# Rud2007-final

Document Sample

```					Partially speciﬁed probabilities (PSP)

Ehud Lehrer
Tel Aviv University
Tel Aviv, Israel

www.tau.ac.il/∼lehrer

RUD2007, Tel Aviv

Ehud Lehrer – Partially speciﬁed probabilities – p. 1/
A-A (vN-M)                                   Fully Bayesian

Partially-specified probabilities
(probability of events in sub-algebra)   Partially Bayesian

Partially-specified probabilities
Partially
(expectation of random variables)
Bayesian
Choquet EUM
(Schmeidler 1989)

Multiple prior
(Gilboa-Schmeidler 1989)
Ellsberg urn

Ehud Lehrer – Partially speciﬁed probabilities – p. 1/?
Ellsberg urn
• An urn contains 90 balls of three colors

Ehud Lehrer – Partially speciﬁed probabilities – p. 1/?
Ellsberg urn
• An urn contains 90 balls of three colors

• 30 are Red and the other 60 are either White or Black

Ehud Lehrer – Partially speciﬁed probabilities – p. 1/?
Ellsberg urn
• An urn contains 90 balls of three colors

• 30 are Red and the other 60 are either White or Black

• One ball is randomly drawn from the urn

Ehud Lehrer – Partially speciﬁed probabilities – p. 1/?
Ellsberg urn
• An urn contains 90 balls of three colors

• 30 are Red and the other 60 are either White or Black

• One ball is randomly drawn from the urn

• A decision maker (DM) is asked to choose between two
lotteries:

Ehud Lehrer – Partially speciﬁed probabilities – p. 1/?
Ellsberg urn
• An urn contains 90 balls of three colors

• 30 are Red and the other 60 are either White or Black

• One ball is randomly drawn from the urn

• A decision maker (DM) is asked to choose between two
lotteries:

• X: obtain \$100 if Red.
Y: obtain \$100 if White

Ehud Lehrer – Partially speciﬁed probabilities – p. 1/?
Ellsberg urn
• An urn contains 90 balls of three colors

• 30 are Red and the other 60 are either White or Black

• One ball is randomly drawn from the urn

• A decision maker (DM) is asked to choose between two
lotteries:

• X: obtain \$100 if Red.
Y: obtain \$100 if White

• W: obtain \$100 if Red or Black.
Z: obtain \$100 if White or Black

Ehud Lehrer – Partially speciﬁed probabilities – p. 1/?
Ellsberg urn
• An urn contains 90 balls of three colors

• 30 are Red and the other 60 are either White or Black

• One ball is randomly drawn from the urn

• A decision maker (DM) is asked to choose between two
lotteries:

• X: obtain \$100 if Red.
Y: obtain \$100 if White

• W: obtain \$100 if Red or Black.
Z: obtain \$100 if White or Black

• If you prefer X over Y and Z over W you are not an
expected utility maximizer                   Ehud Lehrer – Partially speciﬁed probabilities – p. 1/?
Ellsberg urn - cont.

Ehud Lehrer – Partially speciﬁed probabilities – p. 2/?
Ellsberg urn - cont.
• The DM ought to take decisions while having only partial

Ehud Lehrer – Partially speciﬁed probabilities – p. 2/?
Ellsberg urn - cont.
• The DM ought to take decisions while having only partial

• From the information DM obtained she can infer that
P(Red) = 1/3 and I
I                 P(White or Black) = 2/3.

Ehud Lehrer – Partially speciﬁed probabilities – p. 2/?
Dynamic urn

Ehud Lehrer – Partially speciﬁed probabilities – p. 3/?
Dynamic urn
• Suppose that on day 2 the number of White balls is
doubled

Ehud Lehrer – Partially speciﬁed probabilities – p. 3/?
Dynamic urn
• Suppose that on day 2 the number of White balls is
doubled

• What is now the distribution of colors?

Ehud Lehrer – Partially speciﬁed probabilities – p. 3/?
Dynamic urn
• Suppose that on day 2 the number of White balls is
doubled

• What is now the distribution of colors?

• The probability of no non-trivial event is known: the prob.
of Red is no longer 1/3; that of White or Black is no longer
2/3

Ehud Lehrer – Partially speciﬁed probabilities – p. 3/?
Dynamic urn
• Suppose that on day 2 the number of White balls is
doubled

• What is now the distribution of colors?

• The probability of no non-trivial event is known: the prob.
of Red is no longer 1/3; that of White or Black is no longer
2/3

• So, after all what the DM knows?

Ehud Lehrer – Partially speciﬁed probabilities – p. 3/?
Dynamic urn
• Suppose that on day 2 the number of White balls is
doubled

• What is now the distribution of colors?

• The probability of no non-trivial event is known: the prob.
of Red is no longer 1/3; that of White or Black is no longer
2/3

• So, after all what the DM knows?

• DM knows the expectation of some, but not all, random
variables

Ehud Lehrer – Partially speciﬁed probabilities – p. 3/?
Dynamic urn - cont.

Ehud Lehrer – Partially speciﬁed probabilities – p. 4/?
Dynamic urn - cont.
1
• Consider the random variable (r.v.): X = [1(R); 6 (W ); 0(B)]

Ehud Lehrer – Partially speciﬁed probabilities – p. 4/?
Dynamic urn - cont.
1
• Consider the random variable (r.v.): X = [1(R); 6 (W ); 0(B)]

1
E(X) =
• We’ll see that I        3

Ehud Lehrer – Partially speciﬁed probabilities – p. 4/?
Dynamic urn - cont.
1
• Consider the random variable (r.v.): X = [1(R); 6 (W ); 0(B)]

1
E(X) =
• We’ll see that I        3

• Denote by ni the number of color i balls at day one

Ehud Lehrer – Partially speciﬁed probabilities – p. 4/?
Dynamic urn - cont.
1
• Consider the random variable (r.v.): X = [1(R); 6 (W ); 0(B)]

1
E(X) =
• We’ll see that I        3

• Denote by ni the number of color i balls at day one
R      W B
(nW + nB = 60)
nR = 30 nW nB

Ehud Lehrer – Partially speciﬁed probabilities – p. 4/?
Dynamic urn - cont.
1
• Consider the random variable (r.v.): X = [1(R); 6 (W ); 0(B)]

1
E(X) =
• We’ll see that I        3

• Denote by ni the number of color i balls at day one
R      W B
(nW + nB = 60)
nR = 30 nW nB
1
nR             n
3 W       1
nR +nW +nB   =   nW    =   3

Ehud Lehrer – Partially speciﬁed probabilities – p. 4/?
Dynamic urn - cont.
1
• Consider the random variable (r.v.): X = [1(R); 6 (W ); 0(B)]

1
E(X) =
• We’ll see that I        3

• Denote by ni the number of color i balls at day one
R      W B
(nW + nB = 60)
nR = 30 nW nB
1
nR             n
3 W       1
nR +nW +nB   =   nW    =   3

• On day 2 there are 2nW White balls

Ehud Lehrer – Partially speciﬁed probabilities – p. 4/?
Dynamic urn - cont.
1
• Consider the random variable (r.v.): X = [1(R); 6 (W ); 0(B)]

1
E(X) =
• We’ll see that I        3

• Denote by ni the number of color i balls at day one
R      W B
(nW + nB = 60)
nR = 30 nW nB
1
nR              n
3 W       1
nR +nW +nB   =    nW    =   3

• On day 2 there are 2nW White balls

1nR + 1 2nW +0nB                1
nR + 3 nW                 1
E(X) =
I                 6
nR +2nW +nB      =   nR +nW +nB +nW        =       3
Ehud Lehrer – Partially speciﬁed probabilities – p. 4/?
Dynamic urn - conclusion

Ehud Lehrer – Partially speciﬁed probabilities – p. 5/?
Dynamic urn - conclusion
• On day 2 the expectation of two random variables is
known:

Ehud Lehrer – Partially speciﬁed probabilities – p. 5/?
Dynamic urn - conclusion
• On day 2 the expectation of two random variables is
known:

• X = [1(R); 6 (W ); 0(B)] (expectation= 1 ) and
1
3
[1(R); 1(W ); 1(B)] (expectation=1).

Ehud Lehrer – Partially speciﬁed probabilities – p. 5/?
Dynamic urn - conclusion
• On day 2 the expectation of two random variables is
known:

• X = [1(R); 6 (W ); 0(B)] (expectation= 1 ) and
1
3
[1(R); 1(W ); 1(B)] (expectation=1).

• The expectation of any r.v. in the algebra they generate is
also known.

Ehud Lehrer – Partially speciﬁed probabilities – p. 5/?
Noisy signals

Ehud Lehrer – Partially speciﬁed probabilities – p. 6/?
Noisy signals
• There are three states of nature: x, y and z

Ehud Lehrer – Partially speciﬁed probabilities – p. 6/?
Noisy signals
• There are three states of nature: x, y and z
• There are two payoffs, 5 and 7 – stochastically depend on
the state:

Ehud Lehrer – Partially speciﬁed probabilities – p. 6/?
Noisy signals
• There are three states of nature: x, y and z
• There are two payoffs, 5 and 7 – stochastically depend on
the state:
x              y             z
[1(5); 0(7)] [ 1 (5); 1 (7)] [0(5); 1(7)]
2      2

Ehud Lehrer – Partially speciﬁed probabilities – p. 6/?
Noisy signals
• There are three states of nature: x, y and z
• There are two payoffs, 5 and 7 – stochastically depend on
the state:
x             y             z
[1(5); 0(7)] [ 1 (5); 1 (7)] [0(5); 1(7)]
2      2
• Suppose it is repeated many times and dist. over x, y and
z is uniforms and i.i.d.

Ehud Lehrer – Partially speciﬁed probabilities – p. 6/?
Noisy signals
• There are three states of nature: x, y and z
• There are two payoffs, 5 and 7 – stochastically depend on
the state:
x             y             z
[1(5); 0(7)] [ 1 (5); 1 (7)] [0(5); 1(7)]
2      2
• Suppose it is repeated many times and dist. over x, y and
z is uniforms and i.i.d.
• DM observes 50% the payoff 5 and 50% the payoff 7

Ehud Lehrer – Partially speciﬁed probabilities – p. 6/?
Noisy signals
• There are three states of nature: x, y and z
• There are two payoffs, 5 and 7 – stochastically depend on
the state:
x             y             z
[1(5); 0(7)] [ 1 (5); 1 (7)] [0(5); 1(7)]
2      2
• Suppose it is repeated many times and dist. over x, y and
z is uniforms and i.i.d.
• DM observes 50% the payoff 5 and 50% the payoff 7
• Asymptotically, all DM knows is that the expectation of
[1(x), 0(y), -1(z)] is 0.

Ehud Lehrer – Partially speciﬁed probabilities – p. 6/?
Noisy signals
• There are three states of nature: x, y and z
• There are two payoffs, 5 and 7 – stochastically depend on
the state:
x             y             z
[1(5); 0(7)] [ 1 (5); 1 (7)] [0(5); 1(7)]
2      2
• Suppose it is repeated many times and dist. over x, y and
z is uniforms and i.i.d.
• DM observes 50% the payoff 5 and 50% the payoff 7
• Asymptotically, all DM knows is that the expectation of
[1(x), 0(y), -1(z)] is 0.

• In an interactive model when players get only noisy
signals about others, players face partially-speciﬁed
probability (PSP)

Ehud Lehrer – Partially speciﬁed probabilities – p. 6/?
Example by Machina (2007)

Ehud Lehrer – Partially speciﬁed probabilities – p. 7/?
Example by Machina (2007)
• An urn contains 20 balls of four different colors a,b,c, and d.
10 are either a or b and the other 10 are either c or d.

Ehud Lehrer – Partially speciﬁed probabilities – p. 7/?
Example by Machina (2007)
• An urn contains 20 balls of four different colors a,b,c, and d.
10 are either a or b and the other 10 are either c or d.

• A decision maker (DM) chooses an act, then a ball is
randomly drawn and a reward (utility) is given. The following
table summarizes the rewards related to four acts.

Ehud Lehrer – Partially speciﬁed probabilities – p. 7/?
Example by Machina (2007)
• An urn contains 20 balls of four different colors a,b,c, and d.
10 are either a or b and the other 10 are either c or d.

• A decision maker (DM) chooses an act, then a ball is
randomly drawn and a reward (utility) is given. The following
table summarizes the rewards related to four acts.
a     b     c    d
f1    0    200   100   100
f2    0    100   200   100
f3   100   200   100   0
f4   100   100   200   0

Ehud Lehrer – Partially speciﬁed probabilities – p. 7/?
Example by Machina (2007)
• An urn contains 20 balls of four different colors a,b,c, and d.
10 are either a or b and the other 10 are either c or d.

• A decision maker (DM) chooses an act, then a ball is
randomly drawn and a reward (utility) is given. The following
table summarizes the rewards related to four acts.
a     b     c    d
f1    0    200   100   100
f2    0    100   200   100
f3   100   200   100   0
f4   100   100   200   0

• Notice: f3 is a mirror image of f2 and f4 of f1

Ehud Lehrer – Partially speciﬁed probabilities – p. 7/?
Example by Machina (2007)
• An urn contains 20 balls of four different colors a,b,c, and d.
10 are either a or b and the other 10 are either c or d.

• A decision maker (DM) chooses an act, then a ball is
randomly drawn and a reward (utility) is given. The following
table summarizes the rewards related to four acts.
a     b     c    d
f1    0    200   100   100
f2    0    100   200   100
f3   100   200   100   0
f4   100   100   200   0

• We expect (as long as symmetry is kept) that if f2                          f1
then f3 f4
Ehud Lehrer – Partially speciﬁed probabilities – p. 7/?
Example by Machina (2007)
• An urn contains 20 balls of four different colors a,b,c, and d.
10 are either a or b and the other 10 are either c or d.

• A decision maker (DM) chooses an act, then a ball is
randomly drawn and a reward (utility) is given. The following
table summarizes the rewards related to four acts.
a     b     c    d
f1    0    200   100   100
f2    0    100   200   100
f3   100   200   100   0
f4   100   100   200   0

• We expect (as long as symmetry is kept) that if f2 f1
then f3 f4 , but this is inconsistent with Choquet EUM
Ehud Lehrer – Partially speciﬁed probabilities – p. 7/?
Example by Machina (2007) - cont.

Ehud Lehrer – Partially speciﬁed probabilities – p. 8/?
Example by Machina (2007) - cont.
a     b     c    d
f1    0    200   100   100
f2    0    100   200   100
f3   100   200   100   0
f4   100   100   200   0

Ehud Lehrer – Partially speciﬁed probabilities – p. 8/?
Example by Machina (2007) - cont.
a     b     c        d
f1    0    200   100       100
f2    0    100   200       100
f3   100   200   100       0
f4   100   100   200       0
1
P(a, b) = I
• It is known that I         P(c, d) =      2

Ehud Lehrer – Partially speciﬁed probabilities – p. 8/?
Example by Machina (2007) - cont.
a     b     c    d
f1    0    200   100   100
f2    0    100   200   100
f3   100   200   100   0
f4   100   100   200   0

• It is known that I           P(c, d) = 1
P(a, b) = I         2
1
E(X1 ) =
• Equivalently, X1 = [1, 1, 0, 0], I            2   and
1
E(X2 ) = 2
X2 = [0, 0, 1, 1], I

Ehud Lehrer – Partially speciﬁed probabilities – p. 8/?
Example by Machina (2007) - cont.
a     b     c    d
f1    0    200   100   100
f2    0    100   200   100
f3   100   200   100   0
f4   100   100   200   0

• It is known that I           P(c, d) = 1
P(a, b) = I         2
1
E(X1 ) =
• Equivalently, X1 = [1, 1, 0, 0], I            2   and
1
E(X2 ) = 2
X2 = [0, 0, 1, 1], I
• Belief: P(b, c) is positive.

Ehud Lehrer – Partially speciﬁed probabilities – p. 8/?
Example by Machina (2007) - cont.
a     b     c    d
f1    0    200   100   100
f2    0    100   200   100
f3   100   200   100   0
f4   100   100   200   0

• It is known that I           P(c, d) = 1
P(a, b) = I          2
1
• Equivalently, X1 = [1, 1, 0, 0], IE(X1 ) = 2 and
1
E(X2 ) = 2
X2 = [0, 0, 1, 1], I
1
• Belief: P(b, c) is positive. It implies IE([0, 1, 1, 0]) ≥            20

Ehud Lehrer – Partially speciﬁed probabilities – p. 8/?
Example by Machina (2007) - cont.
a     b     c    d
f1    0    200   100   100
f2    0    100   200   100
f3   100   200   100   0
f4   100   100   200   0

• It is known that I           P(c, d) = 1
P(a, b) = I           2
1
• Equivalently, X1 = [1, 1, 0, 0], IE(X1 ) = 2 and
1
E(X2 ) = 2
X2 = [0, 0, 1, 1], I
1
• Belief: P(b, c) is positive. It implies I E([0, 1, 1, 0]) ≥            20
1
• Let’s assume that I  E([0, 1, 1, 0]) = 20

Ehud Lehrer – Partially speciﬁed probabilities – p. 8/?
Example by Machina (2007) - cont.
a     b     c    d
f1    0    200   100   100
f2    0    100   200   100
f3   100   200   100   0
f4   100   100   200   0

• It is known that I           P(c, d) = 1
P(a, b) = I           2
1
• Equivalently, X1 = [1, 1, 0, 0], IE(X1 ) = 2 and
1
E(X2 ) = 2
X2 = [0, 0, 1, 1], I
1
• Belief: P(b, c) is positive. It implies I E([0, 1, 1, 0]) ≥ 20
1
• Let’s assume that I  E([0, 1, 1, 0]) = 20
• Now the expectations of X1 , X2 and [0, 1, 1, 0] are known.

Ehud Lehrer – Partially speciﬁed probabilities – p. 8/?
Partially-speciﬁed probability (PSP)

Ehud Lehrer – Partially speciﬁed probabilities – p. 9/?
Partially-speciﬁed probability (PSP)
P,
• Partially-speciﬁed probability is a pair (I Y)

Ehud Lehrer – Partially speciﬁed probabilities – p. 9/?
Partially-speciﬁed probability (PSP)
P,
• Partially-speciﬁed probability is a pair (I Y)

P
• I – probability

Ehud Lehrer – Partially speciﬁed probabilities – p. 9/?
Partially-speciﬁed probability (PSP)
P,
• Partially-speciﬁed probability is a pair (I Y)

P
• I – probability

• Y – a set of random variables

Ehud Lehrer – Partially speciﬁed probabilities – p. 9/?
Partially-speciﬁed probability (PSP)
P,
• Partially-speciﬁed probability is a pair (I Y)

P
• I – probability

• Y – a set of random variables

EP         E(Y ), Y ∈ Y
• DM is informed only of I I (Y ) = I

Ehud Lehrer – Partially speciﬁed probabilities – p. 9/?
How one can use PSP?

Ehud Lehrer – Partially speciﬁed probabilities – p. 10/?
How one can use PSP?
P,
• Let (I Y) be PSP (partially-speciﬁed probability)

Ehud Lehrer – Partially speciﬁed probabilities – p. 10/?
How one can use PSP?
P,
• Let (I Y) be PSP (partially-speciﬁed probability)

• X is a random variable

Ehud Lehrer – Partially speciﬁed probabilities – p. 10/?
How one can use PSP?
P,
• Let (I Y) be PSP (partially-speciﬁed probability)

• X is a random variable

P,
• Deﬁne the expectation of X w.r.t (I Y),

Ehud Lehrer – Partially speciﬁed probabilities – p. 10/?
How one can use PSP?
P,
• Let (I Y) be PSP (partially-speciﬁed probability)

• X is a random variable

P,
• Deﬁne the expectation of X w.r.t (I Y),

EP
I (I ,Y) (X) = max{      E(Y );
αY I         αY Y ≤ X and αY ∈ R}

Ehud Lehrer – Partially speciﬁed probabilities – p. 10/?
Evaluating acts with PSP

Ehud Lehrer – Partially speciﬁed probabilities – p. 11/?
Evaluating acts with PSP
P,
• Suppose that (I Y) is a PSP over S (the state space)

Ehud Lehrer – Partially speciﬁed probabilities – p. 11/?
Evaluating acts with PSP
P,
• Suppose that (I Y) is a PSP over S (the state space)

• and u is an afﬁne (utility function) deﬁnes over ∆(L)

Ehud Lehrer – Partially speciﬁed probabilities – p. 11/?
Evaluating acts with PSP
P,
• Suppose that (I Y) is a PSP over S (the state space)

• and u is an afﬁne (utility function) deﬁnes over ∆(L)

• Note that if f is an act, then u ◦ f is a random variable
deﬁned over S

Ehud Lehrer – Partially speciﬁed probabilities – p. 11/?
Evaluating acts with PSP
P,
• Suppose that (I Y) is a PSP over S (the state space)

• and u is an afﬁne (utility function) deﬁnes over ∆(L)

• Note that if f is an act, then u ◦ f is a random variable
deﬁned over S

• PSP and u induce a complete order over acts. Let f and g
be acts, then

Ehud Lehrer – Partially speciﬁed probabilities – p. 11/?
Evaluating acts with PSP
P,
• Suppose that (I Y) is a PSP over S (the state space)

• and u is an afﬁne (utility function) deﬁnes over ∆(L)

• Note that if f is an act, then u ◦ f is a random variable
deﬁned over S

• PSP and u induce a complete order over acts. Let f and g
be acts, then

f        EP                  EP
g iﬀ I (I ,Y) (u ◦ f ) ≥ I (I ,Y) (u ◦ g)

Ehud Lehrer – Partially speciﬁed probabilities – p. 11/?
Back to Machina’s Example

Ehud Lehrer – Partially speciﬁed probabilities – p. 12/?
Back to Machina’s Example
a     b     c    d
f1    0    200   100   100
f2    0    100   200   100
f3   100   200   100   0
f4   100   100   200   0

Ehud Lehrer – Partially speciﬁed probabilities – p. 12/?
Back to Machina’s Example
a     b     c    d
f1    0    200   100   100
f2    0    100   200   100
f3   100   200   100   0
f4   100   100   200   0
1                  1
• Recall, IE(X1 ) = I                E(X3 ) =
E(X2 ) = 2 and I            20
P,
(X3 = [0, 1, 1, 0]). This deﬁned PSP (I Y)

Ehud Lehrer – Partially speciﬁed probabilities – p. 12/?
Back to Machina’s Example
a     b     c    d
f1    0    200   100   100
f2    0    100   200   100
f3   100   200   100   0
f4   100   100   200   0
• f1 corresponds to Y = [0, 200, 100, 100] and f2 to
Z = [0, 100, 200, 100] (u identity – the reward in utile terms)

Ehud Lehrer – Partially speciﬁed probabilities – p. 12/?
Back to Machina’s Example
a     b     c    d
f1    0    200   100   100
f2    0    100   200   100
f3   100   200   100   0
f4   100   100   200   0
• f1 corresponds to Y = [0, 200, 100, 100] and f2 to
Z = [0, 100, 200, 100] (u identity – the reward in utile terms)
• Z = [0, 100, 100, 0] + [0, 0, 100, 100] and
I
E(Z) = 100I  E(X3 ) + 100I X2 = 100 + 100
E            20    2

Ehud Lehrer – Partially speciﬁed probabilities – p. 12/?
Back to Machina’s Example
a     b     c    d
f1    0    200   100   100
f2    0    100   200   100
f3   100   200   100   0
f4   100   100   200   0
• f1 corresponds to Y = [0, 200, 100, 100] and f2 to
Z = [0, 100, 200, 100] (u identity – the reward in utile terms)
• Z = [0, 100, 100, 0] + [0, 0, 100, 100] and
I
E(Z) = 100I  E(X3 ) + 100I X2 = 100 + 100
E            20    2
• The best approximation from below to Y with linear
combinations of X1 , X2 and X3 is 100X2

Ehud Lehrer – Partially speciﬁed probabilities – p. 12/?
Back to Machina’s Example
a     b     c    d
f1    0    200   100   100
f2    0    100   200   100
f3   100   200   100   0
f4   100   100   200   0
• f1 corresponds to Y = [0, 200, 100, 100] and f2 to
Z = [0, 100, 200, 100] (u identity – the reward in utile terms)
• Z = [0, 100, 100, 0] + [0, 0, 100, 100] and
I
E(Z) = 100I  E(X3 ) + 100I X2 = 100 + 100
E            20    2
• The best approximation from below to Y with linear
combinations of X1 , X2 and X3 is 100X2
EP             EP
• Thus, I (I ,Y) (Z) > I (I ,Y) (Y ), which implies f2 f1

Ehud Lehrer – Partially speciﬁed probabilities – p. 12/?
Back to Machina’s Example
a     b     c    d
f1    0    200   100   100
f2    0    100   200   100
f3   100   200   100   0
f4   100   100   200   0
• f1 corresponds to Y = [0, 200, 100, 100] and f2 to
Z = [0, 100, 200, 100] (u identity – the reward in utile terms)
• Z = [0, 100, 100, 0] + [0, 0, 100, 100] and
I
E(Z) = 100I  E(X3 ) + 100I X2 = 100 + 100
E            20    2
• The best approximation from below to Y with linear
combinations of X1 , X2 and X3 is 100X2
EP             EP
• Thus, I (I ,Y) (Z) > I (I ,Y) (Y ), which implies f2 f1
• From similar reasons, f3 f4

Ehud Lehrer – Partially speciﬁed probabilities – p. 12/?
Axiomatization

Ehud Lehrer – Partially speciﬁed probabilities – p. 13/?
Axiomatization
• Pretty much like Anscombe-Aumann with a different
version of the independence axiom

Ehud Lehrer – Partially speciﬁed probabilities – p. 13/?
Axiomatization
• Pretty much like Anscombe-Aumann with a different
version of the independence axiom

• Deﬁnition: An act f is fat free (FaF) if

f (s)   g(s) for every s ∈ S

with at least one strict preference, implies
f g

Ehud Lehrer – Partially speciﬁed probabilities – p. 13/?
Axiomatization
• Pretty much like Anscombe-Aumann with a different
version of the independence axiom

• Deﬁnition: An act f is fat free (FaF) if

f (s)   g(s) for every s ∈ S

with at least one strict preference, implies
f g
• Deﬁnition: An act f is strongly fat free (SFaF) if for every
constant act c and every α ∈ (0, 1), αf + (1 − α)c is fat free

Ehud Lehrer – Partially speciﬁed probabilities – p. 13/?
Axiomatization
• Pretty much like Anscombe-Aumann with a different
version of the independence axiom

• Deﬁnition: An act f is fat free (FaF) if

f (s)   g(s) for every s ∈ S

with at least one strict preference, implies
f g
• Deﬁnition: An act f is strongly fat free (SFaF) if for every
constant act c and every α ∈ (0, 1), αf + (1 − α)c is fat free

• SFaF independence: if f and g are acts such that f g
and h is SFaF, then αf + (1 − α)h αg + (1 − α)h for every
α ∈ (0, 1]

Ehud Lehrer – Partially speciﬁed probabilities – p. 13/?
Relation to Choquet EUM (Schmeidler 1989)

Ehud Lehrer – Partially speciﬁed probabilities – p. 14/?
Relation to Choquet EUM (Schmeidler 1989)

• If in PSP Y consists of the indicators of events in a
sub-algebra, then a convex capacity can be deﬁned.

Ehud Lehrer – Partially speciﬁed probabilities – p. 14/?
Relation to Choquet EUM (Schmeidler 1989)

• If in PSP Y consists of the indicators of events in a
sub-algebra, then a convex capacity can be deﬁned.

• On convex capacities the concave integral (Lehrer, 2005)
coincides with Choquet integral

• Thus, when Y consists of the indicators of events in a
sub-algebra the PSP model is strictly between A-A and
Choquet EUM

Ehud Lehrer – Partially speciﬁed probabilities – p. 14/?
Relation to Choquet EUM (Schmeidler 1989)

• If in PSP Y consists of the indicators of events in a
sub-algebra, then a convex capacity can be deﬁned.

• On convex capacities the concave integral (Lehrer, 2005)
coincides with Choquet integral

• Thus, when Y consists of the indicators of events in a
sub-algebra the PSP model is strictly between A-A and
Choquet EUM

Ehud Lehrer – Partially speciﬁed probabilities – p. 14/?
Relation to the Multiple Prior Model (Gilboa-Schmeidler 1989)

• From duality: estimating the expectation of a random
variable by members of Y is equivalent to minimizing wrt to
P
probability distributions that agree with I on Y

Ehud Lehrer – Partially speciﬁed probabilities – p. 15/?
Relation to the Multiple Prior Model (Gilboa-Schmeidler 1989)

• From duality: estimating the expectation of a random
variable by members of Y is equivalent to minimizing wrt to
P
probability distributions that agree with I on Y
P,
• Thus, (I Y) induces a set of probability distributions that
are consistent with the information available. DM minimizes
over these distributions

Ehud Lehrer – Partially speciﬁed probabilities – p. 15/?
Relation to the Multiple Prior Model (Gilboa-Schmeidler 1989)

• From duality: estimating the expectation of a random
variable by members of Y is equivalent to minimizing wrt to
P
probability distributions that agree with I on Y
P,
• Thus, (I Y) induces a set of probability distributions that
are consistent with the information available. DM minimizes
over these distributions
• This set is an afﬁne space of distributions (intersection of
an afﬁne space with the simplex of distributions)

Ehud Lehrer – Partially speciﬁed probabilities – p. 15/?
Relation to the Multiple Prior Model (Gilboa-Schmeidler 1989)

• From duality: estimating the expectation of a random
variable by members of Y is equivalent to minimizing wrt to
P
probability distributions that agree with I on Y
P,
• Thus, (I Y) induces a set of probability distributions that
are consistent with the information available. DM minimizes
over these distributions
• This set is an afﬁne space of distributions (intersection of
an afﬁne space with the simplex of distributions)
• The PSP model is strictly between A-A and the Multiple
Prior model

Ehud Lehrer – Partially speciﬁed probabilities – p. 15/?
Relation to the Multiple Prior Model (Gilboa-Schmeidler 1989)

• From duality: estimating the expectation of a random
variable by members of Y is equivalent to minimizing wrt to
P
probability distributions that agree with I on Y
P,
• Thus, (I Y) induces a set of probability distributions that
are consistent with the information available. DM minimizes
over these distributions
• This set is an afﬁne space of distributions (intersection of
an afﬁne space with the simplex of distributions)
• The PSP model is strictly between A-A and the Multiple
Prior model
• The PSP model is information-based: the set of priors is
determined by the information structure and the real
distribution

Ehud Lehrer – Partially speciﬁed probabilities – p. 15/?
Relation to the Multiple Prior Model (Gilboa-Schmeidler 1989)

• From duality: estimating the expectation of a random
variable by members of Y is equivalent to minimizing wrt to
P
probability distributions that agree with I on Y
P,
• Thus, (I Y) induces a set of probability distributions that
are consistent with the information available. DM minimizes
over these distributions
• This set is an afﬁne space of distributions (intersection of
an afﬁne space with the simplex of distributions)
• The PSP model is strictly between A-A and the Multiple
Prior model
• The PSP model is information-based: the set of priors is
determined by the information structure and the real
distribution
• This fact is essential when considering interactive models.
Players play actual strategies. The beliefs are determined
by the information player get about them      Ehud Lehrer – Partially speciﬁed probabilities – p. 15/?
Non-cooperative games

Ehud Lehrer – Partially speciﬁed probabilities – p. 16/?
Non-cooperative games
• A game consists of:

Ehud Lehrer – Partially speciﬁed probabilities – p. 16/?
Non-cooperative games
• A game consists of:

• N – a ﬁnite set of players

Ehud Lehrer – Partially speciﬁed probabilities – p. 16/?
Non-cooperative games
• A game consists of:

• N – a ﬁnite set of players

• Ai – A ﬁnite set of player i’s actions

Ehud Lehrer – Partially speciﬁed probabilities – p. 16/?
Non-cooperative games
• A game consists of:

• N – a ﬁnite set of players

• Ai – A ﬁnite set of player i’s actions

• ui – player i’s utility function.

R
ui : ×Ai → I

Ehud Lehrer – Partially speciﬁed probabilities – p. 16/?
Partially-speciﬁed equilibrium

Ehud Lehrer – Partially speciﬁed probabilities – p. 17/?
Partially-speciﬁed equilibrium
• Each player plays a pure or a mixed strategy

Ehud Lehrer – Partially speciﬁed probabilities – p. 17/?
Partially-speciﬁed equilibrium
• Each player plays a pure or a mixed strategy

• Each player obtains partial information about other
players’ strategies

Ehud Lehrer – Partially speciﬁed probabilities – p. 17/?
Partially-speciﬁed equilibrium
• Each player plays a pure or a mixed strategy

• Each player obtains partial information about other
players’ strategies

• Each player maximizes her payoff against the worst
(uncoordinated/independent) strategy consistent with her
information

Ehud Lehrer – Partially speciﬁed probabilities – p. 17/?
Example

Ehud Lehrer – Partially speciﬁed probabilities – p. 18/?
Example
• Consider the coordination game
L     M     R
T   3,3   0,0   0,0
C   0,0   2,2   0,0
B   0,0   0,0   1,1

Ehud Lehrer – Partially speciﬁed probabilities – p. 18/?
Example
• Consider the coordination game
L    M     R
T   3,3   0,0   0,0
C    0,0   2,2   0,0
B   0,0   0,0   1,1

3             3
• There are three equilibria: a. p = ( 2 , 5 , 0), q = ( 2 , 5 , 0); b.
5               5
2 3 6
p = (0, 0, 1), q = (0, 0, 1); and c. p = ( 11 , 11 , 11 ),
2 3 6
q = ( 11 , 11 , 11 ).

Ehud Lehrer – Partially speciﬁed probabilities – p. 18/?
Best response

Ehud Lehrer – Partially speciﬁed probabilities – p. 19/?
Best response
• Suppose that player i is informed only of the variables
over A−i in Yi

Ehud Lehrer – Partially speciﬁed probabilities – p. 19/?
Best response
• Suppose that player i is informed only of the variables
over A−i in Yi

• p1 is a best response to p2 (w.r.t. Y1 ) if p1 maximizes
I (p2 ,Y1 ) ( payoff of player 1)
E

Ehud Lehrer – Partially speciﬁed probabilities – p. 19/?
Best response
• Suppose that player i is informed only of the variables
over A−i in Yi

• p1 is a best response to p2 (w.r.t. Y1 ) if p1 maximizes
I (p2 ,Y1 ) ( payoff of player 1)
E

• Deﬁnition: (p1 , p2 ) is partially speciﬁed probability w.r.t. to
the information Y1 and Y2 if pi is a best response to p−i
(w.r.t Yi )

Ehud Lehrer – Partially speciﬁed probabilities – p. 19/?
Best response
• Suppose that player i is informed only of the variables
over A−i in Yi

• p1 is a best response to p2 (w.r.t. Y1 ) if p1 maximizes
I (p2 ,Y1 ) ( payoff of player 1)
E

• Deﬁnition: (p1 , p2 ) is partially speciﬁed probability w.r.t. to
the information Y1 and Y2 if pi is a best response to p−i
(w.r.t Yi )

• In n-player games: pi is a best response to the
independent p−i (w.r.t Yi )

Ehud Lehrer – Partially speciﬁed probabilities – p. 19/?
Information-based

Ehud Lehrer – Partially speciﬁed probabilities – p. 20/?
Information-based
• The notion of partially-speciﬁed equilibrium is
information-based

Ehud Lehrer – Partially speciﬁed probabilities – p. 20/?
Information-based
• The notion of partially-speciﬁed equilibrium is
information-based

• The information structure, namely, the information
available to each player, determines the set of equilibria

Ehud Lehrer – Partially speciﬁed probabilities – p. 20/?
Information-based
• The notion of partially-speciﬁed equilibrium is
information-based

• The information structure, namely, the information
available to each player, determines the set of equilibria

• No player has a prior belief about other players’
strategies. The information structure is exogenous

Ehud Lehrer – Partially speciﬁed probabilities – p. 20/?
Information-based
• The notion of partially-speciﬁed equilibrium is
information-based

• The information structure, namely, the information
available to each player, determines the set of equilibria

• No player has a prior belief about other players’
strategies. The information structure is exogenous

• The belief a player has about other players is determined
by the actual strategies, as well as by the partial information

Ehud Lehrer – Partially speciﬁed probabilities – p. 20/?
PS correlated equilibrium (PSCE)

Ehud Lehrer – Partially speciﬁed probabilities – p. 21/?
PS correlated equilibrium (PSCE)
• A mediator select (p1 , ..., pn ) according to distribution Q

Ehud Lehrer – Partially speciﬁed probabilities – p. 21/?
PS correlated equilibrium (PSCE)
• A mediator select (p1 , ..., pn ) according to distribution Q

• Players don’t know Q

Ehud Lehrer – Partially speciﬁed probabilities – p. 21/?
PS correlated equilibrium (PSCE)
• A mediator select (p1 , ..., pn ) according to distribution Q

• Players don’t know Q

• The player gets a recommendation to play pi and a
partially-speciﬁed probability about Q(p−i |pi )

Ehud Lehrer – Partially speciﬁed probabilities – p. 21/?
PS correlated equilibrium (PSCE)
• A mediator select (p1 , ..., pn ) according to distribution Q

• Players don’t know Q

• The player gets a recommendation to play pi and a
partially-speciﬁed probability about Q(p−i |pi )

• And he plays a best response

Ehud Lehrer – Partially speciﬁed probabilities – p. 21/?
Learning to play PSCE
• Recall the example with the noisy signals

Ehud Lehrer – Partially speciﬁed probabilities – p. 22/?
Learning to play PSCE

• The game is played repeatedly

Ehud Lehrer – Partially speciﬁed probabilities – p. 22/?
Learning to play PSCE

• The game is played repeatedly

• Players observe noisy signals of past actions

Ehud Lehrer – Partially speciﬁed probabilities – p. 22/?
Learning to play PSCE

• The game is played repeatedly

• Players observe noisy signals of past actions

• Each player plays a conditional regret-free strategy (“I
have no regret for playing p because this is the best I could
do in response to the worst strategy of the others that is
consistent with the signals I received during the times I was
playing p.")

Ehud Lehrer – Partially speciﬁed probabilities – p. 22/?
Learning to play PSCE

• The game is played repeatedly

• Players observe noisy signals of past actions

• Each player plays a conditional regret-free strategy (“I
have no regret for playing p because this is the best I could
do in response to the worst strategy of the others that is
consistent with the signals I received during the times I was
playing p.")

•A recent paper with Eilon Solan (
“Learning to play partially-speciﬁed equilibrium") shows that
conditional regret-free strategies exist and the empirical
frequency of the mixed strategies played converges to PSCE

Ehud Lehrer – Partially speciﬁed probabilities – p. 22/?
A-A (vN-M)                                   Fully Bayesian

Partially-specified probabilities
(probability of events in sub-algebra)   Partially Bayesian

Partially-specified probabilities
Partially
(expectation of random variables)
Bayesian
Choquet EUM
(Schmeidler 1989)

Multiple prior
(Gilboa-Schmeidler 1989)

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