1
A credence is a price
If you would pay 1/2 for a ticket that pays 1 when the coin lands heads, and would equally sell one for 1/2, then your credence in heads is 1/2.
Lesson · Knowledge and Evidence · Credence
Credence and the Dutch Book
A credence is a degree of belief, measured by the price at which a person would buy or sell a bet that pays 1 if the event happens. Credences that break the rules of probability allow a set of bets that loses whatever happens. The lab finds the book and prints the loss.
credence: the price of a bet paying 1
every credence lies between 0 and 1
exclusive events: their credences add
break a rule: a set of bets loses for sure
Prices, not feelings
Core ideas
What this lesson establishes
The rest of the course measures belief, and the first thing to settle is what a number for a belief is and when a set of them is allowed.
2
Coherence is the absence of a sure loss
A set of credences is coherent when no combination of bets at those prices loses in every outcome. The rules of probability are the conditions for that.
3
Coherent is not the same as reasonable
A coherent set can still be absurd, such as 99/100 for heads on a fair coin and 1/100 for tails. The test catches contradictions among credences, not mistakes about the world.
The material
A Dutch book, built by hand
Two credences for a coin, a pair of bets, and a table that shows the loss.
Take a coin that lands heads or tails and nothing else. Someone gives credence 3/5 to heads and 3/5 to tails. At their own prices they should be willing to buy a bet on heads that pays 1 if the coin lands heads, for 3/5, and a bet on tails on the same terms. Together the two bets cost 6/5.
outcome bet on heads bet on tails net
-------------------------------------------
heads +2/5 -3/5 -1/5
tails -3/5 +2/5 -1/5
Each bet pays 1 when it wins, so a winning bet leaves 1 − 3/5 = 2/5 ahead and a losing bet leaves 3/5 behind. Exactly one bet wins in either outcome, so the net is 2/5 − 3/5 = −1/5 whichever way the coin falls. A set of bets with a sure loss is called a Dutch book. The lab reports this as a loss of 1/5 per unit staked.
Definition · Coherent credences
A set of credences is coherent when it obeys three rules: each credence lies between 0 and 1; a certain event gets 1; and for two events that cannot both happen, the credence in “A or B” is the sum of the credences in A and in B.
The coin example breaks the third rule in its simplest form: heads and tails exclude each other and exhaust the outcomes, so their credences should sum to 1. If the sum is below 1, as with 1/3 and 1/3, the book runs the other way: the person sells both bets for 2/3 in total and pays out 1 whichever wins, losing 1/3. The lab computes the size of the loss as the distance of the sum from 1.
The die preset breaks additivity for three events. “Even” has credence 1/2, “one” has 1/6, and “even or one” also has credence 1/2. The first two are exclusive, so the third should be 2/3, and the lab finds the book that the gap of 1/6 allows.
What credence 1/2 means
Credence 1/2 in heads is a statement of price. It is the credence of a person who knows the coin is fair, and the same price can come from a person who knows nothing about it. It does not mean “no idea”, and no one could say no idea about each face of a die by writing 1/2 for each, since six credences of 1/2 sum to 3 and the rules require the six to sum to 1.
The Dutch book result runs one way only. If your credences break a rule you can be exploited. If they obey the rules, the lab finds no book, whether the credences are sensible or not. Coherence is a necessary condition of rational credence, and the lessons that follow add what coherence leaves out.
Interactive
A threshold for belief, and a book
what a threshold accepts, whether it can all be true, and what incoherence costs
Method
Checking credences for a Dutch book
What the lab does, in the order you can do it by hand.
1
Write each credence as a fraction
Use exact fractions. A credence of 0.6 is 3/5, and exact arithmetic keeps the loss exact too.
2
Check the range
A credence below 0 or above 1 is already a book: someone could sell you a bet they must lose.
3
Pair every event with its negation
Add the two credences. The distance of the sum from 1 is the loss per unit that a book on the pair can extract.
4
Check exclusive events against their union
If A and B cannot both occur, the credence in “A or B” should equal the credence in A plus the credence in B. The gap is again the loss.
5
State the bets
A Dutch book is a recipe: if the sum is too high, sell the bets; if too low, buy them. Write the net payoff in each outcome and check that it is the same negative number everywhere.
Prices and books
Score 0 / 0Choose the answer you can justify.
Heads and tails at 3/5 each
The person buys both bets at their own prices.
price paid: 3/5 + 3/5 = 6/5
payout, heads: 1
payout, tails: 1
net in each case: 1 − 6/5 = −1/5
lab: loss 1/5 per unit
Whatever the coin does the person is 1/5 poorer, and no information about the coin could change that. The fault is in the prices, not in any belief about the world.
Common mistakes
1
Reading credence 1/2 as “I have no idea”A credence is a price, and 1/2 is a specific price: pay up to 1/2 for a bet paying 1 on heads, and equally for tails. Ignorance among the six faces of a die cannot be written as 1/2 for each, since the six sum to 3 and the lab’s rules need them to sum to 1. The same number, 1/2, can come from knowing the coin is fair or from knowing nothing, and so the number alone does not say which.
2
Thinking a Dutch book shows that the credences are falseIt shows only that they cannot all be acted on at their own prices. The person with 3/5 on both faces has not made a mistake about the coin; they have made a mistake about each other. And a coherent person can be wrong about the coin, which the coherent preset does not rule out.
3
Assuming coherence is enoughThe rules never mention the world. A person who gives 99/100 to heads on a fair coin and 1/100 to tails passes the test. The next lessons are about how evidence is supposed to move credences, which is what coherence leaves open.
Check your understanding
✓
Finish when you can write the bets, not just say “incoherent”.
You should be able to take credences over a small outcome space, check each against the rules, and for a violation write the bets that lose in every outcome and the amount lost. “Their credences do not add up” without the bets and the size of the loss is not yet the result.
Note: The Dutch book argument is controversial as a foundation for probability: it supposes that credences are prices a person must honour, which real people are not forced to do. It is used here because it is exact and gives the rules a reason you can check.