Expectation Costs in International Disputes: How Words Work Like …people.fas.harvard.edu/~jkertzer/HISC2017/schedule/papers/Walsh.pdf · accountable leaders nonetheless function

Expectation Costs in International Disputes:

How Words Work Like Tanks

for Accountable Leaders

David Hunter Walsh∗

Rutgers University

September 27, 2017

∗Prepared for presentation at the Harvard International Security Conference, Oct.

14-15, 2017. I am indebted for crucial feedback from Jack Levy, Paul Poast, Rick Lau,

Scott Wolford, Giacomo Chiozza, Avi Acharya, Ken Schultz, several anonymous review-

ers, the Political Psychology and Theories of War & Peace graduate seminars at Rutgers,

and attendees of panels at the International Studies Association annual conference 2016

and the American Political Science Association annual conference 2017; all errors and

shortcomings, however, are my own. Draft subject to change. All comments welcome:

[email protected].

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mailto:[email protected]

Expectation Costs

Abstract

Recent research has emphasized that military mobilizations both

sink costs and tie hands, unlike signaling strategies that rely solely

on punishments for bluffing, like audience costs. I argue that pub-

lic reactions to bluffs, and inconsistency more generally, have been

misunderstood in this regard, and that a reconceptualization of

audience costs as prospect-theory-driven “expectation costs” sug-

gests that threats from leaders who are more accountable to some

opinionated audience function as costly signals very much like

military mobilizations. This dynamic can help explain why all but

the most unaccountable leaders consistently act as if bluffing is

costly even though audiences cannot be relied on to punish them

for it; it can also reconcile several contrasting findings regarding

the complicated relationship between domestic accountability and

conflict, especially with regards to the democratic peace alongside

the particular war-proneness of unconsolidated democracies.

Word count: 9,752

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Expectation Costs 1. INTRODUCTION

1 Introduction: Cheap Talk, Democracy,

and Bargaining

One of the most hotly-debated and intensely-researched topics in interna-

tional relations scholarship is that of “cheap talk” - whether or not it really

is cheap, and if not, then why not, when, and for whom. James Fearon’s

(1994) article originating the concept and modeling parameter of audience

costs inspired the conversation even more intensely than Jervis (1970) did

before him: at the time of this writing, there is a grand total of 499 peer-

reviewed articles on ProQuest and EBSCOHost that feature or otherwise

engage with the concept of an audience cost in a non-trivial way, to say

nothing of books.

The overall trajectory of this research, however, may be toward a con-

sensus that mass publics do not consistently punish leaders for bluffing or

for inconsistent behavior more generally, and thus that audience costs of

appreciable magnitude generally do not exist (Kertzer and Brutger 2016;

Levendusky and Horowitz 2012; Mercer 2012; Snyder and Borghard 2011;

Trachtenberg 2012). As Schultz (2012) points out, though, audience costs

were needed to explain self-evident phenomena that will still require expla-

nation if and when we abandon the theory. Leaders threaten very hesitantly

and very rarely – less often in fact than aspiring leaders attempt coups,

which obviously entail the near-certainty of death upon failure – so what

are they afraid of? A number of plausible explanations exist, especially from

Jervis (1970), but the fact remains that leaders shy away from making in-

tense and clearly-worded threats even when they are resolved and desperate

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to prove it without resorting to war. We may consider as examples George

H. W. Bush’s “this will not stand” speech following Iraq’s invasion of Kuwait

(Stein 1992), and General Galtieri’s mostly quiet resignation about going to

war with England after years of pleading with them in closed-door diplomacy

(Lebow 1985).

In these extreme cases, leaders are clearly more averse to issuing a public

threat of maximum intensity than they are to starting the very war that

they would threaten to start. The argument and model introduced in this

article cannot rationalize that extreme degree of rhetorical restraint, but it

can help to resolve this critical tension in the scholarly literature by pro-

viding an alternative, or at least a parallel companion, to audience costs –

one that is already empirically supported, indirectly, by the decades-long re-

search program of prospect theory (Camerer 2000; Kahneman and Tversky

1979). These “expectation costs” also go beyond the simple focus on the net

costs of bluffing: by its nature, the theoretical construct of expectation costs

formally includes “belligerence effects” (Kertzer and Brutger 2016) and gen-

erally carries with it a wider range of observable implications for international

bargaining than do previous theories of inconsistency.

There are many reasons why we should care greatly about the true na-

ture and identity of cheap, or supposedly cheap, political talk. If talk can be

made to be expensive, then it can be a useful, perhaps invaluable tool for re-

solving international disputes and generally carrying on international affairs

in a more efficient, stable, and safe way, as both Fearon’s and Slantchev’s

work makes clear. Moreover, if talk is naturally cheaper for some than it is

for others – or if its costs are more obvious for some, such that others can

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more readily verify that they exist – then unit-level properties gain more

prominence in international relations and, conversely, the international sys-

tem becomes weaker as an explanatory force.

This paper seeks broadly to resolve these three questions, principally via

a retheorizing of the cognition behind the reactions of audiences involved

in international bargaining. The starting point is that prospect theory, as

classically formalized (Kahneman and Tversky 1979), should apply to how

observers value, and in turn react to, the sorts of positional shifts that a

leader engages in when they do or do not issue threats and do or do not back

them up. I model these individual-level reactions, given the individual’s own

assumed preference with regards to hawkish/dovish policies, and then aggre-

gate upward, in a highly generalizable method for quantifying the expected

mass-level reaction to any given policy shift, under these psychology-based

assumptions. Finally, I factor these audience reactions into an ultimatum

game, as appropriate payoffs for an accountable leader who is contemplating

a threat and possible subsequent attack to acquire a desired good. Naturally,

the model is stylized and rests on assumptions that significantly limit its

scope, but it can be compared directly to previous work on these topics (e.g.

Levy et al. 2015; Slantchev 2011), it is most appropriate for the theoretical

questions raised above, and it can form the basis for several relatively obvious

extensions and empirical investigations going forward.

I find that words in this sense always sink costs as well as tie hands –

as operationalized in Slantchev (2005, 2012) – for domestically accountable

leaders, and so-called cheap talk can thus be no less momentous than military

mobilizations in crisis bargaining, escalation, and eventual lock-in to war.

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Audiences who have any consistent opinion of significant prominence about

the country’s dealings with the rival nation act and react in such a way

that they inevitably generate expectation costs for a leader who is in any

way accountable to them and who alters, or fails to maintain, the audience’s

expectations regarding future policies.

These overall patterns of prospect-theory-driven behavior also alter lead-

ers’ payoffs for both successful coercion and escalating to war, which distin-

guishes expectation costs in important ways and leads to novel and particular

predictions for international dynamics. At the same time, it is important not

to overstate – or perhaps understate – the size and influence of this “genuinely

opinionated” segment of the public: it may be argued either way whether they

are generally significant enough to impact crisis bargaining, though the more

fruitful question is undoubtedly when and where they are significant enough.

The most interesting implications of this argument are likely those regard-

ing the role of democracy in crisis bargaining, which has been denigrated by

research such as Slantchev’s, but, more importantly, problematized in nu-

merous ways (Downes and Sechser 2012; Snyder and Borghard 2011; Weeks

2008, 2012). One recent and prominent work has even problematized the

basic link between democratic institutions and meaningful leader account-

ability (Achen and Bartels 2016). While we must define both accountability

and democracy carefully so as to respect these recent findings, the conclusion

to be drawn from the theory presented here is that democratic and otherwise

accountable leaders nonetheless function both as a sort of public good in the

international system, generating peace and stability, and also as a dominant

force in an evolutionary sense, coercing more effectively, more often, and yet

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Expectation Costs 2. KEY CONCEPTS

with less risk of mutually costly war.

2 Clarifying Key Concepts

What I call expectation costs was aptly described (in one specific form) in

a small paragraph of Robert Jervis’ seminal work, The Logic of Images in

International Relations (1970, p. 75-76). After describing what amounts to

using “sticky elite cues” for bargaining advantages (cf. Guisinger and Saun-

ders 2013), Jervis notes that:

[a] related mechanism can be brought into play by decision-

maker’s [sic] signals that lead the public to believe a certain

settlement will probably be attained. If these expectations are

not met the public may turn the leaders out of office. If both the

decision-makers and the other side know this the former are more

apt to live up to their signals and the latter more apt to believe

them. By promising the union members a certain settlement, la-

bor negotiators strengthen their bargaining position by invoking

this process.

Jervis’ story lacked grounding in substantiated psychological theory, be-

cause it predated that theory (Kahneman and Tversky 1979; Levy 2000).

We now know that the driver of such a dynamic is loss aversion, i.e. “losses

loom larger than gains” (Quattrone and Tversky 1988, p. 721). This sim-

ple notion proves surprisingly complex to define and isolate in formal theory

(Abdellaoui, Bleichrodt, and Paraschiv 2007; Kobberling and Wakker 2005)

but is nevertheless very well-established as an empirical regularity of human

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behavior (Camerer 2000; Kahneman and Tversky 2000; Markle et al. 2015).

One’s constituents will punish them particularly hard for failing to deliver

a promised and particularly desirable outcome because the promise frames

that outcome as a loss rather than as the failure to make a gain that it would

otherwise, presumably, be seen as.

This is to say that the promise shifts the reference point, i.e. the hypo-

thetical outcome to which the real outcome is automatically compared in

the individual’s mind. The simplest assumption is that the reference point

is the real status quo, but research shows that this is generally not the case

even in the absence of framing effects per se (Markle et al. 2015; Niven 2000;

Song 2016) and, in fact, can’t be the case if we are to make sense of several

well-established patterns of behavior (Kimball and Patterson 1997; Koszegi

and Rabin 2006; Schmidt, Starmer, and Sugden 2008; Waterman, Silva, and

Jenkins-Smith 2014). The common finding in all this research is that it is

primarily expectations that determine the reference point: as Koszegi and

Rabin (2006, p. 1135) point out, “while an unexpected monetary windfall in

the lab may be assessed as a gain, a salary of $50,000 to an employee who

expected $60,000 will not be assessed as a large gain relative to status-quo

wealth, but rather as a loss relative to expectations of wealth.”

Work such as Waterman, Silva, and Jenkins-Smith (2014) and especially

Elinder, Jordahl, and Poutvaara (2015) substantiate the particular notion

that voters do, for whatever reason, generally allow their expectations and

therefore their reference points to be set by politicans’ “cheap talk”: speeches,

campaign promises, and the like. Thus, Jervis’ story of strategic and observ-

able manipulation of one’s constituents’ reference points has grown far more

8


plausible over the years, even as security studies have focused almost exclu-

sively on audience-cost theories that rely on either the classical rationality

of audiences (Smith 1998) or on no real reason at all as to why audiences

should react negatively to unsuccessful bluffs or other inconsistent behaviors

(Kertzer and Brutger 2016).

Implicit in these discussions of bargaining advantages are the well-known

and often-confused concepts of hand-tying, cost-sinking, and money-burning,

which I will first attempt to clarify somewhat. As shown below, these concepts

are valid and carry the bargaining implications that they are known for, but

they can often mislead in complex strategic situations – they invite us to jump

to conclusions and to cherry-pick which dynamic we assume will predominate

at any point in time.

Fearon (1997, p. 70) authoritatively defined hand-tying as “taking an

action that increases the costs of backing down if the would-be challenger ac-

tually challenges but otherwise entails no costs if no challenge materializes.”

This popular definition, as Slantchev (2011) points out, is unnecessarily lim-

ited: any action that makes you genuinely and observably more likely in the

future to choose one course of action as opposed to its alternative(s) serves a

hand-tying function, and this can result from any of a wide range of changes

in either or both the back-down and stand-firm payoffs. An alternative defini-

tion of a hand-tying action, then, is one that alters the relative attractiveness

of backing down from a threat and going through with it, in favor of the lat-

ter. While the rival never knows for sure if the threatener is “locked in” to

war such that their stand-firm payoff really is higher than the back-down

one, the rival must rationally place a larger probability on this being the

9

Expectation Costs 3. PSYCHOSPATIAL MODEL

case once they observe a hand-tying action, all else equal.

With regards to cost-sinking, Fearon (1997, p. 70) asserts that “sunk-cost

signals are actions that are costly for the state to take in the first place but

do not affect the relative value of fighting versus acquiescing in a challenge.”

This, though, would make it a synonym of money-burning (Avery and Zem-

sky 1994), while “sunk costs” usually refers to costs required to undertake a

given course of action that need not necessarily be taken after part or even

all of its costs have been paid. The latter is also the definition employed in

prospect theory literature (Garland and Newport 1991), and relates most

clearly to sinking the costs of literally going to war – spending the money to

mobilize troops – which similarly incurs immediate and visible costs which

indeed cannot be recouped, but which do clearly and crucially alter the cost

of escalating fully to war from there (Slantchev 2011).

To sink the costs of war in this sense is to pay some of the costs of

going to war in advance and without the possibility of a full refund – to

begin buying the war on layaway – which is a combination of money-burning

and hand-tying. In the model developed here, the central contrast will be

between burning money and tying hands, which, so defined, can be clearly

distinguished from each other while jointly covering a wider and more sub-

stantively interesting range of bargaining possibilities.

3 The Psychospatial Model

The larger expectation costs model consists of two separable parts, the second

of which is the ultimatum game that models international interactions. The

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first is a novel combination of the well-established spatial voting model and

the well-established prospect-theory value function; the marriage of the two

can be considered in its own right, as a “psychospatial” model.

3.1 Adapting the Basic Spatial Model

The spatial voting model, despite its nominal focus on formal elections, can

model a wide range of collective decisionmaking contexts, including leader

selection in autocracies where “votes” take the form of throwing one’s influ-

ence behind your preferred aspirant; here, “voters” are any and all persons

with (domestic) political influence, be they actual voters, rebellious masses,

the officers of a military junta, or the lesser bosses of a political machine

(Weeks 2012). Possible policies and issue positions are modeled as points on

a number line running between two political extremes, i.e. liberal vs. conser-

vative; the enactment of the leader’s chosen policy delivers a payoff to each

voter corresponding directly to how far away that policy is from the voter’s

position and most-preferred policy. In the spatial model, this payoff is:

−φ|pi − x| (1)

where pi and x are the leader’s and voter’s positions, respectively, and φ

measures the issue’s overall importance to the voter in question.

Importantly, voters without real and at least somewhat consistent issue

positions receive no payoffs from anything here and are essentially absent

from the model. Those whose preferences are volatile or manipulable, and es-

pecially those whose preferences merely reflect the cues that their party elites

have most recently sent them (the majority of voters according to Berinsky

11


2009), cannot generate expectation costs. Again, this is a substantive prop-

erty of the scope and purpose of the model: accountability and stable issue

preferences amongst the public interact to produce these effects, and one

can interpret a more accountable leader as being literally more accountable,

as being accountable to a more opinionated and engaged public, or, espe-

cially, both. Unsurprisingly, a focus on inconsistency costs as a punishment

consciously dished out on inconsistent leaders has led to the opposite con-

clusion, that such costs could only ever realistically come from citizens who

consistently have no particular policy preference (Chaudoin 2014; Mcgraw,

Lodge, and Jones 2002) – which, given elite cue theory, would also be a small

segment of any influential public.

While formal prospect theory has expanded in some very sophisticated

ways (Koszegi and Rabin 2006; Schmidt, Starmer, and Sugden 2008), the

classic and relatively simple prospect-theory (PT) value function from Kah-

neman and Tversky (1979) is sufficient for our purposes here, since “voters”

neither make any choices here themselves nor, presumably, is there any con-

sumption utility (direct and non-reference-dependent effects of the leader’s

international security policies on the voter and their situation) to account for.

The PT value function returns the subjective, experienced utility resulting

from an outcome with a given objective value, and compared to a reference

point that is itself an outcome – the expected outcome – with a given objec-

tive value (note that the expected outcome, being the reference point itself,

would have a subjective value of zero). If we define p1 as the leader’s previ-

ous position, which set the voter’s expected policy outcome, and p2, which

12


determines the realized outcome, then the value function becomes:

φv(x, p2|p1) =

(|p1 − x| − |p2 − x|)α if |p1 − x| ≥ |p2 − x|

−λ(|p2 − x| − |p1 − x|)α if |p1 − x| < |p2 − x|(2)

with λ > 1 and 0 < α < 1 in order model loss aversion and diminishing sen-

sitivity, respectively. It is rather standard to assume λ = 2 following Tversky

and Kahneman (1992), though various studies have found loss-aversion coef-

ficients ranging from 1.43 to 4.8 (Abdellaoui, Bleichrodt, and Paraschiv 2007,

p. 1662).

Issue Position

Cha

nge

in V

oter

's A

ppro

val o

f Lea

der

−3 −2 −1 0 1 2 3

− 6φ

− 3φ

0

3φ

6φp1

p2 = −1p2 = 0p2 = 1

Figure 1: PT value functions for three possible shifts away from p1 = −2,

assuming standard parameters

Take for example a leader who is at p1 = −2 and is contemplating a

rightward shift of either 1, 2, or 3 units. Fig. 1 plots the three different

value functions for each of these three shifts, assuming the standard values

13


of λ = 2 and α = 0.88. The p2 of each shift is unlabeled but is evidenced by

the point at which the value function abruptly stops rising. Voters at x = 0

will experience a gain of φ from a shift to either -1 or 1 here, and a gain of

about 1.84φ from a shift right onto their own position at 0.

The S-shape of the PT value function is suggested (though not exactly

present), as is the range of losses exceeding the range of gains in terms of

magnitude for each of the curves. Loss aversion should be interpreted here as

follows: if the distance between the voter’s and leader’s issue positions ends up

larger than expected by some amount z, then the voter’s reaction is negative

and larger in magnitude than the positive reaction that would have resulted

from the difference in their issue positions instead becoming smaller than

expected by z units. Disappointing a voter results in a disproportionately

strong negative reaction, compared to pleasantly surprising them to the exact

same degree in objective terms. Thus there must be substantially more people

who will approve of a given policy shift rather than disapprove of it before

it does more harm than good for the leader, in the simplest sense of the

aggregate public reaction being net positive.

The goal here is to model aggregate public reactions to policy shifts, for-

mulated in the simplest acceptable way yet with a minimum of assumptions

about factors like opinion intensities, electoral structure, and opportunities

for extra-institutional activism. If we have a distribution of voter positions

o(x), called the opinion distribution, and a value function with a given p1

and p2, then the product of the two functions gives us a function that, when

integrated indefinitely over x, gives us the sum total of all voters’ comparable

reactions to a shift from p1 to p2. If o(x) represented only the count of voters

14


at each issue position (or density if it’s a theoretical PDF), then this method

would assume that each voter placed equal weight on the issue in question,

but any method that did account for varying weights would be functionally

identical to transforming o(x) to represent both the locations and intensities

of each voter’s preferences, so we need not model those weights separately.

φ, in fact, then drops out of the model, if we assume that it has been in-

corporated into o(x); I retain it below however as a normalizing parameter

reflecting the weight the public as a whole places on the issue.

Thus we can express the domestic payoff for a leader at p1 shifting to

some p2 as the public’s aggregate reaction to that shift:∫o(x)v(x, p1, p2) dx (3)

Issue Position

Sum

med

Pub

lic R

eact

ion

from

Shi

ft to

x

−3 −2 −1 0 1 2 3

− 2φ

− φ

0

φ

2φp1 m.v.

o(x)

⌠⌡

o(x)v(x)

Figure 2: Aggregate public reactions to possible shifts from -2 to x, given

standard normal opinion distribution

15


If, for example, we assume a standard normal opinion distribution and

a p1 of -2, then fig. 2 plots the leader’s domestic-politics utility for p2 =

x. A shift toward the median voter in and of itself is certainly worthwhile

for this leader, which is unsurprising given her extreme starting position;

however, the utility-maximizing p2 is short of the median voter’s position

(x∗ ≈ −0.31, about 84% of the way from p1 to the median voter). This,

namely, is something I do not explore here.

3.2 Expectation Costs

The primary parameter of interest, expectation costs, can now be defined

as such: a loss of overall popularity due to any shift from one policy posi-

tion to another and then back to the original position after the public has

psychologically renormalized to the new position.

Define the net effect of two such shifts, unweighted by leader accountabil-

ity η, as:

ε =

∫o(x)v(x, p1, p2) dx+

∫o(x)v(x, p2, p1) dx (4)

(assuming that φ is normalized at 1 and omitted). Fig. 3 plots these two value

functions separately, one for a shift from -2 to 0 and the other from 0 to -2;

fig. 4 plots the integral of these summed value functions, ε, for all possible

p2, given that o(x) (after all necessary transformations) is standard normal,

p1 = −2, and the leader shifts to x and then back to -2. The y-axis there is

defined by η, the degree of leader accountability, to indicate how this integral

ultimately becomes a given leader’s payoff for both shifts put together.

Note the crucial assumption here that the second shift, back to p1, occurs

after the entire public has renormalized to p2 being the expected policy;

16


Issue Position

Rea

ctio

n of

Citi

zen

at x

−3 −1 0 1 2 3

− 6φ− 3φ

0

3φ6φ

p1 p2

p1 to p2

p2 to p1

Figure 3: Value functions v(·)

for countervailing shifts

Issue PositionPolit

ical

Cos

t of S

hift

ing

to x

and

Bac

k

ε

−3 −1 0 1 2 3

− 2η

− η0

η2η

p1

o(x)

Figure 4: Approval loss resulting

from a shift to and from x

enough time must pass between the two shifts for even those who saw the

first shift as a loss to renormalize to the lower reference point (indeed, far less

time need pass if the first shift was favored by all, thanks to the endowment

effect: Jervis 1992; Thaler 1980).

Now, leave o(x) undefined beyond the fact that it is non-negative and

globally integrable; whatever conclusions we draw, then, hold no matter what

public opinion is on the issue in question. Factor out the integral in eq. 4:

ε =

∫o(x)[v(x, p1, p2) + v(x, p2, p1)] dx (5)

Given the value function from eq. 2, we can further specify the sum of the

two value functions, in the brackets. The algebra is bulky and so I relegate

it to the appendix, but the following holds true:

v(x,p1, p2) + v(x, p2, p1) = (1− λ)(||p1 − x| − |p2 − x||)α (6)

17


Substituting into eq. 5 gives us:

ε =

∫o(x)(1− λ)(||p1 − x| − |p2 − x||)α dx (7)

= (1− λ)

∫o(x)(||p1 − x| − |p2 − x||)α dx (8)

Given the bounds on λ and α, eq. 6 is strictly negative (ignoring the very

particular case in which the effect is zero because all citizens’ positions are

precisely between p1 and p2). This proves the existence of expectation costs

as such: no one with any opinion on the issue can observe their leader shift

from one position to another and then back again later without experiencing

a net loss, which translates into an inevitable net loss for the leader in terms

of their popularity.

3.3 Decomposing Expectation Costs

This psychospatial model, so defined, may serve as a widely-applicable tool

for modeling mass reactions in a way that accords with prospect theory. Here,

I apply the model to the specific matter of a rational leader deciding whether

or not to issue an ultimatum to a rival state and to make good on the threat

or back down from it if it fails to coerce. The issuance of a threat in the

psychospatial model is a rightward shift away from some status-quo starting

position, and backing down if the threat fails is modeled as a leftward shift

back to that position after the public’s reference point has changed (i.e., after

renormalization has occurred); thus, expectation costs apply just as specified

above. Both successful coercion and escalation to war following failed coercion

are modeled as the leader remaining at their new rightward position, where

expectation costs, so defined, are avoided.

18


As Kertzer and Brutger (2016) aptly demonstrated, it is crucial to ap-

preciate not just the effect of backing down from a threat but also the prior

effect of issuing a threat in the first place. They dub the latter belligerence

costs, generally assuming that the prevailing public reaction to belligerence

will be negative – and indeed it should be, for all but exceedingly hawkish

publics, given prospect theory. However, these belligerence effects must also

be understood and defined in relation to the effect of backing down. Rather

than define this relationship in all of its algebraic complexity, with o(x) left

entirely to vary, I define and discuss the leader’s payoffs in stylized scenarios

where the public is either entirely hawkish, entirely dovish, or polarized with

half at one extreme and half at the other.

To begin with, define these three ranges of x in relation to p1 and p2 as

follows:

1. Dovish positions: x ≤ min{p1, p2}

2. Hawkish positions: x ≥ max{p1, p2}

3. Moderate positions: min{p1, p2} ≤ x ≤ max{p1, p2}

The leader’s (unweighted) payoff for a given shift can be expressed far more

simply if there are no moderate voters. o(x) then simplifies to a quantity

of hawks and a quantity of doves, and v(·) simplifies to the two separate

but uniform reactions of each of those groups. The appendix again contains

the simplification and formulas; fig. 5 graphs the leader’s predicted approval

level, from left to right, as she undertakes her shifts to and from a more

hawkish position, depending on which of the simplified opinion distributions

we assume.

19


Positions Taken Over Time

Lead

er's

Dom

estic

−P

oliti

cs P

ayof

f

●

●

●

●

− 2η

− η

0

η

p1 p2 p1

All hawks Evenly split All doves

Figure 5: Three Paths to the Same Expectation Costs, Depending on Public

Preferences

It would seem that the all-hawk and all-dove paths are boundaries within

which all other shifts would range between, namely such that both h ≤ −12d

and d ≤ −12h always hold even with moderate voters, but this is not yet

proven.

If the public’s preferences are aligned so as to maximize the political ben-

efit of issuing the threat, then they are also aligned so that the political cost

of subsequently backing down will be about twice the magnitude of these

belligerence benefits. If the opinion distribution instead maximizes belliger-

ence costs, then the rapprochement benefits will be only half the magnitude

of these costs. Thus, the positive effect is the same whether it comes from the

hawkish or dovish shift, and the same goes for the negative effect; expectation

costs are suffered just the same either way.

20


Inconsistency costs as usually defined result only from the second of the

two shifts; indeed, Snyder & Borghard’s (2011) otherwise formidable findings

against audience cost theory rely primarily on such a definition. An audience-

cost story has the leader end up at a lower approval level following these two

shifts regardless of public issue positions as it is here, but it usually has her

at a higher level of approval following the first shift than is the case here –

a split public would have no net reaction to the first shift at all, while the

negative reaction of an all-dove public would only be −η rather than −2η.

Expectation costs must be tested for differently, and call for a revisiting of

both qualitative and quantitative evidence against audience costs.

Kertzer and Brutger (2016), who discuss this directly, still imply that

belligerence costs are a direct and opinionated reaction from the public a

la the classic spatial voting model, and they seem to assume accordingly

that there would be equal and opposite rapprochement benefits if not for

the inconsistency costs that result from the rapprochement specifically. The

implication is that the issuance of a threat is not seen as a significant incon-

sistency on the leader’s part, even though it is at any rate a major political

departure from the status quo which the leader was previously allowing to

persist, belligerently or not. This is its own separate question pertaining to

the psychology of consciously identifying and deciding what is and is not

political inconsistency; here, however, the postulated psychological process

is simpler, and non-consciously automatic.

21

Expectation Costs 4. COERCIVE BARGAINING

Status Quo Back Down

Stand Firm

Refuse

Concede

Challenge

S1

S2

S1

(0, 1)

(1 + ηh, 0)

(ηε, 1)

(w1 + ηh, w2)

Figure 6: The Ultimatum Game with Domestic Payoffs from the Psychospa-

tial Model

4 Coercive Bargaining with Expectation

Costs

We can now properly factor the effects of altering public expectations into

a model of coercive bargaining. The model employed here is the standard

and relatively simple ultimatum game, with three binary decision nodes.

Fig. 6 displays the game tree. The formal proof of equilibrium is in the

appendix; here I very briefly recap the construction of the game, provide

intuitive interpretations of the results, and specify the key equations from

which they’re drawn.

This is a two-player game where State 1 (S1) is the challenger, or revision-

ist state, who desires some good possessed by State 2 (S2). The value of the

contested good is standardized at 1. All parameters are public information

except for the war payoffs w1 and w2 – the states’ levels of resolve – which are

22


private information for States 1 and 2, respectively, though they are drawn

at the start of the game from known distributions f1(w1) and f2(w2). Both

of these distributions have a maximum of 1, creating the potential for prior

exogenous lock-in to war for S2, and for S1 if ηh > −1. A crucial construct,

especially for the formal proof, is Fi(x) which is the integral of fi(x) and

therefore quantifies the known probability that wi < x.

It will also prove important to understand that wi is conceived of as the

value of the contested good times the probability pi of winning the war and

thus the contested good, all minus some cost ci. w1 is private knowledge solely

because ci is; pi is common knowledge, as is the distribution from which ci is

drawn, ranging from 0 to some maximum Ci (the maximum Ci results in the

maximum cost, which is subtracted from w1 and therefore minimizes w1).

The other terms are derived directly from the psychospatial model, gen-

erally left to vary along with the distribution of public opinion. S1, the chal-

lenger, is defined as the one state with a domestic constituency to which it is

accountable to some degree, and it is therefore the only one for whom these

psychospatial terms apply. An increase in accountability here constitutes as

an increase in η and therefore in the magnitude of the domestic incentive

terms h, d, and, by extension ε.

To be clear, o(x) is not assumed here to be made only of strict hawks

and/or doves as it was in the previous section – the proven equilibria pre-

sented here hold with o(x) left entirely variable. h and d here represent what-

ever the aggregate public reaction is to a hawkish and dovish shift, respec-

tively, given any o(x) of any kind, and not just the extreme cases in the

previous section.

23


ε is used here in lieu of h+ d in the spirit of audience-cost models which

denote that payoff as −a; it is significant that (as proven above and in the

appendix) it is always negative as are audience costs, but nothing else is

assumed here beyond what has been proven with o(x) left to vary.

First recall that the signs of h and d vary based on the relative hawkish-

ness and dovishness of the public to which the leader of State 1 is accountable;

they do not vary without constraint, however, as they cannot both be pos-

itive, and if one is positive, then the other must be negative and have at

least twice the magnitude of the positive parameter. It is also sensible to

assume that in the majority of real cases, both will be negative; the exact

requirement to make either h or d positive varies depending on the intensity

of loss aversion as well as the size of the shift (so it also matters somewhat

how out-of-character the threat is/would be), but we can generally assume

that the genuinely opinionated public, i.e. those who resist elite cues, will be

neither overwhelmingly hawkish nor overwhelmingly dovish. It is also impor-

tant to identify the effects of accountability as interacting with, but separate

from, the assumed distribution of issue opinions, lest we read too much into

results driven tautologically by what the public is assumed to prefer and how

accountable to them the leader is assumed to be.

4.1 Bargaining Concepts and Prospect-Theoretic

Audience Reactions

The earlier proof of the fact that ε < 0 clearly means that in the model so

constructed, accountability does in fact monotonically and linearly increase

the cost of bluffing. If not, namely, for the dynamics of prospect theory, opin-

24


ionated audiences would just return to their previous level of leader approval

following a decision to back down, as they often do in the sterile and cerebral

conditions of a one-shot survey experiment (Chaudoin 2014; Levendusky and

Horowitz 2012; but cf. Trager and Vavreck 2011). Given prospect theory and

the assumption that the public at least partially renormalizes to the leader’s

post-threat position, however, public threats do tie hands for democratic or

otherwise accountable leaders in the classic Fearonian sense.

Do they always tie hands in the more general sense, however, where back-

ing down becomes less attractive compared to war? Consider the general

condition for S1 being locked in to war, remembering that ε = h+ d:

w1 + ηh > ηε (9)

w1 > ηd (10)

Thus accountability really ties hands whenever the public is not so dovish

that there are rapprochement benefits, which, again, will be the clear major-

ity of cases. Expectation costs functioning as audience costs in this narrow

sense is then generally confirmed. Intuitively, though, a more accountable

leader with a very dovish public is visibly tempted to back down and gar-

ner the positive reaction for doing so. At the same time, a highly dovish

public makes the threat more costly to make in the first place, thereby feed-

ing its credibility, which illustrates the need to consider all these questions

simultaneously.

Money-burning, in the sense of what Fearon defines as cost-sinking, is even

more clearly present than is hand-tying: since expectation costs always exist,

they always work to screen out would-be bluffers, and to a greater degree as

accountability increases. In general, also, there are belligerence costs, which

25


further screens out bluffers by making successful coercion less profitable. At

the same time, belligerence benefits, when present, do the opposite, making

it unclear at first blush what the net effect of accountability is on bluff rates

in that case.

Lastly, there is the question of cost-sinking, properly defined in a way

befitting military mobilizations: paying a part of the cost of war without the

possibility of fully recouping what is paid if one backs down. With belligerence

costs generally present, threats generally do sink those exact costs: backing

down can only ever recoup part of what was lost by the issuance of the

threat, when it doesn’t just incur further costs in itself, while making good

on the threat will merely be seen as “the other shoe dropping” and incur no

additional cost following the public’s renormalization to the leader’s post-

threat policy position. The idea is that going to war was going to incur a

public outcry from the start, and the threat incurs this outcry early, with

prospect theory again being responsible for the known impossibility of fully

reversing this. Again, though, one extreme of public opinion alignment will

invert this calculus: an extremely hawkish public means belligerence benefits

instead of costs to sink, and it means an additional incentive to go to war as

well as win concessions.

4.2 Accountability and Threat Credibility in

Equilibrium

First a note on the leaders, which I treat here as classical utility-maximizers

rather than prospect-theoretic actors. This is meant only as a simplifying

assumption so that the analytical focus here stays on the significance of

26


prospect-theoretic audiences. Modeling leaders as PT actors has proven very

interesting and fruitful (Berejikian 2002; Schaub 2004), especially in Butler

(2007) which found very different bargaining dynamics associated with dif-

ferent possible placements of the two bargaining leaders’ reference points.

The separate findings of that strain of research and this one should prove

clearly compatible – the reference points of audience members will have the

implications identified in this study, while the reference points of leaders will

have their separate and independent impact. In any given case, the effect of

the leader’s and audience’s reference points may move in the same direction

or different directions in terms of the chance of war, crisis, etc., but it seems

clear that their logics will not interact or depend on one another in any par-

ticular way. Namely, this is because public reactions ultimately factor into

leaders’ outcome payoffs alongside the value of the contested good, the costs

of war, etc., so the gain/loss weighting that the leader might apply to any of

those other incentives would also apply in any given case to their valuation

of public support, and the substantive interpretation would be the same as

in the aforementioned models.

There is technically just one perfect Bayesian equilibrium in this game;

it is standard, however, to present it as two separate equilibria, since both

players follow two distinct and very different sets of moves depending on

whether or not the known chance of an exogenously resolved S2 (i.e., 1−F2(0))

is low enough for it to be rational for S1 to ever bluff. I follow this convention

of discussing these as separate equilibria.

Equilibria are best interpreted by defining cutpoints along the continuum

of possible values of wi. ηd can be thought of as the cutpoint for S1’s being

27


resolved to war: S1 will stand firm following a threat rather than back down

if and only if w1 ≥ ηd. Similarly, S1 will threaten in the bluffing equilibrium

only if w1 exceeds some value b, which is less than ηd but is otherwise quite

complicated and difficult to interpret algebraically.

Fig. 7 provides some illustrative cases of the b and ηd cutpoints along

the range of w1, and how they shift as accountability increases. These are

computed states of the game, accurately scaled. Specifically here: p = 0.5;

C1 = 2.5; C2 = 1.5; ε = −1; η = 0.01 in the low-accountability state; and

η = 0.25 in the high-accountability state. I also assumed here that fi(·) is

uniform, such that Fi(·) is increasing at a constant rate and the probabilities

of drawing a wi within given ranges map directly onto this figure.

We can first focus on the ratio of the distance from ηd to p and the distance

from b to p, as this constitutes the percentage of threats that are genuine,

and not bluffs. Accordingly, this ratio increasing means more credible threats,

and more concessions from S2. The relationship between η and b is actually

only guaranteed to be monotonic, however, in the special case where there

are rapprochement benefits (d > 0), in which case the relationship is positive

and accountability decreases both bluffs and genuine threats as shown here.

Making the public more hawkish consistently works to weaken this relation-

ship, but it may or may not reverse it, even at the all-hawk extreme. While

the evenly-split-public example happens to exhibit a barely-perceptible de-

crease in b as accountability increases here, this is a function, for one, of

the assumed Ci. Thus, this approach is also insufficient to draw conclusions

about the relationship between accountability and threat credibility, and the

relationship must be uncovered within the formal equilibrium solution.

28


p− C1 p

p− C1 p

p− C1 p

p− C1 p

p− C1 p

p− C1 p

All-Dove Public

Evenly Split Public

All-Hawk Public

Low Accountability

Low Accountability

Low Accountability

High Accountability

High Accountability

High Accountability

b

b

b

b∗

b∗

b∗

ηd

ηd

ηd

ηd∗

ηd∗

ηd∗

Figure 7: Effects of accountability on resolve and bluff rates

29


In the bluffing equilibrium, S2 concedes at such a rate as to make an un-

resolved S1 (who would back down rather than go to war) indifferent between

bluffing and not bluffing. If we define s as the refusal rate – making 1 − s

the crucial chance of concession – then an unresolved S1 is indifferent about

threatening when:

sε+ (1− s)(1 + ηh) = 0 (11)

s =1 + ηh

1− ηd(12)

Accountability (η)

Prob

. of C

once

ssio

n (1

−s)

0 0.5ε ε 1.5ε 2ε

0.0

0.2

0.4

0.6

0.8

1.0

All doves Evenly split All hawks

Figure 8: Positive relationships between accountability and winning conces-

sions

Fig. 8 plots the chance of concession as a function of accountability for

the three illustrative types of public opinion distributions of all-hawk, all-

dove, and evenly split. Algebraically as well, s is only ever decreasing as

30


accountability increases; thus, the “democratic” advantage (including the ad-

vantage of highly-accountable and non-personalistic autocracies a la Weeks

2008, 2012) is predicted by this theoretical framework. The magnitude of this

effect, however, strongly depends on the opinion distribution. Hawkishness in

the public consistently dampens the credibility-generating effect of account-

ability; this indirectly proves that while the bluff-encouraging dynamic of

belligerence benefits is consistently outweighed by the accompanying hand-

tying and money-burning dynamic of expectation costs when it comes to the

target’s appraisal of the threat, the former dynamic is substantial and can

rob accountability of almost all of its credibility-generating power. By com-

parison, accountability actually has a hand-loosening effect when the public

is extremely dovish, but this is strongly and unambiguously outweighed by

the cost-sinking as well as the money-burning that accompanies it.

It is also notable that the effect size of accountability depends on both its

current level and the opinion distribution. With a very dovish public, a given

increase in accountability will add the most credibility when accountability

is already high, while in the majority of cases, the increase will make the

most difference for a challenger who was entirely unaccountable before. This

begins to weigh in on the question of whether expectation costs can help

explain the relative belligerence of countries in the middle of the democracy

scale, though a direct answer is provided below.

Crucially, there is also a range of game states that can be interpreted

as representing the democratic peace, albeit in a blunt and monadic way.

Assuming that public opinion is, if not generally dovish, then generally not

overwhelmingly hawkish, then h is negative, and sufficiently accountable lead-

31


ers (η > −1h

) will never issue a threat, much less initiate war (see Poznansky

and Scroggs 2016, on the dearth of inter-democratic threats). This is evident

in fig. 8 where the curve with an all-dove public reaches and would exceed

1; even if concession were certain, an unresolved leader with such high ac-

countability would strictly prefer not to bluff. When there are belligerence

costs rather than benefits and accountability is high enough, then the payoff

for winning a concession is negative for S1, coercion simply doesn’t pay, and

the status quo is unconditionally preferable. On the other hand, belligerence

benefits coupled with high accountability can invert this and make a threat

both guaranteed and 100% credible; thus, the implication here is that, in the

rare (perhaps even entirely hypothetical) event that a democratic public is

clamoring for war with another democracy, war is possible, but the chance

that the rival sees this clamoring and makes concessions to avert war is also

maximized, so war is still generally unlikely, if not particularly so.

s is also related to the scope condition of the bluffing and non-bluffing

equilibria, in that S2 may not be able to rationally concede at a high enough

rate to make bluffing ever rational for S1, given the rate of resolved S2s who

simply prefer war to concession. Specifically, bluffing can only occur while:

1− F2(0) < s =1 + ηh

1− ηd(13)

Given that s is strictly decreasing with accountability, increased accountabil-

ity also works against this condition holding – it pushes the game toward the

no-bluffing equilibrium, in which all threats are 100% credible, S2 only ever

stands firm if it prefers war to concession, and even some S1s who would

prefer war to backing down opt not to threaten in the first place. While a

threat is very likely to result in war under such circumstances because an ac-

32


ceptable and enforceable division of the good may not be locatable, a threat

is also far less likely to be made in the first place. For this reason as well,

further investigation of the general relationship between accountability and

the overall probability of war is called for.

4.3 Accountability and the Probability of War in

Equilibrium

Schultz (1999) investigated the overall probability of war as a function of

both the costs of war and of inconsistency in a model with this same game

structure, and Morrow (1999) similarly analyzed the relationship between

trade and the outbreak of war by treating disruption of the former as a cost

of the latter. The main conclusion of both works in this regard was that

the chance of war was generally non-monotonically related to the parameters

of interest, which were also negatively related to the chance of a resolved

challenger and/or a recalcitrant target. The consistent result is an inverted-U-

shaped relationship between the cost parameter and war, whether or not the

former is related to domestic incentives; thus, my ultimate conclusions here

are more optimistic, in that they find this relationship to be more negative

than positive in the vast majority of conceivable circumstances. An initial

positive relationship between accountability and war can be found only under

very specific conditions.

War between rational actors occurs here if S1 threatens and would rather

go to war than back down at the same time that S2 prefers, in expectation,

not to concede. Thus the chance of war, m, is the joint probability of these

two conditions. These conditions, in turn, differ significantly between the

33


0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0

Accountability (η)

Prob

. of W

ar (m

)

p = 0.5 p = 0.25 p = 0.05

Figure 9: With all-hawk public

0.0 0.5 1.0 1.5 2.0

0.00

0.10

0.20

0.30

Accountability (η)

Prob

. of W

ar (m

)

p = 0.5 p = 0.25 p = 0.05

Figure 10: With evenly split

public

bluffing and no-bluffing equilibria. A fuller treatment is in the appendix; the

revealing statics and conclusions are as follows

Whenever the public is overwhelmingly dovish, to such a degree that there

are rapprochement benefits as well as belligerence costs, accountability cer-

tainly drives down the chance of war: not only do targets concede more, but

challengers both bluff and stand firm less overall. For illustrative purposes,

figs. 9 and 10 plot the overall chance of war as a function of accountability for

when, respectively, the public is instead all hawks or is evenly split between

hawks and doves; in each case, the relationship is plotted separately for when

S1’s common-knowledge chance of winning, p, is 12, 1

4, and 1

20. C1 is fixed at

2.5 here throughout, as it was in fig. 7; not shown here is that the effect of C1

is almost entirely to shift these curves up or down, so it proves, surprisingly,

to be of little interest. Namely, C1 has no significant relationship in any case

to the shape of the curve and the direction of its slope at any point, which

34


rules out any particular significance within this theoretical framework of the

real and material costs of war rising over time.

Fig. 9 illustrates a monotonic positive relationship between accountability

and war; this relationship is generally less strong as p increases, but remains

substantially positive (even at p = 0.99 here [not shown], the curve starts at

m ≈ 0.6 and steadily approaches 1 as η approaches 2). Fig. 10, on the other

hand, illustrates that the relationship between accountability and war is posi-

tive only when the public is not predominantly hawkish and the accountable,

challenging state is highly unlikely to win this war if it occurs.

If we assume that the wars a challenger can threaten to start are evenly

distributed along the continuum of the challenger’s chance of victory, and

we look specifically at wars that the challenger starts, then we would ex-

pect based on this model to see a substantially lower rate of victories for

challengers in the middle of the autocracy-democracy scale. We cannot make

such a clear prediction about any increase in the overall rate of war, however.

We would only expect to see a higher overall rate of war initiation for such

states if there was in the first place, and for some reason, a greater number of

potential wars that the challenger is very likely to lose. Importantly, the wars

that become more likely within even a small range of very low accountability

are only those where the probability of victory is generally below 25%. Un-

der almost any set of assumptions, we would expect the overall relationship

between accountability and war to be almost consistently positive, albeit

consistently weaker at low starting levels of accountability.

Clearly, though, this raises concerns about democratization efforts for

states that are particularly weak, militarily. Such a state is in a position

35


where their threats are highly incredible and very likely to fail; if the state

is fully autocratic, then it is likely both bluffing and backing down at very

high rates. North Korea pops to mind. Some increase in accountability (to

a not-extremely-dovish public) that does not constitute full and sudden de-

mocratization will make the state’s threats marginally more credible, but

more importantly, the state now has reason to follow through on some of

the threats from which it would previously just back down. Paradoxically, if

the state were stronger, then its resolve would not be so open to question in

the first place, the increased rate of concessions would compensate for the

increased rate of lock-in to war, and the newly-accountable state would end

up in war less often rather than more.

Another assumption was made above that must be addressed and dis-

cussed further, namely that the distribution of possible war costs and the

corresponding continuum of resolve values is uniform (the same assumption

was made earlier to generate fig. 7). The shape of this distribution is pivotal

because it can completely and radically alter how much the likelihood of a

resolved S1 changes with any given change in S1’s payoffs: if there’s a large

proportion of possible challengers within the span between the old and new

cutpoints, then the corresponding change in the chance of war is momentous,

but with F1(·) left entirely to vary, that proportion could be literally zero

and the change would have no effect at all on the chance that S1 is resolved

to war.

This distribution deserves its own research program, or at least a larger

one than it currently has: along the range of possible levels of resolve that a

challenger (or target) state may have, what reasons might we have to think

36

Expectation Costs 5. CONCLUSION

that there will be a disproportionate number of highly-resolved states, or

weakly-resolved states, or perhaps moderately-resolved states with only a few

at either extreme? A substantial literature exists on the escalating costs of

war ensuring peace, which would perhaps imply that the distribution should

be skewed toward the low-resolve extreme, yet this argument is generally

limited to great-power relationships. A solid theory of resolve distributions

would at any rate have to account for the economic and technological de-

velopment of the category of states to which the random state observably

belongs.

5 Conclusion

It is argued here that an underappreciation for the relevance of prospect

theory to opinionated audiences reactions has led to an excessive focus on

an oversimplified and arguably negligible form of audience costs. The model

presented here establishes deductively that otherwise cheap talk becomes

expensive for accountable leaders in much the same way that military mobi-

lizations are for all leaders; it can thus indeed serve as an additional and pow-

erful tool in the ways Slantchev (2011) identified, but only for democratic and

otherwise accountable leaders. “Otherwise accountable’ here mainly refers to

Jessica Weeks (2008; 2012) landmark findings on accountable democracies;

influential elite audiences are human and also subject to prospect theory, and

so her findings fit and further motivate the theoretical framework presented

here.

This model should serve as a starting point for both theoretical and em-

37

Expectation Costs 5. CONCLUSION

pirical research that is still sorely needed, despite all that has already been

done with regards to audience costs. Survey experiments have rarely if ever al-

lowed for the setting and shifting of reference points; qualitative studies have

focused on audience reactions at the time that inconsistency is revealed and

not across the whole span of shifting audience expectations; and a panoply

of theoretical models have relied upon audience cost parameters, but only

ever as a negative in the back-down terminal node independent of all other

parameters. Aside from revisiting this work, there is also the clear need to

expand upon the present model, especially to account for the time frame in-

volved and for the fact that reference points shift downward far more slowly

and stubbornly than upward. Implications for domestic politics would almost

certainly require a more explicit and active role for audience members, or “the

audience” as a whole, making explicit choices to keep or replace the leader –

though this would virtually require the more sophisticated and challenging

PT formalizations mentioned earlier. From what is presented here, however,

we should be able to conclude that it would be premature, at least, to dis-

regard the reactions of domestic arguments as drivers of major international

bargaining phenomena.

38

Expectation Costs 6. APPENDIX

6 Appendix

6.1 Derivation of Expectation Costs

Proposition 1: Given any constants p1 and p2, and given that

v(x, p1, p2) =

(|p1 − x| − |p2 − x|)α, if |p1 − x| ≥ |p2 − x|

−λ(|p2 − x| − |p1 − x|)α, if |p1 − x| < |p2 − x|(14)

it is true that

v(x, p1, p2) + v(x, p2, p1) = (1− λ)(||p1 − x| − |p2 − x||)α (15)

where v(x, p2, p1) denotes v(x, p1, p2) with the values for p1 and p2 switched.

Proof. The value function for shift 2, given that that shift’s p1 is shift 1’s p2

and vice versa, is:

v(x, p2, p1) =

(|p2 − x| − |p1 − x|)α, if |p2 − x| ≥ |p1 − x|

−λ(|p1 − x| − |p2 − x|)α, if |p2 − x| < |p1 − x|(16)

Some basic rearrangements make it easier to see this function’s relationship

to the standard PT value function in eq. 14. We can reverse the directions of

both scope-condition inequalities, swap the order of the two cases as written,

and change which of the conditions covers |p2 − x| = |p1 − x| (since y = 0 in

either case) to arrive at:

v(x, p2, p1) =

−λ(|p1 − x| − |p2 − x|)α, if |p1 − x| ≥ |p2 − x|

(|p2 − x| − |p1 − x|)α, if |p1 − x| < |p2 − x|(17)

39


This shows how the scope conditions are identical to those in eq. 14. We can

then express both cases of this second-shift value function in terms of the

standard function used for the first shift:

v(x, p2, p1) =

−λv(x, p1, p2), if |p1 − x| ≥ |p2 − x|

− 1λv(x, p1, p2), if |p1 − x| < |p2 − x|

(18)

The sum of these two shifts, given fixed values for p1 and p2 as proposed, can

then be expressed purely in terms of the standard PT value function, and

then simplified:

v(x,p1, p2) + v(x, p2, p1) (19)

=

v(x, p1, p2)− λv(x, p1, p2), if |p1 − x| ≥ |p2 − x|

v(x, p1, p2)− 1λv(x, p1, p2), if |p1 − x| < |p2 − x|

(20)

=

(1− λ)v(x, p1, p2), if |p1 − x| ≥ |p2 − x|

(1− 1λ)v(x, p1, p2), if |p1 − x| < |p2 − x|

(21)

=

(1− λ)(|p1 − x| − |p2 − x|)α, if |p1 − x| ≥ |p2 − x|

(1− 1λ)(−λ)(|p2 − x| − |p1 − x|)α, if |p1 − x| < |p2 − x|

(22)

=

(1− λ)(|p1 − x| − |p2 − x|)α, if |p1 − x| ≥ |p2 − x|

(1− λ)(|p2 − x| − |p1 − x|)α, if |p1 − x| < |p2 − x|(23)

Due to the scope conditions, the base of α is nonnegative in either case; it is

40


therefore equivalent to its own absolute value, and we can substitute:

v(x,p1, p2) + v(x, p2, p1) (24)

=

(1− λ)(||p1 − x| − |p2 − x||)α, if |p1 − x| ≥ |p2 − x|

(1− λ)(||p2 − x| − |p1 − x||)α, if |p1 − x| < |p2 − x|(25)

All terms within the bounds of an absolute value can be multiplied by -1

without changing the value; thus, the bases of α in each case are alternative

and equivalent expressions of each other, which makes both cases entirely

equivalent, and we can simplify to:

v(x, p1, p2) + v(x, p2, p1) = (1− λ)(||p1 − x| − |p2 − x||)α (26)

Q.E.D.

6.2 Decomposing Expectation Costs

For doves and hawks, as defined with regard to a given shift, the shift is either

good or bad, entirely and unequivocally; thus, we can simplify those individ-

uals’ value functions. With p1 as reference point r and p2 as the objective

payoff, a hawk’s value function is:

v(p1, p2|x ≥ max{p1, p2}) =

(p2 − p1)a, if p2 ≥ p1

(−λ)(p1 − p2)a, if p2 < p1

(27)

The value function for doves can be derived from this by reversing the sign

of p1 and p2 throughout, most importantly in the case conditions:

v(p1, p2|x ≤ min{p1, p2}) =

(p1 − p2)a, if p2 ≤ p1

(−λ)(p2 − p1)a, if p2 > p1

(28)

41


x has no effect on these two function values and only serves to delineate

which of the two functions applies; thus, all hawks have the same reaction

to any given shift, and the same goes for doves, both groups defined as

they are in relation to the shift itself. It can be verified as well that these

are merely conditional simplifications of the spatial PT value function – the

latter returns the same function values given the same parameters.

Define σ as the proportion of hawks in this population; the leader’s payoff

from a hawkish shift (p2 > p1) is then

h = σ(p2 − p1)a + (1− σ)(−λ)(p2 − p1)a (29)

while the payoff for a subsequent dovish shift back to p1 is

d = σ(−λ)(p2 − p1)a + (1− σ)(p2 − p1)a (30)

If the public is entirely hawkish, then σ = 1, both payoffs simplify, and the

ratio between them becomes:

d

h=

(−λ)(p2 − p1)a

(p2 − p1)a(31)

= −λ (32)

meaning also that

d = −λh (33)

If the public is instead entirely hawkish, then σ = 0 and:

d

h=

(p2 − p1)a

(−λ)(p2 − p1)a(34)

=−1

λ(35)

42


Lastly, when the public is split evenly between hawks and doves (again, with

no moderates among those who have some position on the issue), then σ = 12.

Eqs. 29 and 30 then both simplify to:

1

2(p2 − p1)a +

1

2(−λ)(p2 − p1)a =

1

2(1− λ)(p2 − p1)a (36)

Thus, the same expectation costs are paid in the end here too, but half when

the threat is issued and half when it is retracted.

6.3 Coercive Bargaining with Expectation Costs

Proposition 2: The following strategies and beliefs constitute the perfect

Bayesian equilibrium to the ultimatum game with expectation costs.

(P2.1) S1 plays:

CH,SF if w1 ≥ max{ηd, t},

SQ, SF if t > w1 ≥ ηd,

CH,BD if ηd ≥ w1 > b, and

SQ,BD otherwise,

where

t =F2(0) + ηh

F2(0)− 1, and

b = F−11

[F−12

(ηε

ηd− 1

)[1− F1(ηd)] + F1(ηd)

]

43


(P2.2) Let q denote S2’s posterior belief of the probability that w1 > ηd;

q = min

{1− F1(ηd)

1− F1(b), 1

}, if F1(b) 6= 1 and S2 observes CH,

q = 1, if F1(b) = 1 and S2 observes CH,

q = max

{F1(t)− F1(ηd)

F1(t), 0

}, if F1(t) 6= 0 and S2 observes SQ, and

q = 0, otherwise.

(P2.3) S2 plays:

RF, ifw2 >q − 1

qor if q = 0, and

CD otherwise.

Proof. S1 prefers SF to BD when:

EUS1(SF ) > EUS1(BD)⇒ w1 > ηd (37)

Thus this is a condition for when S1 plays SF here, both on and off the

equilibrium path.

q follows Bayes’ rule, given S1’s strategy and the possible range of game

states. The major relevant distinction between game states is that t > ηd

and ηd > b cannot hold simultaneously, and it is therefore not possible in

any given game state for S1 to play both CH, BD and SQ, SF with positive

probability. First consider t > ηd, remembering that fi(·) is defined such that

Fi(0) < 1:

If t > ηd:

F2(0) + ηh

F2(0)− 1> ηd (38)

F2(0) <ηε

ηd− 1(39)

44


and if ηd > b:

ηd > F−11

[F−12

(ηε

ηd− 1

)[1− F1(ηd)] + F1(ηd)

](40)

F2(0) >ηε

ηd− 1(41)

which precludes eq. 39.

Lemma 1: Within the equilibrium defined in Proposition 2, any given game

state falls, observably and a priori, into one and only one of two categories:

bluffing games, in which S1 never plays SQ, SF, and no-bluffing games, in

which S1 never plays CH, BD.

In the bluffing state, S2 knows that ηd > max{t, b} and so all S1s for

whom w1 > ηd play CH, SF, but these S1s are only a subset of those who

play CH, for whom w1 > b. Bayes’ rule then dictates following CH that:

q =1− F1(ηd)

1− F1(b)(42)

When the game is in the no-bluffing state and those who play CH are instead

a subset of those who play SF, then eq. 42 is an incorrect formulation of Bayes’

rule and returns a value greater than 1; the proper application amounts to

1−F1(t)1−F1(t)

and simplifies to 1, so the minimum (min) operator in S2’s belief

following CH adjusts the strategy accordingly.

The definition of q following SQ follows the same logic, instead utilizing

a maximum (max) operator as appropriate. q in the no-bluff state is the

proportion of SQ-playing S1s who would play SF off the equilibrium path:

q =F1(t)− F1(ηd)

F1(t)(43)

45


which evaluates to a negative in the bluffing state when all who play SF also

play CH, and it is then set at zero in an implicit reformulation of Bayes’ rule.

F1(t) = 0 is possible, namely if S2’s war costs are expected to be very high

relative to S1’s. In this case, q = 0 following SQ because we are either in the

bluffing state of the game and q = 0 is correct, or we aren’t, all S1s should

play CH, and we are off the equilibrium path such that no belief is irrational

to have. It is also possible that F1(b) = 1, in which case q = 1 following CH

for analogous reasons: either we are in the no-bluffing equilibrium, F1(b) is

not part of Bayes’ rule properly applied, and the proper application always

returns 1, or no S1s can rationally play CH (because ηh < −1), we are off the

equilibrium path, and no belief is irrational for S2. S2’s belief and subsequent

move in this case do not and cannot incentivize S1 to change their strategy

of playing SQ rather than CH when ηh < −1.

S2’s strategy follows from his condition for preferring RF to CD in expec-

tation given q:

q(w2) + (1− q)(1) > 0 (44)

w2 >q − 1

q(45)

This condition also holds when q = 0 (because 1 > 0), hence the defined

special case.

Since q from eq. 42 can be substituted into eq. 45, S2 plays RF after CH

in the bluffing game state when:

w2 >F1(b)− F1(ηd)

1− F1(ηd)(46)

and in the no-bluff game state, where q = 1, when

w2 > 0 (47)

46


which happens with probability

1− F2(0) (48)

Whether S1 prefers CH to SQ in expectation depends on which outcome

she will choose if S2 plays RF. Define s as the probability of S2 playing RF

following CH. A resolved S1 (w1 > ηd) prefers CH when:

s(w1 + ηh) + (1− s)(1 + ηh) > 0 (49)

w1 >s− ηh− 1

s(50)

In the no-bluffing game state, s is defined by the condition in eq. 48 and so

eq. 50 simplifies to:

w1 >[1− F2(0)]− ηh− 1

1− F2(0)=F2(0) + ηh

F2(0)− 1(51)

which defines t and establishes the dual requirement for CH, SF that w1 >

max{ηd, t}. In the bluffing state as per Lemma 1, it is the case that ηd > t,

eq. 51 holds whenever EUS1(SF ) > EUS1(BD), and it is not then rational

for any S1 to play CH, BD.

All unresolved S1s (w1 ≤ ηd) are indifferent between CH and SQ when:

sε+ (1− s)(1 + ηh) = 0 (52)

s =1 + ηh

1− ηd(53)

As implied by Lemma 1, this s is irrational for S2 in the no-bluffing game

state, which defines that state and the absence of unresolved S1s that play CH

as per S1’s strategy. Specifically, S2 never rationally plays CD when w2 > 0,

47


so 1− F2(0) sets a lower bound for s in equilibrium and eq. 53 cannot hold

when:

1− F2(0) >1 + ηh

1− ηd⇒ F2(0) <

ε

ηd− 1(54)

Inequality 53 must hold in the bluffing game state, so S1 must bluff at

such a rate as to make it hold given S2’s rational beliefs and strategy (eq.

46):

1 + ηh

1− ηd= 1− F2

(F1(b)− F1(ηd)

1− F1(ηd)

)(55)

b = F−11

[F−12

(ηε

ηd− 1

)[1− F1(ηd)] + F1(ηd)

](56)

Thus b is so defined.

It is possible for eq. 56 to require F1(b) < 0. While F−1i (·) is defined to

then ensure that F1(b) = 0, it results in such cases of the proposed equilibrium

that eq. 56 does not hold; the ratio of bluffs to genuine threats is smaller than

it would be if either or both states’ possible war costs were larger, and so

S2 plays RF with less probability than he otherwise would. Unresolved S1s

in such cases strictly prefer to bluff and resolved S1s prefer CH to SQ by a

larger margin, but neither of these properties violate equilibrium criteria as

both players are still simultaneously playing their best replies. Q.E.D.

6.4 Accountability and the Probability of War in

Equilibrium

Proposition 3: η in this game is negatively related to the game ending with the

SF outcome, except for when h < 0 in the no-bluffing equilibrium, or when

d < 0 and p1 is sufficiently low in the bluffing equilibrium.

48


In the no-bluffing equilibrium, all threats are credible, so S2 refuses to

concede simply when his resolve is positive; and whether or not she would

prefer war to backing down, S1 issues a threat in the first place if and only

if her resolve is high enough that threatening has a positive expected value,

which implies a separate cutpoint t:

t =F2(0) + ηh

F2(0)− 1(57)

Thus the chance of war in the no-bluffing equilibrium is:

m|(F2(0) <

ε

ηd− 1

)=

[1− F1

(F2(0) + ηh

F2(0)− 1

)][1− F2(0)] (58)

= F1

(F2(0) + ηh

F2(0)− 1

)(F2(0)− 1)− F2(0) + 1 (59)

Consider how t, being the argument to F1(·) here, changes as η changes:

t′ =d

dη

(F2(0) + ηh

F2(0)− 1

)=

h

F2(0)− 1(60)

Since F2(0) − 1 is negative, the sign of t′ is the opposite of the sign of h;

the cutpoint is increasing and accountability means less threats if there are

belligerence costs, but more threats if there are belligerence benefits. In the

overall m equation, F1(·) (which equals the rate of S1s that don’t threaten)

is multiplied by F2(0) − 1, which, again, is negative, so the higher t in the

presence of belligerence costs means a lower chance of war, but it means a

higher chance of war if and when there are belligerence benefits. In either

case, the effect of accountability is monotonic.

When bluffing is possible, the relevant cutpoint for S1 is the one that

resolves them to war, ηd, rather than the one where they choose to threaten.

S2’s cutpoint for conceding is a function of S1’s domestic incentives, and also

49


of b, but in equilibrium, we can simply use s as the overall probability that

they do not concede, which is after all how their behavior can lead to war.

The joint probability is then:

m|(F2(0) >

ε

ηd− 1

)= [1− F1(ηd)]

(1 + ηh

1− ηd

)(61)

When d > 0, η decreases m: s is negatively related to η, so if 1−F1(ηd) is also

decreasing, as it is when there are rapprochement benefits, then the effect is

unambiguous. When there are rapprochement costs, however, as there will

be for most opinion distributions, then the effect is ambiguous.

An increase in accountability, which in the case we’re considering de-

creases the chance of a refusal but increases the chance of a resolved chal-

lenger, is most likely to be positive – that is, to increase the chance of war

– when resolved challengers were already rare and the rate of refusal was

already high. s is strictly decreasing with accountability, but if d < 0 and

the corresponding drop in F1(ηd) is large enough, then m will increase, and

with F1(·) left completely to vary, then any drop or even no drop at all is

theoretically possible so long as the chance of a resolved challenger ranges

between zero and one.

If fi(x) is a uniform distribution ranging from p− Ci to p, then:

fi(x) =

1

Ci+p, if p− Ci < x < p

0, otherwise

(62)

50

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and the CDF is:

Fi(x) =

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1, if x > p

(63)

Eq. 61 then becomes:

m =

[1−

(ηd+ C1 − pC1 + p

)](1 + ηh

1− ηd

)(64)

=2p− ηdC1 + p

(1 + ηh

1− ηd

)(65)

Naturally, C1 drives down the rate of resolved challengers while p drives

it upward; if we are considering when an increase in the rate of resolved

challengers will have the largest upward effect onm and possibly outweigh the

decrease in s, then we would look to when Ci is high, p is low, and the starting

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effect in absolute terms, however, when the denominator is minimized, which

is when both Ci and p are low; this makes the overall effect of Ci ambiguous,

while reinforcing the fact that a low p is most likely to mean a strong positive

effect of ∆ηd on m. Q.E.D.

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