Life's a game

If you haven't watched ITV's Goldenballs before, don't worry. All we need to worry about is the end bit, which is shown in the video above. (This saves about an hour of watching Jasper Carrot).

In the final part of the game, the contestants have amassed a prize pool, and can either choose to "split" or "steal". If both contestants choose split, they split the prize money equally. If they both "steal" they get nothing. If one steals, and the other splits, the stealer takes all.

What's the best tactic?

The answer is very similar to the "Prisoner's dilemma", which goes like this:

John and Ron are big-time criminals, and they've both been arrested and kept in separate cells. The police don't have enough evidence to convict them for the big bank job they just pulled off, but they've got enough other crimes to put them in prison for a year each. The police have a cunning plan. They go to John and offer him a deal: if he grasses on Ron, they'll ignore the lesser charges. John gets to walk free, but they can charge Ron for the bank job and put him in prison for ten years. They offer the same deal to Ron. There's one catch though: if they both confess then they'll both get five years in prison.
We can summarise the offers, in terms of the number of years in prison.

The pay-offs in the table show John's pay-off first, then Ron's. So if John confesses and Ron stays quiet (top right), then John gets freedom and Ron gets ten years in prison.
The dominant strategy, also a Nash equilibrium, is for both players to choose confess. This is easy to explain.

Suppose you're John. If Ron stays quiet, then you can either "confess" and get freedom or "stay quiet" and get a one year sentence. Freedom is better, so the best choice here is to confess.
If Ron confesses, then you can either "confess" and get a five year sentence, or "stay quiet" and get a ten year sentence. Five years in prison is better than ten, so the best choice is confess.
The same logic can be applied to Ron (since the case is absolutely symmetrical).
That is the basic underlying principle to game theory.


Returning to Goldenballs, there are four scenarios in the game, and they are laid out below.

Suppose the final prize money is £100, and that you're player one. Like John in the prisoner's dilemma you look at the options facing you. If player two splits, the best option is to steal and take all the money. If player two steals, you have no choice - you will always leave empty handed. So you may as well choose steal.

Theory tells us that steal is the only option to go for, so why did the guy choose to split? We'll leave that for another time.

The economics of humble pie

You can't have football without pie.

At the start of the season, I made a bet with my brother that Manchester City would finish above their local rivals, Manchester United. £10, winner takes all. Not a huge amount of money, and it never was about the money. I was the older, wiser brother - this was about pride.

28 games down the line, and it's United who sit on the top of the league. There's only 10 more games to go, and City's fixtures are looking harder. And there's a crunch derby game in late April which could effectively decide our bet. United are suddenly favourites.

Here's the conundrum: the bookies are offering odds of 8/13 on Manchester United winning the league.
The question: to bet or not to bet?

If I don't bet, I have the option of either getting £10 or losing £10. And that's losing £10 to my brother.
I could, instead, hedge my bets by betting on United. This would offset the potential loss if United win, but would mean less money if City manage to beat their rivals.

For example, if I bet £5 on United I have the following two scenarios:

Scenario 1 - United win: I pay the £10 owed, but get a profit from the bookies of £3, so I lose £7
Scenario 2 - City win: I get £10 from my brother, but lose the bet with the bookies. Overall, up £5.

Some other bets and returns in each scenario are summed up in the following table:

What to do hinges on what I think the two associated probabilities are. Expected returns are given as

E(£) = p1S1 + p2S2

where p is the probability associated with each scenario, S1 the loss in scenario 1 and S2 the winnings in scenario 2. S1 will be negative if S2 is positive.

We can plug in some probabilities and see what the expected return is.

If the probabilities are 50/50 (so it's equally likely for city or united to win) then the expected return of the original bet is £0, since

E(£) = (0.5 x -£10) + (0.5 x £10) = 0

The table below has some other probabilities and their associated expected returns.

So if I think the probability of United winning is 80%, a bet of £10 reduces the expected loss from £6 to £3.08.

What have I gone for, and what do I think the probability is?

I think the probability is around 80/20, so I've bet £7.65. This might look odd, since I could reduce my expected loss to £3.08 by betting more. But I've put another condition in: I don't want to lose money if City win. £7.65 allows me to minimise my expected losses whilst still being able to celebrate a City win.

Remember, you can't have football without pie. It's my turn to eat humble pie, accept that maybe my brother could be right, and hedge my bets.