Theory of Moves

Adding n dynamic dimension to game theory allows players to look ahead

before making moues, thereby ueating a mlre renlistic game

Steven J. Brams

J-\uring the Cuban missile crisis jnl-zt October 7962, the Kennedy admin-istration demanded that the SovietUnion remove its missile bases fromCuba. The Soviets acquiesced, but onlyafter the world teetered for days be-tween peace and disaster. Theodore C.Sorenson, special counsel to PresidentKennedy, later recalled, "We discussedwhat the Soviet reaction would be toany possible move by the United States,what our reaction with them wouldhave to be to that Soviet reaction, and soon, trying to follow each of those roadsto their ultimate conclusion."

The Cuban missile crisis is a classic,albeit high-stakes, example of strategicgame-playing. Like chess players,world leaders in conflict situationsmake carefully considered moves andcountermoves. But the outcomes are notalways what the players or onlookersexpect; in particular, it is sometimeshard to understand why players chooseconflict over cooperation.

A body of theory, called game theoryhas been developed and applied overthe past half-century to analyze mathe-matically the strategic behavior of peo-ple in situations of conflict. The theoryfacilitates reconstruction of past situa-tions and modeling of possible futureones, which can explain how rationaldecision makers arrive at outcomes thatare often puzzhng at first glance.

Stez:en l. Brams is professor of politics at New

York Unittersity, iohere he lns tnught since 1969.

He is the autlrcr or coauthor of 17 books thnt

hnolae npplications of game theory or social-choice theory to ztoting and elections, interrntion-al relations, the Bible and theology. Theory ofMoves, the book on zuhich this article is based,

will be published by Cambridge Uniuersity Press

in lanuary. Address: Department of Politics,

New York Uniaersity, Nezu York, NY 10003.

562 American Scientist, Volume 81

Game theory approaches conflicts byasking a question as old as games them-selves: How do people make "optimal"choices when these are contingent onwhat other people do? The seminalwork was done in the 1940s by mathe-maticianlohn von Neumanrt and econ-omist Oskar Morgenstern, both ofPrinceton University, who discoveredthat they held similar ideas about strate-gies in games. They realized, first, thatstrategies are interdependent: Playerscannot make unilaterally optimal deci-sions, because one player's best choicedepends on the choices of other play-ers. Von Neumann was responsible formost of their theoretical work, whereasMorgenstern pushed the applicationstoward economic questions.

Their collaboration led to a monu-mental and difficult treatise, Tlrcory ofGames and Economic Behaaior (7944),which was revised in 1.947 and thenagain in 1953. Over the next severaldecades investigators applied game the-ory to strategic situations ranging fromthe evolution of animal behavior to therationality of believing in God.

According to the classical theory,players choose strategies, or courses ofaction, that determine an outcome. VonNeumann and Morgenstem called theirtheory "thoroughly static" because itsays little about the dynamic processesby which players' choices unfold toyield an outcome.

I have developed what I call the "the-ory of moves" to add a dynamic dimen-sion to classical game theory. Like theclassical theory the theory of moves fo-cuses on interdependent strategic situa-tions in which the outcome depends onthe choices that all players'make. But itradically alters the rules of play, enablingplayers to look ahead-sometimes sev-eral steps-before making a move.

These modifications lead to differentstable outcomes, or equilibria, fromthose of classical game theory and newconcepts of power. In this article, I shalldescribe informally ideas underlyingthe theory of moves and illustrate someof its concepts in several games-thelast being one that models the Iranhostage crisis that began tn 1979.

Making MovesBefore considering the theory of moves,it is worth noting some basic elementsof classical game theory. Von Neumannand Morgenstern defined a game as"the totality of ruies of play which de-scribe it," which includes a startingpoint and a list of legal moves that play-ers can make. The game of tic-tac-toe,for example, begins when one playermakes a mark on a three-by-threeboard. The rules state that a player canmark either an X or an O, but only in anunmarked block. After the first playermakes a mark, the second player does,with the players then alternating inmaking marks on the board. The gameends when one player gets three marksin a row or all the blocks are filled.

Most games can be described in twodifferent ways. The "extensive form" isgiven by a game tree, with play begin-ning at the first fork in the tree. Oneplayer selects one side of this fork,which moves the game to another fork.Then the other player selects a side ofthat fork, and so on until the game ends.This form of a game provides a full de-scription of its sequentiai moves.

By contrast, the "normal form" is giv-en by a payoff makix, in which playerschoose strategies simultaneously o4 ifnot, at least independently of each other.(A strategy gives a complete plan ofpossibly contingent choices-if you dothis, I will do that, etc.) Thus, if a game

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Figure 1. Game theory evaluates behavior in conflict sifuations. During the Cuban missile crisis, President Kennedy demanded that the Soviet Union re-

move its missiles from Cuba. Kennedy and his advisors, shown here in an Excom (Executive Commiftee) meeting considered all their possible moves, thepossible countermoves by the Soviets and the possible counter<ountermoves by the United States. U.S. decision makers used their knowledge of the past

and predicted future moves, which classical game theory tends to treat myopically. The author's theory of moves adds a dynamic component to classical

game theory enabling players to look ahead and select shategies that yield "nonmyopic equilibria." (Photograph courtesy of the John E Kennedy Library.)

has two players, each with two possiblestrategies, it can be represented by atwo-by-two payoff matrix. One player'sstrategy choices are given by the tworows; the other's are given by the twocolumns. Each row-and-column inter-section defines an outcome, where pay-offs are assigned to the two players.

The theory of moves combines the ex-tensive and normal forms of classicalgame theory. A theory-of-moves gameis played on a payoff matrix, like a nor-mal-form game. The players, however,can move from one outcome in a payoffmatrix to another, so the sequentialmoves of an extensive-form game arebuilt into the more economical normalform. In large part, I shall concentrate ontwo-player games in which each playerhas two strategies; more complicated

games become quite intractable after justa few moves, although in principle thetheory of moves is applicable to rt-per-son games in which each of the n play-ers has a finite number of strategies.

Beyond the structure of a game (nor-mal or extensive form), one can makeother modifications in the definition ofwhat constilutes a rational choice. A ra-tional choice depends on, among otherthings, hon, far players look ahead as

they contemplaie each other's possiblemoves and countermoves. In addition,moves are influenced by the capabilitiesof the players and their informationabout each other.

The theory of moves incorporates allthese features. It is dynamic becauseplayers do not make choices de nouo.In-stead, their choices depend on the past

and present as well as the future, whichplayers can anticipate at least in partand about which I assume they canmake rational calculations.

In the theory of moves, I assume thatplayers can rank the possible outcomesfrom best to worst. These payoffs, how-ever, are only ordinal: They indicate anorder of preference, but not the degreeto which a player prefers one outcomeover another. (Although other forms ofdecision-making theory indicate the de-gree of preference in payoffs, I have cho-sen ordinal payoffs to simplify theanaiysis and make it more applicable toreal-life strategic situations.) In addition,the theory allows for power differencesamong players by assuming, for exam-ple, that one player may have the abilityto carry out threats when necessary. Fi-

1993 November-December

op{ion 1 optron 2

player B player B

option option Z

outcome outcorne 2 outcome 3 outcorne 4

Figure 2. Extensive form of classical game theory is given by a game tree. This game involves twoplayers, with each player able to choose one of two options. The game begins when Player A se-

lects an option, Then Player B selects an optiory which leads to one of four possible outcomes.This form highlights the sequential nature of moves.

playeF B

strategy 1 strdtegy 2

oqtcorne

(x,,Y,)

oulcome

( x,,y.)

oqtcorne 2

(xr., y1)

outCorn€ 4

(xa,/r)

Figure 3. Normal form of classical game theory is given by a payoff matrix. In a two-player game inwhich each player has two strategies, the matrix is two-by-two. Player ,{s strategies are represent-ed by the two rows, and Player B's are represented by the two columns. Players independently se-

lect strategies that lead to an outcome. Each outcome is assigned payoffs, which are given in x-ycombinations such that x; is the payoff to the row player (Player A) and y; is the payoff to the col-umn player (Player B), where i and I are given by the players' shategies (either 1 or 2).

nally, the theory is information-depen- Rule 2 says that either player can switchdent, meaning that players do not al- to a new strategy, thereby generating aways share the same informatiory mak- new outcome; the fust player to move ising misperception and deception called Player 1. According to Rule 3, thepossible. other playeq, Player 2, can then move.

The theory of moves includes six ba- A game's end is determined by Rule 4:sic rules. Rule 1 states that a game starts The players respond alternately untilat an "initial state," which is a row-and- neither switches strategies. The result-column intersection of a payoff matrix. ing outcome is the "final state," which is

564 American Scientist, Volume 81

r

cr)qJ+JoL{{r/)

player AN

ul(lJ+rro\+rb

the only point at which the players ac-crue payoffs.

The remaining rules, which I call ra-tionality rules, explain the reasons formoving or not moving. Rule 5 statesthat a player will not make a move un-

less it leads to a preferred outcome/based on his or her anticipation of the fi-nal state. Rule 6, which I call the two-sidedness rule, says that a player con-siders the rational calculations of theother players before moving, taking intoaccount their possible moves, the possi-ble countermoves of the other players,their own counter-countermoves, andso on. Thus, a player may do immedi-ately better by moving first accordingto Rule 5; but if this player can do evenbetter by letting the other player movefirst, and it is rational for that player todo so, then the first player will awaitthis move, according to Rule 6.

TruelsSome of the differences between classi-cal game theory and the theory ofmoves arise in an imaginary confronta-tion situation called a truel. A truel islike a duel, except there are three play-ers. It illushates nicely the applicabilityof the theory of moves to games withmore than two players.

In the truel I posit, a player has twochoices: either to fire or not to fire at oneof the other two players. Each playerhas one bullet and is a perfect shot. Theplayers cannot communicate, whichprevents the selection of a common tar-get. I assume that a player's primarygoal is to survive, and his or her sec-ondary goal is to survive with as fewother players as possible.

In this rather gruesome situation, thetheory of moves suggests a differentoutcome than does classical game theo-

ry. In fact, the theory of moves providesa resolution that is more satisfactory forall the players.

If the players must make simultane-ous strategy choices, they will all fire ateach other according to classical gametheory. They do so because their ownsurvival does not depend one iota onwhat they do. Since they cannot affectwhat happens to themselves but can af-fect only how many others survive (thefewer the bettel, according to the pos-tulated secondary goal), they should allfire at each other.

Such a scenario generates two possi-ble results: Either one player survivesor no players survive. Players A and Bmightboth fire at Player C, who fires at

one of them, say Player A. This leaves a

single survivor, Player B. On the otherhand, each player may fire at a differentplayer, leaving all players dead.

If each player has an equal probability of firing at one of the other two play-ers, there is only a 25 percent chance

that any player will survive. The reasonis that Player A will be killed if fired atby Player B, Player C or both (three cas-

es); the only case in which Player A willsurvive is if Players B and C fire at each

other, which gives Player A one chance

in four of surviving. Although this cal-

culation implies a 75 percent chancethat some player will survive, an indi-vidual player will be more concernedwith his or her own low chance (25 per-cent) of survival.

The theory of moves offers a differentperspective. lnstead of assuming simul-ianeous strategy choices, it asks each

player: Given your present situation andthe situation that you anticipate will en-sue if you fire first, should you fire? Atthe start of a truel, all the players are

alive, which satisfies their primary goalof survival but not their secondary goalof surviving with as few others as possi-

ble. Player A now contemplates shoot-ing Player B to reduce the number ofsun.ivors. Bv looking ahead, hott'ever,Player A realizcs ihat firir"rg ai Piaycl Dwill cause Player C subsequently to fireat him or her (Player A). This would be

in Player C's interest, because it wouldmake C the sole survivor.

hstead of firing, therefore, Player Awill, thinking ahead, not shoot at any-body. By symmetry, the other playerswill choose the same strategy, so all willsurvive. This longer-term perspectiveleads to a better outcome than that pro-vided by classical game theory, in whicheach player's primary goal is satisfiedorly 25 percent of the time when play-ers make simultaneous strategy choices

without looking ahead.The purpose of the theory of moves,

however, is not to generate a better out-come but to provide a more plausiblemodel of a strategic situation that mim-ics what people might think and do.The players in a tn-rel, artificial as such a

shoot-out might be, would be motivat-ed to look ahead, given the dire conse-

quences of their actions. To be sure, clas-

Jical game theory can also provide thisoutcome if one player (say, A) were des-

ignated to move first. Then Player Awould rationally choose not to fire, lest

he or she be killed subsequently by the

sole surwiving player (either B or C).

strategy 1

\A/hat classical game theory does notask is whether it is rational for one play-er, if afforded the opportunity, to movefirst. This is specified by the rules in the

classical theory instead of being madeendogenous-that is, incorporated intothe theory as a question to be an-su'ered-as in the theon'of moves.

Ch..ngiiig tlie lulcs oi play may terrerate still different outcomes. For exam-ple, permit the players of a truel the ad-ditional option of firing in the air,thereby disarming themselves, andspecify the order of play, such as PlayerA goes first, foliowed by Players B andC going simuitaneously. Given theserules, Player A will fire in the air, and

outcome 1

(x,,,Y,)

m

#outcome 3

( xr,/r)

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player B

StrategY 2

then Players B and C will shoot eachother. The disarmed Player A is, afterall, no threat, so he or she would not be

shot by Player B or Player C. On theother hand, Players B and C will fire im-mediately at each other; otherwise, theywiil have no chance of surviving to get

in the last shot. In the end, Plaver A willbe il-ie sole survivor under these rulesthat give a player the option of firing inthe air.

Prisoners'DilemmaGame theory's most famous game is

called Prisoners'Dilemma. It starts withthe following scenario: Two persons,suspected of being partners in a crime,

f

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player A^j>.{f)C/Fro!

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Figure 4. Theory of moves embeds a game tree in a payoff matrix. Players can move from an ini-tial outcome or state to another one by either moving vertically (Player A) or horizontally (Play-

er B). The matrix shows the strategies of the players and their payoffs at various outcomes. The

arrows within the matrix reveal how players might move-in this case counterclockwise-be-

tween different outcomes through a sequence of moves.

cFigure 5. Tiuel is a three-person duel. Here each player is assumed to have a single bullet and be

a ierfect shot. A playerCprimary goal is surviving and the secondary goal is surviving with as

few others u, pogibi". If ihe players must make simultaneous choices, a player cannot affect his

or her own survival but can iffect how many others survive; consequently, each player is moti-

vated to shoot another. Classical game theory indicates two possible outcomes: either no one sur-

vives (Ieft),or one player-Player B in this case-suwives iight)'

1993 November-December 565

negot tat ion

Khorneini

obstuction

Iran Hostage CrisisAlthough a rational player should lookahead before acting, that advice workswell only if the players have completeinformation about their opponents.The United States apparently lacked

such information about Iran in No-vember 7979, when Iranian militantsseized personnel at the U.S. embassy.By analyzing the news reports of thetime and the later writings of somegovernment officials, Walter Mattli ofthe University of Chicago and I recon-structed the strategic thinking of deci-sion makers in this crisis. As I shallshow, it explains well why the crisistook so long to resolve.

During the crisis, the military capa-bilities of the two opponents were al-most irrelevant. ln April 1980 ihe Unit-ed States attempted a rescue that costeight American lives and freed nohostages, but the conflict was never re-ally a military one. The crisis canbebestrepresented as a game in which Presi-dent |immy Carter misperceived thepreferences of Ayatollah RuhollaKhomeini. In desperation, Cartersought a solution in the wrong game.

Why did Khomeini sanction thetakeover of the American embassy bymilitant students? Doing so providediivo aciv aniages. First, by cleatiilg a con-frontation with the United States,Khomeini was able to sever the manylinks that remained between Iran andthe "Great Satan" from the days of theshah. Second, the takeover mobilizedsupport for extremist revolutionary ob-jectives just at the moment when secularelements in Iran were challenging theprinciples of the theocratic state thatKhomeini had installed.

Carter's primary goal was immedi-ate release of the hostages. His sec-ondary goal was holding discussionswith Iranian religious authorities aboutresolving the differences that hadstrained relations between the UnitedStates and Iran. Of course, if thehostages were killed, the United Stateswould likely defend its honor, probablythrough a military strike on Iran.

Carter considered two strategies: ne-gotiation and military intervention. Be-cause the seizure of the embassy had1ed to a severing of diplomatic relations,negotiation could be pursued onlythrough the United Nations SecurityCouncii, the World Court or informaldiplomatic channels. Military interven-tion could have taken the form of a res-cue mission, as it did, or punitive strikes

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'f&rMFigure 11. Carter apparently misperceived the structure of the Iran hostage crisis by believingthat Khomeini prefened compromise to a confrontation that Carter thought might end in a dis-aster. Carter's misperceived payoff matrix shows that he gets a better payoff by selecting ne-gotiation, regardless of Khomeini's choice. According to this payoff matrix, Khomeini's beststrategy depends on Carter's selection. If Carter selects military intervention, Khomeini shouldselect negotiation. If Carter selects negotiatiory Khomeini should select obstruction, resultingin the outcome called "Carter surrenders," which is the equilibriurn outcome (blue). ll theplayers moved and countermoved around the matrix, the moves would be clockwise, because

in that direction no player ever moves from his best payoff.

Khomeini

negotialton obstruction

Figure 12. Real-game payoff matrix, taking into account Iran's internal politics as revealed byevents and analysis, shows that Khomeini has a dominant strategy of selecting obstruction,which is better for him regardless of Carter's strategy. Like the misperceived-game payoff ma-trix (Figure 11), Catler has a dominant strategy of selecting negotiation. These strategies leadagain to a negotiation-obstruction outcome, which is an equilibrium outcome (blue), nowcalled "Khomeini succeeds" because Khomeini ranks the other outcomes differently than inthe misperceived-game payoff matrix. Cycling in this matrix would be clockwise.

568 American Scientist, Volume 81

co;ro

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CI>rHro \g

.:L;AJIiJ c

Carter

Khomeini succeeds

2" 4

Carter succeed5

Khome'ni:adanr antCarter adarn ant

Cart€r

against selected targets, such as refiner-ies, rail facilities or power stations.

Khomeini also had two strategies: ne-gotiation or obstruction. His negotiatingdemands included a retum of the shah'sassets and ending U.S. interference inIran's affairs. On the other hand, a re-fusal to negotiate was sure to block aresolution of the crisis.

The two players and their two strate-gies generate a two-by-two payoff ma-trix. Each cell in the matrix has an asso-ciated payoff for each player. As inPrisoners' Dilemma, I assume thatCarter and Khomeini can rank the fouroutcomes frombest (4) to worst (1).

Carter obtains a better payoff bychoosing negotiation, which wouldsave him from the overwhelming diffi-culties of military intervention, whatev-er Khomeini does. In December 1979those difficulties were compounded bythe Soviet invasion of Afghanistan,which eliminated the Soviet Union as apossible ally in seeking concerted actionfor the release of the hostages. More-over, the Soviet troops next door inAfghanistan made the strategic envi-ronment for military intervention any-thing but favorable.

Carter initially believed that his selec-tion of negotiation lt'ould appeal toKhomeini as well. 1'he president per-ceived that Khomeini faced seriousproblems in Iran, such as demonstra-tions by the unemployed and Iraqi in-cursions across Iran's westem border. InCarter's 1982 memoir, Keeping Fsith,hereported his belief that a U.S. choice ofnegotiation rvould give Khomeini a dig-nified way out of the impasse.

The president also believed thatKhomeini preferred a U.S. surrenderthat would result from the obshuctionof negotiations. That result, Carterthought, would give Khomeini his bestpayolf of 4, whereas Khomeini wouldget his next best payoff of 3 if both sidesselected negotiation. And finally, Cartersaw Khomeini getting inferior payoffsof 2 and 1 if the United States selectedmilitary intervention.

Carter's MiscalculationsUnfortunately for Carter, he misper-ceived the strategic situation and, hence,played the wrong game. Khomeiniwanted the total Islamization of Iraniansociety; he viewed the United States as" a global Shah-a personification ofevil" that had to be cut off from anycontact with Iran. Khomeini abjured hisnation never to "compromise with any

po\\'er ... [and] to topplc from tl'Le posi-tion of power anyone in any positionwho is inclined to compromise with theEast and West."

For Khomeini to have selected nego-tiation would have rveakened his un-compromising position. Iranian lead-ers who tried negotiating, includingPresident Abolhassan Bani-Sadr andForeign Minister Sagdegh Ghotbzadeh,lost in the power struggle. Bani-Sadrrvas forced to flee for his life to Paris,and Ghotbzadeh was arrested and iat-er executed.

In the "real game"-the actual strate-gic situation-Khomeini most pre-ferred obstruction (4 and 3), regardlessof the U.S. strategy choice. Doubtless,he preferred that the United Stateschoose negotiation (4) over military in-tervention (3).

\A/hat does classical game theory sayabout the rational choices of the playersin the misperceived game and the realgame? In both games, Carter's domi-nant, or unconditionally best, strategyis negotiation. Regardless of whatKhomeini chooses, Carter's payoff fromnegotiation is better than his payofffrom military intervention. Lr the mis-perceived game, for example, if Khome-ini chooses negotiation, Carter gets a

payoff of -1 bv choosing ncgotiaiion anda payoff of 3 by choosing military inter-vention; if Khomeini chooses obstruc-tion, Carter receives apayoff of 2bychoosing negotiation and a payoff of 1

by choosing military intervention.Although Carter's dominant strategy

in both games is independent ofKhomeini's choice, Khomeini's bestchoice in the misperceived game de-pends on what Carter selects. If Carterchooses negotiation, which he shouldbecause it is dominant Khomeini, an-ticipating this, does better by choosingobstruction, which gives him a payoffof 4, rather than choosing negotiation,r,vhich gives him a payoff of 3. So in themisperceived game, Khomeini shouldchoose obstruction, leading to the nego-tiation-obstruction outcome. That out-come, which I call "Carter surrenders,"gives Carter a payoff of 2 and Khomeiniapayoff of 4.

Game theory calls this outcome-Carter chooses negotiation and Khome-ini chooses obstruction-rational in thereal game as well, because both playershave dominant strategies associatedwith it. In the real game, I call this out-come "Khomeini succeeds." (The otherthree outcomes in the real game areranked differently by Khomeini from

Figure 13. Carter tried military intervention even though negotiation was his dominant shategy inboth games. The attempted rescue mission left one U.S. helicopter destroyed and another aban-doned. Classical game theory makes such a strategy appear irrational in both game makices,whereas the theory of moves offers a rational explanation for Carter's action. In the misperceived-game matrix (Figure 11), Carter believes that selecting military intervention--or threatening itsuse-will force Khomeini to select negotiation in order to improve his payoff, at which pointCarter can also choose negotiation to obtain his best payoff at the "compromise" outcome, which isalso better for Khomeini. In the real-game payoff matrix (Figte 12),I{homeini cannot be swayedftom obstruction. The crisis, in fact, remained at a negotiation-obstruction outcome until thehostages were released on January 20,7987.

1993 November-December 569

those in the misperceived game, whichis why I give them different shorthanddescriptions.) In the real game, the ra-fionality of the (2, 4) outcome is rein-forced by Khomeini's dominant strate-

$)'/ of obstruction associated with it;obstruction is not dominant in the realgame but, instead, Khomeini's best re-sponse if Carter chooses his own domi-nant shategy of negotiation.

Given that Carter does belter in bothgames by choosing negotiation, whywould he consider, much less try, mili-tary intervention? Classical game theorydoes not give a reasory but the theory ofmoves suggests the basis for his miscal-culation. Carter might have thought-with some justification in the misper-ceived game-that by threateningKhomeini with military intervention hewould induce him to choose negotia-tion, giving Carter the opportunity, bychoosing negotiation himself, to obtainhis best payoff.

The reasoning underlying this calcu-lation goes as follows: In the misper-ceived game, a negotiation-negotiationoutcome gives Carter his best payoff of4 and gives Khomeini his next-bestpayoff of 3. A threat by Carter tochoose military intervention, if carriedout, would inflict upon Khomeini histwo worst outcomes in the misper-ceived game: a payoff of 2 if he chosenegotiation and a payoff of 1 if hechose obstruction. Since Khomeiniwould prefer a payoff of 2 over 1, hewould choose negotiation, givenCarter's threat were credible. Howeveq,because both players do better bychoosing "compromis e" at (4, 3) ratherthan "Khomeini surrenders" at (3,2),Khomeini should choose negotiationwhen Carter does, assuming that hetakes seriously Carter's threat of mili-tary intervention.

There are two problems with this rea-soning. First, it is not clear that Carterhad what I call the "threat power"needed to induce a compromise out-come in the misperceived game. Moreimportant, that was not the game beingplayed. In the real game, Khomeini hadno reason to accede to a threat fromCarter, because his political positionwas stronger if he refused to compro-mise. Regardless of Carter's choice,Khomeini does better by selecting ob-struction in the real game.

Nonetheless, Carter hied threats. Hedispatched the aircraft carrier USS KitfyHawk and its supporting battle groupfrom the Pacific to the Arabian Sea. The

570 American Scientist, Volume 81

carrier USS Midlvay and its battlegroup were already in the area. Thosetwo battle groups created the largestU.S. naval force in the Indian Oceansince World War II. But this vast array offirepower proved useless, at least for the

purpose o{ inducing Khomeini to selectnegotiation.

The failed rescue operation in April1980 kept the situation at the negotia-tion-obstruction outcome for anothernine months. This was so despite thefact that Iranian leaders had concludedin August 1980-after the installation ofan Islamic govemment consistent withKhomeini's theocratic vision-thatkeeping the hostages was a net liability.

Further complicating Iran's positionwas the attack by Iraqi forces in Sep-tember 1980. It was surely no accidentthat the hostages were set free on theday of Carter's departure from the\ /hite House on January 20,798L. Al'though Gary Sick claimsinOctober Sur-prise (1991) that the hostages were notreleased before the November 1980presidential election because of a secretdeal that Iran made with Ronald Rea-gan's supporters, later congressional in-vestigations disputed Sick's claim, atleast regarding the involvement ofGeorge Bush.

Perhaps Carter should not be judgedtoo harstrly for misperceiving the strate-gic situation. If he had correctly foreseenthe real game from the start, both gametheory and the theory of moves agreethat he could not have moved awayfrom an outcome that gave him a payoffof 2 and Khomeini apayoff of 4. Whatthe theory of moves explains, and gametheory does not, is why Carter mighthave thought that he could implementthe compromise outcome through theexercise of tfueats.

The theory of moves also shows howa series of moves and countermoves inthe misperceived game can induce thisoutcome if Carter has what is called"moving power." Assume that ihe play-ers move and countermove in a clock-wise direction on the misperceived-game payoff matrix. In that direction,neither player ever moves from his bestoutcome (Carter vertically or Khomeinihorizontally). In a counterclockwise di-rection, by contrast, players do movefrom their best outcomes: Khomeinimoves from a payoff of 4 at (2,4) whenhe switches from obshuction to negoti-ation, and Carter moves from a payoffof 4 at (4,3) when he switches from ne-gotiation to military intervention. So, if

there is cycling, it must be in a clock-wise direction.

If Carter believed that he had mov-ing power-the ability to force Khome-ini to stop in the move-countermoveprocess-Carter could force Khomeini

to stop at the negotiation-negotiationoutcome or the military interven-tion-obstruction outcome, which arethe two outcomes where Khomeini hasthe next move. Khomeini would pre-fer the formet which gives him a pay-off of 3, rather than the latter, whichgives him a payoff of 1. In the realgame, however, these outcomes giveKhomeini payoffs of 2 and 3 respec-tively, so he would choose to stay at themilitary intervention-obstruction out-come. As a consequence, Carter'shoped-for negotiation-negotiation out-come in the misperceived game be-came, in April 1980, a military inter-vention-obstruction outcome in thereal game.

The theory of moves formally incor-porates into the framework of gametheory an initial state in a payoff matrix,possible moves and countermoves fromit to try to reach a nonmyopic equilibri-um, and threat and cycling to weardown an opponent. It also allows forthe possibility that players possess onlyincomplete informatiory as I iliustratedin the case of the Iran hostage crisis,which can lead to misperception. As atheory that assumes that players canrank outcomes but not necessarily at-tach utilities to them, it is eminently ap-plicable to the way we contemplate thestrategic choices of others as we try tomake our ownbest choices in a dy:ram-ic environment.

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