Christmas Games

FOOD for THOUGHT: CHRISTMAS GAMES

Christmas is almost upon us, with all its merry traditions: mince pies, mistletoe, Mariah, and – to crown it all - Monopoly with your in-laws. Oh yes, that wonderful hour of festive fighting, when everyone’s had a bit too much Christmas pudding and Uncle Andy has had to go for a ‘lie down’.

And have you ever noticed – of course, you have! And it annoys the dickens out of you every year! – that there is one particular member of the family who ‘always’ wins, due to their ‘superior strategy’ and generally smug demeanour?

In truth, it’s probably just that they care wa-a-ay too much about winning, when everyone else is just trying to have a bit of family fun. But if they really do win a lot, (whisper it quietly) it’s possible that you’re dealing with a secret mathematician!
Game Theory is that branch of mathematics that analyses strategic interactions between different parties. Yes, that’s right – there’s a whole branch of maths dedicated to game situations. How do mathematicians have the time to fritter away on investigating such stuff, I hear you cry? Well, remember they have no social life to speak of.

Actually, Game Theory is not just interesting and entertaining – it’s incredibly important in helping experts in different fields to understand the complex web of interactions that take place all around us, economics being just one example.
And even if you’ve never heard of Game Theory, believe me, we are all, on a basic level, making the sort of decisions that this branch of mathematics analyses on a daily basis, as we try to exert some sort of control on the professional and personal relationships that entangle us:

“But if I say that, he will only say this – it’s the way the argument always goes!”

“If I ask for that shift off, I’ll be first in line for the next bad task that comes along!”

“I don’t want to look greedy, but there are only 3 Brussels sprouts left, and if I offer them round, I might not get one myself.”

People who think like this too obviously and too often are usually accused of being political – but the awful truth is that most of us have thoughts like this much of the time – it’s part of being human.
And it’s also a vital aspect of the sort of strategic thinking necessary to many professions – and, what’s more, it is the source of a great deal of fun.
So, to keep us all amused at Christmas, here are three of my favourite brain teasers in the field of Game Theory. See what you think…

1) Splitting the Christmas Chocolates:
The Baker family Christmas chocolate tin contains a thousand little sweet treats of joy, and we have three children who could eat chocolate ad infinitum. But they have also been very well brought up, naturally, and so none of them wishes to appear greedy.
Therefore, when they split the chocolates up, no child will wish to have the highest number of chocolates – that would look terribly selfish! (For the purposes of the exercise, joint highest counts as highest.)
However, they are children – they certainly don’t want the least chocolates either – that would make them sad! (Again, joint lowest counts as lowest.)
Within the above conditions, they want the greatest number of chocolates they can get.

Baker children are very rational beings, as you would expect, so they will rationally act in accordance with those two motivations, in order of priority:

1) Take neither the highest nor the lowest number of chocolates;
2) Get as many chocolates as you can.

The children take turns in order of age to choose a number of chocolates from the tin, youngest first. They know the others have the same motivations as they do, but they cannot communicate with each other before or during the share-out.
How many chocolates does each child end up with?

2) Resolving the Family Feud:
The traditional argument over the turkey has turned nasty (Uncle Andy brought up Brexit), and consequently you find yourself in a three-way gun fight with two of your in-laws. You know you are not a great shot: in fact, if you aim at something, you only have a 1/3 chance of hitting it. Uncle Andy is a better shot than you: he hits his target half the time. Your mother-in-law, however, is a crack shot: you’ve seen her take out a cyclist on the pavement from a hundred yards with just her scowl. She’s even better with a gun – she literally never misses.
You will all take turns to take a shot, with the shooting continuing until only one person remains alive. With uncharacteristic generosity, your in-laws allow you to go first, with Andy next, and the mother-in-law last of all.
Who do you take a shot at?
(Note that you all make rational decisions and all know each other’s level of accuracy.)

3) Playing Cards:
One of the children won a mini-card game in their cracker, and traded it for the whistle that fell out of yours (a trade that you instantly regret!). Over coffee and a headache, you examine the game. It is remarkably simple.
There are 9 cards with the numbers 1-9 written on them. You and Auntie Audrey take turns to choose a card and add it to your hand. The first player to gain any set of three cards that total to 15 wins the game. If the cards run out, the game is a draw.
Auntie Audrey (who enjoys her sherry) boasts, “I’ve played this game before. If you let me go first, I can always win.”
Could she be right? If so, what’s her strategy? If not, how do you prove no such winning strategy exists?

Merry Christmas, when it comes!

DPB