Temptation and Self-control

Transcription

1 Temptation and Self-control Frank Gul & Wolfgang Pesendorfer Econometrica, 2001, 69(6),

2 1. Introduction In the morning, an agent want to decide what to eat at lunch, a vegetarian dish ( x ) or a hamburger ( y ); At lunch time, such an agent is craving for hamburger. In order to lessen such impacts, she may limit the options at lunch time; She may choose a vegetarian restaurant or if not possible, she will exercise self-control to choose vegetarian meal when facing the option of a hamburger.

3 x y Consumption point a vegetarian dish a hamburger? x y x, y T Morning Afternoon The singleton sets x & y : committed to choose only one of them; x, y : choosing between two elements, x & y.

4 No choice: x {y}; Alternative y is tempting if it is on the menu: x {x, y}; Succumbing to temptation: y {x, y}; Self-control at x, y : x, y {y}; Temptation preference with self-control: x {x, y} {y}; temptation self control

5 Preference presentation with self-control: U A max u x x A utility under commitment [max y A v y v x ] cost of self control u & v : von Neumann-Morgenstern utility functions over lotteries. u : agent s commitment ranking (the ranking of singleton set) over lotteries; v : agent s temptation ranking over lotteries(the urges in period 2); Agent chooses a lottery to maximize U A

6 Preference presentation without self-control: U A max x A u x Subject to, v x v y, for y A. The agent gives in to her temptation, i.e., choosing the lottery to maximize v, but evaluates these choices by using u.

7 Extended preference Combination of choice behavior in periods 1 and 2; For example (the self-control preference): U A, x u x + v x max y A v y In period 2, given the choice A in period 1, the relation induces a preference over lotteries x A to maximize U A, ; In period 1, the agent evaluates sets anticipating her choice in the second period and agent chooses A to max x A {u x + v x } max y A v y ;

8 Related literature Strotz (1955): preferences are dynamically inconsistent; benefits of commitment are from strategic behaviors; based on the situation where all selves are better off, welfare comparison are allowed. This paper: preferences are dynamically consistent; Set betweenness accommodates temptation and the desire for commitment to the dynamically consistent preferences; Temptation without self-control model has the same behavioral implications as Strotz (1995); When ex ante undesirable temptations are not allowed, the decision-maker is unambiguously better off.

9 Related literature Kreps (1979): This paper follows Kreps (1979) by studying the preferences over alternatives. Dekel et al, (2001): This paper follows Dekel et al, (2001) by modeling the set of alternatives as lotteries and utilizing the resulting linear structure by imposing von Neumann-Morgenstern axioms.

10 2. A model of self-control

11 2. A model of self-control Here αa + 1 α C {z = αx + 1 α y: x A&y C} Remark: Axiom 3 can be interpreted as: the occurrence of C should not interfere with the preference for A over B.

12 2. A model of self-control For the standard decision-maker: Axioms 1-3 are satisfied; His only interest is the best element of a set, i.e., A B A A B Utility presentation: U A max x A u x

13 2. A model of self-control The preference of decision-maker featured with temptation with self-control satisfies Axioms 1-4.

14 2. A model of self-control x A y z B (choice of B) (choice of A) (most tempting) A A B A B B

15 2. A model of self-control Remarks on Axiom 4: Axiom 4 is an implication of temptation, which is utility decreasing; Further assumptions: Lotteries can be ranked according to how tempting they are; Only the most tempting option available affects the agent s utility;

16 2. A model of self-control Violation of Set Betweenness (excluded by this paper): larger sets leading to greater temptation, i.e., A B A B Randomization of temptation (depending on the commitment ranking): A = x(0.5), y(0.5), B = z(0.5), w(0.5), A B = {w, x(0.5), y, z(0.5)} A B A, A B B, A A B, B A B

17 2. A model of self-control

18 2. A model of self-control Remarks: A singleton set presents the case where the agent commits in period 1 to a consumption choice in period 2; u presents the commitment ranking of lotteries and refer to u(x) as the commitment utility of choice x; v presents the temptation ranking of lotteries and v x max v y as the disutility of self-control; y A In period 2, the agent chooses x A to maximize u + v.

19 2. A model of self-control x A y z B (choice) (most tempting) A A B If y A, A A B A B B If x B, A B B

20 2. A model of self-control

21 2. A model of self-control Distinguishing commitment & self-control: x {x, y} {y} Commitment: choose x at period 1, the planning period; Self-control: if committing to x is not possible, choose x from {x, y} at period 2, the consumption period.

22 3. Preference with and without selfcontrol U A max x A u x + k[v x max y A v y ] When k is sufficiently large, temptation cannot be resisted. k w, strong continuity should be modified.

23 3. Preference with and without selfcontrol {x, y} & {x, y }: y & y are very close; Assume that u x > u(y), v x > v y & u x > u y, v y > v(x); When the agent cannot resist temptation, {x, y} x y {x, y }; y & y are very close, but their utilities are not! Hence, the notion of continuity should be weakened.

24 3. Preference with and without selfcontrol

25 3. Preference with and without selfcontrol

26 3. Preference with and without selfcontrol Set betweenness and no self-control A A B or B A B

27 3. Preference with and without selfcontrol Let (u, v) represent ; Preference for no commitment: u is constant or v is constant or u is a positive affine transformation of v Maximal preference for commitment: A { x } for x A or has a preference for commitment at A; Regular preference relation: It has some preference for commitment but does not have the maximal preference for commitment;

28 3. Preference with and without selfcontrol

29 4. Extended preferences, temptation and second period choice G {(A, x) A : x A}; Given the first period choice A, the relation induces a preference over lotteries x A; In the first period, the agent evaluates sets anticipating her choice in the second period. In the second period, the agent chooses x A; The induced first period preference: 1 : A 1 B x A s.t. A, x 1 B, y, for all y B

30 4. Extended preferences, temptation and second period choice For the self-control preference: U A, x u x + v x max v y y A In period 2, the agent chooses x A to maximize U A, ; In period 1, according to max the agent chooses A. x A {u x + v x } max y A v y,

31 4. Extended preferences, temptation and second period choice For the preference without self-control : m min x A u(x) In period 2, the agent chooses x A to lexicographically maximize first u and then v ; 1 is the overwhelming temptation preference (u, v).

32 4. Extended preferences, temptation and second period choice The extended preference representation is not unique.

33 4. Extended preferences, temptation and second period choice Theorem 6 indicates if the induced first period preference is a temptation preference, then, v y > v x y attemps x

34 4. Extended preferences, temptation and second period choice

35 5. Measures of preference for commitment and self-control

36 5. Measures of preference for commitment and self-control

37 6. Discussion Temptation may be the cause of a preference for commitment; This paper introduces self-control preferences and quantify the cost of self-control as a utility penalty; The paper distinguishes between self-control & commitment;

38 6. Discussion The model of temptation with self-control implies: an irrelevant alternative may worse off decision maker s welfare; decision makers will expend resources to remove tempting alternatives from the choice sets. The model of temptation with self-control implies: Similar predictions as Strotz (1955); Removing temptation makes agents unambiguously better off.

39 Thank You