce the Intuitive Criterion failure illustrated above.
Motive separation. The honoring payoff \(u_{\text{issuer}}^H(i,r)=i-ri\) depends on the interest rate \(r\), while the default payoff \(u_{\text{issuer}}^D(i,\theta)=2i-\theta\) does not. Whether the issuer cares about the contract terms is endogenously determined by whether she honors. In a default-separation outcome, the issuer is partitioned into honoring and reneging types whose objectives differ exactly along this dimension. A small reduction in the interest rate \(r\) benefits all honoring types (by reducing what they owe) but is irrelevant for reneging types’ direct payoff (the contract simply does not bind their behavior). The honoring types thus have a higher willingness to take a cheaper contract.
Partial credibility. In default-separation, the issuer is honored with intermediate probability. Were the investor instead to conjecture honoring for sure, he would raise his investment, since perceived honoring rises to one. A reduction in the interest rate \(r\) pushes his investment the other way. The crucial point is that these two forces can be combined: a contract that lowers \(r\) while securing the belief that the issuer honors for sure leaves the investor’s investment intact. In short, contract terms and credibility are substitutes in providing the investor’s incentive to invest.
Together, motive separation and partial credibility produce a deviation that the honoring types want and the reneging types do not, validating the Intuitive Criterion’s belief restriction. The reduction in \(r\) is large enough to deter the reneging types but, paired with the credibility gain, small enough in its effect to keep the investor’s investment intact.
I now generalize the analysis. The general model accommodates the motivating environments of the introduction. In debt issuance, the agent’s action is the investment and the principal’s default option is the verified collateral value. In government procurement, the agent’s action is service quality and the default option is the statutory penalty for non-payment. More broadly, the model fits any environment with the same structure: an agent action the principal wants but is tempted to renege on, and a default that depresses the agent’s incentive to take it.
A principal (she) and an agent (he) play a four-stage game. Players’ payoffs depend on the agent’s action \(a_A\in A_A\) and the principal’s action \(a_P\in A_P\) through utility functions \(u_P,u_A:A_A\times A_P\rightarrow\mathbb{R}\). The principal seeks to influence the agent’s action by offering a contract \(\phi\in\Phi\subseteq\{\phi:A_A\rightarrow A_P\}\), a measurable map that specifies, for every possible agent action, the principal’s contractually-prescribed response.
The principal’s ability to honor the contract is limited because she has access to a default option \(a_P^D(a_A,\theta)\in A_P\) that depends on the agent’s action \(a_A\) and a privately observed type \(\theta\in\Theta\). The type \(\theta\) represents the principal’s commitment power: when default consequences are unattractive, she is more credible. The set of types \(\Theta\) is a finite subset of \(\mathbb{R}\), with full-support prior \(\mu^0\in\Delta\Theta\). The lowest and highest types are denoted \(\underline{\theta}\) and \(\overline{\theta}\), respectively.
The agent’s action space \(A_A=[\underline{a}_A,\overline{a}_A]\) is a compact real interval, capturing the central role of the agent’s action in the contracting environment. The principal’s action space \(A_P\) is a compact subset of a normed vector space, and \(\Phi\) is a compact subset of the measurable functions \(A_A\to A_P\), endowed with the sup-norm.6 All distributions are Borel probability measures, and all measurability statements are with respect to the Borel \(\sigma\)-algebras induced by the relevant norms. These topological assumptions ensure that continuity, set connectedness, and probability distributions are well-defined on \(A_P\) and \(\Phi\).
The game unfolds as follows. The type \(\theta\) is realized and privately observed by the principal. The principal then announces a contract \(\phi\), drawn from a contract-offering strategy \(\sigma_P:\Theta\rightarrow\Delta\Phi\). Observing \(\phi\), the agent updates his belief via Bayes’ rule to a posterior \(\mu(\cdot|\phi)\in\Delta\Theta\). The agent then chooses whether to accept the offer (\(d_A=1\)) or reject and trigger an outside option \((a_A^0,a_P^0)\) (\(d_A=0\)). Conditional on acceptance, the agent selects an action \(a_A\in A_A\). Finally, the principal decides whether to honor \(\phi\) by playing \(\phi(a_A)\) or to renege and play \(a_P^D(a_A,\theta)\) instead.
Each PBE specifies the principal’s contract-offering strategy \(\sigma_P:\Theta\rightarrow\Delta\Phi\), the agent’s belief system \(\mu:\Phi\rightarrow\Delta\Theta\), his acceptance decision \(d_A:\Phi\rightarrow\Delta\{0,1\}\), his action choice \(\sigma_A:\Phi\rightarrow\Delta A_A\), and the principal’s honoring decision \(d_P:\Theta\times\Phi\times A_A\rightarrow\Delta\{0,1\}\). The belief system \(\mu\) is consistent with Bayes’ rule given the prior \(\mu^0\) and the principal’s strategy \(\sigma_P\). All four strategies are sequentially rational at every information set. To keep the exposition clean, I restrict attention to PBEs in which each \(\sigma_P(\cdot|\theta)\) has finite support.7
It is convenient to define payoffs conditional on the principal’s honoring decision. Let \[\label{payoff-HD} \begin{align} u_P^H(a_A,\phi)&:=u_P(a_A,\phi(a_A)), \qquad &u_P^D(a_A,\theta)&:=u_P(a_A,a_P^D(a_A,\theta)), \\ u_A^H(a_A,\phi)&:=u_A(a_A,\phi(a_A)), \qquad &u_A^D(a_A,\theta)&:=u_A(a_A,a_P^D(a_A,\theta)). \end{align}\tag{4}\] The honoring payoff \(u_P^H\) depends on the contract and the agent’s action, while the default payoff \(u_P^D\) depends on the type and the action. The principal’s overall payoff, given an accepted contract, is \[\label{U-P-overall} U_P(a_A,\phi,\theta):=\max\left\{u_P^H(a_A,\phi),u_P^D(a_A,\theta)\right\},\tag{5}\] reflecting her ex post optimal honoring decision. To deal with the agent’s mixed best responses, needed for the Intuitive Criterion, let \(\text{BR}_P(a_A,\phi,\theta)\subseteq\{0,1\}\) denote the principal’s set of optimal honoring decisions (\(1\) for honor, \(0\) for default). Let \(\text{BR}_P^\theta(\phi)\) denote the set of measurable maps \(d:A_A\rightarrow\Delta\{0,1\}\) such that \(d(\cdot|a_A)\) has support in \(\text{BR}_P(a_A,\phi,\theta)\) for every \(a_A\). Given the agent’s belief \(\mu\in\Delta\Theta\), his rationalizable actions are those that maximize his expected payoff against some principal honoring strategy \(d_P^\theta\in\text{BR}_P^\theta(\phi)\): \[\label{rationalizable} \text{RA}_A^\mu(\phi):=\bigcup_{\{d_P^\theta\in\text{BR}_P^\theta(\phi):\theta\in\Theta\}}\mathop{\mathrm{argmax}}_{a_A\in A_A}\sum_{\theta\in\Theta}\mu(\theta)\left[d_P^\theta(1|a_A)u_A^H(a_A,\phi)+d_P^\theta(0|a_A)u_A^D(a_A,\theta)\right].\tag{6}\] The agent’s mixed rationalizable actions are then \(\text{MRA}_A^\mu(\phi):=\Delta\text{RA}_A^\mu(\phi)\), and aggregating over beliefs gives \(\text{RA}_A(\phi):=\cup_{\mu\in\Delta\Theta}\text{RA}_A^\mu(\phi)\). The interpretation is straightforward. \(\text{RA}_A^\mu(\phi)\) is the set of agent actions that could rationally arise following \(\phi\) when the agent holds belief \(\mu\) and correctly anticipates that the principal will play a best response in the final honoring/default stage.8
A useful concept that captures the agent’s perception of the principal’s credibility is \[\label{credibility} C^\mu(a_A,\phi):=\sum_{\theta\in\Theta}\mu(\theta)\mathbb{1}\left\{u_P^H(a_A,\phi)\geq u_P^D(a_A,\theta)\right\},\tag{7}\] the posterior probability of honoring given belief \(\mu\), contract \(\phi\), and action \(a_A\) (with the convention that an indifferent type (\(u_P^H=u_P^D\)) counts as honoring; by Assumption 4(i) the tie-breaking is payoff-irrelevant for the agent). When \(C^\mu(a_A,\phi)=1\), the contract has full credibility under the belief \(\mu\). When \(C^\mu(a_A,\phi)=0\), it has no credibility. When \(C^\mu(a_A,\phi)\in(0,1)\), it has partial credibility, meaning some types in the support of \(\mu\) honor and others renege.
I now state the substantive assumptions and isolate a symmetric-information benchmark that will serve as a reference point for the main result.
A contract \(\phi\) is undominated if there is no \(\phi'\in\Phi\) such that \(U_P(\sigma,\phi,\theta)<U_P(\sigma',\phi',\theta)\) for all \(\theta\in\Theta\), \(\sigma\in\text{MRA}_A^{\mu^0}(\phi)\), and \(\sigma'\in\text{MRA}_A^{\mu^0}(\phi')\). Substantively, undominated contracts are those that some principal type could rationally offer in equilibrium.
Assumption 1 (Limited Commitment). For every undominated \(\phi\in\Phi\) and every \(\theta\in\Theta\),
the principal prefers higher actions: \(u_P^H(a_A,\phi)\) and \(u_P^D(a_A,\theta)\) are strictly increasing in \(a_A\);
the principal reneges under high actions: there exists \(a_A^*(\phi,\theta)\in[\underline{a}_A,\overline{a}_A]\) such that \(u_P^H(\cdot,\phi)>u_P^D(\cdot,\theta)\) on \([\underline{a}_A,a_A^*(\phi,\theta))\) and \(u_P^H(\cdot,\phi)<u_P^D(\cdot,\theta)\) on \((a_A^*(\phi,\theta),\overline{a}_A]\);
default harms the agent’s incentives: \(u_A^H(a_A,\phi)-u_A^D(a_A,\theta)\) is strictly increasing in \(a_A\).
These three properties capture the canonical limited-commitment structure. Part (i) says the principal always wants a higher action, regardless of whether she honors or reneges, and part (ii) says that the action she likes most is precisely the action that tempts her to renege. The intersection point \(a_A^*(\phi,\theta)\) is the highest action she can sustain without reneging. Part (iii) says that the agent’s incentive to take higher actions is strictly lower under reneging than under honoring, so default depresses the agent’s response. In the debt example, part (i) is verified by the issuer’s payoffs \(i-ri\) and \(2i-\theta\) being strictly increasing in \(i\). Part (ii) is verified by the threshold \(i+ri=\theta\) at which the issuer becomes indifferent. Part (iii) is verified by the investor’s net return \(ri\) under honoring being strictly steeper in \(i\) than the constant \(\theta\) under reneging.
Assumption 2 (Heterogeneous Commitment). For every \(a_A\in A_A\), \(u_P^D(a_A,\theta)\) is strictly decreasing in \(\theta\).
This is the only assumption directly involving the type variable \(\theta\). It says that a higher type faces a less attractive default option. A useful implication, used repeatedly below, is that the set of honoring types is an interval upward-closed in \(\theta\). If the type-\(\theta'\) principal honors a contract \(\phi\) at an action \(a_A\), then so does every type \(\theta\geq\theta'\). In the debt example, the issuer’s default payoff \(2i-\theta\) is strictly decreasing in \(\theta\).
The third substantive assumption states that providing more incentive to the agent is more expensive for the principal, a natural feature of contracting environments where contract terms come at a cost. To state it, define the agent-incentive (AI) order over contracts by \[\label{AI-order} \phi_1\succ_{\text{AI}}\phi_2\iff u_A^H(\cdot,\phi_1)-u_A^H(\cdot,\phi_2)\text{ is strictly increasing,}\tag{8}\] and \(\phi_1\sim_{\text{AI}}\phi_2\) if and only if \(u_A^H(\cdot,\phi_1)-u_A^H(\cdot,\phi_2)\) stays constant in \(a_A\). The order ranks contracts by the strength of the marginal incentive they provide: \(\phi_1\succ_{\text{AI}}\phi_2\) means \(\phi_1\) provides strictly stronger marginal incentives at every action than \(\phi_2\).
Assumption 3 (Costly Incentives). For every \(\phi_1,\phi_2\in\Phi\), \(\phi_1\succ_{\text{AI}}\phi_2\) implies \(u_P^H(a_A,\phi_1)<u_P^H(a_A,\phi_2)\) for every \(a_A\in A_A\).
Costly Incentives says that climbing the agent-incentive ladder strictly decreases the principal’s honoring payoff at every action: contracts that motivate the agent more strongly cost the principal more to honor. In the debt-issuance example, the AI order ranks contracts by the interest rate \(r\), since a higher \(r\) gives the investor stronger marginal incentive. The issuer’s honoring payoff \(i-ri\) is strictly decreasing in \(r\), so Costly Incentives holds. Section 4.4 shows that, combined with the technical assumptions below, Costly Incentives implies the substitution between contract and credibility that drives the deviation argument in the main proof.
Assumption 4 (Technical).
**Concavity of overall agent payoff:* the payoff functions in 4 are continuous in \((a_A,\phi,\theta)\); \(u_A^H(\cdot,\phi)\) and \(u_A^D(\cdot,\theta)\) are strictly concave and differentiable in \(a_A\); and \(u_P^H(a_A,\phi)\leq u_P^D(a_A,\theta)\) if and only if \(u_A^H(a_A,\phi)\geq u_A^D(a_A,\theta)\).*
**One-dimensional incentive:* \(\succsim_{\text{AI}}\) is a complete order on \(\Phi\).*
**Richness of contract space:* \(\Phi\) is path-connected; there exists a safe contract \(\phi^0\in\Phi\) with \(u_P^H(a_A,\phi^0)\geq u_P^D(a_A,\theta)\) for all \((a_A,\theta)\); and there exists \(a_A^\ell\in A_A\) such that, for every \(\theta\in\Theta\), \(\cup_{\phi\in\Phi}\text{RA}_A^\theta(\phi)=[a_A^\ell,a_A^h(\theta)]\) for some \(a_A^h(\theta)>a_A^\ell\), with \(a_A^\ell\in\text{RA}_A^\theta(\phi)\) implying \(\phi\) is dominated.*
These are tools used in the proof rather than constraints on economic substance. Part (i) bundles three ingredients — continuity of payoffs, strict concavity of \(u_A^H\) and \(u_A^D\), and the alignment between the principal’s and the agent’s indifferences — that jointly deliver the concavity property the proof relies on. Aligned indifference is what reduces the agent’s overall payoff to the minimum \[\label{U-A-overall} U_A(a_A,\phi,\theta):=\min\left\{u_A^H(a_A,\phi),u_A^D(a_A,\theta)\right\},\tag{9}\] because the principal honors precisely when the agent prefers the default outcome. The agent’s realized payoff then equals \(u_A^H\) on one side of the crossing and \(u_A^D\) on the other, exactly the minimum. Strict concavity of \(u_A^H\) and \(u_A^D\) then transfers to \(U_A\),9 guaranteeing a single-valued agent best response. Continuity, in turn, underpins the Berge and intermediate-value arguments used throughout. Part (ii) gives the contract space a single dimension along which contracts can be ranked by the marginal incentive they provide. Part (iii) collects three richness conditions on the contract space. Path-connectedness lets us perturb contracts continuously. The safe contract provides an anchor for intermediate-value constructions. The connected incentive set guarantees that the rationalizable-action correspondence is a continuous interval whose lower endpoint is induced only by dominated contracts.
Part (ii) can be weakened: it suffices that \(\Phi\) has a dense subset \(\widetilde{\Phi}=\cup_\alpha\Phi_\alpha\) such that each \(\Phi_\alpha\) satisfies the relevant parts of the assumption set in place of \(\Phi\). This covers cases where the contract space contains contracts with different functional forms (e.g., linear vs. concave wage schedules) but each functional family is itself well-ordered by \(\succ_{\text{AI}}\).
A direct consequence of Assumption 4 is that the four-stage game is equivalent to a standard two-stage signaling game with messages \(\phi\in\Phi\) and receiver actions \(a_A\in A_A\), principal payoff \(U_P(a_A,\phi,\theta)\), and agent payoff \(U_A(a_A,\phi,\theta)\). The Intuitive Criterion of chkr1987? thus applies directly.
I next show that under symmetric information, the model admits a clean benchmark monotone in the principal’s type.
Proposition 1 (Symmetric-information benchmark). Suppose Assumptions 1–4 hold. Consider the game with commonly known type \(\theta\in\Theta\). A subgame-perfect equilibrium (SPE) exists, and the principal’s payoff is the same across all SPEs; denote it by \(U_P^{\text{SI}}(\theta)\). Moreover, \(U_P^{\text{SI}}(\theta)\) is weakly increasing in \(\theta\).
Under symmetric information, PBE degenerates to SPE: there is nothing to signal because the type is publicly known. The monotonicity of \(U_P^{\text{SI}}(\theta)\) in \(\theta\) formalizes the role of commitment power as a contracting asset. A higher type can always replicate a lower type’s behavior, and Assumption 2 ensures she does at least as well. Let \(\Phi^{\text{SI}}(\theta)\) denote the set of contracts the type-\(\theta\) principal offers on the equilibrium path in some SPE of the symmetric-information game.
The monotonicity proof in Appendix 9.1 is more involved than a benchmark claim suggests. The complication is that, although the principal type changes monotonically, the agent’s best response to a fixed contract may not. Assumption 2 restricts only \(u_P^D\) and leaves the dependence of \(u_A^D\) on \(\theta\) unrestricted. The proof therefore constructs a default-proof variant of any benchmark contract, a contract under which the agent’s best response is independent of belief and type. Higher types can replicate the lower type’s benchmark payoff by offering this variant. The default-proof contract is the natural tool for replicating benchmark payoffs across the type space, and it reappears in the converse part of the main theorem (Section 5).
The substantive assumption of Costly Incentives does not directly mention the deviation argument used in the proof of the main theorem. The next proposition bridges this gap. Under the substantive and technical assumptions, Costly Incentives implies a substitution property between contract and credibility, namely the existence of an alternative contract that “buys” credibility at the cost of slightly lower agent action.
Proposition 2 (Contract-credibility substitution). Suppose Assumptions 1–4 hold. For every undominated \(\phi\in\Phi\) and every belief \(\mu\) rationalizing an action \(a_A\in\text{RA}_A^\mu(\phi)\) with partial credibility \(C^\mu(a_A,\phi)\in(0,1)\), there exists \(\phi'\in\Phi\) such that
every action rationalized by the best belief under \(\phi'\) is strictly lower than \(a_A\): \(\text{RA}_A^{\overline{\theta}}(\phi')\subseteq[\underline{a}_A,a_A)\);
some such lower action makes the honoring type strictly better off: \(u_P^H(a_A',\phi')>u_P^H(a_A,\phi)\) for some \(a_A'\in\text{RA}_A^{\overline{\theta}}(\phi')\).
In words, whenever a contract is honored only by some types in the population, there exists an alternative contract such that, when the agent treats this alternative as offered by the most committed type, (i) it induces a lower agent action, but (ii) the honoring types of the principal are strictly better off. The contract \(\phi'\) thus represents a deviation that “buys” credibility (by triggering the best belief) at the cost of slightly lower action. The proof, in Appendix 9.2, constructs \(\phi'\) by moving along a continuous path in \(\Phi\) that decreases the agent-incentive order. The path’s existence comes from Assumption 4(iii) (path-connectedness and the connected incentive set), the one-dimensional ordering comes from Assumption 4(ii), and the strict improvement in the honoring payoff comes from Costly Incentives (Assumption 3).
I first define the Intuitive Criterion in the context of the model. Recall from Section 4.2 that the four-stage game reduces to a standard signaling game with principal payoff \(U_P\) and agent payoff \(U_A\), to which the Intuitive Criterion applies directly.
Definition 1 (Intuitive Criterion). Fix a PBE \((\sigma_P,d_A,\sigma_A,d_P,\mu)\) and let \(U_P^*(\theta)\) denote the type-\(\theta\) principal’s expected equilibrium payoff. For any contract \(\phi'\in\Phi\) off the equilibrium path, define \[P(\theta|\phi'):=\{\sigma\in\Delta A_A:\sigma\in\text{MRA}_A^\nu(\phi')\text{ for some }\nu\in\Delta\Theta\text{ and }U_P(\sigma,\phi',\theta)\geq U_P^*(\theta)\},\] the set of mixed agent responses to \(\phi'\) that are rationalizable under some belief and that make the deviation weakly profitable for \(\theta\). The PBE survives the Intuitive Criterion if, for every off-path \(\phi'\), the existence of some \(\theta'\in\Theta\) with \(P(\theta'|\phi')\neq\emptyset\) implies \[\mu(\{\theta\in\Theta:P(\theta|\phi')=\emptyset\}\,|\,\phi')=0.\] A PBE surviving the Intuitive Criterion is called an intuitive PBE.
In plain language, after observing an off-path contract \(\phi'\), the agent should put zero probability on any type \(\theta\) who is strictly worse off from \(\phi'\) than her equilibrium payoff under every rationalizable agent response. The requirement applies only when some other type \(\theta'\) could weakly benefit from \(\phi'\) under some rationalizable response. The content of the Intuitive Criterion is to rule out off-path beliefs that “punish” deviators who would never deviate.
The Intuitive Criterion is the weakest of the standard signaling refinements, which makes the paper’s positive result especially strong. By Assumption 4 and Assumption 1(i), the equivalent signaling game is monotonic in the sense of chso1990?: the principal’s payoff \(U_P(a_A,\phi,\theta)\) is strictly increasing in the agent action \(a_A\). In monotonic signaling games, chso1990? establish that the Intuitive Criterion coincides with universal divinity, never-a-weak-best-response, and strategic stability. Theorem 2 therefore holds under every standard refinement, not just the Intuitive Criterion. Using the weakest refinement is a feature, not a limitation.
I now state the main theorem. Two objects from earlier sections appear in the statement. Here the lowest type \(\underline\theta\) is the weakest type of the introduction, the one with the least commitment power. \(U_P^{\text{SI}}(\underline\theta)\) is its principal payoff in the symmetric-information benchmark (Proposition 1). \(\Phi^{\text{SI}}(\underline\theta)\) is the set of contracts offered on path by the lowest type in some SPE of that benchmark. Two new objects are needed to state the converse. Let \(U_P^{\text{IE}}(\theta)\) denote the type-\(\theta\) principal’s expected payoff in a given intuitive PBE. A contract \(\phi\) is default-proof if \(U_A(\cdot,\phi,\underline\theta)\) and \(u_A^H(\cdot,\phi)\) have the same maximizer. Let \(a_A^H:=\arg\max_a u_A^H(\cdot,\phi)\) denote this common maximizer. By Assumption 4(i), \(\underline\theta\) honors \(\phi\) at \(a_A^H\). By Assumption 2, every \(\tau\geq\underline\theta\) also honors there, so under any belief on \(\{\tau\in\Theta:\tau\geq\underline\theta\}\) the agent’s expected payoff at \(a_A^H\) equals \(u_A^H(a_A^H,\phi)\). At any other action \(a\), types that honor at \(a\) contribute \(u_A^H(a,\phi)\) and types that renege contribute \(u_A^D(a,\tau)\leq u_A^H(a,\phi)\) (by aligned indifference, since reneging requires \(a>a_A^*(\phi,\tau)\)), so the convex combination is bounded above by \(u_A^H(a,\phi)<u_A^H(a_A^H,\phi)\) (strict concavity of \(u_A^H\)). Hence the agent’s best response is \(a_A^H\) and the principal’s payoff is \(u_P^H(a_A^H,\phi)\), belief-independent and type-independent across \(\{\tau\geq\underline\theta\}\). Let \(\Phi^{\text{DP}}\) denote the set of default-proof contracts.
Theorem 2 (Main result). Suppose Assumptions 1–4 hold. Then:
In every intuitive PBE, every type \(\theta\)’s equilibrium payoff equals the lowest-type symmetric-information payoff: \[U_P^{\text{IE}}(\theta)=U_P^{\text{SI}}(\underline\theta),\quad\text{for every }\theta\in\Theta.\] Moreover, every contract offered on path lies in \(\Phi^{\text{SI}}(\underline\theta)\).
Conversely, every principal contract-offering strategy \(\sigma_P:\Theta\rightarrow\Delta(\Phi^{\text{SI}}(\underline\theta)\cap\Phi^{\text{DP}})\) is supported in some intuitive PBE. In particular, \(\Phi^{\text{SI}}(\underline\theta)\cap\Phi^{\text{DP}}\neq\emptyset\), so an intuitive PBE exists.
Theorem 2 has two implications worth stressing. First, hidden commitment power generates the maximal possible contracting distortion. Compared to the symmetric-information benchmark, every type behaves as if she were the lowest type, and no type obtains a payoff above the worst-case benchmark through contractual choice. Second, the prediction is sharp. Without refinement, the game sustains a vast range of outcomes: a continuum of default-separation equilibria with widely varying contracts, actions, and payoffs, as the debt example shows. The Intuitive Criterion collapses this entire range to one: a single payoff, the lowest type’s benchmark, even though several contracts may still support it.
The result also overturns a natural intuition that signaling should at least sometimes help. In standard signaling problems, higher-quality senders incur costly signaling actions to separate and extract surplus from uninformed receivers. Here, by contrast, a higher type who tries to break away from the worst-case pool through default-separation is overturned by the Intuitive Criterion, while full pooling on a contract the weakest type can honor is left intact. The asymmetry leaves only the worst pooling outcome.
The proof of part (i) proceeds in three steps. First, I show that every on-path contract must be honored with full credibility (\(C^\mu=1\)) in any intuitive PBE. Second, I argue that the lowest type’s payoff in any intuitive PBE cannot exceed the lowest-type benchmark \(U_P^{\text{SI}}(\underline\theta)\). Third, I show that every type can guarantee at least \(U_P^{\text{SI}}(\underline\theta)\) by mimicking a benchmark offer of the lowest type, so \(U_P^{\text{IE}}(\theta)\geq U_P^{\text{SI}}(\underline\theta)\) for every \(\theta\). The chain of monotonicity inequalities then forces equality.
The first step is the heart of the argument and the source of the powerlessness conclusion. Suppose for contradiction that some on-path contract \(\phi\) is honored with partial credibility, that is, some types in the support of \(\mu\) honor \(\phi\) while others renege. By Proposition 2 (Costly Incentives plus the technical assumptions), there exists an alternative contract \(\phi'\) that, under the best belief \(\overline{\theta}\), induces a strictly lower action \(a_A'<a_A\). The same \(\phi'\) delivers a strictly higher honoring payoff for the principal. I show that, after \(\phi'\), the intuitive belief must concentrate on the honoring types of \(\phi\), and these types do honor \(\phi'\) as well. The honoring types are therefore strictly better off by deviating to \(\phi'\) (because \(\phi'\) has strictly higher honoring payoff), contradicting the equilibrium. A separate argument handles the case where no one honors \(\phi\) on path. In that case I construct an intermediate contract that the highest pooling type just barely honors, and reduce to the partial-credibility case.
A fully rigorous, step-by-step proof is given in Appendix 9.
The contracting result has a direct implication for information design before contracting begins. Consider a designer who commits to a Bayes-plausible signal \(\pi\in\Delta(\Delta\Theta)\), a distribution over posterior beliefs \(\mu\) that averages to the prior \(\mu^0\). The designer reveals the posterior publicly before the principal-agent game begins. By Theorem 2, the contracting outcome under posterior \(\mu\) depends only on \(\min\mathop{\mathrm{supp}}\mu\), the lowest type in the posterior’s support, so designer welfare is \[\label{designer} \int_{\Delta\Theta}V(\min\mathop{\mathrm{supp}}\mu)\,d\pi(\mu)-C(\pi),\tag{10}\] where \(V(\theta):=U_P^{\text{SI}}(\theta)\) is the symmetric-information benchmark payoff (or any weakly increasing welfare measure) and \(C(\pi)\geq 0\) is the cost of implementing the signal.
Corollary 1 (Worst case is the only handle). Designer welfare 10 depends on each posterior only through \(\min\mathop{\mathrm{supp}}\mu\). A signal inducing a posterior with full support yields that posterior’s welfare component \(V(\underline\theta)\), which is the no-disclosure payoff. Strictly improving designer payoff over no disclosure requires inducing at least one posterior with \(\min\mathop{\mathrm{supp}}\mu>\underline\theta\).
Corollary 1 delivers the policy lesson: only measures that improve the worst case have value. In the absence of disclosure cost, full revelation is optimal. In the presence of disclosure cost, the designer often prefers coarser signals. The leading institutional example is credit rating, where rating agencies aggregate information about issuer creditworthiness and publish a discrete grade. Suppose disclosure cost is a complexity cost: \(C(\pi)\) is weakly increasing in \(|\mathop{\mathrm{supp}}\pi|\), the number of distinct signal realizations. The motivation is that each additional category imposes operational, governance, and monitoring costs that scale with category count.
Call a signal \(\pi\) monotone-partitional if there exist thresholds \(\underline\theta=\theta_1<\theta_2<\cdots<\theta_n<\theta_{n+1}=\overline{\theta}\) such that, conditional on \(\theta\in[\theta_j,\theta_{j+1})\), \(\pi\) realizes the restricted posterior \(\mu^0|_{[\theta_j,\theta_{j+1})}\).
Proposition 3 (Monotone-partitional optimality under complexity cost). Suppose \(V\) is weakly increasing and \(C\) is a complexity cost. Then the designer’s problem 10 admits an optimal signal, and there exists an optimal signal that is monotone-partitional. If, in addition, \(V\) is strictly increasing on \(\Theta\) and \(C\) is strictly increasing in \(|\mathop{\mathrm{supp}}\pi|\), then every optimal signal is monotone-partitional.
The proof, in Appendix 9, constructs the dominating monotone partition directly. The result provides a contracting-theoretic foundation for the threshold structure of long-term credit rating scales: each rating category is anchored by its lower bound, which determines the contracting outcome for all issuers in the band. This is exactly the structure observed in S&P and Moody’s grades.
This section discusses extensions of the main analysis. I sketch modified assumptions and conclusions for several extensions and argue why the conclusion holds in each. Full proofs are not provided, because the technical details differ from the main result mainly in bookkeeping rather than in conceptual structure.
The strong prediction of Theorem 2, full pooling at the lowest-type benchmark, relies on the bare setup: a one-shot contracting interaction, no public information about \(\theta\) at the time of contracting, and no principal-side enforcement device other than the contract itself. Real-world contracting routinely involves institutional remedies for each of these. Public information: rating agencies, audited financials, and observable market conditions provide partial disclosure about commitment power before contracting. Section 6 shows how such disclosure restores scope for separation. Repeated interaction: when the principal and agent expect future trade, reputation effects substitute for the contract as a signal of the principal’s willingness to honor, and the powerlessness result is dulled. Principal-side enforcement devices: collateral, covenants, sinking funds, and third-party guarantors convert part of the principal’s default option into a contractible asset, observable to the agent, that bounds the privately-known dimension. The theorem identifies the baseline force that these institutions must overcome. Observing that bond markets work, that high-credibility central banks exist, and that procurement contracts are signed is not evidence against the mechanism, but evidence that institutions have evolved to defuse it. The empirical content of the paper is therefore this. In environments where the institutional remedies are weak or absent, hidden commitment should generate the threshold-like pooling the theorem describes. Where remedies are strong, the theorem’s prediction is correspondingly attenuated.
Theorem 2 is stated for finite \(\Theta\), which is the standard domain for the Intuitive Criterion. Extending to continuous \(\Theta\subset\mathbb{R}\) raises two issues: existence of intuitive PBEs and definition of the Intuitive Criterion at off-path messages. Both can be handled by approximation.
Given a continuous \(\Theta=[\underline\theta,\overline{\theta}]\) with prior \(\mu^0\) admitting a continuous density, consider a sequence of finite approximations \(\Theta_n\subset\Theta\) with \(\Theta_n\to\Theta\) in the Hausdorff metric. Let the induced finite priors \(\mu^0_n\) converge weakly to \(\mu^0\). Theorem 2 applies to each \((\Theta_n,\mu^0_n)\), yielding an intuitive PBE with every type’s payoff equal to \(U_{P,n}^{\text{SI}}(\underline\theta_n)\), where \(\underline\theta_n=\min\Theta_n\). By continuity of \(U_P^{\text{SI}}\) in \(\theta\) (which follows from continuity of payoffs and Berge’s theorem in the symmetric-information benchmark), \(U_{P,n}^{\text{SI}}(\underline\theta_n)\to U_P^{\text{SI}}(\underline\theta)\). The natural limit of these intuitive PBEs is a strategy profile in which every type plays a contract supporting the lowest-type benchmark. The formal definition of an “intuitive PBE” in the continuous-type model requires care, since the Intuitive Criterion’s belief restriction has measure-theoretic subtleties at off-path messages. Even so, the limit profile inherits the powerlessness conclusion: hidden commitment power is powerless along the limit.
The analysis assumes contracts are accepted on the equilibrium path. If the outside option \((a_A^0,a_P^0)\) is itself rationalizable, then the agent may reject some contracts. Suppose there exists \(\phi^r\in\Phi\) such that the agent rejects \(\phi^r\) regardless of his belief, that is, \(u_A(a_A^0,a_P^0)>U_A(a_A,\phi^r,\mu)\) for every \(a_A\in\text{RA}_A(\phi^r)\) and every \(\mu\). Then \(\phi^r\) is dominated for the principal as long as the outside option gives her at most \(U_P^{\text{SI}}(\underline\theta)\). The main result thus generalizes: \(U_P^{\text{IE}}(\theta)=\max\{u_P(a_A^0,a_P^0),U_P^{\text{SI}}(\underline\theta)\}\) for every \(\theta\). The argument is identical to the proof of Theorem 2 with the deviation contract \(\widetilde{\phi}^*\) from Step 3 replaced by the better of \(\widetilde{\phi}^*\) and the outside option. In settings where the outside option strictly exceeds the lowest-type benchmark, hidden commitment can shut down the market for higher types, with every principal forced into a contract that yields exactly the outside-option payoff.
The aligned-indifference component of Assumption 4(i) can fail in an important class of applications in which reneging carries a type-dependent cost. There the principal’s default payoff takes the form \(u_P^D(a_A,\theta)=u_P(a_A,\bar a_P)-c(\theta)\), for a fixed default option \(\bar a_P\) and a type-dependent cost \(c(\theta)\) strictly increasing in \(\theta\). Beyond the paper’s two leading applications, one such setting is monetary policy under intervention risk. A central bank (principal) announces a policy, and the public (agent) responds. The announced policy plays the role of the contract, and an override is reneging: the bank honors only when the temptation to override, the benefit of the fixed default \(\bar a_P\) net of its cost \(c(\theta)\), is small. A more politically insulated bank faces a larger override cost, so greater independence is greater commitment power. The friction is thus again an ex-post enforcement failure governed by the size of the default temptation, now with a type-dependent cost. In such settings, \(u_A^H(\cdot,\phi)\) and \(u_A^D(\cdot,\theta)\) remain strictly concave in \(a_A\), but the principal’s crossing of \(u_P^H\) and \(u_P^D\) no longer coincides with the agent’s crossing of \(u_A^H\) and \(u_A^D\). The principal’s action then jumps between \(\phi(a_A)\) and the fixed default \(\bar a_P\) at a point where the agent strictly prefers one regime to the other, and the agent’s overall payoff 9 jumps with it. The minimum in 9 therefore fails to represent the realized payoff. The realized payoff is non-concave at the principal’s crossing point, and the agent may have multiple best responses. The Intuitive Criterion alone is then not sharp enough to deliver the powerlessness result, but strengthening the refinement to the D1 criterion (chkr1987?) restores it. In Sub-step 1a of the proof of Theorem 2, the deviation \(\phi'\) can be chosen so that the set of agent best responses making the deviation profitable is strictly larger for the highest honoring type than for any reneging type. D1 then forces the posterior at \(\phi'\) to concentrate on the highest honoring type, and the remaining proof steps are unchanged.
The discussion in Section 2.3 described two natural PBE structures, full pooling and default-separation. A richer pattern combining pooling and partial separation also arises.
Example. Take \(\theta_L=\tfrac{1}{25}\), \(\theta_H>\tfrac{1}{4}\), and \(\mu^0_H=0.9\). The following is a PBE.
The low type offers \(r_1=0.5\) with probability one.
The high type offers \(r_1=0.5\) with probability \(\tfrac{4}{9}\) and \(r_2=\tfrac{1-\sqrt{0.2}}{2}\approx0.276\) with probability \(\tfrac{5}{9}\).
On the path induced by \(r_1=0.5\), the two types are pooled with default-separation: only the high type honors. On the path induced by \(r_2\approx0.276\), only the high type offers, with the agent’s posterior degenerate at \(\theta_H\). Off-path beliefs assign probability one to \(\theta_L\) at every other rate, deterring deviation by either type.
By the Intuitive Criterion (and Theorem 1), this PBE does not survive refinement.
This appendix collects rigorous proofs of all formal results. Throughout, fix the model primitives of Section 3 and Assumptions 1–4. As in the body, for any contract \(\phi\) and type \(\theta\), let \(a_A^*(\phi,\theta)\in A_A\) denote the unique crossing point of \(u_P^H(\cdot,\phi)\) and \(u_P^D(\cdot,\theta)\) (Assumption 1(ii)). Extend it to \(\overline{a}_A\) when \(u_P^H>u_P^D\) everywhere and to \(\underline{a}_A\) when \(u_P^H<u_P^D\) everywhere. By Assumption 4(i), \(a_A^*(\phi,\theta)\) is also the unique crossing of \(u_A^H(\cdot,\phi)\) and \(u_A^D(\cdot,\theta)\). Below \(a_A^*(\phi,\theta)\) the principal honors and the agent satisfies \(u_A^H\leq u_A^D\), so \(U_A=u_A^H\). Above \(a_A^*(\phi,\theta)\) the principal reneges and \(U_A=u_A^D\). Let \(a_A(\phi,\theta):=\mathop{\mathrm{argmax}}_{a_A\in A_A}U_A(a_A,\phi,\theta)\) denote the agent’s best response under symmetric information about \(\theta\), which is unique by strict concavity (Assumption 4(i) and the footnote following it). By Berge’s maximum theorem, \(a_A(\phi,\theta)\) is continuous in \(\phi\).
Under symmetric information about \(\theta\), the agent’s posterior is degenerate at \(\theta\) regardless of the contract offered. The game thus reduces to a sequential-move problem in which the principal offers \(\phi\), the agent picks \(a_A(\phi,\theta)\), and the principal honors if \(u_P^H(a_A(\phi,\theta),\phi)\geq u_P^D(a_A(\phi,\theta),\theta)\) and reneges otherwise. Her resulting payoff is \[\label{benchmark-V} V_P(\phi,\theta):=U_P(a_A(\phi,\theta),\phi,\theta)=\max\left\{u_P^H(a_A(\phi,\theta),\phi),u_P^D(a_A(\phi,\theta),\theta)\right\}.\tag{11}\] By Assumption 4(i), the agent’s best response \(a_A(\phi,\theta)\) is single-valued (strict concavity) and continuous in \(\phi\) (Berge’s theorem of the maximum), so \(V_P(\cdot,\theta)\) is continuous in \(\phi\). Since \(\Phi\) is compact (Section 3), the maximum \(\max_\phi V_P(\phi,\theta)\) is attained. Define \(U_P^{\text{SI}}(\theta):=\max_\phi V_P(\phi,\theta)\). By construction, \(U_P^{\text{SI}}(\theta)\) is the principal’s payoff in every SPE of the symmetric-information benchmark.
It remains to show that \(U_P^{\text{SI}}(\theta)\) is weakly increasing in \(\theta\). Fix \(\theta'>\theta\) in \(\Theta\), and let \(\phi^*\in\mathop{\mathrm{argmax}}_\phi V_P(\phi,\theta)\) with induced action \(a^*=a_A(\phi^*,\theta)\). I show that type \(\theta'\) can obtain at least \(U_P^{\text{SI}}(\theta)\) by offering a suitably modified default-proof contract \(\widetilde{\phi}^*\).
Step (i): Construction of \(\widetilde{\phi}^*\). I claim there exists \(\widetilde{\phi}^*\in\Phi\) such that (a) \(a^*\) is the unique maximizer of \(u_A^H(\cdot,\widetilde{\phi}^*)\) over \(A_A\), (b) type \(\theta\) honors \(\widetilde{\phi}^*\) at \(a^*\), and (c) \(u_P^H(a^*,\widetilde{\phi}^*)=V_P(\phi^*,\theta)\).
If \(\phi^*\) itself satisfies (a)–(c), set \(\widetilde{\phi}^*:=\phi^*\) and proceed. Otherwise, consider the following two sub-cases.
Sub-case (i.1): Type \(\theta\) honors \(\phi^*\) at \(a^*\), but (a) fails. Then \(V_P(\phi^*,\theta)=u_P^H(a^*,\phi^*)\). The failure of (a) means that the unique maximizer of \(u_A^H(\cdot,\phi^*)\), call it \(a_A^H(\phi^*)\), satisfies \(a_A^H(\phi^*)\neq a^*\). Since the agent’s best response under symmetric information \(\theta\) is the unique maximizer of \(U_A(\cdot,\phi^*,\theta)=\min\{u_A^H(\cdot,\phi^*),u_A^D(\cdot,\theta)\}\) and type \(\theta\) honors at \(a^*\) (so \(a^*\leq a_A^*(\phi^*,\theta)\)), the relevant region for the agent’s optimum is \([\underline{a}_A,a_A^*(\phi^*,\theta)]\) where \(U_A=u_A^H\). So \(a_A^H(\phi^*)>a_A^*(\phi^*,\theta)\), i.e., \(u_A^H\) keeps increasing past the crossing. Now consider varying \(\phi\) from \(\phi^*\) toward the safe contract \(\phi^0\) along the path-connected set \(\Phi\) (both from Assumption 4(iii)). At \(\phi^0\), \(u_P^H(a,\phi^0)\geq u_P^D(a,\theta)\) for all \(a\), so \(a_A^*(\phi^0,\theta)=\overline{a}_A\) and the agent’s optimum coincides with the unconstrained maximum of \(u_A^H(\cdot,\phi^0)\). By continuity (Assumption 4(i)), as \(\phi\) moves from \(\phi^*\) to \(\phi^0\), \(a_A^*(\phi,\theta)\) varies continuously upward (eventually exceeding \(a_A^H(\phi)\)), so at some intermediate \(\widetilde{\phi}^*\), the unique maximizer of \(u_A^H(\cdot,\widetilde{\phi}^*)\) equals \(a^*\).10 At this \(\widetilde{\phi}^*\), type \(\theta\) still honors at \(a^*\) (since moving toward \(\phi^0\) only weakens the temptation to default), and by continuity, \(u_P^H(a^*,\widetilde{\phi}^*)\) can be tuned via further perturbation to match \(V_P(\phi^*,\theta)\) exactly. (Optimality of \(\phi^*\) in the type-\(\theta\) benchmark prevents \(u_P^H(a^*,\widetilde{\phi}^*)>V_P(\phi^*,\theta)\), so the match is from below.)
Sub-case (i.2): Type \(\theta\) reneges on \(\phi^*\) at \(a^*\). Then \(V_P(\phi^*,\theta)=u_P^D(a^*,\theta)\). Along the connected path from \(\phi^*\) to \(\phi^0\), \(u_P^H(a^*,\phi)\) varies continuously from \(u_P^H(a^*,\phi^*)<u_P^D(a^*,\theta)\) (since \(\theta\) reneges) to \(u_P^H(a^*,\phi^0)\geq u_P^D(a^*,\theta)\) (Assumption 4(iii), safe contract). By the intermediate-value theorem applied to the continuous function \(\phi\mapsto u_P^H(a^*,\phi)\) on the connected set \(\Phi\), there exists \(\widetilde{\phi}^*\) on the path with \(u_P^H(a^*,\widetilde{\phi}^*)=u_P^D(a^*,\theta)=V_P(\phi^*,\theta)\). At this \(\widetilde{\phi}^*\), type \(\theta\) is exactly at the crossing \(a_A^*(\widetilde{\phi}^*,\theta)=a^*\), so she honors at \(a^*\). To ensure (a), further perturb \(\widetilde{\phi}^*\) as in Sub-case (i.1).
In both sub-cases, the constructed \(\widetilde{\phi}^*\) satisfies (a)–(c).
Step (ii): \(\widetilde{\phi}^*\) is default-proof and agent-best-responds with \(a^*\) regardless of belief. By (a), \(a^*\) is the unique maximizer of \(u_A^H(\cdot,\widetilde{\phi}^*)\). By (b) and Assumption 2, every type \(\tau\geq\theta\) honors \(\widetilde{\phi}^*\) at \(a^*\), so \(a^*\leq a_A^*(\widetilde{\phi}^*,\tau)\) for all \(\tau\geq\theta\). For any belief \(\mu\in\Delta\Theta\) with support contained in \(\{\tau\in\Theta:\tau\geq\theta\}\), the agent’s payoff \(U_A(a^*,\widetilde{\phi}^*,\mu)=u_A^H(a^*,\widetilde{\phi}^*)\) (because \(a^*\) is in the honoring region of every type in \(\mathop{\mathrm{supp}}\mu\)). For any other action \(a\neq a^*\) in the honoring region of all types in \(\mathop{\mathrm{supp}}\mu\), \(U_A(a,\widetilde{\phi}^*,\mu)=u_A^H(a,\widetilde{\phi}^*)<u_A^H(a^*,\widetilde{\phi}^*)\) by (a). For actions \(a\) in which some type in \(\mathop{\mathrm{supp}}\mu\) reneges, the agent’s payoff equals a convex combination of \(u_A^H\) (for honoring types) and \(u_A^D(\cdot,\tau)\) (for reneging types \(\tau\)); this convex combination is at most \(u_A^H(a,\widetilde{\phi}^*)\) at such \(a\), hence at most \(u_A^H(a^*,\widetilde{\phi}^*)\). Therefore \(a^*\) is the agent’s best response to \(\widetilde{\phi}^*\) under any belief supported on \(\{\tau\geq\theta\}\).
Step (iii): Type \(\theta'\)’s payoff. Type \(\theta'\) in the symmetric-information benchmark can offer \(\widetilde{\phi}^*\). The agent (with belief \(\delta_{\theta'}\) supported on \(\{\tau\geq\theta\}\)) responds with \(a^*\), and type \(\theta'\) honors \(\widetilde{\phi}^*\) at \(a^*\) (by Step (ii) and Assumption 2). Her payoff is \(u_P^H(a^*,\widetilde{\phi}^*)=V_P(\phi^*,\theta)=U_P^{\text{SI}}(\theta)\). Hence \(U_P^{\text{SI}}(\theta')\geq U_P^{\text{SI}}(\theta)\). \(\blacksquare\)
Fix an undominated \(\phi\in\Phi\) and a belief \(\mu\in\Delta\Theta\) rationalizing some \(a_A\in\text{RA}_A^\mu(\phi)\) with partial credibility \(C^\mu(a_A,\phi)\in(0,1)\). I construct a contract \(\phi'\in\Phi\) satisfying parts (i) and (ii) of the proposition.
By Assumption 4(iii) (richness of contract space), the set \(\cup_{\phi\in\Phi}\text{RA}_A^{\overline{\theta}}(\phi)\) is a (closed) interval \([a_A^\ell,a_A^h(\overline{\theta})]\), and any \(\phi\in\Phi\) inducing the lowest action \(a_A^\ell\) under belief \(\overline{\theta}\) is dominated. Since \(\phi\) is undominated, \(a_A>a_A^\ell\). By the same assumption, there exists \(\widetilde{\phi}\in\Phi\) such that \(\text{RA}_A^{\overline{\theta}}(\widetilde{\phi})\ni a_A^\ell\) (taking, e.g., any contract that induces \(a_A^\ell\) as agent best response under belief \(\overline{\theta}\)). The function \(\phi\mapsto\text{RA}_A^{\overline{\theta}}(\phi)\) is continuous in \(\phi\) (Berge’s theorem, using Assumption 4(i)) and single-valued by the strict concavity in Assumption 4(i). Hence, along a continuous path in the path-connected set \(\Phi\) (Assumption 4(iii)) from \(\phi\) to \(\widetilde{\phi}\), the agent’s best response under \(\overline{\theta}\) varies continuously from \(\text{RA}_A^{\overline{\theta}}(\phi)\) to \(a_A^\ell<a_A\).
By Assumption 4(ii) (one-dimensional incentive), \(\succsim_{\text{AI}}\) is a complete order on \(\Phi\), and the path strictly descends it: any contract strictly past \(\phi\) on the path satisfies \(\phi'\prec_{\text{AI}}\phi\) (in particular, \(\widetilde{\phi}\prec_{\text{AI}}\phi\), since \(\widetilde{\phi}\) induces a strictly lower action under \(\overline{\theta}\)). The real-valued map \(\phi\mapsto\text{RA}_A^{\overline{\theta}}(\phi)\) is continuous on the connected path (Berge with Assumption 4(i)), varying from \(a_A(\phi,\overline{\theta})\geq a_A\) at one end to \(a_A^\ell<a_A\) at the other (since \(\overline{\theta}\) honors \(\phi\) at \(a_A\) by \(C^\mu(a_A,\phi)>0\) and \(\overline{\theta}\) being the highest type). By the intermediate-value theorem, there exists \(\phi'\) on the path with \(\phi'\prec_{\text{AI}}\phi\) and \(\text{RA}_A^{\overline{\theta}}(\phi')\subsetneq[a_A^\ell,a_A(\phi,\overline{\theta}))\), in particular \(\text{RA}_A^{\overline{\theta}}(\phi')\subseteq[\underline{a}_A,a_A)\), verifying part (i).
For part (ii), by Assumption 3 (Costly Incentives), \(\phi'\prec_{\text{AI}}\phi\) implies \(u_P^H(a,\phi')>u_P^H(a,\phi)\) for all \(a\in A_A\). In particular, at the (unique) action \(a_A'\in\text{RA}_A^{\overline{\theta}}(\phi')\), \(u_P^H(a_A',\phi')>u_P^H(a_A',\phi)\). As \(\phi'\to\phi\) along the path, \(a_A'=\text{RA}_A^{\overline{\theta}}(\phi')\to\text{RA}_A^{\overline{\theta}}(\phi)=a_A\) by Berge’s theorem (continuity of the best-response function), and so by continuity of \(u_P^H\) in \((a_A,\phi)\), \(u_P^H(a_A',\phi)\to u_P^H(a_A,\phi)\). Choosing \(\phi'\) sufficiently close to \(\phi\) ensures \(u_P^H(a_A',\phi')>u_P^H(a_A,\phi)\). This verifies part (ii). \(\blacksquare\)
Theorem 1 is a corollary of Theorem 2 applied to the debt-issuance example. I verify Assumptions 1–4 and identify the relevant quantities.
Verification of Assumption 1. The issuer’s honoring payoff is \(u_P^H(i,r)=i-ri\) and default payoff is \(u_P^D(i,\theta)=2i-\theta\). Part (i): both are strictly increasing in \(i\) for \(r\in[0,1]\) (derivatives \(1-r\) and \(2\) respectively). Part (ii): \(u_P^H-u_P^D=i-ri-(2i-\theta)=\theta-i-ri\), which is strictly decreasing in \(i\) and equals zero at \(i=\theta/(1+r)=:a_A^*(\phi,\theta)\); below this point \(u_P^H>u_P^D\) and above it \(u_P^H<u_P^D\). Part (iii): \(u_A^H(i,r)-u_A^D(i,\theta)=(ri-\tfrac12i^2)-(\theta-i-\tfrac12i^2)=ri+i-\theta=(1+r)i-\theta\), strictly increasing in \(i\) for any \(r\geq 0\).
Verification of Assumption 2. \(u_P^D(i,\theta)=2i-\theta\) is strictly decreasing in \(\theta\).
Verification of Assumption 3. Identify the contract \(\phi:i\mapsto ri\) with the scalar \(r\in[0,1]=:\Phi\). The AI order \(\succ_{\text{AI}}\) ranks contracts by the marginal incentive \(\partial u_A^H/\partial i=r-i\), hence by \(r\); specifically, \(r_1>r_2\) iff \(\phi_{r_1}\succ_{\text{AI}}\phi_{r_2}\) (the slope \(r\) is strictly higher and the level difference \(u_A^H(i,r_1)-u_A^H(i,r_2)=(r_1-r_2)i\) is strictly increasing in \(i\)). Higher \(r\) gives strictly lower issuer honoring payoff: \(u_P^H(i,r_1)=i-r_1i<i-r_2i=u_P^H(i,r_2)\) for \(r_1>r_2\). Hence Costly Incentives holds.
Verification of Assumption 4. (i) All payoffs are continuous (polynomial in \(i\), \(r\), \(\theta\)); \(u_A^H(i,r)=ri-\tfrac12i^2\) and \(u_A^D(i,\theta)=\theta-i-\tfrac12i^2\) are strictly concave and differentiable in \(i\); and \(u_P^H-u_P^D=\theta-i-ri\) and \(u_A^H-u_A^D=(1+r)i-\theta\) are negatives of each other up to sign, so \(u_P^H\leq u_P^D\) iff \(u_A^H\geq u_A^D\). (ii) The AI order ranks \(\Phi=[0,1]\) by \(r\) (as verified for Assumption 3), so \(\succsim_{\text{AI}}\) is a complete order on \(\Phi\). (iii) \(\Phi=[0,1]\) is path-connected; the safe contract is \(r=0\), which gives \(u_P^H(i,0)=i\geq 2i-\theta=u_P^D(i,\theta)\) iff \(\theta\geq i\), which holds for rationalizable \(i\) (with optimal \(i\leq r\) small), and alternatively extending the contract space to allow \(r<0\) gives an obvious safe contract; finally, the set of rationalizable agent actions under symmetric information \(\theta\) is \(\cup_r\{a_A(r,\theta)\}=[0,i^{\text{SI}}(\theta)]\), a connected interval starting at \(i=0\), induced by \(r=0\), a dominated contract since it yields the issuer the lowest possible payoff.
Symmetric-information benchmark. For each \(\theta\), the type-\(\theta\) benchmark SPE involves the issuer offering the highest \(r\) she can honor, equal to \(r^{\text{SI}}(\theta)=\frac{\sqrt{4\theta+1}-1}{2}\) for \(\theta\leq\tfrac{3}{4}\) and \(r^{\text{SI}}(\theta)=\tfrac12\) for \(\theta\geq\tfrac{3}{4}\). The investor’s best response is \(i^{\text{SI}}(\theta)=r^{\text{SI}}(\theta)\), and the issuer’s payoff is \(U^{\text{SI}}(\theta)=r^{\text{SI}}(\theta)-r^{\text{SI}}(\theta)^2\).
Applying Theorem 2. Since all assumptions hold, Theorem 2 applies, yielding \(U_P^{\text{IE}}(\theta)=U^{\text{SI}}(\underline\theta)\) for all \(\theta\), and every on-path contract lies in \(\Phi^{\text{SI}}(\underline\theta)=\{r^{\text{SI}}(\underline\theta)\}\). The investor’s response is \(i^{\text{SI}}(\underline\theta)=r^{\text{SI}}(\underline\theta)\). This is Theorem 1. \(\blacksquare\)
The proof proceeds in five steps. Step 1 is the heart of the argument: it shows that every on-path pooling contract is honored with full credibility, by constructing a profitable deviation for honoring types whenever credibility is partial. Steps 2–5 then follow by bookkeeping: Step 2 establishes the upper bound \(U_P^{\text{IE}}(\underline\theta)\leq U_P^{\text{SI}}(\underline\theta)\), Step 3 the matching lower bound for every type via a default-proof deviation, Step 4 concludes part (i), and Step 5 proves the converse part (ii).
Claim. In every intuitive PBE, every on-path contract \(\phi\) offered by multiple types satisfies \(C^\mu(a_A,\phi)=1\) at the on-path action \(a_A\), where \(\mu\) is the on-path posterior given \(\phi\).
Suppose for contradiction that \(C^\mu(a_A,\phi)<1\). Two sub-cases arise.
Case 1: \(C^\mu(a_A,\phi)\in(0,1)\). By Proposition 2, there exists \(\phi'\in\Phi\) such that (i) \(\text{RA}_A^{\overline{\theta}}(\phi')\subseteq[\underline{a}_A,a_A)\), and (ii) \(u_P^H(a_A',\phi')>u_P^H(a_A,\phi)\) for some \(a_A'\in\text{RA}_A^{\overline{\theta}}(\phi')\). By Assumption 4(i), \(\text{RA}_A^{\overline{\theta}}(\phi')\) is a singleton; denote its element by \(a_A'\). Without loss, \(\phi'\) is off the equilibrium path: if \(\phi'\) is on path, then by continuity (Assumption 4(i)) and path-connectedness (Assumption 4(iii)) we can perturb \(\phi'\) to an off-path contract while preserving the strict inequalities of Proposition 2; the on-path set is finite by the PBE restriction (Section 3).
I now apply the Intuitive Criterion to \(\phi'\).
Sub-step 1a: Every reneging type of \(\phi'\) at \(a_A'\) is strictly worse off after deviating to \(\phi'\). Let \(\widetilde{\theta}\) be any type with on-path offer \(\widetilde{\phi}\) and on-path action \(\widetilde{a}_A\), such that \(\widetilde{\theta}\) reneges given \(\phi'\) at \(a_A'\). By the monotonic-signaling structure of the game (Assumption 1(i) and Assumption 2; cf.chso1990?), the agent’s best response is non-decreasing in the belief’s stochastic order, so \(\max\bigcup_{\nu\in\Delta\Theta}\text{MRA}_A^\nu(\phi')=a_A'\), achieved at \(\nu=\delta_{\overline{\theta}}\) (Proposition 2). Hence \(\widetilde{\theta}\)’s deviation payoff is bounded above by \(u_P^D(a_A',\widetilde{\theta})\) (she reneges at any \(\sigma\in[\underline a_A,a_A']\cap(a_A^*(\phi',\widetilde{\theta}),\overline{a}_A]\), and \(u_P^D\) is strictly increasing in \(a_A\) by Assumption 1(i)). By the same Assumption 1(i) and \(a_A'<a_A\), \(u_P^D(a_A',\widetilde{\theta})<u_P^D(a_A,\widetilde{\theta})\). The latter is bounded by \(\widetilde{\theta}\)’s payoff from imitating \(\phi\) at \(a_A\), namely \(\max\{u_P^H(a_A,\phi),u_P^D(a_A,\widetilde{\theta})\}\), which by equilibrium optimality is no greater than her equilibrium payoff \(U_P^*(\widetilde{\theta})\): \[u_P^D(a_A',\widetilde{\theta})<u_P^D(a_A,\widetilde{\theta})\leq\max\{u_P^H(a_A,\phi),u_P^D(a_A,\widetilde{\theta})\}\leq U_P^*(\widetilde{\theta}).\] The deviation payoff for \(\widetilde{\theta}\) at \(\phi'\) is thus strictly less than \(U_P^*(\widetilde{\theta})\), regardless of the agent’s belief, so \(P(\widetilde{\theta}|\phi')=\emptyset\).
Sub-step 1b: Some honoring type of \(\phi\) at \(a_A\) exists in \(\Theta':=\{\theta\in\Theta:\theta\text{ honors }\phi'\text{ at }a_A'\}\), hence \(\Theta'\) is non-empty and contains \(\overline{\theta}\). Since \(C^\mu(a_A,\phi)>0\), some on-path type \(\widetilde{\theta}\) honors \(\phi\) at \(a_A\), so \(u_P^D(a_A,\widetilde{\theta})\leq u_P^H(a_A,\phi)\). Then \[u_P^D(a_A',\widetilde{\theta})<u_P^D(a_A,\widetilde{\theta})\leq u_P^H(a_A,\phi)<u_P^H(a_A',\phi'),\] where the first inequality uses Assumption 1(i), and the third uses Proposition 2(ii). Hence \(\widetilde{\theta}\) honors \(\phi'\) at \(a_A'\), so \(\widetilde{\theta}\in\Theta'\). By Assumption 2, every type \(\tau\geq\widetilde{\theta}\) (including \(\overline{\theta}\)) also honors \(\phi'\) at \(a_A'\), so \(\overline{\theta}\in\Theta'\).
Sub-step 1c: Under any belief supported on \(\Theta'\), the agent’s best response to \(\phi'\) is \(a_A'\). The argument splits on whether \(a_A'\) lies strictly inside \(\overline{\theta}\)’s honoring region.
Case A: \(a_A'<a_A^*(\phi',\overline{\theta})\). I first claim \(a_A'=a_A^H(\phi')\), where \(a_A^H(\phi'):=\arg\max_{a\in A_A}u_A^H(a,\phi')\) (unique by strict concavity in Assumption 4(i)). If \(a_A^H(\phi')>a_A'\), then by strict concavity \(u_A^H\) is strictly increasing on \([a_A',a_A^H(\phi')]\); the open interval \((a_A',\min\{a_A^H(\phi'),a_A^*(\phi',\overline{\theta})\})\) is nonempty, and any action \(a\) in it lies in \(\overline{\theta}\)’s honoring region and yields \(U_A(a,\phi',\overline{\theta})=u_A^H(a,\phi')>u_A^H(a_A',\phi')=U_A(a_A',\phi',\overline{\theta})\), contradicting \(a_A'\) being the agent’s best response under \(\delta_{\overline{\theta}}\). Similarly, if \(a_A^H(\phi')<a_A'\), then \(u_A^H(a_A^H(\phi'),\phi')>u_A^H(a_A',\phi')\) by strict concavity, and \(a_A^H(\phi')\in[\underline a_A,a_A')\subseteq[\underline a_A,a_A^*(\phi',\overline{\theta}))\) lies in \(\overline{\theta}\)’s honoring region, again contradicting the best-response property. Hence \(a_A^H(\phi')=a_A'\).
Now take any belief \(\nu\) supported on \(\Theta'\). By Assumption 2, \(a_A'\leq a_A^*(\phi',\min\Theta')\leq a_A^*(\phi',\tau)\) for every \(\tau\in\Theta'\), so every \(\tau\in\Theta'\) honors \(\phi'\) at any \(a\leq a_A^*(\phi',\min\Theta')\). In particular, \(U_A(a_A',\phi',\nu)=u_A^H(a_A',\phi')\). For \(a\neq a_A'\):
If \(a\leq a_A^*(\phi',\min\Theta')\) and \(a\neq a_A'\): every \(\tau\in\Theta'\) honors at \(a\), so \(U_A(a,\phi',\nu)=u_A^H(a,\phi')<u_A^H(a_A',\phi')\) by strict concavity of \(u_A^H\) around its maximizer \(a_A^H(\phi')=a_A'\).
If \(a>a_A^*(\phi',\min\Theta')\): partition \(\Theta'\) at \(a\) into \(H(a):=\{\tau\in\Theta':a\leq a_A^*(\phi',\tau)\}\) and \(D(a):=\Theta'\setminus H(a)\). Then \[U_A(a,\phi',\nu)=\nu(H(a))\,u_A^H(a,\phi')+\sum_{\tau\in D(a)}\nu(\tau)\,u_A^D(a,\tau).\] For each \(\tau\in D(a)\), \(a>a_A^*(\phi',\tau)\), so by aligned indifference (Assumption 4(i)) \(u_A^D(a,\tau)\leq u_A^H(a,\phi')\). Hence \(U_A(a,\phi',\nu)\leq u_A^H(a,\phi')<u_A^H(a_A',\phi')\) by strict concavity of \(u_A^H\) past its maximizer \(a_A'\).
Therefore \(a_A'\) is the unique maximizer of \(U_A(\cdot,\phi',\nu)\).
Case B: \(a_A'=a_A^*(\phi',\overline{\theta})\). Every \(\tau\in\Theta'\) satisfies \(a_A'\leq a_A^*(\phi',\tau)\), that is, \(a_A^*(\phi',\overline{\theta})\leq a_A^*(\phi',\tau)\). By Assumption 2, \(a_A^*(\phi',\cdot)\) is strictly increasing in \(\tau\), forcing \(\tau=\overline{\theta}\). So \(\Theta'=\{\overline{\theta}\}\), the only belief supported on \(\Theta'\) is \(\delta_{\overline{\theta}}\), and the agent’s best response to \(\phi'\) under \(\delta_{\overline{\theta}}\) is \(a_A'\) by hypothesis.
Sub-step 1d: Contradiction. By Sub-step 1a, \(P(\theta|\phi')=\emptyset\) for every reneging type \(\theta\notin\Theta'\). By Sub-step 1b, the honoring type \(\widetilde{\theta}\) identified there satisfies \(u_P^H(a_A',\phi')>u_P^H(a_A,\phi)\), and by on-path optimality for \(\widetilde{\theta}\) at \(\phi\), \(u_P^H(a_A,\phi)\geq U_P^*(\widetilde{\theta})\); combining, \(\widetilde{\theta}\)’s deviation payoff with action \(a_A'\) strictly exceeds her equilibrium payoff, so \(a_A'\in P(\widetilde{\theta}|\phi')\). The Intuitive Criterion therefore requires the agent’s belief at \(\phi'\) to assign positive probability only to types in \(\Theta'\). The agent’s best response is then \(a_A'\) by Sub-step 1c. The honoring type \(\widetilde{\theta}\) identified in Sub-step 1b strictly prefers \(\phi'\) to her equilibrium offer, contradicting equilibrium. This rules out Case 1.
Case 2: \(C^\mu(a_A,\phi)=0\).11 Let \(\Theta'\) be the support of \(\mu(\cdot|\phi)\), and \(\widetilde{\theta}:=\max\Theta'\). Since \(\widetilde{\theta}\) reneges on \(\phi\) at \(a_A\), \(a_A>a_A^*(\phi,\widetilde{\theta})\). Define \(f(a):=\sum_{\tau\in\Theta'}\mu(\tau|\phi)u_A^D(a,\tau)\), strictly concave by Assumption 4(i).
I claim \(a_A^*(\phi,\widetilde{\theta})\) is weakly below the unique maximizer of \(f\). Suppose not. Then for every \(\tau\in\Theta'\) with \(\tau<\widetilde{\theta}\), \(a_A^*(\phi,\tau)<a_A^*(\phi,\widetilde{\theta})\) by Assumption 2. At \(a_A>a_A^*(\phi,\widetilde{\theta})\), all types in \(\Theta'\) renege, so \(U_A(a_A,\phi,\mu)=f(a_A)\). Since \(f\) has its max below \(a_A^*(\phi,\widetilde{\theta})<a_A\), \(f\) is strictly decreasing at \(a_A\), so the agent strictly prefers slightly lower actions, contradicting \(a_A\in\text{RA}_A^\mu(\phi)\). Hence the maximizer of \(f\) is at or above \(a_A^*(\phi,\widetilde{\theta})\). By Assumption 4(i), the maximizer of \(U_A(\cdot,\phi,\mu)\) is precisely \(a_A\), and since below \(a_A^*(\phi,\widetilde{\theta})\) we have \(U_A=u_A^H<u_A^D\), the equality \(U_A(a_A,\phi,\mu)=f(a_A)\) persists.
Now construct \(\widetilde{\phi}\in\Phi\) such that \(a_A^*(\widetilde{\phi},\widetilde{\theta})=a_A\). By Assumption 4(iii) (safe contract \(\phi^0\)), \(u_A^H(a_A,\phi^0)\geq u_A^D(a_A,\widetilde{\theta})\) (i.e., \(a_A\leq a_A^*(\phi^0,\widetilde{\theta})\)). By contrast, \(u_A^H(a_A,\phi)<u_A^D(a_A,\widetilde{\theta})\) (since \(\widetilde{\theta}\) reneges on \(\phi\) at \(a_A\)). The function \(\phi\mapsto u_A^H(a_A,\phi)\) is continuous (Assumption 4(i)) on the path-connected set \(\Phi\) (Assumption 4(iii)). By the intermediate-value theorem along a continuous path from \(\phi\) to \(\phi^0\), there exists \(\widetilde{\phi}\in\Phi\) on the path with \(u_A^H(a_A,\widetilde{\phi})=u_A^D(a_A,\widetilde{\theta})\), hence \(a_A^*(\widetilde{\phi},\widetilde{\theta})=a_A\). Under \(\widetilde{\phi}\) and belief \(\mu\), the agent’s optimum is still \(a_A\) (since \(f\) is unchanged and the new crossing equals \(a_A\)). Now \(\widetilde{\theta}\) honors \(\widetilde{\phi}\) exactly at \(a_A\), while lower types still renege, so \(C^\mu(a_A,\widetilde{\phi})\in(0,1)\). Applying Case 1’s argument with \(\widetilde{\phi}\) in place of \(\phi\) yields a contradiction. This rules out Case 2.
This completes Step 1.
In any intuitive PBE, the lowest type \(\underline\theta\) either separates (offers a contract by herself) or pools with other types. By Step 1, any pooling contract has \(C^\mu=1\). Two cases:
Case A: \(\underline\theta\) separates. Then her on-path contract induces the agent’s belief \(\delta_{\underline\theta}\). Her payoff is bounded by what she can achieve under symmetric information about \(\underline\theta\), that is, \(U_P^{\text{SI}}(\underline\theta)\).
Case B: \(\underline\theta\) pools with other types on \(\phi\), with \(C^\mu(a_A,\phi)=1\). Every type in the support of \(\mu\) honors \(\phi\) at \(a_A\). In particular, \(\underline\theta\) honors, so \(a_A\leq a_A^*(\phi,\underline\theta)\). By the same argument as Sub-step 1c (with \(\widetilde{\theta}_*=\underline\theta\) now playing the role of the lowest honoring type), the agent’s best response to \(\phi\) under any belief supported on the honoring types is \(a_A\). Concentrating belief on \(\delta_{\underline\theta}\) thus still yields agent best response \(a_A\) and gives \(\underline\theta\) payoff \(u_P^H(a_A,\phi)\). This is achievable under symmetric information about \(\underline\theta\), so \(u_P^H(a_A,\phi)\leq U_P^{\text{SI}}(\underline\theta)\).
Both cases yield \(U_P^{\text{IE}}(\underline\theta)\leq U_P^{\text{SI}}(\underline\theta)\).
I show that every type can guarantee at least \(U_P^{\text{SI}}(\underline\theta)\) by a one-shot deviation to a suitable contract.
By the proof of Proposition 1 (Step (i) construction), there exists \(\widetilde{\phi}^*\in\Phi\) such that \(\underline\theta\) honors \(\widetilde{\phi}^*\) at \(\check a_A:=a_A(\widetilde{\phi}^*,\underline\theta)\), the agent’s best response under \(\delta_{\underline\theta}\) is the unique maximizer \(\check a_A\) of \(u_A^H(\cdot,\widetilde{\phi}^*)\), and \(u_P^H(\check a_A,\widetilde{\phi}^*)=U_P^{\text{SI}}(\underline\theta)\). This contract is default-proof in the sense of Section 5: under any belief \(\mu\), the agent’s best response to \(\widetilde{\phi}^*\) is \(\check a_A\). To see this, note that by Assumption 2, every type \(\tau\geq\underline\theta\) honors \(\widetilde{\phi}^*\) at \(\check a_A\), so \(\check a_A\leq a_A^*(\widetilde{\phi}^*,\tau)\) for all \(\tau\). The agent’s expected payoff under \(\mu\) at action \(\check a_A\) is \(u_A^H(\check a_A,\widetilde{\phi}^*)\), the maximum of \(u_A^H(\cdot,\widetilde{\phi}^*)\). For other actions, the same argument as Sub-step 1c Case A shows that the expected payoff is strictly lower.
Now consider any intuitive PBE and any type \(\theta\). If \(\widetilde{\phi}^*\) is off the equilibrium path, \(\theta\) can deviate to it; if \(\widetilde{\phi}^*\) is on path, a small perturbation within the path-connected \(\Phi\) (Assumption 4(iii)) yields an off-path contract with the same default-proof property. The Sub-case (i.1) construction picks \(\widetilde{\phi}^*\) on the path from \(\phi^*\) toward \(\phi^0\), along which \(a_A^H(\phi)\) strictly decreases (by 8 and strict concavity in Assumption 4(i)) while \(a_A^*(\phi,\underline\theta)\) strictly increases (by Costly Incentives and Assumption 1(ii)); since \(\widetilde{\phi}^*\) sits on this path, \(\check a_A=a_A^H(\widetilde{\phi}^*)<a_A^*(\widetilde{\phi}^*,\underline\theta)\) strictly. Continuity of \(a_A^H(\cdot)\) and \(a_A^*(\cdot,\underline\theta)\) in \(\phi\) (Berge with Assumption 4(i)) preserves this strict inequality under small perturbations, so the resulting off-path contract retains default-proofness, and we may assume \(\widetilde{\phi}^*\) is off path. Under any agent belief at \(\widetilde{\phi}^*\), the agent’s response is \(\check a_A\). Type \(\theta\) honors at \(\check a_A\) (Assumption 2), so her deviation payoff is \(u_P^H(\check a_A,\widetilde{\phi}^*)=U_P^{\text{SI}}(\underline\theta)\). By sequential rationality, her equilibrium payoff \(U_P^{\text{IE}}(\theta)\geq U_P^{\text{SI}}(\underline\theta)\).
Combining Steps 2 and 3, \(U_P^{\text{IE}}(\underline\theta)=U_P^{\text{SI}}(\underline\theta)\). For \(\theta>\underline\theta\), take her on-path offer \(\widetilde{\phi}(\theta)\) inducing \(\widetilde{a}_A(\theta)\). Since \(\widetilde{\phi}(\theta)\) is on path, the agent’s response under Bayes’ rule is \(\widetilde{a}_A(\theta)\); by sequential rationality of \(\underline\theta\) (she cannot profitably imitate \(\widetilde{\phi}(\theta)\) in equilibrium), \[U_P^{\text{IE}}(\underline\theta)\geq U_P(\widetilde{a}_A(\theta),\widetilde{\phi}(\theta),\underline\theta)=\max\{u_P^H(\widetilde{a}_A(\theta),\widetilde{\phi}(\theta)),u_P^D(\widetilde{a}_A(\theta),\underline\theta)\}.\] By Assumption 2, \(u_P^D(\widetilde{a}_A(\theta),\underline\theta)\geq u_P^D(\widetilde{a}_A(\theta),\theta)\), so \[\max\{u_P^H(\widetilde{a}_A(\theta),\widetilde{\phi}(\theta)),u_P^D(\widetilde{a}_A(\theta),\underline\theta)\}\geq\max\{u_P^H(\widetilde{a}_A(\theta),\widetilde{\phi}(\theta)),u_P^D(\widetilde{a}_A(\theta),\theta)\}=U_P^{\text{IE}}(\theta).\] Therefore \(U_P^{\text{IE}}(\theta)\leq U_P^{\text{IE}}(\underline\theta)=U_P^{\text{SI}}(\underline\theta)\). Combined with Step 3, \(U_P^{\text{IE}}(\theta)=U_P^{\text{SI}}(\underline\theta)\).
For the contract claim, take any \(\theta\in\Theta\) offering \(\phi\) on path (inducing \(a_A\)). If \(\underline\theta\) also offers \(\phi\) on path, then by Step 1, the pooling has \(C^\mu=1\), so \(\underline\theta\) honors \(\phi\) at \(a_A\). By the Case-B argument of Step 2, even under symmetric information about \(\underline\theta\), \(\phi\) yields payoff \(U_P^{\text{SI}}(\underline\theta)\), so \(\phi\in\Phi^{\text{SI}}(\underline\theta)\). If \(\underline\theta\) does not offer \(\phi\) on path, suppose \(\underline\theta\) cannot honor \(\phi\) at \(a_A\). Then \(u_P^D(a_A,\underline\theta)>u_P^H(a_A,\phi)\), and by Assumption 2, \(u_P^D(a_A,\underline\theta)>u_P^D(a_A,\theta)\). So \[u_P^D(a_A,\underline\theta)>\max\{u_P^H(a_A,\phi),u_P^D(a_A,\theta)\}=U_P^{\text{IE}}(\theta)=U_P^{\text{IE}}(\underline\theta),\] which would let \(\underline\theta\) profit from deviating to \(\phi\), a contradiction. Hence \(\underline\theta\) honors \(\phi\) at \(a_A\), and by the Case-B argument of Step 2 applied with the belief \(\delta_{\underline\theta}\), the agent’s response under symmetric information about \(\underline\theta\) to \(\phi\) is also \(a_A\), giving \(\underline\theta\) payoff \(u_P^H(a_A,\phi)=U_P^{\text{SI}}(\underline\theta)\). Therefore \(\phi\in\Phi^{\text{SI}}(\underline\theta)\).
I show that every contract-offering strategy \(\sigma_P:\Theta\to\Delta(\Phi^{\text{SI}}(\underline\theta)\cap\Phi^{\text{DP}})\) is supported in an intuitive PBE.
First, \(\Phi^{\text{SI}}(\underline\theta)\cap\Phi^{\text{DP}}\neq\emptyset\). Indeed, by the construction in Step (i) of the proof of Proposition 1, applied to type \(\underline\theta\), there exists a default-proof contract \(\widetilde{\phi}\) achieving \(U_P^{\text{SI}}(\underline\theta)\) in the symmetric-information benchmark for \(\underline\theta\), so \(\widetilde{\phi}\in\Phi^{\text{SI}}(\underline\theta)\cap\Phi^{\text{DP}}\).
Given \(\sigma_P\), construct a PBE: (a) On the equilibrium path, every type’s offer \(\phi\) is default-proof, so the agent’s response is \(a_A(\phi)\) regardless of belief, and the principal’s payoff is \(u_P^H(a_A(\phi),\phi)=U_P^{\text{SI}}(\underline\theta)\) (since every default-proof contract in \(\Phi^{\text{SI}}(\underline\theta)\) yields this payoff to every type, by Assumption 2). The agent’s belief at each on-path \(\phi\) is consistent with Bayes’ rule given \(\sigma_P\). (b) For any off-path \(\phi'\in\Phi\), assign the agent’s belief at \(\phi'\) probability one to \(\underline\theta\). Under this belief, the agent’s best response is \(a_A(\phi',\underline\theta)\), and the principal’s payoff at \(\phi'\) is \(U_P(a_A(\phi',\underline\theta),\phi',\underline\theta)\leq U_P^{\text{SI}}(\underline\theta)\) (by definition of \(U_P^{\text{SI}}(\underline\theta)\) as the maximum payoff in the symmetric-information benchmark for \(\underline\theta\)). Hence no type profits from deviating to any off-path \(\phi'\). By Assumption 2, \(U_P(a_A(\phi',\underline\theta),\phi',\cdot)\) is weakly decreasing in \(\theta\), so no higher type can do strictly better than \(\underline\theta\) at \(\phi'\) under belief \(\delta_{\underline\theta}\).
Finally, verify the Intuitive Criterion. By Assumption 2, for any \(\sigma\in\Delta A_A\) the principal’s payoff \(U_P(\sigma,\phi',\theta)\) is weakly decreasing in \(\theta\) (since \(u_P^D\) is decreasing in \(\theta\) and \(u_P^H\) does not depend on \(\theta\)). All on-path payoffs equal \(U_P^{\text{SI}}(\underline\theta)=U_P^*(\theta)\) for every \(\theta\), so if some \(\theta\) satisfies \(P(\theta|\phi')\neq\emptyset\) via some \(\sigma\in\text{MRA}_A^\nu(\phi')\), then with the same \(\sigma\), \(U_P(\sigma,\phi',\underline\theta)\geq U_P(\sigma,\phi',\theta)\geq U_P^*(\theta)=U_P^*(\underline\theta)\), so \(\sigma\in P(\underline\theta|\phi')\) and hence \(P(\underline\theta|\phi')\neq\emptyset\). Assigning belief \(\delta_{\underline\theta}\) at \(\phi'\) thus satisfies the criterion: it places probability on a type with \(P(\cdot|\phi')\neq\emptyset\).
This completes the proof. \(\blacksquare\)
I first establish a dominance lemma that constructs, for any signal \(\pi\), a monotone-partitional signal \(\widehat\pi\) with weakly higher welfare and weakly lower complexity cost. The proposition then follows.
Setup. Let \(\Theta=\{\theta_1,\theta_2,\ldots,\theta_K\}\) with \(\theta_1<\theta_2<\cdots<\theta_K\), and let \(\mu^0\) be the prior with \(\mu^0(\theta_k)>0\) for every \(k\). A signal \(\pi\in\Delta(\Delta\Theta)\) is a Bayes-plausible distribution over posteriors, that is, a distribution satisfying \(\sum_{\mu\in\mathop{\mathrm{supp}}\pi}\pi(\mu)\mu(\theta)=\mu^0(\theta)\) for every \(\theta\in\Theta\). For a posterior \(\mu\), let \(S(\mu):=\mathop{\mathrm{supp}}\mu\) and \(a(\mu):=\min S(\mu)\in\Theta\). The welfare component of the designer’s payoff is \[W(\pi)=\sum_{\mu\in\mathop{\mathrm{supp}}\pi}\pi(\mu)V(a(\mu)),\] and the cost component is \(C(\pi)\), weakly increasing in \(|\mathop{\mathrm{supp}}\pi|\).
A signal \(\pi\) is monotone-partitional if there exist indices \(0=k_0<k_1<k_2<\cdots<k_n=K\) defining blocks \(B_j=\{\theta_{k_{j-1}+1},\ldots,\theta_{k_j}\}\) for \(j=1,\ldots,n\), such that \(\pi\) has \(n\) realizations \(\mu_1,\ldots,\mu_n\) with \(S(\mu_j)=B_j\) and \(\mu_j\) equal to the restriction of \(\mu^0\) to \(B_j\) (normalized), and \(\pi(\mu_j)=\mu^0(B_j)\).
Dominance lemma. For every signal \(\pi\) with \(|\mathop{\mathrm{supp}}\pi|=n\), there exists a monotone-partitional signal \(\widehat\pi\) with \(|\mathop{\mathrm{supp}}\widehat\pi|\leq n\) and \(W(\widehat\pi)\geq W(\pi)\).
Proof of dominance lemma. Let \(\mathop{\mathrm{supp}}\pi=\{\mu_1,\ldots,\mu_n\}\), \(a_j:=a(\mu_j)\in\Theta\). Let \(A:=\{a_1,\ldots,a_n\}\) (with \(|A|\leq n\) if there are repeats), and order \(A\) as \(\alpha_1<\alpha_2<\cdots<\alpha_m\) where \(m=|A|\leq n\). Note \(\alpha_1=\theta_1\) since the supports of \(\pi\) must cover \(\Theta\) (Bayes plausibility forces \(\mu^0(\theta)>0\) to imply some posterior with \(\theta\) in its support; in particular the lowest \(\theta_1\) must be in some support, so \(\theta_1\in A\)).
Define the monotone partition \(\widehat\pi\) with blocks \(B_j:=\{\theta\in\Theta:\alpha_j\leq\theta<\alpha_{j+1}\}\) for \(j=1,\ldots,m-1\), and \(B_m:=\{\theta\in\Theta:\theta\geq\alpha_m\}\). Then \(\{B_j\}_{j=1}^m\) partitions \(\Theta\). Let \(\widehat\pi\) be the partitional signal that, conditional on \(\theta\in B_j\), deterministically realizes the posterior \(\mu^0\) restricted to \(B_j\) (normalized). By construction, \(|\mathop{\mathrm{supp}}\widehat\pi|=m\leq n\).
To compute welfare, define for each \(\theta\in\Theta\) the index \(\widehat j(\theta)\) such that \(\theta\in B_{\widehat j(\theta)}\). Then \(W(\widehat\pi)=\sum_\theta\mu^0(\theta)V(\alpha_{\widehat j(\theta)})\).
Decomposing \(W(\pi)\) by type: \[\begin{align} W(\pi)&=\sum_{j=1}^n\pi(\mu_j)V(a_j)=\sum_{j=1}^n\pi(\mu_j)\sum_{\theta\in S(\mu_j)}\mu_j(\theta)V(a_j) \\ &=\sum_\theta\sum_{j:\theta\in S(\mu_j)}\pi(\mu_j)\mu_j(\theta)V(a_j) =\sum_\theta\mu^0(\theta)\cdot E\big[V(a_J)\,\big|\,\theta\big], \end{align}\] where the expectation is over the random realization \(J\), conditional on \(\theta\), with weights \(\pi(\mu_j)\mu_j(\theta)/\mu^0(\theta)\) on those \(j\) such that \(\theta\in S(\mu_j)\) (Bayes plausibility ensures these weights sum to 1).
For each \(\theta\) and each \(j\) with \(\theta\in S(\mu_j)\), \(a_j=\min S(\mu_j)\leq\theta\). Since \(a_j\in A\) (it is the minimum of some posterior’s support, hence one of \(\alpha_1,\ldots,\alpha_m\)) and \(\alpha_{\widehat j(\theta)}\) is by construction the largest element of \(A\) at or below \(\theta\), \(a_j\leq\alpha_{\widehat j(\theta)}\). By weak monotonicity of \(V\), \(V(a_j)\leq V(\alpha_{\widehat j(\theta)})\). Taking expectations, \[E\big[V(a_J)\,\big|\,\theta\big]\leq V(\alpha_{\widehat j(\theta)}).\] Summing over \(\theta\) with weights \(\mu^0(\theta)\) gives \(W(\pi)\leq W(\widehat\pi)\). This establishes the dominance lemma.
Conclusion of Proposition 3. By the dominance lemma, for any signal \(\pi\), the monotone-partitional signal \(\widehat\pi\) satisfies \(W(\widehat\pi)\geq W(\pi)\) and \(|\mathop{\mathrm{supp}}\widehat\pi|\leq|\mathop{\mathrm{supp}}\pi|\). Since \(C\) is weakly increasing in \(|\mathop{\mathrm{supp}}\pi|\), \(C(\widehat\pi)\leq C(\pi)\). Combining, the designer’s payoff \(W(\widehat\pi)-C(\widehat\pi)\geq W(\pi)-C(\pi)\). Hence the supremum of the designer’s payoff over all signals equals its supremum over monotone-partitional signals. Existence: the set of monotone partitions of \(\Theta\) is finite (since \(\Theta\) is finite), and the designer’s objective is finite on this set; hence a maximum is attained on this finite set, which is therefore an optimal signal.
Strict version. Suppose, in addition to the standing assumptions, that \(V\) is strictly increasing on \(\Theta\) and \(C\) is strictly increasing in \(|\mathop{\mathrm{supp}}\pi|\). Let \(\pi\) be any signal that is not monotone-partitional. There are two cases. First, suppose \(|\mathop{\mathrm{supp}}\pi|=m\) (the same count as \(\widehat\pi\)). Then \(\pi\) and \(\widehat\pi\) have the same cost. Since \(\pi\) is not monotone-partitional but \(\widehat\pi\) is, the two signals differ on the structure of their supports, and there exists at least one \(\theta\) such that some \(j\) with \(\pi(\mu_j)\mu_j(\theta)>0\) has \(a_j<\alpha_{\widehat j(\theta)}\). By strict monotonicity of \(V\), the expectation inequality \(E[V(a_J)|\theta]<V(\alpha_{\widehat j(\theta)})\) is strict, hence \(W(\pi)<W(\widehat\pi)\). Second, suppose \(|\mathop{\mathrm{supp}}\pi|>m\). Then \(C(\pi)>C(\widehat\pi)\) by strict cost monotonicity, and \(W(\widehat\pi)\geq W(\pi)\) by the dominance lemma. In either case, \(W(\pi)-C(\pi)<W(\widehat\pi)-C(\widehat\pi)\), so \(\pi\) is not optimal. Therefore, under strict \(V\) and strict \(C\), every optimal signal is monotone-partitional. \(\blacksquare\)
Hongcheng Li: Department of Economics, Yale University (e-mail: hongcheng.li@yale.edu).↩︎
The author is grateful to Yeon-Koo Che, Tan Gan, Nima Haghpanah, Marina Halac, Ravi Jagadeesan, Navin Kartik, Elliot Lipnowski, Paulo Natenzon, Daniel Rappoport, Collin Raymond, Andy Skrzypacz, João Thereze, Nicholas Wu, Mu Zhang, and Jidong Zhou for their helpful discussions and comments. I also thank the participants of the seminars at Yale.↩︎
The notion is kin to the imperfect commitment of best2001?, where the principal commits to a contract’s contractible terms but retains ex-post discretion over the payoff-relevant action, here whether to honor or take the default option. Leaning on this lineage, I also call the same friction imperfect contract enforceability, or refer to it by the size of the default temptation.↩︎
The cost captures the operational, governance, and monitoring costs that scale with category count. Section 6 develops this implication for credit rating.↩︎
The investor’s expected payoff in default-separation is \(\mu^0_H(ri-\tfrac{1}{2}i^2)+\mu^0_L(\theta_L-i-\tfrac{1}{2}i^2)\), whose first-order condition with respect to \(i\) gives \(\mu^0_H r-i-\mu^0_L=0\), i.e., \(i=\mu^0_H r-\mu^0_L\). Since this requires \(r>\mu^0_L/\mu^0_H\) for \(i>0\), I focus on rates in this range, which capture all default-separation PBEs of interest.↩︎
Compactness in the sup-norm imposes implicit regularity on \(\Phi\) (such as equicontinuity, by Arzelà-Ascoli for continuous functions). In the canonical applications, \(\Phi\) is a finite-dimensional family parameterized by contract terms (e.g., the interest rate in debt issuance), and the regularity is automatic.↩︎
This restriction is benign: it rules out only the artificial case where the contract space contains infinitely many “names” for behaviorally equivalent contracts.↩︎
The argmax in 6 is attained because, by Assumption 1(ii), the principal’s honoring decision is strict except at a single action \(a_A^*(\phi,\theta)\), and the agent’s payoff (combined with Assumption 4(i)) is continuous in \(a_A\) as a result. The compact \(A_A\) then guarantees the maximum.↩︎
The minimum of two strictly concave functions is strictly concave. For \(a'=\lambda a_1+(1-\lambda)a_2\) and (without loss) \(h(a')=f(a')\), we have \(h(a')=f(a')>\lambda f(a_1)+(1-\lambda)f(a_2)\geq\lambda h(a_1)+(1-\lambda)h(a_2)\).↩︎
More precisely: define \(g(\phi):=a_A^H(\phi)-a^*\in\mathbb{R}\), continuous in \(\phi\). At \(\phi^*\), \(g(\phi^*)>0\). The AI order 8 combined with strict concavity in Assumption 4(i) guarantees that as \(\phi\) descends \(\succsim_{\text{AI}}\), the unconstrained maximizer of \(u_A^H(\cdot,\phi)\) strictly decreases; since the path from \(\phi^*\) to \(\phi^0\) strictly descends \(\succsim_{\text{AI}}\) (Assumption 4(ii)–(iii)), \(g\) strictly decreases along it, and by the intermediate-value theorem, there exists \(\widetilde{\phi}^*\) with \(g(\widetilde{\phi}^*)=0\).↩︎
This case is a technical completeness step. In canonical applications, where contracts that no type honors are dominated by an exit option, the case is vacuous. Including it ensures that the theorem covers the full PBE space without an additional regularity assumption ruling out fully-reneged pools.↩︎