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.