Algorithms for hierarchical power

Algorithms

Power

for Hierarchical

Mitali De School of Business and Economics Wilfrid Laurter University Waterloo, Ontario, Canada, N2L 3C5 Keith W. Hipel Departments of Systems Design Engineering and Statistics and Actuarial Science University of Waterloo Waterloo, Ontario, Canada, N2L 3Gl and D. Marc Kilgour Department of Mathematics Wilfrid Laurier University Waterloo, Ontario, Canada, N2L 3C5 and Department of Systems Design Engineering University of Waterloo Waterloo, Ontario, Canada, N2L 3Gl

Transmitted by L. Duckstein

ABSTRACT A flexible noncooperative strategic focuses

concept

of hierarchical

interactions

of the decision

of a decision

maker to choose

computing

the hierarchical-power

developed,

and a computational

on the rules

of hierarchical

hierarchical method power

within

the framework

approach

having asymmetric Hierarchical

when

there

for assessing

is also given. involved

AND COMPUTATZON

0 Elsevier Science Publishing Co., Inc., 1990 655 Avenue of the Americas, New York, NY 10010

to capture

involves

choice.

a procedure

are two decision programs

in both these applications.

for

makers

the stability of outcomes

39:21-36 (1990)

the

the ability

of the opponent’s

Computer

of

roles. The definition

power

in the knowledge

of

to the analysis

of the model,

power in a game are studied,

outcome

to ease the computations

APPLIED MATHEMATICS

positions.

a strategy

of incorporating

developed

makers

than the structure

makers’

The effects

is developed

the systems-theory

among decision

on the rules of play, rather

asymmetry

power

game theory to extend

have

is

based been

The use of

21 0096-3003/90/$03.50

M. DE, K. W. HIPEL, AND D. M. KILGOUR

22

these stability concepts indicates that hierarchical power tends to stabilize cooperative behavior in many noncooperative situations. An exploratory study of two wellknown classes of games involving two decision makers shows that hierarchical power can provide a new technique for analyzing conflicts within hierarchical systems of decision makers.

1.

INTRODUCTION Many social disputes,

in which conflicting

interest

groups control possible

scenarios or outcomes, can best be modeled by suitable games of strategy. But virtualiy all techniques of conflict analysis based on noncooperative game theory assume that decision makers or players have equal capacities to act, even if the specific conflict gives some player(s) a structural advantage relative to others. Recently, however, the idea of strategic advantage for decision maker(s) has been formalized in various concepts of asymmetric power [l-3, 6, 12, 141. Models of an asymmetric power relationship are necessary for a realistic analysis of conflicts in many instances. Recent work in applied game theory has stressed the importance of maintaining a distinction between the structural characteristics of interactive situations and the rules that govern the actions of the players [5, 111. It is this analytical separation of structure from rules that has facilitated the introduction of power asymmetries

into game models through new definitions

of rules

of play. A strategic advantage for a decision maker, or power asymmetry, often reflects the existence of information. Specifically, if a decision maker has prior information about an opponent’s strategy choice, then the outcome of the conflict will almost certainly be affected. Furthermore, a model of how a strategic advantage affects the outcome can provide a measure of the value of prior information. In general, power concepts enhance the strategic analysis of a conflict [S, 9, 16, 171, which considers the choices made by decision makers to further their own individual interests. These decisions are made in the knowledge of the existence, available choices, and preferences of the other decision makers involved in the conflict. Strategic analysis can help decision makers determine what behavior others are likely to adopt and suggest which courses of action should be chosen to bring about a particular outcome or result. A major tool for the strategic analysis of conflicts is noncooperative game theory. The earlier concept of staying power, introduced by Brams and Hessel [2], was further developed and improved by De [6] and Kilgour et al. [12, 131 into

Algorithms for Hierarchical Power

23

hierarchical power (HP). Brams and Hessel [2] proposed staying power to model the advantage of a player who can defer his strategy choice. Hierarchical power is the ability of a player to choose an action in the knowledge of the action(s) chosen by the opponent(s) in a move-response sequence. (A detailed description of the differences between HP and staying power is contained in Section 3.) Because analysis based on hierarchical power considers a definite sequence of play, each action is considered to be chosen in the knowledge of all previous strategy choices. Selections can continue until the situation stabilizes, in the sense of convergence to a definite outcome no matter how long this convergence takes; or selections can be curtailed at fixed horizon shortsighted (myopic) decision makers are to be modeled.

if

In the following sections, concepts from two-person game theory are briefly reviewed; then various definitions pertaining to the concept of HP are explained and their properties developed. An explanation on how HP may be used to analyze a hierarchical system of decision makers is also given. The development focuses on conflicts involving exactly two decision makers, which are general enough to cover many important applications. The logic involved in FORTRAN programs that determine the final outcomes for an arbitrary starting position (with or without HP), and calculate the stability of such outcomes, is also indicated. As an illustration, HP analysis is applied to a well-known game called Chicken, which is game 66 of Rapoport and Guyer

m31. 2.

TWO-PERSON A two-person

GAMES game

is a model

of a situation

involving

two decision

makers, called player 1 and player 2. Each player has a limited set of available actions and must select one-the players’ choices jointly determine the outcome (state) of the conflict. Each player’s preferences over the possible outcomes are represented by his payoffs for the outcomes. In some applications, the payoffs are taken to be cardinal utilities, but here payoffs are ordinal and represent preference order only: a player strictly prefers outcome u to outcome w iff the payoff is greater at u than at w, but no measure of the amount or degree of this preference can be inferred from the numerical values of u and w. Of course, any analysis based on ordinal payoffs also applies to the cardinal case. The normal form of a two-person game is a matrix representation of the conflict, in which the rows correspond to player l’s strategies (available actions) and the columns to player 2’s. A two-person game with m strategies for player 1 and n for player 2 is called an m X n game. Each of the mn cells

24

M. DE, K. W. HIPEL, AND D. M. KILGOUR TABLE 1 CAME

OF

CHICKEN

Player 2

Player 1

11

cj

is thus an outcome (state), and is usually labeled using the payoffs of the two players, player l’s first. An example of a 2 X 2 game is the game of Chicken [18] shown in Table 1. Recall that the players’ preferences over the four cells (outcomes) are given as ordinal payoffs (4 = best, 3 = next-best, 2 = next-worst, 1 = worst). The first entry in each cell represents the preference of player 1; the second entry, the preference of player 2. Notice that in Chicken, outcome (3,3) is second-most preferred by both players, and outcome (1,l) is least preferred by both. One way to analyze normal-form games that is very relevant to power definitions is through the stability of outcomes. A cell in a game matrix is a

stable outcome (position)

for a player if that player is content to stay at that position, given that the game is already there [g-11]. The concept of stability depends on the ability of a player to change strategies so as to move the game from one position to another. Of course, a rational player’s decision whether to stay at a status quo outcome depends on his payoff at the eventual outcome he believes his departure would lead to. Different beliefs about how

the game will continue after a departure correspond to different types of stability. An outcome which possesses a specific type of stability for both players is an equilibrium of that type; it is usually interpreted as a possible resolution to the conflict. An outcome is Nash stable (individually rational) for a player if that player cannot unilaterally achieve a more preferred outcome [15]. Nash stability models a player who believes that his opponent will make no response to his move. An outcome which is Nash stable for both players is a Nash equilibrium. For example, in Chicken outcomes (4,2) and (2,4) are Nash equilibria. Notice that, from outcome (4,2), player 1 disimproves if he changes his strategy from r2 to rr, thereby moving the game from (4,2) to (3,3). Likewise, a unilateral move from (4,2) to (1,l) by player 2 produces an immediate disadvantage to player 2. Because neither player can unilaterally improve from (4,2), this outcome is individually rational for both players, and is therefore a Nash equilibrium.

25

Algorithms for Hierarchical Power

In contrast to the concept of Nash equilibrium are those related to nonmyopic equilibria and extended nonmyopic equilibria (XNM equilibria) [4, 111.These stability definitions refer to a player’s rational decision to stay at a particular position when he anticipates that his move may trigger a response by his opponent, then his own counterresponse, etc., in alternating sequence [Ill. A player who can move from a status ~JO position views himself as the first player in a departure game, an extensive game in which each player, in sequence, decides whether to stay at the current outcome or depart to a new outcome. The sequence ends as soon as some player chooses to stay, or as soon as the number of moves reaches some predetermined maximum-this maximum length is a parameter called the horizon, h. Departure games are finite extensive games of perfect information and can be solved by backward induction. In solving these games, the inertia principle outcome

[19] is generally applied: a player prefers to depart from the current only if the anticipated final outcome is strictly preferred. The

outcomes of all possible departure games from a given game can be displayed conveniently as sequences of matrices of outcomes, as illustrated for Chicken in Table 2 and explained in detail below. Some notation is required to describe the outcomes of departure games. If the original game, considered as a matrix of outcomes, is G (of size m X n), denote by MA(G) th e m X n matrix whose (i,j) entry, [ML(G)]i,j, is the outcome resulting from a departure game with horizon h, beginning with a possible move by player 1 from cell (i,j). Similarly let M;(G) represent the outcome of the analogous game with initial move by player 2. It is convenient to write M;(G) and M:(G) M:(G) = G by convention. Although

side by side, as in Table 2. Note that M:(G)

each entry of the matrices

ML(G)

and M:(G)

is defined

=

to be

the outcome of a specific extensive game, there is a much more efficient method to generate the entire sequence of bimatrices M, = (M;(G), M:(G)), h=12 , ).... If x0,x ,“.‘, x,, is a list of outcomes, define P,(x,; xi, x2,. . .,x,) to be player l’s

. . , x,) = x0 unless at least one of x1,x2 ,..., x, is strictly preferred by player 1 to x0. A similar definition is used for P, and player 2’s preference. The matrices ML(G) and M:(G) can be determined follows:

inductively

by using componentwise

relations

as

M. DE, K. W. HIPEL, AND D. M. KILGOUR

26

TABLE 2 BIMATRIX

REPRESENTATION UP TO LENGTH

OF DEPARTURE h =

4,

FROM

GAME

OUTCOMES,

CHICKEN

Player 2

f-1

Player 1

h

0

1

2

f-2

Cl

CP

(3,3) (4,2)

(2,4) Cl,11

Mlt (373)

(2,4)

(3,3)

(2,4)

(472)

(Ll)

(4,2)

(~1)

(42) (42)

(2,4) (f&4)

(2,4) (42)

(2,4) (42)

(472)

(4>2)

(4,2)

(2,4)

I1:::I

m 4

(373)

(4,2)

(4,2)

(2,4)

Note that this calculation incorporates the inertia principle: a player prefers Gi, j unless an alternative (i’ # i or j’ # j) provides a strictly greater payoff. Table 2 illustrates the bimatrices of final outcomes of departure games, up to length h = 4, for the 2 x 2 game of Chicken (Table 1). As an example, the final outcome of a departure by player 1 from outcome (3,3) of length h = 2 is obtained by comparing the payoffs of player 1 for the outcomes G,,, and [M:(G)],,, [i.e. (3,3) and (4,2)]. In other words, the question is whether player 1 prefers to stay at (t-r, c,) and receive (3,3), or to move to (rZ, c,), giving player 2 the first move in a departure game of length h = 1 from that point. But M&(G) indicates that this latter departure game has outcome (4,2), which is preferred by 1 to (3,3); therefore 1 moves to (ra, c,), and the final outcome is [M;(G)],,, = (4,2). This induction method is in fact used to analyze all of the examples

in this paper.

27

Algorithms for Hierarchical Power The outcomes

of these

departure

games

can be used to define

types of stability. Of particular importance is nonmyopic stability [ll]. Gi,j is determinate for player 1 if there exists a t such that

and Gi j is nonmyopically

stable for player 1 iff it is determinate

various A cell

and

G,,j = [ M:(G)]i,j. The definitions for player 2 are analogous. For example, in the game of Table 2, cell G,,,, containing (1, l), is determinate for player 1 (with t = l), because [M:(G)],, = (2,4) for t = 1,2,3,4. Also notice that cell G,, i, containing (3,3), is nonmyopically stable for player 1 (with t = 3), because [M:(G)] = (3,3) for t = 3,4. To define other types of stability, some other properties of the sequence of bimatrices M,, M,, . . . , must be specified. If there exist t and r such that, for all h > t,

M,,+.(G)= W,(G), and if t and r are the smallest positive integers with this property, then convergence is said to have taken place at horizon t and G is said to have an r-cycZe. (Two bimatrices are equal iff all corresponding entries are equal. Notice that the definition requires that the departure game outcomes for both players must exhibit the r-cycle.) The characterization (1) implies that the sequence M,,M,,M,, ... must eventually cycle. In fact, in every m X n game known to the authors, either a l-cycle, a 2-cycle, or a 4-cycle occurs, and it is conjectured that these are the only possibilities. A l-cycle is called a fixed point. For instance, Table 2 shows that the iteration of Chicken converges to a I-cycle, or fixed point, at length h = 3; the bimatrices are equal from this point on. While the above rules are sufficient to solve the departure games resulting from most games, a problem arises in games which are larger than 2 X 2 and nonstrict ordinal. In a nonstrict ordinal game, one or both players may be indifferent between distinct outcomes; thus, a player may sometimes have to choose among outcomes which are equally preferred to himself but not his opponent. If this occurs, the inertia principle is applied first. If the inertia principle does not completely determine the player’s selection, it is assumed that, among equally preferred outcomes, a player chooses the one least

28

M. DE, K. W. HIPEL, AND D. M. KILGOUR

preferred by his opponent. This assumption is conservative, not only from the point of view of the opponent, but also for the player himself, in anticipating what his opponent will think. The above definition essentially describes the method of convergence, wherein a departure game is allowed to continue to any length until it terminates in either a fixed point, a e-cycle, or a 4-cycle. Hierarchical power imbedded within the method of convergence helps end departure games that cycle. Hierarchical power allows a designated player-the player with HP-to end the departure game at the outcome most favorable to himself (when it is his turn to move) in a cycle. The method of convergence allows the move-countermove sequence to stabilize naturally. Briefly, if the outcomes of departure games converge to a fixed point (as in Table 2 from h = 2 onwards), the players are assumed to have sufficient foresight to take the eventual outcomes into account. If the outcomes of departure games do not converge but instead cycle, the player with HP is allowed to terminate the game at whichever outcome of the cycle he prefers. These rules ensure that every move-countermove sequence comes to a conclusion, and also give a slight advantage to the player with HP in terminating such a sequence. The procedure applies to any finite general ordinal game. A FORTRAN program was developed in bimatrix form until convergence.

3.

DEFINITION

OF HIERARCHICAL

to iterate any m X n game

POWER

Until recently most methods for the analysis of strategic conflicts considered decision makers to be equal in the sense that if the strategic choices and preferences were reassigned the outcome would not change. However, in many real-life situations asymmetry (interaction among unequals) is a crucial feature. In international politics, for instance, powerful decision makers are few and the meek are many; in oligopolistic markets relatively few price leaders influence the decisions of price followers. In these areas and many others, equality is the exception rather than the rule. An understanding of the nature and significance of power asymmetry is obviously important for a better understanding of the strategic problems represented by these realworld conflicts. In the following sections the theoretical methods of Section 2 will be applied to conflicts which are structurally asymmetric. Several recent conflict analysis methods have distinguished between structural characteristics of a conflict and the rules that govern the actions of the players [l-3, 5, 6, 12-141. The resulting models, in which rules of play are independent of structure, form a natural setting for the definition of

Algorithms for Hierarchical Power

29

power asymmetries. Other than HP, the different forms of power that have been defined within this framework include holding power [14], threat power [3], moving power [l], and staying power [2]. Hierarchical power (HP) is the ability of a player to choose

an action in

the knowledge of the action(s) chosen by the opponent(s) in a conflict. De [6] and Kilgour et al. [12] revised some aspects of the original definition of staying power [2] and introduced some additional features in formulating the concept of hierarchical power. The rules associated with HP allow the initial outcome to be prespecified or to be obtained as a result of strategy choices by the players, such that the player with HP chooses in the knowledge of his opponent’s choice. In their original

definition

of staying power for 2 X2

games,

Brams

and

Hessel [2] used terminal rationality to force every departure game to terminate. According to terminal rationality, the player without staying power may move from an initial outcome process terminates before it returns

if and only if he can ensure that the to the initial outcome and a new cycle

commences. The player with staying power will move from a subsequent outcome if either he can ensure his most preferred outcome before cycling, or he can force a return to the initial outcome through cycling. These assumptions inherent in terminal rationality make the player without staying power responsible for the departure sequence not cycling. In HP, terminal rationality is replaced by the method of convergence. This change permits the players to at least consider the possibility of cycling a move-response sequence back to an outcome attained previously, a choice expressly forbidden by terminal rationality. (Presumably the players will not rationally choose to do this, but there is no reason why they cannot consider doing it.) These definitions permit HP to be applied in any m X n game, and even suggest how to define HP in situations with three or more players. De [6] also defines some variations on HP which make it distinct from other power concepts. The following definitions are made on the assumption that player 1 has HP-when 2 has HP, the definitions are analogous. The single-shot HP outcome for 1 is the outcome resulting from rational play when both players know that 2 must choose his strategy first, and then 1 may choose 1 is the outcome resulting from rational play when 2 must choose his strategy first, then 1, then 2 may change his strategy, then 1 may change, then 2 again, etc. The sequence ends as soon as a player chooses not to change strategy, or 1 (but not 2) chooses to end the sequence immediately after he has changed strategy if the outcome is cycling. Each decision is made in the knowledge of all previous decisions, and both players are fully informed of the rules. These rules are consistent with the method of convergence. A FORTRAN program for determining the HP outcome has been developed.

M. DE, K. W. HIPEL, AND D. M. KILGOUR

30 The HP can affect known, HP conflict. It

outcome assists in asserting the belief that power asymmetries the results of a conflict. When power asymmetries are already outcome shows how power asymmetries affect the outcome of the can also be inferred that power asymmetries exist when the

outcome of a game depends on who the players are (i.e., a player is with or without HP) and on who starts the game.

4.

STABILITY As defined

ANALYSIS earlier,

USING

an outcome

HIERARCHICAL

POWER

is stable for a player if it is not advanta-

geous for the player to move away from the outcome. Stability analysis under HP can be carried out for any status quo outcome. This models the situation in which both players are aware that one player has HP, but the initial position may be any game outcome; it does not arise as part of the model. If the player with HP starts the departure, the departure-game outcomes are examined at odd length to determine whether the initial outcome is stable; if the player without HP starts the departure, then these outcomes are examined at even length. Another variant is a stability definition for an arbitrary outcome. When a particular player, say i, has HP, an outcome may be hierarchical power (i) stable [or HP(i) stable] for either player. Thus, HP is used to define a stability concept, which can be applied in any m X n game. Further, an outcome is an HP(i) equilibrium if it is HP(i) stable for both players in the hierarchy. A simple flow chart for finding the HP stability of outcomes is shown in Figure 1. Using this procedure, the stability of outcomes converging in fixed points or 2 or 4 cycles can be determined.

of games

To determine HP(l) stability for player k, consider departures in which player k moves first and player 1 moves last. If k = 1, these are Mi where h is odd, and if k = 2, they are Ml where h is even. A given cell of G has the indicated stability if its outcome is player l’s most preferred outcome among those in the corresponding cell of any eligible matrix, provided h is large enough that cycling has begun. For example, in Table 2 cycling begins at length h = 3 when the game converges to a fixed point. Outcomes (3,3) and (4,2) are HP(l) stable for player k = 1 [for outcome (3,3) compare the entries in row 1 and column 1 in MA and Mi, and similarly for (4,2) compare the entries of row 2 and column 1 of MJ and Mi]. Outcomes (3,3) and (2,4) are HP(l) stable when k = 2 [compare the entries of row 1 and column 1 of Mt and Mj for outcome (3,3), and those of row 1 column 2 of Mi and Mf for outcome (2,411, indicating that (3,3) is an HP(l) equilibrium of Chicken. The flow chart in Figure 1 indicates that the game is iterated until convergence. Then the

Algorithms

for Hierarchical

31

Power

0

Torn

FIG. 1.

Flow chart for determining

HP stability of outcomes.

32

M. DE, K. W. HIPEL, AND D. M. KILGOUR TABLE 3 PRISONER

S

DILEMMA

Player 2

Player 1

appropriate

matrix (eligible

1

m

matrix), depending

on which player has HP and

who moves first, is located, and the stability of outcomes in game determined. If the game converges in a 4-cycle, there are additional involved in determining the eligible matrix (see De [6]).

G is steps

Using these stability concepts, it is shown that HP tends to stabilize cooperative behavior in a noncooperative situation. In games like Chicken and Prisoner’s Dilemma, HP can single out and stabilize a Pareto-superior outcome which is reasonably preferable for both sides. This can be particularly valuable in games without much natural stability. For example the game of Prisoner’s Dilemma (see Table 3) is a well-known model of duopoly. In this model, the players are firms, 1 and 2, whose strategies are prices, H (high) and L (low); the payoffs reflect the firms’ profit at each of the four possible outcomes. Using the algorithm developed in this research, it was found that the Pareto-optimal outcome (3,3) is the HP outcome for both players. Furthermore, (3,3) is an HP equilibrium. However, the unique Nash equilibrium of this duopoly model is (2,2), which is Pareto inferior. Outcome (3,3) is obtained when both firms choose a high pricing policy. Therefore, HP indicates that the cooperative outcome (3,3) can be obtained in a noncooperative environment without actual collusion. It appears that the duopolists can charge higher prices and attain higher profits when they choose their actions sequentially, according to a well-defined hierarchy of knowledge and action. This suggests that HP may be of major importance the understanding of real-life conflicts. 5.

APPLYING

HIERARCHICAL

in

POWER

For a two-person game, when the player with power has been identified, a power concept such as HP can be used to calculate the outcome of the game. These power outcomes reflect how the specific definition models the effects of power in the game, and constitute the primary point of comparison among the definitions.

33

Algorithms for Hierarchical Power

describing the effects of Following De [6], some important definitions power definitions can now be formulated. These definitions create a new classification of the 78 strict ordinal 2 X2 games [18] and the 726 general ordinal 2 x 2 games [7]. A game is power neutral if the power outcomes for 1 and 2 are identical; otherwise it is power effective. In other words, a particular form of power is effective in a game of it makes a difference who holds it. A power-effective game is positively effective if each player does at least as well, and at least one does strictly better, with power. It is negatively effective if each player does no better, and at least one does worse, with power. In a power-effective game which is neither positively effective nor negatively effective, both players strictly prefer that a specific player, say k, has power. In this case, the game is said to have power consensus on k. As, an illustration, the HP outcome of Chicken is (3,3) no matter which player has HP; the game is therefore HP neutral. A computer-assisted application of HP to all 726 general ordinal 2 X 2 games [7] and the well-known class of 78 strict ordinal 2 X 2 games [18] is now described. Of all ordinal 2 X 2 games, 645 (89%) are HP neutral. In 623 games (97%) the HP-neutral outcomes occur in cells which are HP(l) stable and HP(2) stable for both players and hence constitute HP equilibria. The remaining 81 general ordinal games (11%) are HP effective. In 71 of these games (88%) HP is positively effective; in the remaining 10 (12%) there is HP consensus. Figure 2 shows the classification of all ordinal 2 X2 games. Notice that HP is not negatively effective in any of the 726 games.

I

HP

with XNM 623

Neutral

I

HP Effective

without XNM 22

FIG 2. Classification

2 HP consensus

71 HP positively effective a HP consensus

of the 726 general ordinal 2 X 2 games

34

M. DE, K. W. HIPEL, AND D. M. KILGOUR

;

12

55

FIG.3. Classification

The

class of 78 strict

games (86%). Figure 3 games which is analogous that a game is HP neutral tested in the class of 726

ordinal

none

11 HP positively effective

of the 78 strict ordinal 2 X 2 games.

2 X 2 games

[18] contains

67 HP-neutral

shows the classification of the strict ordinal 2 X2 to Figure 2. Prior to this study, it was conjectured iff it has an XNM equilibrium. This possibility was general ordinal games and the subclass of 78 strict

ordinal games. It was found to be true in the latter case, but not in the former (see De [6]). There are a number of other ways in which power definitions can be applied, in addition to the calculation of power outcomes. If the initial outcome is prespecified, a stability definition (such as HP stability or holding-power stability [ 141) can be used to assess how likely that outcome is to persist. Another line of research centers on the question of whether power tends to stabilize a situation, no matter who possesses it. In addition, some authors [2, 3, 61 have argued that power may lead to an outcome which is better for both players than would be achieved under strictly noncooperative play.

6.

DISCUSSION

AND CONCLUSIONS

A form of power asymmetry called HP has been developed for modeling conflict dynamics in hierarchical systems of decision makers. HP is applicable in two different situations: the first when the initial outcome is formed according to the rules of HP, leading to the HP outcome of the game, and the second when the HP stability of an initial outcome, given exogenously, is

35

Algorithms for Hierarchical Power

determined. The types of HP outcome that result when analyzing a 2 X2 game allow the entire set of 2 X 2 games to be classified according to power neutrality or effectiveness. Furthermore, the algorithms for computing HP outcome and HP stability have been outlined. The results obtained in this study indicate that the concept of hierarchical power provides a new method of gaining insight into the outcomes of conflicts in hierarchies. Proper use of the methodology of HP analysis can augment the ability of decision makers to anticipate the responses of competitors, leading to improved strategic decision making and superior performance.

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