A When it comes to Christmas
presents, do you give as good as you get? Most people think they do. Even
President Barack Obama is on record saying he goes one better. 'Here's the
general rule: I give nicer stuff than I get,' he told Oprah Winfrey in a
pre-Christmas interview last year. That may seem ungrateful, but consider the
implications. Most people believe that the gifts they get are not as good as the
ones they give. No wonder Christmas is so often a crushing
disappointment. B There is a better way: abandon the
ritual of mutual gift-giving in favour of a much more rational system called
secret Santa. The beauty of this is that you only have to buy one present for
each social circle you belong to, rather than one for everyone you
know. C In the original version of secret Santa each
member of a group—colleagues, say—is anonymously assigned to buy a gift for
another and give it to them at the Christmas party. Thankfully, that game has
evolved into something more Machiavellian: thieving Santa, also known as dirty
Santa or the Grinch game. As its name suggests, this revolves around theft and
dirty tricks. In its simplest version, everybody buys a present costing between,
say, ￡10 and ￡20. They then secretly deposit it, gift-wrapped, into a sack. To
start the game, numbers are drawn out of a hat to decide the order of
play. D Now the horse-trading begins. The first player
must take a present from the sack and open it. The second player then has a
choice—open a new present, or steal the already opened one. If they choose to
steal, player 1 gets to open another present, but they are not allowed to steal
their present straight back. Player 3 now enters the fray, either opening a new
present or stealing an opened one, whereupon the victim gets to play again,
either stealing a different present or opening another new one. And so it goes
on until everybody has had a turn and there are no more unopened
presents. E The system is not without its flaws, however.
For example, if the first player opens a poor present they are likely to be
stuck with it, while the player picked to go last has a good chance of getting a
really good present, perhaps the best. For that reason there are many variants
designed to spread the pain. One is to allow dispossessed players to steal a
present back, although this tends to lead to endless rounds of tit-for-tat
larceny. Another is to set a limit on how many times an individual present can
be stolen. F If you have never played thieving Santa,
give it a go. It's fun. Fun, though, can be overrated. What you really want is
to win—and that means ending up with the best possible present. So how should
you 8o about getting it? Imagine you are playing a game in which a present can
only be stolen once and it is your turn. There are three opened presents on the
table and four in the sack. One of the opened ones is not bad, and if you steal
it you can keep it. But there may be even better ones in the sack, so why not
gamble? Then again, if you open a really good present somebody is certain to
steal it from you, and you risk ending up with something really terrible. What
to do? G Here is where a strategy developed by game
theorists Arpita Ghosh and Mohammad Mahdian of Yahoo Research in Santa Clara,
California, can help. 'I heard about this game at a New Year's party, from
somebody who had just been playing it at Christmas,' says Ghosh. 'I thought it
would be fun to analyse.' H Ghosh and Mahdian decided to
play a simplified version of the game. Assuming certain things—that the players
are sober, for example, and that everybody puts the same value on the same
presents. They wanted to work out how to 'maximise the expected utility'. Or, in
English, to work out what you theoretically expect to get out of a transaction
before it has happened. They started by thinking about the game's final round,
where all but one of the players has had a turn and there is just one unopened
present left in the sack. In this case the strategy is pretty obvious. If all
the presents are worth somewhere between ￡10 and ￡20, the expected value of the
final unopened present is ￡15. The rational strategy, therefore, is to look
around the table and steal any present worth more than that. But remember, if
the top present has been unwrapped it is likely to have been stolen already so
you won't be able to have it. So if there isn't a present worth more than ￡15
that hasn't already been stolen, open the one in the sack. This means that the
final player has an expected utility of at least ￡15, which is about as good as
it gets. Expected utility, of course, is not the same as what you actually get.
You might think that the final present is trash, in which case the strategy
wasn't much help. But at least you can console yourself that it was correct in
theory. I Ghosh and Mahdian then started to work
backwards through each player's turn—a process known as backward induction—to
derive a general strategy. Their next stop was the last-but-one player, where
there are two unopened presents. This is a bit more complicated than before, as
you have to take into account the possibility of opening a really good present
which is immediately stolen. This possibility means that the last-but-one player
must have a lower steal threshold than the final player. Ghosh and Mahdian's
calculations show that a player in this position should steal any present worth
￡13.75 or more. If there is no such present available, they should open a new
one. This 'threshold strategy' turns out to work for all players, except the
first, who has no choice but to open an unopened gift. The steal threshold
itself rises as each player takes their turn because the fewer people left to
pick a present, the fewer opportunities there are for someone to steal yours.
What this means is the steal threshold starts low; in an 8-player game it is
approximately 11.56 for the second player. But, surprisingly, it delivers an
expected utility of slightly more than ￡15 for all players—except poor old
player 1, of course, who is more or less guaranteed to 'take one for the team'
and get something pretty lousy. J 'When your turn
arrives, have a look at the gifts that you can possibly steal,' says Ghosh. 'If
the best of those is good enough, where "good enough" depends on how many
unopened gifts remain, steal it. If the best of those is not good enough, open a
new gift. What this is really all about is making sure you get the best of the
rubbish.' This should perhaps be known as 'minimising your futility', Ghosh
said. So what of the first player, who seems doomed to pick the short straw?
This is an acknowledged problem in real-world thieving Santa, and is usually
solved by giving the first player a chance to steal right at the end. In this
case all the players have an expected utility of exactly ￡15. With all this
strategy at her fingertips, you would expect Ghosh to be arranging secret Santa
games every year. 'Actually no,' she says. 'I've never played it.' Typical
theorist.
—New Scientist
填空题
Do the following statements agree with the claims of the writer in Reading Passage 2?
In boxes on your answer sheet, write
YES if the statement agrees with the claims of the writer
NO if the statement contradicts the claims of the writer
NOT GIVEN if it is impossible to say what the writer thinks about this
People need to buy gifts for each other in Secret Santa Game.
填空题
Secret Santa is more reasonable than thieving Santa.
填空题
The disadvantage of thieving Santa is that the first player always gets the worst gift.
填空题
It is possible that expected utility is effective theoretically.
填空题
Ghosh and Mahdian can get final strategy to thieving Santa through backward induction.
填空题
The threshold in the 'threshold strategy' becomes higher along with the process of the game.
填空题
Complete each sentence with the correct ending, A-M, below.
Write the correct letter, A-M, in boxes on your answer sheet.
A only play the simplified version of the game.
B will shape players into thefts.
C uses 'backward induction' to analyse the game.
D is more interesting than the traditional game.
E has the same opportunity of choosing as others.
F can calculate the value of gifts each player gets accurately.
G has one more chance to play, which may avoid the inequality of the game.
H can succeed finally.
I becomes the sacrifice of the game which can not be changed any more.
J played thieving Santa at a party.
K make people design variants.
L is that the final player can get the worst gift.
M can use 'threshold strategy' to estimate the value of the gifts left in order to make a smarter choice.
Thieving Santa
填空题
The drawback of thieving Santa
填空题
Ghosh and Mahdian
填空题
The final player
填空题
The second player
填空题
The first player
单选题
Choose the correct letter, A, B, C or D. Write the
correct letter in box on your answer sheet. Which of the
following is the main idea of Reading Passage 2?
A. the relationship between secret Santa and thieving Santa
B. the suggestions for the first player in the thieving Santa
C. what the secret of Santa games is and the improvement suggested