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D. Krpelík and T. Basu
Example 2.5 (continued)
We also have additional information that C got injured before the race and
A won the previous race. However, we have no information for B.
2.5.1.1 Axioms
Definition 2.16 (Desirability) We call a gamble desirable if we accept it
depending on the available information about the outcomes.
In our example in Table 2.2, we can see that for g 1 , we will lose 1 if C wins.
However, based on our information, C is injured and less likely to win the race;
therefore, we can accept this gamble. Conversely, we see that for g 2 , we can lose
some amount if B wins. But because A won the previous race, we may accept this
gamble.
Besides accepting g 1 and g 2 , we accept or reject a gamble depending on some
rationality criteria. We can see that g 3 < 0, that is, we can’t win anything, and
eventually, we may lose some amount if C wins the race. Therefore, we will not
choose this gamble as we cannot win anything; i.e. we want to avoid sure loss.
Here, by g < 0, we mean that g(x) ≤ 0 ∀x ∈ Ω and that there exists at least
one x ∈ Ω, such that g(x) < 0.
Contrarily, for g 5 , we won’t be losing anything, and we can actually win some if
A or B wins the race. Therefore, g 5 is a safe bet for gambling, and we will accept
g 5 ; i.e. we want to accept sure gain.
Now, from the example, it can be also seen that g 4 = 2g 2 and g 6 = g 1 + g 5 .
Therefore, if we are committed to accepting g 2 , then we should also accept g 4 ;
similarly, if we are committed to accepting g 1 and g 5 , then we should also accept
g 6 .
We can see the abovementioned rationality criteria as desirability axioms for
gambling. We list these axioms as follows:
1. Avoiding sure loss: For any gamble g, if g < 0, then we don’t accept g.
2. Accepting sure gain: For any gamble g, if g ≥ 0, then we accept g.
3. Positive homogeneity: If we accept a gamble g, then we accept λg for any λ ≥ 0.
4. Combination: If we accept two gambles, g and g , then we will accept g + g .
From the abovementioned axioms, we can prove the following statement: If we
accept g and g ≥ g, then we accept g . That is, if g is a bounded gamble that
dominates the accepted bounded gamble g, then we accept the bounded gamble g
too. We can view this as a monotonicity property of desirability [39].
D. Krpelík and T. Basu
Example 2.5 (continued)
We also have additional information that C got injured before the race and
A won the previous race. However, we have no information for B.
2.5.1.1 Axioms
Definition 2.16 (Desirability) We call a gamble desirable if we accept it
depending on the available information about the outcomes.
In our example in Table 2.2, we can see that for g 1 , we will lose 1 if C wins.
However, based on our information, C is injured and less likely to win the race;
therefore, we can accept this gamble. Conversely, we see that for g 2 , we can lose
some amount if B wins. But because A won the previous race, we may accept this
gamble.
Besides accepting g 1 and g 2 , we accept or reject a gamble depending on some
rationality criteria. We can see that g 3 < 0, that is, we can’t win anything, and
eventually, we may lose some amount if C wins the race. Therefore, we will not
choose this gamble as we cannot win anything; i.e. we want to avoid sure loss.
Here, by g < 0, we mean that g(x) ≤ 0 ∀x ∈ Ω and that there exists at least
one x ∈ Ω, such that g(x) < 0.
Contrarily, for g 5 , we won’t be losing anything, and we can actually win some if
A or B wins the race. Therefore, g 5 is a safe bet for gambling, and we will accept
g 5 ; i.e. we want to accept sure gain.
Now, from the example, it can be also seen that g 4 = 2g 2 and g 6 = g 1 + g 5 .
Therefore, if we are committed to accepting g 2 , then we should also accept g 4 ;
similarly, if we are committed to accepting g 1 and g 5 , then we should also accept
g 6 .
We can see the abovementioned rationality criteria as desirability axioms for
gambling. We list these axioms as follows:
1. Avoiding sure loss: For any gamble g, if g < 0, then we don’t accept g.
2. Accepting sure gain: For any gamble g, if g ≥ 0, then we accept g.
3. Positive homogeneity: If we accept a gamble g, then we accept λg for any λ ≥ 0.
4. Combination: If we accept two gambles, g and g , then we will accept g + g .
From the abovementioned axioms, we can prove the following statement: If we
accept g and g ≥ g, then we accept g . That is, if g is a bounded gamble that
dominates the accepted bounded gamble g, then we accept the bounded gamble g
too. We can view this as a monotonicity property of desirability [39].