UNRAVELLING BUENO DE MESQUITA’S

GROUP DECISION MODEL

Jason B. Scholz, Gregory J. Calbert and Glen A. Smith

C3I Division, Defence Science and Technology Organisation, PO Box 1500, Edinburgh, South Australia

Keywords: Multi-agent theory, Group decision making, Expected utility, Alliance, Risk, Influence.

Abstract: The development of societies of human and machine agents should benefit from an understanding of human

group decision processes. Political Scientist and Professor, Bruce Bueno De Mesquita has made significant

claims for the predictive accuracy of his computational model of group decision making, receiving much

popular press including newspaper articles, books and a television documentary entitled “The New

Nostradamus”. Despite these and many journal and conference publications related to the topic, no clear

elicitation of the model exists in the open literature. We expose and present the model by careful navigation

of the literature and illustrate the soundness of our interpretation by replicating De Mesquita’s own results.

We also discuss concerns regarding model sensitivity and convergence.

1 INTRODUCTION

There is little doubt that some of the greatest social

challenges for the future of mankind include

terrorism, war, climate change, poverty, and

economics. So, the pursuit of an integrated theory

capable of explanation and prediction of group

decision outcomes is a worthy endeavour. Such

efforts, often classed under the realm of

computational political science, aim to form testable

yet tractable models for human agency (Kollman et

al, 2010). Bueno De Mesquita (herein abbreviated to

BDM) has laid claims to such an achievement. An

example prediction was “…the ability to dominate

Iran’s politics resides with Khamenei and

Rafsanjani. And between these two – though the

contest is close – the advantage seems to lie with

Khamenei.” (BDM, 1984, p. 233)

The accuracy of this prediction is demonstrated

by the fact that Khamenei succeeded Khomeinei as

Supreme Leader of Iran in June 1989 and Rafsanjani

became the fourth president of Iran in August 1989.

BDM’s model of group decision making

considers conflict and agreement, and is based on

expected utility theory. BDM (1997) states:

The model itself depicts a game in which

actors simultaneously make proposals, and

exert influence on one another. They evaluate

options and build coalitions by shifting

positions on the 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 into why the model has never been

fully disclosed:

…Bueno de Mesquita does not publish the

actual computer code of his model. (Bueno

de Mesquita cannot do so because his former

firm owns the actual code, but he counters

that he has outlined the math behind his

model in enough academic papers and books

for anyone to replicate something 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 apparent to anyone who tries 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 agent software that reproduces his published

results to an adequate level of accuracy.

2 EVOLUTION OF THE MODEL

BDM’s predictions depend on two parts. First, his

Scholz J., Calbert G. and Smith G..

UNRAVELLING BUENO DE MESQUITA’S GROUP DECISION MODEL .

DOI: 10.5220/0003121500180030

In Proceedings of the 3rd International Conference on Agents and Artiﬁcial Intelligence (ICAART-2011), pages 18-30

ISBN: 978-989-8425-40-9

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

method of data collection and interpretation from

human subjects; second, the computational model

which he applies to that data.

The first part is significant, but has not been

described by BDM in the open literature and so

evades current interpretation or analysis. The second

part, the computational model we examine further.

The 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 x

represents each nation’s

preferred date, measured in years, by which

emission standards should be applied to medium-

sized automobiles as revealed at the outset of

discussions on the issue.” (BDM, 1994, p.77). We

will continue with this example later in the results

section. A number of ‘actors’ (i=1,2,…,n) exist, each

of which hold a single ‘position’ (x

) with regard to

the issue, represented by their assignment to a

location.

Each ‘actor’ is also considered to possess some

‘capability’ (c

) 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 value represents an actor’s level of

influence with regard to the issue.

Lastly, each ‘actor’ is also considered to possess

some ‘salience’ (s

) with respect to the ‘issue’.

‘Salience’ is sometimes interchangeably referred to

as ‘importance’, ‘priority’, ‘attention’ by BDM. Like

‘position’ and ‘capability’, ‘salience’ is given a

value on a metric scale. This value represents an

actor’s level of energy with regard to the issue.

Table 1 from BDM (1994, p. 78) illustrates.

Table 1: Example input data for the computational model.

The issue is ‘the date (years) of introduction of emission

standards for medium-sized automobiles’.

Actor (

) Capabilit

)

Position

)

Salience

)

etherlands 0.08 4 80

Belgiu

0.08 7 40

Luxembour

0.03 4 20

German

0.16 4 80

France 0.16 10 60

Ital

0.16 10 60

UK 0.16 10 90

Irelan

0.05 7 10

Denmar

0.05 4 100

Greece 0.08 7 70

BDM’s model decomposes the social fabric into

pairwise ‘contests’ between actors with support or

otherwise of third-party alliances. Based on actor i’s

perception of expected utility, actor i considers

whether or not to challenge each other actor j, in an

attempt to convince them to adopt i’s position. The

expected utility includes an assessment of the level

of third-party support for actor i’s challenge. If actor

i’s expected utility of challenging actor j versus not

challenging is greater than zero, actor i will

challenge actor j, otherwise it will not. This model of

mind or agency is confrontational and wholly self-

interested.

Not surprisingly, BDM has adapted the model

over the years. So it is necessary to clearly identify

which version we are using when considering its

form and results.

BDM (1980) provides the earliest form, which is

repeated in BDM (1981). The notation is later

revised in BDM (1985), and includes a modification

to include a risk exponent; however the basic

expected utility calculations remain the same from

1980 to 1985. We are readily familiar with the

expected value of a random variable Z, with various

states Z

each with probability P

of occurring as:









ZPzE

Expected utility follows the same structure in

that the utilities of different contest outcomes are

estimated along with the associated probabilities.

An apparent motivation for BDM’s expected

utility model was predicting the outbreak of war as

per BDM (1981). It is thus not surprising to find a

confrontational mentality to the basic form of the

model. BDM considers an actor i to choose to

‘challenge’ a rival or opponent actor j. Thereby the

expected utility for i to challenge j is:





fiisii

UPUPUE  1

Where U

refers to the utility for actor i if it

succeeds and U

is the utility for actor i if it fails.

BDM (1985, p. 158) extends this with a third

term relating to the third-party contribution to i’s

expected utility (using BDM’s notation):

















jik

kijkik

UUPP

UPUPUE

))(1(

(1)

If actor i does not challenge j, i stays at the same

position and 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

UNRAVELLING BUENO DE MESQUITA'S GROUP DECISION MODEL

better for i or worse for i. The expected utility for i

not challenging is then, BDM (1985, p.158):



 

UQUQQ

UQUE





(2)

Where Q

refers to the probability of status quo and

refers to a switch (value either 1 or 0) depending

on whether the outcome was better or worse. These

and other issues will be explained fully later.

The full form of the expected utility difference

combines (1) and (2):

  

UEUEUE 

(3)

In BDM and Lalman (1986) a problem with the

following term in (1) is identified:





(1)()

iii

ij ik jk ki kj

kij

EU P P U U







(4)

The problem is described in BDM and Lalman

(1986):

Because of the manner in which third parties

are treated in earlier studies (Bueno De

Mesquita, 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 values in the expected utility models set

out earlier. This is a serious shortcoming in

that it makes comparison of a single nation’s

utility scores in different years difficult. …

The new formulation fixes 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 incorporating a more complex

form of calculation of the probability P

. In BDM

(1985, p.161) the probability

P refers to “P

= i’s

probability of succeeding in a bilateral contest with

j”.

From 1986 onwards, the definition of the

probability is changed to account for multilateral

contributions to the contest between i and j. The new

form is denoted

P . This will be defined later.

The form of the expected utilities as stated in

BDM and Lalman (1986, p. 1118) are:









UEUEUE 

(5)















sij

UPUPsUE





(6)





 



UTTUQ

QUUE





(7)

The notable change is the inclusion of salience s

and

an extra term in the expected utility for challenge.

Otherwise, notation changes are minimal.

This latter structure remains throughout BDM

(1994, 1997, and 2002). In BDM (2009b) a new

structure of model is announced, however we do not

consider this new model further. As a result of the

multi-lateral scaling issue with the model in its pre-

1986 form, we focus on the model structure and

results for 1986 and later, using the form from

equations 5, 6 and 7.

3 UTILITIES

BDM (1997, p.242-243) uses equations (5), (6) and

(7) though with different notation. 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, p.161), corresponding to a stoic opposition

and a value of Q=0.5 is assumed in both BDM and

Lalman (1986, p.1122) and BDM (2009a, p.5),

corresponding to a maximally uncertain outcome of

whether the actor j will move or stay in position. No

explicit value for Q is specified in other papers.

3.1 Base Utilities

To find the expected utility, we need to calculate the

basic utilities:

UUUUU ,,,,

(8)

These utilities are a function of the policy position of

actors, x

and x

. BDM (1997, p.264) tells us that “…

i’s utility for

, is a decreasing function of

the distance between the proposal and i’s preferred

resolution, so that





Ikk

xxfxu 

. ”

(9)

ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence

The notation

is not the same as that used in (8),

so some transformation is probably required. Thus

we interpret (9) as the general class of model only. It

is worth pointing out that a specific class is stated in

BDM (1997, p.245):

jij

xxxu

1 

(10)

However, (10) is inconsistent with the more detailed

earlier explanations, as the following will now

reveal.

One clue to the utility calculations is given in

BDM (1994):

Should i succeed, then i will derive the utility

associated with convincing j to switch from

its current policy stance to that supported by

i. This is denoted by

dxu





, which equals

)(

xxu 

. Should i fail, then it confronts

the prospect of having to abandon its

objectives in favour of those pursued by j,

denoted by

).(|

xxudxu 



(p. 84)

Once again, BDM introduces additional notation

)(

xxu 

and

)(

xxu 

which remains

undefined. However, BDM (1985, p.158) gives

utility for i’s success which is of the form of a

difference between positions i and j:



















(11)

Also, the utility for i’s failure, which is of the form

of a difference between positions j and i:



















(12)

Noting also in BDM (1985):

The reason for the transformations by 2’s and

4’s is to preserve the original scale of

numbers while avoiding the generation of

imaginary numbers. Because r

can be less

than 1.0, the absence of transformations

would mean that for negative values of, for

instance U

, no real root would exist. This

problem is eliminated with the introduction

of these transformations. (p. 158)

U and

U are defined by BDM (1985):

With

U being equal to the value i attaches

to his own policy portfolio (Both

U and

U are assumed to equal 1.0, with

U and

U ranging between possible values of 1.0

and -1.0), and with

U 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,

1

UUUU

(13)

And to satisfy the stated range requirement we

propose,

minmax





(14)

Equation (14) is consistent with the statement and

equation at (9). Note that -1≤

≤1. Where,

max

−x

min

is the range of positions. Note the

maximum value of +1 occurs when policy positions

of i and j coincide and is at its minimum of -1 when

the positions are maximally separated.

Summarising, so far we have now accounted for

(8) parts a and b, which simplify to:

















minmax

5.05.042

(15)

















minmax

5.05.042

(16)

Note the ranges

2)5.0(42 

and

U )5.0(422 

are consistent with the

diagram in BDM (1985, p.159).

BDM (1985) does not explicitly define

, however, we are given in BDM (1985, p.158):







24 2 4

iiiii

qi ii ij ii ij

tn t

UUUUU













(17)

UNRAVELLING BUENO DE MESQUITA'S GROUP DECISION MODEL

The subscripts t0 and tn are not defined, but in

BDM (1981, p.48) these correspond to before and

after j’s policy change, respectively. The utility

subscript q is usually signifies status quo, but we

believe this is an error in (17) and should instead

refer to j making a policy change (or move) which

gains or betters the situation for i. We are led to

believe this by BDM (1981, p.47-48):





UU 

=i’s perception of what may be

gained by succeeding in a bilateral conflict

with j…





UU 

= i’s perception of what

may be lost by failing in a bilateral conflict

with j…

So we adopt the subscript, b indicating ‘better’

consistent with (8). Furthermore, we note a problem

with scaling for (17), as it will become undefined

(negative number raised to a power less than 1.0).

Thus, corrections are also required to realign scaling.

The result is as follows:



UUUU

















(18)

Similarly we expect that j’s movement may

potentially result in a worse condition for i:



UUUU

















(19)

Note this adjusted scaling ensures

2)5.0(42 

and

U )5.0(422 

We are given a clue that some relation to the

median voter position is important by

“

]||[ dxuanddxu

i 



are approximated by

comparing the value actor i attaches to the current

median voter prediction to the value i attaches to the

median anticipated if i accepts j’s preferred

outcome.” (BDM 1997, p. 248)

BDM (2009a) expresses this most clearly:

… they are anticipated to move towards the

median voter position if they make an

uncoerced move. This means that if B lies on

the opposite side of the median voter from A,

then A anticipates that if B moves

(probability=0.5), then B will move in such a

way as to come closer to the policy outcome

A supports and so A’s welfare will improve

without A having to exert any effort. If B lies

between the median voter position and A,

then whether A’s welfare improves or

worsens depends on how far B is expected to

move compared to A. The same is true if A

lies between B and the median. (p. 6)

We interpret this to mean that for no challenge

(uncoerced), and B (or j) moves, that A (or i)

expects B (or j) will move to the median position.

The cases are illustrated in figures 1 to 4.



Figure 1: Case 1:



between i and j



utility for i gets

better as a result of

j moving.



Figure 2: Case 2:

j between i and



utility for i gets worse

as a result of

j moving.



Figure 3: Case 3A:

i between j and



utility for i gets

better as a result of

j moving.



Figure 4: Case 3B:

i between j and



utility for i gets

worse as a result of

j moving.

For any of these cases then we expect,





minmax





(20)





minmax







(21)

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





jii

xxx



















minmax

25.05.042



(22)

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

ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence



jii

xxx



















minmax

25.05.042



(23)

We note further, that these utilities must be applied

in the appropriate cases. Thus if case 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 in

equation (7), though no known publication by the

author states this explicitly. So, cases 1 and 3A

correspond to T=1 and cases 2 and 3B correspond to

T=0. This helps explain the description in BDM

(1986, p. 1122).

Lastly, for the situation of no change in policy

(status quo) i does not challenge and j does not

move, BDM (1985, p.158) defines:



U 5.042

)02(

42 

















(24)

We expect this corresponds to (8) part e.

To determine the calculations for actor j, BDM

(1985) notes:

Of course, the

UU ,

and

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 equations with a j

superscript, j’s risk taking propensity is used

to estimate what j perceives to be the value of

success, failure, or no challenge for i in

accordance with the equations delineated

below. (p. 158)

Thus, the two main equations become:



(1 ) (1 )

iiiii i

ij j i si i fi j si

iii

qbiwi

EU s PU PU sU

QU Q TU T U



 

(25a)



(1 ) (1 )

jjjjjj

ji j j sj j fj j sj

jjj

sq bj wj

UsPU PU sU

QU Q TU T U



 

(25b)

3.2 The Median Voter Position

In the previous section, it became clear that the

“median voter position” must be determined in order

to calculate the utility terms in (22) and (23).

BDM defines comparative votes in direct propor-

tion to utility difference, capability and salience. The

votes ‘cast’ by agent i in comparing positions

and

is given as BDM (1997, p.239):





iikj

xuxuscxxv ,|

(26)

We emphasise, that these votes cast to k may

indeed be negative, if for example, agent i prefers j

.k To map this notation to that used previously,

we interpret:

ijj

Uxu 

ikk

Uxu 

(27)

Using (14) we get:



















minmax

2,|

xxxx

scxxv

jiki

iikj

(28)

According to BDM (1997):

The prospect that a proposal will succeed is

assumed to depend on how much support can

be mustered in its favour as compared with

the feasible alternatives. This is calculated as

the sum of “votes” across all actors in

comparison between x

and x

. (p. 240)





(29)

In general, this pairwise determination is termed a

Condorcet Method of voting. A Condorcet winner is

the candidate whom voters prefer to each other

candidate, when compared to them one at a time.

Black’s Median Voter theory now comes into

play, so “the decision adopted by the committee

becomes determinant as soon as the position of the

one optimum – which we can refer to conveniently

enough as the median optimum – is given.” (Black,

1948).

That is, in a majority election where a voter’s

attitude is represented as a point in one dimension, if

all voters vote for a candidate closest to their own

preference and there are only two candidates, then if

the candidates want to maximise their votes they

should commit 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 may be determined.

UNRAVELLING BUENO DE MESQUITA'S GROUP DECISION MODEL

4 ALLIANCE PROBABILITY

The probabilities of equations (25) are determined

by the 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 normalising. BDM (1997, p.244) states the

estimator as:







xuxuk

vvP

(30)

When more agents are ‘for’ than ‘against’, this raises

the probability of winning the bilateral contest. As

per previous derivation of votes, substitute and

expand (31):











ikjkkk

ifk

ikjkkk

xxxxsc

0arg

)(

(31)

5 RISK PROPENSITY

As seen in the previous section, utility calculations

involve a risk exponent. This risk exponent is in turn

derived from the expected utility. BDM (1985,

p.157) is first to describe the basis for risk

calculation,

I define each nation’s security level as



ij

UE )( . The greater the sum, the more

utility i believes its adversaries expect to

derive from challenging i. … as this sum

decreases, i’s relative security increases, so

that i is assumed to have adopted safe

policies ...

BDM (1985, p.157) goes on to define:

max min

2() () ()

() ()

ji ji ji

ji ji

EU EU EU

EU EU







 



(32)

Note that BDM (1997, p.247) reverses the subscripts

of the above, which is inconsistent with his

conceptual basis of security. Further, BDM

(1997,p.247) provides an inconsistent transformation

formula which would not accommodate the range

11 

. Thus, we choose the earlier conversi-

on formula from BDM (1985, p.157):

3/1







(33)

The purpose of the formula according to BDM

(1985, p.157) is to ensure that r

ranges between 0.5

and 2, noting that the divisor of 3 appears arbitrary,

but effects curvature. Equation (33) is illustrated in

figure 5.

0.5

1.5

2

‐1

‐0.6

‐0.2 0

0.2

0.6



Figure 5: Scaling conversion formula.

The following equation (34), expresses (32)

more precisely:

1, 1, 1,

1, 1,

2()max ()min ()

max ( ) min ( )

nn n

ii i

ji ji ji

jji jji jji

ji ji

jji jji

EU EU EU

EU EU

  

 









 



 









 



(34)

We still need to know, however, how to calculate the

expected utilities in (25), which use a modified

notation. BDM (1985) describes:

Thus, the risk terms are calculated by

manipulating the alliance portfolios used as

the policy indicator through simulation to

locate the best and worse portfolios for any

given nation, where the best and worst are

defined in terms of the sum of expected

utilities of all others vis-à-vis the nation in

question under the assumption that utilities

are strictly a function of similarities 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 the worst and best case alliance

strategy for each nation each year, using the

original, linear utility functions to define the

ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence

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 utility functions.)

(p. 167-168)

This implies a process to first determine the

expected utilities of equations (25) using r

=1, then

apply (34) and (33) to estimate r

and lastly apply the

estimates to re-estimate the expected utilities of

equations (25).

6 DECISION

6.1 Offer Categories

The expected utilities





and



are used

to classify the ‘offers’ between all actor pairs into

categories according to potential outcomes as

illustrated in figure 6. An actor may expect to

conflict, compromise, capitulate, or stalemate with

another. Unfortunately no single publication by

BDM explains how to quantify these.

Confrontation +

j moves to i

Confrontation -

i moves to j

i’s Expected

Utility

j’

xpected Utility

Compromise +

j moves part

way to i

Compromise -

i moves part

way to j

Compel -

i moves to j

CONFLICT

STALEMATE

COMPROMISE

CAPITULATE

Status Quo

i stays put

Compel +

j moves to i









Figure 6: Classifying the outcome of challenges according

i’s viewpoint.

Conflict. Actors i and j conflict if

0)( 

and

0)( 

. So “If both i and j believe that they

have the upper hand in the relationship, then conflict

is likely and that conflict has an uncertain outcome.”

BDM (1997, p.244)

BDM (1984, p. 230) labels for the

“Confrontation-” octant, “Challenger Favored” and

for the “Confrontation+” octant “Favoring Focal

Group”. We interpret this to mean i moves to j and j

moves to i respectively, as shown in figure 6.

Compromise. Actor i has the upper hand if

0)( 

0)( 

and

)()(

UEUE 

Actor j has the upper hand if

0)( 

and

)()(

UEUE 

BDM (1997) describes:

… both players agree that i has the upper

hand. In this instance, j is expected to be

willing to offer concessions 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’s expectation and j’s.

(p. 243-244)

However, the weighted average is not clear. BDM

(1994) states (presumably with regard to i having the

‘upper hand’):

… the concession is assumed to equal the

distance on R

between x

and x

multiplied by

the ratio of the absolute value of j’s expected

utility to i’s expected utility. This treats the

compromise as the weighted average of the

perceived enforceability of the demand… (p.

96)

We might interpret this literally as:



)(

xxx 

(35)

Noting this relates only to the octant labeled

“Compromise +” in figure 6. Considering the

boundary conditions in this octant, if

)()(

UEUE 

then

x

and actor j does not

move from x

and if

)()(

UEUE 

then

x

and actor j moves from x

to x

. For the octant

labelled “Compromise -” we use:



)(

xxx 

(36)

Acquiescence and Stalemate. The states for

acquiescence and stalemate are illustrated in figure 6

and require no further explanation.

6.2 Offer Selection

Given that each actor has chosen who to challenge

and to remain silent for those not to be challenged,

UNRAVELLING BUENO DE MESQUITA'S GROUP DECISION MODEL

then each actor will have received ‘challenge offers’

from other actors. How does an actor come to a

decision on it 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 to the extent that it can. Those better

able to enforce their wishes than others can

make their proposals stick. Given equally

enforceable proposals, players move the least

that they can. … When the players finish

sorting out their choices among proposals,

each shifts to the position contained in the

proposal it accepted. (p. 251-252)

If all offers are equally enforceable, we would

propose to order these according to an actor’s

preferred choice as follows, so that actor i moves

‘the least that it can’. Thus, summarising the order

of decision choice for actor i is as follows:

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 stalemate

with actor j. Actor i status quo.

Thus, for example if actor i is in conflict with

several other actors, each of which have greater

expected utility than i, then the agent will need to

concede to the one that allows i to move the least.

If all offers are not equally enforceable, then we

might expect an actor to be more likely to concede

to the most powerful actor. Thus, in the prior

example, actor i concedes to the actor with highest

expected utility.

7 RESULTS

BDM (1994) provides an example. The data for this

was introduced in table 1. BDM (1994) provides

three graphs of results. These compare expected

utility for Belgium versus the others, France versus

the others and the Netherlands versus the others.

Figures 7 and 8 compare the result using our

interpretation of the algorithm as given in section 6,

compared directly with the results published in

BDM (1994, p.91). No value for Q was given. We

chose Q=1.0.

Figure 7: Comparison of results for BDM (1994) (top) and

our interpretation (bottom), view from Belgium.

Note that some countries are not shown on

BDM’s graphs. In figure 7, our expected utility

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 expected

utilities, we can only reasonably assume the

quadrants where the names are labelled corresponds

to the location of each locus. The correspondence of

our results to this level of accuracy (within a

quadrant) is 100%. We note that if the expected

utilities were derived randomly, the probability of

getting any one of these points located in the correct

quadrant is one in four. In order to get all nine

results in the correct quadrants for any one graph of

the two graphs above would constitute a probability

of (1/4)

~ 4x 10

-6

. We therefore assert that BDM’s

results have effectively been reproduced.

ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence

Figure 8: Comparison of results for BDM (1994) (top) and

our interpretation (bottom), view from France.

We came across an issue with trying to

reproduce the result given by BDM for the

Netherlands as compared in figure 9. We assert that

the result published by BDM was in error. Our result

showing ‘Others’ as



against





for

i=Luxembourg is compared with BDM’s quoted

result in figure 10. This shows 100%

correspondence in terms of quadrant accuracy as for

the previous two results.

BDM (1994) summarises the final result:

The dominant outcome would be, as

indicated above, a lag of 8.35 years.

However, if the participants were prepared to

bear the costs of slightly prolonged

negotiations, then the model’s predicted

dominant outcome rises to 9.05 years and

stabilizes at that point. … The actual

resolution was for a delay of 8.833 years.

(p.98)

We found the median voter position at the end of

the first round to be 8.4 years. At the end of the

second, third, fourth and fifth rounds the median

voter position was for each 9.9 years.

Figure 9: Comparison of results for BDM (1994) (top) and

our interpretation (below), view from Netherlands.

8 DISCUSSION

As identified earlier, we chose Q=1.0 to reproduce

the results above. We observed that a value of Q=0.5

produces very different results. The following figure

11 provides an illustrative example.

In figure 11, the positions of Greece and

Belgium change entire quadrants if Q=1 or Q=0.5 is

chosen.

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 later papers.

Despite the fact that BDM (1994) and BDM (1997)

differ only in the detailed example used, it is

UNRAVELLING BUENO DE MESQUITA'S GROUP DECISION MODEL

perplexing that we were unable to reproduce the

results from the 1997 paper. Indeed attempts to

apply this algorithm (using either Q=0.5 or Q=1.0)

to the 1997 “Sultan” problem yielded wildly

different results to those published.

Figure 10: Comparison of results for BDM (1994) ‘view

from Netherlands’ and our interpretation, view from

Luxembourg.

Further insight on the evolution of the median

voter position over rounds is also warranted. This is

shown in table 2.

Table 2 shows that the median appears to

stabilise, but then continues to change. Calculation

of the mean voter position provides insight. The

coarseness of the median voter position becomes

evident. Indeed given the fact that the “compromise”

state allows for intermediate valued positions (as per

equation 36) it is surprising that BDM would want to

continue with median over mean values. In general

the results do not stabilise. There is no reason from

examining the algorithm to expect that they should.

Figure 11: Example for Netherlands comparing results

using

Q=1.0 (top) and Q=0.5 (bottom).

Table 2: Evolution of the median and mean voter positions

for ‘the date of introduction of emission standards for

medium-sized automobiles’ problem in BDM (1994).

Round 1 2 3 4 5 6 7 8

Median 8.4 9.9 9.9 9.9 9.9 7.4 8.8 9.6

Mean 7.4 7.5 7.6 7.3 7.3 7.4 7.5 7.6

9 CONCLUSIONS

The algorithm outlined (and summarised in the

Appendix) has for the first time exposed and

provided independent means of replicating the

results of BDM’s computational model. This opens

BDM’s model, method and claims to scientific

discussion.

The correctness of the interpretation was

illustrated using the example from BDM (1994). We

note the chance of replicating to this level of

accuracy by random selection would be much less

ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence

than one in one million. This fulfils Bueno De

Mesquita’s own prediction that enough material is

available so that “anyone (may) replicate something

close to his work”!

Concerns with regard to the model’s sensitivity

and convergence have been identified.

Given these concerns, we conclude that adoption

of BDM’s model for agent development would be

premature at this time.

REFERENCES

Black, D. (1948). On the Rationale of Group Decision-

making.

Political Economy 56, No 1, Feb: 23-34.

Congleton, R. D. (2003). The Median Voter Model.

The

Encyclopedia of Public Choice.

Kluwer Academic

Press: 382-386.

Bueno De Mesquita, B. B. (1980). An Expected Utility

Theory of International Conflict.

The American

Political Science Review

, Vol. 74, No. 4 (Dec), pp.

917-931.

Bueno De Mesquita, B. B. (1981). The War Trap. Yale,

CT: Yale University Press.

Bueno De Mesquita, B. B. (1984). Forecasting Policy

Decisions: An Expected Utility Approach to Post-

Khomeini Iran, The American Political Science

Review

, Vol. 17, No. 2, pp. 226-236.

Bueno De Mesquita, B. B. (1985). The War Trap

Revisited: A Revised Expected Utility Model.

American Political Science Review 79, No 1, 156-177.

Bueno De Mesquita, B.B. and Lalman, D. (1986). Reason

and War, The American Political Science Review, Vol.

80, No. 4, (Dec), pp. 1113-1129.

Bueno De Mesquita, B. B. (1994). Political Forecasting:

An Expected Utility Method,

European Community

Decision Making

, Yale University Press, Stockman, F.

(ed.), Chapter 4, pp 71-104.

Bueno De Mesquita, B. B. (1997). A Decision Making

Model: Its Structure and Form.

International

Interactions

23: No 3-4 235-266.

Bueno De Mesquita, B. B. (2002).

Predicting Politics,

Ohio State University Press.

Bueno De Mesquita, B. B. (2009a).

A New Model for

Predicting Policy Choices: Preliminary Tests

, 50

Meeting of the International Studies Association, New

York, Feb 15-18.

Bueno De Mesquita, B. B. (2009b).

The Predictioneer’s

Game

ed.). New York: Random House.

Kollman, K, Miller, J. H. and Page, S.E. eds (2010)

Computational models in political economy. MIT

Press.

APPENDIX

The following summarises the full procedure.

1. Given i=1,2,…,n actors, initial positions for each actor

(t=0), c

, s

and number of rounds=



2. Let r

3. Calculate the pairwise votes:























jiki

xxxx

scv

minmax

Then find the maximum value which corresponds to

the Condorcet winner position or median =



4. Calculate basic utilities,

















minmax

5.05.042

















minmax

5.05.042





jii

xxx



















minmax

25.05.042







jii

xxx



















minmax

25.05.042





U 5.042 

5. Calculate probabilities:











ikjkkk

ifk

ikjkkk

xxxxsc

0arg

)(

6. Let Q=0.5 (or 1.0).

7. Calculate:















sij

ijij

UTTUQQU

UsUPUPsUE

)1()1(



















jji

UTTUQQU

UsUPUPsUE

)1()1(





If second pass (used the calculated values of r

) then,

go to step 11.

8. Calculate:

































































































ijj

UEUE

UEUEUE

,1,1

,1,1,1

)(min)(max

)(min)(max)(2

UNRAVELLING BUENO DE MESQUITA'S GROUP DECISION MODEL

Calculate:

3/1







10. Go to step 4, using calculated values of r

11. Determine new position decisions

x, based on rules in

section 5 for octant of

(i) vs E

(j).

12. Increment the rounds, t=t+1

13. If t=



then stop.

ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence