Intransitive Preferences

Suppose these are your preferences:

#1: Large Apple > Orange

#2: Orange > Small Apple

#3: Small Apple > Large Apple

This seems strange but it tells you what to choose in any two-way choice situation. Why does RCT deem it irrational?

Because one can construct a “money pump” to take money away from you.

Incoherent Probabilities

Suppose you’re at a cockfight and you’re discussing who will win.

“Pip looks good,” you say. “Odds are 9 to 1 that he wins. Just look at how scrawny his opponent, Squeak, is. I’d give him only one chance in four of winning.”

Your probability assignments are

as follows:

Prob (Pip wins) = 9/10

Prob (Pip loses) = 1/10

Prob (Squeak wins) = 1/4

Prob (Squeak loses) = 3/4

But either Pip wins or Squeak wins so those probabilities should sum to 1.

Dutch Book Argument

Once again we note that your probabilities are pretty bizarre but exactly why does RCT deem them irrational?

Because if you are willing to act on them an opponent can force you to lose money every time, no matter which chick wins!

This is done by constructing a series of bets called a “Dutch Book.”

First let’s set up a series of fair bets associated with your odds.

If you think that the odds on Pip are 9:1, you should be willing to undertake either of the following:

a) You bet on Pip and I pay you10 cents if he wins; you pay me 90 cents if he loses. Or:

b) You bet against Pip and I pay you 90 cents if he loses; you pay me 10 cents if he wins.

c) If you think the odds are only 1:3 that Squeak will win, these are fair bets:

c) You bet on Squeak and I pay you 75 cents if he wins; you pay me 25 cents if he loses.

d) d) You bet against Squeak and I pay you 25 cents if he loses; you pay me 75 cents if he wins.

How You Lose No Matter What Happens

Here’s a Dutch Book against you:

I ask you to bet on Pip and Squeak under conditions (a) and (c).

If Pip wins, you win 10 cents by (a) but lose 25 cents by (c).

If Pip loses, you lose 90 cents by (a) although winning 75 cents by (c)

In either eventuality you lose 15 cents.

Are there other series of bets that constitute a Dutch Book?

How a “Money Pump” Works

Suppose you have an Orange. By #1, you would prefer to have a Large Apple. So I say “I’ll trade you a Large Apple for your Orange if you’ll toss in a penny.”

If #1 is true then you should accept the trade; you should be prepared to give me a penny (or some other token that is less than your differential evaluation of the pair).

You now have a Large Apple, but by #3 you should be willing to pay an iota to get a Small Apple instead.

But now by #2 you should pay to get the Orange back. And so I continue to go around the circle of your preferences and pump pennies out of you!