Time-Dependent Strategies in Repeated Asymmetric Public Goods Games

Hübner, Valentin; Hilbe, Christian; Staab, Manuel; Kleshnina, Maria; Chatterjee, Krishnendu Article — Published Version Time-Dependent Strategies in Repeated Asymmetric Public Goods Games Dynamic Games and Applications Provided in Cooperation with: Springer Nature Suggested Citation: Hübner, Valentin; Hilbe, Christian; Staab, Manuel; Kleshnina, Maria; Chatterjee, Krishnendu (2025) : Time-Dependent Strategies in Repeated Asymmetric Public Goods Games, Dynamic Games and Applications, ISSN 2153-0793, Springer US, New York, NY, Vol. 15, Iss. 5, pp. 1617-1645, https://doi.org/10.1007/s13235-025-00627-5 This Version is available at: https://hdl.handle.net/10419/330232 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Dynamic Games and Applications (2025) 15:1617–1645 https://doi.org/10.1007/s13235-025-00627-5 Time-Dependent Strategies in Repeated Asymmetric Public Goods Games Valentin Hübner1·Christian Hilbe2·Manuel Staab3·Maria Kleshnina4· Krishnendu Chatterjee1 Accepted: 26 January 2025 / Published online: 7 February 2025 © The Author(s) 2025 Abstract The public goods game is among the most studied metaphors of cooperation in groups. In this game, individuals can use their endowments to make contributions towards a good that benefits everyone. Each individual, however, is tempted to free-ride on the contributions of others. Herein, we study repeated public goods games among asymmetric players. Previous work has explored to which extent asymmetry allows for full cooperation, such that players contribute their full endowment each round. However, by design that work focusses on equilibria where individuals make the same contribution each round. Instead, here we consider players whose contributions along the equilibrium path can change from one round to the next. We do so for three different models – one without any budget constraints, one with endowment constraints, and one in which individuals can save their current endowment to be used in subsequent rounds. In each case, we explore two key quantities: the welfare and the resource efficiency that can be achieved in equilibrium. Welfare corresponds to the sum of all players’ payoffs. Resource efficiency relates this welfare to the total contributions made by the players. Compared to constant contribution sequences, we find that time-dependent contributions can improve resource efficiency across all three models. Moreover, they can improve the players’ welfare in the model with savings. Keywords Social dilemmas ·Public goods games ·Inequality ·Direct reciprocity Mathematics Subject Classification 91A05 ·91A06 ·91A10 ·91A20 Maria Kleshnina and Krishnendu Chatterjee have contributed equally to this work. BValentin Hübner v[email protected] 1Institute of Science and Technology Austria, 3400 Klosterneuburg, Austria 2Max Planck Research Group Dynamics of Social Behavior, Max Planck Institute for Evolutionary Biology, 24306 Plön, Germany 3School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD 4000, Australia 4School of Economics, The University of Queensland, Brisbane, QLD 4067, Australia 1618 Dynamic Games and Applications (2025) 15:1617–1645 1 Introduction Cooperation is typically conceptualised as a behaviour that is costly for the individual but beneficial to the group [39]. Examples of cooperation abound, ranging from small favours among friends to collective efforts to mitigate climate change. These cooperative interactions can, and have been, described with game theory [8,31]. This literature has produced rich predictions about potential mechanisms that can sustain cooperation [27,34]. One such mechanism is direct reciprocity [5,7,40,44]. Here, individuals are assumed to engage in the same interaction repeatedly, over many rounds. Repeated interactions allow players to condition their behaviour on the previous history of play. In this way, they can enforce mutual cooperation despite any short-run temptations to free-ride [13,14]. Traditionally, many models of direct reciprocity, especially in the evolutionary game theory literature, assume that interactions are symmetric [4,12,16,19,22,25,26,30,35,37, 38,41,42,46]. This means that players are completely interchangeable with respect to their actions and feasible payoffs. More recently, however, the evolution of cooperation among asymmetric players has received more attention [1–3,9,11,24,29,32,33,45]. This interest has also been spurred by empirical studies that explore the role of inequality in controlled experiments [6,10,17,21,28,36,43,47]. Oftentimes, these studies are based on some variation of the linear public goods game. In this game, players obtain their fixed endowments in the beginning of each round. Then they independently decide how much of their endowment they wish to contribute to the public good. Contributions are multiplied by some productivity factor, and the resulting amount is evenly split among all group members. There are various ways to allow for asymmetry in this game. For example, players may have unequal endowments, unequal productivities, or both. The main takeaway from the above-mentioned studies is that endowment inequality tends to be detrimental to cooperation. However, as shown by [20]and[23], there can be exceptions. If individuals already differ in their productivity, it can become easier to sustain full cooperation if they also differ in their endowments. As a rule of thumb, a player’s endowment ought to be larger the more productive that player is. However, the studies of [20]and[23] consider a rather restricted question. They ask: Under which conditions are there subgame perfect equilibria in which all players contribute their full endowment in every round? In particular, they thereby only consider equilibria whose resulting contribution sequence along the equilibrium path is constant. Instead, in the following we are interested in contribution sequences that can vary in time. We ask: Once individuals have the ability to make time-dependent contributions along the equilibrium path, to which extent can they achieve outcomes that are infeasible with constant contribution sequences? To this end, we study three different but related models of public good provision. The first is most convenient from a mathematical perspective. Here, players can make arbitrary (non-negative) contributions each round. In particular, contributions are not constrained by any endowments that individuals might have in that round. Instead, we only require that the players’ overall discounted contributions over the entire game are bounded (with the upper bound being arbitrary). We refer to this model as the ‘base model’. The second model reproduces typical public goods game models, such as the ones considered in [20]and[23]. Here, a player’s contribution each round is bounded by the player’s assigned endowment. Accordingly, we speak of the ‘endowment model’. Finally, the third model is a hybrid of the first two. Here, players obtain a fixed and constant endowment each round. But now they can decide to deposit some of this endowment into a savings account, which is then Dynamic Games and Applications (2025) 15:1617–1645 1619 1 2 (a) Player 1 contributes an amount of 1 12 (a) Player 2 contributes an amount of 2 (b) Player 1’s contribution is scaled up by 1 (b) Player 2’s contribution is scaled up by 2 (c) Together, the scaled contributions form the public good (d) Player 1 receives one half of the public good (d) Player 2 receives one half of the public good (e) Payoff is equal to share from public good minus contribution (e) Payoff is equal to share from public good minus contribution − 1=2=− 1 1 Fig. 1 A one-round asymmetric public goods game. To illustrate the base model, we consider n=2 players. They freely choose the size of their contributions, c1and c2. Contributions are enhanced by the individual productivity factors, which are r1=1.5andr2=1.1 in this example. The size of the public good is the sum of all enhanced contributions, rc=r1c1+r2c2. Each player receives an equal share of this sum. The payoff of each player then equals their share of the public good minus their contribution available in the next round. Under this assumption, contributions each round are not bounded by the players’ endowments anymore. Instead, they are bounded by the players’ accumulated endowments up to that point. We call this the ‘savings model’. For all three models, we consider the equilibrium outcomes that can be achieved with time-dependent contributions. We compare them to the possible equilibrium outcomes when players are required to make a fixed and constant contribution along the equilibrium path. We make this comparison based on two key quantities. One quantity is the group’s welfare in equilibrium (the total sum of the players’ payoffs). As we discuss in more detail below, this quantity is particularly relevant in the endowment model and in the savings model. The other quantity is an equilibrium’s resource efficiency (the ratio of the group’s welfare relative to the players’ total contributions). This quantity is relevant for all three models. We characterise under which condition time-dependent contributions allow for equilibria with larger resource efficiency (compared to equilibria based on constant contributions). We do so for arbitrary discount factors. In particular, we do not require that players are sufficiently patient, as often done in the classical folk theorem literature [13,14]. Our results depend on the group size and on the number of players with the highest productivity. In particular, we find that when there is a unique player with maximum productivity, time-dependent contributions provide an advantage. With respect to welfare maximisation, we find a similar result – but only for the savings model. 2 The Base Model 2.1 Model Setup We start with the base model (as illustrated in Fig. 1), which we will use to derive our first results. These results will also have important implications for the other two models studied subsequently. 1620 Dynamic Games and Applications (2025) 15:1617–1645 In the base model, a group of n≥2 players interacts for an indefinite sequence of rounds. In every round t, each player idecides which non-negative amount ci(t)to contribute towards the public good. There is no limit on how much players may contribute. Hence, ci(t)∈R≥0. These individual contributions can be collected in a vector, c(t)=(c1(t),...,cn(t)).We refer to a sequence c(t)tof contribution vectors as a contribution sequence, or as a ‘play’ of the game. If c(t)=c(0)for all times t, the contribution sequence is called constant. Otherwise, it is time-dependent. Contributions of each player iare multiplied by their productivity factor riand added to the public good. The total public good is then evenly shared among all players. This is a slight generalisation of the standard formulation of the public goods game, according to which every player has the same productivity. In the following, we use r=(r1,...,rn)to denote the vector of all productivities. Based on this notation, we can write payer i’s payoff πi(t)in round tas πi(t)=1 nrc(t)−ci(t). (1) We assume productivities satisfy 1<ri<nfor all players i. The first inequality ri>1 ensures that the group’s total payoff (across all members) is increasing in each player’s contributions. The second inequality, ri<n, on the other hand, ensures that each individual is tempted to give as little as possible. Together, these two inequalities render the game a social dilemma. For each player, there is a conflict between their private interest and the collective interest of the group. To define the players’ payoffs over the entire repeated game, we assume players value each subsequent round at a discount of δ, with 0<δ<1. Accordingly, when the contribution sequence c(t)tis bounded, we define the total payoff of each player as the weighted sum of their payoffs each round, ˆπi=(1−δ) ∞  t=0 δtπi(t). Here, the term 1 −δserves as a normalising factor. It ensures that repeated-game payoffs are comparable to the game’s one-shot payoffs. Because we assumed the contribution sequence to be bounded, the above sum is guaranteed to converge. We do not define a total payoff for unbounded contribution sequences. In a common alternative interpretation of repeated games with discounting, which is applicable to our base model and model variation I, but not model variation II, the number of rounds is finite and random, with δbeing not a discount factor but the round-wise continuation probability. This means that after each round, with probability δthe game continues for at least one more round and with probability 1 −δit ends. In that interpretation, all rounds have equal value given that they are played, and the expected number of rounds is 1/(1−δ). Intuitively, this suggests that higher values of δ, which mean longer games, are conducive to cooperation. Notation 1 In this base model, a game is fully specified by the players’ productivities rand by the discount factor δ. We denote the corresponding game as B(r,δ). For our subsequent analysis, it will be useful to consider the weighted sum of a player’s contributions after a given time t. Formally, these continuation contributions of player iare Dynamic Games and Applications (2025) 15:1617–1645 1621 defined as ¯ci(t)=(1−δ) ∞  τ=0 δτci(t+τ). (2) One can also define a sequence that collects the respective continuation contributions for each round, ¯ c(t)t=¯ c(0), ¯ c(1), ¯ c(2),....Wecall¯ c(t)tthe continuation contribution sequence associated with contribution sequence (c(t))t. Every contribution sequence uniquely specifies a continuation contribution sequence and vice versa. Analogously, we can also define continuation payoffs at time t, ¯πi(t)=(1−δ) ∞  τ=0 δτπi(t+τ). We write ¯ π(t)=¯π1(t),..., ¯πn(t)for the respective vector, and note that ˆ π=¯ π(0). Similarly, we write ˆ c=¯ c(0)for the continuation contributions at time zero. We call ˆ cthe total contribution vector. By linearity of the one-round payoffs (1), we have ˆπi=1 nrˆ c−ˆci.(3) That is, each player’s total payoff is uniquely determined by the total contribution vector. By definition, this payoff ˆπiis the quantity that player iaims to maximise. Players make their decisions based on their strategies. A strategy σifor player iis a function that assigns to each initial contribution sequence c(0), c(1),...,c(t−1)anext contribution value ci(t)=σi(c(τ))τ<t∈R≥0. A strategy is called bounded if it always produces a bounded contribution sequence (irrespective of the co-players’ strategies). An assignment of one strategy to each player (σi)iis called a strategy profile. A strategy profile is bounded if all its strategies are bounded. 2.2 Sustainable Contribution Sequences In the following, we are particularly interested in those strategy profiles that form a subgame perfect equilibrium [SPE, or ‘equilibrium’, see15]. We say a bounded strategy profile is in equilibrium when no player has an incentive to deviate, after no finite sequence of moves. Formally, (σi)iis in equilibrium if there is no initial contribution sequence c(0),...,c(t−1) such that some player jcould get a larger payoff by deviating towards another bounded strategy σ∗ jafter that time t.ForagivengameB(r,δ), we call a contribution sequence sustainable if it is the contribution sequence of some equilibrium strategy profile. A total contribution vector ˆ cis sustainable if it is the total contribution vector of a sustainable contribution sequence. To derive our main results, we make extensive use of the previously published Theorem 1below. This theorem gives us a comfortable characterisation of sustainable contribution sequences. Theorem 1 ([23]) For a given game B(r,δ), define an associated n×nmatrixD=(Dij), called the productivity matrix in zero-diagonal form, by Dij =rj/(n−ri)if i= j 0ifi=j.(4) 1622 Dynamic Games and Applications (2025) 15:1617–1645 Then a contribution sequence (c(t))tis sustainable if and only if the associated continuation contributions satisfy ¯ c(t)≤δD¯ c(t+1)for all t.(5) With Theorem 1, we can determine whether a given contribution sequence is sustainable by checking if the associated continuation contribution sequence (¯ c(t))t,asdefinedby(2), satisfies (5)forallt. The two following corollaries are immediate consequences of this theorem. Corollary 2 For any game B(r,δ), the set of sustainable contribution sequences (c(t))t,the set of sustainable continuation contribution sequences (¯ c(t))t, and the set of sustainable total contribution vectors ˆ care closed under addition and multiplication by a non-negative scalar (that is, they are convex cones). This convexity result implies that whether or not a contribution sequence is sustainable only depends on the relative magnitude of the players’ contributions. This result holds because payoffs depend linearly on contributions. Therefore, scaling all contributions up or down by the same positive factor does not affect whether or not the equilibrium conditions are satisfied. Corollary 3 In a given game B(r,δ), a constant contribution sequence of (ˆ c)tis sustainable if and only if ˆ c≤δDˆ c.(6) So we have a set of nlinear constraints that defines the set of feasible constant contribution sequences. We can use Corollary 3to derive a version of the folk theorem, applied to our setup. The folk theorem famously relates the possible equilibrium payoffs in the repeated game to the properties of the one-shot payoffs [13,14]. To state our version, we note that in our public goods game, players can always guarantee a non-negative payoff (by not contributing anything). Hence, we say a contribution vector ˆ cfor the one-shot game is individually rational if it yields a non-negative payoff to each player. Theorem 4 (Folk theorem of repeated games) Constant contributions (ˆ c)t∈R≥0are sustainable in the game B(r,δ)for sufficiently large δif and only if ˆ cis individually rational. The condition for a contribution vector ˆ cto be individually rational can be written as max iˆci≤1 nrˆ c.(7) An equivalent formulation of the folk theorem is therefore: the constant contribution sequence (ˆ c)tis sustainable for sufficiently large δif and only if Eq. (7) holds. The above results allow us to characterise the properties of sustainable contribution sequences. Perhaps one of the most important properties is whether or not the contribution sequence entails at least some cooperation. More specifically, we define a play to be nondefective if at least one player makes a positive contribution in at least one round (i.e., ˆ c=0). Otherwise we call the play defective. It is easy to see that for non-defection to be sustainable, not just one, but at least two players have to make positive contributions. This is because a hypothetical lone non-defector would benefit from deviating towards full defection. If non-defection is sustainable in a given game B(r,δ),thenwesaythatB(r,δ)allows for nondefection. Any productivity vector r(satisfying the general requirement ri>1foralli) allows for non-defection when δis sufficiently large [20, SupplementaryInformation,Proposition 2]. Dynamic Games and Applications (2025) 15:1617–1645 1623 2.3 Welfare and Resource Efficiency While the binary distinction between defection and non-defection is useful, not all forms of non-defection are equally desirable. After all, even non-defective contribution sequences might result in payoffs arbitrarily close to the full defection payoff of zero. Therefore, in the following we introduce two other key metrics of interest. The first metric is the (overall) welfare Wof a given play, which equals the sum of all payoffs, W= n  i=1 ˆπi.(8) This welfare can be expressed as a function of the total contribution vector ˆ cas W(ˆ c)=(r−1)ˆ c.(9) By this equation, non-defective plays have W>0, whereas defective plays have W=0. By Corollary 2, any non-defective contribution sequence can be scaled arbitrarily without affecting their sustainability. It follows that our base model allows for arbitrary welfares. As an alternative metric that is still relevant with unlimited resources, we measure how efficiently the players are able to use them. The resource efficiency Eof a non-defective play is defined as the sum of all payoffs divided by the sum of all contributions, E=n i=1ˆπi n i=1ˆci .(10) Resource efficiency, too, is a function of the total contribution vector: E(ˆ c)=rˆ c 1ˆ c−1.(11) The first term on the right hand side of Eq. (11) can be slightly rewritten, as rˆ c 1ˆ c= n  i=1 ci c1+...+cn ·ri. This representation allows us to interpret this term as a weighted mean of the players’ productivities; the weights correspond to the players’ contributions. In particular, this observation implies that E(ˆ c)is always in between miniri−1andmax iri−1. Whether or not the upper bound (or equivalently, the lower bound) can be realised depends on which players contribute in equilibrium. For example, the upper bound can be realised if and only if there is an equilibrium in which only those players iwith ri=maxjrjmake a contribution. That is only possible if there are multiple players with maximum productivity. Example It is instructive to illustrate these concepts with a two-player game (which we continue to use throughout this article). Consider the game B(1.5,1.1),0.9.Thatis, there are n=2 players with productivities r1=1.5andr2=1.1(asinFig.1), and the discount factor is δ=0.9. In this example, the value of the matrix Dis D=0r2/(2−r1) r1/(2−r2)0=011 /5 5 /30≈02.2 1.666 0 . Suppose player 1 makes a constant contribution ˆc1=7 in every round, whereas player 2 makes the constant contribution ˆc2=5. It follows that the total size of the public good is rˆ c=1.5·7+1.1·5=16. Thus, each player’s share of the public good is 8, and their 1624 Dynamic Games and Applications (2025) 15:1617–1645 payoffs according to Eq. (3)are ˆπ1=8−7=1and ˆπ2=8−5=3. Because both payoffs are non-negative, the respective constant contribution sequences are individually rational. Hence, by the folk theorem, they are sustainable in the repeated game for sufficiently large δ. For this case of n=2, Eq. (6) in Corollary 3takes the form of the following system of inequalities: ˆc1≤δr2 2−r1 ˆc2(12) ˆc2≤δr1 2−r2 ˆc1(13) With these, we can verify that the given discount factor δ=0.9 is indeed sufficiently large. According to Eq. (8), the resulting welfare is W=3+1=4, and according to Eq. (10), resource efficiency is E=4/12 ≈0.333. If contributions were ten times larger, welfare would increase to 40 but the resource efficiency would remain the same. Instead, suppose now that the two players contribute equal constant amounts, say ˆc1= ˆc2=6. By the inequalities (12–13), this contribution vector is also sustainable. It yields a payoff of 1.8 for each player. Thus, the welfare is W=3.6, whereas resource efficiency is E=0.3. We conclude that equal contributions are less resource efficient, compared to the previous example with unequal contributions. This is intuitive because in the previous example, the more productive player 1 contributed a larger share. In light of these observations, it is natural to ask what the optimal ratio of the two player’s contributions is, if we aim to maximise resource efficiency in equilibrium. Again from the inequalities (12–13), we see that when r1>r2(as in our example), this ratio is given by ˆc1 ˆc2 =δr2 2−r1 . If for example ˆc1=10, then ˆc2=500/99 ≈5.051, which yields payoffs of ˆπ1≈0.278 and ˆπ2≈5.227. The resulting welfare is W≈5.505 and resource efficiency is Ec sup ≈0.366. Note that this maximum resource efficiency depends on the discount factor δ.Ifδwere larger, even higher efficiencies would be possible. In contrast, if δwere lower, either Ec sup would be lower, or the game might not allow for non-defection at all. In the above example, we only considered the simple case of constant contributions. We now address the question of whether we can achieve higher resource efficiency when contribution sequences are allowed to be time-dependent. 2.4 Efficiency with Time-Dependent Contributions If we only consider the binary distinction between whether or not a game allows for nondefection, then constant and time-dependent contributions are equally effective. Specifically, if a game has any non-defective equilibrium at all, then it also has a non-defective equilibrium with a constant contribution sequence [Corollary 6 of23]. However, below we show that within the space of non-defective equilibria, time-dependent contributions can indeed enable outcomes that are not sustainable otherwise. To this end, we first introduce some notation. Notation 2 For two vectors vand w,wewritev≤1wif vi≤wifor all iand vi=wifor at most one i.Inotherwords,v≤1wcorresponds to v≤wwith equality in at most one component. Using this notation, we can characterise which total contribution vectors are sustainable with time-dependent contribution sequences. Dynamic Games and Applications (2025) 15:1617–1645 1631 Round 0 Round 3 Round 4Round 2Round 1 Round 5 Contributed ( ) Consumed ( ) Deposited ( ) ×−1 ×−1 ×−1 ×−1 ×−1 Fig. 5 The savings model. Here, we depict the first six rounds of a player’s gameplay. In each round, the player not only has their endowment eiavailable, but also whatever resources they deposited in the previous round, plus interest at the rate δ−1−1. They decide which part of that they want to contribute, which part to consume, and which part to deposit to the savings account 4 Model Variation II: A Model with Savings The savings model builds on the earlier endowment model. Again, each player iobtains a fixed endowment eievery round. However, now players have three options for how to spend their endowment, rather than two. They can contribute to the public good, consume parts or all of the endowment privately, or they can make a deposit into a savings account. Savings pay interest at the rate of (δ−1−1)per round – which exactly corresponds to the time value of money at the discount factor δ. These savings can then be spent in future rounds, either to contribute to the public good or for private consumption. More specifically, each round proceeds as follows. In the beginning of each round t, players receive an endowment ei. In addition, they have access to an amount of si(t)on their savings account (in the very first round, savings are set to zero). Players then decide which amount pi(t)to consume privately, which amount ci(t)to contribute to the public good, and which amount di(t)to deposit into the savings account. These variables need to satisfy the budget constraint ei+si(t)=pi(t)+ci(t)+di(t). Contributions to the public good and private consumption directly enter the player’s payoff function, πi(t)=1 nrc(t)+pi(t). Deposits, on the other hand, determine a player’s savings in the beginning of the next round, si(t+1)=δ−1di(t). At time t+1, the process is repeated; individuals again have to decide how much to consume, to contribute, and to save, see Fig. 5. Total payoffs (across all rounds), welfare, and resource efficiency are then defined as in the previous two models. Notation 4 The savings model uses the same parameters as the endowment model: productivities r, endowments e, and the discount factor δ. We denote the game as S(r,e,δ). We make the following observations about the savings model: First, in our implementation of this model, savings are payoff-neutral: The interest earned over one round is exactly offset by a player’s discounting of future rewards. Second, without the opportunity for saving, 1632 Dynamic Games and Applications (2025) 15:1617–1645 this model recovers the endowment model as discussed in the previous section and studied in [20]. Third, the savings model is equivalent to saying that endowments only apply as a constraint to the cumulative contributions. That is, each player iis required to play such that t τ=0δτci(t)≤t τ=0δτeifor all t, but without the stronger requirement that ci(t)≤eifor all t. Finally, since resource efficiency is equal to surplus welfare divided by total contributions, maximising resource efficiency in the base model is equivalent to maximising welfare in the savings model when the endowments are also an optimisation variable. To state our main results for this section, we first define the notion of a welfare supremum with and without savings. For given parameters r,e,δ, the welfare supremum with savings Ws sup(e)is the supremum of welfare over all equilibria of the game S(r,e,δ). It quantifies the maximum value that the group can derive from cooperation. Similarly, we define the welfare supremum without savings, Wsup(e), as the supremum of welfare over all equilibria that satisfy di(t)=0foralliand t. It quantifies the maximum value that the group can derive without ever saving any amount. Equivalently, it corresponds to the welfare supremum of the endowment model, E(r,e,δ). We interpret the difference Ws sup(e)−Wsup(e)as the (positive or zero) advantage that savings can provide. Theorem 12 Let (r,δ) allow for non-defection. Take any endowment distribution e.Then savings provide no advantage (i.e. Ws sup(e)=Wsup(e)), if and only if e≤δDe.(18) Without savings, ˆ c=erequires constant contributions. So by Corollary 3,Eq.(18)is equivalent to ˆ c=ebeing sustainable without savings. Therefore, Theorem 12 states that exactly one of the following is the case: Either full contributions are sustainable without savings, or savings provide an advantage for welfare. In particular, for any rand e,when δis sufficiently low, savings provide an advantage. Alternatively, for any rand δ,when eis sufficiently unequal, savings provide an advantage ( [20], Supplementary Information Proposition 3). Savings also provide an advantage from the perspective of a social planner who chooses an endowment distribution with the aim of maximising welfare. To assess this, we consider thesupremaofWs sup(e)and Wsup(e)over all possible endowment distributions e.From Theorem 8, we can derive the following result: Theorem 13 Let (r,δ)allow for non-defection. Let rmax =maxiri, and let m be the number of players with productivity rmax.ThensupeWs sup(e)>supeWsup(e)if and only if 1+δ(m−1)·rmax <n.(19) To understand the theorem intuitively, consider again the case that there is only a single player with maximum productivity rmax, i.e. that m=1. Then Theorem 13 states that strictly better welfare is possible when players are permitted to save part or all of their endowment for later rounds, compared to when they are not. This is because with saving, they can play a more resource-efficient time-dependent contribution sequence and still productively use all of their endowment, whereas without, they are restricted to make constant contributions in order to maximise welfare. Conversely, as another special case of Theorem 13, we conclude that savings never provide a welfare advantage if all players have the same productivity. Example We revisit the same example as before, now in the savings model: S((1.5,1.1),(0.2,0.8),0.9). In the endowment model, we had to scale the optimally resource-efficient contribution sequence to c1(0)=0.2 so that players do not exceed their Dynamic Games and Applications (2025) 15:1617–1645 1633 endowment limits. With savings, it is enough that at no point in time their cumulative contributions exceed the endowment limit. A simple example of a superior contribution sequence is as follows. In round 0, player 1 contributes and consumes nothing (c1(0)=p1(0)=0) and deposits everything (d1(0)=e1=0.2). In round 1, with the interest received, player 1 has savings of approximately 0.222 and again receives an endowment of 0.2, which makes for a total available amount of 0.422. Of this, player 1 contributes c1(1)≈0.222 and again deposits d1(1)=0.2. In round 2, savings with interest again make up 0.222, and player 1 continues with contributing 0.222 and depositing 0.2 in every subsequent round. Player 2, on the other hand, from the beginning simply contributes c2(t)≈0.333 in every round and privately consumes the rest, p2(t)≈0.467, without depositing anything. The welfare of this contribution sequence is W≈1.164, which is more than the optimal value without saving, W=1.13. Theorem 12 predicts that saving provides an advantage like this as long as full contributions are not sustainable, which is the case here. The optimal welfare over all endowment distributions, supeWs sup(e), requires the endowment distribution e=(11 /16,5 /16). This is exactly the endowment distribution at which the maximally resource efficient contribution sequence can be played in such a way that all endowments are eventually contributed: Player 1 contributes c1(t)=11/16 in every round. Player 2 deposits everything in round 0 (d2(0)=e2=5/16). Thereafter, player 2 contributes 50/(16 ·9)and deposits 5/16 in every round. (The sequence of contributions is identical to that of the earlier example in Sect.2, up to rescaling by a factor of 160/11, and thus also maximally resource efficient.) In this sequence, all resources are used productively and none are consumed privately, which means that the optimal resource efficiency also translates to optimal welfare. Indeed, the welfare is W=supeWs sup(e)=1+Esup =1.375. The fact that this is greater than the optimum without saving over all endowment distributions, which by Proposition 11 is supeWsup(e)=1+Ec sup ≈1.366, is predicted by Theorem 13. 5 Discussion The repeated public good game is one of the major models in (evolutionary) game theory to understand cooperation in groups. This literature describes how individuals can use conditionally cooperative strategies to sustain outcomes that are infeasible in one-shot encounters. Yet when describing the possible equilibrium outcomes, many previous studies implicitly restrict their analysis to the case that players make the same constant contribution each round [e.g.20,23]. Instead, here we study the effect of time-dependent contributions. Contrary to many other models of reciprocity, we allow players to select their actions from a continuum between full defection and full cooperation. We explore to which extent individuals can obtain better outcomes (e.g., a better resource efficiency or welfare) when they are able to vary their contributions along the equilibrium path. From the outset, it is not clear whether time-dependent contributions provide any substantial advantage at all. After all, suppose players could achieve a superior outcome with contributions c(t)tthat vary in time. Then players might achieve just the same outcome by instead making a constant contribution ˆ ceach round, where ˆ cis the appropriate (timediscounted) average contribution per round, ˆ c=(1−δ)tδtc(t). With respect to their payoff implications, the two sequences c(t)tand (ˆ c)tare identical. After all, by Eq. (3), payoffs only depend on the players’ total contributions across all rounds. As a result, the two sequences generate the same resource efficiency and welfare. However, as we show in this article, the two sequences may differ in their sustainability. There are instances in which the 1634 Dynamic Games and Applications (2025) 15:1617–1645 time-dependent sequence c(t)tcan be realised by a subgame perfect equilibrium, whereas the constant sequence (ˆ c)tcannot. To make this point, we study three different models: a base model, a model with endowment constraints, and a model with savings. In the base model, players are allowed to make arbitrary contributions each round (the only requirement is that the sequence of contributions does not diverge). This setup imposes minimal constraints on the players’ behaviour, and it is convenient to work with mathematically. In contrast, the other two models are perhaps more realistic (and hence they have been studied more frequently). For example, the endowment model corresponds to the classical setup that is also frequently used in experiments [e.g.6,10, 21,28]. Here, contributions are constrained by the endowments that the players receive each round. The savings model is similar, but in addition it allows players to (payoff-neutrally) transfer some of their endowments to future rounds. Interestingly, many of our results for these last two model variants are directly related to our findings in the base model. As an example, with Theorem 13, we characterise under which circumstances savings provide a welfare advantage in the savings model. The respective result is directly related to whether or not time-dependent contributions provide an advantage in the base model, Theorem 8.These similarities between those theorems highlight how several findings in the more abstract base model carry over to more applied settings. Interestingly, the respective theorems also suggest that for our results, some asymmetry among players is crucial. Specifically, when players are identical with respect to their productivities, Theorem 8shows that time-dependent contributions do not grant any advantage. Any resource efficiency that can be sustained with time-dependent contributions can already be sustained with constant contributions. But once players differ in their productivities, it becomes fairly easy for time-dependent contributions to be superior. In fact, such an advantage is guaranteed when the group contains a single player whose productivity exceeds everyone else’s. Overall, our findings highlight the impact of variable contributions on resource efficiency and, more generally, on the sustainability of cooperation. They suggest that by focussing solely on constant contributions, we may overlook important equilibria that can arise in dynamic settings. Appendix A Proofs A.1 Proof of Theorem 5 Proof of Theorem 5Clearly ˆ c=0is sustainable, so it is sufficient to show the statement for ˆ c= 0. First we will show that any sustainable ˆ c= 0satisfies ˆ c≤1Dˆ c. By Theorem 1,wehave for any sustainable ¯ c(t)that δ¯ c(1)≤¯ c(0)≤δD¯ c(1). (A1) Left-multiplying with the non-negative matrix Don both sides of the first inequality, we obtain δD¯ c(1)≤D¯ c(0) with equality in the ith component exactly if δcj(1)=cj(0)for all j= i. Dynamic Games and Applications (2025) 15:1617–1645 1635 Together with the second inequality of (A1), this gives ¯ c(0)≤D¯ c(0), or equivalently ˆ c≤Dˆ c,(A2) where equality in the ith component requires δcj(1)=cj(0)for all j= i. If (A2) has equality in at least two components, then δcj(1)=cj(0)for all j,soδc(1)= c(0). Analogously, equality in two components requires δc(2)=c(1), etc., so the sequence (c(t))tdiverges and is not a valid contribution sequence. That is a contradiction, so we can have ˆci=(Dˆ c)ifor at most one i. Now we will show that if some ˆ c= 0satisfies ˆ c≤1Dˆ c,thenˆ cis sustainable with a continuation contribution sequence (¯ c(t))tthat satisfies ˆ c≤¯ c(t)for all t. Assume first that the stronger condition 0<ˆ c<Dˆ cholds. Take ε>0 such that 1 +ε<δ −1and (1+ε)x≤Dx.Letvbe the Perron eigenvector of D, scaled so that v≤x. Let finally T= δmin maxixi vi−minixi vi ε(1−δmin) or T=0, whichever is larger. Let x=ˆ cfor a given ˆ cwith 0<ˆ c<Dˆ c. We define a continuation contribution sequence (¯ c(t))tby ¯ c(t)=((1+ε)δ)−t(x+εtv) for all 0 ≤t≤Tand ¯ c(t)=(1+ε)−Tδ−(T+1)min i xi vi +εTv for all t>T. If we show that it obeys (5)forallt, we know it is sustainable. Since ¯ c(0)=x, that is enough to prove the first statement of the present theorem. We also see that (¯ c(t))tis non-decreasing. Therefore, ˆ c=¯ c(0)≤¯ c(t)for all t, which is the second statement of the theorem. For 0 ≤t<T, the first inequality, δ¯ c(t+1)≤¯ c(t), follows from v≤x as follows. First, we multiply with εand add ε2tv, which is non-negative, on the right-hand side: εv≤εx+ε2tv We add x+εtvon both sides and factor out: x+ε(t+1)v≤(1+ε)(x+εtv) We multiply with δ((1+ε)δ)−(t+1)on both sides: δ((1+ε)δ)−(t+1)(x+ε(t+1)v)≤((1+ε)δ)−t(x+εtv) This is equivalent to δ¯ c(t+1)≤¯ c(t) 1636 Dynamic Games and Applications (2025) 15:1617–1645 by the definition of ¯ c(t). The second inequality, ¯ c(t)≤δD¯ c(t+1), follows from v=δmin Dv. We replace δmin with δ, which is larger or equal, on the right-hand side: v≤δDv By construction, 1 −ε<δ −1, so we can write: (1+ε)v<Dv Multiply by εton the left-hand side and by ε(t+1)on the right-hand side: (1+ε)εtv<ε(t+1)Dv Sum with the inequality (1+ε)x≤Dx, which also holds by construction of ε: (1+ε)(x+εtv)<D(x+ε(t+1)v) Multiply by δ((1+ε)δ)−(t+1)on both sides: ((1+ε)δ)−t(x+εtv)<δ((1+ε)δ)−(t+1)D(x+ε(t+1)v) This is equivalent to ¯ c(t)≤δD¯ c(t+1). Next, we consider t=T: The first inequality follows from x+εTv≤(1+ε)(x+εTv). First, we replace xby minixi viv, which is at most as large in each component: min i xi vi v+εTv≤(1+ε)(x+εTv) Then, we multiply by δ((1+ε)δ)−(T+1)on both sides: δ((1+ε)δ)−(T+1)min i xi vi +εTv≤((1+ε)δ)−T(x+εTv) This is by definition equivalent to δ¯ c(T+1)≤¯ c(T). The second inequality follows from δmin maxixi vi−minixi vi ε(1−δmin)≤T, which is by definition of T. We remove the ceiling function and multiply by ε(1−δmin)on both sides: δmin max i xi vi −min i xi vi ≤(1−δmin)εT We move minixi vito the right-hand side, δminεTto the left, and then multiply by δ−1 minv: max i xi vi v+εTv≤min i xi vi +εTδ−1 minv Dynamic Games and Applications (2025) 15:1617–1645 1637 Now, we can replace maxixi vivby x, which is at most as large in each component, and δ−1 minv by Dv, which is equal: x+εTv≤min i xi vi +εTDv We multiply by ((1+ε)δ)−Ton both sides: ((1+ε)δ)−T(x+εTv)≤δD(1+ε)−Tδ−(T+1)min i xi vi +εTv This is by definition equivalent to ¯ c(t)≤δD¯ c(t+1). Finally, for t>T,wehave¯ c(t)=¯ c(t+1). The first inequality is trivially true, and the second inequality is true since ¯ c(t)is a multiple of v.Sowehaveshownthatˆ cis sustainable if 0<ˆ c<Dˆ c. Now we will reduce the more general case of ˆ c= 0and ˆ c≤1Dˆ cto this in two sequential steps. First, let some ˆ c= 0have 0<ˆ c≤1Dˆ cwith equality of the weak inequality in exactly one component i.Letx(ε) =ˆ c−εui,whereuiis the ith standard unit vector. Since x(ε) →ˆ c as ε→∞and ˆcj<(Dˆ c)jfor all j= i, we can choose ε>0 sufficiently small such that ˆcj=xj(ε) < (Dx(ε))jfor all j= ias well, and 0 <xi(ε).Wethenhavexi(ε) < ˆci,but (Dx(ε))i=(Dˆ c)i,soalsoxi(ε) < (Dx(ε))i. Hence x(ε) satisfies 0<x(ε) < Dx(ε) and ˆ c≤Dx(ε). By the above result, we can choose a continuation contribution sequence (c∗(t))t such that ˆ c∗=x(ε). Now consider the sequence (ˆ c,δ−1¯ c∗(0), δ−1¯ c∗(1), δ−1¯ c∗(2),...).By definition, ¯ c∗(0)=x(ε), and we have δ(δ−1x(ε)) ≤ˆ c≤δD(δ−1x(ε)), so by Theorem 1, the sequence we constructed is a sustainable continuation contribution sequence. It begins with ˆ c, so the statement of the Lemma holds for ˆ c. Now, the only case that remains is ˆ c≤1Dˆ cand 0<ˆ c= 0. In the definition of the public goods game, we imposed the conditions 1 <riand ri<nfor all i.Weusedri<nin the proofs about sustainability (it is necessary for Dbeing well defined and positive), but 1 <ri was only used to prove statements about maximal welfare. So we can consider games that only satisfy r>0instead of r>1, and the same results about sustainabililty will apply, including the ones from this proof. Let ˆ c= 0be any total contribution vector satisfying ˆ c≤1Dˆ c.Letnbe the number of nonzero components of ˆ c. W.l.o.g. let these be the first ncomponents of ˆ c. Necessarily n>1. Let ˆ cbe the first ncomponents of ˆ c,andletrbe the n-vector defined by r i=nn−1rifor all 1 ≤i≤n. Consider the game B(r,δ) and its zero-diagonal productivity matrix D. We have D ij =Dij for all 1 ≤i,j≤n.Sinceˆ csatisfies ˆ c≤1Dˆ c, consequently ˆ calso satisfies ˆ c≤1Dˆ c.Butˆ cadditionally satisfies 0<ˆ c. So by the case handled above, there is a sustainable continuation contribution sequence (c(t))tstarting with ˆ c.Let(c(t))tbe the n-player sequence with c i(t)=ci(t)for all i≤nand all t,andc i(t)=0foralli>nand all t.UsingEq.5, we can see that sustainability of (c(t))tfollows trivially from sustainability of (c(t))t. So we have found a continuation contribution sequence starting with ˆ c. We have thus shown in the most general case that if some ˆ c= 0satisfies ˆ c≤1Dˆ c,then ˆ cis sustainable. Together with the converse, which we already showed, this completes the proof.  1638 Dynamic Games and Applications (2025) 15:1617–1645 A.2 Proof of Theorem 8 Proof of Theorem 8Theorem 5says that a total contribution vector of ˆ cis sustainable exactly if ˆ c≤1Dˆ c. Corollary 3says that constant contributions of ˆ care sustainable exactly if ˆ c≤δDˆ c. Let Fbe the region defined by ˆ c≤1Dˆ c.Let Esup := sup ˆ c∈F\{0} E(ˆ c)=sup ˆ c∈F\{0} E(ˆ c), where Fis the closure of F, which is given by ˆ c≤Dˆ c. Firstly, note that Esup is well defined, since E(ˆ c)is bounded above by maxiri. We observe that E(ˆ c)attains a maximum on F\{0}: We have {E(ˆ c)|ˆ c∈F\{0}} = {E(ˆ c/ˆ c)|ˆ c∈F\{0}} = {E(ˆ c)|ˆ c∈F∩Sn−1}, where Sn−1={x∈Rn|x=1}.SinceFis closed, F∩Sn−1is a compact set. So we can write Esup =max ˆ c∈F\{0} E(ˆ c). Now we are ready to prove the statement of the theorem. But instead of showing that a sustainable time-dependent contribution sequence that is more resource-efficient than all constant contribution sequences exists exactly if (δ(m−1)+1)rmax <n, as stated in the theorem, we will show the following equivalent statement: A constant contribution sequence with resource efficiency Esup exists exactly if n≤(δ(m−1)+1)rmax.(A3) The following statements are equivalent for ˆ c∈Rn: ˆ c∈F ˆ c≤Dˆ c ∀iˆci≤ j=i rj n−ri ˆcj(A4) ∀inˆci≤ j rjˆcj.(A5) Take any ˆ c∈F\{0}such that E(ˆ c)=Esup that is sustainable with a constant contribution sequence, which we assume exists. This ˆ cmaximises E(ˆ c)over all ˆ c= 0satisfying the above inequality (A5). This implies the following statement (∗): There are no i1,i2such thatri1>ri2 and nˆci1<jrjˆcjand ˆci2>0. Otherwise, we could increase ˆci1and decrease ˆci2by equal amounts to increase E(ˆ c)while also staying within F\{0}. Dynamic Games and Applications (2025) 15:1617–1645 1639 Since ˆ cis sustainable with a constant contribution sequence, we have ˆ c≤δDˆ c,whichis equivalent to ∀iˆci≤δ j=i rj n−ri ˆcj.(A6) At least two ˆcimust be strictly positive, so j=i rj n−riˆcj>0foreveryi.Sinceδ<1, (A6) implies ∀iˆci< j=i rj n−ri ˆcj. So the statement (∗) simplifies to: There are no i1,i2such that ri1>ri2and ˆci2>0. This means that for any i,ifˆci>0, then ri=rmax. From (A6), we get ∀i(n−(1−δ)ri)ˆci≤δ j rjˆcj,(A7) where  j rjˆcj=rmax  j ˆcj. Exactly mcomponents of ˆ care non-zero. So choose some isuch that ˆci>0and ˆci≥1 m j ˆcj. Inserting into (A7), we get (n−(1−δ)rmax)1 m j ˆcj≤δrmax  j ˆcj. The ˆcjsum to 1. We multiply with mon both sides and get n−(1−δ)rmax ≤δrmaxm. By simple rearrangement, this is equivalent to (A3). So (A3) being false is a necessary condition for the existence of a constant contribution sequence with resource efficiency Esup, which was our only assumption. Now instead assume conversely that (A3) holds. Let ˆci=1forallisuch that ri=rmax, and let ˆci=0 for all other i. We can check easily from the definitions that ˆ csatisfies ˆ c≤δDˆ c, so it is sustainable with a constant contribution sequence. The resource efficiency of ˆ cis E(ˆ c)=rmax, so it is maximal. Therefore, (A3) is also a sufficient condition, and hence an exact condition, for the existence of a constant contribution sequence with resource efficiency Esup. A.3 Proof of Theorem 12 Proof of Theorem 12 We will show that if e≤δDe,thenWs sup(e)=Wsup(e),andife≤ δDe, then Ws sup(e)>Wsup(e). Firstly, if e≤δDe, then by Corollary 3, the constant contribution sequence (c(t))t=(e)t is sustainable. So the total contribution vector ˆ c=eis sustainable without saving. Since e 1640 Dynamic Games and Applications (2025) 15:1617–1645 is an upper bound on the total contribution vector, reis also an upper bound on welfare in general, and we have Ws sup(e)=Wsup(e)=re. Now, assume that e≤ δDe. By Proposition 11, there is always a total contribution vector ˆ cthat satisfies W(ˆ c)=Wsup(e)and is sustainable with a constant contribution sequence in E(r,e,δ), i.e. without saving. Take such a ˆ c. By Corollary 3,wehaveˆ c≤δDˆ c.Soˆ c= e, meaning there is some isuch that ˆci<ei. Fix such an i. Define ˆ c(ε) =ˆ c+ε(e−ˆ c)for ε≥0. Clearly ˆ c(ε) →ˆ cas ε→0. Since ˆ c≤δDˆ c, since Dˆ cis a positive vector, and since δ<1, we have ˆ c<Dˆ c. So we can choose ε>0 sufficiently small such that ˆ c(ε) < Dˆ c(ε) as well. By Theorem 5,ˆ c(ε) is thus a sustainable total contribution vector in S(r,e,δ).Butˆ c(ε) ≥ˆ cand ˆc(ε)i>ˆci.SoW(ˆ c(ε)) > W(ˆ c). Consequently, Ws sup(e)>Wsup(e). A.4 Proof of Theorem 13 Proposition 14 Take any game B(r,δ)that allows for non-defection. Then for any ˆ c∈Rn ≥0 satisfying n i=1ˆci≤1, the following are equivalent: 1. A total contribution vector of ˆ cis sustainable in the game B(r,δ). 2. There exists an endowment distribution esuch that a total contribution vector of ˆ cis sustainable in the game S(r,e,δ). In a game S(r,e,δ), a contribution sequence (c(t))tis called sustainable if there is a play that results in contribution sequence (c(t))t. It is easy to check that a contribution sequence (c(t))tis sustainable in S(r,e,δ)if and only if ˆ c≤δte+δt+1¯ c(t+1)(A8) for all t≥0. Inequality A8 simply states that every player ihas enough resources in every round tin order to make a contribution of ci(t)as long as they never consume any of their available resources privately. In the game B(r,δ), the Grim strategy profile G((c(t))t)for a contribution sequence (c(t))tis the pure strategy profile G((c(t))t)=(σi)idefined as follows: In each round tand for each ithe strategy σicontributes ci(t)if all players have so far also played according to (c(t))t, but otherwise contributes 0. In the base model, a contribution sequence (c(t))tis sustainable in a given game if and only if its associated Grim strategy profile G((c(t))t)is a SPE of the game [23]. In the game B(r,e,δ), the Grim strategy profile Gs((c(t))t)for a contribution sequence (c(t))tis defined as follows. First, we recursively construct a deposit sequence (d(t))t. For each tand each i,letdi(t) be the minimal dsuch that there is a play that results in contribution sequence (c(t))tand an initial deposit sequence for player iof di(0),…,di(t−1),d. We can check that a minimum is indeed attained. Then there also exists a play that results in contribution sequence (c(t))tand deposit sequence (d(t))t. This also uniquely determines the private consumption sequence (p(t))t. Now the Grim strategy Gs((c(t))t)ifor player iplays as follows. In each round t,ifall players have so far played according to Gs((c(t))t), then player icontributes ci(t), privately consumes pi(t), and deposits di(t). Otherwise, playericontributes 0, deposits 0, and privately consumes the entire available amount.