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A Game-Theoretic Interpretation of Sun Tzu's The Art of WarSTOR

Emerson M. S. Niou; Peter C. Ordeshook

Journal o f Peace Research, Vol. 31, No.2 (May, 1994), 161-174.

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Journal of Peace Research, vol. 31, no. 2, 1994, pp. 161-174

A Game-Theoretic Interpretation of Sun Tzu'sThe Art of War*

EMERSON M. S. NIOUDepartment of Political Science,Duke University

PETER C. ORDESHOOKDivision of Social Sciences,California Institute of Technology

Over 25 hundred years ago the Chinese scholar Sun Tzu, in The Ar t of War, attempted to codify thegeneral strategic character of conflict and, in the process, offer practical advice about how to winmilitary conflicts. His advice is credited with having greatly influenced both Japanese military andbusiness practices, as well as Mao Tse-Tung's approach to conflict and revolution. Th e question,however, is whether or to what extent Sun Tzu anticipated the implications of the contemporary theoryof conflict - game theory. Th e thesis of this essay is that he can be credited with having anticipated theconcepts of dominant, minmax, and mixed strategies, b ut tha t he failed to intuit the full implications of

the notionof

equilibrium strategies. Thus, while he offers a partial resolutionof

'he-thinksthat-I-think'regresses, his advice remains vulnerable to a more complete strategic analysis. In judging Sun Tzu'scontribution to ou r understanding of strategy, however, we should keep in mind that resolving circularreasoning in some circumstances requires the use of advanced principles of probability theory andmathematics, and so we should not be surprised to learn that Sun Tzu's treatment of information isincomplete. Indeed, we should marvel at the fact that he understood intuitively as much as he did.

1. IntroductionTh e formulation of general strategic principles - whether applied to war, parlor

games such asGo, or

politics - has longfascinated scholars. For some, such as theChinese strategist Sun Tzu, this fascinationis motivated by the necessity for formulatingimmediate practical advice, while for othersthe motivation is merely by intellectual curiosity. Regardless of motivation, however,the study of strategic principles is of interestbecause it grapples with fundamental factsof human existence - first, people's fates areinterdependent; second, this interdependence is characterized generally by conflict

ing goals; and, finally as a consequence ofthe first two facts, war is not accidental butis the purposeful extension of a state'spolicies and must be studied in a rationalway.

Sun Tzu's The Art o f War, written morethan 25 hundred years ago, is, of course, ou rfirst written record of the attempt to understand strategy and conflict in a coherent and

This research was supported by NSF grant SES9223185 to Duke University and NSF grant SES-8922262 to the California Ins titute of Technology.

general way. Its age, however, is less important to us than the fact that it was writtenat a time of prolonged conflict within anemerging China - in an era in which theleaders of competing kingdoms possessedconsiderable experience in the conduct ofdiplomacy, strategic maneuver, and in theart of war. As such, then, we should presume that The Ar t of War codifies theinsights of an er a skilled at strategy and tactics. Th e better we understand it, the betterwe understand not only the era in which SunTzu wrote, but also the essentials of conflicttoday.

But just as we might suppose that Sun Tzu

offers insight into the past and the present,we cannot suppose that ou r understandingof strategy has not progressed in 25 centuries. New modes of analysis, including thedevelopment of decision theory and theapplication of mathematics, have all enteredthe domain of strategic analysis to generalize and refine ou r thinking. Thus, forscholars interested in understanding SunTzu's particular contribution, we mustaddress the issue of how best to interpretand analyze his writings in the context ofthese advances, since it is in this way that we


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162 Emerson M. S. Niou & Peter C. Ordeshook

maximize The Art o f War's contemporaryrelevance. This is the issue with which thisarticle deals. Rather than begin ou r discussion with Sun Tzu himself, we begin with

the theory of strategic behavior developedduring the latter part of this century calledgame theory. Briefly, game theory, whichwe can view as either a branch of mathematics or of political science and economics(Ordeshook, 1986), seeks to isolate general,abstract principles of decision-making whenthe outcomes of people's choices depend onwhat others decide and when everyone isaware of their mutual interdependence.Thus, while we might interpret much of TheAr t of War as modeling and analyzing a particular type of game - interactions of pureconflict - game theory is concerned with thegeneral issue of interdependent decisionmaking, including the possibility that peoplemay choose to cooperate so as to achievemutually beneficial ends. Hence, gametheory's applications include not only strategic military planning, but also an analysisof the decisions confronting the heads ofbusiness firms as they compete for profits ormarket share, of political candidates whomust compete in order to win elections, ofmembers of committees who compete toform alternative coalitions, and of nationstates who compete to secure advantageouspositions in alliances.

Game theory and The Ar t of War, then,each bring something different to ou r understanding of strategy. Game theory offersgenerality and mathematical precision and itallows us to ascertain the logical coherenceof our ideas about strategic interaction; SunTzu provides a specific application of

general principles, and demonstrates the artof rendering logical and abstract reasoningpractical. The plan of this essay, then, is toreview the essential components of gametheory in such a way that we can explore theextent to which Sun Tzu's writings are consistent with or illustrate parts of that theory.

Because it is unreasonable to suppose,however, that someone writing more than25 centuries ago could have anticipated allthe nuances of strategic interaction thatformal mathematical reasoning reveals to ustoday, we also want to learn which aspects

of Sun Tzu's writings fail to take accountfully of what we know generally about strategic choice. We begin Section 1 of thisessay with a discussion of 'pre-game' de

cision-making - of situations in which thereis only one decision-maker. Section 2 takesus to the core of game theory and describeshow interdependent choice situations differfundamentally from simple decision problems. In Section 3 we explore more carefullySun Tzu's writings as they apply to a particular class of games in which decision-makersimplement their actions sequentially, as inparlor games like Chess and Go. Section 4looks at games that better model the strategic situation that confronts decisionmakers in battIe - simultaneous move gamesand games of imperfect information, andintroduces the concept of a Nash equilibrium as a general solution for such games.Section 5 considers those situations that donot have simple Nash equilibria, and whichcause us to introduce the idea of a mixedstrategy. Here we argue that mixed strategies - strategies that leave something tochance - are not mere mathematical curiosities, but are in fact a central feature ofSun Tzu's strategic principles. Section 6studies the importance of information manipulation in war, and it is here that we findfar less compelling reasons to suppose thatSun Tzu anticipated fully the analytic structure characterizing contemporary gametheory.

2. Individual Decision-makingIn order to appreciate the lessons and perspectives of game theory as well as of SunTzu's insights, it is necessary first to considera situation in which there is a single decisionmaker whose actions we are trying to understand, and who must choose one action fromsome set of alternative actions. The usualdecision-theoretic representation of suchsituations requires a specification of thefollowing components:

(1) A list of the decision-maker's alternative actions, where this list is exhaustivein the sense that at least one action inthe list must be chosen by the decisionmaker in question and where the actions


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in the list are exclusive in the sense thatone and only one action can be taken.

(2) A list of possible outcomes (consequences of actions taken) that is also

exhaustive and exclusive.(3) A specification of the relationship between actions and outcomes - what outcome prevails after a specific act ischosen.

Appropriately, Chapter 1 of Sun Tzu'stext begins with a specification of those elements of a strategic situation that mediatebetween outcomes and actions - domesticpolitics (moral influence), weather, terrain,and doctrine (organizational efficiency ofthe state and armies). The remaining factor,command (quality of leadership), is a decision variable pertaining to actions andstrategies.

An additional important component ofany decision-making environment is a specification of the goals (preferences over outcomes) of the participants. Of course, thegoals of the king and an Army's leaders areobvious - to win. Indeed, winning is anecessity if the state is to survive: l 'War is a

matter of vital importance to the state; theprovince of life or death; the road to survival or ruin' (I, 1), and 'Victory is the mainobject of war' (II, 3). Refining goals further,Sun Tzu is concerned also that too myopic aview of victory can weaken a state. Thus,'What is essential in war is victory, not prolonged operations ' (II, 21).

More specifically, the first few chapters ofThe Ar t of War contain a number of passages pertaining to protracted conflicts,which we can quite directly interpret to

mean that one ought to avoid conflicts thatdeplete one's resources. 'For to win onehundred victories in one hundred battles isnot the acme of skill. To subdue the enemywithout fighting is the acme of skill' (III, 3).Thus, one's goal should not merely be thatof winning a battle, but to win at minimalcost so as not to deplete one's resources forfuture conflicts. In addition, one should usetactics that maximize the gains from victory.'Generally in war the best policy is to takethe state intact; to ruin it is inferior to this'(III, 1).

A Game-Theoretic Interpretation 163

Sun Tzu does not ignore the goals of thosewho must actually implement plans in warfare - soldiers:

The reason troops slay the enemy is because theyare enraged . . . [and] they take booty from theenemy because they desire wealth . . . . Therefore,when in chariot fighting more than ten chariots arecaptured, reward those who take the first (II, 16-18).

In the simplest decision-making environment, decision-making under certainty, weassume that each action leads to a welldefined, specific outcome. In this instance,we need only know the decision-maker'spreference order over outcomes, since wecan suppose that the action leading to themost preferred outcome will be chosen.However, a more general environment, de-cision-making under risk, allows for thepossibility that we or the decision-maker inquestion are uncertain as to what outcomeprevails after a specific action is taken.

In simple decision theory we assume thatthe consequences of action are contingenton the 'choices' of nature, where by naturewe mean an entity that operates withoutpurpose - an entity that cannot be said to

operate in the pursuit of some goal. Becausenature has no purpose - because it is neitherpurposefully malevolent nor benevolent -we suppose that it takes its form outside theinfluence of ou r actions. Thus, we merelysuppose that we can associate probabilitieswith each of nature's alternative choices.

On e example of this situation is a farmer'sdecision about what crops to plant. Theseeds a farmer might sow are the farmer'salternatives; yields are outcomes; preferences are dictated by the relative profit

ability of outcomes; and nature's choicesmight correspond to weather patterns forthe year. In this instance, we would not normally view nature as either a benevolent ormalevolent creature with goals; instead, wewould merely associate probabilities withalternative weather patterns (based, presumably, on the historical record).

With this conceptualization, the consequence of a specific action is a lottery overoutcomes, and one question that concernsdecision theorists is how people evaluatethese lotteries - how preferences over


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specific outcomes determine preferencesover lotteries over these outcomes. Generally, game theorists assumed that we canassign a value to each outcome - a numeri

cal representation of preference - such thatwe can compute the expected value of eachaction and predict that a decision-maker willchoose the action that yields the greatestexpected value.

To illustrate so that we can later add somestrategic complexity, consider Figure 1,

Figure 1. Decision-making Under Risk

A

B

p

4

Nature

I -p

2

3

which assumes that a decision-maker,person 1, must choose between A and B,that nature has two alternatives which itchooses with pr.obabilities p and 1 - Prespectively, and that the values assigned bythe decision-maker to specific outcomes areas shown. In this instance, then, theexpected value of alternative A, E(A),equals 4p + 2(1 - p) = 2 + 2p; and theexpected value of B, E(B), equals p + 3(1 -p) = 3 - 2p. Thus the decision-makerchooses A if E(A) > E(B), or equivalently,if 2 + 2p > 3 - 2p which requires that p >114. Hence, if p = 114, then the decisionmaker is indifferent between A and B; andif p < 114, then the decision-maker prefers B

toA.On occasion, Sun Tzu offers advice thatwe might interpret as corresponding to thissimple structure. For example:

There ar e five methods of attacking with fire . . . .There ar e suitable times and appropriate days onwhich to raise fires. 'Times' means when theweather is scorching hot; 'days' means when th emoon is in Sagittarius, Alpharatz, I, or Chenconstellations, for these ar e days of rising winds(XII, 1, 4, 5).

But there are only a few instances in The Ar tof War in which he focuses on specific de-

cision-making problems with nature as theprimary adversary. I t is evident that SunTzu realized that a complete specification ofthe strategic structure includes the moves

available to nature, the strategies of the decision-maker in question, and the strategiesavailable to all antagonists, 'Know theenemy, know yourself; your victory willnever be endangered. Know the ground,know the weather; your victory will then betotal' (X, 26). Thus, to explore mattersfurther we must turn to game theory and theanalysis of interdependent choice.

3. The Special Relevance of Game Theory

The preceding discussion of simple decisionmaking merely serves as a point of departure, and, in fact, Sun Tzu offers an earlyadmonition that reveals that he consideredsuch a representation and any discussion ofalternative choices and goals as preliminary:'All warfare is based on deception' (I, 17).We cannot deceive nature, because by definition nature is an unthinking anonymousentity. Thus, all warfare entails the strategicinteraction of two or more people, and thenext issue we must address is the distinction

between decision-making against nature anddecision-making against another person.

To see that this distinction is important,suppose we replace nature in Figure 1 with asecond decision-maker, person 2, who mustalso choose between two alternatives, say Cand D. Thus, the consequences of person 1 sdecisions depend on what 2 does, and viceversa. Suppose, moreover, that this personevaluates the outcomes differently from 1 -specifically, suppose as is most likely thecase in military conflict situations that 2 hasprecisely the opposite preferences of 1.Figure 2 portrays this new situation, wherethe first number in each cell corresponds asbefore to l 's preferences, and where the second number corresponds to 2's preferences.

Figure 2. Two-Person Constant-Sum Game

A

B

C

4,1

1,4

D

2,3

3,2


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There are now two ways in which we cancharacterize the scenario that confrontspersons 1 and 2: (1) perfect information -one person mUSt choose first and the other

chooses second, after learning the choice ofthe first person; and (2) imperfect information - both persons choose either simultaneously or in ignorance of what each otherwill do. With perfect information, case (1),the situation can be analyzed straightforwardly. If , for example, person 1 choosesfirst, then he knows that if he chooses A,person 2 will respond with D, whereas if hechooses B, person 2 will respond with C.Person 1, then, ought to choose A becausethis alternative maximizes his minimumgain.

Matters achieve considerably greatercomplexity with imperfect information, case(2), and it is here that we see the profounddifference between Figures 1 and 2. Specifically, we can now imagine the followingthought process for person 1 as he contemplates his options:

I think that I should choose A, because it offers memy best choice and provides a better guarantee thanB (2 versus 1). But then . . . if person 2 reasons as Ido, he will infer that I will choose A, in which casehe will choose D, in which case I should choose B(since I prefer a payoff of 3 to a payoff of 2). Butthen ... if person 2 reasons as I do further, he willinfer my decision to switch to B in responsc to hischoice of D, in which case he will conclude that C ishis better choice; in which case I ought to choose A.But then, again, if he anticipates my reasoning, hewill conclude that I will choose A, in which case hewill respond with D . . . and so on.

Of course, circular reasoning will also characterize person 2's thinking. So suppose that

person 1, exasperated and perplexed, concludes that person 2 will choose between Cand D with equal probability. In this instance, the expected value from A equals4/2 + 212 = 3, whereas the expected valuefrom B equals 3/2 + 112 = 2. Person 1, then,might decide to choose A. But again, ifperson 1 believes that 2 will anticipate l 'stentative speculation that 2 will choose randomly, I should also anticipate that 2 caninfer 1 s choice of A, in which case 2 will notchoose randomly at all and will insteadchoose D, because it is a best response to A.


But, once again, if 1 anticipates this reasoning himself . . . and so on and so forth.

Such circular reasoning does not arise insimple decision-making scenarios, since by

assumption nature does not reason. Thus,the situation changes drastically once weadmit the possibility that 'nature' is not ananonymous entity, bu t is a person capable ofthe same strategic thought as ou r initial decision-maker. An d this change requires thedevelopment of new tools for thinking aboutrational individual action.

Implicit in ou r argument about circularreasoning, however, is the presumption thatnot only is each decision-maker aware ofthis situation, but each is aware that theother is aware, and so on. Game theoristsrefer to this assumption as the assumption ofcommon knowledge. In the context of SunTzu's analysis, such common knowledge willarise if both sides to a conflict have read TheAr t of War (or have an equivalent source ofadvice), and if both sides are aware that theother side has access to this volume. Inassessing the game theoretic credentials ofThe Ar t of War, then, one question we mustask is whether and to what extent Sun Tzu's

advice accommodates alternative solutionsto the dilemma of circular reasoning thatcommon knowledge admits.

In order to make this evaluation, we mustfirst discuss the general components of agame. Briefly, these components are muchthose of the decision problem we outlinedpreviously, except that now we must makeallowance for the fact that there are two ormore decision-makers. We also must makeallowances for the possibility of complexstrategic interaction in which people interact

over long intervals of time. To illustratethese ideas as we present them, we encourage the reader to keep a parlor game such aschess in mind. A description of any game,then, necessarily includes the following:

(1) A list of relevant decision-makers orplayers. In chess, there are but twoplayers - white and black - whereas incard games there can be many moreplayers.

(2) A description of the strategic situationin terms of who moves at what time, and


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in what order. In chess, white movesfirst, then black, etc. Chess also allowslong sequences of moves. In othergames, however, players may have to

move simultaneously, and they mayonly allow short sequences, includingonly a single choice for each player.

(3) A specification of the alternative actionsthat each person can take at everyopportunity that person has to act. Inchess, White's first move allows tenchoices - each of eight pawns, pluseither knight. This list of alternativeswill expand or decrease as the gameproceeds, as other pieces are freed oreliminated.

(4) A specification of what each personknows about the previous choices ofother players. In chess, a person knowsall the earlier moves of the opponent. Inother games, such as poker, thesemoves may be hidden from view; and inother circumstances, players may haveto make some choices simultaneously.

(5) A description of each player's goals - anevaluation of all of the final outcomesthat the game allows. In chess, as in

war, these outcomes include victory anddefeat, with the presumption that allplayers prefer victory.

The final idea we must introduce is that of astrategy - a plan of action for how to playthe game. This plan - one for each player-specifies the alternative action a playershould choose for each and every contingency that that player can encounter asthe game unfolds. Strategies, then, consistof contingent actions, and take the form: ' I fmy opponent does . . . then I will respondwith ... but if my opponent chooses ...then respond with ... . In chess, a strategyfor white consists of an opening move, aresponse to each of black's potential responses to the opening move, and so on. .

In the following passage, Sun Tzu ISclearly seeking to spell out a complete strategy - a sequence of choices contingent onwhat an opponent chooses:

I f I first occupy constricted ground [then] I mustblock the passes and await the enemy. [But] I f theenemy first occupies such ground and blocks thedefiles [then] I should not follow (attack) him; [But]

if he does not block them completely [then] I maydo so (X, 5).

I t is important to appreciate the centralityof this concept of a strategy to game theory.I f we have modeled the situation well, wecan anticipate every possible contingencythat a player can confront, including thosecontingencies established by nature. Thus, astrategy should allow a player to adjust hischoices in accordance with things learned asthe game proceeds. And because a strategythereby allows for shifting tactics dependingon all possible contingencies, we can thinkof the outcome of a game as being fullydetermined after all players choose their

strategies (up to the uncertainties created bynature), but before those strategies areactually implemented. Once strategies arechosen events unfold according to plans, asthe full character of each player's strategy isrevealed over time.

Unsurprisingly, Sun Tzu seems to havefully appreciated this idea. In addition to hisearly emphasis of correctly assessing situation, Sun Tzu argues that:

. . . what is of supreme importance in war is toattack the enemy's strategy (III, 4).

. . . determine the enemy 's plans and you will knowwhich strategy will be successful and which will not(VI, 20).

And as water shapes its flow in accordance with theground, so an army manages its victory in accordance with the situation of the enemy. And as waterhas no constant form, there are in war no constantconditions. Thus, one able to gain the victory bymodifying his tactics in accordance with the enemysituation may be said to be divine (VI, 28-30).

and as to consequences,

Know the enemy and know yourself; in a hundredbattles you will never be in peril. When you areignorant of the enemy but know yourself, yourchances of winning or losing are equal. If ignorantboth of your enemy and of yourself, you are certainin every battle to be in peril (III, 31-33).

Thus a victorious army wins its victories beforeseeking battle; an army destined to defeat fights inthe hope of winning (IV, 14).

From the above passages, Sun Tzu's generalintent is clear - to analyze the diversity ofinterdependent choice situations in warfareand to deduce efficient strategies - plans ofaction that lead to victory, broadly defined.


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4. Solving Sequential GamesThe issue, then, is how to identify goodstrategies so that we can resolve the circularreasoning that arises from games like the

one we offer in Figure 2. At this point wemust return to the distinction offered earlierbetween two types of situations: (1) situations in which the players make theirchoices in sequence - one after the other -as in parlor games like chess, go, and tic-tactoe; and (2) situations in which players mustformulate strategies and make choicessimultaneously, or, equivalently, in whichchoices must be made in ignorance of theopponent's actions. In this section weconsider the first type of situation, games ofperfect information.

Consider a game like chess in which oneplayer moves first, then the other, then thefirst again, then the second again, and so on.Suppose we can map out all actions insequence so that all possibilities arecovered. Some sequences will end quickly,as when one person makes a series of skilledmoves against an ineffective opponent.However, it is a well-known result of gametheory that, regardless of the complexity of

the situation,in

principal the game's outcome will be determinate and we will beable to specify an unambiguously best strategy (plan of action for playing the game) foreach person.

This result is general. For example, recallthat if the game in Figure 2 is played sequentially with person 1 choosing first, then, aswe have previously argued, the eventualoutcome is (2, 3). Restated in terms of strategies, person 1 confronts only a single contingency - the necessity for choosing first.

Thus, he has only two strategies, A and B.Person 2, on the other hand, confronts twocontingencies - person 1 s two choices - andthereby he has four strategies:

s1: Choose C regardless of what person 1chooses.

s2: Choose C if 1 chooses A, but chooseD if 1 chooses B.

s3: Choose D if 1 chooses A, but chooseC if 1 chooses B.

s4: Choose D regardless of what person 1chooses.


Looking at Figure 2 directly, however, wecan see that if 1 chooses A, then 2 willchoose D, whereas if 1 chooses B, then 2will choose C. Clearly, then, 1 should

choose A and 2 should respond with D.Thus, with sequential decisions, the outcome (2, 3) will prevail. In other words, wecan straightforwardly deduce that person 1will choose the strategy 'A ' and that person2 will choose the strategy 'Choose D if 1chooses A, but choose C if 1 chooses B'.

Game theory tells us, then, that in principle games such as chess can be 'solved'.Practical considerations, of course, render itimpossible to map out all moves in suchgames, which keeps us playing them as testsof skill and experience. On the other hand,a simpler game such as tic-tac-toe, becauseit too can be solved, is interesting only tochildren who have not yet comprehended itsstrategic structure. The skill and experienceof playing chess well, though, is much likethe skill at playing any complex game -using our experience and skills to simplify acomplex strategic structure so that we canlearn strategies that ought to be avoided,and so that we can identify when ou r oppo

nentis

using a good or a poor strategy. Skilland experience also help us reduce a game'scomplexity so that, although its simplifiedform may not match that of tic-tac-toe, itsgeneral principles can be understood andoptimal contingent plans formulated.

And it is evident that the formulation ofoptimal contingent plans characterizes SunTzu's intent. For example. The doctrine ofwar is to follow the enemy situation in orderto decide on battle' (XI, 60).

We should consider one final issue with

respect to games in which the players movesequentially - namely, whether there is anyadvantage to moving first or second. Ofcourse, whether and what type of advantages accrue to the first mover in a game inwhich the players move in sequencedepends on the specific structure of the situation under consideration. For example,Sun Pin, writing one hundred years afterSun Tzu, recounts the now famous story ofthe horse race in which the opponent possessed three horses generally superior toT'ien Chi, the commander and chief of Ch'i.


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Sun Pin's advice was simple: pair the worsthorse against the opponent's best, the bestagainst the opponent's second best, and thesecond best against the opponent's worst. In

this way, two out of three races can be won.Clearly, moving second by having theoption of determining the pairings of thehorses confers the advantage on T'ien Chi.

In order to interpret this example properly, however, we must keep in mind thedistinction between actions and strategies.Strategies are plans - rules for selectingactions as contingencies arise - whereasactions are merely pieces of the plan. SunPin's example is especially simple becauseaction (the assignment of horses) is equivalen t to a strategy. In chess, on the otherhand, as well as in the complex maneuversof war, the equivalence between actions andstrategy is lost.

With respect now to the issue of advantages; for the class of games we are considering here - games of pure conflict - theadvantage belongs to the player who revealshis strategy last. But this does not necessarily mean that the advantage belongs to theplayer who makes a second choice in a

sequential game.In

chess, for example, asin tic-tac-toe, the advantage belongs towhoever moves first, which is different fromrevealing one's strategy. Now consider SunTzu' s view of advantages. In one instance heargues that rather than engage the enemy asthe first move, certain terrain dictates forcing the enemy to move first:

Ground equally disadvantageous for both theenemy an d ourselves to enter is indecisive. Thenature of this ground is such that although theenemy holds ou t a bait I do not go forth but entice

him by marching off. When I have drawn out halfhis force, I can strike him advantageously (X, 4).

But at another point Sun Tzu appears toargue for the advantage of the first move:

Generally, he who occupies the field of battle firstand awaits his enemy at ease; he who comes later tothe scene and rushes into the fight is weary. Andtherefore those skilled in war bring the enemy to thefield of battle and are not brought there by him (VI,1,2).

Ground which both we and the enemy can traversewith equal ease is called accessible. In such ground,he who first takes high sunny positions convenientto his supply routes can fight advantageously (X, 2).

This apparent confusion, however, isresolved if we keep in mind the differencebetween strategy and action. In the firstcited passage, the positions of the players

have already been determined - the circumstances of the game are fixed - and from thatpoint it is best to reveal one's strategy second. In contrast, in the second cited passage, the field of battle has yet to be determined, and Sun Tzu is in fact arguing that itis better to be the one who dictates whichgame is to be played or , equivalently, whichplayer is to be assigned which position in thegame. In this second passage, then, Sun Tzuis referring to the first move in the game.Thus, there is no confusion, and we can conclude that Sun Tzu does in fact appreciatethe advantages of choosing one's strategysecond.

5. Games With Imperfect InformationPrewar preparations - evaluating one'sdomestic power, recruiting skilled commanders, training troops, and choosingwhether or not to engage in war - proceedsequentially so that one's character, as wellas that of an enemy, is revealed as eventsunfold. Tactics are chosen by differentrules. Th e success of battIe tactics dependson contingencies, including the opponent'spreparations and tactics, that often becomeapparent only after it is too late to conditionon them, and decisions must be made withimperfect information. In some instances,this imperfect information arises becausechoices are made simultaneously by all antagonists, whereas at other times choices arenot simultaneous but information is imper

fect because choices are hidden from view.Regardless of its source, the game theorist'sapproach to imperfect information is toidentify classes of games for which this factmatters little. Referring to Figure 3, notice

Figure 3. Two-Person Zero-Sum Game with DominantStrategy

A

B

c

4,1

2,3

D

3,2

1,4


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that each player has a dominant choice - Afor person 1 and D for person 2 - where bydominant we mean an action that is betterthan all others, regardless of what other

persons choose. In Figure 3, A is better thanB for person 1 regardless of what 2 chooses,whereas D is better than C for 2, regardlessof what 1 chooses. When decision-makershave dominant choices the analysis of ultimate decisions and outcomes avoids the circular reasoning that we applied to Figure 2.

We should emphasize that the possibilityof a dominant choice is not altogether unrelated to the issue of simultaneous versussequential choice. Indeed, a game withsequential choices such as chess also occasions dominant strategies, and thus wecan view the concept of dominance as a generalization of previous analysis. To see whatwe mean by this, recall that if the game inFigure 2 is played sequentially with person 1choosing first and person 2 choosing second,person 1 has two strategies, A and B,whereas person 2 has four strategies whichwe denoted sl, s2, s3, and s4. Consider nowthe game portrayed in Figure 4, which de-

Figure 4. Normal Form of a Sequential Move Game

A

B

s l

4,1

1,4

s2

4,1

3,2

s3 s4

2,3 2,3

1,4 3,2

scribes the outcome that prevails for each ofthe eight possible joint choices of strategiesby persons 1 and 2. Notice in particular that

although person 1 does not have a dominantstrategy, s3 is dominant for person 2 - it isnever worse and is sometimes better than sland s4, and it is uniformly better than s2.Thus, person 2 should choose s3. But, sinceperson 1 is also assumed to be cognizant ofthis game, person 1 should be able to inferthat 2 will use s3, in which case person 1should choose A. This reasoning, of course,leads to precisely the same outcome wededuced earlier - person 1 chooses A, 2chooses D in response, and the outcome (2,3) prevails.


Sequential choice, then, is sufficient togenerate dominant strategies (but notnecessary), and although, as we argue later,Sun Tzu seems to have understood the

consequences of moving from a game ofsimultaneous choice to one with sequentialchoices, much of his strategic analysis consists of identifying dominant and dominatedstrategies. First, with respect to dominatedstrategies - strategies that ought to beavoided in any context:

Yo u should not encamp in low lying ground . . . .Yo u should not linger in desolate ground . . . . Thereare some roads not to follow; some troops not tostrike; some cities not to assault; and some groundwhich should not be contested (VIII, 1-7).

. . . do not ascend to attack .... When an advancingenemy crosses water do not meet him at the water'sedge . . . . Do no t take positions downstream (IX, 2-6).

And with respect to dominant strategies,

In enclosed ground resourcefulness is required. Indeath ground, fight (VIII, 5, 6).

Encamp on high ground facing th e sunny side; Fightdownhill. ... After crossing a river you must movesome distance away from it. It is advantageous toallow half his force to cross and then strike . . . .Cross salt marshes speedily (IX, 1-8).

We have introduced this notion of a dominant strategy not only because it helps usinterpret parts of The Ar t of War, but alsobecause the existence of such strategieshelps us avoid circular reasoning. Specifically, notice the special feature of the cellcharacterized by the joint choice of (A , D)in Figure 3 - once 'at' such a cell, neitherplayer has any incentive unilaterally tochange his decision. Indeed, it is this characteristic of (A , D) that terminates circularreasoning. The existence of dominant strategies, however, is not essential to offering asolution to a game. Consider Figure 5 and

Figure 5. Constant-Sum Game with a Nash Equilibrium

A

B

C

D

8,1

7, 2

2,7

E

3,6

5, 4

4,5

F

1,8

6,3

9,0


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the cell corresponding to the joint choice of(B, E). Notice first that neither player has adominant choice - for example, A is best for1 if 2 chooses D, B is best if 2 chooses E, and

C is best if 2 chooses F. Nevertheless, onceat (B, E) neither player has a unilateralincentive to defect to some other choice.Thus, (B, E) terminates circular reasoningas well - in other words, the existence ofdominant choices is sufficient to terminatesuch reasoning, but they are not necessary.

Because it terminates circular reasoning,cells such as (B, E) - called Nash equilibrium after the theorist who proved a numberof important results about them for differentclasses of games - are profoundly importantas an idea about ultimate choices. Consequently, game theorists have devoted considerable efforts analyzing the properties ofequilibria, refining Nash's original formulation, applying those refinements tostations of far greater complexity than thesimple games we describe here, and testingideas in empirical contexts. I f there is a critique that can be directed against Sun Tzu'sanalysis of war, though, it is that he seems topay too little heed to the necessity of resolving the dilemma of circular reasoning, andas a consequence he fails to infer this concept of an equilibrium as a solution to interdependent decision-making. Put differently,he fails to take full account of the possibilitythat, in addition to the king he is advising,enemy kings have also read The Art of War,and the fact of this common knowledge isitself common knowledge. In that event,only the notion of an equilibrium can beused to formulate strategic plans and toresolve circular reasoning.

Despite this criticism, however, we canfind within game theory itself a reason forsupposing that this failure does not necessarily negate the value of Sun Tzu's adviceand analysis. Specifically, one fact aboutNash equilibria that applies to the types ofgames that especially concerned Sun Tzu -games of pure conflict in which one person'sgain is another person's loss (called zerosum or constant-sum games) - is that decision-makers are assured of achieving anequilibrium if they abide by a simple rule ofthumb in choosing their actions. Suppose a

person assumes that his opponent is as intelligent as he is and is capable of anticipatinghis thoughts. In this event, the decisionmaker whose actions we are studying should

'assume the worst' - should assume thatregardless of what action he takes, his opponent will take best advantage of him. Barring the assumption that one's opponent issomehow less capable than oneself - alwaysa dangerous supposition and likely to lead tounpleasant surprises - a person should thenchoose the strategy that maximizes one'sminimum gain, or equivalently, minimizesone's maximum loss (called a minmax strategy).

Applying this argument to the game inFigure 5, notice that the minimum gainperson 1 associates with A is 1, from B it is5, and from C it is 2. Thus, person 1 shouldchoose B. Similarly, the minimum gain toperson 2 from D is 1, for E it is 4, and fromF it is O. Thus, person 2 should choose E.More interestingly, notice now that thisreasoning leads to the joint choice of (B, E) ,which is the game's Nash equilibrium.Hence, in the case of games of pure conflict,a prudent strategy leads to actions that areconsistent with the strategic imperativesproscribed by game theory.

Although we cannot find any directreference in Sun Tzu to the notion of anequilibrium as a means of terminating circular strategic reasoning, it is not unreasonable to suppose that he nevertheless graspedthe essence of a strategy designed to minimize one's losses, but which would nevertheless take advantage of an unskilledopponent.

Anciently the skillful warriors first made themselvesinvincible and awaited the enemy's moment of vulnerability (IV, 1).

Therefore the skilled commander takes up a position in which he cannot be defeated and misses noopportunity to master his enemy (IV, 13).

Invincibility depends on one's self; the enemy's vulnerability on him. It follows tha t those skilled in warcan make themselves invincible but cannot cause anenemy to be certainly vulnerable. Therefore it issaid that one may know how to win, but cannotnecessarily do so (IV, 2, 3, 4).

Sun Tzu, then, takes us at least part of theway towards the notion of an equilibrium in


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that if both sides to a conflict heed theadvice of adopting strategies that minimizeone's potential losses against a skilled opponent but that are also viable against a less

skilled player, then an equilibrium prevails.Thus, as long as the game under consideration is constant-sum or zero-sum, and aslong as an equilibrium exists in simple strategies, Sun Tzu provides us with the requisitetools for achieving an equilibrium and fortaking advantage of an opponent who failsto act accordingly.

6. Mixed StrategiesOne consequence of the preceding discussion is that whether or not a game ischaracterized by perfect or imperfect information matters little if there is a dominantchoice or a well-defined equilibrium for agame. Players can 'solve' the game andarrive at determinate outcomes. However,military conflicts typically have a differentstrategic character. Specifically, they generally do not have an equilibrium in simplestrategies. Thus, we must be careful aboutou r arguments for the relevance of the

notion of an equilibrium, because if gamesdo not always have equilibria then this nonexistence precludes the possibility of scientific generality.

With this in mind, let us return to thegame in Figure 2, which we used to illustratecircular reasoning and which, at first glance,does not appear to possess an equilibrium.Recall our suggestion that person 1 mightdecide, out of exasperation, that person 2will make a random choice. We tentativelyrejected this idea because such an assump

tion did not avoid the circular reasoning wesought to avoid. However, notice that weassumed that person 2 used a particular lottery, 50-50, and we did not check whetherall such lotteries led us in a cycle.

So suppose more generally that person 2chooses between alternatives C and D withprobabilities p and 1 - p, and suppose alsothat person 1 chooses between A and B withprobabilities q and 1 - q. Notice now that ifperson 1, given 2's probabilities of p and 1 -p, is not indifferent between the lotteriesprovided by A and B, then he should switch


to one or the other with certainty. That is,person 1 would be willing to stay with q and1 - q if and only if the expected value heassociates with A equals the expected value

he associates with B, as those expectedvalues are determined by person 2's strategy. For the game in Figure 2, this equalityrequires that

4p + 1(1 - p) = p + 3(1 - p)which we can solve, and conclude that p =114. The same argument applies, of course,to person 1 - 1 should be willing to give Aand B some weight if and only if 2 is indifferen t between C and D, which requires that,

q+

4(1 - q) = 3q+

2(1 - q),which we can solve to give q = 1/2.

The conclusion we reach here, then, isthat if person 1 chooses randomly betweenA and B, and if person 2 chooses C withprobability 114 and D with probability 3/4,then neither person has any incentive toshift unilaterally to any other lottery(including degenerate lotteries in which Aor Band C or D are chosen with certainty).Thus, for the situation portrayed in Figure

2, there exists a mixed strategy Nash equilibrium.What gives this notion of a mixed strategy

equilibrium special relevance is the important theorem proved by Von Neumann &Morgenstern (1944), which establishes thatevery n-person game in which each decisionmaker has a finite number of choices has atleast one equilibrium in either mixed orpure strategies. 2 Thus, the potential scientific generality of the Nash equilibrium concept is established.

However, notice that a mixed strategy (1)minimizes one's vulnerability to an equallystrategic opponent, and (2) takes advantageof an opponent who errs. Moreover, the useof a random device in particular ensures thatone's tactics do not fall into a pattern thatan opponent can detect. Admitting mixedstrategies into one's arsenal of choices hasthe consequence of taking a finite number ofpure strategies and rendering one's choicesinfinite in number. Moreover, the specialcharacter of a mixed strategy is that evenafter pure choices are revealed, an oppo-


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nent cannot be certain that a choice thatappears inferior in the short term is not partof a grander, more all-encompassing plan -they preclude the possibility that an enemy

can infer a future choice with certaintybased on previous actions.We cannot say whether Sun Tzu ever ex

plicitly advocated the use of random devicesto disguise strategic intent, nor do the commentators on his writings offer clarification.Nevertheless, we can offer a possible interpretation of his distinction betweennormal (cheng) and extraordinary forces(ch'i) that renders the notion of a mixedstrategy a centerpiece of his analysis: 'Inbattle there are only the normal and extraordinary forces, but their combinations arelimitless; none can comprehend them all'(V, 8-11).

In Chinese, the character Ch'i meansuncommon, unusual, or atypical. A strategyis ch'i if you think your opponent does notanticipate it. Consequently, no strategy isalways ch'i or always cheng. A cheng can bea ch'i, and vice versa. I t is whatever youthink your opponent thinks about yourstrategy. So the concept of ch'i and cheng

already reflects the strategic thinking between the antagonists. 3 I f we interpret ch'ias a strategy that your opponent thinks youare unlikely to use and cheng as a strategythat you are more likely to use, then tomake sure that any strategy you use will beregarded as ch'i by your opponent , you mustrandomize your choices to keep the enemyguessing. By interpreting ch'i and cheng inthis way, we can give fuller meaning to SunTzu's admonition that:

It is according to the enemy's shapes that I lay th eplans for victory, but the multitude does no t comprehend this. Although everyone can see theoutward aspects, none understands the way inwhich I have created victory. Therefore, when Ihave won a victory, I do not repeat my tactics bu trespond to circumstances in an infinite variety ofways (VI, 25, 26).

Admittedly, there is an alternative way ofinterpreting ch'i and cheng: a strategy isregarded by your opponent as ch'i if youropponent has no idea whether it is possibleor feasible. I f we accept this interpretation,then we cannot so easily (if at all) set this

part of Sun Tzu's analysis into a gametheory mold, since game theory assumesthat all players are aware of all strategiesavailable to an opponent. Nevertheless,

having interpreted ch'i and cheng as evidence that Sun Tzu appreciated the role ofmixed strategies, we can offer one criticismof his analysis. Recall that in the previoussection we also cited evidence that Sun Tzuunderstood the role of minmax pure strategies. The difficulty, however, is that if agame possesses an equilibrium only inmixed strategies, minmax pure strategiescannot yield an equilibrium and, thus, theycannot terminate cyclical reasoning. And,unfortunately, what we cannot discover inThe Ar t of War are any clear guidelines forascertaining when a strategist should chooseminmax strategies and when he shouldabide by mixed strategies. In this sense,then, Sun Tzu's analysis is incomplete.

7. Secret AgentsSun Tzu ends his text with a chapter onsecret agents and with the admonition that:

... only th e enlightened sovereign and the worthygeneral who are able to use the most intelligentpeople as agents ar e certain to achieve great things.Secret operations are essential in war; upon themth e army relies to make its every move (XIII, 23).

The emphasis Sun Tzu places on secretagents is understandable owing to the enormous strategic advantage to be gained fromknowing an opponent's strategy beforehand. In effect, the role of the secret agentis to allow a decision-maker to condition hisactions on a richer information base and to

render moves sequential rather than simultaneous. To see the advantages of thischange, consider, for example, the game inFigure 2. I f choices are made simultaneouslyor, equivalently, if no player in the game hasperfect information, and if each person useshis equilibrium mixed strategy, then theexpected payoff to each person is 2.5. Onthe other hand, as we have already seen, ifperson 2 moves after 1, but knows personl 's choice beforehand, then the final payoffsare 2 to person 1 and 3 to person 2. Moreover, if person 1 is unaware of 2's better


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information and continues to employ amixed strategy, then 2's expected payoffrises higher to 3.5, and person 1 s drops to1.5: 'The enemy must not know where I

intend to give battle. For if he does notknow where I intend to give battle he mustprepare in a great many places' (VI, 14).

The role of secret agents, therefore, isclear - to allow a decision-maker to condition on his opponent's choice and to proceed through the play of the game as if itwere a sequential game:

Now the reason the enlightened prince and the wisegeneral conquer the enemy whenever they moveand their achievements surpass those of ordinarymen is foreknowledge. What is called 'foreknowledge' cannot be elicited from spirits, nor from gods,nor by analogy with past events, nor from calculations. It must be obtained from men who knowthe enemy situation (XIII, 3, 4).

For the particular class of games withwhich Sun Tzu is concerned, choosing one 'sstrategy with foreknowledge of an opponent's choices is always advantageous -regardless of whether the opponent is awareof one's foreknowledge or not. The issuewith which Sun Tzu seems to have the

greatest difficulty, however, is the possibility that both sides will use agents to feedeach other false information. He advisesthat:

It is essential to seek out enemy agents who havecome to conduct espionage against you and to bribethem to serve you. Give them instructions and carefor them. Thus double agents are recruited and used(XIII, 17).

But what of the possibility that doubleagents are triple agents, and so forth? Once

this question is admitted, we return again tothe dilemma of circular reasoning that gametheory tries to resolve. Now, however, thegame is characterized not only by imperfectinformation but also by incomplete information, and the players must be concernedwith what others believe about them - abouttheir capabilities and their intent - as a function of their choices. Thus, choices must beselected not only to manipulate outcomesdirectly, but also to manipulate them indirectly through the manipulation of beliefs.And when all players know that such ma-

A Game Theoretic Interpretation 173

nipulations are possible, the analysis ofstrategy becomes doubly complex.

Earlier we interpreted Sun Tzu's admonition that 'all warfare is based on deception'

as implying that game theory as opposed tosimple decision theory is required to understand his analysis. Certainly, mixed strategies are one form of deception. However,it is also evident that deception meant moreto Sun Tzu than merely randomizing one'schoices:

. . . when capable, feign incapacity; when active,inactivity. When near, make it appear that you arefar away; when far away, that you are near. Offerthe enemy bait to lure him; feign disorder and strike

him (I , 18-20).

Pretend inferiority and encourage his [the enemygeneral's] arrogance (I, 23).

Once we accept this view of deception, however, a variety of new questions arise, suchas: Is it possible to deceive an opponentwhen that opponent is aware of one's intentand opportunities? Can players deceiveeach other simultaneously when each knowsthat the other is trying to deceive, and wheneach knows that the other knows?

In judging Sun Tzu's contribution to ou runderstanding of strategy, however, weshould keep in mind that although gametheory is a field that has undergone nearlyfifty years of development, with hundreds ofresearchers participating in the process, wehave only recently learned how to treat themanipulation of information as part of aplayer's strategic arsenal (e.g. Kreps, 1990;Myerson, 1991). Understanding the stra

tegic manipulation of information is especially difficult because it deals with thepotential for altering an opponent's view ofthe game that is being played when thatopponent knows it is in your interest thus tomanipulate. Resolving circular reasoning inthis circumstance requires the use of advanced principals of probability theory andmathematics, and so we should not be surprised to learn that Sun Tzu's treatmentof information is incomplete. Indeed, weshould marvel at the fact that he understoodintuitively as much as he did.


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NOTES1. Throughout this essay ou r quotations of Sun Tzu's

The Art of War are taken from Samuel B. Griffith's(1963) translation. The notation following eachquote refers to chapter and paragraph as indexed byGriffith. To ensure that Griffith's translation is consistent with other more contemporary interpretations, we consulted Shih I Chia Chu Sun Tzu(Eleven Schools of Thought on Sun Tzu), interpreted by Kuo Hwa Zu o (1978) and The Art o f Wartranslated by Thomas Cleary (1988).

2. By pure strategies we mean the strategies formed bya direct analysis of the game's structure, whereasmixed strategies correspond to lotteries over thesepure strategies.

3. Notice that the concept of ch'; and cheng was elaborated in The Art of War written by Sun Pin a hundredyears later. Ou r interpretation of this conceptfollows from Sun Pin.

REFERENCESChang, Cheng-Tse, 1984, Sun Pin Pin Fa Chao L i [A

Reorganization and Rearrangement of The Art ofWar by Sun Pin]. Beijing: Chong-Hwa Press.

Cleary, Thomas, 1988. The Art o f War: Sun Tzu (translation). Boston, MA: Shambhala Publications.

Griffith, Samuel, B. , 1963. Sun Tzu: The Art of War(translation). London: Oxford University Press.

Kreps, David M., 1990. A Course in MicroeconomicTheory. Princeton, NJ: Princeton University Press.

Kuo, Hwa Zuo, 1978. Shih I Chia Chu Sun Tzu [ElevenSchools of Thought on Sun Tzu]. Shanghai: ShanghaiAncient Book Publishing House.

Myerson, Roger B. , 1991. Game Theory: Analysis ofConflict. Cambridge, MA: Harvard University Press.

Ordeshook, Peter c., 1986. Game Theory and PoliticalTheory. Cambridge, MA: Cambridge UniversityPress.

von Neumann, John & Oskar Morgenstern, 1944.

Theory of Games and Economic Behavior. Princeton, NJ: Princeton University Press.

EMERSON M. S. NIOU, b. 1958, PhD in Political Science (University of Texas at Austin, 1987);Associate Professor of Political Science at Duke University. He specializes in formal theory, international relations, political economy, and Chinese politics. He is the coauthor of The Balance ofPower (Cambridge University Press, 1989). His most recent research focuses on the formation anddissolution of alliances in anarchic international systems.

PETER C. ORDESHOOK, b. 1942, PhD in Political Science (University of Rochester, 1969);Professor of Political Science at the California Institute of Technology. He is the author of GameTheory and Political Theory and A Primer in Political Theory, coauthor of Positive Political Theory(with William Riker) and The Balance of Power (with Emerson M. S. Niou and Gregory Rose). Hisprimary research interest at present is the study and design of electoral, constitutional, and international institutions, with special emphasis on those things that facilitate cooperation and coordination in complex environments.

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