Carrots, Sticks, and Words:

Comparing Forms of Power in International Relations

Alexander Theisen

November 25, 2019

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Abstract

We analyze a classical three-way distinction between means of influence: the

use of threats to deter, the use of promises to compel, and the use of information to

persuade. We present a model of a two-stage interaction which incorporates costly

threats, costly promises, and asymmetric information. We go on to characterize the

relative value of three different types of commitment power: the ability to commit

to specific threats, the ability to commit to specific promises, and the ability to

commit to information disclosure. We prove that, if all three of these tools are

both available to an actor and fully effective on their own, an optimal policy will

still use only one of the three. We also present specific historical cases to which

this model applies, and demonstrate that the parameters and central concepts in

the model can be given concrete interpretations in these settings.

1 Introduction

In the study of power in international relations, a long-standing distinction has been

made between the power to threaten costly sanctions, offer positive incentives, and to

persuade through the use of information.

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The ability to inflict costs upon an opponent allows an actor to engage in deterrence;

by making a credible threat of either economic or military costs contingent upon their

counterpart’s failure to comply with one’s own wishes, one can induce said counterpart

to comply by making non-compliance sufficiently cosly.

Similarly, the ability to make promises of future benefits to a counterpart, in the form

of either direct payments or some more disguised form of transfer (for example, an offer

to purchase military equipment or infrastructure on favored terms), allows said actor to

induce compliance by making compliance sufficiently beneficial. Both of these forms of

power involve taking actions and utilizing resources which directly affect the payoffs of

both actors, thereby falling under the domain of what Joseph Nye (1990) termed “hard

power.”

“Soft power”, in contrast, was described as “[That] which occurs when one country

gets other countries to want what they want,” through either producing some intrinsic

desirability of complying with one’s own wishes, or through persuasion. Persuasion, as

defined by Nye, is

“the use of argument to influence the beliefs and actions of others without

the threat of force or the promise of payment. Persuasion almost always

involves some degree of manipulation, with some points being emphasized

and others neglected. Dishonest persuasion can go so far as to involve fraud

. . . But most arguments involve assertions about facts, values, and framing

that depend upon some degree of trust that the source is credible.” (Nye

2004, p. 93)

There are numerous channels by which the sharing of information and the making of

arguments can allow one international actor to influence the actions of another. In some

cases, it may be that one country possesses verifiable information about the desirability

of some course of action. For example, the U.S.’s (largely failed) attempt to achieve buy-

in among the international community for an invasion of Iraq in 2002 and 2003 rested

almost entirely upon presenting intelligence on the Hussein regime’s pursuit of weapons

of mass destruction, and hinged crucially upon the credibility of its claims.

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In other cases, it may be that a significant share of the returns to cooperation with

a country depend on said country’s willingness to behave conducively relatively far into

the future, or at least past the point at which either country involved can successfully

tie their own hands. The decision of one’s “opponent” to comply with one’s wishes

often comes down to whether or not said opponent expects one to, at some unspecified

point in the future, take advantage of their compliance in a way that harms their own

interests. In situations such as these, controlling the beliefs and information available to

the other country is as important as direct policy concessions or threats. It may even be

that controlling beliefs matters more to achieving success than could be achieved by any

plausible threat.

For example, during Adolf Hitler’s initial program of rearmament and expansion from

1933-1939, a decisive factor in the decision-making of the European powers in response

to Hitler’s aggression lay in their own uncertainty as to the goals Hitler was pursuing. In

the event that Hitler was simply attempting to reconstitute the historical German state

and to achieve full parity with his fellow European powers, the costs of a confrontation

with Hitler in order to directly enforce the Treaty of Versailles would have been too high

relative to the benefits. On the other hand, given Hitler’s (in retrospect, true) objective

of massive expansion into Western Europe, the concessions of Britain, France, and other

European powers in the 1930s, culminating in the Munich Agreement of 1938, would

prove to be devastating.

In the end, maintaining and exploiting this uncertainty via diplomacy proved to be

a crucial asset for Germany in the build-up to World War II (indeed, given that the

very issue at stake was whether or not to permit a rebuilding of German capacity and

industrial strength, direct threats and promises on the Nazis’ part were almost entirely

off the table). Joseph Goebbels, in a 1940 briefing, after hostilities with the Western

powers had begun in earnest, drew a direct connection between Germany’s success and

its skill at influencing “the enemy’s” beliefs:

“Up to now we have succeeded in leaving the enemy in the dark concerning

Germany’s real goals, just as before 1932 our domestic foes never saw where

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we were going . . . They could have arrested a few of us in 1925 and that

would have been that, the end. No, they let us through the danger zone . .

. In 1933 a French premier ought to have said (and if I had been the French

premier I would have said it): ‘The new Reich Chancellor is the man who

wrote Mein Kampf, which says this and that. This man cannot be tolerated

in our vicinity. . .’ But they didn’t do it.” (quoted in Kissinger (1994), p.

295)

As Henry Kissinger (1994, p. 294) would later ruefully describe the situation, “The

tuition fee for learning about Hitler’s true nature was tens of millions of graves.”

In cases such as this, the ability of one country to gain what it wanted depended

as much on its ability to successfully shape the beliefs of its counterparts and convince

them of its trustworthiness as it was in its ability to directly grant concessions or to make

threats; in Hitler’s case, no direct threats or material concessions were ever given in any

of the agreements, beyond largely toothless guarantees of future good behavior. This

influencing of behavior indirectly through beliefs and trust rather than through direct

extrinsic incentives is an example of the power of “soft power.”

Since the origin of the concept of soft power, the idea has taken on currency in pop-

ular discussions of foreign policy. The term was frequently used to characterize Obama

administration policy (Hallams 2011, Rayman 2014), as well as Xi Jinping’s cultural out-

reach initiatives (Albert 2018). Defending a proposed 28% cut to the State Department’s

budget, Office of Management and Budget director Mick Mulvaney explicitly cited Nye’s

distinction between hard and soft power, saying that “[i]t is not a soft-power budget.

This is a hard-power budget, and that was done intentionally. The president very clearly

wants to send a message to our allies and to our potential adversaries that this is a

strong-power administration” (Berman 2017).

In spite of the expansive scope of the idea of soft power, the idea of communication and

the shifting of beliefs as a crucial element of foreign policy and international relations has

lagged behind in its application to formal game theory. Part of this failure lies in the fact

that discussions of “soft power” as an explicit element of foreign affairs tend to eschew

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formal modeling or description in the language of game theory. This paper attempts

to render Nye’s concept of persuasive soft power in formal terms, put them on an even

footing with more traditional studies of military force and economic power, and in so

doing bring the practioner’s terms of international relations into a better correspondence

with the formal approaches that have proved so useful in economics.

In what follows, we present a stylized model of international interaction as a par-

ticularly simple form of Principal-Agent problem with a privately informed Principal,

who may also opt to take costly moves contingent upon an Agent’s dichotomous action.

When viewed this way, one can reframe the question of “soft” and “hard” power as a

study of the most effective means by which one international actor (the Principal), who

is privately informed about the value of a particular course of action, can influence the

behavior of another (the Agent), subject to certain constraints.

“Soft” power, in our framework, can be characterized as the ability to shift one’s

counterpart’s beliefs through the provision of information (“words”), while “hard” power

can be characterized as the ability to use one’s own physical capabilities to carry out

promises (“carrots”) or threats (“sticks”). Given this distinction, one can then ask ques-

tions about the relative value of these different forms of power, ie., when is shifting one’s

opponent’s beliefs more valuable than directly committing to threats or payments? Are

these forms of power “substitutes” or “complements”, from the Principal’s perspective?

Using the literature started by Myerson (1983) on generalized Principal-Agent prob-

lems, as well as recent literature on information design (Kamenica and Gentzkow 2011,

Bergemann and Morris 2016, 2017) we provide results on when one form of power “beats”

another one in our model, and how closely each approximates a hypothetical “first-best”,

in a simple three-stage game of incomplete information. For example, we demonstrate

that in a wide range of cases, all three forms of power function as direct substitutes, with

“words”, for example, sometimes vitiating the need for “carrots” or “sticks” at all, and

vice versa. Moreover, we demonstrate conditions under which neither form of power can

be effective without the other. We then go on to demonstrate analogies between our own

model’s results and important cases in the past.

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In our conclusion, we describe some plausible extensions of the model, and some

fruitful questions raised by this research.

2 Previous Literature

Our tripartite distinction between “carrots”, “sticks”, and “words” is not novel to this pa-

per; as noted above, Nye (1990) highlighted this distinction, but numerous other writers

in political science, sociology, and economics have made similar distinctions. Boulding’s

(1989) categories of “economic”, “threat”, and “integrative” power fits roughly with this

distinction, as does Galbraith’s (1983) classification of “compensatory”, “condign”, and

“conditioning” power. Parsons (1963) distinguishes between “inducement”, “coercion”,

and “persuasion”, and Harsanyi (1962) in a discussion of measuring power in game-

theoretic terms distinguishes between setting up “rewards” and “punishments” and the

supply of “information” as methods of wielding power. With regards to the first two cate-

gories, Schelling (1960) contains extensive discussions of the different strategic properties

of “compellence” and “deterrence”, which Klein and O’Flaherty (1993) characterize in

terms of commitment power in a sequential-move game. None of these authors, however,

have attempted to render the question of the optimal deployment of different forms of

commitment power in the style of this paper.

The bulk of the theoretical contribution of this paper is based upon work originating

with Myerson (1983), who details a generalization of classical Principal-Agent problems,

which allow for both private information and private actions on the part of the Prin-

cipal, as well as the Agent. More recently, Kamenica and Gentzkow (2011) make use

of a concavification result from Aumann and Maschler (1995) to characterize the value

of commitment in a sender-receiver game. This “Bayesian Persuasion” modeling envi-

ronment corresponds to our “soft power” setting in the paper. Bergemann and Morris

(2016) generalize Kamenica and Gentzkow’s information design problem using a different,

linear-programming based method, and Bergemann and Morris (2017), in an overview of

the field, note the analogies between their approach and that of Myerson (1983). The

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only application of Bayesian Persuasion to combine the revelation of information with

the ability to directly affect incentives through transfers is Li (2017), who proves that

limitations on the level of transfers provided by a sender of information can induce the

sender to provide more information while Kolotilin et al (2015) have proved results on

Bayesian Persuasion with a privately informed receiver (Agent). In both of the previously

mentioned papers, the setting is one with a single dichotomous decision, as in this paper.

The Bayesian Persuasion literature has seen numerous applications in the field of po-

litical economy, including Goldstein and Huang (2016), studying regime change among

dictators, and Alonso and Camara (2016a, 2016b) on persuading voters. Information de-

sign has yet to be applied to an international relations setting, to the author’s knowledge.

In the field of international relations, this paper builds off of the wide literature

making use of game theory to study international conflict. Fearon (1995) on rationalist

explanations for war notes “asymmetric information” and “lack of commitment” as two

key explanatory factors in wars; both of these play a key role in our model as well. Our

work is most closely related to work on strategic communication in international settings,

including Baliga and Sjostrom (2004, 2008, 2012) and Jung (2007), who give applications

of the cheap talk literature to arms races, nuclear proliferation, hawk-dove games, and

media manipulation. Sartori (2002) uses cheap talk in a repeated games setting to study

diplomacy and reputation in international negotiations.

3 Baseline (No-Commitment, No-Communication) Game

In this section we describe a stylized model, inspired by Schelling (1960) and Dixit and

Skeath’s (2015) stylized models of compellence and deterrence in international relations,

with the introduction of asymmetric information. This baseline game is intended to

represent the underlying structure (in terms of information and physical capabilities) of a

situation, independent of any agreements, commitments or deals made between countries.

We will first describe this baseline, sequential-move Bayesian game, and then proceed to

examine the welfare implications (for the Principal) from differing assumptions regarding

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the Principal’s commitment power in this game, and use these examples to study the

comparative costs and benefits of “hard” and “soft” power.

Consider two countries, labeled “Principal” and “Agent”. The Agent must choose

whether or not to comply with the Principal’s desired goal. The Principal receives a direct

(commonly known) payoff of 1 in the event that the Agent complies, and 0 otherwise. The

Agent receives a payoff of tpG from complying, where tp ∈ T p = {0, 1} is privately known

to the Principal and reflects the “trustworthiness” of the Principal. With commonly

known probability πp the Principal is trustworthy (tp = tph = 1), while with probability

1 − πp, the Principal is not trustworthy (tp = tpl = 0). The Agent’s payoff from not

complying, on the other hand, is 1 + taB, where ta ∈ T a = {0, 1} is privately known by

the Agent, and represents whether the Agent is “reasonable”. With probability πa, the

Agent is reasonable (ta = tah = 0), while with probability 1−πa the Agent is unreasonable

(ta = tal = 1).

To motivate the payoffs, one could interpret the “trustworthiness” of the Principal

as reflecting either some verifiable, privately known information about shared interests

(for example, whether some mutual adversary has or is seeking weapons of mass destruc-

tion), or it could represent the Principal’s preferences in an unmodeled later stage of

the game, beyond the point at which the Principal could plausibly make any commit-

ments. Depending on the Principal’s type, the Principal may be in a position to take

advantage of the Agent’s cooperation in the first stage in a way that harms the Agent’s

interests (for example, if “compliance” represents disarmament on the part of the Agent,

the Principal’s private type may represent its likelihood of behaving aggressively in the

future).

The Agent’s type can similarly be seen as possibly reflecting information about how

directly opposed to the Principal’s interests the Agent is (an “unreasonable” Agent may

be one for whom any cooperation with the Principal would be seen as unacceptable, either

to the leader of the country in question themselves or to certain constituencies within

the country), or as also reflecting some preference for opportunistic behavior in a later

stage (the Agent may not want to disarm or submit to an international body because the

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Figure 1: Structure of Foreign Policy Game

Agent knows that this will block their ability to pursue opportunistic behavior in this

later stage).

In the event that the Agent chooses not to comply, the Principal has the option to

choose between “Stick” and “No Stick”. If the Principal plays “Stick”, then the Principal

incurs a cost Dp, while the Agent incurs a cost Da. If the Principal plays no stick, then

neither receives any additional cost.

If the Agent chooses to comply, the Principal now may choose, in the second stage, to

offer a costly “Carrot” to the Agent. This concession imposes a cost Cp on the Principal,

and yields a benefit Ca for the Agent.

The game and payoffs unfold according to the game tree in Figure 1. Note that,

contrary to normal convention, at terminal nodes, the Principal’s payoff is listed first, in

spite of the fact that the Agent moves first in the structure of the overall game. This

is mostly to draw attention to the fact that the Principal’s welfare will be the primary

focus of our analysis.

Before continuing, we will first note that we will from here on out assume that 1+B−

Da > Ca +G, that is, that the Agent, if unreasonable, is incapable of being incentivized

to comply. This matches our characterization of ta as representing whether the Agent is

“reasonable”.

It may seem as if the assumption of which nodes are affected by each player’s private

type is not without loss of generality; this is not the case. In particular, one could switch

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the assumptions on which node is affected by the individual types by simply shifting the

Agent’s payoffs by a constant −B−G. The crucial fact is how the different players’ types

affect the difference in the Agent’s payoff from complying as opposed to not complying.

The main assumption of our model, in terms of our later results, is our assumption of

additive separability : the payoffs to each player from the move taken in the first stage does

not depend directly upon subsequent moves, and neither player’s payoffs from actions in

the second stage is affected by their private type. We feel this is the obvious baseline

assumption, as a first pass at capturing the core strategic properties of the different forms

of influence one player in such a situation might bring to bear upon another.

Before proceeding to what commitments and communications might be made on top

of this underlying game, we should first notice that this game always has a unique and

straightforward Perfect Bayesian Equilibrium, solvable by backwards induction. In this

equilibrium, the Agent chooses Comply if and only if πp > 1G

and ta = 0, and the Principal

never plays either “Carrot” or “Stick” at the nodes at which those are available. In this

situation, the Principal is entirely at the mercy of the Agent’s beliefs, due to a lack of

any external commitment or ability to communicate information. Furthermore, in the

extreme case where G < 1, the Agent refuses to Comply regardless of their belief in the

Principal’s trustworthiness.

4 A “Revelation Principle” Approach

Instead of treating the situation described above simply as a game of incomplete infor-

mation, in what follows we analyze it as what Myerson (1991) calls a “Bayes Incentive

Problem”. A Bayes Incentive Problem, like a game, involves both private information

(t), and private actions under the control of each player, but adds an additional set of

“public” or “enforceable” actions. These actions can be interpreted as actions that are

either carried out by some public authority, or, more metaphorically, as actions that may

be publically committed to ex ante via some coordinating mechanism. Myerson uses this

definition to introduce a general formulation of the mechanism design problem, ie., the

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design of a communication protocol that maps messages from players into actions, both

“private” and “public”, subject to constraints due to the Agents’ private incentives.

In order to make claims about what may be achieved by an arbitrary mechanism,

Myerson makes use of the device of a “direct revelation” mechanism, personified as a

“mediator”. This mediator elicits reports from each player of their own “type”, issues

(sometimes probabilistic) recommendations for private actions to each player, and car-

ries out (again, sometimes probabilistically) some public action. Myerson’s “revelation

principle” states that any Bayesian equilibrium outcome of any mechanism can also be

achieved as a Bayesian equilibrium outcome of some direct revelation mechanism. Rather

than being interpreted as an actual mediator existing in reality, we interpret the medi-

ator as being an abstract representation of all possible institutions for communication

and commitment-making on the part of the Principal. Assumptions about the power

of this mediator are then assumptions about the power of the Principal to both reveal

information and to commit themselves to courses of actions.

In what follows, we generalize Myerson’s direct revelation mechanism to also allow it to

be endowed with its own “quasi-public”1 information, ie., information that can be directly

“observed” by some mechanism and conditioned upon when issuing recommendations

and taking “public” actions. This generalization was first noted by Bergemann and

Morris (2018), who point out that the “information design” problem with which they

and Kamenica and Gentzkow (2011) concern themselves is simply a special case of this

generalized Bayes Social Problem that only allows the mediator to make use of their

information.

There are several ways of interpreting the exact nature of the mediator in this context.

The “mediator” could be interpreted as a reduced-form representation of an equilibrium

in an infinitely repeated game with reputation. In games such as these, the Principal

interacts (both via communication) over a long time horizon with a sequence of short-

lived Agents, and there is some probability that the Principal has some set of “behavioral”

1We describe this information as “quasi-public” even though, to maintain the analogy with “private”and “public” actions in Myerson’s terminology, it might be better to refer to this information as “public”.We do so because the term “public” would imply that this information is observable by all parties when,crucially, in this case, the information in question is only necessarily observable by the mediator.

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types who respond with a fixed behavioral rule, either in communication or in their actual

behavior. Assumptions on different types of commitment power for a mediator would then

be interpreted as differing assumptions on the information structure of this larger game,

and therefore assumptions on the extent to which the Principal can credibly mimic a

behavioral type.

Alternately, the mediator could be taken as representing public and private com-

munications between the Principal and the Agent in a one-shot setting, but with some

additional costs incurred afterwards for failure (on the Principal’s part) to follow through

on commitments or to truthfully convey information. Audience costs similar to those

described by Fearon (1994) or costs to miscalibrating information as in Guo and Shmaya

(2018) supply microfoundations for such costs in the case of our “hard” and “soft” power

respectively.2

Unlike previous authors, who have focused upon the implications of some fixed capabil-

ities for a mediator for what can possibly be achieved, we are interested in understanding

how varying the capabilities of a Myerson-style mediator can be used to compare the im-

portance of different aspects of a player’s capabilities in a strategic setting. In particular,

we are interested in using this device to compare the value of credibly communicating

information in order to shape beliefs, with the value of credibly committing to actions in

order to shape extrinsic incentives.

To define terms, let Sp = {Carrot, No Carrot}×{Stick, No Stick} = {CS,CNS,

NCS,NCNS} be the set of all pure strategies available to the Principal3, and let Sa =

{Comply, Don′t Comply} be the set of all pure strategies available to the Agent. Let

SC ( Sp = {CS,CNS} be the set of all pure strategies that involve playing “Carrot”

at the relevant node, while SS ( Sp = {CS,NCS} is the set of all pure strategies that

involve playing “Stick” at the relevant node. Let T = T p×T a and S = Sp×Sa. We

can then define a general direct revelation mechanism or mediator as m : T → ∆(S), a

2At the end of this section we will discuss how these different interpretations of the mediator interactwith our assumptions on the mediator’s commitment power in the game.

3Note that these strategies are not functions of the Principal’s type, as a strategy would often bedefined in a Bayesian game, and are instead physical descriptions of the Principal’s contingent behaviorin the game itself.

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mapping from the Principal and Agent’s types to a distribution over joint (pure) strategies

for both players, and label the set of all possible mediators as M . Finally, let up, ua :

T×S → R represent the payoffs of each player, as described in the previous section, and

let upm be the expected utility to the Principal from a mechanism m.

With these terms defined, we can now consider constraints a mediator might face in

implementing outcomes.

The most obvious constraint, given that we are interested in how the Principal might

be able to compel the Agent to Comply, is what we will call an Agent-Obedience con-

straint. If we interpret the output of the mediator as being a recommendation, it must

be the case that the Agent lacks an incentive to deviate from that recommendation.

Let δ : Sa → Sa represent some “deviation rule” that the Agent might adopt, mapping

recommended pure strategies into actual pure strategies. In order for the Agent to be

properly induced to obey the mediator, it must be the case that:

(πp)∑S

m(sp, sa|tph, ta)ua(sp, sa, tph, t

a) + (1− πp)∑S

m(sp, sa|tpl , ta)ua(sp, sa, tpl , t

a) ≥

(πp)∑S

m(sp, sa|tph, ta)ua(sp, δ(sa), tph, t

a) + (1− πp)∑Sp,Sa

m(sp, sa|tpl , ta)ua(sp, δ(sa), tpl , t

a)

∀δ : Sa → Sa, ta ∈ T a

(1)

Going forward, we will assume that the mediator always faces this constraint in any

possible mechanism we consider.

Similarly, the Agent must be induced to truthfully reveal their type to the mediator4.

4We also allow the Agent to lie about their type and then deviate from the recommendation of themediator

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This constraint may be expressed as:

(πp)∑S

m(sp, sa|tph, ta)ua(sp, sa, tph, t

a) + (1− πp)∑S

m(sp, sa|tpl , ta)ua(sp, sa, tpl , t

a) ≥

(πp)∑S

m(sp, sa|tph, ta)ua(sp, δ(sa), tph, t

a′) + (1− πp)∑S

m(sp, sa|tpl , ta)ua(sp, δ(sa), tpl , t

a′)

∀δ : Sa → Sa, ta, ta′ ∈ T a

(2)

We will also assume that every possible mediator faces these constraints.

We can also consider an even more extreme constraint on the mediator; we might

require that the mediator must satisfy obedience constraints (analogous to those defined

for the Agent above) for the Principal as well. In this case, the mediator is incapable of

exercising any direct control over the Principal’s actions. We may also want to distinguish

between two different Constraints on the mediator exercising control over the Principal.

In one of these cases, the mediator is unable to force the Principal to carry out threats,

and is therefore constrained from forcing the Principal to play “Stick” in the subgame fol-

lowing the Agent choosing “Don’t Comply”. We express this Threat-Obedience constraint

as follows:

m(CS, sa|tp, ta) = m(NCS, sa|tp, ta) = 0

∀sa ∈ Sa, (tp, ta) ∈ T(3)

In one of these cases, the mediator is unable to force the Principal to carry out

promises, and is therefore constrained from forcing the Principal to play “Carrot” in the

subgame following the Agent choosing “Comply”. We express this Promise-Obedience

constraint as follows:

m(CS, sa|tp, ta) = m(CNS, sa|tp, ta) = 0

∀sa ∈ Sa, (tp, ta) ∈ T(4)

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These two constraints, when applied jointly, imply that the mechanism is constrained

to recommend the Principal only play the strategy NCNS with probability 1.

Finally, we can consider one more possible constraint upon the mediator. Suppose

that the mediator is required to induce the Principal to tell the truth about their own

trustworthiness. This would be a case wherein the Principal lacks any direct credibility

in revealing information, and faces an incentive to lie about their own type. In this case,

the mediator would face the following classical truth-telling incentive constraint, from

traditional “hidden information” mechanism design problems:

(πa)∑S

m(sp, sa|tp, tah)up(sp, sa, tp, tah) + (1− πa)∑S

m(sp, sa|tp, tal )up(sp, sa, tp, tal ) ≥

(πa)∑S

m(sp, sa|tp′, ta)up(sp, sa, tp, tah) + (1− πa)∑S

m(sp, sa|tp′, tal )up(sp, sa, tp, tal )

∀tp, tp′ ∈ T p

(5)

For the remainder of this and the following section, we will examine the implications

for what can be achieved by a mediator on behalf of the Principal by varying the above

constraints, particularly constraints (3)-(5). These constraints represent limits on the

“soft” and “hard” power of the Principal in this strategic situation.

Depending on whether we include constraints on either truth-telling or Principal-

obedience, we have the four combinations presented in Table 1. Varying along the rows

are the presence or absence of constraints on the verifiability of the Principal’s type (which

we interpret as the Principal’s credibility in diplomacy or “soft power”), while varying

along the columns we have the presence or absence of constraints on the mediator’s ability

to commit the Principal to costly actions (which we interpret as the Principal’s credibility

in exercising “hard power”). One can then work one’s way through the cells of this matrix

to determine the properties of different types of mediator.

The upper-left quadrant represents maximal power for the Principal, in which the

mediator can both verify the Principal’s type, and can commit the Principal to actions in

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Partial Control over No Control OverPrincipal’s Actions Principal’s Actions

Types verifiable “First-Best” Mediator “Soft Power” Mediatorby Mediator (This Paper) (Kamenica-Gentzkow 2011)

Types not verifiable “Hard Power” Mediator “Toothless” Mediatorby Mediator (Myerson 1983) (Forges 1993)

Table 1: Varying capabilities of a mediator in this framework

the second stage of the game. By definition, any outcome that could be implemented in

the other quadrants can be implemented here, and hence we employ the (slightly abusive)

terminology of “first best” to describe this environment. We denote the maximum payoff

achievable for the Principal under this environment with V ∗, defined as:

V ∗ = maxm∈M

upm

s.t. (1), (2)

This payoff will serve as a benchmark for welfare under our later cases, as both of these

constraints are, in the cases with which we are concerned, essential and unavoidable

aspects of the problem facing a Principal.

Moving to the lower-left quadrant, we have the case in which the mediator may

control the Principal’s actions in the second stage of the game. This mediator, however,

is incapable of directly conveying any information about the Principal’s type to the Agent,

due to its inability to verify any of the Principal’s reports. This turns our problem into a

problem of mechanism design with a privately informed Principal, of the kind analyzed in

Myerson (1983). Consistent with our above distinction between commitment to threats

and commitment to promises, we can distinguish between a “compellent” mediator and a

“deterrent” mediator. We denote the payoffs achievable under a “compellent” mediator

as V c, defined below as:

V c = maxm∈M

upm

16

s.t. (1), (2), (3), &(5)

while the payoffs achievable under a deterrent mediator are:

V d = maxm∈M

upm

s.t. (1), (2), (4), &(5)

In the upper-right quadrant, in contrast, the mediator is incapable of forcing the

Principal to make hard policy concessions. Instead, the mediator is empowered to verify

the Principal’s type and to make probabilistic recommendations to China. This environ-

ment corresponds to Kamenica and Gentzkow (2011)’s “Bayesian Persuasion” or “cheap

talk with commitment” environment, or which Bergemann and Morris (2017) refer to

as “Information Design,” noting the relationship between Myerson’s framework outlined

above and Kamenica and Gentzkow’s results. We can associate this environment with

Nye’s “soft power”, in that influence is wielded by altering a counterpart’s beliefs through

communication and persuasion, rather than on using concrete actions to induce said coun-

terpart. We denote the payoffs achievable by a “soft power” mediator as V s, defined as

follows:

V s = maxm∈M

upm

s.t. (1), (2), (3), &(4)

Finally, in the lower-right quadrant, we have a mediator who faces all of the above

constraints on behavior. This mediator cannot concretely affect any payoffs or reveal any

information to any of the players, and therefore functions solely as a method of “cheap

talk” on the part of participants. This type of mediator corresponds to the “communi-

cation equilibrium” as defined by Forges (1993). We denote the payoff achievable by this

“toothless” mediator by V , defined as:

V = maxm∈M

upm

17

s.t. (1), (2), (3), (4), &(5)

Stepping back to our earlier-mentioned interpretations of the mediator, the presence

or absence of any of these constraints can be interpreted in numerous ways. For example,

if we take a repeated-game, reputational approach to the problem, we can interpret

these constraints as reflecting assumptions on the information structure of the repeated

game. A mediator facing constraint (4) would be a representation of a game wherein

the Principal’s choice of “Carrot” in any given repetition of the game is unobservable

to Agents in later plays of the game, while a mediator facing constraint (5) would be a

representation of a repeated game in which the Principal’s type is unobservable to Agents

in later plays of the game. These constraints would then limit the ability of the Principal

to credibly mimic the behavior of a “behavioral” type.5

By framing the comparison of hard and soft power directly in terms of a mediator

with varying capacities, we can take advantage of several advantages to this approach.

First, on a purely conceptual level, it places each type of power on the same footing; in

this framework, we can view the choice between hard and soft power as simply being the

choice between two different sets of constraints on an optimization. Moreover, given this

framework, we can now directly compare the relative value of one type of power in this

strategic setting to another, and establish how closely each approximates our “first-best”.

For example, it is clearly trivially true that V ∗ ≥ V c, V d, V s ≥ V , but what more can we

say about the relation between these three values?

In the remaining theoretical sections, we first establish conditions under which the

Principal’s “compellent”, “deterrent”, and “soft power” payoffs (V c, V d and V s) differ

from the Principal’s baseline payoffs under a “toothless” mediator. We then move onto

our main results: we present conditions under certain payoffs are higher than one another

and vice versa, and under which the first-best payoffs can be achieved through only one

or the other.

5The repeated games interpretation of this mediator is slightly more complicated in the case ofprobabilistic recommendations; in particular, in the repeated games setting, there must be some ability toenforce or observe randomized communication. For more on some of the difficulties posed by randomizedcommunication of the type exploited by Kamenica-Gentzkow in a repeated games setting, see Best andQuigley (2017).

18

5 Comparing Equilibrium Payoffs Under Different

Mediators

5.1 Payoffs under a Toothless Mediator

Our first result is straightforward and establishes that our “toothless” mediator adds

nothing in terms of welfare for the Principal, in spite of the seeming advantages of being

able to communicate.

Proposition 1. The maximum payoff achievable by a toothless mediator is exactly equal

to the equilibrium payoff of the Bayesian game without communication.

To see this, notice first that in the case where πp > 1G

, the mediator is unnecessary,

and the Principal receives their highest possible payoff without any mediator at all (since

when ta = 1 there is no hope of persuading the Agent at all). We can then restrict

ourselves to the case where πp < 1G

. First, as mentioned above, constraints (3) and (4)

pin down the Principal’s strategy to only be NCNS. In order to induce the Agent to play

“Comply” with positive probability while still satisfying Constraints (3) and (4), then,

the mediator must recommend “Comply” with a higher probability when tp = tph than

when tp = tpl . In this case, however, the Principal would then face an incentive to always

report that tp = tph and then deviate in the final stage when tp is actually tpl . Thus, the

only mechanism that a mediator can implement in this case is one in which the mediator

recommends “Don’t Comply,” regardless of the Principal’s report.

Having established a “worst-case” payoff of V for the toothless mediator, we can now

characterize “best-case” payoffs under different constraints, and compare them to that

payoff.

5.2 Compellence compared to a Toothless Mediator

We begin with the payoffs to “compellence”, represented here by the ability of our medi-

ator to commit the Principal to enforceable costly actions in the second stage of the game

19

(playing “Carrot”), conditional on the Agent’s compliance. The mediator must therefore

balance the incentives of the Agent to Comply and truthfully reveal their type

We can establish the following proposition:

Proposition 2. The Principal strictly gains from compellence relative to a toothless

mediator (ie., V c > V ) if 1) 1− πpG < Ca, 2) (1− πpG)Cp

Ca < 1, and 3) πp < 1G

. Under

these conditions, V c ≥ πa − πa(1− πpG)(Cp

Ca )

The conditions in the result above under which the Principal benefits ex ante from

compellence also have a straightforward interpretation. In particular, we can view these

as feasibility, cost-benefit, and necessity conditions:

1. Feasibility (1 − πpG < Ca): It must be case that the “carrot” that the Principal

wields is sufficiently tempting to the Agent to serve as an effective incentive.

2. Cost-benefit ((1 − πpG)Cp

Ca < 1): It must be that the Principal gains enough from

the Agent’s compliance that it is willing to incur the cost of playing carrot.

3. Necessity (πp < 1G

) The Agent must be sufficiently suspicious of the Principal that

it requires additional inducement to Comply.

Finally, the equation representing V c also has a direct and intuitive interpretation.

The first term represents the “benefits” from exercising compellence, namely the proba-

bility that the Agent complies (πa), while the second term represents the costs to inducing

compliance, which is proportional to the probability that the Agent complies (πa), the

“gap” that the Principal has to make up in terms of incentives (1− πpG), and the “con-

version rate” between the Principal’s disutility and the Agent’s utility Cp

Ca .

All of the terms introduced above will play heavily into our discussion of comparisons

of different types of mediator with one another and with the first-best.

5.3 Deterrence compared to a Toothless Mediator

Our results for deterrence are directly analogous to our results for compellence:

20

Proposition 3. The Principal strictly gains from deterrence relative to a toothless me-

diator (ie., V d > V ) if 1) 1− πpG < Da, 2) (1− πpG)Dp

Da <πa

1−πa , and 3) πp < 1G

. Under

these conditions, V d ≥ πa − (1− πa)(1− πpG)(Dp

Da )

Many properties of this deterrence mechanism are exactly analogous to a compel-

lence mechanism. The mediator will never recommend the Principal play “Carrot” by

similar backwards induction reasoning to that given above, and is unable to induce the

“unreasonable” Agents to play “Comply”.

The key difference lies in how the “unreasonable” or “low-type” Agents matter for

the mechanism (recall that they were barely relevant for describing any of the properties

of a compellent mechanism). Although, as before, it is never feasible for a mediator to

recommend “Comply” to an “unreasonable” Agent, this very fact introduces a cost to

deterrence. In particular, if the Principal never plays “Stick” against an “unreasonable”

Agent, a “reasonable” Agent then faces an incentive to lie and claim to be “unreasonable”,

because such a lie would secure a payoff of 1 as opposed to πpG < 1. In order for

any threat against a reasonable Agent to be effective, then, the Principal is required to

treat unreasonable Agents as if they were reasonable, and had simply deviated. In this

case, then, the presence of “good” Agents actually imposes a cost upon “bad” Agents,

precisely because “good” Agents are amenable to incentives, and because any incentive

mechanism cannot distinguish between genuinely unreasonable Agents and opportunistic

yet reasonable Agents.

The interpretation of Proposition 3’s conditions are directly analogous to our inter-

pretation of Proposition 2, in terms of feasibility, cost-benefit, and necessity conditions,

but with a few interesting differences.

1. Feasibility (1 − πpG < Da): It must be case that the “stick” that the Principal

wields is sufficiently tempting to the Agent to serve as an effective incentive.

2. Cost-benefit ((1 − πpG)Dp

Da < πa

1−πa ): The odds ratio of a trustworthy Agent must

be high enough that the Principal’s expected gains from the reasonable Agent’s

compliance are worth the expected cost of playing “Stick” against an unreasonable

21

Agent.

3. Necessity (πp < 1G

) The Agent must be sufficiently suspicious of the Principal that

it requires additional inducement to Comply.

Finally, the equation representing V d also has an analogous interpretation to that of

the compellent mediator. The first term represents the same “benefit” as before (the

probability that the Agent complies (πa)), while the second term represents the costs to

inducing compliance, which is proportional to the probability that the Agent is unreason-

able and therefore requires “Stick” to be played against it (πa), the “incentive gap” to be

made up (1 − πpG), and the “conversion rate” between the Principal’s and the Agent’s

respective disutilities Dp

Da .

5.4 Soft Power Compared to a Toothless Mediator

Our soft power environment, wherein the Principal is capable, via a mediator, of con-

veying credible information regarding their own trustworthiness, is analogous precisely

to Kamenica and Gentzkow’s (2011) “Bayesian persuasion” setting. In this setting, one

can establish the following result:

Proposition 4. The Principal strictly gains from soft power (ie., V s > V ) relative to a

toothless mediator if and only if πp < 1G≤ 1. Moreover, V s = πa − πa(1− πpG)

Under this optimal “soft power” mechanism, the mediator never recommends either

“Carrot” or “Stick” from the Principal, and also never recommends that an unreasonable

Agent Comply. The only tool by which the Mediator achieves higher expected payoffs

from the Principal, then, comes from conveying information to the Agent by conditioning

its recommendation on the Principal’s type.

The mediator, by randomizing, can on average achieve a higher payoff for the Princi-

pal, by sometimes revealing (by virtue of recommending Don’t Comply) that the Principal

is untrustworthy to the Agent. By committing to revealing this bare minimum of infor-

mation, the mediator is then able to induce the Agent to obey their recommendation to

Comply, even though there is a positive probability (1− 1G

) that the Principal is actually

22

untrustworthy. In fact, under this optimal policy, the Agent is at exactly the level of

uncertainty that leaves them precisely indifferent between obeying and not obeying the

mediator’s recommendations.

The condition on probabilities can be observed by noting that, as long as there exists

some probability π = 1G

of the Principal being trustworthy that would induce the Agent

to cooperate if it held such a probability, and as long as πp is below that level, that

means that, at this probability V is a convex function of πp at πp. Therefore it is possible

for the mediator to secure a strictly higher payoff for the Principal by recommending

Comply with probability 1 given t = tph, and with probability πp(G−1)1−πp given t = tpl , while

recommending Don’t Comply with probability 1− πp(G−1)1−πp G when t = tp1.

We can give concrete interpretations to these conditions on probabilities. Whereas

both forms of “hard power” had feasibility, cost-benefit, and necessity conditions, here

we only have feasibility and necessity conditions:

1. Feasibility ( 1G≤ 1): It must be case that the gains from the Principal being trust-

worthy are sufficiently high to the Agent that there is some belief the Agent could

hold in the Principal’s trustworthiness that would make the Agent willing to Com-

ply.

2. Necessity (πp < 1G

): The Agent must be sufficiently suspicious of the Principal that

they require additional information to induce them to Comply.

The lack of a cost-benefit condition makes sense; in this setting, the Principal is taking

no costly actions, and does not directly face a cost-benefit trade-off. The primary costs

of soft power lie in the fact that, in order to maintain credibility, the mediator must

sometimes provide truthful and harmful information. The Principal can be interpreted

as having some fixed budget of lies that it cannot exceed without sacrificing credibility.

This “cost of credibility” interpretation is why our expression V s = πa − πa(1− πpG) is

written in a seemingly awkward format: the “cost term” of our soft power formula comes

from the foregone compliance that comes with revealing credible information.

23

Maximum Payoffs “Costs of Credibility”Compellence πa − πa(1− πpG)(C

p

Ca ) πa(1− πpG)(Cp

Ca )Deterrence πa − (1− πa)(1− πpG)(D

p

Da ) (1− πa)(1− πpG)(Dp

Da )Soft Power πa − πa(1− πpG) πa(1− πpG)

Table 2: Comparing Forms of Power

5.5 Comparing forms of power to one another

In this section, we compare the payoffs under soft power, compellence, and deterrence

(V s, V c, and V d) to one another, and establish conditions under which one is better for

the Principal than the other. We then present results on how these payoffs compare to

those achievable by a maximally powerful mediator (V ∗).

In Table 2, we can directly compare the payoffs achievable from each form of power,

and note certain similarities and differences. All forms of power have costs that come

from maintaining credibility. With compellence, these costs come in the form of playing

“Carrot” with some probability when the Agent actually complies, while with deterrence,

these costs come from having to actually punish an unreasonable Agent. The costs of

soft power, as discussed above, come from needing to actually reveal harmful information,

thereby sacrificing some compliance in order to achieve credibility.

One can then observe that the relative value of each type of power can be reduced

down to a comparison of 1, Cp

Ca , and 1−πa

πa (1− πpG)Dp

Da . Depending on which is lower, soft

power, compellence, or deterrence would be the least-costly form of power.

One intriguing subset of cases are those wherein (1−πpG)Cp

Ca ,1−πa

πa (1−πpG)Dp

Da < 1 <

Cp

Ca ,1−πa

πa (1−πpG)Dp

Da . In these cases, under a hard power mediator, the Agent will always

Comply, whereas, under a soft power mediator, the Agent only complies with probability

πa(1−πpG) < 1, yet V s > V c. In other words, a policy that occasionally recommends the

Agent to not Comply (for the sake of the Principal’s credibility) is sometimes preferable

to one that entirely eliminates any uncertainty as to the Agent’s actions. Another way

of viewing this is that the Principal would rather face uncertainty than pay out a carrot,

or deploy a stick when such an incentive is unnecessary.

24

One can also note that, in terms of payoffs, soft power is indistinguishable from

being able to offer a carrot with a perfect “conversion rate” of 1; unsurprisingly, then,

compellent offers of carrot are only strictly better than soft power when playing carrot

yields a greater utility for the Agent than it yields disutility for the Principal. In addition,

deterrence is the only form of power whose value relative to other forms of power depends

upon πa; that is, the cost of deterrence is crucially dependent upon how likely the Agent

is to respond to deterrence.

5.6 First-Best Payoffs

In fact, building off of this discussion, one can demonstrate a possibly surprising result,

namely that these forms of power, if all are effective, operate purely as strong substitutes

for one another:

Proposition 5. Suppose V c, V d, V s > V = 0. Then, it must be the case that one of the

three are equal to the first-best achievable payoff.

To grasp the intuition behind this result, suppose that we begin with one of our

individual optimal mechanisms, such as the optimal compellence mechanism. If the

mediator were to introduce a small amount of “deterrence” (ie., increase the probability

of playing “stick”), that would allow the mediator to leave the probability of compliance

unchanged, but to lower costs from playing “carrot”. In order to do this while satisfying

the Agent truth-telling constraint, however, the Principal would need to also play “stick”

with the same probability against unreasonable Agents. The only way such a move would

be an improvement in the Principal’s expected utility is if the gains from reducing the

probability of playing “carrot” outweigh the losses from playing “stick”, but in this case,

any substitution from “carrot” to “stick” must be beneficial, meaning that the optimal

deterrence mechanism payoff-dominates any mechanism that plays both. By deploying

analogous arguments for each optimal mechanism, one can then prove the result.

Remarkably, then, we have that if all forms of power are capable of achieving results

on its own, one of those forms of power must be sufficient to achieve the first-best on its

own.

25

5.7 Feasibility Conditions

There are, however, cases wherein hard and soft power do act as complements, and the

Principal does gain from the ability to both faithfully convey information about their type

and to commit themselves to concrete actions. In particular, when particular mechanisms

fail to achieve positive payoffs on their own, not because of cost-benefit considerations,

but because of feasibility conditions, then it is possible for a mediator who has access to

multiple forms of commitment power to gain higher payoffs than a mediator with access

to only one.

Proposition 6. Suppose that G < 1, and Ca, Da < 1− πpG, but that G+ Ca +Da > 1

and Ca + 1−πa

πa Da < 1. Then, V ∗ > V d = V c = V s = V = 0.

Intuitively, the idea here is that no form of power may be enough to sway the Agent

on its own, but as long as all of the Principal’s tools are both effective and sufficiently

“cheap” (from a cost-benefit perspective) overall, it is possible to use them in tandem to

produce a superior outcome. For example, it might be that even if the Agent knew with

probability 1 that the Principal was trustworthy, the Agent would still not be persuaded

to play Comply. The Principal is then able to secure a higher payoff by using both forms

of power in tandem in order to tip the Agent “over the edge”. In this event, even though

no individual form of power yields a positive payoff, it is possible to construct a mecha-

nism which still yields a positive payoff, by following a procedure of selecting the form of

power with the lowest relative “cost of compliance” (out of Cp

Ca ,1−πa

πa (1 − πpG)Dp

Da , and 1

for Compellence, Deterrence, and Soft Power, respectively), setting the probability (con-

ditional upon the Agent’s compliance) of either playing the relevant strategy (“Carrot” or

“Stick”) or of being a trustworthy type to 1, and then repeating this process for the next

lowest-cost option until the payoff differential for the Agent between playing Comply and

Don’t Comply has been reduced to zero. This result suggests why one might still observe

the use of multiple forms of power in tandem in a given interaction or dispute; the model

would predict this complementary use of forms of power in cases where the Principal

faces a binding resource constraint relative to the Agent’s incentive to not Comply.

26

6 Historical Examples

6.1 Compliance Through Soft Power: Hitler, 1933-38

In the wake of World War I and the Treaty of Versailles, German military capacity had

been left in ruins, both due to the direct costs of military conflict and the penalties levied

upon Germany in peacetime. The Nazi platform on foreign policy largely consisted of

a rejection of Versailles, a commitment to rearmament, and a restored and territorially

expanded German state. Upon gaining power, Hitler set about implementing his pro-

gram of renewed German power: in 1933 he withdrew from the Disarmament Conference

established by Versailles, and in 1934 formally announced a policy of rearmament, in

direct violation of German commitments.

These early moves towards rearmament were not met with an aggressive response by

British and French politicians however, who continued to pursue a program of diplomatic

rapprochement with Hitler. In part this attitude was driven by domestic fears of a renewed

war, and an unwillingness to commit resources to defending an agreement without an

assurance that other parties would carry out their part in enforcing that same agreement,

as well as a sense that a rearmed Nazi Germany might serve as a useful counterweight

to the spread of Communism. Even more important than uncertainty about their allies’

intentions and motives, however, was an underlying uncertainty about Hitler’s underlying

intent.

Hitler’s initial aggression was equally consistent with multiple models of his own

future behavior. Hitler professed repeatedly to be restoring Germany to a position of

parity with respect to other European states, with a goal of being recognized as a nation

with equal standing in European politics. In Mein Kampf, however, Hitler had explicitly

laid out an additional program: territorial expansionism into Eastern Europe, and the

claiming of Eastern territories for German Lebensraum. Even in this case, however, it was

unlikely that those ambitions in themselves would fundamentally harm British or French

interests in themselves, particularly if they functioned as a check on Soviet power. If

Hitler was ultimately embarking on a program of broader European expansion, however,

27

a confrontation with Hitler was both inevitable and better conducted sooner than later.

As Kissinger (1994, p.294) would later note, “[H]ad the democracies forced a showdown

with Hitler early in his rule, historians would still be arguing over whether Hitler was a

misunderstood nationalist or a madman bent upon world domination.”

Hitler and his circle were acutely aware of this uncertainty and actively worked to

exploit it, as Goebbels proudly proclaimed in the quote given in our introduction. The

success of this approach is underlined even more by the striking lack of direct concessions

made by Hitler. To the extent that any “concessions” were made, they took the form of

caps upon future expansion and military capacity, and non-aggression pacts. Even at a

later date, Hitler largely refrained from direct use of “hard power” strategies against the

Western European powers. In the immediate run-up to his seizure of the Sudetenlands

of Czechoslovakia, Hitler “worked to magnify hysteria about an imminent war without,

in fact, making any specific threat” (Kissinger 1994, p. 312). In the early stages it might

be argued that this was due to the fact that Hitler was coming from a place of lower

direct bargaining power- in order to issue direct threats, Hitler would have to commit a

still-growing arsenal to aggression against the powers to his west. Even at the height of

his military power, a large element of the success of a “soft power” strategy came from

the structure of his enemies’ preferences, rather than any direct budgetary or resource

constraints, as our framework helps to illustrate.

Recall our discussion above: hard power can frequently be inferior as a method of

attaining one’s goals, even when it may still be cost-effective on its own terms in the

absence of any explicit “soft power” strategy. This comes less from the direct costs of

ones’ promises themselves, and more from the way in which the effectiveness of this hard

power varies depending upon the information of one’s opponent. In particular, if hard

power in unnecessary in favorable states of the world, and prohibitively costly under

unfavorable states of the world, it serves one’s goals better to act directly upon beliefs

rather than through the explicit use of hard power. As we shall see, this case aligns

precisely with the situation unfolding in Europe in the 1930’s.

British preferences as to the power and influence of Germany were expressly limited to

28

whether Germany encroached upon its western neighbors, and in particular as to whether

it encroached upon France. “Determined never to fight for Eastern Europe, it perceived

no vital British interest in a demilitarized Rhineland serving as a kind of hostage in

the West.” As such, all of Hitler’s explicitly claimed goals approached right upon the

margin of what Britain viewed as an acceptable loss. Britain was not even interested in

keeping Germany uncertain regarding these commitments; following Hitler’s entrance into

the Rhineland, the British Secretary of State for War assured the German ambassador

directly,

“[T]hough the British people were prepared to fight for France in the event

of a German incursion into French territory, they would not resort to arms

on account of the recent occupation of the Rhineland . . . they did not care

’two hoots’ about the Germans reoccupying their own territory” (quoted in

Kissinger (1994), p. 304)

These qualms came partly from the objective costliness of a war, but also from a

sense of moral ambiguity as to the status of Versailles itself. Germany remaining in a

subordinate position to the rest of Europe was not in and of itself a position that was

seen as requiring military intervention and enforcement. As Kissinger notes:

“[I]t was argued that Germany would be satisfied as soon as it had been

conceded the right to defend its own national borders, something every other

European nation simply took for granted. Did British and French leaders have

the moral right to risk their peoples’ lives in order to maintain a so blatantly

discriminatory state of affairs?” (Kissinger (1994), p. 301)

Motivated both by a sense of unfairness as well as an extreme sensitivity to public

opinion regarding military interventionism, the question of Germany’s ultimate motives

was explicitly front-and-center in determining whether or not an aggressive stance against

German expansionism passed a cost-benefit test for the Western European powers. As

such, the exploitation and manipulation of this uncertainty gave Hitler the room he

needed to prevent either France or Britain from checking his ambitions.

29

Hitler’s strategic manipulation of his enemies’ beliefs maps well to our own charac-

terization of the optimal soft power policy. Recall that the optimal information policy is

such that, given a recommendation of Comply, the Agent is precisely indifferent between

obeying the recommendation and not obeying. Hitler’s implicit policy of manipulating

the beliefs of British and French leaders as to his own future goals was such that, given

Britain and France’s own interests, the Western powers were only just willing to concede

Germany’s claims, and were, in fact, close to that exact margin of uncertainty as to

whether or not such a policy was a good idea.

In an ironic twist, this very element of Germany’s policy meant that the Munich

agreement involving the annexation of the Sudetenlands marked the high water mark of

German diplomatic success. When Germany went on to falsify their own claims about

their limited strategic goals, Kissinger claims, Germany had also exhausted the limits of

what it could sustainably extract from the other world powers.

“All of Hitler’s great foreign policy triumphs occurred in the first five years

of his rule, 1933-1938, and were based upon his victims’ assumption that his

aim was to reconcile the Versailles system with its purported principles. Once

Hitler abandoned the pretense of rectifying injustice, his credibility vanished.

Embarking on naked conquest for its own sake made him lose his touch.”

(Kissinger (1994), p. 289)

7 Conclusions, Plausible Extensions, and Areas of

Future Research

In applying what is likely the simplest possible game-theoretic model to incorporate the

possibility of compellence, deterrence, and persuasion, we have been able to derive a

number of insights into the nature and relative costs and benefits of these different types

of power. We restate them here, in intuitive terms:

1. Soft power and hard power differ crucially in the source of their costs; hard power

relies upon material commitments, while soft power relies on the willingness to, for

30

the sake of credibility, reveal unfavorable information about oneself, and thereby

sacrifice some potential compliance in some situations.

2. The deployment of hard power relies upon direct cost-benefit considerations that

do not explicitly factor into the use of soft power (Propositions 2, 3 and 4).

3. Hard Power can often be a means of taking advantage of uncertainty; in a situa-

tion where, regardless of their type, a country would be willing to pay the cost of

making a short-term policy decision to induce compliance, it might be better for

that country to eschew communication-based policies in favor of direct incentives,

whether those be through threats or through promises.

4. Conversely, in situations where, in the absence of uncertainty, the use of direct in-

centives would be either unnecessary or prohibitively costly, a “soft power” approach

that emphasizes diplomacy and the manipulation of beliefs can achieve everything

that a compellence- or deterrence- based approach can and more.

5. Deterrence, unlike other forms of power, depends for its effectiveness on the proba-

bility that the other country is responsive to incentives, because effective deterrence

requires one to commit to punishing both reasonable and unreasonable counterparts

alike.

6. In a wide range of circumstances (countries with high capacity and low costs for

multiple forms of power), one should expect to observe countries deploying an all-or-

nothing approach: the different forms of power described here operate as substitutes

for one another. (Proposition 5)

7. In situations where one or both of “hard power” and “soft power” are ineffective,

it may be possible to use both types of commitment power in tandem to achieve

better outcomes than could be achieved alone (Proposition 6).

Our framework can help to make more precise claims about situations in which one

form of power or the other may be more or less effective or decisive in international affairs.

Nye’s discussion of “Smart Power” as the coordinated use of hard and soft power policies

31

fits well with our 7th and point, regarding situations where the effective use of power

requires one to use both in tandem.

Moreover, historical situations can often be productively illuminated in light of these

points. In Hitler’s case, persuading the European powers to allow the annexation of the

Sudetenlands was a coup for a certain brand of soft power policy. Hitler at the time

was not in a position to make direct threats to European powers, and in the presence

of absolute certainty about his motives, Great Britain and France could not have been

successfully incentivized by anything Hitler had to offer. As we saw above, situations like

these are ones in which soft power strategies like those pursued by Hitler are maximally

beneficial.

There are also several suggestive areas for future research. We have been interpreting

the mediator as having either perfect commitment power or none at all. One way to

interpret this is that our soft power and hard power payoffs should be interpreted as

upper bounds on what can be achieved by different types of mechanisms. This is in

keeping with a tradition in mechanism design that is primarily interested in providing

welfare bounds on what is possible across all mechanisms, as in Myerson and Satterthwaite

(1983). In the interest of characterizing everything that could happen, however, more

work on what it would mean in our case to have “imperfect” credibility or “imperfect”

commitment power would complement the work done here nicely. Two recent papers,

Guo and Shmaya (2018) and Lipnowski, et al. (2018) have developed of “imperfect

commitment” in communication that converge to “Bayesian Persuasion” in the limit.

Finding some way to incorporate these microfoundations into our framework would help

to better characterize the value of less-than-maximal soft and hard power.

The model presented here is a very simplified one. In part, this was because the goal

of the paper was to establish a model that had the minimal ingredients necessary to

represent common problems of international relations of the type that we are interested

in, and with easily interpretable results. Many of our results appear to be driven by

separability assumptions on the Principal’s type, the costs and benefits of carrots and

sticks to both the Principal and the Agent, and the Principal’s gains from compliance.

32

A more general model in which these assumptions can be made explicit would help to

make the theoretical contributions of the paper better founded.

References

[1] Albert, E. (2018), “China’s Big Bet on Soft Power.” Council on Foreign Relations.

Retrieved from https://www.cfr.org/backgrounder/chinas-big-bet-soft-power

[2] Alonso, R. and O. Camara (2016a): “Persuading Voters,” American Economic Re-

view, 106(11), 3590-3605.

[3] Alonso, R. and O. Camara (2016b): “Political Disagreement and Information in

Elections,” Games and Economic Behavior, 100, 390-412.

[4] Aumann, R., and M. Maschler (1995): Repeated Games with Incomplete Information.

MIT Press.

[5] Baliga, S. and T. Sjostrom (2004): “Arms Races and Negotiations,” Review of Eco-

nomic Studies, 71, 351-369

[6] Baliga, S. and T. Sjostrom (2008): “Strategic Ambiguity and Arms Proliferation,”

Journal of Political Economy, 116(6), 1023-1057

[7] Baliga, S., and Sjostrom, T. (2012): “The Strategy of Manipulating Conflict,” The

American Economic Review, 102(6), 2897-2922.

[8] Bergemann, D., and S. Morris (2016): “Bayes Correlated Equilibrium and the Com-

parison of Information Structures in Games,” Theoretical Economics, 11, 487-522.

[9] Bergemann, D., and S. Morris (2017): “Information Design: A Unified Perspective,”

Journal of Economic Literature, 57(1), 1-57.

[10] Berman, R. (2017): “Trump’s Budget Slashes Popular Domestic Pro-

grams to Spend More on National Security,” The Atlantic, Retrieved from

33

https://www.theatlantic.com/politics/archive/2017/03/president-trumps-hard-

power-budget/519702/

[11] Best, J. and D. Quigley (2017): “Persuasion for the Long Run,” Working Paper.

[12] Boulding, K. (1989): Three Faces of Power. Newbury Park: Sage Publications.

[13] Dixit, A. K., Skeath, S., and Reiley, D. H. (2015): Games of strategy. New York.:

W. W. Norton.

[14] Fearon, J. (1994): Domestic Political Audiences and the Escalation of International

Disputes. The American Political Science Review, 88(3), 577-592.

[15] Fearon, J. (1995): “Rationalist Explanations for War,” International Organization,

49(3), 379-414.

[16] Forges, F. (1993): “Five Legitimate Definitions of Correlated Equilibrium in Games

with Incomplete Information,” Theory and Decision, 35, 277-310.

[17] Galbraith, J. K. (1983): The Anatomy of Power. Boston: Houghton Mifflin.

[18] Goldstein, I., and C. Huang (2016): “Bayesian Persuasion in Coordination Games.”

American Economic Review, 106, 592-96.

[19] Guo, Y., and E. Shmaya (2018): “Costly Miscalibration,” Working Paper.

[20] Harsanyi, J. C. (1962): “Measurement of social power, opportunity costs, and the

theory of two-person bargaining games,” Behavioral Science, 7(1), 67-80.

[21] Hallams, E. (2011): “From Crusader to Exemplar: Bush, Obama and the Rein-

vigoration of America’s Soft Power,” European Journal of American Studies, 6(1),

doi:10.4000/ejas.9157

[22] Kamenica, E., and M. Gentzkow (2011): “Bayesian Persuasion,” American Economic

Review, 101, 2590-2615.

[23] Jung, H. (2007): “Strategic Information Transmission through the Media,” Working

Paper, Lahore University.

34

[24] Kissinger, H. (1994): Diplomacy, Simon and Schuster.

[25] Klein, D. B. and O’Flaherty, B. (1993): “A game-theoretic rendering of promises

and threats,” Journal of Economic Behavior and Organization, 21(3), 295-314.

[26] Kolotilin, A., M. Li, T. Mylovanov, and A. Zapechelnyuk (2015): “Persuasion of

Privately Informed Receiver,” Discussion paper.

[27] Li, C. (2017): “A Model of Bayesian Persuasion with Transfers,” Economics Letters,

161, 93-95.

[28] Lipnowski, E., D. Ravid, and D. Shishkin (2018): “Persuasion via Weak Institu-

tions,” Working Paper.

[29] Myerson, R. (1982): “Optimal Coordination Mechanisms in Generalized Principal-

Agent Problems,” Journal of Mathematical Economics, 10, 67-81.

[30] Myerson, R. (1983): “Mechanism design by an informed Principal,” Econometrica,

51, 1767-1797.

[31] Myerson, R. and M. Satterthwaite (1983): “Efficient Mechanisms for Bilateral Trad-

ing,” Journal of Economic Theory, 29(2), 265–281.

[32] Nye, J. (1990): “Soft Power,” Foreign Policy, 80, 153-171. doi:10.2307/1148580

[33] Nye, J. S. (2004): Soft power: the means to success in world politics. New York:

Public Affairs.

[34] Nye, J. S. (2011): The future of power. New York: Public Affairs.

[35] Parsons, T. (1962): “On the concept of influence,” The Public Opinion Quarterly,

27(1), 37-62.

[36] Rayman, N. (2014): “President Barack Obama Promotes Soft Power Foreign Policy

in Interview,” Time, Retrieved from http://time.com/135645/obama-cooperation-

soft-power/

35

[37] Sartori, A. (2002): “The Might of the Pen: A Reputational Theory of Communication

in International Disputes,” International Organization, 56(1), 121-149.

[38] Schelling, T. (1960): The Strategy of Conflict. Cambridge: Harvard University Press.

[39] St. John, R.B. (2017): Libya: From colony to revolution. London: Oneworld Publi-

cations.

[40] Vandewalle, D. J. (2012): A history of modern Libya. Cambridge: Cambridge Uni-

versity Press.

Appendix A Proof of Proposition 1

First, notice that, by assumption, B > Da+Ca+G−1, meaning that no m satisfying (2)

for an “unreasonable” Agent can have m(sp, Comply|tp, tal ) > 0, ∀sp ∈ Sptp ∈ T p. This is

because even in the best case scenario, wherein the probability of playing both “Carrot”

and “Stick” is 1, and a recommendation of “Comply” is maximally informative (that is,

m(CS,Comply|tph, tal ) = 1 and m(sp, Comply|tpl , tal ) = 0,∀sp ∈ Sp), the “unreasonable”

Agent would still be better off not complying.

We then have two cases. In the first case, πp ≥ 1G

, meaning that an “reasonable”

Agent would always play Comply, and therefore the Principal receives a payoff of πa in

the No-Communication game. By the above observation, no mechanism satisfying Agent-

Obedience can change the “unreasonable” Agent’s choice, and therefore V = πa = V ∗.

In the second case, πp < 1G

, the payoff under the No-Communication game is 0. We

now must establish that V = 0.

Since any toothless mechanism m satisfies (3) and (4), we straightforwardly have that

m(NCNS, sa|tp, ta) = 1, ∀sa ∈ Sa, (tp, ta) ∈ T .

We now only have to establish that m(NCNS,Comply|tp, ta) = 0 for all m satisfying

constraints (1)-(5).

Suppose there were somem such that this were not true (that is, m(NCNS,Comply|tp, ta) >

0 for some (tp, ta) ∈ T , while still satisfying constraints (1)-(5) ). Notice that a “reason-

36

able” Agent is able to guarantee themselves a payoff of 1 by simply playing Don’t Comply,

meaning that the Agent-Obedience constraint for the “reasonable” Agent straightfor-

wardly reduces to:

(πp)m(NCNS,Don′t Comply|tph, tah) + (1− πp)m(NCNS,Don′t Comply|tpl , t

ah)

+(πp)m(NCNS,Comply|tph, tah)G ≥ 1

with the left-hand side of the inequality representing the expected payoff from obeying

the mechanism m. By straightforward algebra, this reduces further to the following

inequality:

(πp)m(NCNS,Comply|tph, tah)G(πp)m(NCNS,Comply|tph, tah) + (1− πp)m(NCNS,Comply|tpl , tah)

≥ 11

G> πp

with the last two inequality following from our above assumptions that πp < 1G< 1.

The above inequality simply states that the Bayesian posterior of the Agent that the

Principal is trustworthy, conditional upon receiving a recommendation of Comply, must

be higher than πp. By the logic of Bayesian updating, then, it must be the case that

a recommendation of Comply is more likely to occur when the Principal is trustwor-

thy than when the Principal is untrustworthy, that is, that m(NCNS,Comply|tph, tah) >

m(NCNS,Comply|tpl , tah). This would in turn imply that (πa)m(NCNS,Comply|tpl , tah) <

(πa)m(NCNS,Comply|tph, tah), meaning that the trustworthy Principal receives a higher

payoff than the untrustworthy Principal, thereby violating Principal-Truthtelling and

yielding a contradiction.

We then have that m(sp, Comply|tp, ta) = 0, ∀(tp, ta) ∈ T, sp ∈ Sp, yielding a payoff

of 0 for the Principal, as in the No Communication game.

37

Appendix B Proof of Proposition 2

Suppose all of the 3 conditions given hold. Consider the following “pooling” mechanism

m:

∑Sp

m(sp, Comply|tpl , tah) =

∑Sp

m(sp, Comply|tph, tah) = 1

m(sp, Comply|tpl , tal ) = hatm(sp, Comply|tph, t

al ) = 0 ∀sp ∈ Sp

m(CNS, Comply|tph, tah) = m(CNS, Comply|tpl , t

ah) =

1− πpGCa

m(CS, sa|tp, ta) = m(NCS, sa|tp, ta) = 0∀ (tp, ta) ∈ T

We know that such a mechanism is feasible (1−πpG

Ca < 1) by our condition 1. (5) is

trivially satisfied, since both Principal types receive the same expected payoff, as is (4).

First, we observe that a reasonable Agent who obeys the mechanism m receives a

payoff of:

(πp)m(NCNS, Comply|tph, tah)(G) + (πp)m(CNS, Comply|tph, t

ah)(G+ Ca)

+(1− πp)m(CNS, Comply|tpl , tah)(C

a)

= πpG+1− πpGCa

(Ca) = 1

Since 1 is exactly the payoff received by a reasonable Agent that always disregards

any recommendation to Comply, we can see that Agent-Obedience is satisfied. Since this

is also the payoff received by a reasonable Agent that lies about their type, we can see

that Agent-Truthtelling is also satisfied.

Having checked that both truth-telling and obedience are satisfied for the Agent, we

38

need only check that the expected payoff from this mechanism for the Principal is greater

than 0 (the payoff for the Principal under a toothless mediator when πp satisfies condition

3). Observe that the expected payoff to the Principal from m is:

(πa)[(πp)m(NCNS, Comply|tph, tah) + (1− πp)m(NCNS, Comply|tpl , t

ah)

+(πp)m(CNS, Comply|tph, tah)(1− Cp) + (1− πp)m(CNS, Comply|tpl , t

ah)(1− Cp)]

= (πa)[(1− (1− πpGCa

))(1) +1− πpGCa

(1− Cp)] = (πa)(1− (1− πpGCa

)Cp)

The term in parentheses, by condition (3), is positive, meaning that the expected

payoff to the Principal is also positive.

Because V = 0 from our condition 3, and the mechanism described above yields a

positive payoff while satisfying all of our constraints, we have established that V c > V .

Appendix C Proof of Proposition 3

Suppose all of the 3 conditions given hold. Consider the following “pooling” mechanism

m:

∑Sp

m(sp, Comply|tpl , tah) =

∑Sp

m(sp, Comply|tph, tah) = 1

m(sp, Comply|tpl , tal ) = m(sp, Comply|tph, t

al ) = 0, ∀sp ∈ Sp

m(NCS, Comply|tph, tah) = m(NCS, Comply|tpl , t

ah) =

1− πpGDa

m(NCS, Don′t Comply|tph, tal ) = m(NCS, Don′t Comply|tpl , t

al ) =

1− πpGDa

39

m(CS, sa|tp, ta) = m(CNS, sa|tp, ta) = 0∀ (tp, ta) ∈ T

We know that such a mechanism is feasible (1−πpG

Da < 1) by our condition 1. (5) is

trivially satisfied, as both Principal types receive the same expected payoff, as is (3).

To check (1), observe that any reasonable Agent who deviates from a recommendation

to comply, or who lies about their type, receives an expected payoff of:

(πp)m(NCNS, Comply|tph, tah) + (1− πp)m(NCNS, Comply|tpl , t

ah)

+(πp)m(NCS, Comply|tpl , tah)(1−Da) + (1− πp)m(NCS, Comply|tpl , t

ah)(1−Da)

= 1− 1− πpGDa

(Da) = (πp)(G)

This, however, is exactly the payoff that a reasonable Agent would receive from obey-

ing m, meaning that both Agent-Obedience and Agent-Truthtelling are satisfied.

Having checked that both truth-telling and obedience are satisfied for the Agent, we

need only check that the expected payoff from this mechanism for the Principal is greater

than 0 (the payoff for the Principal under a toothless mediator when πp satisfies condition

3). Observe that the Principal’s expected payoff is:

(πa) + (1− πa)[(πp)m(NCS, Don′tComply|tpl , tal )(−Da)

+(1− πp)m(NCS, Don′tComply|tpl , tal )(−Da)]

= (πa)− (1− πa)((1− πpGDa

)Dp)

By our condition 2, this is positive.

Because V = 0 from our condition 3, and the mechanism described above yields a

positive payoff while satisfying all of our constraints, by the definition of V d it must be

the case that V d > V .

40

Appendix D Proof of Proposition 4

From (3) and (4), we immediately have that the Principal’s strategy is restricted to only

be NCNS, while from (1) we immediately have that m(sp, Comply|tp, tal ) = 0, ∀sp ∈

Sp, tp ∈ T p. This means we can restrict ourselves to considering only our selection of the

choice probabilities m(NCNS,Comply|tph, tah) and m(NCNS,Comply|tph, tah), subject to

Agent-Obedience (if Agent-Obedience is satisfied, Agent-Truthtelling is satisfied as well,

since the Agent’s payoff from deviating to Don’t Comply is exactly the same as that

achievable by misreporting their type).

The “only if” portion of the proof is trivial: if 1G> 1, then even in the case of a

maximally informative recommendation of “Comply” (ie., m(NCNS,Comply|tph, tah) = 1

and m(NCNS,Comply|tph, tah) = 0), a “reasonable” Agent would still achieve a higher

payoff from deviating to playing “Don’t Comply”, and therefore any mechanism which

recommended “Comply” with positive probability would violate (1). Similarly, if 1G≤ πp,

the toothless mediator would yield an expected payoff of πa to the Principal, which cannot

be improved on.

To show the “if” portion of the proof, we assume that both our conditions hold; that

is, πp < 1G< 1.

Observe that the expected payoff to the Principal is simply equal to πa[(πp)m(NCNS,

Comply|tph, tah) + (1− πp)m(NCNS,Comply|tph, tah)], the total probability of compliance.

One can also observe that the Principal’s expected payoff is bounded below by the payoff

from the mechanismm such thatm(NCNS,Comply|tph, tah) = 1 andm(NCNS,Comply|tph,

tah) = 0. Such a mechanism clearly satisfies both Agent-Obedience and -Truthtelling, be-

cause it yields a payoff of (πp)G + (1 − πp), which is strictly greater than 1 by our

assumption that πp < 1G< 1. The expected payoff from said mechanism is just πaπp > 0,

and therefore in any optimal mechanism the total probability of compliance must also be

greater than 0.

Next, notice that in any optimal mechanism satisfying Agent-Obedience, it must

be the case that m(NCNS,Comply|tph, tah) = 1, meaning that, when the Principal is

trustworthy, a reasonable Agent is recommended to comply with probability 1. For

41

any mechanism satisfying Agent-Obedience such that this were not the case, it would be

possible to strictly increase the Principal’s payoff by increasing m(NCNS,Comply|tph, tah),

while also increasing the left-hand side of the inequality in Agent-Obedience.

As in our proof of Proposition 1, we can rewrite constraint Agent-Obedience as:

(πp)m(NCNS,Comply|tph, tah)G(πp)m(NCNS,Comply|tph, tah) + (1− πp)m(NCNS,Comply|tpl , tah)

≥ 1

which, given our finding above that m(NCNS,Comply|tph, tah) = 1, can be rewritten

as:

(πp)G ≥ (πp) + (1− πp)m(NCNS,Comply|tpl , tah)

We can then observe that in any optimal mechanism, this inequality must be an

equality, that is, it must be the case that:

m(NCNS,Comply|tpl , tah) =

πp(G− 1)

1− πp

which is both positive (because 1G< 1) and less than zero (because πp < 1

G).

Under this optimal mechanism, wherem(NCNS,Comply|tph, tah) = 1 andm(NCNS,Comply|tpl , tah) =

πp(G−1)1−πp , we then have that the Principal’s expected payoff is

πa[(πp)m(NCNS,Comply|tph, tah) + (1− πp)m(NCNS,Comply|tph, t

ah)]

= πa[(πp) + (1− πp)πp(G− 1)

1− πp]

= πa[(πp) + πp(G− 1)]

= πa[πpG]

= πa − πa(1− πpG)

As in our stated Proposition.

42

Appendix E Proof of Proposition 5

First, we can define the following objects for every mechanism m:

Let

ρ =

(πp)∑SC

m(sp, Comply|tph, tah) + (1− πp)∑SC

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)

be the probability of playing Carrot conditional upon a recommendation of playing Com-

ply, let

σ =

(πp)∑Sp

m(sp, Comply|tph, tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)

be the probability, conditional upon a recommendation of Comply, that the Principal is

trustworthy, let

φ =

(πp)∑SS

m(sp, Comply|tph, tah) + (1− πp)∑SS

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)

be the probability of playing Stick conditional upon a recommendation of playing Comply

to a reasonable agent, let

ψ =

(πp)∑SS

m(sp, Don′t Comply|tph, tah) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)

be the total probability of playing Stick against a reasonable Agent who’s been recom-

mended to play Not Comply, divided by the total probability of recommending Comply

to a reasonable Agent, let

τ =

(πp)∑SS

m(sp, Don′t Comply|tph, tal ) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tal )

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)

be the total probability of playing Stick against an unreasonable agent divided by the

43

total probability of recommending Comply to a reasonable agent, and let

q = (πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)

∑Sp

m(sp, Comply|tpl , tah)

be the total probability of recommending Comply to a reasonable Agent.

We can write our Agent-Truthtelling and Agent-Obedience constraints directly in

terms of these objects. Consider first the Agent-Truthtelling constraint for a reasonable

Agent:

(πp)∑Sp

m(sp, Don′t Comply|tph, tah) + (1− πp)

∑Sp

m(sp, Don′t Comply|tpl , tah)

+ (πp)∑SS

m(sp, Don′t Comply|tph, tah)(−Da)+(1−πp)

∑SS

m(sp, Don′t Comply|tpl , tah)(−Da)

+ (πp)∑SC

m(sp, Comply|tph, tah)C

a + (1− πp)∑SC

m(sp, Comply|tpl , tah)C

a

+ (πp)∑Sp

m(sp, Comply|tph, tah)G

≥ 1+(πp)∑SS

m(sp, Don′t Comply|tph, tal )(−Da)+(1−πp)

∑SS

m(sp, Don′t Comply|tpl , tal )(−Da)

⇐⇒ (1−q)+(πp)∑SS

m(sp, Don′t Comply|tph, tah)(−Da)+(1−πp)

∑SS

m(sp, Don′t Comply|tpl , tah)(−Da)

+ (πp)∑SC

m(sp, Comply|tph, tah)C

a + (1− πp)∑SC

m(sp, Comply|tpl , tah)C

a

+ (πp)∑Sp

m(sp, Comply|tph, tah)G

≥ 1+(πp)∑SS

m(sp, Don′t Comply|tph, tal )(−Da)+(1−πp)

∑SS

m(sp, Don′t Comply|tpl , tal )(−Da)

⇐⇒ ρCa + σG− ψDa ≥ 1− τDa

with the last inequality coming from subtracting 1−q from both sides of the inequality

and dividing both sides by q.

Similarly, consider the Agent-Truthtelling constraint for an unreasonable agent, specif-

44

ically the constraint that an unreasonable Agent not be incentivized to misreport their

type and then deviate to always play Don’t Comply6:

B+1+(πp)∑SS

m(sp, Don′t Comply|tph, tal )(−Da)+(1−πp)

∑SS

m(sp, Don′t Comply|tpl , tal )(−Da)

≥ B+1+(πp)∑SS

m(sp, Don′t Comply|tph, tah)(−Da)+(1−πp)

∑SS

m(sp, Don′t Comply|tpl , tah)(−Da)

+ (πp)∑SS

m(sp, Comply|tph, tah)(−Da) + (1− πp)

∑SS

m(sp, Comply|tpl , tah)(−Da)

⇐⇒ 1− τDa ≥ 1− ψDa − φDa

with the last inequality coming from subtracting B + 1 − q from both sides of the

inequality and dividing both sides by q.

Next, consider the Agent-Obedience constraint for a reasonable Agent:

(πp)∑Sp

m(sp, Don′t Comply|tph, tah) + (1− πp)

∑Sp

m(sp, Don′t Comply|tpl , tah)

+ (πp)∑SS

m(sp, Don′t Comply|tph, tah)(−Da)+(1−πp)

∑SS

m(sp, Don′t Comply|tpl , tah)(−Da)

+(πp)∑SC

m(sp, Comply|tph, tah)C

a + (1− πp)∑SC

m(sp, Comply|tpl , tah)C

a

+(πp)∑Sp

m(sp, Comply|tph, tah)G

≥ 1 + (πp)∑SS

m(sp, Comply|tph, tah)(−Da) + (1− πp)

∑SS

m(sp, Comply|tpl , tah)(−Da)

+ (πp)∑SS

m(sp, Don′t Comply|tph, tah)(−Da)+(1−πp)

∑SS

m(sp, Don′t Comply|tpl , tah)(−Da)

⇐⇒ 1−q+(πp)∑SS

m(sp, Don′t Comply|tph, tah)(−Da)+(1−πp)

∑SS

m(sp, Don′t Comply|tpl , tah)(−Da)

6Because B > Ca +Da +G− 1, under no possible mechanism would an unreasonable Agent have anincentive to misreport their type and obey a recommendation to comply.

45

+(πp)∑SC

m(sp, Comply|tph, tah)C

a + (1− πp)∑SC

m(sp, Comply|tpl , tah)C

a

+(πp)∑Sp

m(sp, Comply|tph, tah)G

≥ 1 + (πp)∑SS

m(sp, Comply|tph, tah)(−Da) + (1− πp)

∑SS

m(sp, Comply|tpl , tah)(−Da)

+ (πp)∑SS

m(sp, Don′t Comply|tph, tah)(−Da)+(1−πp)

∑SS

m(sp, Don′t Comply|tpl , tah)(−Da)

⇐⇒ ρCa + σG− ψDa ≥ 1− ψDa − φDa

with the final inequality again coming from subtracting 1 − q from both sides and

dividing both sides by q.

Finally, observe that we can write the expected payoff to the Principal from any

mechanism in terms of q, ρ, ψ, and τ :

(πa)[q(1− ρCp − ψDa)] + (1− πa)[q(−τDp)]

Observe that as long as the two Agent-Truthtelling constraints given above are sat-

isfied, Agent-Obedience is satisfied. When considering first-best mechanisms, then, we

need only to consider the two Agent-Truthtelling constraints.

Having recast the problem of finding a first-best mechanism in terms of q, ρ, φ, ψ, σ,

and τ , we now prove some properties that any first-best mechanism must have.

First, consider any mechanism m such that both of the Truthtelling constraints held,

and such that ψ ≥ 0. Observe that any such mechanism yields a weakly lower ex-

pected payoff to the Principal than a mechanism m which sets ψ = 0 and φ = τ =

max{1−ρCa−σGDa , 0}, while leaving ρ, σ, and q unchanged. This is because such a mecha-

nism, relative to our original mechanism m, has a weakly lower ψ and τ , both of which

enter the Principal’s expected payoff negatively. Moreover, we can see that such a mech-

anism m is both possible (because V d > V = 0, meaning that 1−ρCa−σGDa ≤ 1−πpG

Da < 1,

by the feasibility condition for deterrence) and (by our definition of τ and φ) guaranteed

to satisfy our constraints. Any first best mechanism, then, must be such that ψ = 0 and

φ = τ = max{1−ρCa−σGDa , 0}.

46

Furthermore, observe that among such mechanisms, it could never be the case that,

in a first-best mechanism, 1−ρCa−σGDa < 0. If that were the case, it would be possible to

either lower ρ (by lowering the probability of playing Carrot) or lower σ (by increasing

the probability of recommending Comply) and thereby increase the expected payoff to

the Principal without weakening any of the constraints (due to our restrictions on τ and

φ). Therefore, in any first-best mechanism, it must be the case that φ = τ = 1−ρCa−σGDa

Finally, observe that any mechanism in which Don’t Comply is recommended with

positive probability for a trustworthy Principal cannot be a first-best mechanism; one

could keep the probability of recommending Don’t Comply for an untrustworthy Princi-

pal constant while changing the probability of recommending Don’t Comply for a trust-

worthy Principal to zero, thereby increasing both the Principal’s expected payoff and σ

(slackening all of the above constraints). From this condition and the law of iterated

expectations, we then have that qσ = πp, or q = πp

σ.

Because the above conditions are all ultimately in terms of just ρ and σ, guarantee the

satisfaction of both Truthtelling constraints, and must hold in any first-best mechanism,

we can now reduce the problem of finding the payoff to a first-best mechanism to the

following problem of finding an optimal ρ and σ:

maxρ∈[0,1],σ∈[πp,1]

(πa)(πp)[(1− ρCp)]1

σ− (1− πa)(πp)[(1− ρCa − σG

DaDp)]

1

σ

s.t. ρCa + σG ≤ 1

with the constraint ρCa + σG ≤ 1 simply reflecting the nonnegativity of τ .

By collecting all of the terms multiplying 1σ

and ρσ, removing any constant terms, and

multiplying by the positive constant 1πpCa , we can once more rewrite this problem as:

maxρ∈[0,1],σ∈[πp,1]

1

Ca[(πa)− (1− πa)(D

p

Da)]

1

σ+ [(1− πa)(D

p

Da)− (πa)

Cp

Ca]ρ

σ

s.t. ρCa + σG ≤ 1

To prove Proposition 5, we need to prove that the only solutions to this problem are

47

such that either (σ = 1G, ρ = 0), (σ = πp, ρ = 1−πpG

Ca ), or (σ = πp, ρ = 0). We proceed

case by case.

E.1 Case 1: 1 < 1−πa

πaDp

Da ,Cp

Ca

In this case, the first bracketed term must be negative, while the second is ambiguous.

We then have that either 1−πa

πaDp

Da >Cp

Ca or 1−πa

πaDp

Da <Cp

Ca .

In the first instance, we have that the expected payoff to the Principal is strictly

decreasing in both ρ and 1σ

for any ρ and σ. The optimal mechanism must then be such

that ρ is at its minimum and σ is at its constrained maximum, that is, ρ = 0 and σ = 1G

.

This is just our optimal “Soft Power” mechanism, however, which means that V s = V ∗.

In the second instance, we have that the expected payoff to the Principal is strictly

increasing in ρ, while it is ambiguous in 1σ. In this case, we know that any “interior”

mechanism can be improved upon in terms of expected payoff by leaving σ unchanged

and setting ρ = 1−σGCa . Substituting this into the Principal’s expected payoff, we then

have:

(πa)(πp)[(1− (1− σGCa

)Cp)]1

σ

= (πa)(πp)[(1− Cp

Ca)]

1

σ+ (πa)(πp)

GCp

Ca

which is strictly decreasing in 1σ. This implies that we can again improve the Princi-

pal’s payoff by setting σ equal to its constrained maximum of σ = 1G

. This again is our

optimal “Soft Power” mechanism, which means that V s = V ∗.

E.2 Case 2: Cp

Ca < 1, 1−πa

πaDp

Da

In this case, the second bracketed term must be positive, while the first is ambiguous.

We then have that either 1−πa

πaDp

Da > 1 or 1−πa

πaDp

Da < 1.

In the first instance, we have that the expected payoff to the Principal is strictly

increasing in both ρ and 1σ

for any ρ and σ. The optimal mechanism must then be such

that σ is at its minimum and ρ is at its constrained maximum, that is, ρ = 1−πpGCa and

48

σ = πp. This is just our optimal compellence mechanism, however, which means that

V c = V ∗.

In the second instance, we have that the expected payoff to the Principal is strictly

increasing in ρ, while it is ambiguous in 1σ. In this case, we know that any “interior”

mechanism can be improved upon in terms of expected payoff by leaving σ unchanged

and setting ρ = 1−σGCa . Substituting this into the Principal’s expected payoff, we then

have:

(πa)(πp)[(1− (1− σGCa

)Cp)]1

σ

= (πa)(πp)[(1− Cp

Ca)]

1

σ+ (πa)(πp)

GCp

Ca

which is strictly increasing in 1σ. This implies that we can again improve the Principal’s

payoff by setting σ equal to its minimum of σ = πp. This again is our optimal compellence

mechanism, which means that V c = V ∗.

E.3 Case 3: 1−πa

πaDp

Da < 1, Cp

Ca

In this case, the first bracketed term must be positive, while the second is negative.

We then have that the expected payoff to the Principal is strictly decreasing in ρ,

while it is ambiguous in 1σ. In this case, we know that any “interior” mechanism can be

improved upon in terms of expected payoff by leaving σ unchanged and setting ρ = 0.

Substituting this into the Principal’s expected payoff, we then have:

(πa)(πp)1

σ− (1− πa)(πp)(D

p

Da)1

σ+ (1− πa)(πp)(G)(

Dp

Da)

= (πp)[(πa)− (1− πa)(Dp

Da)]

1

σ+ (1− πa)(πp)(G)(

Dp

Da)

which is strictly increasing in 1σ. This implies that we can again improve the Principal’s

payoff by setting σ equal to its minimum of σ = πp. This, however, is our optimal

deterrence mechanism, which means that V d = V ∗.

49

Appendix F Proof of Proposition 6

Suppose that G < 1, and Ca, Da < 1−πpG, but that G+Da+Ca > 1 and Cp+ 1−πa

πa Dp <

1. By our assumption that G < 1 and Ca, Da < 1 − πpG, it must be the case that

V d = V c = V s = v = 0.

In order to demonstrate that V ∗ > 0, we proceed case by case, and construct mech-

anisms that satisfy our Agent-Obedience and Agent-Truthtelling constraints and which

yield a positive expected payoff for the Principal.

For any proposed mechanism m, let φ, ψ, ρ, σ, τ , and q be as defined above in

our proof of Proposition 5, and recall from that proof that the Agent-Obedience and

Agent-Truthtelling constraints are satisfied as long as

ρCa + σG− ψDa ≥ 1− τDa

and

1− τDa ≥ 1− ψDa − φDa

and that the Principal’s expected payoff is just

(πa)[q(1− ρCp − ψDa)] + (1− πa)[q(−τDp)]

.

F.1 Case 1: 1 < 1−πa

πaDp

Da <Cp

Ca

Either G+Da > 1 or not.

In the first instance, consider the mechanism such that:

∑Sp

m(sp, Comply|tpl , tah) = 0,

∑Sp

m(sp, Comply|tph, tah) = 1

m(sp, Comply|tpl , tal ) = m(sp, Comply|tph, t

al ) = 0, ∀sp ∈ Sp

50

∑SS

m(sp, Comply|tph, tah) =

1−GDa

∑SS

m(sp, Don′t Comply|tph, tal ) =

1−GDa

∑SC ,Sa

m(sp, sa|tp, ta) = 0∀(tp, ta) ∈ T

We know that such a mechanism is feasible (ie, 1−GDa < 1) under the assumption that

G+Da > 1.

For this mechanism, we have that

q = (πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)

∑Sp

m(sp, Comply|tpl , tah) = πp

σ =

(πp)∑Sp

m(sp, Comply|tph, tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 1

ρ =

(πp)∑SC

m(sp, Comply|tph, tah) + (1− πp)∑SC

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 0

φ =

(πp)∑SS

m(sp, Comply|tph, tah) + (1− πp)∑SS

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)=

1−GDa

τ =

(πp)∑SS

m(sp, Don′t Comply|tph, tal ) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tal )

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)=

1−GDa

and

51

ψ =

(πp)∑SS

m(sp, Don′t Comply|tph, tah) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 0

Checking our first Truthtelling constraint, we have that:

ρCa + σG− ψDa = G

= 1− 1−GDa

Da = 1− τDa

and for our second, we have that:

1− τDa = 1− 1−GDa

Da

= 1− φDa

Finally, the expected payoff to the Principal is just:

(πa)[q(1− ρCp − ψDa)] + (1− πa)[q(−τDp)]

= (πa)(πp)− (1− πa)(πp)1−GDa

(Dp)

Dividing by (πa)(πp) gives us that the above is positive if and only if:

1− πa

πa1−GDa

(Dp) < 1

From our assumption that G+Da > 1 and that Cp + 1−πa

πa Dp < 1, we then have that:

1− πa

πa1−GDa

(Dp) <1− πa

πa(Dp) < Cp +

1− πa

πa(Dp) < 1

Since all of our constraints are satisfied by our mechanism and the Principal’s expected

payoff is positive, we then have that V ∗ must also be positive, and we are done.

In the second instance, consider the mechanism such that:

52

∑Sp

m(sp, Comply|tpl , tah) = 0,

∑Sp

m(sp, Comply|tph, tah) = 1

m(sp, Comply|tpl , tal ) = m(sp, Comply|tph, t

al ) = 0, ∀sp ∈ Sp

∑SS

m(sp, Comply|tph, tah) = 1

∑SS

m(sp, Don′t Comply|tph, tal ) = 1

∑SC

m(sp, Comply|tph, tah) =

1−G−Da

Ca

We know that such a mechanism is feasible (ie., 1−G−Da

Ca < 1) under the assumption

that G+Da + Ca > 1.

For this mechanism, we have that

q = (πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)

∑Sp

m(sp, Comply|tpl , tah) = πp

σ =

(πp)∑Sp

m(sp, Comply|tph, tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 1

ρ =

(πp)∑SC

m(sp, Comply|tph, tah) + (1− πp)∑SC

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)=

1−G−Da

Ca

φ =

(πp)∑SS

m(sp, Comply|tph, tah) + (1− πp)∑SS

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 1

53

τ =

(πp)∑SS

m(sp, Don′t Comply|tph, tal ) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tal )

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 1

and

ψ =

(πp)∑SS

m(sp, Don′t Comply|tph, tah) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 0

Checking our first Truthtelling constraint, we have that:

ρCa + σG− ψDa =1−G−Da

CaCa +G

= 1−Da = 1− τDa

and for our second, we have that:

1− τDa = 1−Da

= 1− φDa

Finally, the expected payoff to the Principal is just:

(πa)[q(1− ρCp − ψDa)] + (1− πa)[q(−τDp)]

= (πa)(πp)(1− 1−G−Da

CaCp)− (1− πa)(πp)(Dp)

Dividing by (πa)(πp) gives us that the above is positive if and only if:

1−G−Da

CaCp +

1− πa

πa(Dp) < 1

From our assumption that Ca +G+Da > 1 and that Cp + 1−πa

πa Dp < 1, we then have

that:

54

1−G−Da

CaCp +

1− πa

πa(Dp) < Cp +

1− πa

πa(Dp) < 1

Since all of our constraints are satisfied by our mechanism and the Principal’s expected

payoff is positive, we then have that V ∗ must also be positive, and we are done.

F.2 Case 2: 1 < Cp

Ca <1−πa

πaDp

Da

Either G+ Ca > 1 or not.

In the first instance, consider the mechanism such that:

∑Sp

m(sp, Comply|tpl , tah) = 0,

∑Sp

m(sp, Comply|tph, tah) = 1

m(sp, Comply|tpl , tal ) = m(sp, Comply|tph, t

al ) = 0,∀sp ∈ Sp

∑SC

m(sp, Comply|tph, tah) =

1−GCa

∑SC

m(sp, Don′t Comply|tph, tal ) =

∑SC

m(sp, Don′t Comply|tpl , tal ) = 0

∑SS ,Sa

m(sp, sa|tp, ta) = 0∀(tp, ta) ∈ T

We know that such a mechanism is feasible (ie., 1−GCa < 1) under the assumption that

G+ Ca > 1.

For this mechanism, we have that

q = (πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)

∑Sp

m(sp, Comply|tpl , tah) = πp

55

σ =

(πp)∑Sp

m(sp, Comply|tph, tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 1

ρ =

(πp)∑SC

m(sp, Comply|tph, tah) + (1− πp)∑SC

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)=

1−GCa

φ =

(πp)∑SS

m(sp, Comply|tph, tah) + (1− πp)∑SS

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 0

τ =

(πp)∑SS

m(sp, Don′t Comply|tph, tal ) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tal )

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 0

and

ψ =

(πp)∑SS

m(sp, Don′t Comply|tph, tah) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 0

Checking our first Truthtelling constraint, we have that:

ρCa + σG− ψDa =1−GCa

Ca +G

= 1 = 1− τDa

and for our second, we have that:

1− τDa = 1

= 1− φDa

Finally, the expected payoff to the Principal is just:

56

(πa)[q(1− ρCp − ψDa)] + (1− πa)[q(−τDp)]

= (πa)(πp)(1− 1−GCa

Cp)

Dividing by (πa)(πp) gives us that the above is positive if and only if:

1−GCa

(Cp) < 1

From our assumption that G+Ca > 1 and that Cp + 1−πa

πa Dp < 1, we then have that:

1−GCa

(Cp) < Cp < Cp +1− πa

πa(Dp) < 1

Since all of our constraints are satisfied by our mechanism and the Principal’s expected

payoff is positive, we then have that V ∗ must also be positive, and we are done.

In the second instance, consider the mechanism such that:

∑Sp

m(sp, Comply|tpl , tah) = 0,

∑Sp

m(sp, Comply|tph, tah) = 1

m(sp, Comply|tpl , tal ) = m(sp, Comply|tph, t

al ) = 0,∀sp ∈ Sp

∑SS

m(sp, Comply|tph, tah) =

1−G− Ca

Da

∑SS

m(sp, Don′t Comply|tph, tal ) =

1−G− Ca

Da

∑SC

m(sp, Comply|tph, tah) = 1

We know that such a mechanism is feasible (ie., 1−G−Ca

Da < 1) under the assumption

that G+Da + Ca > 1.

For this mechanism, we have that

57

q = (πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)

∑Sp

m(sp, Comply|tpl , tah) = πp

σ =

(πp)∑Sp

m(sp, Comply|tph, tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 1

ρ =

(πp)∑SC

m(sp, Comply|tph, tah) + (1− πp)∑SC

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 1

φ =

(πp)∑SS

m(sp, Comply|tph, tah) + (1− πp)∑SS

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)=

1−G− Ca

Da

τ =

(πp)∑SS

m(sp, Don′t Comply|tph, tal ) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tal )

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)=

1−G− Ca

Da

and

ψ =

(πp)∑SS

m(sp, Don′t Comply|tph, tah) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 0

Checking our first Truthtelling constraint, we have that:

ρCa + σG− ψDa = Ca +G

= 1− 1−G− Ca

DaDa = 1− τDa

and for our second, we have that:

1− τDa = 1− 1−G− Ca

DaDa

58

= 1− φDa

Finally, the expected payoff to the Principal is just:

(πa)[q(1− ρCp − ψDa)] + (1− πa)[q(−τDp)]

= (πa)(πp)(1− Cp)− (1− πa)(πp)1−G− Ca

Da(Dp)

Dividing by (πa)(πp) gives us that the above is positive if and only if:

Cp +1− πa

πa1−G− Ca

Da(Dp) < 1

From our assumption that G+Ca +Da > 1 and that Cp + 1−πa

πa Dp < 1, we then have

that:

Cp +1− πa

πa1−G− Ca

Da(Dp) < Cp +

1− πa

πa(Dp) < 1

Since all of our constraints are satisfied by our mechanism and the Principal’s expected

payoff is positive, we then have that V ∗ must also be positive, and we are done.

F.3 Case 3: Cp

Ca < 1 < 1−πa

πaDp

Da

Either G+ Ca > 1 or not.

In the first instance, consider the mechanism such that:

∑Sp

m(sp, Comply|tpl , tah) =

( G1−Ca − 1)πp

1− πp,∑Sp

m(sp, Comply|tph, tah) = 1

m(sp, Comply|tpl , tal ) = m(sp, Comply|tph, t

al ) = 0,∀sp ∈ Sp

∑SC

m(sp, Comply|tpl , tah) =

( G1−Ca − 1)πp

1− πp,∑SC

m(sp, Comply|tph, tah) = 1

59

∑SC

m(sp, Don′t Comply|tpl , tal ) =

∑SC

m(sp, Don′t Comply|tph, tal ) = 0

∑SS ,Sa

m(sp, sa|tp, ta) = 0∀(tp, ta) ∈ T

We know that such a mechanism is feasible (ie.,( GCa−1

−1)πp

1−πp < 1) under the assumption

that πpG+ Ca < 1.

For this mechanism, we have that

q = (πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)

∑Sp

m(sp, Comply|tpl , tah)

= (1− πp)(( G1−Ca − 1)πp

1− πp) + πp =

πpG

1− Ca

σ =

(πp)∑Sp

m(sp, Comply|tph, tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)=

1− Ca

G

ρ =

(πp)∑SC

m(sp, Comply|tph, tah) + (1− πp)∑SC

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 1

φ =

(πp)∑SS

m(sp, Comply|tph, tah) + (1− πp)∑SS

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 0

τ =

(πp)∑SS

m(sp, Don′t Comply|tph, tal ) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tal )

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 0

and

60

ψ =

(πp)∑SS

m(sp, Don′t Comply|tph, tah) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 0

Checking our first Truthtelling constraint, we have that:

ρCa + σG− ψDa = Ca +1− Ca

GG

= 1 = 1− τDa

and for our second, we have that:

1− τDa = 1

= 1− φDa

Finally, the expected payoff to the Principal is just:

(πa)[q(1− ρCp − ψDa)] + (1− πa)[q(−τDp)]

= (πa)(πpG

1− Ca(1− Cp))

which is positive if and only if:

Cp < 1

Which must be true given our assumption that Cp + 1−πa

πa Dp < 1.

Since all of our constraints are satisfied by our mechanism and the Principal’s expected

payoff is positive, we then have that V ∗ must also be positive, and we are done.

In the second instance, consider the mechanism such that:

∑Sp

m(sp, Comply|tpl , tah) = 0,

∑Sp

m(sp, Comply|tph, tah) = 1

61

m(sp, Comply|tpl , tal ) = m(sp, Comply|tph, t

al ) = 0,∀sp ∈ Sp

∑SS

m(sp, Comply|tph, tah) =

1−G− Ca

Da

∑SS

m(sp, Don′t Comply|tph, tal ) =

1−G− Ca

Da

∑SC

m(sp, Comply|tph, tah) = 1

We know that such a mechanism is feasible (ie., 1−G−Ca

Da < 1) under the assumption

that G+Da + Ca > 1.

For this mechanism, we have that

q = (πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)

∑Sp

m(sp, Comply|tpl , tah) = pip

σ =

(πp)∑Sp

m(sp, Comply|tph, tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 1

ρ =

(πp)∑SC

m(sp, Comply|tph, tah) + (1− πp)∑SC

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 1

φ =

(πp)∑SS

m(sp, Comply|tph, tah) + (1− πp)∑SS

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)=

1−G− Ca

Da

τ =

(πp)∑SS

m(sp, Don′t Comply|tph, tal ) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tal )

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)=

1−G− Ca

Da

62

and

ψ =

(πp)∑SS

m(sp, Don′t Comply|tph, tah) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 0

Checking our first Truthtelling constraint, we have that:

ρCa + σG− ψDa = Ca +G

= 1− 1−G− Ca

DaDa = 1− τDa

and for our second, we have that:

1− τDa = 1− 1−G− Ca

DaDa

= 1− φDa

Finally, the expected payoff to the Principal is just:

(πa)[q(1− ρCp − ψDa)] + (1− πa)[q(−τDp)]

= (πa)(πp)(1− Cp)− (1− pia)(pip)1−G− Ca

Da(Dp)

Dividing by (πa)(πp) gives us that the above is positive if and only if:

Cp +1− πa

πa1−G− Ca

Da(Dp) < 1

From our assumption that G+Ca +Da > 1 and that Cp + 1−πa

πa Dp < 1, we then have

that:

Cp +1− πa

πa1−G− Ca

Da(Dp) < Cp +

1− πa

πa(Dp) < 1

Since all of our constraints are satisfied by our mechanism and the Principal’s expected

payoff is positive, we then have that V ∗ must also be positive, and we are done.

63

F.4 Case 4: Cp

Ca <1−πa

πaDp

Da < 1

Either Da + Ca > 1− πpG or not.

In the first instance, consider the mechanism such that:

∑Sp

m(sp, Comply|tpl , tah) =

∑Sp

m(sp, Comply|tph, tah) = 1

m(sp, Comply|tpl , tal ) = m(sp, Comply|tph, t

al ) = 0,∀sp ∈ Sp

∑SC

m(sp, Comply|tph, tah) =

∑SC

m(sp, Comply|tpl , tah) = 1

∑SC

m(sp, Don′tComply|tph, tal ) =

∑SC

m(sp, Don′tComply|tpl , tal ) = 1

∑SS

m(sp, Comply|tph, tah) =

∑SS

m(sp, Comply|tpl , tah) =

1− πpG− Ca

Da

∑SS

m(sp, Don′tComply|tph, tal ) =

∑SS

m(sp, Don′tComply|tpl , tal ) =

1− πpG− Ca

Da

We know that such a mechanism is feasible (ie., 1−πpG−Ca

Da < 1) under the assumption

that Da + Ca > 1− πpG.

For this mechanism, we have that

q = (πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)

∑Sp

m(sp, Comply|tpl , tah) = 1

σ =

(πp)∑Sp

m(sp, Comply|tph, tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= πp

64

ρ =

(πp)∑SC

m(sp, Comply|tph, tah) + (1− πp)∑SC

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 1

φ =

(πp)∑SS

m(sp, Comply|tph, tah) + (1− πp)∑SS

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)=

1− πpG− Ca

Da

τ =

(πp)∑SS

m(sp, Don′t Comply|tph, tal ) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tal )

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)=

1− πpG− Ca

Da

and

ψ =

(πp)∑SS

m(sp, Don′t Comply|tph, tah) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 0

Checking our first Truthtelling constraint, we have that:

ρCa + σG− ψDa = Ca + πpG

= 1− 1− πpG− Ca

DaDa = 1− τDa

and for our second, we have that:

1− τDa = 1− 1− πpG− Ca

DaDa

= 1− φDa

Finally, the expected payoff to the Principal is just:

(πa)[q(1− ρCp − ψDa)] + (1− πa)[q(−τDp)]

65

= (πa)(1− Cp)− (1− pia)1− πpG− Ca

Da(Dp)

Dividing by πa gives us that the above is positive if and only if:

Cp +1− πa

πa1− πpG− Ca

Da(Dp) < 1

From our assumption that Da + Ca > 1 − πpG and that Cp + 1−πa

πa Dp < 1, we then

have that:

Cp +1− πa

πa1− πpG− Ca

Da(Dp) < Cp +

1− πa

πa(Dp) < 1

Since all of our constraints are satisfied by our mechanism and the Principal’s expected

payoff is positive, we then have that V ∗ must also be positive, and we are done.

In the second instance, consider the mechanism such that:

∑Sp

m(sp, Comply|tpl , tah) =

( G1−Ca−Da − 1)πp

1− πp,∑Sp

m(sp, Comply|tph, tah) = 1

m(sp, Comply|tpl , tal ) = m(sp, Comply|tph, t

al ) = 0,∀sp ∈ Sp

∑SC

m(sp, Comply|tpl , tah) =

( G1−Ca−Da − 1)πp

1− πp,∑SC

m(sp, Comply|tph, tah) = 1

∑SC

m(sp, Don′tComply|tpl , tal ) =

∑SC

m(sp, Don′tComply|tph, tal ) = 0

∑SS

m(sp, Comply|tph, tah) =

( G1−Ca−Da − 1)πp

1− πp,∑SS

m(sp, Comply|tpl , tah) = 1

66

∑SS

m(sp, Don′tComply|tph, tal ) =

( G1−Ca−Da − 1)πp

1− πp,∑SS

m(sp, Don′tComply|tpl , tal ) = 1

We know that such a mechanism is feasible (ie.,( G1−Ca−Da−1)πp

1−πp < 1) under the assump-

tion that πpG+Da + Ca < 1.

For this mechanism, we have that

q = (πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)

∑Sp

m(sp, Comply|tpl , tah)

= (1− πp)(( G1−Ca−Da − 1)πp

1− πp) + πp =

πpG

1− Ca −Da

σ =

(πp)∑Sp

m(sp, Comply|tph, tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)=

1− Ca −Da

G

ρ =

(πp)∑SC

m(sp, Comply|tph, tah) + (1− πp)∑SC

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 1

φ =

(πp)∑SS

m(sp, Comply|tph, tah) + (1− πp)∑SS

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 1

τ =

(πp)∑SS

m(sp, Don′t Comply|tph, tal ) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tal )

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 1

and

ψ =

(πp)∑SS

m(sp, Don′t Comply|tph, tah) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 0

67

Checking our first Truthtelling constraint, we have that:

ρCa + σG− ψDa = Ca +1− Ca −Da

GG

= 1−Da = 1− τDa

and for our second, we have that:

1− τDa = 1−Da

= 1− φDa

Finally, the expected payoff to the Principal is just:

(πa)[q(1− ρCp − ψDa)] + (1− πa)[q(−τDp)]

= (πa)1− Ca −Da

G(1− Cp)− (1− πa)1− Ca −Da

G(Dp)

Dividing by (πa)1−Ca−Da

Ggives us that the above is positive if and only if:

Cp +1− πa

πa(Dp) < 1

which is true by assumption.

Since all of our constraints are satisfied by our mechanism and the Principal’s expected

payoff is positive, we then have that V ∗ must also be positive, and we are done.

F.5 Case 5: 1−πa

πaDp

Da < 1 < Cp

Ca

Either G+Da > 1 or not.

In the first instance, consider the mechanism such that:

∑Sp

m(sp, Comply|tpl , tah) =

( G1−Da − 1)πp

1− πp,∑Sp

m(sp, Comply|tph, tah) = 1

68

m(sp, Comply|tpl , tal ) = m(sp, Comply|tph, t

al ) = 0,∀sp ∈ Sp

∑SS

m(sp, Comply|tph, tah) =

( G1−Da − 1)πp

1− πp,∑SS

m(sp, Comply|tpl , tah) = 1

∑SS

m(sp, Don′tComply|tph, tal ) =

( G1−Da − 1)πp

1− πp,∑SS

m(sp, Don′tComply|tpl , tal ) = 1

∑SC ,Sa

m(sp, sa|tp, ta) = 0∀(tp, ta) ∈ T

We know that such a mechanism is feasible (ie.,( G1−Da−1)πp

1−πp < 1) under the assumption

that πpG+Da < 1.

For this mechanism, we have that

q = (πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)

∑Sp

m(sp, Comply|tpl , tah)

= (1− πp)(( G1−Da − 1)πp

1− πp) + πp =

πpG

1−Da

σ =

(πp)∑Sp

m(sp, Comply|tph, tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)=

1−Da

G

ρ =

(πp)∑SC

m(sp, Comply|tph, tah) + (1− πp)∑SC

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 0

φ =

(πp)∑SS

m(sp, Comply|tph, tah) + (1− πp)∑SS

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 1

69

τ =

(πp)∑SS

m(sp, Don′t Comply|tph, tal ) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tal )

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 1

and

ψ =

(πp)∑SS

m(sp, Don′t Comply|tph, tah) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 0

Checking our first Truthtelling constraint, we have that:

ρCa + σG− ψDa =1−Da

GG

= 1−Da = 1− τDa

and for our second, we have that:

1− τDa = 1−Da

= 1− φDa

Finally, the expected payoff to the Principal is just:

(πa)[q(1− ρCp − ψDa)] + (1− πa)[q(−τDp)]

= (πa)1−Da

G− (1− πa)1−Da

G(Dp)

Dividing by (πa)1−Da

Ggives us that the above is positive if and only if:

1− πa

πa(Dp) < 1

Which is true by our assumption that Cp + 1−πa

πa Dp < 1.

Since all of our constraints are satisfied by our mechanism and the Principal’s expected

70

payoff is positive, we then have that V ∗ must also be positive, and we are done.

In the second instance, consider the mechanism such that:

∑Sp

m(sp, Comply|tpl , tah) = 0,

∑Sp

m(sp, Comply|tph, tah) = 1

m(sp, Comply|tpl , tal ) = m(sp, Comply|tph, t

al ) = 0,∀sp ∈ Sp

∑SS

m(sp, Comply|tph, tah) = 1

∑SS

m(sp, Don′tComply|tph, tal ) = 1

∑SC

m(sp, Comply|tph, tah) =

1−G−Da

Ca

We know that such a mechanism is feasible (ie., 1−G−Da

Ca < 1) under the assumption

that G+Da + Ca > 1.

For this mechanism, we have that

q = (πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)

∑Sp

m(sp, Comply|tpl , tah) = πp

σ =

(πp)∑Sp

m(sp, Comply|tph, tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 1

ρ =

(πp)∑SC

m(sp, Comply|tph, tah) + (1− πp)∑SC

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)=

1−G−Da

Ca

71

φ =

(πp)∑SS

m(sp, Comply|tph, tah) + (1− πp)∑SS

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 1

τ =

(πp)∑SS

m(sp, Don′t Comply|tph, tal ) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tal )

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 1

and

ψ =

(πp)∑SS

m(sp, Don′t Comply|tph, tah) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 0

Checking our first Truthtelling constraint, we have that:

ρCa + σG− ψDa =1−G−Da

CaCa +G

= 1−Da = 1− τDa

and for our second, we have that:

1− τDa = 1−Da

= 1− φDa

Finally, the expected payoff to the Principal is just:

(πa)[q(1− ρCp − ψDa)] + (1− πa)[q(−τDp)]

= (πa)(πp)(1− 1−G−Da

CaCp)− (1− πa)(πp)(Dp)

Dividing by (πa)(πp) gives us that the above is positive if and only if:

1−G−Da

CaCp +

1− πa

πa(Dp) < 1

72

From our assumption that G+Da +Ca > 1 and that Cp + 1−πa

πa Dp < 1, we then have

that:

1−G−Da

CaCp +

1− πa

πa(Dp) < Cp +

1− πa

πa(Dp) < 1

Since all of our constraints are satisfied by our mechanism and the Principal’s expected

payoff is positive, we then have that V ∗ must also be positive, and we are done.

F.6 Case 6: 1−πa

πaDp

Da <Cp

Ca < 1

Either Da + Ca > 1− πpG or not.

In the first instance, consider the mechanism such that:

∑Sp

m(sp, Comply|tpl , tah) =

∑Sp

m(sp, Comply|tph, tah) = 1

m(sp, Comply|tpl , tal ) = m(sp, Comply|tph, t

al ) = 0,∀sp ∈ Sp

∑SS

m(sp, Comply|tph, tah) =

∑SS

m(sp, Comply|tpl , tah) = 1

∑SS

m(sp, Don′tComply|tph, tal ) =

∑SS

m(sp, Don′tComply|tpl , tal ) = 1

∑SC

m(sp, Comply|tph, tah) =

∑SC

m(sp, Comply|tpl , tah) =

1− πpG−Da

Ca

We know that such a mechanism is feasible (ie., 1−πpG−Da

Ca < 1) under the assumption

that Da + Ca > 1− πpG.

For this mechanism, we have that

q = (πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)

∑Sp

m(sp, Comply|tpl , tah) = 1

73

σ =

(πp)∑Sp

m(sp, Comply|tph, tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= πp

ρ =

(πp)∑SC

m(sp, Comply|tph, tah) + (1− πp)∑SC

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)=

1− πpG−Da

Ca

φ =

(πp)∑SS

m(sp, Comply|tph, tah) + (1− πp)∑SS

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 1

τ =

(πp)∑SS

m(sp, Don′t Comply|tph, tal ) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tal )

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 1

and

ψ =

(πp)∑SS

m(sp, Don′t Comply|tph, tah) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 0

Checking our first Truthtelling constraint, we have that:

ρCa + σG− ψDa =1− πpG−Da

CaCa + πpG

= 1−Da = 1− τDa

and for our second, we have that:

1− τDa = 1−Da

= 1− φDa

Finally, the expected payoff to the Principal is just:

74

(πa)[q(1− ρCp − ψDa)] + (1− πa)[q(−τDp)]

= (πa)(1− 1− πpG−Da

CaCp)− (1− pia)(Dp)

Dividing by πa gives us that the above is positive if and only if:

1− πpG−Da

CaCp +

1− πa

πa(Dp) < 1

From our assumption that Ca + Da > 1 − πpG and that Cp + 1−πa

πa Dp < 1, we then

have that:

1− πpG−Da

CaCp +

1− πa

πa(Dp) < Cp +

1− πa

πa(Dp) < 1

Since all of our constraints are satisfied by our mechanism and the Principal’s expected

payoff is positive, we then have that V ∗ must also be positive, and we are done.

In the second instance, consider the mechanism such that:

∑Sp

m(sp, Comply|tpl , tah) =

( G1−Ca−Da − 1)πp

1− πp,∑Sp

m(sp, Comply|tph, tah) = 1

m(sp, Comply|tpl , tal ) = m(sp, Comply|tph, t

al ) = 0,∀sp ∈ Sp

∑SC

m(sp, Comply|tpl , tah) =

( G1−Ca−Da − 1)πp

1− πp,∑SC

m(sp, Comply|tph, tah) = 1

∑SC

m(sp, Don′tComply|tpl , tal ) =

∑SC

m(sp, Don′tComply|tph, tal ) = 0

75

∑SS

m(sp, Comply|tph, tah) =

( G1−Ca−Da − 1)πp

1− πp,∑SS

m(sp, Comply|tpl , tah) = 1

∑SS

m(sp, Don′tComply|tph, tal ) =

( G1−Ca−Da − 1)πp

1− πp,∑SS

m(sp, Don′tComply|tpl , tal ) = 1

We know that such a mechanism is feasible (ie.,( G1−Ca−Da−1)πp

1−πp < 1) under the assump-

tion that πpG+Da + Ca < 1.

For this mechanism, we have that

q = (πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)

∑Sp

m(sp, Comply|tpl , tah)

= (1− πp)(( G1−Ca−Da − 1)πp

1− πp) + πp =

πpG

1− Ca −Da

σ =

(πp)∑Sp

m(sp, Comply|tph, tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)=

1− Ca −Da

G

ρ =

(πp)∑SC

m(sp, Comply|tph, tah) + (1− πp)∑SC

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 1

φ =

(πp)∑SS

m(sp, Comply|tph, tah) + (1− πp)∑SS

m(sp, Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 1

τ =

(πp)∑SS

m(sp, Don′t Comply|tph, tal ) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tal )

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 1

and

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ψ =

(πp)∑SS

m(sp, Don′t Comply|tph, tah) + (1− πp)∑SS

m(sp, Don′t Comply|tpl , tah)

(πp)∑Sp

m(sp, Comply|tph, tah) + (1− πp)∑Sp

m(sp, Comply|tpl , tah)= 0

Checking our first Truthtelling constraint, we have that:

ρCa + σG− ψDa = Ca +1− Ca −Da

GG

= 1−Da = 1− τDa

and for our second, we have that:

1− τDa = 1−Da

= 1− φDa

Finally, the expected payoff to the Principal is just:

(πa)[q(1− ρCp − ψDa)] + (1− πa)[q(−τDp)]

= (πa)1− Ca −Da

G(1− Cp)− (1− πa)1− Ca −Da

G(Dp)

Dividing by (πa)1−Ca−Da

Ggives us that the above is positive if and only if:

Cp +1− πa

πa(Dp) < 1

which is true by assumption.

Since all of our constraints are satisfied by our mechanism and the Principal’s expected

payoff is positive, we then have that V ∗ must also be positive, and we are done.

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