COOPERATION WITH TIME-INCONSISTENCY Exended Absrac for LMSC09 By Nicola Dimiri Professor of Economics Faculy of Economics Universiy of Siena Piazza S. Francesco 7 53100 Siena Ialy Dynamic games have proven o be a powerful framework o concepualize social and economic repeaed ineracions, among sraegic agens. Wihin dynamic games, repeaed sraegic form games have played a major role, boh in he heory and in he applicaions (Mailah - Samuelson; 2006). Well known achievemens of such games are he celebraed Folk Theorems, a class of resuls showing ha cooperaive behaviour can obain as a Nash Equilibrium in an infiniely repeaed Prisoner s Dilemma. The main inuiion behind hese heorems, namely ha cooperaion can emerge even among selfish players as long as hey assign sufficienly high weighs o fuure payoffs, lies behind incenives design in a number of real life repeaed relaionships. The heory of infiniely repeaed games (IRG) has been developed assuming ha players are ime-consisen. Consisency implies, for example, ha an acion planned o be aken a a fuure dae is effecively aken when ha dae comes. In game and decision heory ime consisency is formalized by assigning exponenial weighs o fuure payoffs. In IRG a major implicaion of ime consisency (exponenial discouning) is ha when players choose a sraegy a he beginning of he game hey do no need o reconsider i a laer daes, as he game evolves. This is because, wih exponenial discouning, a any dae he welfare level generaed by an infinie sream of payoffs is he same, from any poin in ime, which implies ha as he game unfolds here is no incenive o revise a sraegy. In recen years however empirical evidence showed ha ime consisency is ofen violaed by individuals, as in he classical example of preference reversal (Ainslie, 2001). More specifically, preference reversal occurs when an individual facing a choice beween, say, 5 in seven days from now and 10 in weny days from now, he firs chooses he laer (higher) 10 payoff and hen, as he firs payoff ges closer in ime, he reverses his own choice oping for he earlier, hough lower, 5 sum. When players, in an IRG, are ime inconsisen heir welfare level a various poins in ime along he game migh change, and herefore heir sraegy choice in he game canno be analyzed by simply considering he welfare, generaed by he infinie sream of available payoffs, a he beginning of he game. As diffused as ime inconsisen behaviour migh be, so far no aemp seems o have been made o invesigae how he heory of IRG could be reformulaed o ake ino accoun individuals inconsisency. More in general, he poin ha we are ineresed in invesigaing is he following: o wha exen problems relaed o ime consisency can be miigaed, or eliminaed, when players ake ino proper accoun heir own fuure inconsisency? In his paper we shall provide a firs conribuion in his direcion, by considering one of he main models proposed in he lieraure o formalize inconsisency, he so called Quasi-Hyperbolic
(someimes also called Quasi-Exponenial) discouning funcion (Phelps and Pollack, 1968; Laibson, 1997, Frederick e al (2002)). In wha follows, afer having suggesed how he sandard IRG heory could be amended o accommodae ime inconsisen players, in he case of he Infiniely Repeaed Prisoner s Dilemma we compare condiions under which cooperaion could emerge, when individuals are ime consisen vs ime inconsisen, where inconsisency is indeed formalized by Quasi-Hyperbolic discouning. 1. Preliminaries In his secion we se he basics of he analysis. Le =0,1,2,,T, be he ime index and x=(x(0),x(1),..,x()..), y=(y(0),y(1),,y(),..) wo alernaive sreams of payoffs available o a decision maker (DM), which for convenience we assume o be moneary amouns. Moreover, le d(), wih d(0)=1 and d() d(+1) be he DM discouning funcion. We also assume ha he DM chooses a =0 and can neiher borrow nor lend, so ha a each ime =0,1,2, he can ge exacly wha s specified iniially in he payoff profile. A ime, he payoff vecor x generaes he following welfare level w (x)= i=0 + d(i )x(i) analogously, for he payoff sream y we have x(i) i=+1 (1) w (y)= y(i) i=0 + d(i )y(i) i=+1 (2) Formulae (1)-(2) specify how, a ime =0,1,2,,T,., he individual welfare is compued by capializing up o dae he payoffs obained prior o reaching i, and discouning a he payoffs available afer ha dae. In repeaed games i is sandard o evaluae he payoff sream x, associaed o a sraegy profile, by considering =0; namely w 0 (x)= i=0 d(i)x(i) (3) This is because, for all i=1,2, he discouning funcion d(i) is ypically assumed o follow an exponenial paern, namely d(i) = d(1) i, which implies ha for all =0,1,2, w 0 (x) = i=0 d(i)x(i) = d()w (x) = d(1) w (x) Hence, if w 0 (x) > w 0 (y) hen, clearly, also w (x) = d(1) w 0 (x) > d(1) w 0 (y) = w (y). More in general, wih exponenial discouning he welfare ordering of wo alernaive payoff sreams is he same a all daes. This is no so wih non-exponenial discouning funcions, so ha he welfare ordering may change a differen imes. For example i could be ha w 0 (x) > w 0 (y) and, for some T>0, w T (x) < w T (y), which would enail ime inconsisency since a decision aken a =0 migh be revised laer a =T. Therefore, if he decision maker is aware of his own being ime inconsisen, how would he proceed?
2. Evaluaing Welfare wih Time Inconsisency Assuming he individual o be aware of his own inconsisency, namely ha a fuure daes his welfare ordering over wo alernaive payoff profile may change, we imagine ha he will evaluae vecor x (and analogously for vecor y) by he following welfare aggregaor W 0 (x) = W(x)= =0 d()w (x) (4) namely ha he decision as o which payoff sream is preferable is sill aken a =0, like in sandard game heoreic approach wih exponenial discouning, however based upon a discouned sum of fuure welfare levels, raher han simply upon payoffs. The underlying moivaion is simple. If discouning of fuure payoffs may lead o inconsisency, and hen o a possible ambiguiy for he individual on which acion o ake a =0 hen i is naural o hink ha he decision a =0 could be aken by weighing, discouning, fuure welfare levels raher han payoffs. We could observe ha (4) represens a firs order sophisicaed reasoning on he par of he decision maker, who anicipaing his own inconsisency discouns fuure welfare levels raher han payoffs. Bu if along he lines of (4) he would also consider, and compare, aggregaors of fuure welfare levels a ime such as W (x)= w i (x) i=0 + d(i )w i (x) i=+1 (5) hen his level of sophisicaion would increase, making his decision more ariculaed since an issue of ime inconsisency could also concern now he welfare aggregaors W (x). In he nex secion we shall confine our aenion o he order of sophisicaion embodied in (4), leaving o fuure research inconsisency issues relaed o (5). In paricular, we shall do so wihin he celebraed Prisoner s Dilemma game, where inconsisency will be modelled by he Quasi Hyperbolic (QHD) discouning funcion d(0) = 1 and d() = β δ, wih β > 0, 0 < δ < 1 and =1,2,... More specifically we shall be ineresed in comparing he condiions on he parameer δ, for he cooperaive oucome o obain, in boh QHD and in he Exponenial Discouning (ED) model d() = δ, wih = 0,1,2,.. We hink such comparison could provide some iniial elemens owards an undersanding of wheher or no awareness of one s inconsisency could eliminae inconsisency. 3. The Infiniely Repeaed Prisoner s Dilemma The heory of infiniely repeaed games (IRG) has clarified how economic, poliical and social oucomes ha could no be obained as a Nash Equilibrium of he one-sho game, could insead be achieved as a Nash Equilibrium when players sraegic ineracion is infiniely repeaed. A mos noable example of his is he infiniely repeaed Prisoner s Dilemma (IRPD), such as ha in Table 1 below
Table 1 C D C 2,2, 0,3 D 3,0 1,1 Sar considering ED, and assume (for simpliciy) boh players o be characerized by he same δ. Then, i is well known ha if δ > 1 a pair of modified rigger sraegies (Osborne, 2003) 2 would suppor he cooperaive oucome (2,2) a every, namely a every repeiion of he game. We now wan o discuss wha would happen if boh players have he same QHD funcion, raher han he same ED funcion. 3.1 Quasi Hyperbolic Discouning Consider now (1) and (2), where x =(2,2,2,.) is he payoff vecor when he cooperaive oucome realizes a every repeiion, and y =(3,1,1,.) he payoff vecor when a players decides o deviae a he beginning and he one-sho-game Nash Equilibrium is played from hen on. Le s firs consider he generic erm of (4), for he payoff sream x βδ w (x) = 2βδ 1 βδ + 1 1 +.. + βδ 1 βδ + 1 + βδ + βδ2 + βδ 3 +. = = 2 (1 δ ) (1 δ) + βδ (1 + βδ (1 δ) ) Analogously βδ w (y) = βδ 3 βδ + 1 1 +.. + βδ 1 βδ + 1 + βδ + βδ2 + βδ 3 +. = = 2 + (1 δ ) (1 δ) + βδ (1 + βδ (1 δ) ) Therefore W(y) W(x) = lim T if δ < 1 2 T (1 2δ) + δ (1 δ) (1 δ) βδ 1 + βδ =0 = (6) (1 δ) if δ > 1 2 so ha he condiion for cooperaion o obain, δ > 1, is he same as when he discouning funcion 2 is exponenial, wih generic erm δ. The above conclusion hinges upon he DM preferring o deviae a =0, when he decides o do so. Such preference however depends on he value of he parameers βand δ and here would always be values of β close o one which makes immediae deviaion mos profiable. Condiions for cooperaion would in case be independen of β.
4. Conclusions In he paper we begun invesigaing how ime inconsisen players, in infiniely repeaed games, could ake ino accoun of heir own inconsisency, if hey are aware of i. We suggesed ha hey could do so by discouning fuure welfare levels, raher han fuure payoffs. In an infiniely repeaed Prisoner s Dilemma we hen enquired wheher or no, when players ake accoun of heir fuure inconsisency as modelled by Quasi Hyperbolic Discouning, condiions for cooperaion differ wih respec o exponenial discouning, namely wih respec o when hey are ime consisen. The analysis is a preliminary sep showing ha if Quasi Hyperbolic Discouners discoun all fuure welfare levels hen cooperaion could emerge under he same condiions of exponenial discouning. More explicily, inconsisen and consisen players may decide o cooperae on he same parameer values. This would seem o sugges ha inconsisency could be ackled by appropriaely aking ino accoun one s inconsisency. Full invesigaion of he issue however is lef o fuure research. References Ainslie G., (2001), Breakdown of Will, Cambridge Universiy Press. Frederick S, Loewensein G, O Donoghue T, (2002) Time discouning and ime preferences: A Criical Review, Journal of Economic Lieraure XL: 351-401. Laibson D (1997) Golden Eggs and Hyperbolic Discouning, Quarerly Journal of Economics 42: 861-871. Mailah G., Samuelson L., (2006), Repeaed Games and Repuaions: Long Run Relaionships, Oxford Universiy Press. Osborne M., (2003), An Inroducion o Game Theory, Oxford Universiy Press. Phelps, E. S., R. A. Pollak (1968), On Second-Bes Naional Saving and Game-Equilibrium Growh, Review of Economic Sudies, 35, 185-199. Pollak R (1968), Consisen Planning, Review of Economic Sudies 35: 201-208. Sroz R (1955), Myopia and Inconsisency in Dynamic Uiliy Maximizaion, Review of Economic Sudies 23, 165-180.