I.A.1.2 Mathematical Expectations: Prices or Utilities? Flashcards
1
Q
Expected Value
A
The sum of outcomes weighted by their respecive probabilities
2
Q
Weak law of large numbers
A
The average of the results obtained from a large number of trails should be close to the expeceted value, and will tend to become closer as more trials are performed.
or
As the number of samples increases, the average of these samples is likely to reach the mean of the whole population
3
Q
Daniel Bernoulli (1738)
A
- First mathematician to question the principle of maximising expected value and try to justify departures from it observed in daily life.
- Suggested applying the principle not to cash outcomes but to utilities associated with cas outcomes in order to reconcile common behavior with a maximum-expectation principle (actions should be directed at a maximising expected utility)
- Assumption that - utility is always inversely proportional to existing wealth
- Small change in utility, du, is related to a small change in wealth, dx
4
Q
Utility
A
the ‘personal value’ of an asset
5
Q
Price
A
The ‘exchange value’ of an asset
6
Q
St Petersburg paradox
A
- Peter tosses a coin and continues to do so until it should land ‘heads’. He agrees to give Paul one ducat if he gets ‘heads’ on the very first throw, 2 ducats if he gets it on the second, 4 if on the third, 8 if on the fourth, and so on, so that with each additional throw the nunber of ducats he must pay is doubled. Determine the value of Paul’s expectation.
- Since there is an infinate number of possinle outcomes, Paul’s monetary expectation is infinate. But a gambler is only willing to pay a few ducats for the right to play the game (hence the paradox)
7
Q
Logarithmic utility function
A
8
Q
Logarithmic utility function and the St Petersburg paradox
A
- If the logarithmic utility function is applied to the paradox then it appears no longer to be a paradox.
- In general, the larger the initial wealth of the gambler, the larger his perceived utility of the game
