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Uncertainty and Hyperbolic Discounting P. Dasgupta E. Maskin

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2 Strong evidence in behavioral ecology and economics that animals and humans place less weight on future than current payoffs, i.e., they discount the future more intriguingly: considerable evidence that discount rates increase as time to payoffs shrinks (experimental evidence better for birds— pigeons and starlings— than for humans)

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3 Strotz (1956) People attempt to lay aside money for Christmas early in the year But later on, spend this money on summer vacations or back-to-school clothes They have become more “impatient.”

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4 O’Donoghue and Rabin (1999) If offered choice in February between –painful 5-hour task on April 1 –painful 8-hour task on April 15 people choose April 1. As April 1 approaches, people apt to postpone to April 15 That is, they discount later pain more as time grows short

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5 Both examples accord with hyperbolic discounting (discount rates increase as payoffs approach, so more distant payoffs are discounted at lower per-period rate) has attracted attention in economics literature because sheds light on important economic phenomena (e.g., saving behavior)

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6 Offer evolutionary explanation for hyperbolic discounting Based on hypothesis that human or animal preferences—as manifested in cravings, urges, or instincts—selected to induce animal to make “right” choice in typical situation it faces “right” = survival-maximizing Will show: if typical situation entails uncertainty about when payoffs are realized, get preference reversals consistent with hyperbolic discounting

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7 Rough intuition: suppose DM offered choice between: prospect P - - small payoff, relatively early realization and prospect - - large payoff, relatively late realization small but positive probability of early realization assume DM initially chooses –bigger payoff –might be realized early with time, chance of early realization of declines –true of P too, but pays off early anyway so DM may switch to P - - à la hyperbolic discounting

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8 Although preference reversal here looks like hyperbolic discounting, actually dynamically consistent –optimal for DM to switch from to P But suppose DM faces atypical choice problem –urge to switch may lead her astray If problem recurs, DM may learn how to overcome impatience

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9 Why discount at all? Conventional answer: future payoffs may disappear or depreciate For example, blackbird waits for fruit on raspberry bush to ripen –crows may devour raspberries before ripe –blackbird should discount the payoff of getting fruit –discount rate = hazard rate (probability that crows arrive

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10 Apparently cheap explanation for hyperbolic discounting: hazard rate declining in time But no particular reason why this should be so (if crow-arrival time is Poisson, hazard rate and hence discount rate is constant— i.e., discounting is “exponential”) Doesn’t explain preference reversals: if over P at time 0, will do so at time t

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11 Another reason for discounting: waiting costs –waiting entails using up energy (physiological cost) –waiting entails passing up other prospects (opportunity cost) Waiting costs immediately explain another empirical regularity: large payoffs discounted less

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12 Two prospects: Suppose Evidence suggest that, for k >1 big enough,

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13 Phenomenon follows immediately from waiting costs Let c be waiting cost per unit time, Suppose (earlier prospect preferred) Then (later prospect preferred when payoffs increased). Both hazard-rate and waiting-cost stories important in practice Henceforth, stick with hazard rates

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14 Introduce uncertainty about realization time Blackbird may be pretty sure raspberries will ripen by tomorrow morning But some chance they will ripen earlier (or later)

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15 Prospect of raspberries V = calories in ripe raspberries T = likely ripening (realization) time, say, tomorrow morning

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16 At any time t < T, there is probability that V will be realized in q = probability density of “early realization” r = hazard rate (rate of crow arrival) expected payoff =

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17 Suppose there is another prospect (blackberry bush) bird can monitor only one bush at a time (must choose ) (more calories/volume in blackberries) (blackberries likely to ripen later) Some chance of early ripening (density q )

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18 Proposition 1: Assume that there exists such that decision maker (DM) indifferent between with and probability density q of early arrival. Then DM prefers Implies “hyperbolic discounting”: when much time remains DM willing to wait for bigger reward. But when horizon short becomes impatient.

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19 Proof : Expected payoffs from (1) and (2) Time derivatives: and

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20 (1) and (2) Can rewrite derivatives as (3) and (4) At sums of first two terms in bracketed expressions are equal Because (3) bigger than (4) Hence, (1) bigger than (2) for (2) bigger than (1) for

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21 Idea: passage of time has 2 marginal effects on expected payoff from prospect (i)brings nearer time at which payoff likely to be realized (ii)reduces probability of early realization effect (i) is proportional to expected payoff - - so same for effect (ii) (negative) is proportional to V for P and to so single-crossing property holds: P’s expected payoff increases faster than that of whenever expected payoffs equal.

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22 2 ways to interpret preference reversal: (i) once-and-for-all choice of at time 0, and then an unexpected opportunity to choose again at time (in which case the choice will be P) (ii) DM can switch at any time, in which case will choose and switch to

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23 Proposition 1 pertains to positive payoffs, but tax story entails negative payoffs Proposition 1*: Maintain hypotheses of Proposition 1 except assume Then if DM indifferent between prefers

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24 Propositions assumed (i)all early realization times equally likely (ii)probability density of early realization same for both prospects (iii)no probability of late arrival –Can relax these assumptions –However, cannot dispense with all assumptions about uncertainty

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25 Example –Consider –In interval has zero probability of early realization but P has positive probability –Then at ’s chances for early realization are the same as before, but P’s chances are dimmer –So is more attractive (by comparison with P) than before, contradicting hyperbolic discounting

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26 Suppose Then is density of P being realized at conditional on t being reached with no previous realization

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27 Similarly is conditional density for Assume (5) and (6) Condition (6) is MLRP condition Condition (7) rules out above example

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28 Proposition 2: Under (5) and (6), if there exists such that DM is indifferent between Then DM prefers

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29 Example Suppose and Then (5) and (6) hold for sufficiently small

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30 –But still one important ingredient to hyperbolic discounting not yet accounted for –Propositions show that preference reversals consistent with hyperbolic discounting follow from uncertainty about realization times –However, choices are dynamically consistent –The DM at would not choose to tie her hands to prevent herself from switching at

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31 Yet Strotz notes people learn to save for Christmas by putting money in illiquid low-interest savings accounts: “Christmas accounts” In experiments, pigeons learn to commit themselves not to switch from “patient” to “impatient” choices must show how this arises naturally in a evolutionary model

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32 Can explain dynamic inconsistency and self-commitment as follows: –Suppose that, over evolutionary time, species faced prospects of type discussed: ( V, T, q ) –Suppose, in any given choice problem between P = (V,T, q ) and individual DM from species can observe (V, T ) and –Then best DM can do is to choose between –Evolution should endow her with urges and inclinations that induce right choice. –If satisfy (5) and (6), then may get preference reversal from

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33 –Important that uncertainty unobservable, otherwise evolutionarily desirable to make discount rates depend on uncertainty from problem to problem –Now suppose DM in a situation where uncertainty differs significantly from “typical.” Specifically, suppose –Given inherited preferences, she will switch from - - but this is dynamically inconsistent

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34 But (thanks to evolution) many species can learn from experience Suppose, DM faces choice between (V,T, 0 ) and recurrently –each round, DM chooses a behavior probabilistically –successful behaviors more likely to be chosen provided DM does not have urge to behave otherwise

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35 How does this model apply to experiments with pigeons? large literature starting with Rachlin (1972) and Ainslee (1974) stylized protocol: –At pigeon makes choice (by pecking key) between (V, T, 0) and – later at pigeon can switch choices –Third key available at - - if pecked, disables switching option at

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36 Stylized findings: (i)pigeon makes same choice in most rounds (ii) if chose P at unlikely to choose more likely to switch if chose (iii)unlikely to peck disabling key in early rounds (iv)if peck disabling key in later rounds, more likely to have chosen (v) if switch from in early rounds, more likely to peck disabling key in later rounds

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37 finding (i) –stable preferences –consistent with bird maximizing something (e.g., discounted calories) finding (ii) –switch from is consistent with model, but from is not finding (iii) –takes experience to discover that uncertainty is atypical findings (iv) and (v) –birds that switch from are the ones that gain from disabling switch option

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38 Know something about pigeons, starlings, and humans –all have preference reversals consistent with hyperbolic discounting Perhaps other species don’t exhibit preference reversals –if so, can try to correlate reversals with nature of uncertainty they face

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