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鈥攔andom 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鈥攖hat鈥檚 the trouble with novel ideas鈥攁nd
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鈥攂aked beans and spam鈥攁nd the bar was a shed piled to the roof
with crates of root beer. Thankfully only one member of the
society鈥攂esides the chairman, treasurer, and secretary鈥攖urned up, so
the usual four thousand would not witness the fiasco.
Trouble was, without them there wasn鈥檛 very much to talk about. By the second
day of the three-week conference, the participating member had delivered her
paper on 鈥淪uboptimal strategies for inequitable pay-off distributions in
infinitesimal gamelets鈥� to an attentive audience of three, and the committee had
finished all its pressing business. The four of them sat dolefully in the
station鈥檚 tiny but comfortable Edgar P. Frogblender Weedbird Dietary Analysis
Cubicle, eating one of the secretary鈥檚 inventions鈥攂aked-bean kebabs washed
down with a cocktail of root beer and tabasco sauce (found behind a filing
cabinet).
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鈥淚 don鈥檛 think I can face two and a half more weeks of this,鈥� said the
chairman.
鈥淣onsense!鈥� said the participating member, who always had a positive attitude
to everything, no matter how dire. 鈥淲hat we need to do is make our own
别苍迟别谤迟补颈苍尘别苍迟!鈥�
鈥淏ut we鈥檝e already seen the 鈥楬ow the Lesser Crested Weedbird Feeds
Regurgitated Squid to its Young鈥� video 16 times,鈥� protested the treasurer.
鈥淵ou haven鈥檛 seen what I鈥檝e found,鈥� 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.
鈥淚t鈥檚 a Monopoly set,鈥� the participating member pointed out helpfully.
鈥淚 can see that,鈥� snapped the chairman, nettled.
鈥淢ost of the upmarket property seems to have been eaten by rats,鈥� the
secretary observed.
鈥淎nd mice have been nesting in the Chance cards,鈥� the treasurer added.
鈥淵es, but we鈥檝e still got nearly all the property cards鈥攍ook, I鈥檝e
taped them back together鈥攁nd the borer weevils have only eaten a few of
the hotels.鈥�
The chairman gave the participating member a strange look. 鈥淲hat do you
propose we do with this . . . museum piece?鈥�
鈥淲hy don鈥檛 we play a game?鈥�
There was a long silence, broken by the secretary, who said, in a high,
squeaky voice, 鈥淏ut we鈥檙e game theorists.鈥�
The rest of the committee nodded.
鈥淪o?鈥� inquired the participating member.
鈥淪o 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.鈥�
The treasurer looked out of the window at the scudding clouds and the barren
rocks. 鈥淚 suppose we could give it a try,鈥� he said.
鈥淭here鈥檚 no money,鈥� the treasurer pointed out.
鈥淭hat鈥檚 all right,鈥� said the chairman. 鈥淲e can use our own. More point to the
game that way. Unless you guys are chicken.鈥�
鈥淚 never refuse a challenge,鈥� said the secretary. 鈥淐ount me in.鈥� The others
nodded their assent.
鈥淗ow do we play?鈥� asked the chairman.
鈥淚f it was a two-person game,鈥� the treasurer pointed out, 鈥渁nd if it could
only terminate in a win for one or other player鈥攏o draws, I
mean鈥攖hen there would always be a winning strategy for exactly one
辫濒补测别谤.鈥�
鈥淓xcellent,鈥� the secretary said encouragingly. 鈥淏ut this is a four-person
驳补尘别.鈥�
鈥淎丑.鈥�
鈥淚 suppose we could take turns playing in pairs. So what would this winning
strategy be?鈥�
鈥淣o idea. The theory doesn鈥檛 say. It just proves there is one.鈥�
鈥淲辞苍诲别谤蹿耻濒.鈥�
鈥淲hat I meant was鈥攕omebody tell me the rules,鈥� said the chairman. He
picked up two red and blue objects with dots on them. 鈥淔or instance, what the
devil are these things?鈥�
鈥淯m . . . dice, chairman.鈥�
鈥淎nd what, secretary, are dice?鈥�
鈥淯h, mechanical randomising devices, chairman.鈥�
The chairman huffed. 鈥淚 only work on deterministic games, as you well know.
Stochastic games are not my speciality. In fact, old man, they鈥檙e yours.鈥�
鈥淵es, I know that,鈥� the secretary persisted, 鈥渂ut you don鈥檛 have to study
this game. All you have to do is play it.鈥�
鈥淎h, but is it fair?鈥� the treasurer asked.
鈥淚s life fair? Is the Universe fair?鈥�
鈥淣o, I wasn鈥檛 speaking philosophically. That was a technical inquiry.鈥�
鈥淭he convenor of the technical inquiries division is currently eating roast
turkey in Abingdon, treasurer. Otherwise we could ask.鈥�
鈥淣o, what I mean is, is each square equally probable?鈥�
鈥淒on鈥檛 see why not,鈥� said the participating member.
鈥淣ot a proof,鈥� said the chairman flatly.
鈥淲hat worries, me, you see,鈥� said the treasurer, 鈥渋s that when you throw two
dice, some totals are more likely than others.鈥�
鈥淥h, right,鈥� said the participating member.
鈥淭he most likely total is 7.鈥�
鈥淪o the first player is most likely to land on the seventh square . . . 鈥�
said the treasurer.
鈥� . . . which is Chance,鈥� the secretary pointed out.
鈥淕reat,鈥� said the chairman. 鈥淪o the most likely first move is to draw a
Chance card. Making the damned thing even more stochastic. I think I鈥檒l go and
watch the regurgitation video again.鈥�
鈥淚鈥檝e got a proof that all squares are equally probable!鈥� shouted the
secretary. 鈥淚 can represent the board as a circle of 40 states鈥斺��
鈥淏ut it鈥檚 a square and they鈥檙e streets, not states鈥斺��
鈥淭opologically, 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鈥斺��
鈥淲hat the deuce are you trying to say?鈥�
鈥淭he chance of landing on any given square, in the long run, is 1/40.鈥�
鈥淣o, it isn鈥檛,鈥� objected the participating member. 鈥淵ou don鈥檛 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鈥檚 possible to
move up to 35 squares.鈥�
鈥淗辞飞?鈥�
鈥淏y throwing double 6, double 6, then a 6 and a 5.鈥�
鈥淵es, but the probabilities of moving more than 29 squares are incredibly
迟颈苍测.鈥�
鈥淭hat may be so,鈥� said the chairman, 鈥渂ut you shouldn鈥檛 neglect them just
because they hardly ever happen. Remember my game-theoretic analysis of the
鈥榯iger economies鈥�? It implied that the chance of them going into recession was
very small, too鈥攂ut look what happened.鈥�
鈥淲recked his chance of a Nobel,鈥� the secretary whispered to the
treasurer.
鈥淚t gets very complicated,鈥� said the participating member. 鈥淐hance and
Community Chest cards can send a player to Jail or to some other position on the
产辞补谤诲.鈥�
鈥淭rue,鈥� said the secretary. 鈥淏ut I can still use the Markov chain method. The
matrix of transition probabilities gets messy, that鈥檚 all . . . give me a few
minutes on a laptop.鈥�
鈥淟ook, why don鈥檛 we just play?鈥� 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.
鈥淥K,鈥� said the secretary. 鈥淭he treasurer can go first.鈥�
The treasurer threw a double six. 鈥淎h. Throw again. Gosh, another double six!
And another鈥斺��
鈥淵ou鈥檙e in Jail,鈥� said the secretary. My turn. Good, seven. Pick up a Chance
card . . . oh. Well, I don鈥檛 think we really need to鈥斺��
鈥淟et me see that,鈥� said the chairman. 鈥淎h. 鈥楪o to Jail. Go directly to Jail.
Do not pass GO. Do not collect鈥�'鈥�
鈥淎ll right, no need to go on about it. Well, treasurer, looks like we鈥檙e both
颈苍鈥斺赌�
Their eyes met.
鈥淧risoner鈥檚 Dilemma!鈥� they both yelled at once.
鈥淪orry?鈥� asked the participating member.
鈥淵ou don鈥檛 know?鈥�
鈥淣辞.鈥�
鈥淚t鈥檚 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?鈥�
鈥淜eep mum?鈥�
鈥淲ell . . . what if the other one rats? They鈥檙e bound to consider that
辫辞蝉蝉颈产颈濒颈迟测.鈥�
鈥淥h. Right. Both rat.鈥�
鈥淧recisely. A non-optimal strategy.鈥�
鈥淔ascinating. So how鈥斺��
鈥淧lay the blasted game!鈥� yelled the chairman. 鈥淭here isn鈥檛 any district
prosecutor in Monopoly!鈥�
鈥淚sn鈥檛 there? Then how do we get out?鈥�
鈥淏uy your way out or throw a double,鈥� said the participating member. 鈥淣ot
yet! Wait for your turn. The chairman goes next.鈥�
鈥淎h, yes. So I take these cubey things and throw鈥斺��
鈥淒ice. You roll them.鈥�
鈥淵es, roll. And then I move this boat thingy鈥斺��
鈥淣o, you鈥檙e the hat.鈥�
鈥淥f course, silly me. I鈥檒l just build six hotels on `Just
痴颈蝉颈迟颈苍驳鈥欌赌斺赌�
鈥淣o, you can鈥檛 do that!鈥�
鈥淗ow about collecting rent to put in the Community Chest, then?鈥�
The participating member sighed. 鈥淟et me explain the rules just once more . . . 鈥�
. . . The secretary stared morosely out of the window at the clouds, which
were still scudding by. The treasurer joined him. 鈥淗ow did the old so-and-so
do that?鈥�
鈥淲in all the money, you mean?鈥�
鈥淭hat鈥檚 exactly what I mean.鈥�
鈥淐lassic con. He鈥檚 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.鈥�
鈥淒o you think the participating member was in on the scam?鈥�
鈥淗ad to be. She suggested we play the 驳补尘别.鈥�
鈥渊别补丑.鈥�
They continued to stare out of the window.
鈥淗ey! Look, over there, beside that stunted bush. Isn鈥檛 it . . . 鈥�
鈥淎 mother lesser crested weedbird feeding its chick regurgitated squid? Yes,
it is.鈥�
鈥淔unny, it looks exactly like the video.鈥�
鈥淚t does. They must have arrived early.鈥�
鈥淥r late.鈥�
鈥淥ne of those.鈥�
They stared at each other.
鈥淚 wonder,鈥� said the treasurer, 鈥渨hat the optimal strategy is. Let me see. If
the weedbirds arrive early, then the pay-off would be . . . 鈥�