On the Ellsberg paradox Keiran Sharpe School of
On the Ellsberg paradox Keiran Sharpe School of Business (ADFA campus), University of New South Wales, Canberra, ACT, 2600. Australia. k. sharpe@adfa. edu. au
0 introduction This paper proposes a model in which decision makers use a ring of (hyper-)complex numbers in order to express their beliefs and attitudes towards ‘ambiguity’ and risk. Decision makers whose beliefs are formed on the ring and who maximize expected utility on it, are shown to behave in ways that are predicted by the Ellsberg and Machina paradoxes.
1. 0 the ring
1. 1 Matrix version of numbers
1. 2 right angle (grid) addition
1. 3 numbers are partially ordered éℝ z ℝ
2. 0 beliefs
Case 1: the Ellsberg 3 -colour problem: 30 balls 60 balls red black yellow $100 $0 $100 $0 $100
Beliefs: Expected values:
Case 1: the Ellsberg 3 -colour problem: éℝ g f’ f g’ ℝ
Case 5: the Blavatskyy twist of the Machina reflection example: black white heads $4, 000 tails $0 $0 black white heads $4, 000 tails $x $0 black white heads $4, 000 $0 tails $4, 000 $0 black white heads $4, 000 $x tails $4, 000 $0
Beliefs: Expected values:
Case 5: the Blavatskyy twist: éℝ g 2 g 1 f 2 f 1 ℝ
3. 0 decisions
3. 2 Ellsberg’s formula
4. 0 paradoxes The model solves: Ellsberg: 1. 2. 3. 4. The 3 colour problem The 2 urn problem The 4 colour problem The n colour problem (assuming the fanning out hypothesis) Machina: 1. The reflection example 2. The reflection example – Blavatskyy’s twist 3. The 50: 51 example Allais: 1. The common consequence effect 2. The common ratio effect
Case 2: the Ellsberg 2 -urn problem: Urn II 100 balls 50 balls red black $100 $0 - - $100 $0 $0 $100 - - $0 $100
Beliefs: Expected values:
Case 2: the Ellsberg 2 -urn problem: éℝ f’ = f g = g’ ℝ
Case 3: the Ellsberg 4 -colour problem: 100 balls 50 balls red black green yellow $100 $0 $0 $100 $0 $0 $100
Beliefs: Expected values:
Case 3: the Ellsberg 4 -colour problem: éℝ f’ = g f = g’ ℝ
Case 4: the Machina ‘reflection’ example: 50 balls E 1 E 2 E 3 E 4 $4, 000 $8, 000 $4, 000 $8, 000 $0 $0 $8, 000 $4, 000 $0 $4, 000 $8, 000 $4, 000
Beliefs: Expected values:
Case 4: the Machina ‘reflection’ example: éℝ f = g’ f’ = g ℝ
Case 6: the 50: 51 example: 50 balls 51 balls E 1 E 2 E 3 E 4 $8, 000 $4, 000 $12, 000 $8, 000 $4, 000 $0 $12, 000 $4, 000 $8, 000 $0
Beliefs: Expected values:
Case 6: the 50: 51 example: éℝ g g’ f’ f ℝ
5. 0 Beliefs reprise normalize diagonalize
5. 1 idempotent basis beliefs
5. 1 beliefs are belief functions Since the real value of beliefs is monotonic with event union, the beliefs that decision makers hold are totally monotone capacities – i. e. , constitute a vector -valued belief function:
6. 0 conclusions If decision makers form beliefs that are (vector-valued) belief functions, and if they maximize expected utility on the ring of numbers that allows them so to express their beliefs, and if they filter these values by consistently applying their well-formed attitude to ambiguity, then we can explain: the Ellsberg paradox and the Machina paradox; furthermore, we can also explain the paradox of Allais.
Case A 1: the Allais paradox: A: 1% B: 89% C: 10% $1, 000, 000 $1, 000 $0 $5, 000, 000 $1, 000 $0 $0 $5, 000 $0
Utilities: Beliefs: Expected values:
Case A 1: the Allais paradox: éℝ g’ g f’ f ℝ
Case A 2: the Ellsberg n-colour problem: Urn I 10 balls 10 balls 10 balls Red Yellow Black Green Blue Purple White Grey Orange Mauve $0 $100 $100 $100 $0 $0 $0 Orange Mauve Urn II 100 balls Red Yellow Black Green Blue Purple White Grey $0 $100 $100 $100 $0 $0 $0
Beliefs: Expected values:
Case A 2: the Ellsberg n-colour problem: éℝ f’ g’ f g ℝ
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