THE STRATEGY OF CHALLENGES:
Centre for International and
Strategic Studies York University, 1990
Abstract.
I use game
models to analyse two mediaeval tales about re:markable challenges. The goal
is to understand their plots and to clarify in general why challenges are made
and accepted. Children's contests of daring provide a simple context to study
challenging, and I argue that children seek a certain reputation; they want to
be known for placing a high payoff weight on others' estimate of that very
weight. This definition might seem circular, but it leads to well‑specified,
solvable games. The model for the children's dares is modified in steps to fit
the mediaeval stories. Game theory has treated language as a way to transmit
information, but here verbal challenges are speech acts, or
"performatives" in Austin's sense, that trigger the concern for
reputation. The analysis of challenging is relevant to international conflicts
where large powers' make commitments and struggle in the Third World over
"credibility."
The Middle English poem Sir Gawain and the Green
Knight and its Old Irish ancestor The Feast of Bricriu recount a
remarkable stranger's challenge to the hero, in effect, "You can chop off
my head if you'll let me return the blow." Old romances and sagas teem
with challenges, but this one puts the hero in a genuine dilemma. This paper
analyses the situation using the theory of games, to clarify some obscure plot
elements and to show that the best action is strategically interesting, that it
exacts real risks, costs and tradeoffs. A game approach reveals some of the
reasons why their storylines are so successful.
SIR
GAWAIN AND THE GREEN KNIGHT
Sir Gawain and the Green Knight
survives in one small manuscript (British Library, Cotton Nero A.x). Its script
and language indicate that it was written in the late 1300s. Authorship then
conferred less personal celebrity and the poem bears no signature. It would
likely have been read aloud to a gathering, and its length of 2500 lines would
have taken up several nights, like a television mini‑series. The poem was
unknown to the literary world for four and a half centuries, until the
palaeographer Sir Frederic Madden published it in 1839. Since then it has
prompted a great scholarly outpouring: the number of books and papers is
nearing one thousand, and only the writings on Beowulf and Chaucer's
poetry surpass this amount for an English work of such antiquity. It has been
popular with general audiences, and can be read in Breton, Turkish, Japanese
and several other languages, or viewed as a motion picture starring Sean
Connery as the fierce green man.
Laura Hibbard Loomis (1959, p.528)
borrows the Green Knight's epithet for Gawain to praise the poem itself,
"The hero of Gawain and the Green Knight is likened to a pearl
beside a pea (vs. 2354), and so might the poem be reckoned among its
contemporaries. It moves over an almost flawless structure as smoothly as
supple skin moves over the bones of the hand. With the exception of Chaucer's Troilus
and Criseyde, no other Middle English romance approaches its artistic and
spiritual maturity, its brilliant realism, its dramatic vigour, its poetic
sensitivity to nuances of word and mood, its humour, its nobility of
spirit." While most mediaeval works allow us a glimpse into a vanished
past, this one engages modern readers by addressing modern problems. Gawain
tries to maintain his ideals when the world brings them into opposition with
each other, forcing him to sacrifice one or another. The poem recounts the
green stranger's intrusion at Camelot's Christmas feast, his offer to allow any
one to behead him in exchange for a return blow, Gawain's acceptance of the
dare, and the latter's event ful search for the Green Knight to give him his
turn. The challenger is of immense size and entirely green. As Gawain
deliberates whether to accept, he and the reader will ask the same questions:
Why would the Green Knight propose this bizarre game? Is he supernatural or deranged,
or neither? What are the stakes for me, and what are my prospects? Both the
reader and Gawain strive to construct the game.
This paper offers two game analyses of
the poem. The first game, of which I describe three versions, portrays Gawain
as taking up the dare for the sake of his reputation. This thesis seems
reasonable, almost obvious, but it is elliptical: his reputation for what? A
reputation is a general belief about a person and must have content. At first
thought, the reputation is simply for bravery, but this hypothesis gives no
role to the Green Knight. If Gawain had needed to boost his name for risktaking
he could have performed a display without waiting for someone to challenge him.
And he could have devised better proofs of bravery than this beheading match
where he runs the risk for no higher end. The Green Knight questioned the Round
Table's bravery but offered no evidence to put it in doubt, beyond the fact
that the Court was not taking up his bizarre challenge. The Green Knight himself
identifies the issue addressed in this paper. When his first offer produces
silence, he furrows his green brow, draws himself up to his full height, and
speaks to the Round Table:
What,
is this Arthurs hous, quoth the hathel thenne
That
al the rouse rennes thurgh ryalmes so mony?
Where
is now your sourquydrye and your conquestes
Your
gryndellayk and your greme and your grete wordes?
Now
is the reuel and the renoun of the Rounde Table
Ouerwalt
wyth a worde of on wyghes speche
For
al dares for drede withoute dynt schewed. (309‑315)
(Excerpts are from
the edition of Tolkien and Gordon, 1925.)
This is surely not Arthur's house
renowned through many kingdoms, he is declaring. The Round Table's pride and
fierceness and bragging talk have vanished, its fame has been upset by one
man's speech, his words alone, for all are cowering without a blow being
dealt. How could the Green Knight's mere words make them play his game or
suffer disgrace? The paper will try to answer this question.
Gawain's goal, I will argue, is a
reputation for placing a high value on his reputation. This objective seems
circular, but as the first model shows, it is well‑specified in the sense
that it allows Gawain to de cide on an action. Defining payoffs recursively in
this way leads to the consequence that sensible actors sometimes choose to
fight over an issue of no direct value to them whatsoever. Because of this
recursive ness, the goal is confusing when stated in English, but it is easy to
formulate in mathematics, and it is also within the intuitive reach of ordinary
people, even though they may be unable to articulate it. It seems to underly
children's contests of daring. The power of the Green Knight's verbal challenge
in triggering Gawain's display of bravery is explained here through a theory of
daring contests, combined with John Austin's theory of performative utterances,
which describes how certain sentences function to change the social
relationships of the speaker and listener. A dare is one such performative utterance,
and the Green Knight's words infuse Gawain with a new motive to establish a
particular type of reputation. The Green Knight's attempt to set up a
compelling dare explains many text elements. The paper then examines a much
earlier beheading story, the final episode of The Feast of Bricriu.
Events are superficially parallel to the Gawain story, but the underlying game
is different, and the difference explains changes in the dialogue and the
procedure of the beheading. An informal strategic analysis clarifies some
obscure events in the plot. The next section of the paper presents a second
game, one played wholly within Gawain, a conflict between his dutiful nature
and his self‑preserving nature. Finally I add some comments on how game
theory can supplement conventional literary analysis, and how the model of
daring clarifies the motives behind superpower proxy wars in the Third World.
SUMMARY
OF THE BEHEADING SCENE
Arthur was the noblest king of England,
and here is an extraordinary episode, a marvel that befell him, that the
storyteller heard told "in town": On New Year's the Round Table is
celebrating the fifteen‑day Christmas festival. True to his holiday
custom and his youthful spirit, the king will not eat until he has heard of some
wonder or seen some feat of arms. As his courtiers sit down to their feast, an
awesome figure rides into the hall, a large man, perhaps a giant, handsome,
sturdy, finely dressed, and all over bright green. He wears no armour, but
carries a cruel battle‑axe in one hand and a holly bough in the other.
The metal of his axe, even his horse, are as green as he is. Offering no one
greeting, he brusquely asks who is the head of the group. Arthur welcomes him
and invites him to dismount. The stranger says he will not spend time with
them, but has come to propose a Christmas "gomen," a game or sport.
The fame of the Round Table is universal and he hopes his request will be
granted here. Arthur promises that he will have single combat if he wants it. The
Green Knight replies that his is a different quest, that in any case none here
could match him; next to him they are like beardless boys. He challenges any
person to deal him one blow in exchange for another. Anyone who is rash‑headed,
hot‑blooded, enough to accept will use his green axe to strike the first
blow, and keep the axe as a gift. The Green Knight will not resist, and he will
not demand his turn before a year and a day.
When no one steps forward, the Green
Knight taunts the company, declaring that he has now exposed its reputation as
counterfeit. Pricked by shame and anger, Arthur leaps forward to seize the axe.
The Green Knight dismounts and bares his neck, but before the king can strike
his blow, Gawain interrupts from the banquet table. In an elaborate and
courteous speech he asks that he stand in for Arthur, and the counsellors
consent. The Green Knight says he is happy to be facing Gawain, and has him
recite the terms of the bargain. Gawain asks the Green Knight who he is and
where he lives? The Knight promises to answer if he survives the blow, and
Gawain must then seek him out. Of course, if afterwards he can reveal nothing,
Gawain will be free of obligation.
The Green Knight again lifts his long
hair to expose his neck. The axe shears through flesh and through bone. The
head rolls across the floor and into the crowd, blood streams down the green
chest, but the body neither staggers nor falls. It springs after the head and
grabs it. The body mounts the horse, holds the head up by the hair, the eyelids
open, and the head commands Gawain to appear one year hence at the Green Chapel
or be branded a coward. The body rides out of the hall. When the hoof‑beats
have receded into the distance, Arthur and Gawain laugh nervously, stifling
their awe at what they have witnessed.
This is the poem's opening episode. The
narrative moves on to Gawain's search for the Green Knight, his encounter with
the lady of the castle and her tests of his chivalry. He passes them creditably
but not perfectly. In the final scene, the Green Knight feigns preparations to
behead Gawain, but only wounds him with a nick on the neck, which will
thereafter signify his imperfection.
If Loomis is right that the poem fits
its structure like skin over the hand, there ought to be some logically
consistent account of what has happened here. How are we to understand Gawain's
view of the Green Knight's challenge? The poet's narrative style is unusual in
that he reports all of the dialogue but nothing about Gawain's thoughts. The
reader can speculate: Does Gawain think the Green Knight is trying to disgrace
the Round Table, or perhaps to kill him by shaming him into a bargain
tantamount to suicide? Why does Gawain accept?
CHILDREN'S
DARING CONTESTS
Gawain is competing for reputation, I
argue, in an adult version of children's contests of daring. (The fullest
accounts of children's dares are given by Opie and Opie, 1969, and Fowke,
1988.) Under the simplest rules, one child challenges another to perform some
bold or risky feat: knock on a door and run away, hail a cab and walk through
it, make a prank phone call, enter an empty old house, lie down on a railroad
track and let a train pass over you, tell a schoolmate of the opposite sex that
you like them or otherwise. In Bielefeld, children dare each other to swallow
one of the large slugs that crawl out during damp times.
Clearly children accept dares, sometimes
dangerous ones, to promote their reputation, but it makes little sense to speak
of reputation attached to no specific character trait. We have to ask, as we
did for Gawain, what personal attribute do they want their friends to reassess?
Toughness, perhaps? But "reputation for toughness" has no clear
meaning. Taking the case of a dare to eat a slug, the reputation could not be
one of indifference to eating slugs. Having that name would make you a
celebrity, but it would not raise your degree of respect or dominance in the
group. If it became known that you savoured slugs, the dare would not be
regarded as a real test, and the group would simply choose another task. For
the daree's purposes, the audience must know that he or she finds the task
aversive but is strong‑willed enough to do it anyway for the sake of the
reputational goal. Eating the slug functions like a measuring marker, to show
how highly the daree values reputation.
REPUTATION
FOR VALUING REPUTATION
My thesis is that the content of the
reputation at stake in dares can be defined recursively. The reputation is for
valuing reputation. The child is in effect saying, "I don't like to eat
this, but I'll do it. I attach great importance to your estimate of this very
importance. When you see me eat this slug, you raise your estimate of that
importance." In its simplest form I am comparing the payoffs (my
reputational weight) x (Other's estimate of my reputational weight given I
accept)·‑·c versus (my reputational weight) x (Other's estimate of my
reputational weight given I decline), where c is the cost of performing the
challenge. (Other papers where game payoffs depend directly on others' beliefs
in dependent of their actions, are those of Gilboa and Schmeidler, 1988,
Geanakoplos, Pearce and Stacchetti, 1989, and Nalebuff, 1990.) It is surprising
that, while the goal of "valuing reputation" is incomplete, that of
"valuing reputation for valuing reputation" leads to a well‑defined
game.
A reputation for valuing reputation
might be desirable exactly because it is free of links to specific traits.
Onlookers can generalize it to other contexts more readily, in that they infer
that the daree would face other risks and discomforts for the sake of
reputation. Compare a dare with a bribe like eating a slug in
exchange for a dollar bill. If you take the bribe, should I conclude you are
particularly tough, or particularly greedy for money? One interpretation is as
valid as the other. The same argument can be made against an institution of
performing unpleasant chores in hopes of a specific reputation: "Eat this
slug, and I'll tell everyone you can speak French." But when you accept a
dare, the audience can assess you as willing to suffer for your reputation in
many contexts, someone to be reckoned with in general. Children play many types
of the daring contests. The simplest involves a dare leading to a simple
performance or a refusal. Another version is Truth, Promise, or Dare, where the
target chooses either to give an honest answer to an embarrassing question,
which the group will select, for to forfeit some object, or to perform a dare.
A risque version of this game was popular with adults in centuries past (Opie
and Opie, 1969). Follow the Leader requires the darer to act first, after which
the daree may accept or de cline. In Darers Go First the daree can either do
the task, or simply refuse, or answer back, "Darers go first!" Then
the darer must perform the deed or the contest is off. Another custom is Double‑daring.
This concept seems to have different meanings, but from my memory someone who
declines can be double‑dared, and the second challenge carries greater
coercive power. "I double‑dog‑dare you!" is stronger
still. Not all dares fit the pattern I have described, that of building a
reputation for valuing reputation; some are more to prove one's ease in doing
the task. A child who can smoke a cigar without throwing up is proving that he
or she is almost an adult, but this type of dare falls outside the present
model.
DARING
AS A FORMAL GAME: VERSION I
The following game models one type of
daring contest. Following it are Versions II and III, each meant to be
successively closer to the poem. That the three games have non‑trivial
solutions shows that my interpretation of daring, whether right or wrong, is at
least strategically coherent. To simplify the analysis I will take the person
who dares as a given, so there is no issue of me trying to jump in and dare you
before you can dare me. In reality, social rules often limit who can make the
dare: groups are more able to do the challenging, and a younger child or
newcomer is more often the tar get. The task will also be taken as a given. It
would be interesting to examine the darer's strategic choice of a task, but the
paper does not address this question. The contest requires both the darer and
the daree to perform the task if the challenge is accepted. This assumption is
closer to the Green Knight's rules.
The sequence of play is shown in Figure
1. Player K (the Green Knight) chooses to dare or not to dare, and Player G
(Gawain) either accepts or declines. If K does not dare, reputation remains non‑salient
and the two players receive 0. (I could have assumed that K suffers a
reputational loss for failing to dare, but the present approach fits the poem
better since the Green Knight could have simply stayed home and never issued a
challenge.)
If G accepts, both do the deed, and the
each receives a difference‑of‑reputations payoff minus the cost of
the deed: FwK(E[wK|dare]-E[wG|accept]) - 1 and
FwG(E[wG|accept]-E[wK|dare]) - 1,
respectively. Each component of these payoffs will now be explained. (The
corresponding two payoffs in the case that G declines are shown in Figure 1,
and have corresponding meanings.) A cost of 1 is subtracted from their payoffs
as the burden of performing the task. Without loss of generality this cost can
be taken as 1 for both, and it then defines each player's unit of utility. The
coefficients wK and wG, both assumed to be greater than
zero, are the weights that the two players attach to their relative
reputations. The expression E[wK|dare] is the onlookers' and Player
G's estimate of K's weight wK after they observe that K has made the
dare. Likewise, other expectations in the two payoffs involve estimates of
Player G's weight, given an acceptance or a refusal of the dare. To derive
their values the game must be solved. The expression E[wK|dare] - E[wG|accept]
in K's payoff is the difference in reputations. These payoffs portray the
challenge in the poem as competitive, a contest of relative reputations. Also,
in the case of the poem, the reputation must be understood as not just Gawain's
own but that of Arthur's court, and likewise the "on lookers" are
really the Round Table's whole world, everyone who matters. For simplicity I
will talk about personal reputation in the eyes of those present.
FIGURE 1: Sequence of play
for Versions I, II and III.
The remaining element in the players'
payoffs is the strictly positive constant F, which stands for the
"felicity," a term introduced by Austin to analyse performative
utterances like daring. It refers to how effectively the darer sets up a
compelling dare. A dare is felicitous if it has real force. Thus, as F
approaches zero, a dare loses its power to change the daree's motivation.
Felicity is determined by both the content and context of the dare, as will be
discussed later after the game's solution has been derived.
SOLUTION
OF VERSION I
This game involves incomplete
information, since each player is uncertain about the other's value for
reputation. They know all the other factors. A (subgame perfect) equilibrium
solution will comprise a pair of sets: the values of wK for which
Player K will dare, and the values of wG for which Player G will
accept. In the Appendix it is shown that at any equilibrium a player who dares
or accepts a dare for a value of wK or wG, will dare or
accept for all higher values. An equilibrium therefore is specified by a pair
of thresholds, k and g: Player K makes a dare for wK > k, and
Player G accepts for wG > g, and they do not act for lower
values. When the weight is just at the threshold either behaviour is part of an
equilibrium.
It is assumed that before the moves,
Player K holds a distribution on Player G's reputational weight wG
that is uniform between 0 and 1. The same is true for G's distribution on wK.
The onlookers hold the uniform distribution for both weights wK and
wG, but the players know their own values. The Appendix shows that
the threshold values are then k = F(F/2‑1)˝ and g = 21/2/F.
These functions are graphed in Figure 2, which shows that if F is high enough the
dare will be made and performed, a result that stands to reason.
I specified the cost of the dare to be
one unit, but if it were assigned the general value c, F would become F/c in
these formulae, so F should be understood as relative felicity, taken in
comparison to the cost of the action.
Contrary to the assumptions of Version
I, the dare in the poem will apparently be costly either to Gawain or to the Green
Knight, but not to both. The Green Knight knows who will suffer. Either he is
invulnerable, a shapeshifter, or he is vulnerable, a bluffer, who will die by
his own axe, and is challenging in hopes that the Round Table will decline and
lose face. In Version II, the costs of performing the dare are 0 to Gawain and
1 to the Green Knight if the Green Knight is vulnerable, or 1 and 0 if the
Green Knight is invulnerable. The sequence of play and payoffs are the same as
Version I, shown in Figure 1 ‑ ‑ only the information conditions
are different.
Let p be Gawain's and the onlookers'
prior probability that the Green Knight is vulnerable. This probability is
"prior" in the sense that it is based on everything they know about
the Green Knight short of his strategic choice. Within the present model,
Version II, his strategic choice is whether to challenge the Round Table, so p
will reflect information about his appearance, his equipment, his words, etc.
Gawain will update this p to a new value in the light of the fact that the
Green Knight is challenging him. This new probability will be lower, its exact
value determined by the solution of the game.
Let wG, wv and ws
be the reputational weights held respectively by Gawain, the Green Knight if he
is vulnerable, and the Green Knight if he is invulnerable ("strong").
An equilibrium for this game involves three sets: values of w9v9 for which a vulnerable
Knight challenges, values of ws for which a strong Knight
challenges, and values of wG for which Gawain accepts. Again the
equilibrium is essentially unique and involves thresholds. It is calculated in
the Appendix, and shown in Figure 3 for various values of F and p. For p = .5,
for example, a strong Knight always challenges, and a vulnerable one challenges
only if his value of wV is above a threshold, which is much higher
than that of Version I. This rise makes sense since if Gawain refused, a
vulnerable Knight would be indistinguishable from strong Knights of low
reputational weight, so his prize would be less than in Version I. Gawain's
threshold remains roughly the same. Comparing the thresholds for p = .1 and p =
.5, different prior probabilities of vulnerability, Gawain is more likely to
accept, the more he believes the Green Knight is vulnerable.
THE
DARING GAME III: PLAYING BY PRETENDING TO PLAY
This final version of the game most
accurately describes the events in the poem. It posits that the Green Knight
and Gawain know the truth about each other. (This is supported by textual
evidence which I will summarize later.) Gawain knows that the Green Knight is
invulnerable, and knows that for some unexplained motive he wants a contest,
that he has no motive to disgrace him by intimidating him into a refusal.
Gawain regards the Green Knight as having chosen his words, appearance, and the
terms of his offer, in order to recruit someone into playing. On his part, the
Green Knight knows that Gawain's value for his reputation is high. The
audience, however, is as uninformed as it was in Version II, and in fact thinks
that Version II is the game being played; it is uncertain about the value for
reputation held by the Green Knight and by Gawain, and also about the Green
Knight's vulnerability. The view of the on lookers is important since they will
give Gawain his reputational payoff. Like a poker player who has spied on the
other's hand but must hedge the use of this information or be accused of
cheating, Gawain must consider the onlookers' view of the game.
Gawain's optimal choice in Version III
is found by comparing his reputational payoff from accepting minus his
performance cost of 1, with his payoff from declining, both reputational
payoffs being calculated for Version II played with a Green Knight who would be
revealed as invulnerable. These payoffs, and therefore Gawain's action, will be
functions of F and wG for a given p, as Figure 4 shows. For the
purpose of later discussion, Figure 5 graphs his strategy on different axes
than Figure 4, using p and F, the variables the Green Knight is manipulating.
Several sections of the text support the
postulates of Version III that the Green Knight wants his challenge accepted,
unlike a normal daring contest, and that Gawain knows that the Knight is
invulnerable. That the Green Knight wants a contest is consistent with his
goading of Arthur. When the gathering was silent to his first challenge, he
could have ridden out to spread the news. Instead he chides the Round Table and
especially Arthur, working on the king's pride in his name. The Green Knight
seems determined to have an exchange of blows, and his tactic succeeds on the
volatile king. Further, the Green Knight's demeanour while Arthur is preparing
to swing the axe reveals that he wants an exchange. He is as calm as if
"any burne vpon bench hade broght hym to drynk of wyne" (337‑338),
as if someone at the table had approached him with a drink of wine. Previously,
when his challenge was unanswered, he had been restlessly shifting in the
saddle (304‑306), but now events are proceeding as he intended. Finally,
note the Green Knight's remark, made and repeated, that he is happy it is
Gawain who will be striking the blow (387‑391). He leaves us ignorant
about why he prefers Gawain in particular, but he does want a contest.
Gawain seems sure that the Green Knight
will live through the blow. In his request to substitute for Arthur he speaks
of his life as being the least valuable to lose (355). Arthur declares that a
deft blow will end the match (372‑374), but Gawain gives no word of
agreement. When he promises to seek the challenger out (402), his words are
unconditional; he does not qualify them with "if you survive my
blow." Also revealing is the poet's focus when the decapitated body stays
standing. We learn about the onlookers' shock as the head rolls into their midst,
their fear as the body remounts the horse and the head gives orders to Gawain,
but not until the visitor has galloped off are we told of Gawain's reaction.
Although his fate is uniquely tied to the news that the Green Knight has
survived, the poet does not report Gawain's response, and the suggestion is
that he expected the Green Knight to live. Version III incorporates these
ideas, that the Green Knight and Gawain know the truth about each other.
The game analyses of Versions I and II
clarify not the actual contest being played, but the social conventions through
which the Green Knight traps the hero into the bargain. The real planning and
strategizing is close to Version III. The Green Knight's problem is to select
Gawain from the company for his adventure. He presents himself to the Round
Table in a way that invokes the social convention of a challenge, that is not
so effective that everyone volunteers, but strong enough to recruit the
opponent he wants. In making his offers, the Green Knight manipulates the prior
probability p of his vulnerability and the felicity F of his challenge. To
describe how he does this, I first look at the determinants of the felicity of
a dare in general.
AUSTIN'S
THEORY OF PERFORMATIVES
A dare is both a statement and an act;
felicity is its effectiveness, regarding it as an act. When John Austin
delivered the 1955 William James Lectures, the philosophical majority held that
sentences assert possible events. He expounded a theory, to become widely
influential in the years following his death, that some sentences do not say
things, but rather do things (Austin, 1962). Performatives do not
primarily make claims about the facts, but are more like actions, and they
typically cannot be labelled true or false. One example is a promise. If I
promise to loan you my car and never do, perhaps never intended to, you cannot
say my promise was "incorrect," that I claimed to be promising but I
was not. I may have promised insincerely and broken my word, but I did indeed
make a promise.
A promise is one type of performative
and there are many others. Austin estimated one or several thousands. Sentences
that thank, blame, apologize, welcome, find a defendant guilty, pronounce a
couple married, all do things beyond their assertive function. Just what do
they do? Many change the social relationship between actors by inducing new
utilities, duties or rights in that relationship. A jury finding a defendant
guilty gives the legal system the right to punish. Forgiving constitutes a
promise to accord the other respect in your future relationship, in spite of a
transgression.
A performative is judged not by its
truth, but by its effectiveness, and Austin labelled this concept felicity.
Felicity depends on both the immediate context around the speaker and listener,
and on the societal customs supporting the performative. For example, if I make
a promise but at the same time show you my crossed fingers, I am not really
promising. My own action undermines my words. If I declare, "I apologize
for X," but I am reciting a line as an actor in a play, then I am not
really apologizing. Taking an example of Austin's, I come upon a ship ready to
be launched, and seizing the occasion, declare, "I name you the S.S.
Joseph Stalin." For good measure I break a bottle of champagne over
the bow and kick out the chocks. However if I am not the person appointed to
christen the ship, my action is void. Felicity is not all or none: it can be a
matter of degree (promising versus solemnly promising), or it can be debatable
(promising my cat a treat). A performative thus is not a magical incantation
whose mere enunciation does the job. Performatives "come off" to a
greater or lesser extent, depending on their appropriateness.
THE
GREEN KNIGHT'S MANIPULATION OF FELICITY
The requirements of felicity depend on
the particular performative. I will suggest some requirements for a successful
dare. (Neither Austin nor later philosophers seem to have investigated daring.
Greenberg, 1990, gives an interesting account of the language of challenges to
duel in the antebellum American South, but he does not use the framework of
performatives.) First, the two actors must be part of a shared culture
that supports the practice of daring. What goes over with children may not with
adults: if one grownup challenges another to eat a slug, the dare will fail. It
also seems unlikely that an adult could dare a child; daring is not a practice
parents can use to manage their children. The folklorist Roger Abrams (1963)
described black youths in Philadelphia playing the "dozens," a back‑and‑forth
game of sexual insults, "Somewhere between the ages of sixteen and twenty‑six,
playing begins to lose its effect . . . When someone indicates that he wants to
start, the one who is supposed to be insulted may reply, 'Oh man, don't play
with me.' If he needs a more clever retort, he may rely on the proverb, 'I
laugh, joke and smoke, but I don't play.'" (p.51). A particular mode of
daring must have cultural support.
Second, the dare must have apparent
purity of motive. If one child were to dare another to hand over $5, the trick
would surely fail. Dares deal in the commodity of reputation, and a blatant
scheme to make money lacks felicity. Third, for those dares where both parties
are to perform the deed, there must be apparent symmetry of the action.
"Let's run through the boys' locker room, I dare you!" This might
work between little girls, but not from a boy to a girl. Fairness of this type
is necessary to compare the players' weights on reputation. Much of the Green
Knight's self‑presentation is aimed at satisfying the three requirements
of felicity of his dare. Regarding the need for a shared culture that supports
daring, the rules of courtly behaviour do provide such support for challenging,
and the Green Knight himself says he is challenging "bi ryght" (274).
His problem is to establish himself as member of the culture, a bone fide
knight whose challenges compel a response. His fine clothing, his articulate
speech, his horse obedient to the rider's command ("To the gome he wath
ful gayn," 179) serve this end. When he is offered a fight, he replies
that he did not come for that, although at home he has a hauberk, a helmet, a
shield and sharp shining spear (268‑269). Cataloging his equipment can be
taken partly for its face meaning, as an elaboration of how he has not come to
do battle, but he is also letting it be known that he is a fellow knight as he
possesses the accoutrements of one (Burrow, 1965). In line with the need for
purity of motive, the Green Knight stresses the reason for his challenge: it is
a game, for sport (263, 274). He mentions no other goal. His manner is almost
jocular. Only after Gawain accepts does he become businesslike (Davenport,
1973). His size ensures that physical prowess is not the issue, since no one
could stand before him. The Round Table has drawn him here by its reputation,
he says (258), and it will lose face if it does not take up his challenge
(313). The third requirement is apparent symmetry of the action. Here the Green
Knight has a difficult problem. Getting his bizarre terms accepted will not be
easy. He is either mortal and vulnerable, or magical and invulnerable, and in
either case his beheading proposal is unfair. He must obscure this truth, so he
loads the details of the agreement with symmetries that give a facade of
fairness. You strike at me, I strike at you. I offer no resistance, neither
will you. I sought you out at your abode, you seek me out at mine, and come at
this same time next year.
THE
GREEN KNIGHT'S MANIPULATION OF THE APPEARANCE OF MAGIC
Having described the ways through which
the Knight ensures felicity F, I turn to how he sets the perceived likelihood p
that he is vulnerable. Several authors have noted the "richness and
variety of suggestion" in the Green Knight's description (Benson, 1965;
Burrow, 1973). He is "bristling with ambiguity." Within the framework
of Version III, this dual nature stems from the Green Knight's manipulation of
p, the audience's probability that he is invulnerable. The poet is careful not
to place him squarely in this world or the other. He promises to tell us of a
"selly in sight summe men hit holden . . . an outtrage awenture . .
." (28‑29), a marvel, so some men say, an extraordinary
adventure, but he refrains from unequivocally promising a tale of magic. The
Green Knight hovers between the human and the supernatural by the way he
enters, his physical form and his words. He does not appear in a puff of smoke,
but rides in as a mortal could. He is large, the poet says, perhaps a half‑giant,
surely the biggest of men, but in any case handsome. The poet describes his
shape and size, anchoring us in the first impression that he may be human, and
after a dozen lines of detail tells us that punchline, that he is bright green.
In the Green Knight's own words, deliberately chosen, he is a "wyghe”
(315), a term that usually denotes a person but could mean a living being of
any kind (Borroff, 1962, p.112). In battle, he claims, he would defeat his
adversary through armour and weapons, not by dint of magic. He addresses Arthur
with the familiar second person pronoun. Is this the manner of a hostile but
human challenger or that of a being who has no obligation to human kings?
Within Version III's approach, uncertainty over the Green Knight's nature,
measured by p, functions to recruit Gawain but to make his decision hard.
The effect of the Green Knight's
manipulation of F and p can be understood through Figure 5. Depending on the
joint values of F and p, a challenge will select a knight of the Round Table
whose weight on reputation is above a certain value. (A further refinement
would portray the situation as an – player game including all the members of
the court.) By setting his parameters correctly the visitor in duces Gawain, the
pearl among peas, to step forward for the adventure that awaits him.
THE
FEAST OF BRICRIU
The earliest surviving version of The
Feast of Bricriu is found with other important Irish sagas in The Book
of the Dun Cow, transcribed about 1100 A.D. The book's title derives from
the tradition that the ancient champion Fergus, dead for centuries, returned to
life to narrate the deeds of the heroes of Ireland to St. Ciaran who copied
them onto the hide of his cow. The Feast of Bricriu was first written down in the
Eighth Century, as indicated by its language, so it was almost as old when the
Gawain poet lived as his work is now. The tale describes a more warlike, less
refined society than Arthur's court, and probably originated far earlier than
the Eighth Century. Although it is highly regarded, the literature on it is
sparse (e.g., Thurneysen, 1924), and spends little attention on the plot
structure. This summary of the final episode in the story follows the
translations of Henderson (1899) and Meyer (1893).
SUMMARY
OF THE FINAL BEHEADING SCENE IN THE FEAST OF BRICRIU
Bricriu of the Poisonous Tongue is a
gossip and a troublemaker. (His motto: "Clearer to me is a whisper, than
to anyone else a cry.") To foment strife, he invites three of Ulster's
eminent warriors, Loigaire the Triumphant, Conall the Victorious, and
Cuchulainn ("Kuh‑hoo‑lin"), to a feast, maliciously
advertising that only one shall have the champion's portion of the meal, and
his wife preeminence over the women of Ireland. The three warriors refuse to
come for fear of the trouble Bricriu might stir up, but when he threatens to
use his poetic skill to incite slaughter across all Ireland, they relent.
Knowing that the heroes will not permit him to attend his own banquet, he
builds a sun‑room of coloured glass from which he can look down on the
great hall. The three heroes and their wives perform various feats, some
grotesque and bawdy, to gain the champion's share. Cuchulainn wins
consistently, but the other two find excuses for denying his victory. In the
final section of the tale Cuchulainn is publicly proven to be the deserving
one, through the following beheading contest.
The Ulstermen return to the Royal Court
after a day of athletics; the three champions are absent. A bachlach (a
rough herdsman) of great size and ugliness, dressed in an old hide, enters the
hall. From his head protrude ravenous yellow eyes, each the size of a cauldron
to hold an ox. (His description is so fear some that the Eleventh Century monk
recopying the manuscript here inserts into the margin the Sign of the Cross.)
In the bachlach's left hand is a block, in his right hand an axe; "its
handle would require a ploughteam to move it; its sharpness would lop off
hairs, the wind blowing them against its edge." He strides to the high
arched forkbeam by the fire, and Duach of the Chaffer Tongue immediately speaks
up to ridicule his height: "Is the hall lacking in room for you, that you
find no other place than by the fork beam, unless you wish to be a domestic
lightholder? Only sooner will a blaze be to the house than bright ness to the
household." The giant stays calm: "Whatever property may be mine, you
will agree, no matter how big I am, that the whole household will be
brightened, while the hall will not be burnt." Bearing a light is not his
only skill, he says. He has searched the world, yet found no one to give him
fair play. As the Ulstermen excel all, he expects someone among them will grant
his request. The company agrees to grant him fair play. He exempts the king and
his relation from his challenge, but to the others, "Come whoever of you
that wishes to do it, and I may cut off his head tonight, and he will cut off
mine tomorrow night." (This, of course, is the reverse of the Green
Knight's game.) Duach doubts that anyone will volunteer, and Munremar mac
Gerreind springs to the floor to propose that the order of the blows be
switched. "Were that my quest, I could have got that anywhere," the
bachlach objects, but Duach goads him and he accepts, adding the comment,
"strange as that may seem to you." Pledging Munremar to come back for
a return blow, the giant lays his head on the block. Munremar swings the axe,
and the head flies across the hall. Blood streams from the giant's neck and
fills the house. He gathers the head, axe and block to his chest and strides
out.
The giant returns the following night,
but Munremar is not to be found. Loigaire, one of the three contending heroes,
is there, and the giant summons him by name. He beheads the giant, but, like
Munremar, shirks his part of the bargain. Conall is challenged and behaves
identically. Finally the bachlach calls Cuchulainn to play, and despite the
latter's innovation of smashing the head after severing it, bending the rules
as Gawain would later do, the bachlach returns intact next night to demand his
turn. Cuchulainn has fallen into deep gloom, but, unlike the other two, keeps
his word and lays his head on the block. The giant mocks the reputation of the
court and teases Cuchulainn, saying that he cannot kill him yet because his
neck does not reach across the block. Cuchulainn complains that he is being
tormented and asks to be dispatched forthwith. He extends his neck so that a
man might put his foot between each of the verte brae, and it stretches across
the whole block. As the giant raises the axe, the hides that cover him creak
like a forest in a winter storm. He brings it down on Cuchulainn's neck, but
blunt side downwards. He bids him rise and accept the champion's portion. The
giant disappears. He was the magician Curoi mac Daire who came to prove to all
that Cuchulainn was the deserving one.
THE
BACHLACH'S STRATEGY
The narrative seems less polished than Sir
Gawain and the Green Knight's, more like a disconnected dream, but an
informal game viewpoint reveals the functions of some obscure plot elements.
Some of the puzzles are these:
--Arthur's
court falls silent when the Green Knight enters the hall, out of fear and
courtesy according to the poet, but Duach speaks up to insult the bachlach.
What is the function of Duach's remark?
--
Why does the bachlach propose the unfair rule that he strike the first blow,
but quickly agree to reverse the order?
--
Why does the giant include Munremar in list of beheaders when he is not one of
the three contending champions, and in fact he has no other role in the poem?
--
Why does the bachlach summon each champion by name?
The key to these riddles is the
different strategic structure of the contest compared to Gawain's dilemma, and
the different trick the challenger uses to induce the heroes to accept. The
Green Knight strove to make the contest look fair between himself and Gawain,
but the bachlach has a different problem. His aim is to establish fairness
among the three champions who he is comparing publicly, so he must first
ensnare them into the game and then design their tests to be equal.
In inducing the heroes to take up his
challenge, the giant is clever. He first extracts a pledge of fair play,
without revealing the game. He then proposes that he strike the first blow.
When Munremar objects that the order should be switched, the giant readily
accepts. Munremar is trapped; he cannot refuse the bargain since he himself
proposed it. The situation is something like Darers‑Go‑First,
except that we suspect that Munremar made his retort without realizing that he
was falling in with the bachlach's plan. To his chagrin, Munremar learns that
the giant did not issue his initial proposal naively; the giant's remark that
he is ready to go first "strange as that may seem to you," reveals
that he knew his earlier plan had no chance of being accepted. Regarding the
giant's goal of making the contest fair among the three, the difficulty is that
they will play the game sequentially, and this ordering introduces unfairness
in two ways. One is that the person offered the first turn will be more ready
to accept, since he is not sure that the giant can replace his head. Later
players will know about the magic and will be less likely to play.
Consequently, the audience cannot gauge their courage by noting who accepts and
who declines. This first difficulty is solved by introducing Munremar, whose
trial displays the bachlach's wizardry to the later players, the actual rivals
for the champion's portion.
The bachlach's second problem involves the
order in which the champions play. He wishes to show the world who will accept
the game and keep his pledge to return. The giant knows that Cuchulainn alone
is worthy among the three, and must ensure that he is the final player. If
Cuchulainn plays too early, the others will learn that the giant will not
behead them. The giant accomplishes his goal by asking each of them by name to
step forward and play. There is a strategic equivalence between Munremar in The
Feast of Bricriu and Arthur in Sir Gawain and the Green Knight. The
Green Knight incites a first‑taker by angering Arthur. Once his king has
accepted the challenge, Gawain can scarcely maintain that the terms are unfair
and confer no obligation on anyone to accept. The bachlach accomplishes the
same end with a more subtle trick, by maneuvering Munremar into proposing and
accepting the terms of the bargain.
Another strategic correspondence between
the poems is the Green Knight's bough of holly and Duach's initial insult to
the bachlach. Both establish immediately that, despite his threatening
appearance, the stranger has not come to fight. The bachlach and the Green
Knight must shift the issue to one of sport and fair play. The latter cites the
holly as a symbol of his peaceful intentions ‑‑ he carries it in
his hand where one would expect a weapon or shield. The bachlach proves his
peacefulness when he stays unruffled by Duach's suggestion that he could not
hold a light for the household. Each device fits its set ting: Arthur's
assembly could not have insulted the Green Knight without violating its code of
courtly behaviour, and the bachlach would have looked ludicrous holding a holly
twig or a flower.
A
ONE‑PERSON TWO‑PLAYER GAME
Another portrayal of Gawain's dilemma
has the hero playing against himself. Here the focus is on the Green Knight's
enigmatic character and on Gawain's choice of what he should believe about it.
The knight offers plenty of evidence that he is malicious, but plenty that he
is chivalrous and will give Gawain fair play. Gawain has two natures, a dutiful
one and a self‑preserving one. Each can regard the Green Knight as
malicious or chivalrous. If the Green Knight were a clearly malicious being,
his challenge need not be accepted; submitting to him would be suicide. However
if he is a chivalrous fellow knight, Gawain owes him fair play and the Green
Knight will somehow reciprocate it. Neither case is entirely convincing from
the evidence available before decision. Either choice presents a risk, one of
abandoning duty, the other of dying, but the two natures weight these risks
differently. His dutiful half worries more about shirking an obligation, and
his self‑preserving half more about being killed.
If Gawain's two natures disagree on the
Green Knight's character, Gawain will experience tension, an aversive mental
state. They want to agree, but one nature will have to compromise by
overlooking the evidence supporting the other half's worries. The question of
what to believe is here a strategic choice; each nature can focus on part of
the evidence and try to rationalize away the rest. However each nature can
ignore its worries with only partial success, and each would prefer that the
other did the compromising.
This approach is somewhat Freudian, with
its duality of the individual's psyche and its idea of minimizing mental
tension. Freud would call the two natures the superego and the id, and expect
that the superego would be logical and verbal while the id would be intuitive
and inarticulate. This dichotomy fits the poem well. In his speech, the Green
Knight talks like a chivalrous knight in the role of a challenger (Burrow,
1973), but the unearthly aspect of him, his green hue, stays entirely off the
spoken record. Neither he nor the court make any reference to it. Freud's framework
suggests that the dutiful nature of Gawain will listen to the speech, while his
basic nature will focus on the huge green figure. The notion of choosing
beliefs to minimize tension is prominent also in cognitive psychology, as in
the theory of dissonance reduction. Seeking out evidence to confirm what you
have already chosen to believe appears in decision‑making psychology ‑‑
Janis and Mann (1977) call it "bolstering."
FIGURE·6: A game between
Gawain's two natures
These assumptions describe a Battle‑of‑the‑Sexes
game, with payoffs shown in Figure 6. It has two strong equilibria and at each
one or the other nature compromises. Since there are two equilibria the game
has no obvious solution. If one were to substitute cardinal payoffs in the
diagonal cells, one selection solution proposal, justified by axioms (Harsanyi
and Selten, 1988; O'Neill, 1988), would multiply the payoffs in the equilibrium
cells to calculate the Nash products, and choose the higher of the two. However
this rule is rather esoteric, and the essential situation has an outcome that
is equivocal. This indeterminacy makes for a more suspenseful plot.
GAME
THEORY AND LITERARY ANALYSIS
Given the memorable impact of the
beheading scene in Sir Gawain and the Green Knight, it is surprising how
few papers analyse the characters' motives and choices. The conventional
literary repertoire seems to lack a good method to analyse plot per se. The
field of narratology, for example, investigates how the story is told but not
the story itself.
Several formal techniques have been
developed to analyse plot. Vladimir Propp (1928) initiated the "linguistic
method," categorizing the elements in Russian folktales by their abstract
functions, and describing the patterns that appear in the corpus of tales.
Kissam (1977) applied the technique to mediaeval romances. More recent work has
paralleled developments in linguistic theory and has investigated "story
grammars" (e.g., Alker, 1987, and the works he cites). Prince (1973)
tackled the interesting question of what makes a story a story rather than a
mere string of facts. More than Propp's work, this study deals with the causal
and relationships among the elements, and in general these researchers are
moving from a syntactic view of a story toward a semantic one. However they
themselves express frustration in their strivings to get at the
"meaning" of the plot (Ryan, 1981). "Meaning of the plot"
is a linguist's way of putting it; a game theorist would say that they do not
deal with the strategic structure of the game, they do not treat characters'
acts as directed towards goals.
Further formal methods to study plots
have sprung from the "Romanian school of poetics," led by the
mathematical linguist Solomon Marcus (1970; in English, see for example,
Marcus, 1977, 1984, and Brainerd and Neufeldt, 1974). An example is the use of
signed graphs along with the hypothesis that events move towards a state of
"structural balance" (Martindale, 1976; Lalu, 1977). Teodorescu‑Brinzeu
(1984) represents the sequence of thought in soliloquies using symbolic logic.
The statistical analysis of joint presence of pairs of characters on stage is
another theme. Although these techniques are interesting, they work at the
periphery of the plot while game theory aims at the center.
More than a dozen papers have applied
game theory to stories, as Table 1 shows. Oskar Morgenstern presented the
Holmes/Moriarty dilemma non‑mathematically in 1928 and 1935, and later in
his book with von Neumann defined it as a formal game. Rapoport's analysis of Othello
was detailed and advanced for its time. Other papers range from informal game
analogies to Mehlmann's use of differential games. He addresses the puzzle of
why Faust was saved after he had lost his wager with the devil. Mystery and de
tective stories, heavy on plot, are frequent subjects, as are stories with
explicit bargains detailing each possible outcome, e.g., The Merchant of
Venice, Faust, and the present study. One topic not in the table is
informal game models of an author trying to maintain suspense versus the reader
trying to discover the story, e.g. Bruss (1977) and Davey (1984). Brams (1991)
discusses the prospects for game theory in the humanities.
Conventional scholars have ignored these
studies, and some of the papers give them good grounds for doing so. Some
models do not take the literary work seriously; they are inconsistent with
important elements in the story, and do not compare their game interpretation with
details from the text. I will give the two examples that are the easiest to
relate, admitting that my criticism is somewhat unfair since the authors
devised them more to illustrate game theory rather than to study a plot.
|
von
Neumann/Morgenstern 1944 |
The
Final Problem, Conan‑Doyle |
2‑person
zerosum |
|
Williams
1954 |
Merchant
of Venice,
Shakespeare |
2x2
zerosum |
|
Rapoport
1960 |
Othello, Shakespeare |
various |
|
" 1962 |
La
Tosca,
Puccini |
Prisoner's
Dilemma |
|
Vorobev
1968 |
Eugene
Onegin,
Pushkin |
informal,
duels |
|
" |
Gift
of the Magi,
O.Henry |
Battle‑of‑the‑Sexes |
|
Howard
1971 |
The
Caretaker,
Pinter |
metagames |
|
" 1980 |
Dr.
Zhivago,
Pasternak |
“ |
|
Teodorescu‑Brinzeu
1977 |
Othello, Shakespeare |
3x4
non‑zerosum |
|
Steriadi‑Bogdan
1977 |
The
Mousetrap,
Christie |
ext've,
perfect info |
|
Lalu
1977 |
Richard
III,
Shakespeare |
ext've,
perfect info |
|
Brams
1980 |
Old
Testament
|
various
finite games |
|
Hintikkas
1982 |
Silver
Blaze,
Conan‑Doyle |
informal
ext've form |
|
Carlson
1984 |
Numerous
mystery novels |
informal |
|
Mehlmann
1989, 1990 |
Faust, Marlowe, Goethe |
differential
games |
TABLE 1. Game theory studies of
literary plot.
The first is the opera La Tosca treated
as a Prisoner's Dilemma. The Chief of Police Scarpia has condemned Tosca's
lover to death but offers to fake the execution in exchange for Tosca's
favours. She consents, but after he has sent off the order, she seizes a knife
and stabs him. He, however, had already doublecrossed her, and had written an
order that her lover be really shot. The pair come to an inefficient outcome
like the players of a Prisoner's Dilemma, but if we scrutinize the model, flaws
appear. If Scarpia knew that Tosca might stab him, why did he leave a knife
lying out? And if she knew his set of strategies, could she not have demanded
to see her lover safe out of the country first, and so on? Prisoner's Dilemma
is a valid model only if the information conditions are right; ending up at an
inefficiency is not enough. The model cannot explain why they limited their
moves in this way, or why they made the bar gain in the first place. The
analysis of O. Henry's short story The Gift of the Magi also overlooks
the data. A fond husband and wife plan to surprise each other with Christmas
gifts. She cuts her hair to buy him a watchchain, but, ironically, he has sold
his watch to buy a clasp for her hair. Vorobev cast it as a Battle‑of‑the‑Sexes
game, symmetrical with two strong equilibria as in Figure 6. (The
interpretation of the payoffs in this "battle" has each worrying
mainly about the other's material welfare.) The model accurately reproduces
what happens in the plot; since the players in the abstract game have no way to
coordinate their choices, it is reasonable that they might end up at an
inefficient non‑equilibrium. However O. Henry's trademark was the
surprise ending, and an analysis that predicts that his story will turn out as
might be expected, clearly has not captured the story's essence. When they
reveal their gifts to each other, the characters are stunned. This fact is
inconsistent with the simple matrix analysis. We need methods that go deeper,
perhaps using recent work on information and knowledge in games.
Lacking a method for examining
characters' overall strategies, scholarship on the plot of Sir Gawain and
the Green Knight has taken several directions. A few critics have tried to
construct new motives for the actions of Gawain, the Green Knight or Morgan Le
Fay. However most conventional analyses of plot (Bloomfield, 1961) have dealt
with the symbolism of the characters (Does the Green Knight's hue symbolize
nature or Christmas or death or the fairy world?), or with the social meaning
of various elements (Is his manner to Arthur rude, or appropriate to a
challenger? Is Arthur's demeanour seemly for a king?), or with motifs shared
with other stories (Where else do we see a challenge by a stranger, or an
exchange of blows?) This third subject of recurrent themes is prominent in the
critical literature on the poem (Catalini, 1979; Benson, 1965; Kittredge, 1916;
Brewer, 1973). The findings are fascinating; for some deep reason, challenges
by strangers to test one's virtue, exchanges of blows and magical survivals of
deadly wounds abound in folk and mediaeval literature. However the comparative
study of plot elements is limited in what it can reveal. One could not learn
much about checkers and backgammon by comparing the physical shapes of the
pieces. The element can have different functions in different stories, or
conversely, similar functions can be fulfilled by different elements. The Green
Knight's bough of holly and Duach's insult to the giant correspond
strategically, as I argued, but are different physically. Munremar's challenge
to the bachlach's offer has the same role as Arthur's impulsive seizing of the
axe, even though the events themselves are dissimilar. The element of Munremar
confronting a stranger appears in Scela Mucce meac Da Tho, The Tidings
of Mac Datho's Pig, another Old Irish claims problem about the champion's
portion of a meal and the ownership of a marvellous hound (Chadwick, 1927), but
there its strategic role is quite different. Mediaeval writers borrowed and
combined familiar ideas without compunction, and the meaning of an element was
not inherent, but arose from the context. Strategy‑oriented methods can
bring out similarities of function across different settings.
SUPERPOWER
RIVALRY IN THE THIRD WORLD
Mediaeval challenges possibly had a
defensive purpose. A knight ready to fight in single combat would be seen as
more reliable in a battle, so courts looked for ways to display their
willingness to fight. In the language of modern strategy studies, chivalric
challenging may have served to enhance deterrence, and the same could be said
of children accepting dares. Nuclear weapons have reemphasized the problem of
reputation in deterrence, although it is now called the problem of credibility
(Jervis, 1979; Morgan, 1985; Powell, 1990). Because a nuclear crisis could lead
to a catastrophe beyond precedent, each government has sought to convince the
other beforehand that it will resist if its interests are crossed.
A model along the lines of the daring
game can be devised for the international case. It could take this form:
Country 1 commits itself to a given issue or does not commit itself, then
Country 2 fights with it on that issue or does not. If the commitment is made
and challenged, both lose pay a conflict cost, and also receive the
reputational payoff. They have a strong desire for this reputation, a desire
generated through its usefulness in future situations. The added self‑referential
element in a side's payoff, the value for reputation for valuing reputation,
outweighs the cost of resisting in some conflicts and make threats credible.
To check the validity of the model we
can identify its key features and look for them in international interactions.
The important elements are the same three as in the daring model: first, a
language act that triggers the reputational goal; second, the goal of
reputation for valuing reputation, as incorporated in payoffs like reputational
weight x [others' estimate of reputational weight given your action] ‑
conflict cost, or the more competitive form of Figure 1; and third, the model's
consequence that the parties will fight over issues where winning is of no
benefit to them. To the extent that Cold War contests show these three
elements, the model deserves more credence.
The first feature of the model, the performative
utterance, is prominent in the international context. One side often declares
or reasserts its "commitment" to defend some interest. At first it is
puzzling why words should matter in the realpolitik of world struggle, but this
issue was the very one raised by the Green Knight's remarks to Arthur, and the
answer again involves Austin's theory of performatives. A difference between
the mediaeval and the modern case is a dare comes from the disturber of the
status quo, whereas a commitment is issued by the status quo supporter, but
both declarations function as performatives in Austin's sense. Verbal threats
by governments also fall under his theory.
In regard to the second feature of the
model, seeking reputation for valuing reputation, some explanations for U.S.
foreign policy seem to treat reputation, will or credibility as autonomous
entities. James Payne, a supporter of U.S. policy, writes, "Hostile states
tend to oppose moves by each other and each other's allies to alter the status
quo because they desire to avoid the loss in threat coerciveness (or reputation
for determination) which would result from the successful alteration"
(1970, p.117). Robert Tucker, also a supporter, argues, "When engaged in a
contest for global stakes, what may appear as a marginal interest will be
invested with a significance it would not otherwise have, for almost any
challenge is likely to be seen by the challenger and by third parties as a test
of one's will" (1981, pp.144‑145, quoted by Schoultz, 1987). Robert
Johnson, a former U.S. State Department analyst and a critic of these policies,
asserts, "Concern with credibility itself defines U.S. interests and plays
the major role in determining the particular commitments the United States
undertakes and the resolve with which it carries out those commitments; the
specific situation is relatively less important" (1985, p.43). Referring
to U.S. policy in Vietnam and Nicaragua, James Chace writes, "the Soviet
challenge has ignited an obsession with credibility so that concerns about it
determine commitments, which in turn expand U.S. interests" (1988, p.10).
These rationales are puzzling, as they
construe determination, will and credibility as standing by themselves. Payne
stresses reputation for determination, but "determination" to do
what? If the situation involves one's interests only marginally, as Tucker
allows, why is it a "test of one's will"? And what could Johnson's
phrase "credibility itself" mean? These were the types of questions that
led to a recursive definition of reputation in dares. The international model
would state that governments seek reputation for valuing reputation, that they
want to make it credible that they place a high value on that credibility. The
idea is confusing in English, but innately simple and consistent, as the model
shows.
The third feature of the model was the possibility of struggles
over issues of no innate importance. Finding an international question of
absolutely no importance is hard, of course, but based on their involvement in
past policy debates, Johnson and Chace hold that commitments often lead to
interests, rather than the other way around as common sense would expect. The
model explains how this phenomenon could happen. Others have doubted that
international struggles over pure non‑issues could be sensible. Desch
(1989) lists some past writings on the subject and concludes that the idea
deprives credibility of any objective or rational basis (p.95). Jervis (1979,
p.315) states, "Commitment can only be built on a foundation of intrinsic
and strategic interest." As theory, their claims seem to be incorrect; the
present model suggests that a logical government might commit itself and fight
for credibility per se. According to the model, in fact, it does not matter who
wins the struggle, as long as each exerts the effort of struggling.
More conflict over non‑issues
arise in arms building decisions and in arms control negotiations. When each
superpower has over 10,000 strategic nuclear weapons, exact counts are
militarily meaningless, but the US and USSR have increased their arsenals over
the years, first in numbers, then in quality. Negotiations have stalled over
small differences in armaments to be given up, even over the banning of weapons
that neither side possessed or wanted to build, like multiple warhead cruise
missiles (Talbott, 1979). Other instances of conflict over nothing were the
"chicken of the sea" games played by Soviet and American warships,
heading at each other and swerving at the last moment (Zumwalt, 1976, p.322),
or shouldering each other off during refuelling (Lynn‑Jones, 1985).
Before the 1972 Prevention of Incidents at Sea Treaty, this perilous sport went
on one or two dozen times a year, apparently tolerated by naval and civilian
authorities.
The model predicts that players will
sometimes welcome a contest. During the Middle courts went beyond responding to
individual challenges, and held jousting tournaments (Vale, 1981) and pas
des armes, in which a group of knights seized a location such as a hill or
bridge and invited others to dislodge them. Modern governments often seem glad
when commitments are challenged, since they have an opportunity to increase
their credibility.
Does the model mean that the superpowers
have acted sensibly in waging proxy wars in the Third World? Not at all, since
a set of beliefs and actions may be self‑consistent yet invalid. Indeed
some empirically‑based studies have suggested that credibility is
overemphasized (e.g., George and Smoke, 1974; Lebow, 1981; Huth and Russett,
1984, p.519; see also the references given by Jervis, 1989, p.193). Those
holding power worry about credibility, however, and this paper's attempt to
state the argument precisely is a step in scrutinizing it. A further reason
that the model does not rationalize proxy wars, is its narrow scope. It treats
what to do in a single situation, not whether Third World rivalry is sensible
as a continuing practice. Looking at a larger game, the superpowers have moves
that could end this system of "security." Its demands seem ill‑defined,
open‑ended and immensely harmful. Rather than imposing its costs on the
privileged, it inflicts suffering on peoples who have no power to end it.
Acknowledgements:: I would like to thank Steve
van Evera, Gabrielle MacDonald, David Pearce, Mancur Olson, Janice Newton,
Steve Brams, Robert Chibka, Elaine Bennett, Neta Crawford and Catharine and
Dorothy O'Neill for their help and good suggestions. Also appreciated are the
ideas of David Rothman, Donna Gregory, Jeff Smith, Art Stein and other
participants in UCLA's Jacob Marschak Colloquium on Mathematics in the
Behavioral Sciences. This work was done with support from an SSRC/MacArthur
Fellowship in International Security.
APPENDIX. CALCULATING THE EQUILIBRIA.
.Version
I. To show that all equilibria must be of the threshold type: Let the
onlookers' estimates of the difference between Player G's (Gawain's) and Player
K's (the Green Knight's) reputational weights, be designated D and A if G declines
or accepts respectively. Assume, contrary to the threshold claim, that there
are weights w and w' such that a G holding w would accept, a G holding w' would
decline, and w' > w. The payoffs with the lower weight w from accepting and
declining are FwA‑1 and FwD, and the existence of an equilibrium
involving acceptance implies A > D and w > 1/F(A‑D). However
declining at higher weight w' yields w' < 1/F(A‑D), a contradiction.
The existence of a threshold in Player K's equilibrium strategy can be shown
similarly.
Define wK and wG
as K's and G's respective weights and k and g as their thresholds. To derive
the thresholds: By the uniform prior on the weights, if Player K chooses to
dare, the others' expectation for wK is (1+k)/2. Also, if Player G
accepts, others' expectation for weight wG will be (1+g)/2, but a
refusal produces estimate g/2. Substituting these in the payoff expressions in
Figure 1, K and G get FwK(k‑g)/2 - 1 and FwG(g‑k)/2
- 1, respectively, when a dare is accepted, and FwK(1+k‑g)/2,
FwG(g‑k‑1)/2, when one is declined. If wG
equals the threshold g, G is indifferent between accepting and declining, so
Fg(g‑k)/2‑1 = Fg(g‑k‑1)/2, yielding g = 21/2F‑1.
Also, at an equilibrium Player K has
likelihoods 1‑g and g of a dare being accepted or declined, respectively.
Using these probabilities to calculate the expectation from daring, it follows
that when K's weight is at the threshold and K is thus indifferent between the
actions, then (1‑g)[Fk(k‑g)/2 ‑ 1] + gFk(1+k‑g)/2 = 0.
Combining this formula with the expression derived for g, gives k = F(F/2‑1)˝.
Version
II. To
simplify the description of the equilibria, we will assume that the weights are
uniform on the open interval (0,1). Derivation of the threshold property is
similar to Version I. Gawain and the onlookers hold probability p that the
Green Knight is vulnerable and the costs will be 1 for the Green Knight and 0
for Gawain. With probability 1‑p the costs will be 0 and 1. Abbreviate
Gawain's threshold by g, a vulnerable Knight's threshold by v, and a strong
Knight's by s. Everyone else's probability that the Knight is vulnerable given
that he challenges will be p' = (1‑v)p/[1‑s(1‑p)‑vp].
This expression is used to calculate the following payoffs:
If a strong Knight challenges and Gawain
accepts, the Green Knight and Gawain get FwS(s/2‑g/2) and FwG(g/2‑s/2)‑1
respectively.
If a vulnerable Knight challenges and Gawain
accepts: FwV(v/2‑g/2)‑1 and FwG(g/2‑v/2).
If a strong Knight challenges and Gawain declines: FwS[(1‑p')(˝+s/2)+p'(˝+v/2)‑g/2]
and FwG[g/2‑(1‑p')(˝+s/2)‑p'(˝+v/2)].
If a vulnerable Knight challenges and
Gawain declines: FwV[(1‑p')(˝+s/2)+p'(˝+v/2)‑g/2] and FwG[g/2‑(1‑p')(˝+s/2)‑p'(˝+v/2)].
If there is no challenge: 0 and 0.
To calculate a strong Knight's expectations
from challenging, weight the appropriate payoffs above by likelihoods of
accepting or refusing (1‑g) and g. Simplifying the expression yields that
a strong Knight expects (FwS/2)[s(1‑p'g)+p'gv] from
challenging, compared to 0 from not challenging. If the expression in brackets
is strictly positive any type of strong Knight will challenge (that is, s = 0).
For a possible equili brium with s > 0, the expression in brackets must be
zero (it cannot be negative), implying p' = g = 1 and v = 0. Equilibria of the
latter type cannot exist, however, since it can be shown that any strong Knight
would be motivated to challenge given the knowledge that Gawain will surely
decline. Therefore we turn back to investigate the remaining possibility of
equilibria with s = 0. The conditions equating the payoffs for each action for
a vulnerable Knight and Gawain when their weights are just at threshold, are
respectively (1‑g)[Fv(v/2‑g/2)‑1] + gFv[(1‑p')(˝)+p'(˝+v/2)‑g/2]
= 0 and p'Fg(g/2‑v/2) + (1‑p')(Fg2/2‑1) = Fg[g/2‑(1‑p')/2‑p'(˝+v/2)].
The first yields g = (2‑Fv2)/[2‑(1‑p')Fv2],
and the second g = 2(1‑p')/F. Equating these and including the further
condition given s = 0, that p' = (1‑v)p/(1‑vp), gives a quartic
equation in v which can be solved numerically and interpolated for a diagram.
Figure 3 uses the technique of parabolic blending.
Version III. Here Gawain is paid as if
he were playing Game II, but in fact he knows that the Green Knight is
invulnerable. Let A(F,p) and D(F,p) be the onlookers' assessment of his
reputational weight minus a strong Knight's, given he accepts or declines
respectively. Given the above analysis of Version II, A(F,p) = g/2 and D(F,p) =
g/2‑(1‑p')/2 ‑p'(˝+v/2), where p'= (1‑v)p/(1‑vp)
and v and g are the equilibrium values calculated in Version II (thus functions
of F and p). Gawain then expects FwGA(F,p) ‑ 1 from accepting
and FwGD(F,p) from declining. He will accept if the former is
greater, i.e., if wG > 1/F[A(F,p)‑D(F,p)] = 2/F(1+v).
Figures 4 and 5 plot the thresholds.
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