Ordinal dominance and risk aversion Bulat Gafarov Bruno Salcedo Pennsylvania State University 25 th 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 0 Bet for Germany 0 2 Do not bet γ γ 2 / 28
Betting on the World Cup winner Argentina Germany Bet for Argentina 2 0 Bet for Germany 0 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 0 Bet for Germany 0 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 ∈ R A × X is compatible with the environment if a ≻ x b ⇔ u ( a , x ) > u ( b , x ) 5 / 28
Cardinal preferences • u ∈ R A × X is compatible with the environment if a ≻ x b ⇔ u ( a , x ) > u ( b , x ) • u ∈ R A × X is strongly compatible with the environment if ( a , x ) ≻ ( b , y ) ⇔ u ( a , x ) > u ( b , y ) 5 / 28
Cardinal preferences • u ∈ R A × X is compatible with the environment if a ≻ x b ⇔ u ( a , x ) > u ( b , x ) • u ∈ R A × 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 ) = µ ( x ) α ( a ) u a , x x ∈ X a ∈ A 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 α Mb ⇔ 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 α Mb ⇔ 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 α Mb 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 b b b b b u ( a ) u � θ a + (1 − θ ) c � u ( b ) b u ( c ) X = { x , y } A = { a , b , c } a ≻ x b ≻ x c c ≻ y b ≻ y a P = ∅ 8 / 28
b b b b u ( a ) ∆ − b u ( b ) ∆ + b u ( c ) X = { x , y } A = { a , b , c } a ≻ x b ≻ x c c ≻ y b ≻ y a P = ∅ 9 / 28
b b b b µ u ( a ) b u ( b ) µ b u ( c ) X = { x , y } A = { a , b , c } a ≻ x b ≻ x c c ≻ y b ≻ y a P = ∅ 10 / 28
Timidity
Timidity • Risk aversion in discrete settings ≈ Decreasing differences 11 / 28
Timidity • Risk aversion in discrete settings ≈ Decreasing differences � a ≻ x b � � � • Next best thing u − ( a , x ) = max u ( b , x ) � a ∈ A � � � • Best possible payoff ¯ u ( x ) = sup u ( a , x ) τ u ( a , x ) = u ( a , x ) − u − ( a , x ) ¯ u ( x ) − u ( a , x ) 11 / 28
b b b u + ( a , x ) u ( a , x ) u − ( a , x ) 1 − ρ u ( a , x ) = u ( a , x ) − u − ( a , x ) 1 u + ( a , x ) − u ( a , x ) m − 1 m + 1 m Arrow-Pratt – local gain ( u + − u ) is small compared to local loss ( u − u − ) 12 / 28
b b b b b b b u ( x ) ¯ u ( a , x ) u − ( a , x ) τ u ( a , x ) = u ( a , x ) − u − ( a , x ) u ( x ) − u ( a , x ) ¯ m − 1 m + 1 m 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 τ u = 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 τ u = 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 W x ( a ) be those actions which are worse than a given x � a ≻ x b � � � W x ( a ) = b ∈ A Lemma – Given an action a and a pure or mixed action α , if there exists a state x such that � � � � τ u ( a , x ) + 1 · α W x ( a ) ≥ 1 , then a is not M -dominated by α given u . 15 / 28
Timidity and dominance • Let W x ( a ) be those actions which are worse than a given x � a ≻ x b � � � W x ( a ) = b ∈ A Lemma – Given an action a and a pure or mixed action α , if there exists a state x such that � � � � τ u ( a , x ) + 1 · α W x ( 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 τ u ( 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 τ u ( 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 τ u ( 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 ⇒ α Ma for some α with � supp( α ) � ≤ K 16 / 28
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