In a symmetric infinitely-repeated game, where players have constant relative risk aversion (CRRA), or constant elasticity of intertemporal substitution, utility functions, it is shown that the critical discount factor required to sustain full cooperation is decreasing in the coefficient of relative risk aversion (increasing in the elasticity of intertemporal substitution). An application to cooperation in international environmental agreements (IEA) is presented and it is shown that the limit of the critical discount factor as the number of countries goes to infinity is equal to one (zero) if the coefficient of intertemporal inequality aversion is less (greater) than one

Collie, David R.

Online Research @ Cardiff

Economics Letters 221 (2022) 110897Cr(tfcaaraurcifNdaAwDh0Contents lists available at ScienceDirectEconomics Lettersjournal homepage: www.elsevier.com/locate/ecoletCRRA utility and the sustainability of cooperation ininfinitely-repeated gamesDavid R. Collie ∗ardiff Business School, Cardiff University, United Kingdoma r t i c l e i n f oArticle history:Received 28 September 2022Received in revised form 20 October 2022Accepted 21 October 2022Available online 28 October 2022JEL classification:C72C73Q58Keywords:CooperationCritical discount factorNash-reversion trigger strategiesCRRA utility functionInternational environmental agreementa b s t r a c tIn a symmetric infinitely-repeated game, where players have constant relative risk aversion (CRRA),or constant elasticity of intertemporal substitution, utility functions, it is shown that the criticaldiscount factor required to sustain full cooperation is decreasing in the coefficient of relative riskaversion (increasing in the elasticity of intertemporal substitution). An application to cooperation ininternational environmental agreements (IEA) is presented and it is shown that the limit of the criticaldiscount factor as the number of countries goes to infinity is equal to one (zero) if the coefficient ofintertemporal inequality aversion is less (greater) than one.© 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license(http://creativecommons.org/licenses/by/4.0/).(tamte(aγritutogttwItσ1. IntroductionIn the literature on experimental tests of the infinitely-epeated prisoners’ dilemma, surveyed by Bó and Fréchette2018), the link between personal characteristics and coopera-ion has been tested. Sabater-Grande and Georgantzis (2002)ind that risk aversion of players is negatively related toooperation whereas Dreber et al. (2014), Davis et al. (2016),nd Proto et al. (2018) do not find a relationship between riskversion of players and cooperation.This note analyses cooperation in a symmetric infinitely-epeated game where the players have constant relative riskversion (CRRA) utility functions. The parameter of the CRRAtility function can be interpreted as the coefficient of relativeisk aversion or, more relevantly in a deterministic game ofomplete information, as the reciprocal of the elasticity ofntertemporal substitution. It is shown that the critical discountactor required to sustain cooperation as a subgame-perfectash equilibrium using Nash-reversion trigger strategies isecreasing in this parameter. Hence, the more risk aversere the players then the easier it is to sustain cooperation.n application to international environmental agreements (IEA)ith many countries is presented.∗ Correspondence to: Cardiff Business School, Aberconway Building, Columrive, Cardiff, CF10 3EU, United Kingdom.E-mail address: Collie@cardiff.ac.uk. pttps://doi.org/10.1016/j.econlet.2022.110897165-1765/© 2022 The Author(s). Published by Elsevier B.V. This is an open access a2. CRRA utility and the critical discount factorConsider a symmetric game with J ≥ 2 players where a stageor constituent) game is repeated infinitely. In the stage game,he monetary payoff to the identical players is a function of theirctions yj (σ) where σ =(σ1, . . . , σJ)and j = 1, . . . , J . Theonetary payoff (or income) of a player is assumed to be con-inuous and strictly quasi-concave in its own action, which willnsure the existence of a Nash equilibrium. The utility of moneyor income) for the players is given by the constant relative riskversion (CRRA) utility function u(yj)= y1−γj / (1 − γ ) where≥ 0 (γ ̸= 1) is the coefficient of relative risk aversion (or theeciprocal of the elasticity of intertemporal substitution), whichs defined as γ = −u′′y/u′ and is more generally a measure ofhe concavity of the utility function. If γ = 1 then the CRRAtility function is u(yj)= ln(yj). Note that this is in line withhe usual assumptions of game theory, as utility is a functionf the actions of the players, uj (σ) = u(yj (σ)). In the stageame, a player maximising utility is equivalent to maximisinghe monetary payoff yj since u(yj)is an increasing monotonicransformation of yj. If the players cooperate to maximise jointelfare then they all play σC and receive the monetary payoff yC .f the players behave non-cooperatively then the outcome will behe Nash equilibrium (assumed to be unique) where they all playN and receive the monetary payoff yN . If one player deviates bylaying σD when all the others are playing σC then the deviatorrticle under the CC BY license (http://creativecommons.org/licenses/by/4.0/).D.R. Collie Economics Letters 221 (2022) 110897relδbciWwts3iIopgcaapywpσeeceives the monetary payoff yD. Obviously, except in perversexamples, it will be the case that yD > yC > yN .1In the infinitely-repeated game, the stage game is playedindefinitely and the players maximise the discounted sum ofutility where the common discount factor is δ ∈ [0, 1). Thediscount factor may be purely the discount factor of the playeror the probability of the game continuing to the next round inwhich case the players are maximising expected utility. As is wellknown, from Friedman (1971), cooperation can be sustained assubgame-perfect Nash equilibrium using Nash-reversion triggerstrategies if the discount factor is sufficiently large.2 Each playerplays σC provided all players have always played σC . If any playerdeviates by playing σD then all players play σN forever thereafter.Hence, players will cooperate if:11 − δu (yC ) ≥ u (yD) +δ1 − δu (yN) (1)Cooperation can be sustained as a subgame perfect Nash equi-ibrium if the discount factor is greater than the critical value:∗N ≡u (yD) − u (yC )u (yD) − u (yN)=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩y1−γD − y1−γCy1−γD − y1−γN=1 − λ1−γC1 − λ1−γNγ ̸= 1ln (yD) − ln (yC )ln (yD) − ln (yN)=ln (λC )ln (λN)γ = 1(2)where λC ≡ yC/yD < 1, and λN ≡ yN/yD < λC < 1.Proposition 1. The critical discount factor required to sustain fullcooperation using Nash-reversion trigger strategies δ∗N is decreasingin γ .Proof. Differentiating the critical discount factor δ∗N with respectto the parameter γ yields:∂δ∗N∂γ=(λN/λC )γ(λN − λγN)2 [λC (−λN + λγN) ln (λC )−(−λC + λγC)λN ln (λN)]γ ̸= 1(3)This derivative will be negative if the expression in squarerackets is negative, which will be the case if:λN ln (λN)λN − λγN<λC ln (λC )λC − λγCγ ̸= 1 (4)Since λC > λN , this will be case if Z = λ ln (λ)/(λ − λγ ) isincreasing in λ. Differentiating Z with respect to λ yields:dZdλ=λ − λγ (1 + (1 − γ ) ln (λ))(λ − λγ )2γ ̸= 1 (5)This will be positive if λ1−γ > 1 + ln(λ1−γ)for λ ∈ [0, 1),γ ≥ 0 and γ ̸= 1, or equivalently Ω ≡ x − (1 + ln (x)) > 0where x = λ1−γ for x ≥ 0 and x ̸= 1 (since this implies thatγ = 1). Since Ω is convex, d2Ω/dx2 = 1/x2 > 0, and has aminimum at x = 1, dΩ/dx = (x − 1) /x, where Ω = 0, it followsthat Ω > 0 for x ≥ 0 and x ̸= 1. Hence, Z is increasing in λ, andthe derivative (3) is negative for γ ≥ 0 and γ ̸= 1.Finally, note that taking the limit of (3) as γ goes to one yieldsthat limγ→1(∂δ∗N/∂γ)= ln (λC ) ln (λN/λC ) /2 ln (λN) < 0.■1 There are games where the Nash equilibrium and the cooperative outcomeoincide so this assumption does not hold, but sustaining cooperation is not anssue in such a game.2 Nash-reversion trigger strategies are considered for the sake of simplicity.ith optimal-punishment strategies, as in Abreu (1986, 1988), the analysisould be complicated as the punishment would depend upon γ . (2The more risk averse are the players (the larger is γ ) thenhe lower is the critical discount factor δ∗N and the easier it is toustain cooperation even though the game is deterministic.. Application to international environmental agreementsNow consider cooperation between countries (an IEA) in annfinitely-repeated environmental game with many countries.3n the stage game, each country has an identical endowmentf labour, L, and uses labour and emissions of pollution, σj, toroduce a consumption good, yj. The welfare of each country isiven by the CRRA utility function, u(yj), where γ in this contextan be interpreted as a coefficient of intertemporal inequalityversion. The production of the consumption good is negativelyffected by global emissions of pollution,∑Ji=1 σj, where theroduction function of the jth country is assumed to be:j = e−β∑Ji=1 σiL1−ασ αj (6)here α ∈ (0, 1) and β > 0.In the stage game, each country can choose its emissions ofollution by setting the quantity of tradeable emissions permits,j, to maximise its output of the consumption good, yj, which isquivalent to maximising its CRRA utility, u(yj). If the countriescooperate then they choose a common quantity of tradeableemissions permits to maximise joint welfare. Setting σj = σfor all countries, j = 1, . . . , J then choosing σ to maximise yjyields the cooperative quantity of tradeable emissions permitsσC = α/βJ , and the corresponding output of the consumptiongood:yC = e−αL1−ασ αC (7)If the countries set the quantity of tradeable emissions permitsnon-cooperatively then each country will choose σj to maximiseyj given the quantities of tradeable emissions permits chosenby all the other countries. This will yield the Nash equilibriumquantity of tradeable emissions permits σN = α/β , and thecorresponding output of the consumption good:yN = e−αJL1−ασ αN (8)If when all the other countries are setting the cooperativequantity of tradeable emissions permits, σC , a country deviatesthen it will set σj to maximise yj given that all the other countriesare setting σC . This yields the quantity of tradeable emissionspermits when a country deviates σD = α/β , which is equal to σNsince the best-reply function of the jth country does not dependsupon the quantities of tradeable emissions permits chosen by theother countries. The corresponding output of the consumptiongood is:yD = e−α(2−1/J)L1−ασ αD (9)Dividing (7) and (8) by (9) yields λC and λN :λC ≡yCyD= eα(1−1/J)J−α < 1, λN ≡yNyD= e−α(J−1)2/J < 1 (10)To show that λC is greater than λN , divide λC by λN then takelogs, which yields:λCλN= eα(J−1)J−α ⇒ ln(λCλN)= α [(J − 1) − ln (J)] > 0 (11)Hence, as assumed in Section 2, λC > λN . An IEA is sustainableif cooperation is sustainable as a subgame perfect Nash equilib-rium using Nash-reversion trigger strategies, and this will be the3 For a survey of the literature on sustaining cooperation in IEA, see Barrett2005).D.R. Collie Economics Letters 221 (2022) 110897cfδaFig. 1. The critical discount factor versus the number of countries.4paoDAeFaRAABBDFPSase if the discount factor is greater than the critical discountactor in (2). Substituting from (10) into (2) yields:∗N =⎧⎪⎪⎪⎨⎪⎪⎪⎩1 − λ1−γC1 − λ1−γN=1 − eα(1−γ )(1−1/J)J−α(1−γ )1 − e−α(1−γ )(J−1)2/Jγ ̸= 1ln λCln λN=J (ln J − 1) + 1(J − 1)2γ = 1(12)This depends upon α, γ and J , but not on β and L, and, fromProposition 1, it is decreasing in γ . For an IEA, an importantquestion is can cooperation be sustained when the number ofcountries becomes very large especially when there are globalexternalities. This can be answered with this model by looking atthe limit of the critical discount factor as the number of countriesgoes to infinity.Proposition 2. The limit of the critical discount factor required tosustain full cooperation using Nash-reversion trigger strategies as thenumber of countries goes to infinity is equal to one if γ < 1 and zeroif γ ≥ 1.Proof. If γ < 1 or 1− γ > 0 then the limit of the numerator andthe limit of the denominator in (12) as the number of countriesgoes to infinity are equal to one. Hence, the limit of the criticaldiscount factor is equal to one as the number of countries goes toinfinity. If γ = 1 then the numerator and the denominator bothgo to infinity as the number of countries goes to infinity, but thedenominator goes to infinity more rapidly than the numerator sothe limit of the critical discount factor is equal to zero. If γ > 1or 1 − γ < 0 then the numerator and the denominator bothgo to minus infinity as the number of countries goes to infinity,but the denominator goes to minus infinity more rapidly than thenumerator so the limit of the critical discount factor is equal tozero. ■To illustrate this proposition, Fig. 1 shows the critical discountfactor versus the number of countries for γ = 0, γ = 12 , γ = 1,nd γ = 2 when α = 1/2.3. ConclusionsProposition 1 is a quite general result, but Proposition 2 ap-lies only to the specific application, although it also holds forCournot oligopoly with linear demand and many firms. Hence,ne might conjecture that the result is more general.ata availabilityNo data was used for the research described in the article.cknowledgementsI thank an anonymous referee for comments. Any remainingrrors are mine.undingThis research did not receive any specific grant from fundinggencies in the public, commercial, or not-for-profit sectors.eferencesbreu, D., 1986. Extremal equilibria of oligopolistic supergames. J. Econom.Theory 39, 191–225. https://doi.org/10.1016/0022-0531(86)90025-6.breu, D., 1988. On the theory of infinitely repeated games with discounting.Econometrica 56, 383–396. https://doi.org/10.2307/1911077.arrett, S., 2005. Chapter 28 the theory of international environmental agree-ments. In: Mäler, K.-G., Vincent, J.R. (Eds.), Handbook of EnvironmentalEconomics. Elsevier, pp. 1457–1516. https://doi.org/10.1016/S1574-0099(05)03028-7.ó, P.D., Fréchette, G.R., 2018. On the determinants of cooperation in infinitelyrepeated games: A survey. J. Econ. Lit. 56, 60–114.reber, A., Fudenberg, D., Rand, D.G., 2014. Who cooperates in repeated games:The role of altruism, inequity aversion, and demographics. J. Econ. Behav.Organ. 98, 41–55. https://doi.org/10.1016/j.jebo.2013.12.007.riedman, J.W., 1971. A non-cooperative equilibrium for supergames. Rev.Econom. Stud. 38, 1–12.roto, E., Rustichini, A., Sofianos, A., 2018. Intelligence, personality, and gainsfrom cooperation in repeated interactions. J. Polit. Econ. 127, 1351–1390.https://doi.org/10.1086/701355.abater-Grande, G., Georgantzis, N., 2002. Accounting for risk aversion inrepeated prisoners’ dilemma games: an experimental test. J. Econ. Behav.Organ. 48, 37–50. https://doi.org/10.1016/S0167-2681(01)00223-2.

CRRA utility and the sustainability of cooperation in infinitely-repeated games

https://orca.cardiff.ac.uk/id/eprint/153829/4/1-s2.0-S0165176522003718-main.pdf

CRRA utility and the sustainability of cooperation in infinitely-repeated games

Abstract

Similar works

Full text

Available Versions

Online Research @ Cardiff