żìĂš¶ÌÊÓÆ”

GAME ON

It's that time of year again, and the chances of playing Monopoly are extremely high
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IT WAS a novel idea to hold the International Congress of Game Theorists at
Christmas. The computer had indicated that the optimal strategy was a mixed
one—random choice between a Christmas meeting in an exotic locale (97 per
cent) and the usual mid-February bash in Boondock, Kansas (3 per cent). The
treasurer of the World Game Theory Society had obediently prepared 100 numbered
cards, drawn Kansas, rejected the result as a statistical outlier, drawn again,
and settled on a five-star hotel in the Canary Islands.

Unfortunately, when he tried to make the booking he discovered he was a
little late in the season—that’s the trouble with novel ideas—and
the best alternative accommodation that he could find on fartoolate.com was at a
scientific station in Tierra del Fuego normally used as a base for the study of
the dietary habits of the lesser crested weedbird. Also, in the rush to set up
the conference, nobody had explained that the station was normally only open in
August, when the weedbirds flocked in their dozens en route from Kerguelen to
Lanai, so the restaurant facilities consisted of a stock of canned
food—baked beans and spam—and the bar was a shed piled to the roof
with crates of root beer. Thankfully only one member of the
society—besides the chairman, treasurer, and secretary—turned up, so
the usual four thousand would not witness the fiasco.

Trouble was, without them there wasn’t very much to talk about. By the second
day of the three-week conference, the participating member had delivered her
paper on “Suboptimal strategies for inequitable pay-off distributions in
infinitesimal gamelets” to an attentive audience of three, and the committee had
finished all its pressing business. The four of them sat dolefully in the
station’s tiny but comfortable Edgar P. Frogblender Weedbird Dietary Analysis
Cubicle, eating one of the secretary’s inventions—baked-bean kebabs washed
down with a cocktail of root beer and tabasco sauce (found behind a filing
cabinet).

“I don’t think I can face two and a half more weeks of this,” said the
chairman.

“Nonsense!” said the participating member, who always had a positive attitude
to everything, no matter how dire. “What we need to do is make our own
±đČÔłÙ±đ°ùłÙČčŸ±ČÔłŸ±đČÔłÙ!”

“But we’ve already seen the ‘How the Lesser Crested Weedbird Feeds
Regurgitated Squid to its Young’ video 16 times,” protested the treasurer.

“You haven’t seen what I’ve found,” said the participating member brightly.
She put a tattered box on the table, and smiled at the others in triumph.

The chairman squinted at it, picked it up, and tipped the contents on to the
table.

“It’s a Monopoly set,” the participating member pointed out helpfully.

“I can see that,” snapped the chairman, nettled.

“Most of the upmarket property seems to have been eaten by rats,” the
secretary observed.

“And mice have been nesting in the Chance cards,” the treasurer added.

“Yes, but we’ve still got nearly all the property cards—look, I’ve
taped them back together—and the borer weevils have only eaten a few of
the hotels.”

The chairman gave the participating member a strange look. “What do you
propose we do with this . . . museum piece?”

“Why don’t we play a game?”

There was a long silence, broken by the secretary, who said, in a high,
squeaky voice, “But we’re game theorists.”

The rest of the committee nodded.

“So?” inquired the participating member.

“So we theorise about games, Madam. We analyse them, we determine optimal
strategies, we advise governments on how to run their economies, and biologists
on why dinosaurs evolved. What we do not do is play games.”

The treasurer looked out of the window at the scudding clouds and the barren
rocks. “I suppose we could give it a try,” he said.

“There’s no money,” the treasurer pointed out.

“That’s all right,” said the chairman. “We can use our own. More point to the
game that way. Unless you guys are chicken.”

“I never refuse a challenge,” said the secretary. “Count me in.” The others
nodded their assent.

“How do we play?” asked the chairman.

“If it was a two-person game,” the treasurer pointed out, “and if it could
only terminate in a win for one or other player—no draws, I
mean—then there would always be a winning strategy for exactly one
±è±ôČčČâ±đ°ù.”

“Excellent,” the secretary said encouragingly. “But this is a four-person
Č”ČčłŸ±đ.”

“Ałó.”

“I suppose we could take turns playing in pairs. So what would this winning
strategy be?”

“No idea. The theory doesn’t say. It just proves there is one.”

“WŽÇČÔ»ć±đ°ùŽÚłÜ±ô.”

“What I meant was—somebody tell me the rules,” said the chairman. He
picked up two red and blue objects with dots on them. “For instance, what the
devil are these things?”

“Um . . . dice, chairman.”

“And what, secretary, are dice?”

“Uh, mechanical randomising devices, chairman.”

The chairman huffed. “I only work on deterministic games, as you well know.
Stochastic games are not my speciality. In fact, old man, they’re yours.”

“Yes, I know that,” the secretary persisted, “but you don’t have to study
this game. All you have to do is play it.”

“Ah, but is it fair?” the treasurer asked.

“Is life fair? Is the Universe fair?”

“No, I wasn’t speaking philosophically. That was a technical inquiry.”

“The convenor of the technical inquiries division is currently eating roast
turkey in Abingdon, treasurer. Otherwise we could ask.”

“No, what I mean is, is each square equally probable?”

“Don’t see why not,” said the participating member.

“Not a proof,” said the chairman flatly.

“What worries, me, you see,” said the treasurer, “is that when you throw two
dice, some totals are more likely than others.”

“Oh, right,” said the participating member.

“The most likely total is 7.”

“So the first player is most likely to land on the seventh square . . . ”
said the treasurer.

” . . . which is Chance,” the secretary pointed out.

“Great,” said the chairman. “So the most likely first move is to draw a
Chance card. Making the damned thing even more stochastic. I think I’ll go and
watch the regurgitation video again.”

“I’ve got a proof that all squares are equally probable!” shouted the
secretary. “I can represent the board as a circle of 40 states—”

“But it’s a square and they’re streets, not states—”

“Topologically, the board is a circle. I can model the game as a Markov
chain, which necessarily has a unique stationary distribution. However, the
transition matrix is symmetric under rotations of the board, so the stationary
distribution has to assign a probability of 1/40 to each—”

“What the deuce are you trying to say?”

“The chance of landing on any given square, in the long run, is 1/40.”

“No, it isn’t,” objected the participating member. “You don’t just move
according to one throw of the dice. For instance, if you throw a double then you
have to move again. The most likely distance is still 7, but it’s possible to
move up to 35 squares.”

“HŽÇ·É?”

“By throwing double 6, double 6, then a 6 and a 5.”

“Yes, but the probabilities of moving more than 29 squares are incredibly
łÙŸ±ČÔČâ.”

“That may be so,” said the chairman, “but you shouldn’t neglect them just
because they hardly ever happen. Remember my game-theoretic analysis of the
‘tiger economies’? It implied that the chance of them going into recession was
very small, too—but look what happened.”

“Wrecked his chance of a Nobel,” the secretary whispered to the
treasurer.

“It gets very complicated,” said the participating member. “Chance and
Community Chest cards can send a player to Jail or to some other position on the
ČúŽÇČč°ù»ć.”

“True,” said the secretary. “But I can still use the Markov chain method. The
matrix of transition probabilities gets messy, that’s all . . . give me a few
minutes on a laptop.”

“Look, why don’t we just play?” proposed the chairman. And so the
participating member spent the next two hours explaining the rules to him.
Eventually, all four were back at the board.

“OK,” said the secretary. “The treasurer can go first.”

The treasurer threw a double six. “Ah. Throw again. Gosh, another double six!
And another—”

“You’re in Jail,” said the secretary. My turn. Good, seven. Pick up a Chance
card . . . oh. Well, I don’t think we really need to—”

“Let me see that,” said the chairman. “Ah. ‘Go to Jail. Go directly to Jail.
Do not pass GO. Do not collect—'”

“All right, no need to go on about it. Well, treasurer, looks like we’re both
Ÿ±ČÔ—â¶Ä

Their eyes met.

“Prisoner’s Dilemma!” they both yelled at once.

“Sorry?” asked the participating member.

“You don’t know?”

“NŽÇ.”

“It’s a classic of game theory. Two prisoners in jail awaiting trial. Each
goes separately to the district prosecutor, and is offered freedom if he rats on
the other, who will get sent down for a stretch of twenty years. On the other
hand, if they both keep mum, both get sentences of only five years. And if both
rat, they both get twenty years. So what do they do?”

“Keep mum?”

“Well . . . what if the other one rats? They’re bound to consider that
±èŽÇČőČőŸ±ČúŸ±±ôŸ±łÙČâ.”

“Oh. Right. Both rat.”

“Precisely. A non-optimal strategy.”

“Fascinating. So how—”

“Play the blasted game!” yelled the chairman. “There isn’t any district
prosecutor in Monopoly!”

“Isn’t there? Then how do we get out?”

“Buy your way out or throw a double,” said the participating member. “Not
yet! Wait for your turn. The chairman goes next.”

“Ah, yes. So I take these cubey things and throw—”

“Dice. You roll them.”

“Yes, roll. And then I move this boat thingy—”

“No, you’re the hat.”

“Of course, silly me. I’ll just build six hotels on `Just
łŐŸ±ČőŸ±łÙŸ±ČÔȔ’â¶Ä”â¶Ä

“No, you can’t do that!”

“How about collecting rent to put in the Community Chest, then?”

The participating member sighed. “Let me explain the rules just once more . . . “

. . . The secretary stared morosely out of the window at the clouds, which
were still scudding by. The treasurer joined him. “How did the old so-and-so
do that?”

“Win all the money, you mean?”

“That’s exactly what I mean.”

“Classic con. He’s played before. He fooled around to make us think he had no
idea what he was doing. Then he slapped hotels on all the key properties and
cleaned us out.”

“Do you think the participating member was in on the scam?”

“Had to be. She suggested we play the Č”ČčłŸ±đ.”

â€Ôš±đČčłó.”

They continued to stare out of the window.

“Hey! Look, over there, beside that stunted bush. Isn’t it . . . “

“A mother lesser crested weedbird feeding its chick regurgitated squid? Yes,
it is.”

“Funny, it looks exactly like the video.”

“It does. They must have arrived early.”

“Or late.”

“One of those.”

They stared at each other.

“I wonder,” said the treasurer, “what the optimal strategy is. Let me see. If
the weedbirds arrive early, then the pay-off would be . . . “

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