Ordinal dominance and risk aversion Bulat Gafarov Bruno Salcedo - - PowerPoint PPT Presentation

Ordinal dominance and risk aversion

Bulat Gafarov Bruno Salcedo

Pennsylvania State University 25th International Game Theory Conference Stony Brook – Summer 2014

Introduction

Motivation

• Empirical content of rationalizability?

1 / 28

Motivation

• Empirical content of rationalizability?

→ Iterated dominance by pure or mixed strategies (M-dominance) → Depends on cardinal information

1 / 28

Motivation

• Empirical content of rationalizability?

→ Iterated dominance by pure or mixed strategies (M-dominance) → Depends on cardinal information

Given ordinal preferences, there are cardinal specifications for which rationalizability coincides with iterated domi- nance by pure strategies (P-dominance)

• Similar results in Ledyard (1986), Börgers (1993), Epstein (1997), Lo

(2000), Bonanno (2008) and Chen and Luo (2012)

1 / 28

Motivation

• Empirical content of rationalizability?

→ Iterated dominance by pure or mixed strategies (M-dominance) → Depends on cardinal information

Given ordinal preferences, there are cardinal specifications for which rationalizability coincides with iterated domi- nance by pure strategies (P-dominance)

• Similar results in Ledyard (1986), Börgers (1993), Epstein (1997), Lo

(2000), Bonanno (2008) and Chen and Luo (2012), Weinstein (2014)

1 / 28

Betting on the World Cup winner

Argentina Germany Bet for Argentina 2 Bet for Germany 2 Do not bet γ γ

2 / 28

Betting on the World Cup winner

Argentina Germany Bet for Argentina 2 Bet for Germany 2 Do not bet γ γ

• Same ordinal ranking as long as 0 < γ < 2
• γ measures risk attitude (concavity)

2 / 28

Betting on the World Cup winner

Argentina Germany Bet for Argentina 2 Bet for Germany 2 Do not bet γ γ

• Same ordinal ranking as long as 0 < γ < 2
• γ measures risk attitude (concavity)
• There is no P-dominance
• Not betting is M-dominated if and only if γ < 1

2 / 28

Quick glance of results

• 1. When are pure and mixed dominance equivalent?

→ When the agent is sufficiently timid (risk averse) → Because mixed strategies introduce risk of their own

3 / 28

Quick glance of results

• 1. When are pure and mixed dominance equivalent?

→ When the agent is sufficiently timid (risk averse) → Because mixed strategies introduce risk of their own

• 2. Given ordinal preferences, when is there a compatible utility function

generating dominance equivalence?

→ Strong compatibility – in all finite environments → Weak compatibility – in some infinite environments → Because risk aversion is a cardinal property

3 / 28

Quick glance of results

• 1. When are pure and mixed dominance equivalent?

→ When the agent is sufficiently timid (risk averse) → Because mixed strategies introduce risk of their own

• 2. Given ordinal preferences, when is there a compatible utility function

generating dominance equivalence?

→ Strong compatibility – in all finite environments → Weak compatibility – in some infinite environments → Because risk aversion is a cardinal property

• 3. What properties does this utility function has?

→ Level of risk aversion grows unboundedly with the size of the game → Decision rules approximate minimax in some cases

3 / 28

Environment

Ordinal decision problem

• X = {x, y, . . .}

states

• A = {a, b, . . .}

actions

• ≻ strict preferences over A × X
• ≻x denotes preferences over actions contingent on state x

4 / 28

Ordinal decision problem

• X = {x, y, . . .}

states

• A = {a, b, . . .}

actions

• ≻ strict preferences over A × X
• ≻x denotes preferences over actions contingent on state x
• Assumption – The set {c ∈ A | a ≻x c ≻x b} is finite for every x ∈ X

and a, b ∈ A

4 / 28

Cardinal preferences

• u ∈ RA×X is compatible with the environment if

a ≻x b ⇔ u(a, x) > u(b, x)

5 / 28

Cardinal preferences

• u ∈ RA×X is compatible with the environment if

a ≻x b ⇔ u(a, x) > u(b, x)

• u ∈ RA×X is strongly compatible with the environment if

(a, x) ≻ (b, y) ⇔ u(a, x) > u(b, y)

5 / 28

Cardinal preferences

• u ∈ RA×X is compatible with the environment if

a ≻x b ⇔ u(a, x) > u(b, x)

• u ∈ RA×X is strongly compatible with the environment if

(a, x) ≻ (b, y) ⇔ u(a, x) > u(b, y)

• Expected utility from mixed actions α ∈ ∆(A) given beliefs µ ∈ ∆(X)

U(α, µ) =

• x∈X
• a∈A

µ(x)α(a)u

• a, x
• 5 / 28

Dominance relations

• Pure dominance P ⊆ A × A

aPb ⇔ a ≻x b for all x ∈ X

6 / 28

Dominance relations

• Pure dominance P ⊆ A × A

aPb ⇔ a ≻x b for all x ∈ X

• Mixed dominance M ⊆ ∆(A) × A

αMb ⇔ U(α, x) > u(a, x) for all x ∈ X

6 / 28

Dominance relations

• Pure dominance P ⊆ A × A

aPb ⇔ a ≻x b for all x ∈ X

• Mixed dominance M ⊆ ∆(A) × A

αMb ⇔ U(α, x) > u(a, x) for all x ∈ X

• Remarks

→ P is ordinal, only depends on (X, A, ≻) → M is cardinal, also depends on u → If an action is P-dominated, then it is also M-dominated

6 / 28

Dominance equivalence

Pure and mixed dominance are equivalent if and only if αMb implies that aPb for some a ∈ A such that α(a) > 0.

• Which utility functions generate dominance equivalence?
• When does there exist a compatible or strongly compatible utility

function generating dominance equivalence?

7 / 28

b

u(a)

b u(c) b

u(b)

b

u θa + (1 − θ)c X = {x, y} A = {a, b, c} a ≻x b ≻x c c ≻y b ≻y a P = ∅

b b b

8 / 28

b

u(a)

b u(c) b u(b)

∆− ∆+ X = {x, y} A = {a, b, c} a ≻x b ≻x c c ≻y b ≻y a P = ∅

b b b

9 / 28

b

u(a)

b u(c) b u(b)

µ µ X = {x, y} A = {a, b, c} a ≻x b ≻x c c ≻y b ≻y a P = ∅

b b b

10 / 28

Timidity

Timidity

• Risk aversion in discrete settings ≈ Decreasing differences

11 / 28

Timidity

• Risk aversion in discrete settings ≈ Decreasing differences
• Next best thing u−(a, x) = max
• u(b, x)
• a ≻x b
• Best possible payoff ¯

u(x) = sup

• u(a, x)
• a ∈ A
• τu(a, x) = u(a, x) − u−(a, x)

¯ u(x) − u(a, x)

11 / 28

b b b

m − 1 m m + 1 u−(a, x) u(a, x) u+(a, x) 1 1 − ρu(a, x) = u(a, x) − u−(a, x) u+(a, x) − u(a, x) Arrow-Pratt – local gain (u+ − u) is small compared to local loss (u − u−)

12 / 28

b b b b b b b

m − 1 m m + 1 u−(a, x) u(a, x) ¯ u(x) τu(a, x) = u(a, x) − u−(a, x) ¯ u(x) − u(a, x) Timidity – global gain (¯ u − u) is small compared to local loss (u − u−)

13 / 28

Timidity and risk aversion

• CARA preferences over rank n(a, x) have constant timidity τu = K

u(a, x) = − exp

• − log(K) · n(a, x)
• 14 / 28

Timidity and risk aversion

• CARA preferences over rank n(a, x) have constant timidity τu = K

u(a, x) = − exp

• − log(K) · n(a, x)
• Proposition – If the set of mixed actions that are preferred

to a given u and x is contained in the set of mixed actions that are preferred to a given v and x, then u is more timid than v at (a, x)

14 / 28

Timidity and dominance

• Let Wx(a) be those actions which are worse than a given x

Wx(a) =

• b ∈ A
• a ≻x b
• Lemma – Given an action a and a pure or mixed action

α, if there exists a state x such that

• τu(a, x) + 1
• · α
• Wx(a)
• ≥ 1,

then a is not M-dominated by α given u.

15 / 28

Timidity and dominance

• Let Wx(a) be those actions which are worse than a given x

Wx(a) =

• b ∈ A
• a ≻x b
• Lemma – Given an action a and a pure or mixed action

α, if there exists a state x such that

• τu(a, x) + 1
• · α
• Wx(a)
• ≥ 1,

then a is not M-dominated by α given u.

• The condition is tight
• Rest of the talk – guarantee existence of such states

15 / 28

Results

Results

Strong compatibility in finite environments

Finite environments

• Let K = min{A, X}

Proposition – If K is finite and τu(a, x) ≥ K − 1 for all a and x, then u generates dominance equivalence

16 / 28

Finite environments

• Let K = min{A, X}

Proposition – If K is finite and τu(a, x) ≥ K − 1 for all a and x, then u generates dominance equivalence

• Sketch of proof:

→ Suppose towards a contradiction that a ∈ M(A) \ P(A)

16 / 28

Finite environments

• Let K = min{A, X}

Proposition – If K is finite and τu(a, x) ≥ K − 1 for all a and x, then u generates dominance equivalence

• Sketch of proof:

→ Suppose towards a contradiction that a ∈ M(A) \ P(A) → Caratheodory’s theorem ⇒ αMa for some α with supp(α) ≤ K

16 / 28

Finite environments

• Let K = min{A, X}

Proposition – If K is finite and τu(a, x) ≥ K − 1 for all a and x, then u generates dominance equivalence

• Sketch of proof:

→ Suppose towards a contradiction that a ∈ M(A) \ P(A) → Caratheodory’s theorem ⇒ αMa for some α with supp(α) ≤ K → Therefore there would exist b ∈ A such that α(b) ≥ 1/K

16 / 28

Finite environments

• Let K = min{A, X}

Proposition – If K is finite and τu(a, x) ≥ K − 1 for all a and x, then u generates dominance equivalence

• Sketch of proof:

→ Suppose towards a contradiction that a ∈ M(A) \ P(A) → Caratheodory’s theorem ⇒ αMa for some α with supp(α) ≤ K → Therefore there would exist b ∈ A such that α(b) ≥ 1/K → Since ¬(bPa), there exists x such that a ≻x b

16 / 28

Finite environments

• Let K = min{A, X}

Proposition – If K is finite and τu(a, x) ≥ K − 1 for all a and x, then u generates dominance equivalence

• Sketch of proof:

→ Suppose towards a contradiction that a ∈ M(A) \ P(A) → Caratheodory’s theorem ⇒ αMa for some α with supp(α) ≤ K → Therefore there would exist b ∈ A such that α(b) ≥ 1/K → Since ¬(bPa), there exists x such that a ≻x b → Therefore α(Wx(a)) > 1/K and thus (τu(a, x) + 1) · α(Wx(a)) ≥ 1

16 / 28

Finite environments

• Let K = min{A, X}

Proposition – If K is finite and τu(a, x) ≥ K − 1 for all a and x, then u generates dominance equivalence

• Sketch of proof:

→ Suppose towards a contradiction that a ∈ M(A) \ P(A) → Caratheodory’s theorem ⇒ αMa for some α with supp(α) ≤ K → Therefore there would exist b ∈ A such that α(b) ≥ 1/K → Since ¬(bPa), there exists x such that a ≻x b → Therefore α(Wx(a)) > 1/K and thus (τu(a, x) + 1) · α(Wx(a)) ≥ 1 → And, by the lemma, a ∈ M(A)

• 16 / 28

Finite environments

• Suppose A and X are finite
• Let u∗ be the CARA function:

u∗(a, x) = − exp

• − log(K) · n(a, x)
• where

n(a, x) =

• (b, y) ∈ A × X
• (a, x) ≻ (b, y)
• 17 / 28

Finite environments

• Suppose A and X are finite
• Let u∗ be the CARA function:

u∗(a, x) = − exp

• − log(K) · n(a, x)
• where

n(a, x) =

• (b, y) ∈ A × X
• (a, x) ≻ (b, y)
• Proposition – If A and X are finite, then u∗ is strongly

compatible and generates dominance equivalence

17 / 28

n(a, x) n(a, y)

b b b b b b b b b b b b b b b

X = {x, y} A = 15 Rank is strongly compatible . . .

b b b b b b b b b b b b b b b

18 / 28

n(a, x) n(a, y)

b b b b b b b b b b b b b b b

X = {x, y} A = 15 . . . but it doesn’t guarantee dominance equivalence

b b b b b b b b b b b b b b b

19 / 28

u∗(a, x) u∗(a, y)

b b b b b b b b b bbb b b b

X = {x, y} A = 15 u∗(a, x) = 1 − 2−n(a,x) Logarithmic rescaling solves the problem

b b b b b b b b b bbb b b b

20 / 28

Results

Weak compatibility in infinite environments

Example: countable environment

• X = N
• A = {ax | x ∈ X} ∪ {a0}
• Preferences are strongly represented by

u(a, x) =      γ if a = a0 if a = ax 1

• therwise

, γ ∈ (0, 1)

21 / 28

Example: countable environment

• X = N
• A = {ax | x ∈ X} ∪ {a0}
• Preferences are strongly represented by

u(a, x) =      γ if a = a0 if a = ax 1

• therwise

, γ ∈ (0, 1)

• a0 is not P-dominated but it is M-dominated

U(a0, µ) = γ U(ax, µ) = 1 − µ(x) U(a0, µ) ≥ U(ax, µ) ⇒ µ(x) ≥ 1 − γ > 0

21 / 28

Countable environments

• In infinite environments τu ≥ K − 1 is not possible
• But we can have lim τu(a, xn) = ∞ along a sequence (xn)

22 / 28

Countable environments

• In infinite environments τu ≥ K − 1 is not possible
• But we can have lim τu(a, xn) = ∞ along a sequence (xn)

Proposition – If X is countable and for every a

• x∈X

1 1 + τu(a, x) ≤ 1 then u generates dominance equivalence

• Idea of the proof – if a is not P-dominated then
• x∈X

α

• Wx(a)
• ≥ 1 ≥
• x∈X

1 1 + τu(a, x)

22 / 28

Countable environments

• In any countable environment there exist

→ An enumeration of states h ∈ NX → A compatible utility function taking integer values m ∈ ZA×X

23 / 28

Countable environments

• In any countable environment there exist

→ An enumeration of states h ∈ NX → A compatible utility function taking integer values m ∈ ZA×X

• Let u∗∗ be the state-wise CARA utility function

u∗∗(a, x) = − exp

• − h(x) · m(a, x)
• 23 / 28

Countable environments

• In any countable environment there exist

→ An enumeration of states h ∈ NX → A compatible utility function taking integer values m ∈ ZA×X

• Let u∗∗ be the state-wise CARA utility function

u∗∗(a, x) = − exp

• − h(x) · m(a, x)
• One can show that
• x∈X

1 1 + τu∗∗(a, x) ≤

• k∈N

1 ek < 1

23 / 28

Countable environments

• In any countable environment there exist

→ An enumeration of states h ∈ NX → A compatible utility function taking integer values m ∈ ZA×X

• Let u∗∗ be the state-wise CARA utility function

u∗∗(a, x) = − exp

• − h(x) · m(a, x)
• One can show that
• x∈X

1 1 + τu∗∗(a, x) ≤

• k∈N

1 ek < 1 Proposition – If A and X are countable, then u∗∗ is com- patible and generates dominance equivalence

23 / 28

Uncountable environments

Similar results apply to uncountable state spaces

• 1. There exists an absolutely integrable function f : X → R++
• 2. There exists some δ > 0 such that if a is preferred to b in at least one

state, then it is preferred to b in a set of states of measure at least δ

24 / 28

Results

Indifference

Example: indifference

• Let X = {x, y}, A = {a, b, c} and:

a ≻x b ∼x c c ≻y b ∼y a

25 / 28

Example: indifference

• Let X = {x, y}, A = {a, b, c} and:

a ≻x b ∼x c c ≻y b ∼y a

• Given any utility function u and any beliefs µ with µ(x) > 0

U(a, µ) = µ(x)u(a, x) + µ(y)u(a, y) > µ(x)u(b, x) + µ(y)u(b, y) = U(b, µ)

25 / 28

Example: indifference

• Let X = {x, y}, A = {a, b, c} and:

a ≻x b ∼x c c ≻y b ∼y a

• Given any utility function u and any beliefs µ with µ(x) > 0

U(a, µ) = µ(x)u(a, x) + µ(y)u(a, y) > µ(x)u(b, x) + µ(y)u(b, y) = U(b, µ)

• Hence b is not P-dominated and yet it cannot be rational

25 / 28

Indifference and Börgers’ dominance

• With indifference, the right notion of pure dominance lies between weak

and strict dominance (Börgers, 1993)

• a is D-dominated in B if for all Y ⊆ X there exits b ∈ B such that:

b x a for all x ∈ Y b ≻x a for at least one x ∈ Y

26 / 28

Indifference and Börgers’ dominance

• With indifference, the right notion of pure dominance lies between weak

and strict dominance (Börgers, 1993)

• a is D-dominated in B if for all Y ⊆ X there exits b ∈ B such that:

b x a for all x ∈ Y b ≻x a for at least one x ∈ Y

• M-dominance and D-dominance are equivalent if for every B ⊆ A, an

action b is M-dominated in B if and only if it is D-dominated in B

• All our results apply

26 / 28

Closing remarks

Summary

• 1. When an agent is sufficiently timid, dominance by pure and mixed

actions coincide

• 2. When only ordinal preferences are known, rationalizability has no

additional restrictions beyond pure dominance

• 3. The required level or timidity grows linearly with the size of the

environment

27 / 28

Motivation

• Estimating risk aversion relies on strong structural assumptions
• Tightness of our conditions implies that the choice of potentially

M-dominated strategies reveals a lower bound on risk aversion

• Potential robust bounds for risk aversion

28 / 28

Thank you for your attention!

paper available at brunosalcedo.com contact me at bruno@psu.edu