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Sherlock Holmes

and Game Theory

MICHAEL WAINWRIGHT

Although mathematical inter-

pretations of rationality appeal

to the analysis of detective

fiction, literary critics have seldom used mathematics to interrogate narratives in which

logical deductions solve crimes or elucidate mysteries. While “the specificity of narrative

models lies in depicting experiential content, if only by virtue of depicting agents in pur-

suit of humanly recognizable goals,” explains Peter Swirski, the elements of logic in

mathematical models “are valued precisely to the extent they can be voided of subjec-

tivity.” Literary critics have offered “scarcely any commentary to date about the analogies

between mathematics and narrative fiction” because they are “intimidated by such man-

ifest differences” (50).

This essay reanalyzes the game theory interpretation by John von Neumann and Oskar Morgenstern of Arthur

Conan Doyle’s “The Final Problem.” The usefulness of this interdisciplinary hermeneutic is then supplemented

by lateral and philosophical thinking as prompted by subsequent tales involving Doyle’s detective.

Before turning to those mortal and mental aspects of

the matter which present the greatest difficulties, let

the inquirer begin by mastering more elementary

problems.

—Arthur Conan Doyle, A Study in Scarlet

Mosaic 45/3 (September 2012)

82

The prolegomena that follows addresses Swirski’s concern by applying the ele-

mentary principles of Hungarian-born mathematician John von Neumann’s game

theory to a selection of Sherlock Holmes tales from the canon of Arthur Conan Doyle.

Attendant philosophical contentions then help to broaden this application to a con-

text that considers lateral thinking and rational irrationality as valuable interpretive

supplements to the necessarily strict delimitations imposed by game-theoretic rules.

This overall treatment supports the positive side of Ian Ousby’s judgment concerning

the Holmes oeuvre: Doyle’s stories do experience a general decline in standard fol-

lowing World War I, but a game-theoretic reading of Holmes’s adventures supports

the case that this deterioration “is neither total nor entirely uniform” (170). Hence, as

Ousby concedes but fails to contemplate in detail, Doyle’s inventiveness occasionally

shapes Holmes’s later adventures, with Holmes evolving into a thought-provoking

portrayal of human cognition.

As his autobiography testifies, Doyle first studied mathematics at Stonyhurst, the

Jesuit college he attended between 1868 and 1875, where he underwent “the usual

public-school routine of Euclid, algebra and the classics.” In Doyle’s opinion, the

Jesuits “calculated to leave a lasting abhorrence of these subjects” (Memories 10) on

their pupils, but in his case failed; rather, as Doyle recounts in Through the Magic

Door, he resolved to gain “a broad idea” of the sciences and “understand their relations

to each other” (249). Membership of the Society of Authors, a formal group of writ-

ers confederated in 1884, rewarded Doyle’s determination through his acquaintance-

ship with Henry Ernest Dudeney. Dudeney’s mathematical conundrums for various

journals, including the Strand Magazine and Tit-Bits, were somewhat of a novelty.

“Puzzles in periodicals were uncommon at that time in England,” notes Angela

Newing. “Lewis Carroll had a few mathematical puzzles printed as a series in ‘The

Monthly Packet’ from 1880, but that was a magazine for young people” (297).

At one level, the cases undertaken by Doyle’s “elementary” detective are akin to

Dudeney’s conundrums, because they require clear thinking and sustained logic to

solve, but at their core, Holmes’s mysteries often amount to the logical puzzles of

interpersonal relations known as “coordination problems.” In these dilemmas, states

William Poundstone, “one must make a choice knowing that others are making

choices too, and the outcome of the conflict will be determined in some prescribed

way by all the choices made” (6). Not only are coordination problems independent of

participant class and social status, but they also respond to demographics in which

individual actions affect numerous people to a previously unimagined extent as the

population of a region markedly increases. Doyle’s oeuvre, especially his collection of

Michael Wainwright 83

Holmes stories, which date from 1887 to 1927, responds to the demand from this

evolving dynamic. A metropolitan resident but not averse to travelling beyond

London when the call arises, Holmes must deal during the course of his investigations

with a large number and wide range of people from mendicants and dope fiends to

prominent politicians and royalty.

The year 1927 also marked the culmination of von Neumann’s postgraduate

work on axiomatics for the German mathematician David Hilbert at the University of

Göttingen. Although diligent in his primary research—an attempt to extrapolate

beyond the provenance of Euclid’s Elements of Geometry as extended by British math-

ematicians Bertrand Russell and Alfred North Whitehead in their three-volume

Principia Mathematica—von Neumann retained the mathematics of gaming as a sup-

plementary pursuit. Indeed, his first major publication, “Zur Theorie der

Gesellschaftsspiele,” arose from this secondary field. Von Neumann’s study of parlour

games draws on the work of the French mathematician Émile Borel, especially “La

Théorie du jeux et les équations intégrales à noyau symétriques,” but eschews Borelian

complexity in order to found the basics of a new mathematical discipline: the theory

of games of strategy, which is more concisely called “game theory.”

The heart of von Neumann’s paper lies in a simple model of fair division, the

proverbial “Cake Cutting Dilemma,” in which a father wishes his two young sons to

share the remainder of his birthday cake, but is worried about their seemingly unap-

peasable self-interest. He knows that however he divides the cake there will be recrim-

inations over who got the largest piece. A solution to his dilemma suddenly comes to

mind: each son is equally dexterous, so one boy must cut the cake, and then the other

must choose a slice. If the first boy cuts the cake unevenly, then his brother will secure

the larger piece. The only logical decision for the cutter is to divide the remaining cake

into even slices.

This type of subjective soliloquy shows how cost-benefit calculations, which con-

sider the likely overall losses and gains accruing from one of two choices, usually pre-

cede basic either/or decisions. The outcomes for each participant of picking a

particular strategy in such dilemmas can therefore be assigned simulative values (or

utilities). In complex mathematical simulations, a continuous scale provides utilities,

but rudimentary models need only refer to utilities on an ordinal scale. The assigned

utilities for the divided gateau in the Cake Cutting scenario are -1 for a small piece, 0

for an even slice, and 1 for a large piece. This is a “zero-sum” dilemma because a par-

ticipant acquires benefits from his opponent so that no gain or loss accrues in toto.

The matrix in figure 1 shows the possible outcomes:

Mosaic 45/3 (September 2012)

84

Both the soliloquy and the corresponding mathematics indicate the rational outcome:

the logic of self-interest dictates that this zero-sum situation must end in a draw. No

matter how often this type of scenario presents itself, an even distribution of the prize

is the only sensible action, making the choice between portions superfluous.

Von Neumann realized that selection involves minimizing the maximum out-

come left by division. He termed this result, indicated by the minimum column of the

maximum row in the game matrix, the “minimax.” Conversely, division involves max-

imizing the minimum amount left by selection. He denoted this outcome, designated

by the maximum row of the minimum column in the game matrix, the “maximin.”

Von Neumann’s “minimax theorem,” states Poundstone, “demonstrate[s] that any two

rational beings who find their interests completely opposed can settle on a rational

course of action in confidence that the other will do the same” (97, emph.

Poundstone’s). The logical outcomes to such dilemmas derive from the equilibrium

implied by the interdependent self-interest of players with antithetical aims. Like

Borel, von Neumann understood that rationality should underpin the consideration

of coordination problems, but unlike Borel’s arcane mathematics, von Neumann’s

minimax hypothesis offered a straightforward manner in which to interrogate such

interrelations.

In the strictest sense, Cake Cutting is not a coordination problem because the

participants take their decisions successively rather than simultaneously. A minor

adjustment to this model leads to the scenario commonly called “Matching Pennies,”

which is more rigorous in terms of game theory. Matching Pennies simulates an

evenly weighted contest between two players across a single divide. Each participant

has a one-penny coin and must lay this coin face down or face up on a table at the

same time as his opponent. If the orientation of the pennies matches, the first player

wins both coins, but if the orientation of the pennies does not correspond, the second

participant pockets both coins. Setting the utilities for this game at 1 for matched

sides and -1 for unmatched sides results in the matrix of figure 2.

In a further evolution from Cake Cutting, Matching Pennies involves multiple

plays rather than a one-off game, thereby acknowledging the ability of participants

Son 2

Son 1

Choose Large Choose Small

Cut Even 0 0 0 0

Cut Uneven -1 1 1 -1

1. Possible outcomes to the Cake Cutting Dilemma.

Michael Wainwright 85

not only to remember particular opponents, but also to anticipate the forthcoming

decisions of these adversaries: theoretically, any consistent pattern of play is recogniz-

able, so the arbitrary choice of heads or tails is the safest course of action. With this

“pure strategy,” so termed because a player need only repeat a single tactic to obtain

the best result, the expected payoff for randomness over a series of plays is a disap-

pointingly muted 0, but there are no better strategies available.

Edgar Allan Poe’s C. Auguste Dupin, an esteemed forebear of Doyle’s Holmes,

describes an example of Matching Pennies in “The Purloined Letter.” There was a

schoolboy, professes Dupin, “whose success at guessing in the game of ‘even and odd’

attracted universal admiration.” This simple recreation “is played with marbles. One

player holds in his hand a number of these toys, and demands of another whether that

number is even or odd. If the guess is right, the guesser wins one, if wrong, he loses

one.” Dupin’s subject “won all the marbles of the school” (170), but what method did

he use? “I fashion the expression of my face as accurately as possible in accordance

with the expression of my opponent,” the boy explained to Dupin, “and then wait to

see what thoughts or sentiments arise in my mind or heart, as if to match or corre-

spond with the expression” (171). While Doyle acknowledged the perfection of “The

Murders in the Rue Morgue” and “The Gold-Bug,” as Through the Magic Door attests,

he “would not admit perfect excellence to any other of Poe’s stories” (115, emph.

Doyle’s), including “The Purloined Letter.” That the lad’s success stems from dubious

physiognomy rather than pure logic may account for Doyle’s exclusion of this story

from his pantheon.

Although game theory seems unnecessary to an explanation of Matching Pennies,

situations that retain the outline of Matching Pennies but with different utilities are fre-

quent, and these variations can produce less obvious and more serious outcomes. Von

Neumann’s minimax theorem emphasizes such dangers, but credit goes to the German-

born Austrian economist Oskar Morgenstern for illustrating these consequences with

reference to a work of fiction. His analysis in Wirtschaftsprognose: Eine Untersuchung

ihrer Voraussetzungen und Möglichkeiten of Doyle’s “The Final Problem” is, in effect, the

first application of game-theoretic ideas as a literary hermeneutic.

Second Player

First Player

Heads Ta i ls

Heads 1 -1

Tail s -1 1

2. Possible outcomes to Matching Pennies.

Mosaic 45/3 (September 2012)

86

Doyle had intended “The Final Problem,” the closing adventure from The Memoirs

of Sherlock Holmes, to be his character’s swan song. Having fled England to avoid the

murderous intentions of Professor Moriarty, Holmes earns a little respite in Switzerland

before coming face to face with his adversary in Meiringen, their physical tussle at the

Reichenbach Falls apparently pitching both men to their deaths. However, within ten

years of “The Final Problem,” as Luciana Pirè chronicles, Doyle “joked that his bank

account had been dying with his detective” (7-8) and he was therefore grateful, as Owen

Dudley Edwards notes, that “public clamour [had] turned Holmes into an economic

asset that could not be ignored” (vii). Doyle revived his creation in 1904 and the detec-

tive’s final outing in print, “The Adventure of Shoscombe Old Place,” would not appear

until April 1927. The following year, Morgenstern found himself contemplating

whether Holmes was lucky to have survived so long, considering the probabilistic

dilemma of Holmes’s escape to continental Europe that precedes the Swiss dénouement

of “The Final Problem.” Despite no knowledge of the minimax theorem, Morgenstern

recognized the arbitrary nature to Doyle’s resolution of this conundrum; even so,

Morgenstern’s reading of “The Final Problem” might have languished in obscurity had

not political events of a global nature promoted the value of game theory.

Since 1929, when he joined the Institute of Advanced Studies (IAS) at Princeton,

von Neumann had split his time between Europe and New Jersey, but the rise of fas-

cism eventually forced him to settle permanently in America in September 1938. Von

Neumann’s subsequent friendship with Morgenstern, another European émigré at the

IAS, widened the application of mathematical strategy simulation to the Theory of

Games and Economic Behavior. The premise of this volume promotes game theory as

an important intellectual resource, an ideal tool for informing the postbellum fiscal

policies of the Allies. Ironically, however, just as the dissemination of Borel’s “La

Théorie du jeux” had fallen foul of mathematical abstruseness, so too did von

Neumann and Morgenstern’s publication. Fortunately, the private RAND

Corporation (Research and Design) in Santa Monica, which would add von

Neumann to its team of elite analysts in 1952, recognized the importance of the

Theory of Games and Economic Behavior in the late 1940s. Hence, von Neumann and

Morgenstern’s study became, as Poundstone submits, “one of the most influential and

least-read books of the twentieth century” (41). In terms of literary hermeneutics,

Morgenstern’s interpretation of Doyle’s “The Final Problem,” willingly reprised from

Wirtschaftsprognose but in the rigorous terms of von Neumann’s minimax theorem,

had resurfaced only to lose academic currency almost immediately.

Nonetheless, that Moriarty is “a man of good birth and excellent education,

endowed by Nature with a phenomenal mathematical faculty,” which had led him by the

Michael Wainwright 87

age of twenty-one to write “a treatise upon the Binomial Theorem” (252-53), as Holmes

tells Watson in “The Final Problem,” must have intrigued von Neumann and

Morgenstern. Holmes deems Moriarty “a genius, a philosopher, an abstract thinker,” and

nothing less than “a mathematical celebrity”; yet, as if Doyle is explicitly delineating the

intimate connections between game theory and biology, “hereditary tendencies of the

most diabolical kind” have ruined the promise of a “most brilliant career” for the pro-

fessor (253, emph. mine). Like some precociously anachronistic game theorist, Holmes

has been dogging Moriarty for months in a “silent contest” that “would take its place as

the most brilliant bit of thrust-and-parry work in the history of detection” (254, emph.

mine), and has laid a police trap for the professor and his principal associates.

Knowing his life to be in danger from the criminal mastermind until his snare is

sprung, Holmes decides to hide in continental Europe, but to do so he must get to the

south coast of England free of Moriarty. His description of this situation to Watson

again prefigures the rhetoric of game theory. “These are your instructions, and I beg, my

dear Watson,” he insists, “that you will obey them to the letter, for you are now playing

a double-handed game with me against the cleverest rogue and the most powerful syn-

dicate of criminals in Europe” (257, emph. mine). Successfully ensconced on a London

Victoria-to-Dover express train the following day, with Canterbury the only intermedi-

ate stop, Watson assumes he has helped Holmes evade the murderous Moriarty.

Holmes, emphasizing the fact that their train waits at Canterbury for fifteen minutes,

disabuses him of the fact. “My dear Watson,” he sighs, “you evidently did not realize my

meaning when I said that this man may be taken as being quite on the same intellectual

plane as myself” (260). That the professor engages a special (or private) train, which he

will command either to stop at Canterbury or to speed non-stop to Dover, in the hope

of anticipating Holmes’s decision to detrain or not en route, and that Holmes expects

Moriarty to behave in this manner is in keeping with this declaration of relative worth.

A zero-sum two-person game delineating a “mixed strategy,” so named because a player

must deploy a combination of tactics to obtain the best result, has evolved.

“Both opponents,” von Neumann and Morgenstern state, “must choose the place of

their detrainment in ignorance of the other’s corresponding decision” (177).

Unsurprisingly, the Theory of Games and Economic Behavior proceeds to allocate utilities,

calculate the probability of each outcome, and suggest the likelihood of Holmes’s

escape to the European continent. “If,” on the one hand, “they should find themselves,

in fine, on the same platform, Sherlock Holmes may with certainty expect to be killed

by the professor” (177). The detective foresees his own “inevitable destruction”

(“Final” 255) in this instance. “If,” on the other hand, “Holmes reaches Dover

unharmed, he can make good his escape.” This situation, explain von Neumann and

The probabilities associated with Moriarty’s choices are ε1(Dover) and ε2

(Canterbury), while the probabilities attendant on Holmes’s decisions are η1(Dover)

and η2(Canterbury). The minimax theorem provides these four values, with the inter-

mediate steps taken by von Neumann in “Zur Theorie der Gesellschaftsspiele” in estab-

lishing these expressions omitted for the sake of convenience and hermeneutical focus:

Mosaic 45/3 (September 2012)

88

Morgenstern, “has obviously a certain similarity to Matching Pennies” (177).

Moriarty is the player who wants a face-to-face outcome—he craves a confrontation

with Holmes at Canterbury or Dover—while Holmes is the participant who desires

an unmatched dénouement—he wants to detrain at the station Moriarty avoids.

What are the possible outcomes to this dilemma?

The Theory of Games and Economic Behavior denotes Moriarty as Player 1 and

Holmes as Player 2, while Choice 1 designates proceeding to the final destination and

Choice 2 indicates detraining at the intermediate station. Holmes is certain of his own

destruction if Moriarty confronts him in England, so the utility for the professor

catching Holmes is 100. If Holmes gets off at Canterbury and Moriarty continues his

journey to Dover, then Holmes is stuck in England, but at least free from criminal

menace for the time being. Von Neumann and Morgenstern therefore set the utility

for Holmes escaping at Canterbury to 0. Dover offers Holmes the best chance of evad-

ing his adversary, but Moriarty must detrain at Canterbury. Even then, as the Swiss

section to “The Final Problem” avers, the European mainland does not guarantee

Holmes’s survival. Von Neumann and Morgenstern’s utility for the Dover-bound

Holmes avoiding the Canterbury-detrained Moriarty is -50. These imbalanced pay-

outs, by granting the antagonists greater scope for their wits, make the Holmes-

Moriarty situation far more intriguing than Matching Pennies. Game theory

translates these preliminary reckonings into the matrix of figure 3:

Sherlock Holmes is Player 2

Professor Moriarty

is Player 1

1 2

1 100 0

2 -50 100

3. Possible outcomes to the Matching Pennies variant in “The Final Problem.”

ε1= (H(2,2) - H(2,1)) / (H(1,1) + H(2,2) - (H(1,2) - H(2,1))) = (100 + 50) / (200 + 50)

ε1= Moriarty detrains at Dover = 3/5

ε2= (H(1,1) - H(1,2)) / (H(1,1) + H(2,2) - (H(1,2) - H(2,1))) = (100 + 0) / (200 + 50)

ε2= Moriarty detrains at Canterbury = 2/5

Michael Wainwright 89

In this situation, conclude von Neumann and Morgenstern, “Moriarty should go to

Dover with a probability of 60%, while Sherlock Holmes should stop at the intermedi-

ate station with a probability of 60%—the remaining 40% being left in each case for the

other alternative” (178). The probability of the detective’s murder (έ, ή) is equal to ε1η1

+ ε2η2= 0.48. “Sherlock Holmes,” von Neumann and Morgenstern profess, “is as good

as 48% dead when his train pulls out from Victoria Station” (178n1). The circumstances

suggest, as Morgenstern had realized in 1928, and as Theory of Games and Economic

Behavior confirms, a delicate imbalance in possible outcomes, a slight discrepancy in

one person’s favour that resonates with the contradictory tensions that define so many

commonplace coordination problems. No wonder “The Final Problem” of Doyle’s

detective left the public’s desire for stories about Sherlock Holmes unsated.

Whatever the extent of his luck in getting to Europe, Holmes must admit to

Watson, while he muses over the professor’s subsequent escape from the English

authorities, “of late I have been tempted to look into the problems furnished by nature

rather than those more superficial ones for which our artificial state of society is

responsible” (“Final” 263). With this remark, Doyle’s precocious strategist appears to

have been refined according to evolutionary tenets, with Holmes accepting the neces-

sary connections that pertain between the natural (or genetic) model of life and the

ability to simulate (via game theory) the rationality founded on that paradigm. Game

theory, as British sociobiologist Richard Dawkins has shown, can model the behaviour

of The Selfish Gene. This promotion of the instinctual basis to individual behaviour

demotes the importance afforded to psychological repression in the molding of per-

sonal conduct. Interest in the conscious mind attenuates (but does not completely

efface) psychoanalytical precepts. This mitigation may be interesting in light of the

teenage von Neumann’s exposure to psychoanalysis by Sándor Ferenczi, a relative who

Separated diagonals (100 is > than 0, -50) indicate that the preferable strategies are

unique and mixed. The probability matrices for Moriarty (έ= {ε1, ε2}) and Holmes

(ή= {η1, η2}) are therefore:

έfor Moriarty = {3/5, 2/5}; ήfor Holmes = {2/5, 3/5}

η1= (H(2,2) - H(1,2)) / (H(1,1) + H(2,2) - (H(1,2) - H(2,1))) = (100 + 0) / (200 + 50)

η1= Holmes detrains at Dover = 2/5

η2= (H(1,1) - H(2,1)) / (H(1,1) + H(2,2) - (H(1,2) - H(2,1))) = (100 + 50) / (200 + 50)

η2= Holmes detrains at Canterbury = 3/5

Mosaic 45/3 (September 2012)

90

introduced the discipline to Hungary, but the relationship between the theory of

games and Freudianism is only tangential. “Game theory,” as Poundstone emphasizes,

“is about perfectly logical players interested only in winning” (44, emph. Poundstone’s).

Individual human organisms are prime examples of this type of selfishness. “Freud has

told us often enough that he would have to go back to the function of consciousness,”

notes French psychoanalyst Jacques Lacan in The Four Fundamental Concepts of

Psycho-Analysis, “but he never did” (57). Game theory therefore covers an epistemo-

logical domain that remained beyond Freudian reanalysis. That Sherlock Holmes, “as

inhuman as a Babbage’s Calculating Machine,” according to Doyle in 1892, provides an

apposite literary subject for an investigation of that unexplored region should come as

little surprise.1That Doyle’s creation also identifies some nuances to logical thought

that mathematical strategists might commonly overlook, however, is less expected.

Doyle’s wish to evolve as an artist, his desire to consider the human condition more

subtly, has often been overshadowed by the critical interest invested in his evolv-

ing mysticism. Ironically, for an author who “did not intend to be written off as ‘the

Holmes man’” (vii), as Edwards recounts, Doyle’s detective would go some way to ful-

filling this artistic aim. Variations on the theme of rationality rather than metaphysics

chart this development. Holmes’s first reappearance after “The Final Problem” is in

the analeptic The Hound of the Baskervilles, which requires no mention of the events

at the Reichenbach Falls. The detective next arises, seemingly out of nowhere, in the

appropriately entitled “The Empty House.” In this opening tale from what would

become The Return of Sherlock Holmes, the detective explains to Watson that thanks

to his prowess at “baritsu, or the Japanese system of wrestling” (9), he had overpow-

ered Moriarty at the Meiringen waterfalls. In effect, Holmes had capitalized on his

luck in the zero-sum two-person dilemma that had played out in England with a tus-

sle in Switzerland that had relied on physical rather than mental agility. The profes-

sor, Holmes assures Watson, “knew that his own game was up” (9, emph. mine).

Furthermore, the precipice over which the doctor thought his friend had fallen was

not as precipitous as Watson believed: “A few small footholds presented themselves,”

chuckles Holmes, “and there was some indication of a ledge” (10). Holmes had

remained in hiding because Moriarty’s death revealed that the professor’s notorious

confederate, Colonel Sebastian Moran, was also at the Reichenbach Falls. Holmes

made this discovery when the Colonel almost succeeded in forcing him into the abyss

by dislodging “a huge rock” (11) from the cliff face above.

Holmes escapes Moran’s immediate attentions but realizes that he has fallen foul

of another dilemma of the kind he confronted in England against Moriarty. In this

Michael Wainwright 91

latest instance, the Colonel desires the match, craving a chance to eliminate Holmes.

A crack shot, Moran knows that Holmes’s undisguised return to London will mean

the detective’s death. Similarly, Holmes recognizes the Colonel as “the best heavy

game shot that our Eastern Empire has ever produced” (19) and fears that Moran

“would certainly get me” (10) should the detective reappear in the capital. “The Final

Problem” that closed The Memoirs of Sherlock Holmes had put pay to Moriarty as the

detective’s primary antagonist but had placed Moran in his stead. As von Neumann

and Morgenstern’s analysis of “The Final Problem” shows, Holmes was lucky in his

first Matching Pennies dilemma, and “The Empty House” indicates that he is not will-

ing to risk a similar deadly game.

Moran is the preclusion to the unfettered return of Sherlock Holmes to his

homeland. The threat posed by Moran forces Holmes into a twofold strategy of eva-

sion. He hides in foreign lands and under assumed identities. “I travelled for two years

in Tibet,” he tells Watson, “and amused myself by visiting Lhasa, and spending some

days with the head Lama. You may have read of the remarkable explorations of a

Norwegian named Sigerson, but I am sure that it never occurred to you that you were

receiving news of your friend. I then passed through Persia, looked in at Mecca, and

paid a short but interesting visit to the Khalifa at Khartoum” (12). Always an admirer

of Eastern culture—one thinks of his closing remark to Watson in “A Case of Identity”

from The Adventures of Sherlock Holmes: “There is as much sense in Hafiz as in

Horace, and as much knowledge of the world” (48)—Holmes’s foreign sojourn

underpins the evolution of his character.

Doyle had likened his creation to a calculating machine in 1892, but in reviving

his detective for The Return of Sherlock Holmes, the author presents a development of

Holmes’s ratiocinative characteristics. Holmes evolves from a pure reasoning machine

to a logician undaunted by the demand for alternative thinking. Lateral abstraction, a

chain of thought in which Holmes imagines himself standing by as a collateral wit-

ness to his own “murder,” solves the Moran dilemma. Holmes orders “a bust in wax”

of himself from “Monsieur Oscar Meunier, of Grenoble” (“Empty” 15), returns to

London, places the effigy so that the shadow of its prominent profile appears in sil-

houette on a window blind at 221B Baker Street, and waits for Moran to make his

move. Holmes has instructed his landlady to move the bust “once in every quarter of

an hour,” adjusting the head “from the front so that her shadow may never be seen”

(17). Holmes hopes that Moran, or a minion following the Colonel’s orders, will take

advantage of the empty house that faces his rooms at Baker Street. Sure enough,

Moran enters the vacated property while Holmes and Watson stand by in the same

darkened room as witnesses to an act of attempted murder. The Colonel proves his

Mosaic 45/3 (September 2012)

92

reputation as a marksman, repeating the recent prowess of his murderous shooting of

the Honourable Ronald Adair, but much to Moran’s consternation, Holmes then

effects his arrest. “I wonder that my very simple stratagem could deceive so old a

shikari,” Holmes admits to Moran. “It must be very familiar to you. Have you not

tethered a young kid under a tree, lain above it with your rifle, and waited for the bait

to bring up your tiger? This empty house is my tree, and you are my tiger” (20).

Holmes’s analogy, of course, is rather disingenuous. The lateral element of his strat-

egy had come to him after three years in contemplative hiding and certainly elevates

his solution beyond that of choosing from the probabilities emanating from a varia-

tion on the Matching Pennies scenario. That Holmes ensures Moran’s conviction is

for the murder of Adair instead of the affair at Baker Street, one case standing aside

for another, emphasizes the lateral aspect of the detective’s triumph.

Notwithstanding the commercial success of The Return of Sherlock Holmes—

“Doyle,” states Pirè, “received a lucrative offer from America of £4000 for each story”

(8)—Doyle’s British publishers were less enthusiastic about His Last Bow, the next

Holmes compendium. “The war had ushered in a world where figures like Holmes

seemed less relevant than they did in the Edwardian era,” explains Ousby. Holmes was

about to be “succeeded by a hero of a very different type, exemplified by John

Buchan’s Richard Hannay and Sapper’s Bull-dog Drummond.” These protagonists

would “rely more on physical courage and on fast reflexes than they do on refined

speculation” (174). Doctor Greenslade in Buchan’s The Three Hostages offers a con-

temporary summation of this debate. Detective fiction is “not ingenious enough, or

rather it doesn’t take account of the infernal complexity of life,” he tells Hannay. “It

might have been all right twenty years ago, when most people argued and behaved

fairly logically. But they don’t nowadays. Have you ever realised, Dick, the amount of

stark craziness that the War has left in the world?” (15). What is more, argues Ousby,

Doyle’s attempt to make Holmes relevant led to a decline in literary quality—“that del-

icate and playful sense of the bizarre which had distinguished Doyle’s best work gives

way to a cruder sense of the exotic” (170-71). Nevertheless, the deterioration in the

Holmes canon, as Ousby concedes, “is neither total nor entirely uniform” (170), and, as

his repetition of the antagonistic verve of “The Empty House” for the first short story

in his next Holmes collection testifies, Doyle’s inventiveness occasionally reappears.

The Case-Book of Sherlock Holmes, which would be the last set of tales dedicated

to the detective, also begins with a confrontation between Holmes and an archenemy.

Doyle adapted “The Mazarin Stone” from his stage play “The Crown Diamond,” in

which Colonel Moran was again the villain, but realized that the continuity of the

Holmes canon necessitated a new antagonist. Count Negretto Sylvius, as his name

Michael Wainwright 93

suggests, is a blackguard with a sinister (Negretto [dark]) background (Sylvius

[wood]). With the help of the boxer Sam Merton, Sylvius has stolen “the Crown dia-

mond” (6) known as the “Mazarin stone” (8); Merton provided the brawn, Sylvius the

brains. Holmes is certain that the two men are behind the crime but he faces a

dilemma: he cannot definitely prove their guilt. The British authorities, through the

auspices of the Prime Minister, the Home Secretary, and Lord Cantlemere, have com-

missioned Holmes to retrieve the jewel. Recovery of the stone is their utmost concern

and this priority affords Holmes some leverage. In a confrontation with Sylvius at

221B Baker Street, Holmes offers his opponent a deal. On the one hand, admit to the

theft, return the stone, and go free. On the other hand, deny the theft, keep the where-

abouts of the stone a secret, and get a long prison sentence. Sylvius is brazen enough

to risk the second option because he believes Holmes is bluffing. “Now, be reasonable,

Count. Consider the situation,” counsels Holmes. “You are going to be locked up for

twenty years. So is Sam Merton. What good are you going to get out of your diamond?

None in the world. But if you hand it over—well, I’ll compound a felony. We don’t

want you or Sam. We want the stone. Give that up, and so far as I am concerned you

can go free so long as you behave yourself in the future” (14). Holmes is mendacious

with this statement, as the dénouement will evince and as Sylvius suspects, but the

Count’s silence threatens his own bargaining power because mutual defection by

Merton will lessen the testimonial value of each man’s confession. With Watson

(“who has never failed to play the game” [10, emph. mine]) headed to Scotland Yard

for the police, Holmes now calls Merton in from the street where he has been keeping

watch according to Sylvius’s orders. “What’s the game now, Count?” asks Merton (15).

Sylvius shrugs his shoulders and does not talk to Merton in front of Holmes. The

imminent arrival of the police surely signals the suspects’ separation in custody; they

appear to have no opportunity to discuss their plight and figure out a possible solu-

tion. Holmes has forced them into a rational dilemma.

After the Theory of Games and Economic Behavior first caught their attention, strate-

gists at the RAND Corporation, specifically the American mathematician Merrill

M. Flood and his Polish-born American colleague Melvin Dresher, would consider this

type of quandary in detail. Canadian-born Albert W. Tucker, another RAND

Corporation employee, named this paradox the Prisoner’s Dilemma. Game theory

scholar Paul Watzlawick’s recent rendition of Tucker’s initial visualization from 1950 is

reputedly true to the original. “A district attorney,” Watzlawick writes, “is holding two

men suspected of armed robbery. There is not enough evidence to take the case to court,

so he has the two men brought to his office.” The attorney, continues Watzlawick,

Mosaic 45/3 (September 2012)

94

The letters X and Y designate the suspects and the silence that pertains between them

is the “coordination condition” that forces each man to enter a guilty or innocent plea

before learning of his counterpart’s response. The state penal system, the “banker” in

Prisoner’s Dilemma terminology, sets the tariffs for each of the four possible out-

comes. These results are set out in figure 4:

No matter what the other suspect does, each individual achieves a better outcome by

confessing. In this way, each suspect is certain to save himself eighteen years’ impris-

onment. The best and worst individual payoffs occur when one participant confesses

but his counterpart keeps silent. The talkative suspect goes free, but his taciturn coeval

receives the longest possible sentence. If both confess, however, that will be worse for

each suspect than if both keep silent. Simply put, the outcome will be worse for both

if each, rather than neither, does what will be better in self-interested terms.

In the mathematics of game theory, a true Prisoner’s Dilemma must meet a

number of conditions, which constitute an essential set of inequalities. These restric-

tions can be explained by using the number of years of imprisonment for X as a meas-

ure of costing so that the Prisoner’s Dilemma above takes the form shown in figure 5:

Y Confesses Y Keeps Silent

X Confesses Outcome 1:

Both get two years.

Outcome 2:

X goes free.

Y gets twenty years.

X Keeps Silent

Outcome 3:

X gets twenty years.

Y goes free.

Outcome 4:

Both get six months.

4. Possible outcomes from an interrogation in a standard Prisoner’s Dilemma.

tells them that in order to have them convicted he needs a confession: without one he can

charge them only with illegal possession of firearms, which carries a penalty of six months

in jail. If they both confess, he promises them the minimum sentence for armed robbery,

which is two years. If, however, only one confesses, he will be considered a state witness and

go free, while the other will get twenty years, the maximum sentence. Then, without giving

them a chance to arrive at a joint decision, he has them locked up in separate cells from

which they cannot communicate with each other. (98)

Y Confesses Y Keeps Silent

X Confesses Outcome 1: 2 Outcome 2: 0

X Keeps Silent Outcome 3: 20 Outcome 4: 1/2

5. Possible outcomes from a standard Prisoner’s Dilemma for X.

Michael Wainwright 95

Define mutual defection to mean both suspects’ complicity with the banker; a partic-

ipant’s unilateral restraint to mean that one suspect remains silent but his counterpart

does not; and mutual restraint to mean both suspects remain silent. Figure 5 shows

that mutual defection is preferable to a suspect than that defendant’s unilateral

restraint because Outcome 1 is better than Outcome 3. Similarly, unilateral restraint

by one’s counterpart is better for a suspect than mutual restraint: Outcome 2 is better

than Outcome 4. In addition, the punishment for mutual restraint is preferred to the

outcome of mutual defection; Outcome 4 is better than Outcome 1 because mutual

defection implies that both suspects suffer for little or no relative gain. Taken together,

these outcomes establish the essential set of inequalities for a Prisoner’s Dilemma,

which state that Outcome 2 must better Outcome 4, Outcome 4 must better Outcome

1, and Outcome 1 must better Outcome 3.

One of the recommendations of Prisoner’s Dilemma theory is its elegant con-

ciseness, but the circular logic that drives participant choice around this conundrum

guarantees no acceptable result to either suspect. What is worse, as Watzlawick asserts,

“situations of the Prisoner’s Dilemma type are more frequent than one might expect”

(100). “The main ingredient,” agrees Poundstone, “is a temptation to better one’s own

interests in a way that would be ruinous if everyone did it” (125-26). If two (or more)

individuals must reach a joint decision about which they cannot communicate, then

a Prisoner’s Dilemma is likely to arise.

APrisoner’s Dilemma seems to face Negretto Sylvius and Sam Merton in “The

Mazarin Stone” when the count refuses to discuss the whereabouts of the dia-

mond with his accomplice. If the police hold the two suspects on remand, then the

Prisoner’s Dilemma in figure 6 will pertain:

Merton confesses Merton keeps silent

Sylvius confesses

Outcome 1: Each man serves

a short sentence because

mutual defection lessens tes-

timonial value and the stone

is lost to both.

Outcome 2: Sylvius goes

free, Merton is jailed for 20

years, and the stone is lost

to both.

Sylvius keeps silent

Outcome 3: Merton goes free,

Sylvius is jailed for 20 years,

and the stone is lost to both.

Outcome 4:

Each man is jailed for 20

years, but the stone awaits

their release.

6. Possible outcomes from the prospective Prisoner’s Dilemma in “The Mazarin Stone.”

Mosaic 45/3 (September 2012)

96

That none of these options completely satisfies either of the suspects is to be expected

because a Prisoner’s Dilemma is by definition a conundrum without a perfect solu-

tion. Neither is a Prisoner’s Dilemma a zero-sum scenario: the payout for both par-

ties from mutual confession betters that from mutual silence. Von Neumann’s

minimax theorem cannot solve this kind of problem. Similarly, Holmes cannot apply

exactly the same logic as he did against Moriarty, but must alight on a new approach.

Not one of the above four outcomes entirely gratifies the detective because his gen-

uine wish is to regain the Mazarin gem and imprison both men. Holmes must there-

fore continue to display the lateral thinking that Doyle had invested in his protagonist

on Holmes’s return from the East.

That Sylvius values selling the diamond, which is apparently worth “a hundred

thousand quid” (17), over imprisonment and that Merton may prove doggedly faithful

to the Count are two further worries for Holmes (in fact, as the unfolding events

eventually make plain, Merton is as much in the dark over the whereabouts of the

Mazarin diamond as Holmes). His solution, which combines the lateral stratagem he

employed against Colonel Moran with a seemingly irrational move, is an inspiration.

Holmes’s use of a life-size dummy recalls the wax bust deployed in “The Empty

House,” while in a move that apparently contradicts game theory, the detective allows

the two suspects to discuss their troubling situation in private, as he removes himself

to his bedroom and plays the violin. Holmes then makes his lateral manoeuvre by

secretly exchanging places with the dummy that sits behind a screen in the lounge cur-

rently occupied by Sylvius and Merton. During their conspiratorial conversation, a

discussion that Holmes triggers by his seeming naïveté toward the standard Prisoner’s

Dilemma that held before he retired from the room, the Count reveals that the dia-

mond is on his person. He instructs Sam to slip out and give the stone to their courier-

cum-jeweller, who will “be out of England to-night” and who will have the Mazarin

gem “cut into four pieces in Amsterdam before Sunday” (17). While Sam effects this

business, explains Sylvius, he will deal with Holmes: “I’ll see this sucker and fill him up

with a bogus confession” (18). The Count and his accomplice think Holmes naïve to

have left them alone to talk, but as the utilitarian philosopher Derek Parfit argues of

Thomas Schelling’s comparable thought experiment in The Strategy of Conflict—

which counsels one to feign madness when armed robbers threaten one’s family—in

light of “any plausible theory about rationality, it would be rational for [Holmes], in

this case, to cause [himself] to become for a period irrational” (13). Even Buchan’s

Greenslade would have to admit to the ingeniousness or “stark craziness” (15) of

Holmes’s rational irrationality, a tactical move in keeping with the post-war era, and

one that proves its worth when the detective pounces on the diamond that the Count

Michael Wainwright 97

is handing to Merton. Just as in “The Empty House,” Holmes is alongside his prey

when the act that clinches his success occurs, but unlike this earlier tale, “The Mazarin

Stone” takes place during the daytime, not at night. The pathetic fallacy hereby cor-

roborates the sense of Holmes’s “enlightenment” beyond a logician’s rigidly demar-

cated thoughts. With the gramophone record of a violinist still playing in Holmes’s

bedroom, there is “an inrush of police,” the criminals are handcuffed, and they are “led

to the waiting cab” (19). Holmes, anything but a “dummy,” has secured the diamond

and both villains in a result that has traversed a course from the rational to the ration-

ally irrational via lateral thinking. Von Neumann’s mathematical description of ratio-

cination, as developed in the models of game theory, commendably avoids biases of

race and culture when characterizing human logic, while “The Empty House” and

“The Mazarin Stone” can be praised for the game-theoretic Holmes’s development

into a figure possessed of a mind that is more than “a Babbage’s Calculating Machine.”

NOTES

1/ Doyle’s letter of 16 June 1892 to Dr. Joseph Bell is quoted by Ely Liebow (Dr. Joe Bell: Model for Sherlock

Holmes. Madison: Popular P, 2007. 173. Print).

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MICHAEL WAINWRIGHT lectures in American literature at the University of Birmingham.

Darwin and Faulkner’s Novels: Evolution and Southern Fiction, his first monograph, was a Choice

Review of Books Outstanding Academic Title for 2009. His latest monograph is Toward a

Sociobiological Hermeneutic: Darwinian Essays on Literature.