Costly Self-Control and Limited Willpower

Transcription

1 Costly Self-Control and Limited Willpower Simon Grant, Sung-Lin Hsieh and Meng-Yu Liang NASMES, June

2 Temptation and Self-control 1 / 13

3 Temptation and Self-control Strotz (1956) and O Donoghue & Rabin (1999) model intra-personal conflict as a game between current self (in cold state) and future self (in hot state). But growing body of evidence from neuroimaging supports hypothesis that automatic and deliberative modes of thought may coexist and jointly determine behavior. E.g. survey of Monterosso & Luo (2010). Gul & Pesendorfer (2001) [Hereafter GP] Suggest preferences of current self in cold state more sensibly defined over menus ( budget sets ). See explicitly: 1. any preference / desire for commitment 2. any cost associated with self-control 2 / 13

4 GP s model of costly self-control GP s axioms imply over menus in cold state represented by function: U (A) = max[u (x) + v (x)] x A }{{} Compromise utility (maximized in hot state) max y A v (y) Notice for any singleton menu {x}: U({x}) = u(x) + v(x) v(x) = u(x). So, re-arranging ( ) U(A) = max u(x) [maxv(y) v(x)] x A }{{} y A Normative utility }{{} (utility) cost of self-control 3 / 13

5 GP s model of costly self-control GP s axioms imply over menus in cold state represented by function: U (A) = max[u (x) + v (x)] x A }{{} Compromise utility (maximized in hot state) max y A v (y) Notice for any singleton menu {x}: U({x}) = u(x) + v(x) v(x) = u(x). So, re-arranging ( ) U(A) = max u(x) [maxv(y) v(x)] x A }{{} y A Normative utility }{{} (utility) cost of self-control 3 / 13

6 (u( ),v( )) utility pairs in GP s framework 4 / 13

7 Addiction Elster & Skog (1999) note that for something to be considered addictive requires an inability of the DM to resist choosing it in hot state. DM, addicted in this sense, may be: consonant, i.e., one who willingly indulges in his temptation. dissonant, i.e., one who wishes to resist temptation (albeit in vain). Dissonant addict evokes St Paul s sentiment of a wretched man : I do not do the good I want, but the evil I do not want I do. 5 / 13

8 Our Contribution - costly self-control with limited willpower axiomatize preferences over menus that admit representation: [ ( )] U(A) =max u(x) max v(y) v(x) x A y A s.t. maxv(y) v(x) w ( stock of willpower) y A e.g. Suppose However, we have u(0) v(x ) v (0) > u(x ). }{{} =0 U({0,x }) = U({x }) = u(x ), because v(x ) > w! Insufficient willpower to resist tempting cup-of-coffee. 6 / 13

9 Our Contribution - costly self-control with limited willpower axiomatize preferences over menus that admit representation: [ ( )] U(A) =max u(x) max v(y) v(x) x A y A s.t. maxv(y) v(x) w ( stock of willpower) y A e.g. Suppose However, we have u(0) v(x ) v (0) > u(x ). }{{} =0 U({0,x }) = U({x }) = u(x ), because v(x ) > w! Insufficient willpower to resist tempting cup-of-coffee. 6 / 13

10 Our Contribution - costly self-control with limited willpower axiomatize preferences over menus that admit representation: [ ( )] U(A) =max u(x) max v(y) v(x) x A y A s.t. maxv(y) v(x) w ( stock of willpower) y A e.g. Suppose However, we have u(0) v(x ) v (0) > u(x ). }{{} =0 U({0,x }) = U({x }) = u(x ), because v(x ) > w! Insufficient willpower to resist tempting cup-of-coffee. 6 / 13

11 Adding the Willpower Constraint 7 / 13

12 When does one DM exhibits more self-control than another? Related Literature 8 / 13

13 Q. How do we obtain representation? Step 1: Key assumption: set betweenness A B implies A A B B Key Lemma: If U( ) represents that satisfies set betweenness, key lemma :: then for any finite set A : U(A) = max min x A y A U({x,y}) = min y A max x A U({x,y}) 9 / 13

14 Q. How do we obtain representation? Step 1: Key assumption: set betweenness A B implies A A B B Key Lemma: If U( ) represents that satisfies set betweenness, key lemma :: then for any finite set A : U(A) = max min x A y A U({x,y}) = min y A max x A U({x,y}) 9 / 13

15 Q. How do we obtain representation? Step 1: Key assumption: set betweenness A B implies A A B B Key Lemma: If U( ) represents that satisfies set betweenness, key lemma :: then for any finite set A : U(A) = max min x A y A U({x,y}) = min y A max x A U({x,y}) 9 / 13

16 Q. How do we obtain representation? Step 2: Another key assumption for GP is independence A B implies αa + (1 α)c αb + (1 α)c for any α (0,1) Note: αa + (1 α)c = {αx + (1 α)y : x A,y B} But limited willpower involves violations of independence. Let b, y and i respectively denote broccoli, yoghurt and icecream, and suppose we have: {[b]} {[y]} {[y],[i ]} {[i]} {[b] [i ]} Then for sufficiently small α (that is, α(v (i) v (b)) w), U (α{[b],[i ]} + (1 α){[i ]}) = α(u (b) + v (b) v (i)) + (1 α)u (i ) > α(u (y) + v (y) v (i )) + (1 α)u (i ) = U (α{[y],[i ]} + (1 α){[i ]}). 10 / 13

17 Q. How do we obtain representation? Step 2: Another key assumption for GP is independence A B implies αa + (1 α)c αb + (1 α)c for any α (0,1) Note: αa + (1 α)c = {αx + (1 α)y : x A,y B} But limited willpower involves violations of independence. Let b, y and i respectively denote broccoli, yoghurt and icecream, and suppose we have: {[b]} {[y]} {[y],[i ]} {[i]} {[b] [i ]} Then for sufficiently small α (that is, α(v (i) v (b)) w), U (α{[b],[i ]} + (1 α){[i ]}) = α(u (b) + v (b) v (i)) + (1 α)u (i ) > α(u (y) + v (y) v (i )) + (1 α)u (i ) = U (α{[y],[i ]} + (1 α){[i ]}). 10 / 13

18 Q. How do we obtain representation? Step 2: Another key assumption for GP is independence A B implies αa + (1 α)c αb + (1 α)c for any α (0,1) Note: αa + (1 α)c = {αx + (1 α)y : x A,y B} But limited willpower involves violations of independence. Let b, y and i respectively denote broccoli, yoghurt and icecream, and suppose we have: {[b]} {[y]} {[y],[i ]} {[i]} {[b] [i ]} Then for sufficiently small α (that is, α(v (i) v (b)) w), U (α{[b],[i ]} + (1 α){[i ]}) = α(u (b) + v (b) v (i)) + (1 α)u (i ) > α(u (y) + v (y) v (i )) + (1 α)u (i ) = U (α{[y],[i ]} + (1 α){[i ]}). 10 / 13

19 Q. How do we obtain representation? Step 2: Another key assumption for GP is independence A B implies αa + (1 α)c αb + (1 α)c for any α (0,1) Note: αa + (1 α)c = {αx + (1 α)y : x A,y B} But limited willpower involves violations of independence. Let b, y and i respectively denote broccoli, yoghurt and icecream, and suppose we have: {[b]} {[y]} {[y],[i ]} {[i]} {[b] [i ]} Then for sufficiently small α (that is, α(v (i) v (b)) w), U (α{[b],[i ]} + (1 α){[i ]}) = α(u (b) + v (b) v (i)) + (1 α)u (i ) > α(u (y) + v (y) v (i )) + (1 α)u (i ) = U (α{[y],[i ]} + (1 α){[i ]}). 10 / 13

20 Q. How do we obtain representation? Restrict independence to singleton menus & any two-element menu so long as any failure of self-control is not attributable to lack of willpower. 2 nd innovation: show when one can infer exercise of self-control exhausts DM s stock of willpower. motivates our new axiom (Axiom 5) which guarantees stock of willpower is menu-independent. 11 / 13

21 Q. How do we obtain representation? Restrict independence to singleton menus & any two-element menu so long as any failure of self-control is not attributable to lack of willpower. 2 nd innovation: show when one can infer exercise of self-control exhausts DM s stock of willpower. motivates our new axiom (Axiom 5) which guarantees stock of willpower is menu-independent. 11 / 13

22 Related Literature Gul & Pesendorfer (2004), (2007) Chatterjee & Krishna (2009), Dekel et al (2009), Kopylov (2009) and Stovall (2010) generalize GP s model by relaxing Set Betweenness while maintaining Independence model agents who are uncertain about temptation and/or face multiple temptations Noor & Takeoka (2010) Uphill self-control model generalize GP s model by maintaining Set Betweenness and relaxing Independence Convex self-control representation [ ( )] U(A) = max u(x) φ max v(y) v(x) x A y A Masatlioglu et al (2014) Revealed Willpower U(A) =max x A u(x) s.t. maxv(y) v(x) w y A 12 / 13

23 Conclusion Adapted Gul and Pesendorfer s framework to allow for temptation with costly self-control and limited willpower. Given strictly positive willpower w > 0 discontinuity in preferences over menus arises from insufficient willpower to resist tempting alternative. Measures to increase normative utility of unpleasant task will be to no avail ( pushing on a string ). Rather should endeavor to increase temptation utility of task. Strength of individual s willpower may be key indicator of ability to make important trade-offs between small immediate gratifications and large long-term rewards. Our contribution is to provide revealed preference approach to determine strength of willpower. 13 / 13

24 Conclusion Adapted Gul and Pesendorfer s framework to allow for temptation with costly self-control and limited willpower. Given strictly positive willpower w > 0 discontinuity in preferences over menus arises from insufficient willpower to resist tempting alternative. Measures to increase normative utility of unpleasant task will be to no avail ( pushing on a string ). Rather should endeavor to increase temptation utility of task. Strength of individual s willpower may be key indicator of ability to make important trade-offs between small immediate gratifications and large long-term rewards. Our contribution is to provide revealed preference approach to determine strength of willpower. 13 / 13

25 Conclusion Adapted Gul and Pesendorfer s framework to allow for temptation with costly self-control and limited willpower. Given strictly positive willpower w > 0 discontinuity in preferences over menus arises from insufficient willpower to resist tempting alternative. Measures to increase normative utility of unpleasant task will be to no avail ( pushing on a string ). Rather should endeavor to increase temptation utility of task. Strength of individual s willpower may be key indicator of ability to make important trade-offs between small immediate gratifications and large long-term rewards. Our contribution is to provide revealed preference approach to determine strength of willpower. 13 / 13

26 Conclusion Adapted Gul and Pesendorfer s framework to allow for temptation with costly self-control and limited willpower. Given strictly positive willpower w > 0 discontinuity in preferences over menus arises from insufficient willpower to resist tempting alternative. Measures to increase normative utility of unpleasant task will be to no avail ( pushing on a string ). Rather should endeavor to increase temptation utility of task. Strength of individual s willpower may be key indicator of ability to make important trade-offs between small immediate gratifications and large long-term rewards. Our contribution is to provide revealed preference approach to determine strength of willpower. 13 / 13

27 Go back Restrict independence

28 Restrict Axiom 5: menu-independent willpower

29 Restrict Axiom 5: menu-independent willpower

30 Axiom 5: menu-independent willpower Axiom 5 requires { 1 2 a x, 1 2 a y} { 1 2 x a, 1 2 x b} Go back

31 Axiom 5: menu-independent willpower Axiom 5 requires { 1 2 a x, 1 2 a y} { 1 2 x a, 1 2 x b} Go back