Choice under uncertainty
A simple choice problem
Consider the following choice problem:
Safe
Medium
Risky-Hedge
Crisis
0
−3
10
No crisis
2
6
−6
A simple choice problem
Consider the following choice problem:
Safe
Medium
Risky-Hedge
Crisis
0
−3
10
No crisis
2
6
−6
The investor is ignorant and he does not know the
probability of Crisis/No Crisis.
MaxiMin
The investor may choose a strategy that maximizes the
minimal payoffs across the states
MaxiMin
The investor may choose a strategy that maximizes the
minimal payoffs across the states
S
M
R-H
C
0
−3
10
NC
2
6
−6
MaxiMin
0
−3
−6
MaxiMin
The investor may choose a strategy that maximizes the
minimal payoffs across the states
S
M
R-H
C
0
−3
10
NC
2
6
−6
MaxiMin
0
−3
−6
It seems that the investor is very pessimistic (we will come
back to that)
MaxiMax
Alternatively, the investor may choose a strategy that
maximizes the maximal payoffs across the states
MaxiMax
Alternatively, the investor may choose a strategy that
maximizes the maximal payoffs across the states
S
M
R-H
C
0
−3
10
NC
2
6
−6
MaxiMax
2
6
10
MaxiMax
Alternatively, the investor may choose a strategy that
maximizes the maximal payoffs across the states
S
M
R-H
C
0
−3
10
NC
2
6
−6
MaxiMax
2
6
10
It seems that the investor is very optimistic now.
Hurwicz α criterion
The investor may choose a strategy that maximizes a linear
combination of Maximin and Maximax
Hurwicz α criterion
The investor may choose a strategy that maximizes a linear
combination of Maximin and Maximax
S
M
R-H
C
0
−3
10
NC
2
6
−6
Hurwicz
α × 0 + (1 − α) × 2
α × (−3) + (1 − α) × 6
α × (−6) + (1 − α) × 10
Hurwicz α criterion
The investor may choose a strategy that maximizes a linear
combination of Maximin and Maximax
S
M
R-H
I
C
0
−3
10
NC
2
6
−6
Hurwicz
α × 0 + (1 − α) × 2
α × (−3) + (1 − α) × 6
α × (−6) + (1 − α) × 10
Depending on the investor’s optimism or pessimism
level you get different solutions
Hurwicz α criterion
The investor may choose a strategy that maximizes a linear
combination of Maximin and Maximax
S
M
R-H
C
0
−3
10
NC
2
6
−6
Hurwicz
α × 0 + (1 − α) × 2
α × (−3) + (1 − α) × 6
α × (−6) + (1 − α) × 10
I
Depending on the investor’s optimism or pessimism
level you get different solutions
I
For α = 0.5 the criterion in equivalent to Laplace but
only if there are two states
Laplace
The investor may assume that each state is equally probable
and choose a strategy that maximize expected value
Laplace
The investor may assume that each state is equally probable
and choose a strategy that maximize expected value
C
S
M
R-H
0
−3
10
NC
2
6
−6
Laplace
1
2
×0+
1
2
×2=1
1
1
2 × (−3) + 2 × 6 = 1.5
1
1
2 × (−6) + 2 × 10 = 2
Laplace
The investor may assume that each state is equally probable
and choose a strategy that maximize expected value
C
S
M
R-H
0
−3
10
NC
2
6
−6
Laplace
1
2
×0+
1
2
×2=1
1
1
2 × (−3) + 2 × 6 = 1.5
1
1
2 × (−6) + 2 × 10 = 2
But it sounds arbitrary just to assume that the states are
equally probable.
Minimax regret
The investor may choose a strategy that minimizes a
maximal regret across states
Minimax regret
The investor may choose a strategy that minimizes a
maximal regret across states
S
M
R-H
Payoff table
C NC
0
2
−3
6
10 −6
Regret table
C NC
10
4
13
0
0
12
Minimax regret
10
13
12
A zero-sum game
Let’s treat this problem as a game:
I
Investor’s problem: what strategy is the best for me?
A zero-sum game
Let’s treat this problem as a game:
I
I
Investor’s problem: what strategy is the best for me?
Depends on the "strategy" of the market p1 , p2
I
Safe: 0p1 + 2p2
A zero-sum game
Let’s treat this problem as a game:
I
I
Investor’s problem: what strategy is the best for me?
Depends on the "strategy" of the market p1 , p2
I
I
I
Safe: 0p1 + 2p2
Medium: −3p1 + 6p2
Risky-Hedge: 10p1 − 6p2
A zero-sum game
Let’s treat this problem as a game:
I
I
Investor’s problem: what strategy is the best for me?
Depends on the "strategy" of the market p1 , p2
I
I
I
I
Safe: 0p1 + 2p2
Medium: −3p1 + 6p2
Risky-Hedge: 10p1 − 6p2
Investor’s problem: max(0p1 + 2p2 ; −3p1 + 6p2 ; 10p1 − 6p2 )
A zero-sum game
Let’s treat this problem as a game:
I
I
Investor’s problem: what strategy is the best for me?
Depends on the "strategy" of the market p1 , p2
I
I
I
Safe: 0p1 + 2p2
Medium: −3p1 + 6p2
Risky-Hedge: 10p1 − 6p2
I
Investor’s problem: max(0p1 + 2p2 ; −3p1 + 6p2 ; 10p1 − 6p2 )
I
Assume the worst case: the market is trying to get as
much money from you as possible
Vicious market problem
Since the market is vicious, it wants to minimize your payoff:
min
max(0p1 + 2p2 ; −3p1 + 6p2 ; 10p1 − 6p2 )
p1 , p2 ≥ 0
p1 + p2 = 1
Vicious market problem
Since the market is vicious, it wants to minimize your payoff:
min
max(0p1 + 2p2 ; −3p1 + 6p2 ; 10p1 − 6p2 )
p1 , p2 ≥ 0
p1 + p2 = 1
Let’s rewrite it as a Linear Programming Problem:
min
z
p1 ,p2 ≥0, z∈R
s.t. z
≥ 0p1 + 2p2
z
≥ −3p1 + 6p2
z
≥ 10p1 − 6p2
1 = p1 + p2
Rewriting
min
z
p1 ,p2 ≥0, z∈R
s.t. z
≥ 0p1 + 2p2
z
≥ −3p1 + 6p2
z
≥ 10p1 − 6p2
1 = p1 + p2
is equivalent to:
Rewriting
min
z
p1 ,p2 ≥0, z∈R
s.t. z
≥ 0p1 + 2p2
z
≥ −3p1 + 6p2
z
≥ 10p1 − 6p2
1 = p1 + p2
is equivalent to:
min
max
p1 ,p2 ≥0, z∈R
q1 ,q2 ,q3 ≥0, v ∈R
z
+ q1 [0p1 + 2p2 − z]
+ q2 [−3p1 + 6p2 − z]
+ q3 [10p1 − 6p2 − z]
+ v [1 − (p1 + p2 )]
Rewriting
min
max
p1 ,p2 ≥0, z∈R
q1 ,q2 ,q3 ≥0, v ∈R
z
+ q1 [0p1 + 2p2 − z]
+ q2 [−3p1 + 6p2 − z]
+ q3 [10p1 − 6p2 − z]
+ v [1 − (p1 + p2 )]
is equivalent to:
Rewriting
min
max
p1 ,p2 ≥0, z∈R
q1 ,q2 ,q3 ≥0, v ∈R
z
+ q1 [0p1 + 2p2 − z]
+ q2 [−3p1 + 6p2 − z]
+ q3 [10p1 − 6p2 − z]
+ v [1 − (p1 + p2 )]
is equivalent to:
max
min
q1 ,q2 ,q3 ≥0, v ∈R
p1 ,p2 ≥0, z∈R
v
+ p1 [0q1 − 3q2 + 10q3 − v ]
+ p2 [2q1 + 6q2 − 6q3 − v ]
+ z[1 − (q1 + q2 + q3 )]
Rewriting
max
min
q1 ,q2 ,q3 ≥0, v ∈R
p1 ,p2 ≥0, z∈R
v
+ p1 [0q1 − 3q2 + 10q3 − v ]
+ p2 [2q1 + 6q2 − 6q3 − v ]
+ z[1 − (q1 + q2 + q3 )]
is equivalent to:
Rewriting
max
min
q1 ,q2 ,q3 ≥0, v ∈R
p1 ,p2 ≥0, z∈R
v
+ p1 [0q1 − 3q2 + 10q3 − v ]
+ p2 [2q1 + 6q2 − 6q3 − v ]
+ z[1 − (q1 + q2 + q3 )]
is equivalent to:
max
v
q1 ,q2 ,q3 ≥0, v ∈R
v
≤ 0q1 − 3q2 + 10q3
v
≤ 2q1 + 6q2 − 6q3
1 = q1 + q2 + q3
Rewriting
max
v
q1 ,q2 ,q3 ≥0, v ∈R
v
≤ 0q1 − 3q2 + 10q3
v
≤ 2q1 + 6q2 − 6q3
1 = q1 + q2 + q3
is equivalent to:
Rewriting
max
v
q1 ,q2 ,q3 ≥0, v ∈R
v
≤ 0q1 − 3q2 + 10q3
v
≤ 2q1 + 6q2 − 6q3
1 = q1 + q2 + q3
is equivalent to:
min(0q1 − 3q2 + 10q3 ; 2q1 + 6q2 − 6q3 )
max
q1 , q2 , q3 ≥ 0
q1 + q2 + q3 = 1
The investor’s problem
max
min(0q1 − 3q2 + 10q3 ; 2q1 + 6q2 − 6q3 )
q1 , q2 , q3 ≥ 0
q1 + q2 + q3 = 1
I
The market is "vicious" so it tries to minimize the
investor’s payoff min(0q1 − 3q2 + 10q3 ; 2q1 + 6q2 − 6q3 )
The investor’s problem
max
min(0q1 − 3q2 + 10q3 ; 2q1 + 6q2 − 6q3 )
q1 , q2 , q3 ≥ 0
q1 + q2 + q3 = 1
I
The market is "vicious" so it tries to minimize the
investor’s payoff min(0q1 − 3q2 + 10q3 ; 2q1 + 6q2 − 6q3 )
I
And the investor is trying maximize his payoff
Comparison
The problem of the market:
min
max(0p1 + 2p2 ; −3p1 + 6p2 ; 10p1 − 6p2 )
p1 , p2 ≥ 0
p1 + p2 = 1
The problem of the investor:
max
min(0q1 − 3q2 + 10q3 ; 2q1 + 6q2 − 6q3 )
q1 , q2 , q3 ≥ 0
q1 + q2 + q3 = 1
Comparison
The problem of the market:
min
max(0p1 + 2p2 ; −3p1 + 6p2 ; 10p1 − 6p2 )
p1 , p2 ≥ 0
p1 + p2 = 1
The problem of the investor:
max
min(0q1 − 3q2 + 10q3 ; 2q1 + 6q2 − 6q3 )
q1 , q2 , q3 ≥ 0
q1 + q2 + q3 = 1
The two problems are DUAL to each other. You can solve
one and infer the solution of the other.
In practice you solve the problem which is easier to solve.
Back to maximin
It turns out that:
I
the problem of the investor in the zero-sum game
Back to maximin
It turns out that:
I
the problem of the investor in the zero-sum game
I
is the same as
Back to maximin
It turns out that:
I
the problem of the investor in the zero-sum game
I
is the same as
I
the problem of the investor trying to find a Maximin
strategy in an individual choice situation
Back to maximin
It turns out that:
I
the problem of the investor in the zero-sum game
I
is the same as
I
the problem of the investor trying to find a Maximin
strategy in an individual choice situation
if you allow mixed strategies
I
Back to maximin
It turns out that:
I
the problem of the investor in the zero-sum game
I
is the same as
I
the problem of the investor trying to find a Maximin
strategy in an individual choice situation
if you allow mixed strategies
I
I
Choosing strategies not for sure but with certain
probability
Solving for a Maximin strategy
max
min(0q1 − 3q2 + 10q3 ; 2q1 + 6q2 − 6q3 )
q1 , q2 , q3 ≥ 0
q1 + q2 + q3 = 1
The solution is (q1 , q2 , q3 ) = (0, 0.64, 0.36 with Minimax value
of 1.68.
Solving for a Maximin strategy
max
min(0q1 − 3q2 + 10q3 ; 2q1 + 6q2 − 6q3 )
q1 , q2 , q3 ≥ 0
q1 + q2 + q3 = 1
The solution is (q1 , q2 , q3 ) = (0, 0.64, 0.36 with Minimax value
of 1.68.
S
M
R
C
0
−3
10
NC
2
6
−6
MaxiMin
0
−3
−6
Solving for a Maximin strategy
max
min(0q1 − 3q2 + 10q3 ; 2q1 + 6q2 − 6q3 )
q1 , q2 , q3 ≥ 0
q1 + q2 + q3 = 1
The solution is (q1 , q2 , q3 ) = (0, 0.64, 0.36 with Minimax value
of 1.68.
S
M
R
0.64 × M + 0.36 × R
C
0
−3
10
1.68
NC
2
6
−6
1.68
MaxiMin
0
−3
−6
1.68
Extending other decision rules to mixed strategies
The question is: Can we also gain in the case of other
decision rules when we allow mixed strategies?
I
Maximax?:
Extending other decision rules to mixed strategies
The question is: Can we also gain in the case of other
decision rules when we allow mixed strategies?
I
Maximax?: NO, we cannot
I
Laplace?:
Extending other decision rules to mixed strategies
The question is: Can we also gain in the case of other
decision rules when we allow mixed strategies?
I
Maximax?: NO, we cannot
I
Laplace?: NO, we cannot
I
Hurwicz?:
Extending other decision rules to mixed strategies
The question is: Can we also gain in the case of other
decision rules when we allow mixed strategies?
I
Maximax?: NO, we cannot
I
Laplace?: NO, we cannot
I
Hurwicz?: NO, we cannot
I
Minimax regret?:
Extending other decision rules to mixed strategies
The question is: Can we also gain in the case of other
decision rules when we allow mixed strategies?
I
Maximax?: NO, we cannot
I
Laplace?: NO, we cannot
I
Hurwicz?: NO, we cannot
I
Minimax regret?: YES, we can. WHY?
Minimax regret
Payoff table
S
M
R
0.48 × M
+0.52 × R
C
0
−3
10
3.76
NC
2
6
−6
−0.24
Regret table
C
10
13
0
6.24
NC
4
0
12
6.24
Minimax regret
10
13
12
6.24