Uncertainty and Hyperbolic Discounting P. Dasgupta E. Maskin 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 birdsвЂ” pigeons and starlingsвЂ” than for humans) 2 Strotz (1956) вЂў People attempt to lay aside money for Christmas early in the year вЂў But later on, spend this money on summer vacations or back-to-school clothes They have become more вЂњimpatient.вЂќ 3 OвЂ™Donoghue and Rabin (1999) вЂў If offered choice in February between вЂ“ painful 5-hour task on April 1 вЂ“ painful 8-hour task on April 15 people choose April 1. вЂў As April 1 approaches, people apt to postpone to April 15 That is, they discount later pain more as time grows short 4 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) 5 вЂў Offer evolutionary explanation for hyperbolic discounting вЂў Based on hypothesis that human or animal preferencesвЂ”as manifested in cravings, urges, or instinctsвЂ”selected to induce animal to make вЂњrightвЂќ choice in typical situation it faces вЂњrightвЂќ = survival-maximizing вЂў Will show: if typical situation entails uncertainty about when payoffs are realized, get preference reversals consistent with hyperbolic discounting 6 Rough intuition: вЂў suppose DM offered choice between: prospect P - - small payoff, relatively early realization and prospect P п‚ў - - large payoff, relatively late realization вЂў small but positive probability of early realization вЂў assume DM initially chooses P п‚ў вЂ“ bigger payoff вЂ“ might be realized early вЂў with time, chance of early realization of P п‚ў declines вЂ“ true of P too, but pays off early anyway вЂў so DM may switch to P - - Г la hyperbolic discounting 7 вЂў Although preference reversal here looks like hyperbolic discounting, actually dynamically consistent вЂ“ optimal for DM to switch from P п‚ў to P вЂў But suppose DM faces atypical choice problem вЂ“ urge to switch may lead her astray вЂў If problem recurs, DM may learn how to overcome impatience 8 Why discount at all? вЂў Conventional answer: future payoffs may disappear or depreciate вЂў For example, blackbird waits for fruit on raspberry bush to ripen вЂ“ crows may devour raspberries before ripe вЂ“ blackbird should discount the payoff of getting fruit вЂ“ discount rate = hazard rate пЂЅ 1 п‚ґ (probability that crows arrive пЃ„t betw een t and t пЂ« пЃ„ t ) 9 вЂў Apparently cheap explanation for hyperbolic discounting: hazard rate declining in time вЂў But no particular reason why this should be so (if crow-arrival time is Poisson, hazard rate and hence discount rate is constantвЂ” i.e., discounting is вЂњexponentialвЂќ) вЂў DoesnвЂ™t explain preference reversals: if P п‚ў over P at time 0, will do so at time t 10 вЂў Another reason for discounting: waiting costs вЂ“ waiting entails using up energy (physiological cost) вЂ“ waiting entails passing up other prospects (opportunity cost) вЂў Waiting costs immediately explain another empirical regularity: large payoffs discounted less 11 Two prospects: пЂЁ V , T пЂ© an d пЂЁ V п‚ў, T п‚ў пЂ© V пЂјVп‚ў payoffs T пЂјTп‚ў realizatio n tim es Suppose пЂЁ V , T пЂ© favored over пЂЁ V п‚ў, T п‚ў пЂ© Evidence suggest that, for k >1 big enough, пЂЁ kV п‚ў, T п‚ў пЂ© favored over пЂЁ kV , T пЂ© 12 вЂў Phenomenon follows immediately from waiting costs вЂў Let c be waiting cost per unit time, вЂў Suppose V пЂ cT пЂѕ V п‚ў пЂ cT п‚ў (earlier prospect preferred) вЂў Then kV пЂ cT пЂј kV п‚ў пЂ cT п‚ў (later prospect preferred for k big enough when payoffs increased). вЂў Both hazard-rate and waiting-cost stories important in practice вЂў Henceforth, stick with hazard rates 13 Introduce uncertainty about realization time вЂў Blackbird may be pretty sure raspberries will ripen by tomorrow morning вЂў But some chance they will ripen earlier (or later) 14 Prospect of raspberries п‚« P пЂЅ пЂЁ V , T пЂ© V = calories in ripe raspberries T = likely ripening (realization) time, say, tomorrow morning 15 вЂў At any time t < T, there is probability q пЃ„ t that V will be realized in пЂЁ t , t пЂ« пЃ„ t пЂ© вЂў q = probability density of вЂњearly realizationвЂќ вЂў r = hazard rate (rate of crow arrival) вЂў expected payoff = пѓІ T qe 0 пЂ rt d tV пЂ« пЂЁ 1 пЂ q T пЂ© e пЂ rT V 16 вЂў Suppose there is another prospect P п‚ў пЂЅ пЂЁ V п‚ў, T п‚ў пЂ© (blackberry bush) вЂў bird can monitor only one bush at a time (must choose P o r P п‚ў ) вЂў Vп‚ўпЂѕV Tп‚ўпЂѕT (more calories/volume in blackberries) (blackberries likely to ripen later) вЂў Some chance of early ripening (density q ) 17 Proposition 1: Assume that there exists t пЂЄ пЂј T such that decision maker (DM) indifferent between P пЂЅ пЂЁ V , T пЂ© and P п‚ў пЂЅ пЂЁ V п‚ў, T п‚ў пЂ© at tim e t пЂЄ with 0 пЂј V пЂј V п‚ў, T пЂј T п‚ў, and probability density q of early arrival. Then DM prefers пЂЄ пЂЄ P п‚ў to P at t пЂј t and P to P п‚ў at t пЂѕ t . Implies вЂњhyperbolic discountingвЂќ: when much time remains пЂЁ t пЂј t пЂЄ пЂ© , DM willing to wait for bigger reward. But when horizon short пЂЁ t пЂѕ t пЂЄ пЂ© , becomes impatient. 18 Proof : Expected payoffs from пѓІ (1) T e P an d P п‚ў at t пЂј T : пЂ rпЃґ пЂ rT qЛ† пЂЁ t пЂ© d пЃґ V пЂ« пЂЁ 1 пЂ qЛ† пЂЁ t пЂ© пЂЁ T пЂ t пЂ© пЂ© e V пЂ rпЃґ qЛ† пЂЁ t пЂ© d пЃґ V п‚ў пЂ« пЂЁ1 пЂ qЛ† пЂЁ t пЂ© пЂЁ T п‚ў пЂ t пЂ© пЂ© e t and пѓІ (2) Tп‚ў e t V п‚ў, w here qЛ† пЂЁ t пЂ© пЂЅ q пЂ rT п‚ў 1 пЂ qt Time derivatives: пЂe пЂ rt qЛ† пЂЁ t пЂ© V пЂ« qЛ† пЂЁ t пЂ© e пЂ rT пЂ« пЂe qЛ† пЂЁ t пЂ© V п‚ў пЂ« qЛ† пЂЁ t пЂ© e e пЂ rпЃґ t and пЂ rt пЂЁпѓІ T пЂ rT п‚ў пЂ« пЂЁпѓІ Tп‚ў t e dпЃґ пЂ пЂЁT пЂ t пЂ© e пЂ rпЃґ пЂ rT dпЃґ пЂ пЂЁT п‚ў пЂ t пЂ© e пЂ© V пЂ rT п‚ў dqЛ† dt пЂ© Vп‚ў пЂЁt пЂ© dqЛ† dt пЂЁt пЂ© 19 пѓІ (1) and (2) пѓІ T e пЂ rпЃґ пЂ rT qЛ† пЂЁ t пЂ© d пЃґ V пЂ« пЂЁ 1 пЂ qЛ† пЂЁ t пЂ© пЂЁ T пЂ t пЂ© пЂ© e V пЂ rпЃґ пЂ rT qЛ† пЂЁ t пЂ© d пЃґ V п‚ў пЂ« пЂЁ1 пЂ qЛ† пЂЁ t пЂ© пЂЁ T п‚ў пЂ t пЂ© пЂ© e V п‚ў, w here qЛ† пЂЁ t пЂ© пЂЅ t Tп‚ў e t п‚ў q 1 пЂ qt Can rewrite derivatives as (3) T пЂ rпЃґ пЂ rT пЂ rt пѓ© Л† q пЂЁ t пЂ© пѓІ e d пЃґ qЛ† пЂЁ t пЂ© V пЂ« пЂЁ1 пЂ qЛ† пЂЁ t пЂ© пЂЁ T пЂ t пЂ© пЂ© e V пЂ e V пѓ№ пѓЄпѓ« t пѓєпѓ» and (4) п‚ў T пЂ rпЃґ пЂ rT п‚ў пЂ rt пѓ© qЛ† пЂЁ t пЂ© пѓІ e d пЃґ qЛ† пЂЁ t пЂ© V п‚ў пЂ« пЂЁ1 пЂ qЛ† пЂЁ t пЂ© пЂЁ T п‚ў пЂ t пЂ© пЂ© e V п‚ў пЂ e V п‚ў пѓ№ пѓЄпѓ« t пѓєпѓ» вЂў At t пЂЅ t пЂЄ , sums of first two terms in bracketed expressions are equal вЂў Because V п‚ў пЂѕ V , (3) bigger than (4) пЂЄ вЂў Hence, (1) bigger than (2) for t пЂѕ t ; (2) bigger than (1) for t пЂј t пЂЄ 20 Idea: вЂў 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 вЂў effect (i) is proportional to expected payoff - - so same for P and P п‚ў at t пЂЄ вЂў effect (ii) (negative) is proportional to V for P and to V п‚ў for P п‚ў; because V п‚ў пЂѕ V , effect m ore negative fo r V п‚ў th an fo r V вЂў so single-crossing property holds: PвЂ™s expected payoff increases faster than that of P п‚ў whenever expected payoffs equal. 21 2 ways to interpret preference reversal: (i) once-and-for-all choice of P п‚ў o ver P at time 0, and then an unexpected opportunity to choose again at time пЂЄ t пЂѕ t (in which case the choice will be P) (ii) DM can switch at any time, in which case will choose P п‚ў at t пЂЅ 0 and switch пЂЄ to P at t пЂЅ t . 22 вЂў Proposition 1 pertains to positive payoffs, but tax story entails negative payoffs Proposition 1*: Maintain hypotheses of Proposition 1 except assume V п‚ў пЂј V пЂј 0 . пЂЄ п‚ў Then if DM indifferent between P and P at t , prefers P to P п‚ў for t пЂј t пЂЄ and P п‚ў to P for t пЂѕ t пЂЄ 23 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 вЂ“ Can relax these assumptions вЂ“ However, cannot dispense with all assumptions about uncertainty 24 Example вЂ“ Consider пЂЁ V , T пЂ© an d пЂЁ V п‚ў, T п‚ў пЂ© , 0 пЂј V пЂј V п‚ў an d T пЂј T п‚ў вЂ“ In interval пѓ©пѓ« 0, t пѓ№пѓ» , P п‚ў has zero probability of early realization but P has positive probability вЂ“ Then at t , P п‚ў вЂ™s chances for early realization are the same as before, but PвЂ™s chances are dimmer вЂ“ So P п‚ў is more attractive (by comparison with P) than before, contradicting hyperbolic discounting 25 Suppose P пЂЅ пЂЁV , T пЂ© an d P п‚ў пЂЅ пЂЁ V п‚ў, T п‚ў пЂ© q пЂЁ t пЂ© d en sity fo r realizatio n o f P at t пЂЁ t пЂј T o r t пЂѕ T пЂ© q п‚ў пЂЁ t пЂ© d en sity fo r realizatio n o f P п‚ў at t пЂЁ t пЂј T п‚ў o r t пЂѕ T п‚ў пЂ© For t пЂј пЃґ пЂј T , let qЛ† пЂЁ пЃґ , t пЂ© пЂЅ q пЂЁпЃґ 1пЂ пѓІ t 0 пЂ© q пЂЁ x пЂ© dx Then qЛ† пЂЁпЃґ , t пЂ© is density of P being realized at пЃґ conditional on t being reached with no previous realization 26 Similarly qЛ† п‚ў пЂЁпЃґ , t пЂ© пЂЅ q п‚ў пЂЁпЃґ 1пЂ пѓІ t 0 пЂ© q п‚ў пЂЁ x пЂ© dx for t пЂј пЃґ пЂј T is conditional density for P п‚ў Assume (5) qЛ† пЂЁ t , t пЂ© п‚і qЛ† п‚ў пЂЁ t , t пЂ© for all t пЂј T and (6) V qЛ† ( t , t ) пЂј V п‚ўqЛ† п‚ў пЂЁ t , t пЂ© , for all t пЂј T Condition (6) is MLRP condition Condition (7) rules out above example 27 Proposition 2: Under (5) and (6), if there пЂЄ exists t such that DM is indifferent between P an d P п‚ў. Then DM prefers пЂЄ пЂЁ пЂ© пЂЄ P п‚ў to P at t пЂј t and prefers P to P п‚ў at t пѓЋ t , T . 28 Example Suppose q пЂЁt пЂ© пЂЅ and qп‚ў пЂЁt пЂ© пЂЅ пЃЎt T пЃЎt Tп‚ў Then (5) and (6) hold for пЃЎ sufficiently small 29 вЂ“ But still one important ingredient to hyperbolic discounting not yet accounted for вЂ“ Propositions show that preference reversals consistent with hyperbolic discounting follow from uncertainty about realization times вЂ“ However, choices are dynamically consistent вЂ“ The DM at t пЂЅ 0 would not choose to tie her hands to prevent herself from switching at tп‚іt пЂЄ 30 вЂў Yet Strotz notes people learn to save for Christmas by putting money in illiquid low-interest savings accounts: вЂњChristmas accountsвЂќ вЂў In experiments, pigeons learn to commit themselves not to switch from вЂњpatientвЂќ to вЂњimpatientвЂќ choices вЂў must show how this arises naturally in a evolutionary model 31 Can explain dynamic inconsistency and self-commitment as follows: вЂ“ Suppose that, over evolutionary time, species faced prospects of type discussed: ( V, T, q ) вЂ“ Suppose, in any given choice problem between P = (V,T, q ) and P п‚ў пЂЅ пЂЁ V п‚ў, T п‚ў, q п‚ў пЂ© , individual DM from species can observe (V, T ) and пЂЁ V п‚ў, T п‚ў пЂ© , b u t n o t q an d q п‚ў. вЂ“ Then best DM can do is to choose between P an d P п‚ў taking expectations over q and q п‚ў вЂ“ Evolution should endow her with urges and inclinations that induce right choice. вЂ“ If q and q п‚ў satisfy (5) and (6), then may get preference reversal from P п‚ў to P 32 вЂ“ Important that uncertainty unobservable, otherwise evolutionarily desirable to make discount rates depend on uncertainty from problem to problem вЂ“ Now suppose DM in a situation where uncertainty differs significantly from вЂњtypical.вЂќ Specifically, suppose q п‚є q п‚ў п‚є 0. вЂ“ Given inherited preferences, she will switch from P п‚ў to P - - but this is dynamically inconsistent 33 вЂў But (thanks to evolution) many species can learn from experience вЂў Suppose, DM faces choice between (V,T, 0 ) and пЂЁ V п‚ў, T п‚ў, 0 пЂ© recurrently вЂ“ each round, DM chooses a behavior probabilistically вЂ“ successful behaviors more likely to be chosen provided DM does not have urge to behave otherwise 34 How does this model apply to experiments with pigeons? вЂў large literature starting with Rachlin (1972) and Ainslee (1974) вЂў stylized protocol: вЂ“ At t1 , pigeon makes choice (by pecking key) between (V, T, 0) and пЂЁ V п‚ў, T п‚ў, 0 пЂ© вЂ“ later at t 2 , pigeon can switch choices вЂ“ Third key available at t1 - - if pecked, disables switching option at t 2 35 Stylized findings: (i) pigeon makes same t1 choice in most rounds (ii) if chose P at t1 , unlikely to choose P п‚ў at t 2 ; more likely to switch if chose P п‚ў at t1 (iii) unlikely to peck disabling key in early rounds (iv) if peck disabling key in later rounds, more likely to have chosen P п‚ў at t1 (v) if switch from P п‚ў to P in early rounds, more likely to peck disabling key in later rounds 36 вЂў finding (i) вЂ“ stable preferences вЂ“ consistent with bird maximizing something (e.g., discounted calories) вЂў finding (ii) вЂ“ switch from P п‚ў to P is consistent with model, but from P to P п‚ў is not вЂў finding (iii) вЂ“ takes experience to discover that uncertainty is atypical вЂў findings (iv) and (v) вЂ“ birds that switch from P п‚ў to P are the ones that gain from disabling switch option 37 вЂў Know something about pigeons, starlings, and humans вЂ“ all have preference reversals consistent with hyperbolic discounting вЂў Perhaps other species donвЂ™t exhibit preference reversals вЂ“ if so, can try to correlate reversals with nature of uncertainty they face 38 39

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