Practical Applications of Game Theory

Chapter 1

The Prisoner’s Dilemma

“Well, It’s not looking good,” the haggard, over-worked and under-shaved public defender conceded. He dropped a stack of papers onto the bare concrete floor of the holding cell, swore in Latin, then stooped to pick them up, managing to drop a few more.

“You might even say it’s looking bad,” he said as he gathered up his documents. “If you were a pessimist. And you consider several years in a state penitentiary to be a bad thing. But it’s just a matter of how bad, really. You’re definitely going up the river, but it’s up to you how far.”

The prisoner, Remy Martine, a bedraggled, wiry, whiskered rascal, covered with dirt and stinking of guilt, slowly dragged his gaze upwards. The single 40-watt bulb dangling on the other side of the bars behind the attorney struggled to illuminate the barren cell, leaving his face barely visible in the shadow.

“Tell me more,” Martine said, raising one eyebrow with exaggerated interest, falling just shy of obvious, mocking contempt. His accent was an exotic chimera, the aural pastiche of an itinerant vagabond who’d run away from his Louisiana home as a boy and taken up with assorted unsavory rogues in far-flung regions, speaking an eclectic smorgasbord of patois and argots. Yet it was elegant, crisp and sonorous, his enunciation slow and flawless, like a ballerina performing an exquisite legato dévéloppé, and ever so discreetly extending a raised middle finger for the briefest instant as she goes into a pirouette.

“The state has more than enough evidence to convict you and your accomplice on burglary and theft. We can’t beat that rap. They’ve got witnesses, fingerprints. Your partner was actually wearing some of the stolen merchandise when you two were arrested, so that’s pretty much a smoking gun. Oh, and a gun, too. The cops found a gun, but it can’t be directly tied to you guys, which makes it a little bit less like a smoking gun, speaking metaphorically rather than structurally. But the evidence that they have is enough to convict you both. Guaranteed.”

“I see,” Martine said, nodding slowly. “But I get the impression that you have another shoe to drop. Metaphorically.”

“Well, the state can’t prove everything that they suspect that you did. I mentioned that gun they can’t directly tie to you. There’s a small matter of an armed robbery in Groversdale.”

“I—”

The attorney waved his hands to shut Remy up.

“Hey hey! Come on. Be careful what you say! I’m your attorney. I’m not interested in whether you did it or not. Unless you didn’t do it.” He paused for a moment and Remy remained silent. “Like I said, I’m not interested in whether you did it or not. The point is, they can’t prove anything about that armed robbery. But they want to offer you a deal.”

“A deal? Is it from the bottom of the deck?”

“Clever puns. Yeah, I hear juries love those. Listen: if you confess to the robbery and testify against your partner-in-crime, Mr. Vicker” he pronounced the name of Remy’s associate with obvious disgust, as if the word itself had a foul taste, the speaking of which would necessitate extensive gargling. “You do that, and they’ll cut you a break. You’ll only do two years. Without any testimony from either of you, both of you are going up for four years on the other charges. Guaranteed.”

Remy pondered the offer. “And if I testify against Mack? What will he get?”

The lawyer shrugged. “He’ll get ten years. If he stays quiet and you testify he’ll get ten years.”

“But the state is making the same offer to him, aren’t they?”

“Of course they are. Wouldn’t be fair not to give him a tantalizing moral dilemma too. And if you don’t cooperate and he testifies, then you’ll be the one scratching hatch marks on the wall for a decade.”

“What if we both give in, take the state up on its oh-so-generous offer. What then?”

“They’ll knock a couple of years off for the confession. You’ll each get eight years.” As your attorney, I strongly advise you to take them up on the offer and testify.”

He left some paperwork with Remy then turned and called for the officer to come and let him out of the cell. He and the taciturn turnkey began to walk down the hall when the attorney swiveled back around with an afterthought.

“There really isn’t much to think about. You haven’t got a thing to lose by testifying. Doesn’t matter what Vicker does. If he sings and you stay mum, you’ll get ten years, instead of the eight you would have gotten if you’d cooperated. If he keeps quiet and you do too, you’re gonna get four years, but you’ll be out in two if you go stoolie in that case. Think about it.”

“I’ll give it the deepest consideration,” Martine said evenly.

“I guarantee you, Vicker’s already thinking about it. Thinking hard.”

Remy nodded as his lawyer turned again and disappeared down the hallway. After he was out of earshot the prisoner mumbled aloud.

“So another game has begun.”

Chapter 2

Ancient Greek Mathematicians

“Remy, your father is calling you.”

“Un moment.”

His mother opened the door without her usual perfunctory courtesy knock. Her twelve-year-old son was seated on his unmade bed, a hulking text opened in his lap. On the mattress beside him lay a notebook, covered with messily scrawled symbols and diagrams, intimidating and inscrutable.

“What is so important that you have to keep your father waiting, eh?” Her accent had diluted only slightly through the generations since Acadia, while her son would never develop more than a hint of it, flavoring his overall elocution.

“I am reading, mamere,” he announced without the briefest glance toward her. As if to demonstrate the truth of his assertion, his eyes continued to scan a complex proof.

“Ah have two eyes, Remy. Ah kin see that. Stop reading and go see your father in the shop. He wants to show you how to make a bench.”

“Carpentry doesn’t interest me.”

His mother folded her arms across her chest and glared down at the single ungrateful fruit that had somehow managed to spring from her loins. “Well fuh shore! You’ve made no secret of dat! Do you want to break your poor father’s heart? Is that it? Break your father’s heart? Quo’ faire?

“I don’t want to break anyone’s heart,” Remy said, finally looking up. “I just don’t want to learn carpentry. That’s all.”

“Oh, dat’s not good enough for you, eh? It was good enough for your father and your papere, but it’s not good enough for Remy Martine.”

“I have nothing against carpentry, Mother. It just doesn’t interest me.”

“And what does?”

“Mathematics,” he replied, diving back into his text on Diophantine equations.

“And how do you expect to earn a living? Who is going to pay you for mathematics?”

“Plenty of people.” He shrugged, not entirely certain he believed it, but fairly sure that he didn’t care.

“Is that so? Whare are dese people, eh? Ah go to the hospital sometimes. The doctor, he gets some money. I go to make the groceries, eh? The grocer gets some money. But ah never needed a mathematician. You goin’ to set up a little math shop on Maple Street, eh? Or were you plannin’ to go door to door?”

“That’s not—”

“ ‘Excuse me madam,’ ” his mother mocked. “ ‘I’m a wandering mathematician, and ah was wondering if you needed any angles bisected.’ ” She turned to her son. “That’s one of the things you do, oui? Bisect angles?”

“On occasion, Mama,” Remy said with a weary sigh.

“ ‘Or if you have any troublin’ equations lying aroun’, well, madam, I can solve for X, an’ I’ll throw in Y for half price.’ ”

“That’s not the way it works.”

“Well you should start thinking about how you are going to make some money. If you want ta go off to college and study mathematics, it’s going to cost a pretty penny.” She rubbed the tips of her thumb, index and middle fingers together. “You may as well make it doing carpentry. Dere’s math in that. You have to measure tings. And you have angles and straight lines. You would love it.”

“I don’t need college. I can learn everything from studying books.”

“That’s some fantasyland, you live in,” she said, shaking her head. “Everyone just sits around reading books all day. Do you tink you are back in ancient Greece, is that it? You read about dem all the time, but do you know what Remy? They’re all dead.”

“Yes, I’ve heard. It’s a shame.”

“Don’t sass me, boy. And do you know why they are dead, Remy?”

“No,” Remy said dourly. “Why?”

“Because dey were too concerned with mathematics and philosophy, and not enough about the real world. With carpentry to build better catapults and tings like that!”

“There weren’t any catapults back then, mamere, and when the Romans attacked Syracuse, it was Archimedes, the mathematical genius who devised defensive—”

His mother snapped closed the textbook he’d been reading.

“While dose Greeks were trying to find the square of the hypotenuse, the Romans were making spears and chariots. When a Roman legion is marchin down the street, all your trigonometry don’t amount to a hill of beans.”

Remy could not help thinking about triangles, but kept these notions to himself.

“And you know why those Greeks had time to waste with all their points and lines and circles? Because dey had slaves, Remy. Dat’s why. None of them had to sweat and toil in a field or a smith or a shop, so dey had all the time in the world to do their algebra!”

“The Greeks were geometers, Mother,” Remy corrected, risking maternal wrath. “They didn’t have algebra. That was the Arabs.”

“How many times do I have to tell you not to sass me? Answer me dat, my little mathematician!”

“Sorry.”

“Well isn’t that sweet?” his mother replied, cocking her head to the side, batting her eyes in absurd exaggeration. “Alors. The boy is sorry!”

Her face softened a moment later and she reached down, cupping her son’s chin. “Now, Remy you know ahm only lookin out for your best interest. What good is all dis going to do you?” she swept her hand in an arc to reference the equations and calculations scattered around him.

“I’m going to be a famous mathematician.” Remy stated confidently.

She shook her head, smiling sadly, “You’re going to wind up begging or stealing, my p’tit boug.”

Mother was always right.

Chapter 3

Cellular Automata

Clink!

Metal on metal, the latch shut with an echoing, piercing crash, redolent of finality, the sound of freedom vanished. The gate closed and locked behind Remy, the onomatopoetic origin of one of prison’s inveterate nicknames reverberating through the long concrete hallways as a bored corrections officer escorted him into cellblock A.

Neither Remy nor Mack had turned evidence against the other, both receiving four-year sentences at the conclusions of their separate, speedy trials. The authorities sent Mack to another prison on the other side of the state, dividing now in the hopes that they would conquer later, convinced that, separated by many miles, one of them would eventually betray the other. Preferably both.

Their court-appointed public defenders had rolled over and played dead even faster than usual, faced with the overwhelming evidence against their doomed clients. These plea-bargains were for the lesser charges of burglary, trespassing, and theft. In spite of tantalizing circumstantial evidence, nothing concrete linked them to the far more serious armed robbery, but it hung over them like the shank of Damocles. The state could haul either of them out of the big house and into the courthouse to stand trial, if any evidence presented itself. But if they both remained silent, each would spend four years behind bars, Remy here at Longacre State Penitentiary, a maximum-security facility, though he felt far from secure. Taunts, whistles, catcalls, random threats and unbidden blown kisses were hurled at him as he was led to his cell.

“Here’s your new luxury suite, Martine,” quipped the guard, a portly white guy with thick glasses and a thicker neck. His nametag identified him as “Jones,” but like everything and everyone at Longacre, his nametag couldn’t be trusted.

“Normally you’d go into the fish tier first,” Jones said, “but the warden wanted you to go directly into gen pop, and lucky you, we got a nice cell all ready.”

“Why is that?”

“Because he said so. That’s all the reason he needs. And if you want to know the reason behind any of our rules and regulations here, save your breath, because the answer is always going to be the same. You’re still a fish though, so you’re locked down and unclassified. You’ll get a work assignment soon. And common area and yard privileges. And don’t forget that those privileges can be taken away. But you’ll be spending most of your time right here until we get all the paperwork in order.”

Remy stepped to the bottom bunk and dropped down his set-up kit. This consisted of a copy of the prison rules and regulations, a couple of sheets, a tube of toothpaste, and a toothbrush, which, the manual failed to note, it was considered standard procedure to transform into a shank almost immediately. He looked around at his new home, estimating its dimensions. About 6 feet wide, 12 feet long, and 7 feet high. While there were two bunks in the cell—metal slabs sticking out from the wall, each with a thin Bob Barker mattress on it—both were empty. There were no other occupants, though pictures of naked ladies and baseball stars adorned the walls.

“I have this expansive mansion to myself, do I?”

“Of course you don’t. Your cellie’s in the hole. He’ll be back soon.” Jones pulled the cell door closed, the lock engaging with another clinking sound, then disappeared down the hall.

“Ah,” Remy said aloud to no one. “In the hole. Doubtless for strangling his previous cellmate in his sleep.”

He surveyed the 504 cubic feet of space that was to be his home, finding it, as he found everything, full of mathematics. Rough approximations of idealized geometric forms surrounded him. The bunks were rectangles. The bars, vertical parallel line segments, broken up occasionally by intersecting horizontals, forming right angles, favorites of the ancient Greeks. He had at least the next two years to unlock those geometric mysteries. At least two years, and maybe longer.

On the backs of the clothing he’d been issued was a number that would serve as his primary identity for the foreseeable future. He would have been amused by the irony, were he not facing years of a hellish existence. He and all his fellow prisoners had been symbolically reduced to numbers. Many would consider this a degrading and dehumanizing transformation, but Remy had a great deal of respect for numbers. He got along with them far better than he did with most people, and so embraced his new identity with alacrity rarely seen in the other prisoners.

The number on Remy’s back was 547298. This was his identity in the eyes of the state. Remy Martine was, for the time being, a non-person. Only 547298 remained.

Few of the other prisoners spent much time contemplating the number assigned to them. They quickly committed it to memory and gave it little further thought. In contrast, Remy immediately set about subjecting his new identity to rigorous numerical analysis. He held no mystical beliefs that this would provide any significant revelations, as the gematriasts, Pythagoreans and other number magicians had in the past. He was interested solely in amusing himself. He had several years of time to kill, and most of the weapons one might employ for such an assault were tightly controlled contraband. But they could never take numbers away.

The first thing he did with his new identity was reduce it to its prime factors. Unlike toothbrush shankcraft, this was not standard practice at the Longacre State Correctional Institution, but wasn’t forbidden by either the official prison handbook or the far more important unwritten convict rules. He discovered three primes. They were 2, 17, and 16,097. The first he established instantly, the second with a little more work. The third took some concentrated effort, but 16,097 wouldn’t divide evenly by any of the primes up to its square root. Indivisible, it, like Remy, was going nowhere.

He spent his first day behind bars establishing the prime factors of his new name and analyzing it for additional mathematical peculiarities. That left him 729 days left to go, at the very least, if he were to deal with the devil and turn evidence against a steadfastly loyal and tightlipped Mack. 1460 if they both stayed quiet. 2921 if they turned on one another. 3651 if Mack stabbed him in the back.

729 is the only one of these he found particularly interesting. It has but one prime factor, 3, multiplied by itself six times. The others quickly gave up a few prime factors and then stopped performing tricks, so Remy had to seek amusement elsewhere. He was pleased to find some chalk his missing cellmate had used to scratch a few lines of verse on the wall in the corner near the toilet.

Inside these prison walls I sit

Straining just to take a shit

If I should die before it’s out

Hacks will have something to bitch about

Remy did a lot of math in his first week at Longacre, alone in his cell, not yet officially admitted to the general population. The chalk and the wall came to rescue him from boredom as he drew a 5-by-5 tic-tac-toe grid and arbitrarily filled some of the boxes with x’s.

This was cellular automata, a favorite plaything of mathematicians. Each individual box is called a cell. X’s are distributed, by random device or conscious volition, throughout the grid, leaving some cells empty and some occupied. Specific rules are then applied and the resulting conditions are determined. The same rules are again applied to the new configuration, and so on.

In one of the more celebrated versions of cellular automata, an x in a cell, surrounded by empty cells, will die, withering in loneliness and despair. An x in a cell adjacent to three or more occupied cells will expire from overcrowding. An empty cell adjacent to exactly three “live” cells will come alive. Then the rules are applied over and over, the x’s dancing through the cells in complex patterns to the thrill of the mathematicians in the audience, watching the delicate ballet.

_ _ x _ x
x x _ x _
x x x _ _
_ _ _ x _
x x _ _ x

_ x x x x
_ _ _ _ _
_ _ _ x _
_ _ _ x _
x x _ _ x

_ x x x x
_ _ _ _ x
_ _ _ x _
_ _ x x _
x x _ _ x

_ x x _ x
_ _ _ _ _
_ _ x _ x
_ x _ _ x
x x x x x

_ x x _ _
_ x x _ _
_ _ x x x
x _ _ _ _
x _ _ _ x

This example of cellular automata was dubbed the “game of life” by the mathematician who invented it, John Conway. There are many others, but this struck Remy as poetically apropos. The prisoners, in their cells, were reduced, in the best-case scenario of their custodians, to automata, caught between the conflicting forces of loneliness and crowding.

Many years ago, in the early 1800’s, at Philadelphia’s Eastern State Penitentiary, authorities initiated a noble experiment, putting men in solitary confinement for the entire length of their sentences. Not for disciplinary violations, as a punishment, but on the theory that it would provoke a spiritual awakening, spending all that time alone with God. Prisoners had no interaction with anyone else, and nothing to read except a bible. So most of them followed the same basic plan to adapt to the unbearable isolation, which was to go stark raving mad.

This did not evoke a reevaluation of solitary confinement as a therapeutic penological strategy. The prison staff’s keen diagnostic instincts suggested that the madness was a direct result of masturbation. Men confined to a tiny cell for 24 hours a day with absolutely nothing to do, and they expected them not to masturbate.

The smell of stale semen, always conspicuously present in the air of Longacre, made it clear that this coping strategy hadn’t lost its appeal, and the frequent frenzied screaming and sobbing, especially in the middle of the night when it wasn’t buried in the general cacophony of cell block A, was a convincing testament that madness was still a fashionable alternative. Remy still had his math, though. A form of mental masturbation perhaps, but it kept him from going mad. For now.