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http://jtp.sagepub.com/ Journal of Theoretical Politics http://jtp.sagepub.com/content/23/4/510 The online version of this article can be found at: DOI: 10.1177/0951629811418142 2011 23: 510 Journal of Theoretical Politics Jason B Scholz, Gregory J Calbert and Glen A Smith Unravelling Bueno De Mesquita's group decision model Published by: http://www.sagepublications.com can be found at: Journal of Theoretical Politics Additional services and information for http://jtp.sagepub.com/cgi/alerts Email Alerts: http://jtp.sagepub.com/subscriptions Subscriptions: http://www.sagepub.com/journalsReprints.nav Reprints: http://www.sagepub.com/journalsPermissions.nav Permissions: http://jtp.sagepub.com/content/23/4/510.refs.html Citations: What is This? - Oct 19, 2011 Version of Record >> at CAPES on March 26, 2012 jtp.sagepub.com Downloaded from
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Page 1: [Simulation bueno de mesquita] scholz et-all-unravelling-bueno-de-mesquita-s-group-decision-model

http://jtp.sagepub.com/Journal of Theoretical Politics

http://jtp.sagepub.com/content/23/4/510The online version of this article can be found at:

 DOI: 10.1177/0951629811418142

2011 23: 510Journal of Theoretical PoliticsJason B Scholz, Gregory J Calbert and Glen A Smith

Unravelling Bueno De Mesquita's group decision model  

Published by:

http://www.sagepublications.com

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Article

Unravelling Bueno DeMesquita’s group decisionmodel

Journal of Theoretical Politics23(4) 510–531

©The Author(s) 2011Reprints and permission:

sagepub.co.uk/journalsPermissions.navDOI:10.1177/0951629811418142

jtp.sagepub.com

Jason B Scholz, Gregory J Calbert and Glen A SmithAustralian Government, Department of Defence, Edinburgh, Australia

AbstractPolitical scientist Professor Bruce Bueno De Mesquita, has made significant claims for the pre-dictive accuracy of his computational model of group decision making, receiving much popularpress, including newspaper articles, books and a television documentary entitled ‘The New Nos-tradamus’. Despite these and many journal and conference publications related to the topic, noclear elucidation of the model exists in the open literature or can be found in a single place. Weexpose and present the model by careful navigation of the literature and illustrate the soundnessof our interpretation by replicating De Mesquita’s own results. We also raise issues regardingsensitivity and convergence.

Keywordsalliance; expected utility; group decision making; influence; risk

1. IntroductionThere is little doubt that some of the greatest social challenges for the future of mankindinclude terrorism, war, climate change, poverty and economics. So, the pursuit of anintegrated theory capable of explanation and prediction of group decision outcomesis a worthy endeavour. Such efforts, often classed under the realm of computationalpolitical science, aim to form testable yet tractable models for human agency (Kollmanet al., 2010). Bueno De Mesquita (herein abbreviated to BDM) has laid claims to suchan achievement. A sample prediction is ‘…the ability to dominate Iran’s politics resideswith Khamenei and Rafsanjani. And between these two – though the contest is close – theadvantage seems to lie with Khamenei’ (BDM, 1984: 233). The accuracy of this predic-tion is demonstrated by the fact that Khamenei succeeded Khomeinei as Supreme Leaderof Iran in June 1989 and Rafsanjani became the fourth president of Iran in August 1989.

Corresponding author:Jason B Scholz, Australian Government, Department of Defence, PO Box 1500, Edinburgh 5111, AustraliaEmail: [email protected]

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Scholz et al. 511

BDM’s model of group decision making considers conflicts and alliances, and isbased on expected utility theory. BDM (1997) states:

The model itself depicts a game in which actors simultaneously make proposals, and exertinfluence on one another. They evaluate options and build coalitions by shifting positions onthe issue in question. The above steps are repeated sequentially until the issue is resolved.(p. 238)

A New York Times article by Thomson (2009) gives some insight that may explain whythe model has never been fully disclosed:

Bueno de Mesquita does not publish the actual computer code of his model. (Bueno de Mesquitacannot do so because his former firm owns the actual code, but he counters that he has out-lined the math behind his model in enough academic papers and books for anyone to replicatesomething close to his work.)

At first BDM (1997) appears to offer the most promise in elucidating the model; however,first impressions prove misleading. Significant errors and obfuscations become apparentin trying to replicate the model and results from this and later works. In the following,we carefully navigate and interpret earlier works to derive a working model and softwarethat reproduces his published results to an adequate level of accuracy.

The following list provides a chronology of significant events relevant to the evolutionof the mathematical model (it is not a complete chronology of BDM’s works).

• BDM (1980) first introduces a theory of expected utility and uses an application tointerstate threats in an attempt to explain the initiation of conflict.

• The book The War Trap (BDM, 1981) reiterates and further details the model anddiscusses insights that may be derived from it.

• BDM (1984) demonstrates an application of the model by forecasting positions onleadership and the role of government in the economy in a post-Khomeini Iran.

• BDM (1985) revises the model by introducing a conceptual basis and expressionfor risk taking. Prior to 1985 there was no explicit risk component.

• BDM and Lalman (1986) detail a problem with the model that prevents an ability tocompare expected utility across events if the number of actors changes, and proposea revision to fix it.

• BDM (1994) provides a summary of the complete model (accepting the revision)and provide an illustrative case study with results.

• BDM (1997) provides a more readily accessible esposé of the model than BDM(1994) with a different case study.

• Feder (2002: 118–119) is often quoted by BDM as a highly credible source tosupport the accuracy of his forecasts. Feder differentiates two versions of BDM’sforecasting model. The first of these Feder terms the ‘voting model’, which calcu-lates only the median voter position (see Section 3.2), while the second he terms‘the political expected utility theory’ model, which ‘is used in conjunction with thevoting model’ as described in full in this paper. Notably, with regards to its use bythe CIA between 1982 and 1986, Feder provides an endorsement of significance inregard to the ‘voting model’, yet remains silent on the performance of the ‘politi-cal expected utility theory’ model. Note that earlier, Feder (1995) provided furtherspecific success stories.

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512 Journal of Theoretical Politics 23(4)

• The book Predicting Politics (BDM, 2002) provides a highly condensed version ofthe BDM (1997) description and sample results.

• BDM (2009a) is a talk that provides some mathematical description of the model.

In the following, we explain the structure of the model in terms of its basic premises,inputs and outputs and explain the main aspects of its mathematical evolution, focusingon its mature form started by BDM and Lalman (1986). We then explain the model indepth by decomposing it into sections on calculation of the utilities, alliance probabilities,risk propensities, decision process and address the uncertainty of parameters. This yieldsa single integrated interpretation of BDM’s model. We then compare the results of ourinterpretation with BDM’s results for his 1994 case study and briefly discuss stabilityconcerns.

2. Structure of the modelBDM’s predictions depend on two parts. First, his method of data collection and inter-pretation from human subjects; second, the computational model which he applies to thatdata.

The first part is significant, but has not been described by BDM in the open litera-ture and so evades current interpretation or analysis. The second part, the computationalmodel, we examine further.

2.1. Basic premisesThe model deals with a single ‘issue’ decomposed into a metric scale, with ‘position’values (x) corresponding to states of the issue. BDM illustrates:

The term xi represents each nation’s preferred date, measured in years, by which emission stan-dards should be applied to medium-sized automobiles as revealed at the outset of discussionson the issue. (BDM, 1994: 77)

We will continue with this example later in the results section. A number of ‘actors’ (i= 1,2,…,n) exists, each of which hold a single ‘position’ (xi) with regard to the issue,represented by their assignment to a location.

Each ‘actor’ is also considered to possess some ‘capability’ (ci) with respect to the‘issue’. ‘Capability’ is sometimes interchangeably referred to as ‘power’ or ‘resources’by BDM. Like ‘position’, ‘capability’ is given a value on a metric scale. This valuerepresents an actor’s level of influence with regard to the issue.

Lastly, each ‘actor’ is also considered to possess some ‘salience’ (si) with respect tothe ‘issue’. ‘Salience’ is sometimes interchangeably referred to as ‘importance’, ‘prior-ity’ or ‘attention’ by BDM. Like ‘position’ and ‘capability’, ‘salience’ is given a valueon a metric scale. This value represents an actor’s level of energy with regard to the issue.Table 1, from BDM (1994), illustrates this.

BDM’s model decomposes the social fabric into pairwise ‘contests’ between actorswith support or otherwise of third-party alliances. Based on actor i’s perception ofexpected utility, actor i considers whether or not to challenge the other actor j, in anattempt to convince them to adopt i’s position. The expected utility includes an assess-ment of the level of third-party support for actor i’s challenge. If actor i’s expected utility

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Scholz et al. 513

Table 1. Sample input data for the computational model. The issue is ‘the date (years) ofintroduction of emission standards for medium-sized automobiles’.

Actor (i) Capability (ci) Position (xi) Salience (si)

Netherlands 0.08 4 80Belgium 0.08 7 40Luxembourg 0.03 4 20Germany 0.16 4 80France 0.16 10 60Italy 0.16 10 60UK 0.16 10 90Ireland 0.05 7 10Denmark 0.05 4 100Greece 0.08 7 70

of challenging actor j versus not challenging is greater than zero, actor i will challengeactor j, otherwise it will not. This model of mind or agency is confrontational and whollyself-interested.

The outputs of the model feature revised positions of each actor, the median actorposition (in effect the social ‘norm’) and include various other parameters resulting fromcalculation of revised position, such as an actor’s alliances, risk attitude and expectedoffers (coercion) to and from other actors.

2.2. Mathematical evolution of the model

As explained in the introduction, BDM’s model has evolved over the years. We nowexplain this in mathematical terms. We are readily familiar with the expected value of arandom variable Z, with various states Zw each with probability Pw of occurring:

E (Z) =∑

w

PwZw

Expected utility follows the same structure in that the utilities of different contestoutcomes are estimated along with the associated probabilities.

In the following we use the exact notation from BDM. However, the form of thatnotation is not consistent, having shifted several times over the years. So, in order toexplicate the development trail we shall explain the changes pertinent to understandingthat development.

An apparent motivation for BDM’s expected utility model was predicting the out-break of war, as per BDM (1981). It is thus not surprising to find a confrontationalmentality to the basic form of the model. BDM considers an actor i to choose to ‘chal-lenge’ a rival or opponent actor j or not. In the case of a challenge, the expected utility toactor i (superscript i) if i challenges j (subscripts) is:

Ei(Uij

)c= PiU

isi + (1 − Pi) Ui

fi

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514 Journal of Theoretical Politics 23(4)

Figure 1. Transitive utility contribution by third-party actors.

where Uisi refers to the utility for actor i (superscript i), succeeding (subscript s) when i

is the challenger (subscript i), Uifi is the utility for actor i if it challenges and fails and Pi

refers to the probability of successful challenge by actor i.BDM (1981) also considers a contribution from multi lateral actors to this expected

utility. The form of this contribution has been adapted since 1981, so we trace thisdevelopment next. To avoid introducing confusion due to significant notational changesbetween BDM (1981) and later publications, we begin with the form used in BDM(1985):

Ei(Uij

)c= PiU

isi + (1 − Pi) Ui

fi +∑k �=i,j

(Pik + Pjk − 1)(Uiki − Ui

kj) (1)

where Uiki refers to the utility to actor i (superscript) for k challenging i (subscripts) and

Pik refers to the probability of actor i challenging actor k. This new term in (1) recognizesthat each third-party actor k may contribute (positively or negatively) towards the overalleffect of i’s challenge on j. This term derives from an assumption of the transitivity ofutilities, as illustrated in Figure 1.

The derivation of the term Pik + Pjk − 1 is explained clearly in BDM (1985) and willnot be reiterated here.

In the case that actor i does not challenge j, i expects to remain at the same positionand j may either remain where it is (status quo) or j may move to a different position.If j moves, the utility of the outcome may prove either better for i or worse for i. Theexpected utility for i not challenging is then, using BDM’s (1985) notation:

Ei(Uij

)nc

= QiqiU

iqi +

(1 − Qi

qi

) (Qi

biUibi + (

1 − Qibi

)Ui

wi

)(2)

where Uiqi refers to the utility to actor i of status quo and Qi

qi the probability of status quo.Ui

bi refers to the utility to actor i if the outcome is better and Uiwi the utility to actor i if

the outcome is worse. If the outcome is better, then Qibi = 1 and Qi

wi = 0. If the outcomewas worse, then Qi

bi = 0 and Qiwi = 1.

The full form of the expected utility difference combines (1) and (2):

Ei(Uij

) = Ei(Uij

)c− Ei

(Uij

)nc

(3)

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Scholz et al. 515

In BDM and Lalman (1986), a problem with the following term in (1) is identified:

Ei(Uij

)m

=∑k �=i,j

(Pik + Pjk − 1)(Uiki − Ui

kj) (4)

The problem is described in BDM and Lalman (1986):

Because of the manner in which third parties are treated in earlier studies (BDM, 1981, 1985),the operational estimate of expected utility values for any decision maker could vary between(2N-2) and –(2N-2), where N is the total number of nations in the relevant international system.Variations in the size of the international community, then, affected the possible range of valuesin the expected utility models set out earlier. This is a serious shortcoming in that it makescomparison of a single nation’s utility scores in different years difficult …The new formulationfixes the range of values, irrespective of system size, in a theoretically meaningful way. (p.1119)

BDM’s proposed solution involves removing the term (4) from (1) and, instead, incor-porating a more complex form of calculation of the probability Pi. In BDM (1985) theprobability Pi refers to ‘Pi= i’s probability of succeeding in a bilateral contest with j’.

From 1986 onwards, the definition of the probability is changed to account for multi-lateral contributions to the contest between i and j. The new form is denoted Pi

i. This willbe defined later.

The form of the expected utilities, as stated in BDM and Lalman (1986), are:

Ei(Uij

) = Ei(Uij

)c− Ei

(Uij

)nc

(5)

Ei(Uij

)c= sj

(Pi

iUisi + (

1 − Pii

)Ui

fi

)+ (

1 − sj

)Ui

si (6)

Ei(Uij

)nc

= QUisq + (1 − Q)

(TUi

bi + (1 − T) Uiwi

)(7)

In BDM’s notation, the term Q substitutes for Qiqi in (2), T substitutes for Qi

bi and Uisq

substitutes for Uiqi. The only notable change is the inclusion of salience sj and an extra

term in the expected utility for challenge. The form of Equations (6) and (7) may bevisualized as in Figure 2.

This structure remains throughout BDM (1994, 1997, 2002). In BDM (2009b), a newstructure of model is announced. We do not consider this new model. As a result of themulti lateral scaling problem identified with the model in its pre-1986 form, we focus onthe model structure and results for 1986 and later, using the form from Equations (5)–(7).

3. Utilities

3.1. Base utilities

To find the expected utility, we need to calculate the basic utilities:

Uisi, Ui

fi, Uibi, Ui

wi, Uisq (8)

These utilities are a function of the policy position of actors, xi and xj. BDM (1997)tells us that ‘i’s utility for xk , uixk , is a decreasing function of the distance between theproposal and i’s preferred resolution, so that

uixk = f (−|xk − x∗I |) (9)

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516 Journal of Theoretical Politics 23(4)

Start positions

i does NOTchallenge j

i succeedsagainst j

i receivesutility

j succeedsagainst i

i losesutility

i receivesutility for status quo

j does notmove

j moves

Situation isbetter for i

i receivesutility

Situation is worse for i

i losesutility

i succeedsagainst j

i receivesutility

i challenges jwith position

Figure 2. The structure of the social ‘game’.

The notation uixj is not the same as that used in (8), so some transformation is probablyrequired. Thus we interpret (9) as the general class of model only. It is worth pointingout that a specific class is stated in BDM (1997):

uixj = 1 −∣∣∣xi − x∗

j

∣∣∣ri(10)

However, (10) is inconsistent with the more detailed earlier explanations, as the followingwill now reveal. One clue to the utility calculations is given by BDM (1994):

Should i succeed, then i will derive the utility associated with convincing j to switch fromits current policy stance to that supported by i. This is denoted by ui�x+

j |d , which equals

ui(xi − xj) . Should i fail, then it confronts the prospect of having to abandon its objectives in

favour of those pursued by j, denoted by ui�x−j |d = ui(xj − xi). (p. 84)

Once again, BDM introduces additional notation ui(xi−xj) and ui(xj−xi), which remainsundefined. However, BDM (1985) gives utility for i’s success, which is of the form of adifference between positions i and j:

Uisi = 2 − 4

⎡⎣2 −

(Ui

ii − Uiij

)4

⎤⎦

ri

(11)

In addition, the utility for i’s failure, which is of the form of a difference between positionsj and i:

Uifi = 2 − 4

⎡⎣2 −

(Ui

ij − Uiii

)4

⎤⎦

ri

(12)

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Scholz et al. 517

Noting also in BDM (1985):

The reason for the transformations by 2’s and 4’s is to preserve the original scale of num-bers while avoiding the generation of imaginary numbers. Because ri can be less than 1.0, theabsence of transformations would mean that for negative values of, for instance Ufi, no real rootwould exist. This problem is eliminated with the introduction of these transformations. (p. 158)

Uiii and Ui

ij are defined by BDM (1985):

With Uiii being equal to the value i attaches to his own policy portfolio (Both Uii and Ujj are

assumed to equal 1.0, with Uij and Uji ranging between possible values of 1.0 and –1.0), and

with Uiij being equal to the value i attaches to j’s policies as a function of their similarity to the

policies of i. (p. 158)

Thus:Ui

ii = Uijj = Uj

ii = Ujjj = 1 (13)

To satisfy the stated range requirement we propose:

Uiij = Ui

ji = 1 − 2

∣∣∣∣ xi − xj

xmax − xmin

∣∣∣∣ (14)

Equation (14) is consistent with the statement and Equation at (9). Note that −1 ≤ Uiij ≤

1, where xmax − xmin is the range of positions. Note the maximum value of +1 occurswhen policy positions of i and j coincide and is at its minimum of −1 when the positionsare maximally separated.

Summarizing, so far we have now accounted for (8) parts a and b, which simplify to:

Uisi = 2 − 4

[0.5 − 0.5

∣∣∣∣ xi − xj

xmax − xmin

∣∣∣∣]ri

(15)

Uifi = 2 − 4

[0.5 + 0.5

∣∣∣∣ xi − xj

xmax − xmin

∣∣∣∣]ri

(16)

Note that the ranges 2 − 4(0.5)ri ≤ Uisi ≤ 2 and −2 ≤ Ui

fi ≤ 2 − 4(0.5)ri are consistentwith the diagram in BDM (1985: 159).

BDM (1985) does not explicitly define Uibi or Ui

wi; however, we are given the statusquo utility in BDM (1985: 158):

Uiqi = 2 − 4

[(2 −

[(Ui

ii − Uiij

)tn

−(

Uiii − Ui

ij

)t0

] )/4] ri

(17)

The subscripts t0 and tn are not defined, but in BDM (1981: 48) these correspond tobefore and after j’s policy change, respectively. The utility subscript q usually signifiesstatus quo, but we believe this is an error in (17) and should instead refer to j making apolicy change (or move) that gains or betters the situation for i. We are led to believe thisby BDM (1981):(

Uiii − Ui

ij

)t0

=i’s perception of what may be gained by succeeding in a bilateral conflict with

j…(

Uiij − Ui

ii

)t0

= i’s perception of what may be lost by failing in a bilateral conflict with j…

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518 Journal of Theoretical Politics 23(4)

We adopt the subscript b, indicating ‘better’ consistency with (8). Furthermore, we notea problem with scaling for BDM’s (17), as it will become undefined (negative numberraised to a power less than 1.0). Thus, corrections are also required to realign scaling.The result ensures Ui

wi ≤ Uiqi ≤ Ui

bi:

Uibi = 2 − 4

⎡⎢⎣4 −

(Ui

ii − Uiij

)tn

−(

Uiii − Ui

ij

)t0

8

⎤⎥⎦

ri

(18)

Similarly, we expect that j’s movement may potentially result in a worse condition for i:

Uiwi = 2 − 4

⎡⎢⎣4 −

(Ui

ij − Uiii

)tn

−(

Uiij − Ui

ii

)t0

8

⎤⎥⎦

ri

(19)

Note this adjusted scaling ensures 2 − 4(0.5)ri ≤ Ubi ≤ 2 and −2 ≤ Uwi ≤ 2 − 4(0.5)ri .We are given a clue that some relation to the median voter position is important by:

[ui�x+j |d̄ and ui�x−

j |d̄] are approximated by comparing the value actor i attaches to the currentmedian voter prediction to the value i attaches to the median anticipated if i accepts j’s preferredoutcome. (BDM, 1997: 248)

BDM (2009a) expresses this most clearly:

…they are anticipated to move towards the median voter position if they make an uncoercedmove. This means that if B lies on the opposite side of the median voter from A, then A antici-pates that if B moves (probability=0.5), then B will move in such a way as to come closer tothe policy outcome A supports and so A’s welfare will improve without A having to exert anyeffort. If B lies between the median voter position and A, then whether A’s welfare improves orworsens depends on how far B is expected to move compared to A. The same is true if A liesbetween B and the median. (p. 6)

We interpret this to mean that for no challenge (uncoerced), where B (or j) moves, thatA (or i) expects B (or j) will move to the median position. The cases are illustrated inFigures 3–6.For any of these cases, then we expect:

(Ui

ij

)t0

= 1 − 2

∣∣∣∣ xi − xj

xmax − xmin

∣∣∣∣ (20)

(Ui

ij

)tn

= 1 − 2

∣∣∣∣ xi − μ

xmax − xmin

∣∣∣∣ (21)

Substituting (20) and (21) into (18) yields:

Uibi = 2 − 4

[0.5 − 0.25

(|xi − μ| + ∣∣xi − xj

∣∣)|xmax − xmin|

]ri

(22)

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Scholz et al. 519

i jµ j iµFigure 3. Case 1: μ between i and j − i gets better as a result of j moving.

i jµ

j iµ

Figure 4. Case 2: j between i and μ − i gets worse as a result of j moving.

Substituting (20) and (21) into (19) yields:

Uiwi = 2 − 4

[0.5 + 0.25

(|xi − μ| + ∣∣xi − xj

∣∣)|xmax − xmin|

]ri

(23)

We note further, that these utilities must be applied in the appropriate cases. Thus ifcase 1 is true, then the probability of i’s utility improving is 1.0 and, by implication,the probability of i’s utility worsening is 0.0. We believe this defines the probability T inEquation (7), although no known publication by the author states this explicitly. So, cases1 and 3A correspond to T = 1 and cases 2 and 3B correspond to T = 0. This helps explainthe description in BDM and Lalman (1986).

Lastly, for the situation of no change in policy (status quo), i does not challenge andj does not move, BDM (1985) defines:

Uisq = 2 − 4

[(2 − 0)

4

]ri

= 2 − 4 (0.5)ri (24)

We reason this corresponds to (8) part e.To determine the calculations for actor j, BDM (1985) notes:

Of course, the Ujs, U

jf and U

jq terms (with appropriate superscripts) are defined analogously.

These terms vary as a function of whose estimate of expected utility is being calculated (i.e.,who is the superscripted actor) by varying the risk exponent, so that for expected utility equa-tions with a j superscript, j’s risk taking propensity is used to estimate what j perceives to bethe value of success, failure, or no challenge for i in accordance with the equations delineatedbelow. (p. 158)

Lastly, the probability of status quo might be determined in a number of ways; however,a value of Q = 1.0 is assumed in BDM (1985), corresponding to a stoic opposition, anda value of Q = 0.5 is assumed in both BDM and Lalman (1986) and BDM (2009a),corresponding to a maximally uncertain outcome of whether the actor j will move or stayin position. No explicit value for Q is specified in the other papers.

Thus, the two main equations become:

Ei(Uij

) = sj

(Pi

iUisi + (

1 − Pii

)Ui

fi

)+ (

1 − sj

)Ui

si

−QUisq − (1 − Q)

(TUi

bi + (1 − T)Uiwi

) (25a)

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520 Journal of Theoretical Politics 23(4)

j iµ

i jµ

Figure 5. Case 3A: i between j and μ − i gets better as a result of j moving.

j iµ

i jµ

Figure 6. Case 3B: i between j and μ − i gets worse as a result of j moving.

Ej(Uji

) = sj

(Pj

jUjsj +

(1 − Pj

j

)Uj

fj

)+ (

1 − sj

)Uj

sj

−QUjsq − (1 − Q)

(TUj

bj + (1 − T)Ujwj

) (25b)

3.2. The median voter position

In the previous section, it became clear that the ‘median voter position’ must bedetermined in order to calculate the utility terms in (22) and (23).

BDM defines comparative votes in direct proportion to utility difference, capabilityand salience. The votes ‘cast’ by agent i in comparing positions xj and xk is given asBDM (1997: 239):

vjki |xj, xk = cisi

(uixj − uixk

)(26)

We emphasize that these votes cast to k may indeed be negative if, for example, agent iprefers j to k. To map this notation to that used previously, we interpret:

uixj = Uij & uixk = Uik (27)

Using (14) we get

vjki |xj, xk = 2cisi

(|xi − xk| − ∣∣xi − xj

∣∣|xmax − xmin|

)(28)

According to BDM (1997):

The prospect that a proposal will succeed is assumed to depend on how much support can bemustered in its favour as compared with the feasible alternatives. This is calculated as the sumof ‘votes’ across all actors in comparison between xj and xk .(p. 240)

The votes for j versus k, given i �= j, k are:

vjk =n∑

i=1

vjki (29)

In general, this pairwise determination is termed a Condorcet Method of voting. ACondorcet winner is the candidate whom voters prefer to every other candidate, whencompared to them one at a time.

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Scholz et al. 521

Black’s Median Voter theory now comes into play, so ‘the decision adopted by thecommittee becomes determinant as soon as the position of the one optimum – which wecan refer to conveniently enough as the median optimum – is given.’ (Black, 1948).1

That is, in a majority election where a voter’s attitude is represented as a point in onedimension, if all voters vote for a candidate closest to their own preference and thereare only two candidates, then if the candidates want to maximize their votes they shouldcommit to the policy attitude preferred by the median voter.

The median voter’s ideal attitude is always a Condorcet winner (Congleton, 2003).Thus the median voter attitude index and the number of votes at the median attitude maybe determined.

4. Alliance probabilityThe outcomes of the challenge contests are multi lateral, but under the transitivityassumption, the probabilities of Equations (25) are decomposable into bilateral alliances.BDM determines these probabilities by combining, across all pairs, an assessment of‘who is with me’ (positive valued vote) versus ‘who is against me’ (negative valued vote)and normalizing. BDM (1997) states the estimator as:

Pi =∑

k| ukxi>uk xj

vijk

/n∑

k=1

∣∣∣vijk

∣∣∣To maintain notational consistency, we substitute vjk

i for vijk in the above:

Pi =∑

i| uixj>uixk

vjki

/n∑

i=1

∣∣∣vjki

∣∣∣ (30)

When more agents are ‘for’ than ‘against’, this raises the probability of winning thebilateral contest. As per previous derivation of votes, substitute and expand (30):

Pi =

∑k if arg>0

cksk(∣∣xk − xj

∣∣− |xk − xi|)

n∑k=1

cksk

∣∣(∣∣xk − xj

∣∣− |xk − xi|)∣∣ (31)

5. Risk propensityAs seen in the previous section, utility calculations involve a risk exponent. This riskexponent is in turn derived from the expected utility. BDM (1985) is first to describe thebasis for risk calculation:

I define each nation’s security level as∑j �=i

E(Uji). The greater the sum, the more utility i believes

its adversaries expect to derive from challenging i. …as this sum decreases, i’s relative securityincreases, so that i is assumed to have adopted safe policies...

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522 Journal of Theoretical Politics 23(4)

0

0.5

1

1.5

2

-1 -0.6 -0.2 0.2 0.6 1Ri

ri

Figure 7. Scaling conversion formula.

BDM (1985) goes on to define:

Ri =2∑

E(Uji) −∑

E(Uji)max −∑

E(Uji)min∑E(Uji)max −

∑E(Uji)min

(32)

Note that BDM (1997) reverses the subscripts of the above, which is inconsistent withhis conceptual basis of security. Further, BDM (1997) provides an inconsistent transfor-mation formula that would not accommodate the range −1 ≤ Ri ≤ +1. Thus, we choosethe earlier conversion formula from BDM (1985):

ri = 1 − Ri/3

1 + Ri/3(33)

The purpose of the formula according to BDM (1985) is to ensure that ri ranges between0.5 and 2, noting that the divisor of 3 appears arbitrary, but effects curvature. Equation(33) is illustrated in Figure 7.

Equation (34) expresses (32) more precisely:

Ri =2

n∑j=1,j �=i

Ei(Uji) − maxi

⎧⎨⎩

n∑j=1,j �=i

Ei(Uji)

⎫⎬⎭− min

i

⎧⎨⎩

n∑j=1,j �=i

Ei(Uji)

⎫⎬⎭

maxi

⎧⎨⎩

n∑j=1,j �=i

Ei(Uji)

⎫⎬⎭− min

i

⎧⎨⎩

n∑j=1,j �=i

Ei(Uji)

⎫⎬⎭

(34)

We still need to know, however, how to calculate the expected utilities in (25), which usea modified notation. BDM (1985) describes:

Thus, the risk terms are calculated by manipulating the alliance portfolios used as the policyindicator through simulation to locate the best and worse portfolios for any given nation, wherethe best and worst are defined in terms of the sum of expected utilities of all others vis-à-visthe nation in question under the assumption that utilities are strictly a function of similarities

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Scholz et al. 523

in alliance commitments. (Note: That is, temporarily applying the expected utility equations(without risk or uncertainty taken into account) as developed in The War Trap, I identify theworst and best case alliance strategy for each nation each year, using the original, linear util-ity functions to define the range of possible expected gains or losses for each nation. These,then, are utilized to measure risk propensities and thereby to introduce curvature into the utilityfunctions.) (pp. 167–168)

This implies a process to first determine the expected utilities of Equations (25) usingri = 1, then applying (34) and (33) to estimate ri and lastly applying the ri estimates tore-estimate the expected utilities of Equations (25). We adopt this approach.

6. Decision

6.1. Offer categories

The expected utilities Ei(Uij

)and Ej

(Uji

)are used to classify the ‘offers’ between

all actor pairs into categories according to potential outcomes, as illustrated in Figure8. An actor may expect to conflict, compromise, capitulate or stalemate with another.Unfortunately, no single publication by BDM explains how to quantify these.

6.1.1. Conflict. Actors i and j conflict if Ei(Uij) > 0 and Ej(Uji) > 0. So, ‘If both i and jbelieve that they have the upper hand in the relationship, then conflict is likely and thatconflict has an uncertain outcome.’ BDM (1997: 244).

BDM (1984: 230) labels for the ‘Confrontation-’ octant, ‘Challenger Favored’ and forthe ‘Confrontation+’ octant ‘Favoring Focal Group’. We interpret this to mean i movesto j and j moves to i, respectively, as shown in Figure 8.

6.1.2. Compromise. Actor i has the upper hand if Ei(Uij) > 0 and Ej(Uji) < 0 and∣∣Ei(Uij)∣∣ >

∣∣Ej(Uji)∣∣. Actor j has the upper hand if Ei(Uij) < 0 and Ej(Uji) > 0 and∣∣Ei(Uij)

∣∣ <∣∣Ej(Uji)

∣∣. BDM (1997) describes:

both players agree that i has the upper hand. In this instance, j is expected to be willing to offerconcessions to i, although the concessions are not likely to be as large as what i would like.The likely resolution of their exchange is a compromise reflecting the weighted average of i’sexpectation and j’s. (pp. 243–244)

However, the weighted average is not clear. BDM (1994) states (presumably with regardto i having the ‘upper hand’):

the concession is assumed to equal the distance on Ra between xi and xj multiplied by the ratioof the absolute value of j’s expected utility to i’s expected utility. This treats the compromise asthe weighted average of the perceived enforceability of the demand (p. 96)

We might interpret this literally as:

x̂ = (xi − xj

) ∣∣∣∣Ej(Uji)

Ei(Uij)

∣∣∣∣ (35)

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524 Journal of Theoretical Politics 23(4)

Confrontation +j moves to i

Confrontation -i moves to j

i’s Expected Utility

j’s Expected Utility

Compromise +j moves part

way to i

Compromise -i moves part

way to j

Compel -i moves to j

CONFLICTCONFLICT

STALEMATESTALEMATE

COMPROMISECOMPROMISE

COMPROMISECOMPROMISE

CAPITULATECAPITULATE

CAPITULATECAPITULATE

Status Quoi stays put

Compel +j moves to i

( )iji UE

( )jij UE

Figure 8. Classifying the outcome of challenges according to i’s viewpoint.

Note that this relates only to the octant labelled ‘Compromise +’ in Figure 8. Consideringthe boundary conditions in this octant, if

∣∣Ei(Uij)∣∣ >>

∣∣Ej(Uji)∣∣ then x̂ → 0 and actor j

does not move from xj and if∣∣Ei(Uij)

∣∣ → ∣∣Ej(Uji)∣∣ then x̂ → 1 and actor j moves from xj

to xi. For the octant labelled ‘Compromise -’, we use:

x̂ = (xi − xj

) ∣∣∣∣Ei(Uij)

Ej(Uji)

∣∣∣∣ (36)

6.1.3. Capitulate. Actor i has the upper hand if Ei(Uij) > 0 and Ej(Uji) < 0 and∣∣Ei(Uij)∣∣ <

∣∣Ej(Uji)∣∣, compelling actor j to abandon its current position and adopt

actor i’s position. Actor j has the upper hand if Ei(Uij) < 0, Ej(Uji) > 0 and∣∣Ei(Uij)∣∣ >

∣∣Ej(Uji)∣∣, compelling actor i to abandon its current position and adopt actor j’s

position.

6.1.4. Stalemate. The conditions for stalemate are Ei(Uij) < 0 and Ej(Uji) < 0. Neitheractor expects to move from its current position.

6.2. Offer selection

Given that each actor has chosen who to challenge and to remain silent for those not to bechallenged, then each actor will have received ‘challenge offers’ from other actors. Howdoes an actor come to a decision on which challenge offer it should accept? BDM (1997)elucidates:

Each player would like to choose the best offer made to it and each proposer enforces its bid tothe extent that it can. Those better able to enforce their wishes than others can make their pro-posals stick. Given equally enforceable proposals, players move the least that they can. …Whenthe players finish sorting out their choices among proposals, each shifts to the position containedin the proposal it accepted. (pp. 251–252)

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Scholz et al. 525

If all offers are equally enforceable, we would propose to order these according to anactor’s preferred choice as follows, so that actor i moves ‘the least that it can’. Thus, wesummarize the order of decision choice for actor i.

1. Actor i conflicts with actor j and actor j (or with some chance actor i) acquiesces.2. Actor i compromises to actor j. Actor i loses some ground.3. Actor i acquiesces to actor j. Actor i loses most ground.4. Actor i is in stalemate with actor j. Actor i is in status quo.

Thus, for example, if actor i is in conflict with several other actors, each of which hasgreater expected utility than i, then the agent will need to concede to the one that allowsi to move the least.

If all offers are not equally enforceable, then we might expect an actor to be morelikely to concede to the most powerful actor. Thus, in the prior example, actor i concedesto the actor with highest expected utility.

7. ResultsBDM (1994) provides an example. The data for this were introduced in Table 1. BDM(1994) provides three graphs of results. These compare expected utility for Belgiumversus the others, France versus the others and the Netherlands versus the others.

Figures 9 and 10 show the results published in BDM (1994) compared with our inter-pretation of the algorithm as given in Section 6. No value for Q was given in BDM(1994). We chose Q = 1.0.

Note that some countries are not shown on BDM’s graphs. In Figure 7, our expectedutility results for Ireland and Greece were (0,0) and in Figure 8, UK and Italy were at(0,0).

As a result of the fact that BDM does not explicitly plot the point locus of the expectedutilities, we can only reasonably assume the quadrants where the names are labelled cor-respond to the location of each locus. The correspondence of our results to this level ofaccuracy (within a quadrant) is 100%. We note that if the expected utilities were derivedrandomly, the probability of getting any one of these points located in the correct quad-rant is one in four. To get all nine results in the correct quadrants for any one graph of thetwo graphs (Figures 9 and 10) would constitute a probability of (1/4)9 ∼ 4 × 10−6. Wetherefore assert that BDM’s results have effectively been reproduced.

We came across an issue with the result given by BDM for the Netherlands ascompared in Figure 11. We assert that the result published by BDM was in error. Ourresult showing ‘Others’ as Ej

(Uji

)against Ei

(Uji

)for i = Luxembourg is compared with

BDM’s quoted result in Figure 12. This shows 100% correspondence in terms of quadrantaccuracy as for the previous two results.

BDM (1994) summarizes the final result:

The dominant outcome would be, as indicated above, a lag of 8.35 years. However, if the par-ticipants were prepared to bear the costs of slightly prolonged negotiations, then the model’spredicted dominant outcome rises to 9.05 years and stabilizes at that point. …The actualresolution was for a delay of 8.833 years. (p. 98)

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526 Journal of Theoretical Politics 23(4)

Figure 9. Comparison of results for BDM (1994) (a) and our interpretation (b), view fromBelgium.

Figure 10. Comparison of results for BDM (1994) (a) and our interpretation (b), view fromFrance.

Figure 11. Comparison of results for BDM (1994) (a) and our interpretation (b), view from theNetherlands.

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Scholz et al. 527

Figure 12. Comparison of results for BDM (1994) (a) ‘view from Netherlands’ and ourinterpretation (b), view from Luxembourg.

We found the median voter position at the end of the first round to be 8.4 years. At theend of the second, third, fourth and fifth rounds the median voter position was for each9.9 years.

8. DiscussionAs identified earlier, BDM uses Q = 1 or Q = 0.5 in various publications. We chose Q =1.0 to reproduce the previous results. We observed that a value of Q = 0.5 produces verydifferent results from Q = 1. Figure 13 provides an illustrative example.

In Figure 13, the positions of Greece and Belgium change entire quadrants betweenQ = 1 and Q = 0.5.

Recalling that Q relates to the probability of a status quo and is an arbitrary parameter,it is not desirable for results to be so sensitive.

We examined the applicability of the interpretation to other examples from laterpapers. Despite the fact that BDM (1994) and BDM (1997) differ only in the detailedexample used, it is perplexing that we were unable to reproduce the results from the 1997paper. Indeed, attempts to apply this algorithm (using either Q = 0.5 or Q = 1.0) to the1997 ‘Sultan’ problem yielded very different results from those published.

Further insight on the evolution of the median voter position over rounds is alsowarranted in light of claims such as BDM (1994: 98) that ‘the model’s predicted dominantoutcome rises to 9.05 years and stabilizes at that point’. Table 2 illustrates our result. Notethat the zeroth round corresponds to the simple ‘voting model’ (only) account that wouldlikely have been the choice selected by Feder (2002).

Table 2 shows that the median may rise and appear to stabilize early on, but in fact, itcontinues to change. In general the median results do not appear to stabilize. Calculationof the mean voter position provides some insight. The median voter position may flip,but the mean voter positions do not vary much. Given the existence of the ‘compromise’zones (Figure 8), BDM’s own definition allows for continuous valued positions ratherthan the initial specified positions only, so it is surprising that BDM would want to adhereto median rather than the far easier to compute mean valued positions. This observation

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528 Journal of Theoretical Politics 23(4)

Figure 13. Example for the Netherlands comparing results using Q = 1.0 (a) and Q = 0.5 (b).

Table 2. Evolution of the median and mean voter positions for ‘the date of introduction ofemission standards for medium-sized automobiles’ problem in BDM (1994).

Round 0 1 2 3 4 5 6 7 8 9 10Median 7 8.4 9.9 9.9 9.9 9.9 7.4 8.8 9.7 9.7 7.0Mean 7.4 7.4 7.5 7.6 7.3 7.3 7.4 7.5 7.6 7.3 7.2

is further supported in light of Hinich’s (1977) identified limitations of the ‘median voter’model.

9. ConclusionThe algorithm outlined (summarized in the Appendix and freely available from the firstauthor under an open source license agreement) has for the first time exposed and pro-vided independent means of replicating the results of BDM’s computational model. Thisopens BDM’s model, method and claims to scientific critique.

The accuracy of the interpretation was illustrated using the example from BDM(1994). We note the chance of replicating to this level of accuracy by random selectionwould be very small. This fulfils BDM’s own prediction that enough material is availableso that ‘anyone (may) replicate something close to his work’.

Concerns with regard to the model’s sensitivity and convergence have been identifiedas areas of further work.

AppendixThe following summarizes the full procedure.

1. Given i = 1, 2, . . . , n actors, initial positions for each actor xi(t = 0), ci, si andnumber of rounds = t.

2. Let ri = 1.

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Scholz et al. 529

3. Calculate the pairwise votes:

vjk =n∑

i=1

cisi

(|xi − xk| − ∣∣xi − xj

∣∣|xmax − xmin|

)

Then find the maximum value that corresponds to the Condorcet winner position ormedian = μ.

4. Calculate basic utilities:

Uisi = 2 − 4

[0.5 − 0.5

∣∣∣∣ xi − xj

xmax − xmin

∣∣∣∣]ri

Uifi = 2 − 4

[0.5 + 0.5

∣∣∣∣ xi − xj

xmax − xmin

∣∣∣∣]ri

Uibi = 2 − 4

[0.5 − 0.25

(|xi − μ| + ∣∣xi − xj

∣∣)|xmax − xmin|

]ri

Uiwi = 2 − 4

[0.5 + 0.25

(|xi − μ| + ∣∣xi − xj

∣∣)|xmax − xmin|

]ri

Uisq = 2 − 4 (0.5)ri .

5. Calculate probabilities:

Pii =

∑k if arg>0

cksk(∣∣xk − xj

∣∣− |xk − xi|)

n∑k=1

cksk

∣∣(∣∣xk − xj

∣∣− |xk − xi|)∣∣ .

6. Let Q = 0.5 (or 1.0).7. Calculate:

Ei(Uij

) = sj

(Pi

iUisi + (

1 − Pii

)Ui

fi

)+ (

1 − sj

)Ui

si

−QUisq − (1 − Q)

(TUi

bi + (1 − T)Uiwi

)Ej(Uji

) = sj

(Pj

jUjsj +

(1 − Pj

j

)Uj

fj

)+ (

1 − sj

)Uj

sj

−QUjsq − (1 − Q)

(TUj

bj + (1 − T)Ujwj

)If second pass (using the calculated values of ri) then, go to step 11.

8. Calculate:

Ri =2

n∑j=1,j �=i

Ei(Uji) − maxi

⎧⎨⎩

n∑j=1,j �=i

Ei(Uji)

⎫⎬⎭− min

i

⎧⎨⎩

n∑j=1,j �=i

Ei(Uji)

⎫⎬⎭

maxi

⎧⎨⎩

n∑j=1,j �=i

Ei(Uji)

⎫⎬⎭− min

i

⎧⎨⎩

n∑j=1,j �=i

Ei(Uji)

⎫⎬⎭

.

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530 Journal of Theoretical Politics 23(4)

9. Calculate:

ri = 1 − Ri/3

1 + Ri/3.

10. Go to step 4, using calculated values of ri.11. Determine new position decisions x, based on rules in Section 5 for octant of Eij(i)

versus Eji(j).12. Increment the rounds, t = t + 1.13. If t = τ then stop.

Notes

1. Note that Black (1948) assumes that there exists a common ordering of the positions (motions)in which every voter’s preferences are single peaked. To the extent that BDM uses Black’stheorem, this constraint applies to his model.

References

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Political Science Review 74: 917–931.Bueno De Mesquita BB (1981) The War Trap. Yale, CT: Yale University Press.Bueno De Mesquita BB (1984) Forecasting policy decisions: an expected utility approach to post-

Khomeini Iran. American Political Science Review 17: 226–236.Bueno De Mesquita BB (1985) The war trap revisited: a revised expected utility model. American

Political Science Review 79: 156–177.Bueno De Mesquita BB (1994) Political forecasting: an expected utility method. In: Stockman F

(ed.) European Community Decision Making. Yale, CT: Yale University Press, Chapter 4,71–104.

Bueno De Mesquita BB (1997) A decision making model: its structure and form. InternationalInteractions 23: 235–266.

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