Public Choice 45:49-71 (1985). © 1985 Martinus Nijhoff Publishers, Dordrecht. Printed in the Netherlands.

'Putting one over on the boss': The political economy of strategic behavior in organizations

T H O M A S H. H A M M O N D J E F F R E Y H. H O R N *

1. Introduction The chief executive of an organization is often dependent on his subordinates for formulating the set of options he will consider. He may also be dependent on them for advice as to which option to choose. Due to the limits on his time, energy, and attention, he cannot always monitor the subordinates as they formulate the options and settle on their recommendations. He must, to some extent, accept their recommendations as reasonable estimates of what is good organizational policy. But what the subordinates think is good policy often diverges from what the boss thinks is good policy. What is good for the subordinates may also diverge from what is good for their boss. The fact that their views and interests diverge from his gives them a motive to manipulate his choices to their own ends. The fact that he has difficulty monitoring their behavior makes it possible for them to manipulate his choices. If the subordinates want to manipulate his choices and can do so 'without being found out,' it is not unreasonable to expect them to do so: they will tell him what they want him to hear and not what his interests and concerns require him to hear. That choices can be manipulated is a generic feature of all social institutions. As Gibbard (1973) and Satterthwaite 0975) have formally demonstrated, for every nondictatorial way of making social choices there will exist situations in which someone can improve outcomes for himself by voting for - in our terms, recommending to the boss - something other than what he truly most prefers, t Hence the problem of hierarchical distor* Support was provided by National Science Foundation grant SES-8207904. We thank Jane Fraser, Paul Thomas, Mathew McCubbins, Rick Wilson, Jon Bendor, and Elizabeth Hoffman for assistance and advice. We remain responsible for our conclusions and interpretations. Department of Political Science, Michigan State University, East Lansing, MI 48824, and Department of Mathematics, Purdue University, West Lafayette, IN 47907.

50 tion of information and advice in bureaucracies (Tullock, 1965: Ch. 14) appears to be unavoidable. However, it is not well understood precisely what strategy of manipulation subordinates should use to achieve their goals. Nor is it known how the manipulative strategies of various subordinates interact with each other. Moreover, while every nondictatorial way of making organizational decisions may be manipulable, it is not known to what extent subordinates may be able to achieve their goals. Each subordinate is usually only one of many and it seems reasonable to think that if disagreement among subordinates is relatively great, the boss's choices will be left unmanipulated. To investigate these problems we have developed a formal model of strategic behavior in organizations. We examine what strategies a subordinate should use, discuss how subordinates' strategies interact, and determine how much a subordinate can benefit from acting strategically.

2. A model of 'Management by Exception' policymaking in a hierarchy The organization we examine will be a simple tree-like hierarchy with only two levels: a boss and several subordinates. 'Subordinate i' will be our object of attention. We assume that each subordinate has a complete and transitive preference ordering over the options which are available; we allow no indifference between any options. Assume subordinate i prefers option a to any of the others that are available. We will speak of his acting 'sincerely' if he always recommends o~ and speak of his acting 'strategically' ('sophisticatedly') if he considers recommending something other than o~. If acting strategically improves outcomes in his estimation, we assume he will do so; if his choice seems likely to have no impact on the outcome, he will recommend his most preferred option. We assume the boss has a general predisposition in favor of some possible policies rather than others. If asked what option he favors, he might respond, 'Just off the top of my head, I prefer policies of type ~ to policies of type o~, everything else being equal.' Since 'everything else' is seldom equal, it would take time and energy for him to turn his general predispositions into a fully worked-out policy choice. Even telling his subordinates precisely how to go about making a decision which meets his standards would take time and energy. In addition, if the boss has to spend time and energy on this particular problem, he incurs the opportunity costs of ignoring other pressing problems (which we are not considering here). Due to these other demands on his time and energy, then, we assume the boss is not always able to impose his general predispositions on the organization.

51 In fact, even if the boss consciously disagrees with his subordinates' recommendation, he will sometimes do what they want anyway. If the subordinates possess technical talents that are difficult to replace, or if there is no one else in whom the boss places as much trust, the boss may be wary of continually overruling their advice. Otherwise, his aides might conclude that being close to the seat of power is of little value and leave for greener pastures. Thus the boss may defer, even against his own judgments, to the views and desires of his subordinates. As Appleby (1945: 123) once observed. No department head can hold principal executives responsible without going along with them substantially most of the time. Watching their product, he may, after an accumulation of dissatisfaction, displace them, if he can find men he thinks abler - but he must then uphold the new men substantially most of the time.

We assume next that each possible preference configuration - that is, a particular set of preference orderings - for the organization as a whole is as likely to occur as any other configuration. That is, we will be studying the workings of an organization with an 'impartial culture' (Garman and Kamien, 1968). We also make some assumptions about how much information is available to the subordinates about each actor's preference ordering. Two polar cases will be distinguished. In one case we assume that no information about others' preferences is available. Strategic behavior by the subordinates will be impossible here because it requires knowing how the others are likely to behave under various circumstances. In the other case we assume that complete information is available to the subordinates: they know all other subordinates' orderings, plus what the boss is likely to do were he to get involved. Policymaking in the organization proceeds in the following fashion. Step One. The boss assigns to the subordinates responsibility for recommending to him some course of action on a problem the organization faces. Each subordinate is asked to report what he thinks is the best response to the problem. Step Two. A set of feasible options is generated, though how this set is generated lies outside our mOdel. One may assume that the set is imposed from outside the organization, by 'the market' or by 'technological feasibility' for a business firm or by 'the legislature' or 'public opinion' for a public agency. Whatever the origins of the set, any recommendations by the subordinates must come from the set, and the boss's final choice will be from the set also. Step Three. The subordinates consider the set o f options together with what they think the boss would choose if he had to make the decision. Each subordinate then recommends one option to the boss. (Under some circum-

52 stances, to be detailed later, subordinates will find it advantageous to 'abstain' - to recommend no option at all.) We will examine cases in which everyone is sincere in their recommendations, in which everyone is sophisticated, and in which some are sincere and some are sophisticated. In no instance will we allow subordinates to engage in bargaining or to form binding agreements with each other. Step Four. The subordinates' recommendations are considered by the boss and he makes the final decision. The boss is assumed to use what we call a 'Management By Exception' ('MBE') decision rule. If at least 'k' of the 'm' subordinates agree on one of the 'n' options, the boss will accept their recommendation; if fewer than k subordinates agree on any one option, the boss will invest his own time and energy to make the decision himm rn self. We assume throughout that k > ~-. (If k ~< ~ , two or more k-sized blocs of subordinates could end up supporting different options and the boss would still have to get involved. Setting the minimum k at a bare majority of m avoids this problem.) Our use of this MBE rule assumes that subordinates are of equal importance: no one is more persuasive or important than his fellows. In real world organizations a boss using an MBE-like rule would probably place differing degrees o f trust in the recommendations of the various subordinates; subordinates would thus have different 'weights' in the MBE decision-making process. This would unnecessarily complicate matters here. In our model, though, if the boss is inclined in general to distrust his subordinates' judgments, he can set k at a relatively high level (perhaps k = m) and thus accept their joint recommendation only when it is completely unanimous; if he is inclined to trust their judgments, he might reasonably set k at a much lower level. In the analysis which follows we examine several general questions about strategic behavior in this organization. Section 3 asks how often the boss can be expected to pick subordinate i's most preferred option when the subordinates all act sincerely. Section 4 examines i's incentives to engage in strategic behavior when he is sophisticated but everyone else is sincere. Section 5 asks how often the boss can be expected to pick subordinate i's most preferred option when all the subordinates act strategically. Section 6 analyzes the situation subordinate i faces if he acts in a sincere manner but everyone else is sophisticated; will he be forced into strategic behavior just for purely defensive reasons? Section 7 discusses the implications of our results.

53 3. A baseline

standard

of sincere choice

in o r g a n i z a t i o n s

We first determine what happens when all subordinates make decisions in a sincere manner. Only if we know the sincere results can we judge how much difference sophistication makes. In this section, then, each subordinate always recommends his most preferred option to the boss. For any one preference configuration it is straightforward to tabulate what the organization's final choice will be. Assume the boss sets k at some level. If k or more subordinates recommend the same option, the boss accepts their collective judgment and that option becomes the organizational choice. If fewer than k subordinates agree on one option, the boss steps in and makes the decision. Our ultimate goal, however, is to understand the general nature of the outcomes when subordinates act strategically and not just what happens in any one particular situation. That is, we want to know the probability that subordinate i will get each of the options if everyone chooses sincerely, given a wide range of preference configurations. Label the options t~ . . . . . aj, . . . , otn and assume that i most prefers option o~x. We will calculate, for all possible preference configurations, what proportion results either in the boss choosing oq on his own, when no bloc of k or more subordinates forms in support of c~, or in the boss accepting the recommendation ~ of some bloc of k or more subordinates (which includes i). This problem is an exercise in combinatorics. For a two level hierarchy the probability that the boss chooses o~, p~(a~), is expressed in Equation (1):

p~,(O~l)

- • = -nm

m-I ~

-

'lm; l

~=~

( m ; 1 ) ( n - l ) m-~-r ) + n .

r=k-I

m-1 E

(n - l ) ( n

-

1) m-l-r

-

( m ; 1 ) ( n - l ) m-r 1 (1)

r=k-I

For the derivation of (1) see Hammond and Horn (1983). If unanimity among subordinates is required (i.e., k=rn), Eq. (1) reduces to Eq. (2): pb(Oq) =

//,n-~ _ 1 + r /

~/m

when k = m.

(2)

In this case, subordinate i can expect the boss to choose ~1 a proportion n m - ~ -- 1 + r t

F/m

of the time. In a completely degenerate tree in which i is the

only subordinate (i.e., rn = 1), Eq. (2) further reduces to

54 n 1-~ - l + n

n = - = 1, which means that i gets his way all o f the time.

//t

/l

In this case, despite i's position as a ' s u b o r d i n a t e , ' he and not his nominal ' s u p e r i o r ' dictates the organization's choices. Table 1 displays the values o f p b ( ~ l ) for various values o f n, m, and k. As one can see, i's expected success rate varies dramatically, depending on the n u m b e r o f options, on the n u m b e r o f other subordinates, and on the value o f k. Eq. (1) has three n o t e w o r t h y features which are reflected in Table 1. First, for a fixed k and a fixed m > 1, as n --, o~, p0(c~) -+ 0. Second, for a fixed k and n, as m ~ ~ , p o ( ~ l ) ~ ~1. Third, these limits are often a p p r o a c h e d rather rapidly. For example, when k = rn and n = 6, the -1 limit is reached when m = 5. t/

Table I. Probability that Subordinate i Gets His First Choice

When Everybody Is Sincere Two Level Tree Bare Majority Needed for Bkx" to Form N u m b e r of Proposals = n 2

N u m b e r of Subordinates (Including i) =m

3 4 5 6 7 8 9 i0

.750 .688 .688 .656 .656 ,637 .637 .623

3 .630 .4al .531 .443 .480 .419 .447 .402

4 .531 .355 .408 .316 .349 .293 .315 .279

5 .456 .277 .323 .241 .266 ,223 .237 .213

6 .398 .225 .263 .193 .211 .180 .188 .~73

7 .353 .188 .220 .161 .174 .151 .156 .146

8 .316 .161 .188 .138 .148 .130 .134 .127

9

I0

,287 .140 .163 .121 .128 .114 .117 .112

.262 .124 .144 .107 .113 .i02 .104 .i01

Complete Unanimity Needed for Bloc to Form N u m b e r of Proposals = n 2

N u m b e r of Subordinates (Including i) = m

2 3 4 5 6 7 8 9 I0

,750 625 563 531 516 508 504 502 .501

3 •556 .407 .358 .342 •336 •334 •334 •333 ,~

4 .438 .297 •262 .253 .251 •250 . . .... ~,

5 .360 .232 .206 .201 .200 " . . ~,

6 .306 .190 .171 •167 " " . . . " ,~

7

8

9

.265 .234 .210 .160 .139 .122 •145 .127 .112 .143 .125 .iii " " " " " " . . . . . . . '. . . . . ~!

Note: On this and Table 2 a ditto mark (") in a column means that the preceding value of the column is repeated.

i,

i~

i0 .190 .i09 .I01 .i00 " " " i~

55 For a complete understanding of the impact of sophisticated choice on an organization's behavior we also need to know how often each option is the outcome. Eq, (1) tells us how often we can expect option t~l, which subordinate i most prefers, to be the organizational outcome when everyone acts sincerely. If or1 is chosen pb(o~l) of the time, the remaining options are chosen 1-pb(oq) of the time. Since each of the n - 1 other options is equally likely to be chosen when o~1 is not chosen, dividing 1-pb(c~) by n - 1 yields the probability that each of the other options is chosen; that is, po(c~j) -

1-p~(c~a) for j # 1. rt-1

(3)

4. When only i is sophisticated Our next task is to determine what happens when i is sophisticated and he knows that everyone else will be sincere. Can he take advantage of them? By how much can he improve outcomes for himsdf? For the subordinate to impose an option on their boss, a bloc of k or more subordinates must agree on some option. Only when a bloc of exactly k - 1 subordinates supports some option other than e~ can i's sophistication make a difference in the outcome. If i joins a k - 1 bloc, the bloc's preference becomes the organization's preference; if i does not join such a bloc, the superior is able to impose his own judgment. A decision by i to join the k - 1 bloc thus becomes one of deciding whether or not the bloc's preferred choice is better than what the superior would impose. Table 2 shows the proportion of all preference configurations in which i can profit from a sophisticated choice. The key question is: how often can i expect to get his way, given that he acts strategically when it can make a difference in the outcome? It is essential to note that if i's first preference is not the sincere outcome, i will never be able to act in such a way that his first preference becomes the outcome. Sophisticated behavior, though, can sometimes ensure that one of his lower ranking options is chosen, thereby averting the even less preferred option that the boss would choose. Thus sophisticated choice here will affect only the frequencies that i's less preferred options will be the outcome. Eq. (1) and Table 1, in other words, tell us how often i's first choice will be the outcome when i alone is sophisticated as well as when everyone is sincere. How often will the remaining options be chosen when i alone is sophisticated? Let us call i's most preferred option ~1, his second most preferred option c~2, on down to o~n, his least preferred option. Set j - - 2 , 3, 4 . . . . . n. Sophisticated choice by i will cause c~j to be the decision when-

56

Table 2. Proportion of All Preference Configurations in Which Subordinate i Will Switch His Vote When All Other Subordinates Are Sincere Two Level Tree Nunlbt,r of | ) r t ) p . ~ a l ~ : Ii Bare Majority Needed for Bloc to Form n= 3

Number of Stnb~rdinates (Includit~g i) : 111

3 4 5 6 7 8 9 I0

.074 .074 .074 .~55 .064 .OA3 .053 .034

4

5

6

7

8

9

I0

.156 .I05 .135 .066 .094 .043 .064 .029

.224 .115 .165 .061 .09b .034 .055 .020

.278 .I16 .177 .054 .088 .026 .043 .013

.321 .i12 .180 .046 .078 .02U .034 .009

.355 .i08 .178 .039 .06B .015 * .006

.384 ,192 .174 ,034 .060 ,012 * .004

.408 .097 .168 .029 .052 .009 * .003

8

9

I0

.328 .041 .005

.346 .038 .004

.360 .03o .004

Complete Un mi ~fity Needed fi~r Bloc t(~ Form n=

N u m b e r of Subordinates t Including i l = m

3

4

5

6

7

2 3 4

.III .037 .012

.188 .047 .012

.240 .04b .010

.27~ .046 .O08

.3UO .044 .OUO

5

.004

.003

.002

.001

.001

.OUl

.000

.000

6

.001

.001

.000

.OUO

.OUO

.000

.000

.OUO

7

.000

.000

"

"

"

"

"

~

~

~

~,

~

~

b

~

9

1'

"

"

H

~

H

Ii

s,

"

"

"

"

"

"

"

"

10

~

"

'*m = . d d e n t r i e s in th'e table were calculaled through the complete enumeration; these particular values would take ~everal days to calculate. It is apparent that the numbers are quite small.

ever a k - 1 bloc exists in support of us and the boss favors any option ah with h > j. There are n - j values which h can take on, so each option with j = 2, 3, 4, . . . , n becomes the decision in I kn- I: ) ( n - j ) ( n - 1)- -k more cases than when i votes sincerely. However, in each of these cases one of the oq+l, oq+2, oq+3, . . . , an options would have been the sincere choice. Thus each j = 2, 3, 4 . . . . .

n option becomes the outcome in

k-1

( j - 2 ) ( n . 1)'~-k fewer cases. When i is sophisticated in a two level tree, the net gain for option oq is: (~SO

(n-2j+2) /./m

( n - l ) '~-k

(4)

Eq. (4) will produce negative values for roughly half the options under con-

57 sideration. Given the options which i ranks in decreasing order of n preference, when n is even, the ~ more preferred options will be the outn come more often than before, and the ~ less preferred options will be the outcome less often. When n is odd, the median option and all the more n-1 preferred options will be chosen more often, while the remaining T less preferred options will be chosen less often. In either case, o~2 will have the greatest increase, and there is a monotonically decreasing net gain (which includes negative 'net gains' for the less preferred options) down to an, which loses the most. Option ~ has no net change. 2 For each otj (j >~ 2) and each combination of n, m, and k, adding the value of Eq. (4) to the value of Eq. (3) yields the proportion of time one can expect aj to be the outcome. Table 3 displays several representative outcome distributions. As we can see from Table 3, i's greatest impact on outcomes occurs when n and rn are small and k is a bare majority of m. Overall, it helps for i to be sophisticated but the improvements are small.

5. Sophisticated choice by all subordinates The next stage of analysis requires that we examine how much i benefits when all subordinates, including i, act in a strategic manner. Each subordinate knows that one of two general classes o f events will ultimately occur: either k (or more) of the subordinates will agree on an option and impose this option on the boss, or else fewer than k will agree on an option, thus forcing the boss to get involved and make his own choice. For the subordinates, then, the problem of rational choice becomes one o f determining whether there exist any options which at least k subordinates prefer to the option the boss would choose if he had to get involved. Assume the superior most prefers option ~3. If there are no options which k or more subordinates prefer to ~, this means that for each of these other options, there are at least m - k + 1 subordinates who prefer ~ to the option. Each of these m - k + 1 subordinates might prefer some option or options to/~, but each realizes that there is no possibility that enough other subordinates would rationally choose the desired option. If there are one or more options which are preferred to fl, collect them in a set 9 . The problem now becomes one o f whether or not k or more subordinates can agree on which one of these options in xI, to select. If there are k or more subordinates who prefer one option in 't, to all others in x~, its k or more supporters will simply select it and forward it to the boss as their joint recommendation. 3 The subordinates who prefer

58

~ O Q ~ o o o ~

~

0

~ o

Q

OQ,~

•

•

•

~,~O~oo Q

o

o

o

o

o

o

E

o o o o

o o o o

o o o o

:z

~

~ ~ • ,~. , ~ ~

~

~

~

~

0

Og

~,~ -'. ~

•~

~

~ .. ~s ~

,~

,a~

~

~

~ ~ ~N ~ ~ e=~

~

~

.~

~

~ ~

•

~

~o

.~

~g ~

I!

~

~II

?

~

7

~

~

~

~

59

Q

~

0

~

0

*

0

0

*

o

0

~

Q

*

*

~ Q O O 0

~~ ~ E

~

~

~ *

*

~

*

*

*

*

*

o

.

~

oo

~:~ -"~ ~:~ ~

Z

v

II

0

some other element of 9 simply do not have the votes to win. Thus in all possible pairwise comparisons of the options in the set ~, if one of these options gains at least k votes in each comparison, this option becomes the subordinates' choice. If the subordinates cannot agree on a single recommendation from ,I~, we assume that the boss will choose whichever of the original n options he most prefers. If xI, has just one element, the option is the obvious group choice. Table 4. E x a m p l e s of S o p h i s t i c a t e d Choice by S u b o r d i n a t e s T w o Level T r e e Explan~l[il,n: Options are mm~bered l,....m. In the subordinales' preference orderings, the boss's re.st preferred option is circled. If an opti.n lies above the circles k or more times, i~ is in ~'.

Preferences Subordinates

Comments Boss

i j k l m n o p q r n=5 m=4 k=3

I 2 3 4 @

n=lO m=lO k= 7

i i0 9 8 6 I @ 6 2 1 8 9 2 @ 4 7 6 5 3 8 7 2 @ b I0 I0 4 4 i 7 7 I0 3 9 @ 6 21010 2 ~ 3 6 7 @ @ 4 3 2 1 1 0 7 7 @ 1 0 6 8 9 1 6 3 2 8 2 4 4 9 7 7 @ @ 9 9 3 3 3 1 8 9 4 9 4 I 0 9 6 1 3 4 ~ 2 7 @

n=8 m=6 k=4

1 ~ 3 4 5 6 7 8

n=5 m--4 k=3

@ 2 3 4 1

4 1 8 5 3 7 ~ 6

3 1 2 ~ 4

7 6 4 ~ 8 3 5 1

4 ~ 2 I 3

5 7 1 8 6 3 4 ~

5 i 4 3 2

7 4 8 1 3 ~ 6 5

7 5 ~ 8 3 6 4 1

I i0 2 4 8

I 3 ~

5 5 2 10 4 1 9 6 7 3

2 6 4 i 3 7 5 8

I I 24 2 @ I 2 3 24 1 4 4 @ 3

5 i 2 4

@3

3

3@

'~ is empty. :> Doss chooses Option 5.

~

:

{i}

=> Subordinates choose Option I.

~

=

(1,4,7l

Option 7 defeats i by 4:2 Option 7 defeats 4 by 4:1 (Person i does not vote for either) => Subordinates choose Option 7

~ = (1,2,4j Option I ties 2 by 2:2. Option 2 beats 4 by 2:1. (Person j does note vote for either) Option I beats 4 by 3:1. No option in ~ defeats all others with k or more. => Boss chooses Option 5.

61 This discussion can be condensed into a set o f three rules. We continue to assume that the boss most prefers option/3.

Rule 1: For each subordinate and each option, determine whether he prefers that option to/3 or prefers/3 to that option. Rule 2: For each option, determine whether it is preferred to 3 by at least k subordinates. Each option which gains k or more votes in a comparison with/3 enters the set 9 . Rule 3: If 9 = 0, then the boss makes the decision and the o u t c o m e is/3. If 9 has one element, that element is the outcome. If 9 has two or more elements, these elements are all compared pairwise. If there is one element in 9 which gains at least k votes when compared to each other element o f 9 , then that one element is the outcome. 4 If there exists no element of 9 which has k votes in comparisons with each other element o f 9 , then the boss decides and B is the outcome.

Table 5. Probability that Subordinate i Gets His First Choice When Everybody Is Sophisticated Two Level Tree Bare Majority Needed for Number of Proposals = n 2

Number of Subordinates t Including

i)

~ m

3 4 5 6 7 8 9 I0

.748 .680 .684 .654 .654 .633 .634 .614

3 .629 .521 .529 .480 .486 .455 .455 .435

Bloc to

4

5

.551 .425 .438 .376 .394 .350 .354 .332

.489 .350 .362 .297 .310 .275 .281 .260

Complete Unanimity Needed for Number of Proposals 2

Number o f Sulx,rdinates (Including i) :

m

2 3 4 5 6 7 8 9 I0

.748 .622 .556 .526 .5[2 .502 .504 .502 .496

3 .611 .477 .406 .373 .357 .347 .344 .338 .343

Form

4 .523 387 323 292 274 259 253 252 254

Bloc to

6 .443 .298 .317 .256 .270 .231 .242 .214

7 .407 .260 .282 .215 .228 .196 .200 .183

8 .377 .232 .258 .192 .206 .174 .180 .157

9 .353 .206 .225 .169 .182 .147 .156 .134

I0 .329 .184 .205 .147 .166 .138 .146 .124

Form

= n

5 .453 .322 .267 .240 .216 •209 .203 •203 .202

6 .407 .251 .233 .206 •191 .179 .172 .171 •171

7 •368 .251 .205 .[79 .161 .153 • 149 • 142 .142

8 .338 • 226

• 186 • 161 • 140 • 138 • 133 •127 • 126

9 .313 ,211 .172 .142 .134 .123 .115 .116 .II0

i0 .292 .185 .152 .129 .114 .112 .106 .105 .102

62 Examples o f the use of these rules are shown in Table 4. It was not clear how to frame Rules 1 - 3 in terms of equations like Eq. (1), but it was possible to use them in a computer algorithm which could process a given set o f preference orderings as in Table 4. One of us (JHH) wrote a F O R T R A N Monte Carlo simulation which first uses a random number generator to create a set o f preference orderings and then analyzes what the subordinates' sophisticated choices are. For each combination of n, m, and k we analyzed 15,000 sets o f randomly generated preference orderings. Table 5 displays how often subordinate i can expect to get his first choice in a two level tree when all subordinates are sophisticated. 5 Table 6, Probability That Subordinate i's Options Will be Chosen

When Everybody Is Sophisticated Two Level Tree Subordinate i Ranks Options in Decreasing Order of Preference: t~l,t~.,,...,a.,. How Often Is Each Option Chosen? ct 1

a 2

a 3

a4

ct 5

a6

ct 7

a8

ct 9

.086 .088 .091 .095

.085 .089 .095 .096

.081 .087 .092 .091

.082

.069 ,086 .094 .099 102

.070 .082 .093 .096

.072 .079 .089 .096 •094

.069 .073 .085 .093 .096

elO

Bare Majority Needed for Bloc to Form n=3

n=6

m=

n=lO

m=

4: 6: 8: i0:

.521 .480 .455 .435

.270 .291 .296 .299

.209 .228 .249 .266

4: 6: 8: I0:

.298 .256 .231 .214

.185 •182 .175 .174

.141 •155 .160 .159

.127 •136 .145 .153

•123 .137 .142 .153

.125 .135 .147 .147

4: 6: 8: I0:

.184 ,147 .138 .124

.131 .122 .117 .114

.i01 .103 .099 .I01

.090 .096 .095 .096

.084 .094 .093 .093

.077 .087 .088 .097

.087 .092 .093

Complete Unanimity Needed for Bloc to Form

n=3

m=

6: 8: I0:

.611 .408 .357 .344 .343

n=6

2: 4: m = 6: 8: i0:

.407 .233 .191 .172 .171

.168 .192 .179 .171 .167

.I18 .164 .169 .166 .165

2: 4: m = 6: 8: I0:

.292 ,152 .I14 .i06 .102

.125 .131 •116 •107 .105

.086 .116 .109 .102 .i00

2: 4:

n=lO

.221 .316 .328 .325 .324

.168 .275 .314 .331 •333

.163 .165

.i00 134 .155 .162 .167

.i04 .130 .!48 .166 .165

.076 .i05 • 105 .I01 .i01

.072 .091 .099 • 102 .099

.069 .084 .096 .096 .I02

• 104 • 149

.159

•

•

• i00

63 Comparison of Tables 1 and 5 suggests that subordinate i generally benefits from sophisticated behavior, but again in a rather modest way. In fact, the entries in Table 5 show signs of approaching the same limits (0 for a fixed /~andm asn-~

oo and--1 f o r a f i x e d k a n d n n

asm-~

oo) as the entries

in Table 1. Table 5 shows how often i's first choice will be the outcome. Table 6 shows several representative examples of how often each of the options is chosen when everyone is sincere and when everyone is sophisticated. When n and m are small and/¢ is a bare majority of m, sophistication produces an increase in the frequencies of i's most preferred alternatives and a decrease in the frequencies of his least preferred alternatives. In general, c~ has the greatest increase, and there is a decreasing net gain (which includes negative 'net gains' for the least preferred alternatives) down to c~n, which loses the most. 6

6. When subordinate i is sincere and everyone else is sophisticated To fill out the picture of the incentives individual i faces, we need to know what would happen if i alone were sincere but everyone else is sophisticated. The key question is whether i will be forced to be sophisticated, perhaps against his own wishes, due to the strategic behavior of his fellow subordinates. Assume the other subordinates know that i is always going to recommend his most preferred option to the boss, just as we assume in Section 3 that i knows the other subordinates are always going to choose sincerely. The interesting (and to us, unexpected) result is that if i acts sincerely, he can expect to get his first choice m o r e often than if he acts strategically like everyone else. The reason is that if i always acts sincerely (and everyone else knows he does), there are a number of occasions in which i's intransigence breaks the subordinates out of situations in which ~ has two or more options but the subordinates are at loggerheads as to which one of the options to select. By committing himself to one of the options in ~, i forces the others to yield to his wishes if they want to avoid having the boss impose an option which they like even less than i's choice. This kind of commitment by i does have occasional costs. Sometimes an option which is not i's first choice but which i prefers to what the boss would impose fails to enter the set ,I, which the others are considering. The boss might then impose something worse than what i could have gotten had his lower-ranking option become a member of xI, and been selected by/¢ or more of the subordinates. But our simulation demonstrates that, in general, i's occasional failure to ensure that one of his lower ranking op-

tions will be chosen is overshadowed by the fact that his first choice will be the outcome somewhat more often. On the whole, then, i benefits from committing himself to his most preferred option in '~/. For example, if n = 3, m = 2, k = 2, and both subordinates act sincerely, each can expect to get his first choice with a probability of .556 (from Table 1), while if both are sophisticated, each can expect to get his first choice with a probability of ,611 (from Table 5). But if i alone is sincere, the probability that he gets his first choice rises to .667; see Table 7. (To conserve computer time we calculated only selected values.) Table 8 shows several representative examples of how often each of i's options will be the outcome. Subordinate i benefits from commitment only if the others assume he is committed and do not adopt commitment as their own strategy. If the subordinates are unable to agree on some element of xl,, each has an incentive to improve things for himself by warning the others that he is going to recommend his first preference. If one of them commits himself, perhaps by going directly to the boss with his most preferred option, he will get his first choice and the others a less preferred option. The other subordinates, however, would soon realize that the game is one of who canfirst Table 7. Probability that Subordinate i Gets His First Choice When i Is Sincere and Everyone Else Is Sophsiticated T w o Level T r e e Bare Majority Needed for Bloc n~ Form N u m b e r t)f Pn,posals = n n= 3

Number of Sub~rdinates I Including i) =lrl

3 4 5 6 7 8 9 I0

4

5

6

7

8

9

I0

.b57 .566

.blO .491

,539 .402

.47b .341

.430 .298

.517

.434

.331

.204

.229

.483

.403

.292

.227

.184

.466

.375

.265

.203

.158

Complete Unanimity Needed ft~r Bh~: to F~,rm n=

Number t~[ Sul~rdinares

(Including i) ~

m

2 3 4 5 6 7 8 9 I0

3

4

5

6

7

8

9

I0

.667 .504 .419

.620 .435 .342

.577 .376 .269

.558 .346 .238

.546 .329 .211

.355

.273

.196

.151

.127

.347

,250

.169

.132

.i06

.332

.253

.165

.124

.102

65 Table 8, Probability That Subordinate i's Options Will be Chosen

When i Is Sincere and Everyone Else Is Sophisticated Two Level Tree Subordinate i Ranks Opt ons in Decreasing Order of Preference: a~:,...,n'Io, How Often Is Each Option Chosen? al

a2 a3 ~4 a5 a6 Bare Majority NeededforBl~ toForm

4: 6: 8: I0:

.566 .517 .483 .466

.219 .238 .260 .26L

.215 .245 .257 .273

4: 6: 8: I0:

.402 .331 .292 .265

.115 .132 .143 .148

.123 .133 .139 .143

.116 .137 .140 .146

.123 .135 .143 .147

.121 .132 .143 .151

4: 6: 8: ~0:

,298 .229 .[84 .158

.080 ,08[ ,088 .096

.077 .086 .094 .093

.077 .086 .092 .093

.07~ .086 .092 ,093

.080 .086 .093 .097

4: m= 6: 5: 10:

.667 .4[9 .355 .347 .332

.[70 .292 .321 .327 ,334

~163 .290 .324 .326 .334

n=6

2: 4: m = 6: 8: I0:

.577 .269 .~.96 .169 .165

.085 .151 .159 .169 .171

.084 .143 .166 .173 .168

.087 .151 .159 .166 ,167

.083 .143 .160 .163 .161

.084 .143 .161 .160 .168

n=lO

2: 4: m~ 6: 8: 10:

.546 .21t .127 .106 .102

.049 .088 .097 ,097 .099

.048 .086 ,099 .100 .104

.052 .089 .091 .098 .096

.052 .087 .094 .103 .096

.050 .086 ,099 .097 .106

n=3

m=

n=b

m=

n=[O

m=

a7

a8

a9

al0

.078 .084 .091 .092

.082 ,088 .091 .094

.077 .085 ,087 .091

.076 .088 .088 .092

.053 .090 .098 .097 .101

.047 .088 .096 .102 .100

,051 .086 ,099 ,098 .097

.050 .089 .099 .103 .100

Complete Unanimity Needed for Bh~: to Form

n=3

commit himself to recommending only his first preference. This is the kind of problem discussed by Schelling (1960: esp. 24ff.). But if each subordinate anticipates that the others will try to commit themselves first, it might happen that each subordinate ends up committing himself to his most preferred option in ~I,. In these situations where each subordinate has an incentive to commit himself to his most preferred option in ~I, but all subordinates end up doing so, the payoff each gets is the s a m e as if everyone had remained sincere: in either case, the boss makes the decision. 7 In other words, our initial expectation that a subordinate might be forced to engage in sophisticated behavior to protect himself against the sophisticated behavior of the other subordinates appears tO be incorrect:

66 if anything, subordinates are forced into sincere behavior to protect themselves! To the extent, however, that subordinates are able to make binding agreements with each other - perhaps by jointly signing a memorandum in which they make their recommendation to the boss - they can break out of this game and ensure for themselves the intermediate benefits of everyone being sophisticated. What happens if the other subordinate expect i to act strategically but i in fact behaves sincerely? Is i ever better off, given these expectations? It is in this case that our original conjecture is correct: i is never better off. To show this we must look at five possible situations. (i) If 9 is empty, whether i chooses sincerely or strategically will make no difference. (ii) Now consider a • which is not empty. If there is a best element (i.e., preferred to all others in xI, by k or more subordinates) and it is something that i does not prefer to the boss's choice, then the fact that i votes sincerely makes no difference. (iii) If there is no best element of 9 , exactly the same situation prevails: i would vote sincerely because his vote would not make any difference to the outcome. (iv) If there is a best element of xI, and it is w h a t / m o s t prefers, i would of course recommend this best element since it is his first choice. (v) Finally, assume there is a best element of • and that it is something that i prefers to the boss's choice but it is not i's most preferred choice. If i decides to vote sincerely, then either the bloc supporting the best element still wins (in which case the outcome is the same) or the bloc now does not win and the outcome is the boss's choice, and so i is'worse off than if he joined the bloc. So whichever of these five conditions prevails, i is never better off, and is sometimes worse off, if he tries to outfox the others by choosing sincerely when they expect him to choose strategically. Thus our original expectation that i will be forced to act strategically to defend himself is correct only when the other subordinates expect him to act strategically.

7. Subordinate i's choice of a strategy

The preceding four sec.tions can now be assembled into an overall picture of subordinate i's incentives to engage in strategic behavior. Consider the four situations in which i might find himself: (1) (2) (3) (4)

he he he he

might might might might

choose choose choose choose

sincerely as does everyone else, sincerely when everyone else is sophisticated, strategically when everyone else is sincere, and strategically as does everyone else.

Given that i cannot directly control the behavior of his fellow subordinates,

67 should he be expected to choose sincerely or should he be expected to choose strategically? There are several answers to this question, depending on the number of options (n), the number of other subordinates (m), and the extent of agreement (R) required by the boss if he is to accept his subordinates' recommendation. It is easiest to discuss the answers in terms of cases in which these variables are all small, moderate, and large in size. In the top example in Table 9 the values of n, m, and/¢ are relatively small (n = 3, m = 4, and k = 3) and we cannot say in general what i's preferred strategy is going to be. If everyone else chooses sincerely, i is generally better off choosing strategically: while he can expect to get his first option with a probability of .481 no matter what he does, in choosing strategically he can raise the probability of getting his second choice from .259 to .333 and he decreases the probability of getting his last choice. But if everyone else chooses strategically, his chances of getting his first choice are greater if everyone else knows he will choose sincerely (p = .566) than if he chooses strategically (p = .521). Nonetheless, if the other subordinates do not know he will choose sincerely, his best strategy involves sophistication. Table 9. E x a m p l e s of Incentives Faced by Subordinate i T w o Level Tree Entries in each box are the probabilities i can expect to get his 1st through nth most preferred option.

Everyone Else Ch~ses Sincerely

Everyone Else chooses Strategically n= 3, m = 4 , k = 3

Subordinate i Chooses Sincerely

.481,.259,.259

.566,.219,.215

Subordinate iCh~)ses Strategically

.481,.333,.185

.521,.270,.209

n=6, m=6, k=6

Sub.rdinate i Chooses Sincerely

•I~7,.167,.167,.167,.167,.167

.196,.159,.166,.159,.160,.161

Subordinate i Chooses Strategically

.167,.167,.I~7,.167,.167,.167

.191,.179,.169,.159,.155,.148

.

n= IO.m= lO, k= l0 Subordinate i Chooses Sincerely

.100,.100,.1.00,.100,.100,.100, • i00,. I00,. I00,. lO0

.I.02,.099,.1.06,.096,.096,.106, .i01,. 100,.097,. i00

Subordinate i C'h,~ses Strategically

•I 0 0 , . i 0 0 , . I 0 0 , . i 0 0 , . I 0 0 , . I 0 0 , • i00,. I00, . i00,. I00

.102,•105,.I00,.I01,.099,.102, . i02,. i00,. 094, .096

68 In the second example n and m are of moderate size (n = 6, m = 6); we will set k = 6. If the other subordinates choose sincerely, how i chooses has no effect on his payoffs: he can expect to get each option with a prob1 1 ability of .167 (= ~ = ~) in either case. If the other subordinates choose strategically and think he is sincere, he is generally better off choosing sincerely since he has a higher probability of getting his first choice. As before, though, if the other subordinates do not know he will choose sincerely, sophistication is his best strategy. In the third example n, m, and k are all large (n = 10, m = 10, k = 10). Whatever i does and whatever the others do makes no difference in 1 the outcomes: each of the ten options will be chosen ~ of the time. (The entries in the right-hand column of Table 9 here are all essentially indistinguishable from p -- . 100.) Given our model, one must conclude that it is only when n, m, and k are small that any one subordinate would rationally choose to act in a strategic manner. Even then, subordinates in general seldom seem to be able to change things for the better. The biggest increase in the probability of i getting his first choice due to sophistication is nearly always less than • 100. The probability of getting caught and punished by the boss for acting strategically would not have to be very large to overcome this incentive to engage in strategic behavior. Only when the value of a subordinate's first choice greatly exceeds that of his lower-ranking options will the costs of being found out be worth the risk of engaging in strategic behavior. Since many political issues do involve the proverbial 'high stakes,' we should not necessarily discount the likelihood that subordinates will engage in such behavior.'8 Overall, though, these results suggest that sophisticated behavior of the kind described here should n o t be particularly prevalent in organizations. Instead, one might expect to find sincere behavior or perhaps some other kind of strategic behavior (e.g., 'telling the boss what he wants to hear'). To judge whether these conclusions are a reasonable guide to the behavior of real world organizations we need to judge the plausibility of several key assumptions. A first point of concern involves our 'Management By Exception' decision rule. Executives do not, of course, use such formal rules to make decisions. The judgments, beliefs, opinions, and preferences of subordinates in organizations are aggregated in a wide variety of ways. But chief executives must exercise some kind of judgment about when to get involved in a problem and when just to ratify what the subordinates recommend: an executive cannot get involved in everything. The 'intrinsic importance' of a problem is one prime criterion which triggers their attention and involve-

69 ment, but there are often many more such problems than a single executive (e.g., the President) can get involved in, and so some additional screening device seems necessary. The extent of conflict among subordinates is usually an important factor which makes the boss pay attention. Whenever some threshold value of conflict is exceeded, the executive allocates some attention. This is precisely how our MBE rule works, with the specific value of k being the threshold for the boss's involvement. Thus our MBE rule strikes us as an empirically plausible, if still somewhat metaphorical, representation of how organizational decision-making proceeds. Perhaps more troubling for our analysis of strategic behavior is the assumption that when the set ~1/has two or more options the subordinates might nonetheless be unable to select one of them. In the real world the subordinates may generally be able to make some collective choice via bargaining and logrolling; even a random choice from ~1/would be better for at least k members than allowing the boss to make the decision. From this perspective it seems possible that our model understates how beneficial sophisticated behavior is to the subordinates. 9 But when the subordinates cannot easily communicate with each other, as in a large, heterogeneous, and geographically dispersed organization, it may be difficult to form a coalition on a particular option. In this case, the rule used in our paper might well be the more appropriate one. Strategic behavior also requires that the subordinates know each others' preference orderings over the options. For officials who can easily communicate and who have worked together for years, something roughly approximating such knowledge may in fact exist. It would not exist, though, in organizations with many newcomers: think of the first months of a new administration in Washington, D.C. Nor would it be as likely to exist in large, heterogeneous, and geographically dispersed organizations. In these cases, the problem of learning the preferences of the other subordinates would be difficult to overcome and we might not expect to find much strategic behavior. The 'impartial culture' assumption may, at first glance, seem wildly at variance with the real world. In many cases subordinates undoubtedly have similar preferences (in which case their boss would never get involved in decision-making) and in other cases the subordinates are undoubtedly badly divided (and their boss would constantly have to get involved in their disputes). It is an empirical matter whether one of these extreme conditions holds in some real world organization. We are impressed, however, by the fact that even in presidential administrations which are mostly 'liberal' (such as the Carter administration) or mostly 'conservative' (such as the Reagan administration), there remains a very substantial amount o f internecine warfare, given the set of options under consideration in each case. While the conflict is not as extensive as would occur in an administration

70 populated by both Carter liberals and Reagan conservatives, it does lead us to think that the 'impartial culture' assumption may tell us more about the workings of at least some real world institutions than one might expect. A further concern may be that the boss in our model imposes no jurisdictional restrictions on the advice he requests from subordinates. He simply asks everyone for advice on what should be overall organizational policy. Under some circumstances, perhaps involving problems with specialized technical components, an executive might treat subordinates as subjectmatter 'specialists' and ask them for advice on what to do regarding their own particular aspects of the problem; he would not ask them what they think in general is best organizational policy. To the extent that executives rely on specialists for Specialized advice, our model will not be appropriate. Finally, recall that our model completely excludes the problem of where the options come from: we treat them as imposed from outside the organization. Obviously, though, the subordinates' greatest strengths sometimes lie in the very definition and formulation of the options to be considered. However, to the extent that the organization's environment imposes the basic choices that are available - and this is often at least somewhat the case - our model may have some applicability. Clearly, then, alternative versions of our model can be considered. But there do seem to be good reasons for treating this particular version as a plausible one, and thus as a useful starting point for the study of strategic behavior in organizations.

NOTES 1. For studies o f strategic behavior in majority rule committees see, for example, Farquharson (1969), McKelvey and Niemi (1978), and Niemi and Frank (1982). 2. When the hierarchy has three or more levels and subordinate i is at level three or below, his strategic behavior c a n c a u s e a net gain for a~. 3. it might happen that the option preferred by several of these k subordinates to what the boss would choose is their first preference. In this case, these individuals will simply be selecting their first preferences and will thus appear sincere in their behavior. Since this 'sincere' behavior is carefully calculated, we will count it as 'sophisticated' in the analysis which follows. 4. A subordinate who prefers the boss's first preference to each of the two elements being compared will abstain. There are two reasons. First, if there are k other subordinates who prefer one o f the elements in • to each of the others, then an individual who likes nothing in • (compared to what the boss would do) will find his own vote makes no difference in the outcome. Second, if no element o f ~I, has the required k supporters, then the individual who likes nothing in xv (compared to what the boss would do) would not vote for anything in 9 because he can count on the boss making the decision: if he voted for either o f the two elements being compared, he might turn a winning situation (boss makes the decision) into a losing one (something in ~/gains k votes). 5. When there are only two options, officials will always recommend their more preferred option, and thus the sophisticated results for n = 2 will be identical to the sincere results.

71

6.

7.

8.

9.

A comparison of the n = 2 column in Table 5 with the n = 2 column in Table 1 shows that the difference between the sincere probability and the sophisticated probability in each case is only .004 on average. The greatest deviation is .009. Similar sincere-vs-sophisticated comparisons with other data suggest that the simulation's error is rarely larger than approximately .01 or .02. The entries for ct~, . . . , c~n in Table 6 should be monotonically decreasing. That they are not is due to the sampling error inherent in our Monte Carlo simulation. More iterations or an analytical solution would smooth out the trend. The only other possible preference configurations are stably sophisticated: either xI, has only one element or there is an element in ~I, which is preferred by k or more subordinates to every other element in xv. In either case, no member of the bloc would have an incentive to defect from the bloc. We have considered here only two level hierarchies, but most individuals in bureaucracies are further removed from the boss than this. The incentives for a third level subordinate to engage in strategic behavior are in general even less than for a second level subordinate, since any one third-level subordinate's manipulations are mediated by the interests and actions of the middle level managers. We are currently modifying our simulation to make such a random choice from 't, when there is no element which is most preferred by k or more subordinates. Results will be reported in the future.

REFERENCES Appleby, P. (1945). Big democracy. New York: Knopf. Farquharson, R. (1969). Theory of voting. New Haven: Yale University Press. Garman, M.B., and Kamien, M.1. (1968). The paradox of voting: Probability calculations. Behavioral Science 13: 306-316. Gibbard, A. (1973). Manipulation of voting schemes: A general result. Econometrica 41: 587-601. Hammond, T.H., and Horn, J.H. (1983). Sophisticated choice in hierarchies. Presented at the Annual Meeting of the Public Choice Society, Savannah, Georgia, March 24-26, 1983. McKelvey, R.D., and Niemi, R.G. (1978). A multistage game representation of sophisticated voting for binary procedures. Journal of Economic Theory 18:1-22. Niemi, R.G., and Frank, A.Q. (1982). Sophisticated voting under the plurality procedure. In P.C. Ordeshook and K.A. Shepsle (Eds.), Political equilibrium. Boston: Kluwer. Satterthwaite, M.A. (1975). Strategy-proofness and Arrow's conditions: Existence and correspondence theorems for voting procedures and social welfare functions. Journal of Economic Theory 10: 187-217. Schelling, T.C. (1960). The strategy of conflict. Cambridge, Mass.: Harvard University Press. Tullock, G. (1965). The politics of bureaucracy. Washington, D.C.: Public Affairs Press.