I am reading probability from AI point of view from book Artificial Intellegence A Modern approach by Russel and Norving. Below is text snippet from chapter 13.

One argument for the axioms of probability, first stated in 1931 by Bruno de Finetti (and translated into English in de Finetti (1993)), is as follows: If an agent has some degree of belief in a proposition a, then the agent should be able to state odds at which it is indifferent to a bet for or against a. Think of it as a game between two agents: Agent 1 states, “my degree of belief in event a is 0.4.” Agent 2 is then free to choose whether to wager for or against a at stakes that are consistent with the stated degree of belief. That is, Agent 2 could choose to accept Agent 1’s bet that a will occur, offering $6 against Agent 1’s $4. Or Agent 2 could accept Agent 1’s bet that ¬a will occur, offering $4 against Agent 1’s $6. Then we observe the outcome of a, and whoever is right collects the money. If an agent’s degrees of belief do not accurately reflect the world, then you would expect that it would tend to lose money over the long run to an opposing agent whose beliefs more accurately reflect the state of the world.

If Agent 1 expresses a set of degrees of belief that violate the axioms of probability theory then there is a combination of bets by Agent 2 that guarantees that Agent 1 will lose money every time.

Below shows that if Agent 2 chooses to bet $4 on a, $3 on b, and $2 on ¬(a ∨ b), then Agent 1 always loses money, regardless of the outcomes for a and b. De Finetti’s theorem implies that no rational agent can have beliefs that violate the axioms of probability

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My question is how we got values inf last column for example a,b column how we got -6, -7 an 2. Kindly explain.

Thanks for your time


It is rather straight forward: A bets on $\neg{a}$, $\neg{b}$, and on $a \lor b$. Accordingly, A will receive money, if these propositions are true—which can be seen in the table that tabulates all $2\times2$ possible results. B takes the counter position and the stakes simply map the probabilities to odds (e.g. $p(a) = 0.4 \implies 4:6$), which means A will lose 6 CU if $a$ and gain 4 if $\neg{a}$.

This simply is an example for a Dutch book as it gives a reason for consistency in beliefs: Since A believes in his probability assignments, he should be willing to engage in a bet with odds (i.e. stakes) clearly following the probability assumptions. As can be seen here, B can take a side on these bets that will allow him to always win.


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