Policy Design in Long-run Welfare Dynamics
Authors: Jiduan Wu
Rediet Abebe
Moritz Hardt
Ana-Andreea Stoica
Venue: ICLR 2025 Poster (2025)
Keywords: Long-run welfare
policy design
Rawlsian policy and utilitarianism
Primary area: alignment, fairness, safety, privacy, and societal considerations

Abstract: Improving social welfare is a complex challenge requiring policymakers to optimize objectives across multiple time horizons. Evaluating the impact of such policies presents a fundamental challenge, as those that appear suboptimal in the short run may yield significant long-term benefits.  We tackle this challenge by analyzing the long-term dynamics of two prominent policy frameworks: Rawlsian policies, which prioritize those with the greatest need, and utilitarian policies, which maximize immediate welfare gains. Conventional wisdom suggests these policies are at odds, as Rawlsian policies are assumed to come at the cost of reducing the average social welfare, which their utilitarian counterparts directly optimize. We challenge this assumption by analyzing these policies in a sequential decision-making framework where individuals' welfare levels stochastically decay over time, and policymakers can intervene to prevent this decay. Under reasonable assumptions, we prove that interventions following Rawlsian policies can outperform utilitarian policies in the long run, even when the latter dominate in the short run. We characterize the exact conditions under which Rawlsian policies can outperform utilitarian policies. We further illustrate our theoretical findings using simulations, which highlight the risks of evaluating policies based solely on their short-term effects. Our results underscore the necessity of considering long-term horizons in designing and evaluating welfare policies; the true efficacy of even well-established policies may only emerge over time.

Published as a conference paper at ICLR 2025
POLICY DESIGN IN LONG-RUN WELFARE DYNAMICS
Jiduan Wu Rediet Abebe Moritz Hardt!'", and Ana-Andreea Stoica!.
Max Planck Institute for Intelligent Systems. Tübingen and Tübingen Al Center *Depurtment of Computer Science, ETH Züricc *ELIIS Institute, Tübingen
ABSTRACT
Improving social welfare is a complex challenge requiring policymakers to op- timize objectives across multiple time horizons. Evaluating the impact of such policies presents a fundamental challenge, as those that appear suboptimal in the short run may yield significant long-term benefits. We tackle this challenge by analyzing the long-term dynamics of two prominent policy frameworks: Rawl- sian policies, which prioritize those with the greatest need, and utilitarian poli- cies, which maximize immediate welfare gains. Conventional wisdom suggests these policies are at odds, as Rawlsian policies reducing the average social welfare, which their utilitarian counterparts directly optimize. We challenge this assumption by analyzing these policies in a sequen- tial decision- decision-making framework where individuals' welfare levels stochastically decay over time, and policymakers can intervene prevent this decay. Under rea- sonable assumptions, we prove that interventions following Rawlsian policies can outperform utilitarian policies in the long run, even when the latter dominate the short run. We characterize the exact conditions under which Rawlsian policies can outperform utilitarian policies. We further illustrate our theoretical findings using simulations, which highlight the risks of evaluating policies based solely on their short-term effects. Our results underscore the necessity of considering long-term horizons in designing and evaluating welfare policies; the true efficacy of even well-established policies may only emerge over time.
assumed come the cost of
INTRODUCTION
important application of sequential decision making the problem of promoting long-run social welfare through sequence of targeted interventions in a population. Policies for this problem face a two-fold challenge. On the one hand, they must be effective at optimizing the long-term objective. On the other hand, they must appeal to the political and normative expectations of policy makers. In particular, simple policies supported by established moral and political arguments are desirable. Two families of policies have been particularly influential in the context of Western welfare programs. One targets individuals of largest immediate welfare gain. The other targets those most seriously in need. While the former derives from utilitarian moral principles, the latter is associated with Rawl's theory of justice. Many scholars, however, have criticized Rawlsian policy for its presumed failure maximize social welfare.
Indeed, there is no obvious reason why allocating resources to those of lowest welfare should also maximize average welfare in the long run. In this work, we study a stochastic dynamic model of long-term welfare in a population. Surprisingly, under reasonable assumptions on the welfare dynamics, Rawlsian policy turns out outperform an idealized utilitarian policy that chooses the individual of largest treatment effect at each step. This is the case even though the Rawlsian policy suboptimal on short-term horizon.
Although our motivation is social welfare, our results hold a broader lesson for sequential decision making. Simple policies can be highly effective, but their long-run efficiency may not be apparent on short time horizon.
Alphabetical order.

---

Published as a conference paper at ICLR 2025
1.1 OUR CONTRIBUTIONS
We propose mulli--gent stochastic dynamical model to describe long-run welfare in a population of individuals. Our model draws from classical economic theory of industrial project management, extending so-called attention allocation policies (Radner & Rothschild 1975) into social policies.
In our model, each individual i has a welfare level Ug (t) at each timestep t. At each timestep, social planner allocates an intervention to one or more of N agents using some policy и. The welfare values evolve according to a stochastic dynamical system, Absent an intervention, an individual's welfare decays in expectation according to a function y: (U)())) > O. When the social planner allocates an intervention to an individual, however, the individual's 's welfare increases in expectation according to function (t) O. We are interested in comparing Rawlsian and utilitarian policies based on the long-term social welfare they achieve, i.e. the asymptotic individual welfare increase, defined as lim (U) (.(0)) /t, averaged over all individuals. t-000
We make two substantive assumptions about welfare dynamics. The well-known Matthew ef- feet (Merton 1968, Rigney, 2010), or "rich--et-richer" while "poor-get-poorer" dynamic, suggests that inequality amplifies over time. We capture this effect by assuming that the return function fi(-) is increasing with welfare, while the decay function g-(-) decreases with welfare. The other as- sumption is a uniform boundedness assumption: the bounds of the return intervention and decay functions are the same for all individuals. In other words, no individual can achieve a highest/lowest possible level of return or decay that is much higher much lower than anyone else.
Under these assumptions, we find a sufficient condition for comparing policies. This condition states that a policy can, in principle, avoid the decay of any individual's 's welfare below O. We call this survival condition and note that it rests on the functional form and bounds of the return and decay functions. Informally, our main result shows:
Under the survival condition, Matthew effect, and uniform boundedness, Rawl- sian policy will achieve better long-term social welfare than a utilitarian policy almost surely.
We complement this result by characterizing condition in which the reverse is true: under called "ruin condition" (when a policy cannot prevent an individual's 's unbounded welfare decay), a utilitarian policy will achieve better long-term social welfare than Rawlsian policy almost surely.
To prove our results, we present a series of theoretical results that characterize in closed form the rate of growth of individual welfare under Rawlsian and utilitarian policies (Sections 4) Our proof extends the elegant argument Radner & Rothschild 1975), who studied a fully homogeneous case in which the return and decay functions are constant terms. This generalization in turn requires a non-trivial departure from the original proof including a variant of Lundberg's 's classical inequality for submartingale processes. The proof may be of independent interest for similar problems arising in sequential decision making and reinforcement learning.
We illustrate our theoretical results by simulating our model with initial conditions drawn from real data from the Survey of Income and Program Participation (SIPP) of the U.S. Census Bureau in Section 5 We see delayed effect of Rawlsian policy, noting that it obtains lower social welfare in the short-term, yet quickly converges
higher social welfare value than the utilitarian policy. We highlight limitations of our work and directions for future study in Section 6 Finally, we discuss potential extensions of our work (e.g., when the functions fi, 9s violate the uniform boundedness assumption) in the appendix.
1.2 RELATED WORK
Welfare-based social policies have a long history in economics research (Sen) 1979 Kaplow & Shavell 2000 Adler 201T). Although Rawlsian principles are based on distributive justice and egalitarian goals (Harsanyi 1975: Blau & Abramovitz 2010), debates remain regarding their effi- ciency as compared to utilitarian policies (Arrow 1973 Sen 1976). The direct comparison between Rawlsian and utilitarian policies generally remains open area of research, with some empirical and model- comparisons made in the context of optimal taxation policies (Atkinson) 1995) and income inequality (Mongin & Pivato 2021).
2

---

Published as a conference paper at ICLR 2025
We build the model proposed by Radner & Rothschild (1975) in the context of industrial project management, which analyzes the behavior of the system under different attention allocation mechanisms. We generalize and re-purpose their model by equipping it with various functional forms of the return and decay functions that capture societal behaviors and analyzing several additional policies. Our modeling choices include the Matthew effect (Merton 1968): individuals with higher level of welfare may benefit the most from interventions ("rich-gettricherr), whereas individuals with low wealth experience more severe income shocks absent any interventions from the social planner "ipoor-get-poorer
Such effects have been documented in the context of economic inequality (Rigney 2010: Stiglitz 2012) and optimal taxation policy for reducing societal inequality (akkinson|20155
Closely related to our work are recent modeling frameworks for wealth fluctuations and policy de- sign. Two recent papers develop algorithms for selecting the optimal candidates for intervening, subject to different objectives: [Abebe et al.] (2020) analyze two policy objectives in a population that undergoes income shocks and proposes algorithms for allocating subsidies optimally; their ob- jectives aim to minimize the probability of ruin for any given individual. Arunachaleswaran et al. (022) analyze the theoretical complexity and give approximation algorithms for the optimal selec- tion of candidates under a social welfare and a Rawlsian objective, considering a transition matrix of welfare states. In addition, |Heidaii & Kleinberg (202T) study the optimal policy for allocating interventions in a population with two welfare states (advantaged and disadvantaged), over a finite time horizon. AAcaary et al.] (2023) study the effect interventions
welfare-based dynamic sys- tem with feedback loops in societal inequality. Their interventions include allocating subsidies to those among most in need, without a comparison between different types of policies on the social welfare. In contrast, we study the effect of different policies in the long-run run, formulating a sufficient condition for a Rawlsian policy to achieve better welfare than a utilitarian policy.
A related line of work focuses on reinforcement learning algorithms for deriving optimal policies. In particular, Zheng et al. (2020) propose framework for a integrating Al into two-level optimization problem in the context of optimal taxation policy, with subsequent work improving the generality of the model (Curry et al. 2023). Offline and online algorithms have been proposed for finding optimal policies with fairness considerations (Zimmer et al., |021 Zhou, 2024) well as in contexts with strategic agents Liu et al. 2022). The problem of optimal policy selection can be tackled using continuous state MDP under the weward criteria, with early works considering bounded reward rates (Doshil 1976) and subsequent extensions that do not require boundedness (Guo & Rieder 2006). These works find theoretical guarantees for the existence of optimal policies. convergence rates, as well as optimality gaps. Often, such works do not find tractable, closed-form solutions for the optimal policy, but rather build heuristics with theoretical guarantees that can closely approximate an optimal policy.
Finally, the problem of allocating resources through objectives such a maximin rule includes lines of work in fair division (Procaccia & Wang 2014) as well as machine learning, often as constraint in larger optimization problem (Binns) 2018 Heidari et al.) 2019). Other works have studied Rawlsian principles under finite time horizon (Dwork et a 2 Zafar et al.) 2017 Diana jet al., (02T) or as a static optimization problem (Chen Hooker| 2020; Stark et al. 2014). Some works have studied the long-term effect of fair algorithms in the context of hiring (Hu & Chen, 2018) and resource-allocatieo data-driven approaches for optimal assigning subsidies to individuals who experience homelessness; their approach uses a prioritization scheme that aims to minimize the probability of an individual to re-enter homelessness, based on an automated prediction.
(Liu et al.) 2018). Kube et al. (2019) and Azizi et al. (018)) offer
2 A MODEL OF WELFARE DYNAMICS AND SOCIAL POLICIES
Preliminaries. Consider N individuals indexed by i = 1,..., .. Each individual i has welfare value of (t(t) at each timestep 2 0. The initial welfare values (1( are drawn from a distribution (e.g. capped normal distribution; different choices of the initial distribution do not change our results). Here, welfare may represent the household income level, expenditure, monthly income, or other variables that define individual welfare.
An intervention at time t is defined through a vector att) = (a, (t));, where an amount of (t( budget is allocated to individual i by the social planner. The exact decision of who receives an
3

---

Published as a conference paper at ICLR 2025
amount of budget and how much they receive is decided by the social planner through social policy. The social planner has a budget M for allocating interventions at every timestep t 2 0: E; at(t) = M,, for M N, 1 I M & N, and 0 I a,(t) 1. In this first analysis, we consider the case when the social planner can only allocate an integer unit to each individual, so a;(() E (0, 1).
2.1 DYNAMIC MODEL OF WELFARE FLUCTUATIONS.
Absent any intervention, assume that the welfare of individuals fluctuates every timestep according to a function 9i R + R+, defined as function of the welfare value for each individual We denote the function gi(-) as the decay function, capturing the welfare decrease in natural conditions (e.g., income shocks due accidents, economic conditions, natural disasters).
In contrast, we model the impact of interventions on individuals" welfare each timestep through function Si IR + R,, defined for all individuals i. We refer to fe(.) as the intervention return function, capturing the effect of intervening on an individual (e-g., a new job through an employment program, social benefits, cash transfers). Let Fe be alagebbra denoting the space of events up to time step We model the rate change of individual welfare between different timesteps under interventions
+ - -)
[i] 1 (.))) (10,(()) - (1-a(()) (.(((())
(I)
Treatment (a, = 1) in our model has two effects. On the one hand, the treated individual realizes the return (t).. On the other hand, the treated individual avoids the decay - 9i (t). The individual treatment effect of allocating an intervention to individual at time therefore corresponds to the expression
{(U((()) + (.(U(()))
Note that this quantity varies both time and by individual. Conceptually, targeted individuals have positive return, whereas non non-targeted individuals suffer decay in their welfare.
2.2 SOCIAL POLICIES
A policy π selects an individual for treatment at each step. This corresponds to setting the co- efficients {t) each timestep t. We restrict our attention policies that allocate M units of resources to M individuals with each individual receiving exactly one unit at each time step. Let (S) denote the set of s largest elements of
A natural utilitarian policy is the one that chooses the individual of largest treatment effect. We call this the max-fg policy:
11, E Top (IfA(U>(0) + 9a 9x a.(1)
(max-fg)
0. otherwise.
Note that this policy requires full information about individual treatment effects at each time step. This may be unrealistic requirement in many applications. We call this the max-U policy:
1, E Topas (IU(00)E))
(t
(max-U)
0. otherwise.
The max-U policy is welfare- based and requires only welfare measurements for its implementation. This utilitarian velfare-based policy directly contrasts with the Rawlsian policy that chooses the individual of minimum welfare each step. We call this the min-U policy:
f 0,
1.
i Topas ((UL(0)))))
ax(1)
(min-U)
otherwise.
Radner & Rothschild (1975) studied these policies under the names "putting out fires" for min-U and "staying with winner" for max-U with M 1.
4

---

Published as a conference paper at ICLR 2025
We explore variation of the utilitarian policy that only uses knowledge of the intervention return functions f((-), ie. the policy will allocate a unit of effort to the individual with the highest inter- vention return:
1. i E Top (IAAUU((OM))) a;(1)=
(max-f)
0, otherwise.
We call this max-f. In contrast to max-fg, the max-f policy requires only partial information about the interventions, only measured through the return on interventions which may be less costly to measure. By analogy, we consider variant of the Rawlsian policy here that only use knowledge of the decay functions 9i(-). That is, the max-g policy will allocate unit of effort to the individual with the highest decay:
1, (t) = {o. 0, otheewise.
i TMp (Lo((U(())))).
(max-g)
Te-breaking rule: Among individuals with the same welfare, we favor the one with the lowest index i E [M] This applies to all policies. For the policies that use the treatment effect information, max-f and max-fg, we break the tie in favor of the individual with the lowest index. For the max-g policy, among individuals with the same di value, we break the tie in favor of the individual with the lowest welfare value, arguing that this best captures Rawlsian principle. When max-g prioritizes the lowest index individual, results do not qualitatively change (see Appendix E Figure
Policy goal. The goal of a policy is to promote long-term term social welfare. Our main results focus on the long-term social welfare comparison of Rawisian and utilitarian policies. We capture long-term social welfare as the average asymptotic welfare gain among individuals, defined as follows.
Definition (Long-term social welfare). The long-term average social welfare induced by policy π on population of N individuals is defined as
1
R = lim (t(t) - U.(0)
R
Ra.
{
1=1
where R₁ defines the rate of growth of individual i, asymptotically.
Note the welfare level (t) depends on the policy n, as æ determines a(t) at every timestep, and therefore the subsequent U,(t 1) through the model described in Equation |
2.3 MODELING CHOICES
The comparison between Rawlsian and utilitarian policies depends on an important condition, called 'survival' condition. Survival means that no individual in a population will obtain negative welfare. The survival condition is necessary and sufficient to obtain positive probability of survival for all individuals under some policy, as noted by RRadner Rothschild (1975). Such a policy only exists under the survival condition, and in fact, Rawlsian policies are examples as we will show later in Section 3 This is sufficient condition for comparing policies the long run. Formally, the survival condition can be stated in terms of weighted sum of the f((-) and g.(-) function bounds (assuming those exist):
Assumption 1 (Survival condition). We assume E(isT ..... fr), (ot...... .... 9) 0 where Č : RON - R. is defined as
?((x1,..., INN, (ei,...,,,)) = M
V4
(3)
34.
y/
Jíj + Vj
and J+ sup (( 0, Ji inf ((x)) 0, , = sup gi (x)
> 0, 9. in ( 0.
Next, we formally state the modeling conditions that capture a Matthew effect, as motivated in the introduction, as well as uniform boundedness condition.
5

---

Published as a conference paper at ICLR 2025
Assumption (Modeling conditions). (a). (Rich--et-riccer) For i 1,.... ..... N, assume that the function f((x) is non decreeears..g.
(b). (Poor-gets getp--o----oro For = -,,,,.... I,...,N, NN we assume that the function gi(a) is non- aom-increasing.
(c). (Uniform boundedness) For i 1,...,A N, assume fr f ft = ft 94
9 9, = 9" for constants f f 9.9.99' 9"
We note that this assumption does not require that the functions fi. be the exact same for all individuals, but rather just their limits.
Finally, in addition to the two assumptions described above, our results require some standard regu- larity conditions, formalized below. Denote the welfare variation between two consecutive timesteps by (t + 1) == (t + 1) (t), Vi E [N]. We note that (t(t) and (t) are random variables with respect to a stochastic process of welfare fluctuations over time (e.g., income shocks).
Assumption 3 (Regularity conditions). Consider a probability space FIP), where Q is the space of possible outcomes of welfare levels, F is o-algebra denoting the space of events, and F: F - [0, 1] is probability measure function. We assume the following properties:
(a). The welfare random variable (!( is Fe-measurable for Vi E [4], (E, for Fo F an increasing sub o-field of F.
(b). The variation random variable Zz(t + 1) is integer-valued, mutually independent (given a(t)), and uniformly bounded, iee. (t 1)1 b. Vi E [7, E N for some constant
b 0.
(c). There exist constants 2", > 0 with 0 < l < s.t. P(z,t + 1) 2 2" 1 (((z(t+1) Fi) 2 I for any [AV, any Fo. Vt 2 0.
POLICY COMPARISONS IN TERMS OF LONG-TERM SOCIAL WELFARE
Our main result compares the long-term social welfare of Rawlsian and utilitarian policies, under the natural behavioral model of welfare fluctuations described in Section 2.1
Theorem 1 (Main result). For a population of N individuals whose welfare (U:(t)); fluctuates according to the model in equation] 7 under regularity, modeling, and survival conditions (Assump- tions a Rawlsian policy will achieve better long-term social welfare than a utilitarian policy:
BRawisian Rutilicarian
a.8.
where the Rawlsian and utilitarian policies are defined in the same informational contexts, i.e. (min-U, ma)- (max-g, max-f), (max-g, max-fg).
Proof sketch. The proof of Theorem includes series of results on the individual rates of growth for different policies. First, we compute the individual rate of growth under the Rawlsian pol- icy to be equal for all individuals (Theorem 33. The survival condition implies the existence of a policy that prevents any individual's welfare from decaying below 0. In fact, it implies some- thing even stronger: under survival, Rawlsian policy can 'lift' everyone's welfare unboundedly: lim min; U+(!) = oo almost surely. This helps us show that the welfare gap between any two individuals vanishes asymptotically, obtaining the same individual rates of growth for all individu- als. In contrast, a utilitarian policy tends to fixate on single individual and repeatedly allocate an intervention to him, while ignoring the rest of the population. We show this formally in Theorem we leverage a generalization of Lundberg's 's incquality for submartingale processes to lowerbound the probability that a utilitarian policy repeatedly allocates interventions to the same high-welfare individuals. Finally, the individual rates of growth and uniform boundedness allow to compute and compare the long-term social welfare under different policies, see Corollaries 2 Essentially, Rawlsian policy obtains better social welfare in the long-run than utilitarian policies long as lim Ge(z) < lim 9: (z), which is true by our "poor-get get-pporerr" modeling condition. is noteworthy that the result holds regardless of the variation of policies: whether the social planner has knowledge of (fr)), (g:); or not, the policy comparison remains the same under our modeling conditions. Detailed proofs for all results be found in Appendix A

---

Published as a conference paper at ICLR 2025
In cases where the survival condition does not hold, find natural complement for theory: define a "ruin condition" as a state of the model in which no policy can prevent all individuals from decaying below O. Our theory under survival naturally extends for this ruin condition, showing that utilitarian policy will achieve better long-term social welfare (see Appendix for the formal theory).
Theorem (Policy comparison under ruin condition). For a population of N individuals whose welfare (U) fluctuates according to the model in equation [ under regularity, modeling, and ruin conditions (Assumptions 2
+). a utilitarian policy will achieve better long-term social
welfare than Rawlsian policy:
RRawisian S Rutilitarian
where the Rawlsian and utilitarian policies are defined in the same informational contexts, i.e. (min-U, max (max-g, g, max-f), (max-g, max-fg).
INDIVIDUAL WELFARE RATE OF GROWTH UNDER DIFFERENT POLICIES
In this section, we characterize in closed form the rate of growth of welfare under different policies for all individuals, that is, proving that R₂ lim,-+oe (U, (0))) converges to closed-form solutions for E [N]. We then compute the long-term average social welfare achieved by all policies and compare them against baseline defined by random allocation policy.
4.1 INDIVIDUAL WELFARE UNDER THE RAWLSIAN POLICY
Theorem 3. Under regularity (Assumption By. modeling conditions (Assumption (a), and the survival condition (Assumption 7). a Rawlsian policy E {min--, max-g] leads to the following closed-form solution of the individual rates of growth:
R4 = ===f f (oT.
i=1,...,N; a.s.
Corollary 1. With the addition of the uniform boundedness condition from Assumption (c). we simplify the individual rates of growth, obtaining the long-ter social welfare value for the Rawlsian policy:
M N
N
M N
Rmin. Rmax-g
9
0.8.
Proof sketch. Under the survival condition, we prove that the minimum welfare level will be lifted unboundedly over time. We model the welfare gap between the treated and untreated individuals and show that this gap vanishes almost surely by applying the law of large numbers. We conclude by
adapting a convergence argument first introduced by Radner & Rothschild (1975), obtaining that Rawlsian policy achieves the same long-run welfare of everyone under our modeling conditions. O
4.2 INDIVIDUAL WELFARE UNDER THE UTILITARIAN POLICIES
Theorem 4. Under regularity (Assumption 3 and modeling conditions (Assumption (b))
and as long as fi(-) + g:(-) is increasing for all i E [N]] utilitarian policy fmax-U, imax-U,maxe max-fg, max-f} leads to the following closed-form solution of the individual rates of growth:
[ft.iee R& 1 1 ie
a.s.
where J a set of M. random variables with values in [N] whose exact value depends on (J.()), and (g;())); In other words, exactly M individuals achieve an asymptotic rate of growth equal to f+t, whereas all others achieve -g.-
"This assumption states that the return from an intervention should, in principle, be higher than the shock experienced by individual absent intervention. It is only needed for the max-fg policy, since it is the only using knowledge of both the return and decay functions.

---

Published as a conference paper at ICLR 2025
With the addition of the uniform boundedness condition from Assumption (c), can simplify the individual rates of growth, obtaining the social welfare value
Corollary 2.
M
N-M g a.s.
Rinax-U Rmaxxf = Rg
Proof sketch. For the individuals chosen by the utilitarian policy time to. we upperbound the probability of an individual obtaining negative welfare at a finite point in time (a variable that we model as submartingale). for any F, and individual i. We make use of generalized Lundberg's inequality (Lundberg, [903] Cramér] [959 Moriconi, (986) for submartingales, for which we provide an adapted version for our model and new proof. We then use to show that the probability that it will be chosen again afterwards (as = 1, (o) is lower-bounded by some positive constant. Asymptotically, the probability of M individuals being targeted by a utilitarian policy approaches and hence obtain the asymptotic convergence of the individual rates of growth under our modeling conditions.
Random policy. Finally, we compare our policies with baseline policy that randomly chooses M individuals to allocate an intervention at every timestep.
Theorem 5. Under regularity, modeling, and survival conditions (Assumptions 7 the random policy leads to the following closed-form solution of the individual rates of growth and long- term social welfare:
M N
N
M N 9
Ryandom R.
i= i=1,...,N, N, a.s.
Proof sketch: Under the assumption of uniform boundedness, may lowerbound the rate of wel- fare increase at every timestep by a positive quantity, Et + 1) (t(t) Fi] 2 N J (1 N. )g > 0 under the random policy. This allows us to show that, in the limit, the welfare value of every individual will increase unboundedly. domly at each time, the welfare gap between individuals converges to 0, just like in the proof of Theorem We follow a similar proof structure henceforth, detailed in Appendix(A A
M
the same time, since individuals are chosen ran-
Policy comparison: Our results show that Rawlsian and random policies will achieve better long-term social welfare than utilitarian policies under the aforementioned conditions. This concludes argument for the main result in Theorem [ Furthermore, our subsequent results show that the comparison holds no matter the informational context (whether the social planner uses only welfare information defining policies, or also has access to the treatment effect through fú, gd) Furthermore, when the survival condition is not satisfied, we find complementary condition under which a policy reversal occurs. We provide a formal theory for this result in Appendix We extend our results to include different functional forms for the treatment effect function (Appendix DD and allocate proportional interventions at each timestep (Appendix F We explore different combinations of monotonicities of return/decay functions through simulations in Appendix G
5 ILLUSTRATION OF THEORETICAL RESULTS AND MODELING CHOICES
We illustrate our theoretical results through simulations using a real-world dataset. We compare the average social welfare under a finite time horizon for all proposed policies.
5.1 SIMULATIONS OF POLICIES ON REAL DATA UNDER FINITE TIME HORIZON
We use data collected from the Survey of Income and Program Participation (SIPP) (uureau 20233, which a longitudinal survey of households in the U.S. containing variables related economic well-being such as income, employ etc. Among numerous indices, we use the income variable as a proxy for the initial individual welfare level, (O), We group the whole population 39, 720 into 13 bins and treat every 200 samples one individual, and every $1, 000 in our model. In the end, we obtain population of 206 individuals. We simulate an instance of the general model from equation [ with Gaussian noise, specified as:
welfare unit
Ut(t 1) U.(t) at(t) (((()(t)) (1 at)))) 9. (U,(t) Ei(),
Vt 2 0, e [N].
(4)
8

---

Published as a conference paper at ICLR 2025
where ({,(t))., (0, o²) and capped within uniform bounds, for some noise parameter We generate homogeneous bounds f" g 9², and then generate the functions ft(),, gi(-) as segment linear functions. Our results averaged over 100 draws, reporting standard deviation in the error bands. See Appendix
C for further experimental details.
We measure social welfare at timestep as the individual growth rate up time t averaged all individuals (equation 2 up to time t).
The average social welfare (solid lines) converges to the theoretical expected welfare (dashed lines) for all policies (Figure D. Furthermore, Rawlsian policies (min-U and max-g) have a lower short- term social welfare than utilitarian policies (max-U, max-f, max-fg). After few hundreds timesteps, this trend is reversed, showing convergence to the theoretical social welfare value. Rawlsian policies achieve better welfare than utilitarian policies in the long-run, which is implied by Theorem The random policy behaves similarly to the Rawlsian policy (as Theorem 5 would suggest), yet with a slower convergence rate. This disadvantage vanishes as the budget M increases, as indicated by Figure lb Figure in Appendix B illlsttrtes the finite time horizon under the ruin condition, showcasing a reversal the Theorem [ result (for a formal statement, see Theorem 2). We also
showcase the complexity of evaluating policies in heterogeneous cases (i.e., where the bounds of the return and decay functions fi. 9: are non non-unioonn con-uniform) in Appendix H
2740
max-g (Rawisian) minUU (Rowtsian)
max-g (uniitarian)
Social welfare
Social welfare
max-u (Ublliarlan)
tuliitarian)
Random
2700
2:1
(Rawtsian)
Aear-u (lltaraa)))
202
2730
212
Time step
Time step
(a)
(b)
Figure 1: Social welfare as the finite-time timte-time growth rate averaged over all individuals, for all policies (solid lines), well theoretical expected growth rate, asymptotically (dashed lines) for budget M = 1 (a) and M = 10 (b).
All simulations are ran on commodity hardware, using Python 3.8. All code and data used in our simulations is available in this repository
DISCUSSION
The problem of optimal policy design remains highly relevant, several countries continue implement changes in their social benefits allocation schemes. A prominent example is Austria, which has shifted from welfare state approach that targeted those most in need an "inactivity trap" approach that targets those most likely to (re)enter the labor market (Allhutter et al. 2020) Christl & De Poli 202 International). In 2019, Austria introduced the New Social Assistance policy that reduced benefits to individuals with low language skills or larger number of dependents. In 2020, it introduced algorithmic profiling by first predicting individuals' probability of re-entering the labor market, and second by offering most support to those with intermediate chances. Both such policies have the purpose of shifting support from the most in need those with the highest chance of benefiting from such support, measured through their integration in the labor market in the near future. In essence, such policies have shifted from a Rawlsian approach to social welfare (Bufacchi & Garmise 1995) to
utilitarian view of social benefits (Christl & De Poli 2021).
Our work demonstrates that choosing the right policy framework is subtle. In particular, our results motivate the necessity of long-run welfare comparisons of policies that short- analysis will necessarily miss. Whereas on the short-term horizon utilitarian policy prevails, it can result in lower social welfare than Rawlsian approach in the long-run, under reasonable conditions. We

---

Published as a conference paper at ICLR 2025
characterize such conditions in closed allowing long-term policy comparison. In particular, the survival condition is a sufficient condition for Rawlsian policy to achieve better social welfare in the long-run when the population of individuals satisfies homogeneous bounds on the intervention return or welfare decay.
To apply model, the social planner does not need know the exact form of fi. for each individual. Rather, they can estimate general trends and effects of income shocks and interven- tions through small pilot experiments or through acquiring domain knowledge, e.g., through poverty trackers or longitudinal studies of intervention effects on income (Garfinkel] 2021). Experimenta- tion through small pilot experiments is often considered a necessary precursor of policy deploy- ment (Office, [2003] Huitema et al. (2018). rapidly increasing as method for policy design and evaluation (Banerjee et al., 2016; 2017, 7 Webber & Prouse 2018). Then, the estimates for the return and decay functions can be used as plug-in in estimates in the survival or the ruin condition. Future work could combine effective estimation methods for the return and decay functions with long-term policy assessments.
Our analysis rests on several modeling conditions, which could be explored future work. We analyze several variations in the Appendix: we provide complementary theory for the case when the survival condition not satisfied (Appendix [B. and we explore variations of our modeling assumptions: mon-monotoiic treatment effect function f: g. (Appendix D) different tie-breaking rule (Appendix E proportional interventions at each timestep (Appendix F). different combinations of monotonicities of return/decay functions (Appendix Gi), and the of heterogenous limits of the intervention and return functions fi, (Appendix
Overall, our theoretical framework provides versatile tools for exploring different modeling con- ditions as well as policy variations. These contributions open new directions for future work in the context of sequential decision-making and optimal policy design, with applications in social programs evaluation.
7 ACKNOWLEDGMENTS
We thank Florian Dorner, Jessie Finocchiaro, Max Kasy, Jon Kleinberg, Ali Shirali, Screna Wang, and anonymous reviewers for their generous feedhack and detailed discussions. Rediet Abebe was partially supported by the Andrew Carnegie Fellowship Program. Jiduan Wu would like to thank the financial support from Max Planck ETH Center for Learning Systems (CLS).
REFERENCES
Rediet Abebe, Jon Kleinberg, and S Matthew Weinberg. Subsidy allocations in the presence of income shocks. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pp- 7032-7039, 2020.
Krishna Acharya, Eshwar Ram Aruachhlessaaran Sampath Kannan, Aaron Roth, and Juba Ziani Wealth dynamics over generations: Analysis and interventions. In 2023 IEEE Conference on Secure and Trustworthy Machine Learning (SallML), pp. 42-57. IEEE, 2023.
Matthew Adler. Well-being and fair distribution: beyond cost-benefi analysis. Oxford University
Press, 2011.
Doris Allhutter, Florian Cech, Fabian Fischer, Gabriel Grill, and Astrid Mager. Algorithmic profiling of job seekers in austria: How austerity politics are made effective. Frontiers in big data, 3: 502780, 2020.
Kenneth Arrow. Some ordinalist-utilitaria notes on Rawls's theory of justice, 1973.
Eshwar Ram Arunachaleswaran, Sampath Kannan, Aaron Roth, and Juba Ziani. Pipeline interven- tions. Mathematics of Operations Research, 47(4):3207-3236 2022.
Anthony B Atkinson. Inequality: What can be done?, 2015.
Anthony Barnes Atkinson. Public economics in action: the basic income/flat tax proposal. Claren- don Press, 1995.
10

---

Published as a conference paper at ICLR 2025
Mohammad Javad Azizi, Phebe Vayanos, Bryan Wilder, Eric Rice, and Milind Tambe. Designing fair, efficient, and interpretable policies for prioritizing homeless youth for housing resources. In Integration of Constraint Programming. Artificial Intelligence, and Operations Research: 15th International Conference, CPAIOR 2018, Delft, The Netherlands, June 26-29, 2018, Proceedings 15, pp. 35-51. Spriinne 2018.
Abhijit Banerjee. Rukmini Banerji, James Berry, Esther Duñlo, Harini Kannan, Shobhini Mukerji, Mare Shotland, and Michael Walton. From proof of concept to scalable policies: Challenges and solutions, with an application. Journal of Economic Perspectives, 31(4):73-102, 2017.
Abhijit Vinayak Banerjee, Esther Dufto, and Michael Kremer. The influence of randomized con- trolled trials on development economics research and on development policy. The state of Eco- nomics, the state of the world, pp. 482 482-488, 2016.
Roland Benabou. Unequal societies: Income distribution and the social contract. American Eco- nomic Review, 91(1):96-129, 2000.
Reuben Binns. Fairness machine learning: Lessons from political philosophy. In Conference on Fairness, Accountability and Transparency, pp. 149-159. PMLR, 2018.
Joel Blau and Mimi Abramovitz. The dynamics of social welfare policy. Oxford University Press, USA, 2010.
Vittorio Bufacchi and Shari Garmise. Social justice in europe: an evaluation of european regional policy. Government and Opposition, (0(2):179-13) 1995.
U.S. Census Bureau. Survey of income and program participaation (SIPP) 2014 panel. https:// www. census grov/prograat 2023. Accessed: 2024-09-25.
wttpe/////sssip/aaaaa/aaaaaaaaaa p
Violet Chen and John Hooker. just approach balancing rawlsian leximax fairness and utilitar- ianism. In Proceedings of the AAAI/ACM Conference on Al, Ethics, and Society, pp. 221-227, 2020.
Michael Christl and Silvia De Poli. Trapped in inactivity? Social assistance and labour supply in Austria. Empirica, 48(3):661-698 2021.
Harald Cramér. On the mathematical theory of risk. Centraltryckeriet, 1959.
Michael Curry, Alexander Trott, Soham Phade, Yu Bai, and Stephan Zheng. Learning solutions in large economic networks using deep multi-agent reinforcement learning. In Proceedings of the 23rd International Conference on Autonomous Agents and Multiagent Systems, pp. 2760-2762, 2023.
Emily Diana, Wesley Gill, Michael Kearns, Krishnaram Kenthapadi, and Aaron Roth. Minimax group fairness: Algorithms and experiments. In Proceedings of the 2021 AAAI/ACM Conference on AI, Ethics, and Society, pp. 66-76, 2021.
Bharat T Doshi. Continuous time control of markov processes arbitrary state space: average return criterion. Stochastic Processes and their Applications, 4(1):55-77, 1976.
Cynthia Dwork, Moritz Hardt, Toniann Pitassi, Omer Reingold, and Richard Zemel. Fairness through awareness. In Proceedings of the 3rd Innovations' in Theoretical Computer Science Con- ference, pp. 214-226, 2012.
David Freedman. Another note on the Borel-Cantelli lemma and the strong law, with the Poisson approximation by product. The Annals of Probability, 1(6):910-92 1973. ISSN 0091 1798. URL httpht/t///wwt
Irwin Garfinkel. New york city longitudinal survey of well-being (poverty tracker), 2015-2018. 2021.
Xianping Guo and Ulrich Rieder. Average optimality for continuous time markov decision processes in polish spaces. 2006.
11

---

Published as a conference paper at ICLR 2025
John C Harsanyi. Can the maximin principle serve a basis for morality? A critique of John Rawls's 's theory. American political science review, 69(2):594-606, 1975.
Hoda Heidari and Jon Kleinberg. Allocating opportunities in dynamic model of intergenerational mobility. In Proceeding of the 2021 ACM Conference on Fairness, Accountability, and Trans- parency, pp. 15-25, 2021.
Hoda Heidari, Michele Loi, Krishna P Gummadi, and Andreas Krause. A moral framework for understanding fair ml through economic models of equality of opportunity. In Proceedings of the Conference on Fairness, Accountability, and Transparency, pp. 181 181-190, 2019.
Lily Hu and Yiling Chen. short-term intervention for long-term fairness in the labor market. In Proceedings of the 2018 World Wide Web Conference, pp. 1389-1398,, 2018.
Dave Huitema, Andrew Jordan, Stefania Munaretto, and Mikael Hildén. Policy experimentation: core concepts, political dynamics, governance and impacts. Policy Sciences, 51:143-159, 2018.
Amnesty International. Barriers in access to social assistance Austria. Ammesty International Austria. URL https: //ww:. amnesty EURR382232024BNG1IE pdf
Louis Kaplow and Steven Shavell. Fairness welfare. Harv. L. Rev., 114:961, 2000.
Amanda Kuhe, Sanmay Das, and Patrick J Fowler. Allocating interventions based on predicted outcomes: study homelessness services. In Proceedings of the AAAI Conference Artificial Intelligence, volume 33, pp. 622-629, 2019.
Lydia Liu, Sarah Dean, Esther Rolf, Max Simehowitz, and Moritz Hardt. Delayed impact of fair machine learning. In International Conference on Machine Learning, pp. 3150-3158. PMLR, 2018.
Zhihan Liu, Miao Lu, Zhaoran Wang, Michael Jordan, and Zhuoran Yang. Welfare maximization in competitive equilibrium: Reinforcement learning for markov exchange economy. In International Conference on Machine Learning, pp. 13870-13911. PMLR, 2022.
Filip Oskar Ernst Lundberg. Approximerud franstlllnng af samolliheersfjukionen. af kollektivrisker. 1903.
Aterförsäkring
Gregory Mankiw, Matthew Weinzierl, and Danny Yagan. Optimal taxation in theory and practice. Journal of Economic Perspectives, 2344):147-174, 2009.
Robert K Merton. The matthew effect science: The reward and communication systems of science are considered. Science, 159(3810):56-63 1968.
Philippe Mongin and Marcus Pivato. Rawls's difference principle and maximin rule of allocation: a new analysis. Economic Theory, (1(4): 149--1555 2021.
Franco Moriconi. Ruin theory under the submartingale assumption. In Insurance and Risk Theory, pp. 177-188. Springer, 1986.
Cabinet Office. Trying it out: The role of 'pilots' in policy making, 2003.
Ariel D Procaccia and Junxing Wang. Fair enough: Guaranteeing approximate maximin shares. In Proceedings of the fifteenth ACM conference on Economics and computation, pp. 675-692, 2014.
Roy Radner and Michael Rothschild. the allocation effort. Journal of Economic The- ory, 10(3):358-376 1975. ISSN 0022-0531. doi: http:://hoo.nglll000000000200033 90006-X. X.
00220531788 0590003779996XXX
Daniel Rigney. The Matthew effect: How advantage begets further advantage. Columbia University Press, 2010.
12

---

Published as a conference paper at ICLR 2025
Michael Rothschild. Further notes on the allocation of effort. In Richard Groves (eds.), Adaptive Economic Models, pp. 195-220. Academic Press, 1975a. ISBN 978- 0-12-207350-2. doi: https://tbboor//00001188898////0
H. Day and Theodore
URL https://
www. sc encedi rect. com /ttm://c.eenee/arrtcc/e//////p// 287010077777722722250000011
Michael Rothschild. Further notes on the allocation of effort. In Adaptive economic models, pp. 195-220. Elsevier, 1975b.
Amartya Sen. Welfare inequalities and rawlsian axiomatics. Theory and decision, 7(4):243-262 1976.
Amartya Sen. Personal utilities and public judgements: what's wrong with welfare economics. The economic journal, 89(355):537-558 1979.
Oded Stark, Marcin Jakubek. and Fryderyk Falniowski. Reconciling the rawlsian and the utilitarian approaches the maximization of social welfare. Economics Letters, 122(3):439-444, 2014.
Joseph E Stiglitz. The price of inequality: How today's 's divided society endangers our fature. WW
Norton & Company, 2012.
Yiu-Kuen Tse. Nonlife Actuarial Models: Theory, Methods and Evaluation. International Series Actuarial Science. Cambridge University Press, edition, 2023.
Sophie Webber and Carolyn Prouse. The new gold standard: The rise of randomized control trials and experimental development. Economic Geography, 94(2):166-187 2018.
Muhammad Bital Zafar, Isabel Valera, Manuel Rodriguez, Krishna Gummadi, and Adrian Weller. From parity to preference- based notions of fairness in classification. Advances in neural informa- tion processing systems, 30, 2017.
Stephan Zheng, Alexander Trott, Sumil Srinivasa, Nikhil Naik, Melvin Gruesbeck, David C Parkes, and Richard Socher. The Al Economist: Improving equality and productivity with Al-driven tax policies. arXiv preprint arXiv:2004. 13332, 2020.
Angela Zhou. Optimal and fair encouragement policy evaluation and learning. Advances in Neural Information Processing Systems, 36, 2024.
Matthieu Zimmer, Claire Glanois, Umer Siddique, and Paul Weng. Learning fair policies decen- tralized cooperative multi-agent reinforcement learning. In International Conference on Machine Learning, pp. 12967-12978. PMLR, 2021.
13

---

Published as a conference paper at ICLR 2025
A COMPLETE PROOFS
This section contains complete proofs to all results stated in the main paper. First of all, we introduce Iwo bounds that will be repeatedly used the following proofs:
First, consider weighted sum of utilities:
Ú() Σ we U.(1),
WN
Jr +
Σ
5 + gj
(5)
The increment in the weighted utility can be computed in expectation as:
E ()( ((t) Fr Σ B|Z(t+ + 1) Fil
Σ wy (a.(t) f((U((t) (1 - (a)((t)) -a)(())))g ((((()))
Σ w: (a:(t) f. - (1- (t))
where 7(1 + 1) = Ej wi Z.(1 + 1). We note that the last term of equation 6 is solely a function
of (ft), and (g.²), Thus, we obtain
E [U(++ (()) 2 (o) .... 9t)
Similarly, we define weighted sum of utilities using slightly different weights:
( ff + 99
(tt) Σ कं U.(t), 1
5tt + g
(8)
Similarly, we obtain the following upper bound for (():
E + 1) (t) Fi Σ w. (a:(t) (t(U((t)) (1 -1- (1-a,(t)) ((())))-(Uuu)) (U((()))
0. (a:(t) ft (1 a,(t))_.)
We observe that the last term of equation can be written as the function Č with switched parameters as compared equation
EE (((++)) ((t) F.] s E((A ... ft). (N))
(10)
Note that both bounds from equations equation ? and equation 10 only use the assumption on the bounds of fa(.) and (-) in Assumption and hence hold under any of the aforementioned social policies and they will be crucial for the asymptotic behavior of the system.
Proof of Theorem 3 We first prove the theorem for the welfare-based policy min-U, and then adapt the proof for the effect-based policy max-g.
We note that the limit conditions from Assumption II us to follow the conditions stated in Roth-
schild (177bby Jt+ = sup fi()) > J inf (x( > 0, g = sup (*) 0, 9,
inl 9i (x) D. Remember that 7, 1 1) = U,(L 1) (1),, Vi E [N]. Thus, conditioning
on individual getting or not getting an intervention and using the monotonicity assumptions, we obtain from the model in equation] [
E|Z,(t + (t) ((
(11)
-9 > = -9.t Vt.
0.
14

---

Published as a conference paper at ICLR 2025
With these conditions, together with the regularity and survival conditions, substitute (t) in Rothschild (1975a) with (t) defined in equation 5 and apply Theorem from Rothschild
1975a). The lower bound equation [ obtains the first part of the result in Theorem Rothschild 1975a): lim inf Ut()/t 2 a.s. for i E [N], which immediately implies that (-000
lim inni min; (t) + +oo a.s. for i E [N].
Then, need the following lemma.
Lemma 1. Suppose (Yihee(T) are random variables and Fy-mmasuuable, for any 2 1. Suppose
IYl S B and E[[,, [F-1] = Pe with B < S 14 S X
B for VI [T] where > 0, pb are
constants. Then
21= Y,
E
lim inf
lim inf 7
2 p.,
T-000
T
T
Σt Y
Σl=
lim sup
lim sup
sup 220000
S Ss,
T
22000
Proof of Lemma / The first inequality is immediate from Theorem 40 in (Freedman 1973) with
X; + B, M: u4
+ 2 u B and the second inequality is obtained similarly by setting
X; B - Y,, M₁ B pt B instead.
Now we continue with the proof for Theorem We apply Lemmal 1 with Y, where b the upperbound on (t + 1) from the regularity conditions (Assumption 3). and
B
E Σ u. (1(.((00(//000))) ((((()))))((()))))) F
Since lim min; U,(t) = co, we obtain lim Ut( +o for all e [N]. Hence, we get lim (U), = ft .im (U.(t)) = 9 - A simple calculation finds that lim 14 "((fft...,, f.), (or .o.......g9)) a.s.
(12)
1000
Σ7 Me
Since μr is uniformly bounded for Vt 2 we know that lim Pe limr therefore
and
Eto Mt = =(((ft...,, (or a.s.
lim T-200
(13)
T
Hence we obtain:
-z=0 Yi
Ü +
lim
lim
-((CT (oi .T,.).999))
(14)
T
T-200
T
Next, apply Lemma
from
Rothschild
(1975)) and for Vj [N]. we have
ÜT +1
(U:(T)
(T)
(T
U
= lim US(T) T T
lim T-x0
lim Σ ii T-000
T
T T
T
finalizing the proof of Theorem 3 for the welfare-based Rawlsian policy min-U U. Note that Lemma from Rothschild (1975)) essentially shows that the welfare gap between any two individuals con-
verges to 0 over time, have used that lim E. (U.(T) U.(T) O, Vi,j. Intuitively, this is
T-500
natural under Rawlsian policy that always 'lifts' the lowest welfare individuals, under our bounded welfare conditions. Finally, we also used that Σ, W; = 1, by definition.
For the effect-based Rawlsian policy max-g, we note that if g; is strictly decreasing for all i, then the individual targeted at each timestep will the exact same individual min-U and max-g. Our modeling conditions only require that gi is decreasing, but not strictly. Therefore, if the function 9; constant for a set of individuals with welfare values under some threshold r, as long as the targeted individuals will be the one with the actual lowest welfare, min-U and max-g g still coincide. Under the
15

---

Published as a conference paper at ICLR 2025
tie-breaking rule of choosing the individuals with the lowest welfare, the proof for computing R, under max-g reduces to our proof for min-U. Under different tie-breaking rules (e.g., choosing the individual with the smallest index) for max-g, the policies might actually differ in the asymptotic rates of growth. We argue that a tie-breaking rule targeting the individuals with lowest welfare under max-g policy is most natural, since it naturally applies Rawlsian principles when information gathered from Ji does not help differentiate individuals.
Finally, the corollary follows immediately under the uniform boundedness assumptions on the bounds of J:
R2 = -uU .......),, (.))(.......999))) = M
Σ 8
9
(15)
9:
f1
9i
9
)
(+
N -1
M N.
ft g
(16)
ft + g
M ft
N- M
N 9
(17)
N
Proof of Theorem 4 We first note the intuition behind the proof, followed by the detailed technical details. We note that while max-U is also known as the 'staying with a winner' policy in Radner & Rothschild (1755). the proof technique does not generalize under non-constant functions fi, gi- To this end, we introduce novel proof that can characterize the individual rates of growth under any informational contexts and for any functions that follow our regularity and modeling conditions (Assumptions (b) and B
The main proof idea hinges on showing that a utilitarian policy tends to choose the same individuals to whom it initially allocates interventions. While the initial conditions do not change the convergence results, whoever were the first M individuals to obtain an intervention at t O have gained an advantage (a positive drift in the random process), whereas everyone else has a disadvantage (a negative drive in the random process). We bound the probability of a policy to reinforce its earlier preferred choices by the probability that an individual never drop below its initial welfare level while the other individuals never grow below their initial welfare level. Then, asymptotically, the rates of growth will converge in the following way: some fixed subpopulation converges to the maximum welfare f+, whereas everyone else converges to the minimum decay g
Intuition:
First, consider the max-fg policy. Without loss of generality, consider the individual i being chosen at timestep O (0) = 1). We will apply Lemma 3 for the welfare process {X,(t)} under an intervention, i.e., (t) (t)
for all t, by showing that Xt(t) is a submartingale and lower- bounding the probability of the welfare level decaying beyond its initial level, X,) (equal to the welfare initial level (O). This defines a random process given that the individual i will be chosen over and over again (the process is conditioned on at) = 1). First, the process { (X:()) is a submartingale since, conditioned on (t) 1,
(t + 1)F=_] = EtU.(t + 1) (t) = 1] Ui(t) + f((U(()) > U,(t) = Xi(t), (18)
and note the uniform bound for (Z.(t)), in regularity conditions (Assumption 3c) By an easy induction on f, we get that [X,()] S oo, Vt, noting that Xx(O) = U+(0) < oo by definition of the initial conditions.
Given our regularity conditions (Assumption 3) we may now apply Lemma 3 in a particular way: consider (0)
for some u R. and we start the welfare process at 1. Note that (xx(t) (t)) is also a submartingale. Then, instead of bounding the probability of ruin, we bound the probability of (t) falling under the threshold u (where = (,)) We do that by the substitution W. = (t) u. Vt 2 1, and We is still a submartingale. Thus, we can apply Lemma] 3 to obtain
(xx((1)) S exp(-r" (X1(1) - u)),
(19)
where ((x(1)) = P(T( < oo), ((), min 1: xt) < u).
In a similar fashion, we now consider all other individuals j who were not intervened on at the first timestep 1. O. For each of these, the process (Y)()), where Yy(1) == (0))),, (1) = 0 (conditioned
16

---

Published as a conference paper at ICLR 2025
on j not being chosen again) is supermartingale:
(t + 1)] = [t + at)
= 0] - ((.()))) < U.())
Yi(t)
(20)
by applying equation | and noting the all functions (:(+) are positive by definition. In addition, again we know [t)( < 00 by uniform boundedness of [t) in regularity Aanditions(Assumttion ()),
and noting that (0 = U,(0) oo by definition of the initial conditions.
Again, we can apply Lemma for the process {- (xx)(t))22 (which is now a submartingale) and ruin threshold -u to obtain
* (-Y(((1) 2 exp(-r" - Y(()),,
(21)
where vi( ((-Y((1))==(TT)))))
< TY), min)t 1: -1:-)(()
Next, we lower bound the probabability that the individuals who were chosen at timestep 0, denoted set S, will continue to be chosen at every timestep. To do so, we note that this probability is equal to the probability that every i E S is chosen at every subsequent 2 and all other j & S are not chosen every 1. Among all events that comprise this probability, one of them is the event in which (t) 2 and (t) < u, Vj & S (remember here that (t) = (.(t)la((t 1 and (() (a) - 0, for j $ S). Thus,
(ta,(t) = 1 and (t) 0, vj $ S, 2 1) 2 (() 2 u and (t S u, Vj $ S, At 1) (22)
The righthandside consists of independent events w.r.t. L, since we have conditioned already on the intervention, so we can further compute it as
II (() > u, 12 ES
II <,,]t)
jES
(23)
II (1- < <o (T) < EES
II (1 - P(T < s. Y,(T) 2n)
j&S
Finally, we lowerbound equation 23 by the bound we obtained by our Lundberg-type inequality for the welfare process:
II (1 - P(TTT 00 s.t X,(T) u)) II (1-E < s.t. Y,(T) u))
(24)
iES
jES
2 II (1 (I)()))) II (1 (1- (1-v)(-Y(11))
(25)
iES
JES
> II (1 (1-ex (1-exp(-r's ((x(1) -)) m))) II (1 -1- exp(-r--" . (u -u- Y,())))
(26)
iES
jES
Our regularity conditions ensure that equation 26 lowerbounded by some positive constant D > 0: Assumption 3 states that 3z", 1 > s.t. (7+ 1) 2 =*) 2 and P ((Z,((+1)) < -2) > l. Since / > O, this offers a strictly positive lower bound on equation 26 Furthermore, pf does not depend on Fo but it may depend on the initial individual i that was intervened on at timestep O. We take the minimum of pt among all individuals E [N] (since any of them could have been intervened on at timestep 0), and obtain p = min(p;) > 0. Then, note that the probability of individual i being chosen for all t > O also depends the rule of max-fg and the tie-breaking rule of choosing the smallest index, which will ensure the individual being constantly chosen once the (t)) + 9, (t) (U)) + 9, (t) won't be violated for all O. This is true since in addition to the modeling condition that states that (-()) is decreasing, we also assumed that f((-) + s((-) is increasing.
As time grows, the probability of the utilitarian policy fixating on one single individual is lower- bounded by 1 (1 p²) Tm where m denotes the number of times the set of individuals who receive the intervention changes, which converges to 1 as m + oc.
Lastly, we prove that for an individual E S with (t) 1 for all 2 0 a.s., have R, f a.s., and for an individual j with (1) = 0 for all 2 0 a.s., j & S, we have R2 = -9 a.s.
-
In doing so, apply Lemma repeatedly:
17

---

Published as a conference paper at ICLR 2025
First, for individual i S that gets chosen the first timestep, set Y, U, + 1) - (t), μ = f and obtain from Lemma | that lim U((t) = +oc. For the same individual,
apply Lemma | again with μ X = ft, which shows convergence of (Ui(1) - (t(0))/t to f+. We then note that lim (U.(1) - U((0))/t = Rg
Second, for any individual j E S. we set Y = U,(t +1 (t) and A= -9 and obtain from Lemma 1 that lim (t) -c0 Finally, we apply Lemma [ again for all such j with μa = -g which shows convergence of (t,(t) - U,(0))/ to ggt We then note that lim (U₃(1) (0))/t R2 -g
We note that the proof goes through in the exact same way for the max-U and max-f policies, since the only place the functions f4 and 9, play role is in the tie-brea rule: when fa is increasing and the tie-breaking rule always chooses the individual with the lowest index, the probability of policy continuing choose the same set of individuals converges 1, whereas the probability of every choosing another individual j converges to 0.
Proof of Theorem 5 By the weak homogeneity condition (Assumption 3())) and the survival con- dition (Assumption D. we have f > N W-M
M g+. Then, we apply Lemma by setting Y; = U,(t + 1) (t) and r = M N f((.(tt) (1 (1- - ()))).(t)) We note that μe actually evaluates in expectation the rate of welfare increase under the random policy, where E(a,(t) Fi] M for any t 2 0, E [N] and Fe Thus. we obtain that [t + 1) (t) [ f. (L M )g,"
under the random policy. From this we conclude that lim,-to min, U,(t) = 00. and thus every individual's 's welfare will increase unboundedly over time. Since limy (T) - to0, we have limy PT ft (1 N g and since pr bounded, we apply Lemma with Y, = U,(++ (t),), X = μ N
M f- N )g+ Finally we conclude
- (0)
M f - (1 N
M N
R² =
lim 100
g > 0, a.8.
(27)
A.1 LUNDBERG'S INEQUALITY FOR SUBMARTINGALES
In this subsection, we present technical details used in the proof of Theorem
In Moriconi (1986) (page 179), the author briefly mentioned the Lundberg's inequality also holds for submartingales. We provide the proof below for completeness. First, we define the adjustment coefficient for submartingales:
Definition 2. Let (X+}
) be a submartingale, the adjustment coefficient, denoted by r", is the positive value such that (exp(-rt X,)) is martingale, i.e., E[exp(- Z4+1)] where Ze+1 X X
Lemma (Lundberg's 's inequality for submartingales) Let {X,} be a submartingale with Xo > O, r" be the adjustment coefficient of (X.) and assume {2r} are i.i.d. The probability of ultimate ruin is bounded as follows
bbu) < exp(prr"u
where (u)) (T) < oo), T(u) min (121 1 X, S 0, xo= u..
Proof. The proof is similar to the one for analyzing the surplus of an insurance portfolio (refer to Theorem 5.2 in Tse (2023). We prove the result by induction for (); u) PP(T(u) () and
18

---

Published as a conference paper at ICLR 2025
denoting
(11
P(Zz z)dr
exp(r" (-u - (2) = x)dxæ
exp(-r"1)
exp(-rr exp(->"Z))P(22 x)dxx
exp(-rru).
Assume Lundberg inequality holds for any time step less than t and u > and now consider (1 + u,,
(tt + 1;u) =1,, u)) / (Z2 =
(6)
P(Z 2)de + (0 (o((->x)PP(2 = u)dx
11
/ exp(-r' u)) = z)dr
<
exp(- -r (x + + P() = x x) +
= -ep((-r E(exp 22)] (c) exp(-r"u).
where inequality (b) holds by Lundberg's inequality for time step t.
Remark 1. Note in the proof of Lemma 2 using condition E/exp(-77 -20pp-=--2))) S 1 in equality (a).(c) enough, which is a weaker condition than Zg is adjustable.
The following corollary is an immediate result the above lemma.
Corollary 3. Let {Xe} be a submartingale with E[X X.) Xx + c where c 0. Denote Z+41 = X++1 X₁ and {Zt] are i..d. Assume there 3z² > s.1. IP 2 2²) > and
P(Z --²) 0. There exists positive constant r" such that
<b(u) exp(-r"u).
Proof. The proof is immediate by noticing
[[xxp(-rx.+1)) Fel = exp(-rX,) E[cxp(-r((X))) X,)) F.).
Denote (6() == Eeeep(_r*r (X xx)) Fi), which is continuously differentiable, and we obtain (00) = 1, (0) - by computing the closed-form derivative. Since P(Z 2*) 0, P(2+1 < -2") > O, and hence we have -im Φ(r) = +oo. Hence there exists r* such that <(r") 1. Moreover, since Z; are i.i.d. and (X,) is adjustable (i.e., there exists an adjustment
coefficient as defined in Definition 2 that does not depend on (X+)), we can apply Lemmal and we conclude the proof.
The following lemma is an adapted version of Lemma and will become useful in the proof of the main result.
Lemma 3. (Lundberg's 's inequality for a welfare process) Consider random process {Xt(t)) de- fined as X; (t) = U,(t) (t,(t) = 1, with U,(0) = u and (Ut(t)} defined as the welfare process 1...., N. As such, (t) defines a welfare process under an intervention, i.e. (;(1) = 1. Assume there exists x² 0, < l < s.I. P(Z((1 + 1) > x² Fi) 2 I, P(Z + 1) S -z") 2 I for any F, and any i. Then, for an individual i, there exists positive constant r*, such that the probability of ultimate ruin is bounded as follows
model [ for
(u) I exp(-r"u)
(28)
where, by an abuse of notation, vlu) = (T < sc). T(u)= = min (in 222214 1::X(11 & 0,) = n).
19

---

Published as a conference paper at ICLR 2025
Proof. For an individual i and timestep t, denote d(r) = E/exp(-r(Xx(t+ x,(())) Fi), for any r. We observe
((r) E
+ expxx(-rk))
k=
('(r)
Σ
(t =) exp((x-p
(r)))= = Σ
1 AP 74 1) =)))) exp(-rk)).
Notice that (0)) = 1, d(() (1 + 1) Fi] S - and 0 < (((r) S 2bk exp(rb) for any Fz. First of all, we know that there exists a small interval for r near zero, (0, a] for some (0, r(Fi] by applying Rolle's theorem. Next,
r(Fe) 0, such that (xx) s for
claim there exists r" > 0 s.t. o(r*) S 1, that is independent of any Fr. This claim can be easily proved by contradiction: assume for any 6 O, there exists r. S 6, Fc(e) such that Φ(r:)
1. Note that under Fi(e) and since ((0) = 1, there must be an x E (0.re) such that ø' (rre) > O. Making 6 arbitrarily small, we get that r. + and since Φ(0) 1 and o' d0(0) < O. there must exist æ E (0, rel such that (x)x) + +00. This contradicts with the fact that (r(r) < 26k² exp(rb) for any Fz. Hence for our welfare process (1), which not i.i.d. for different t, and not adjustable, but satisfies Ejexp(-r" (,)t) S 1 for some positive constant r" that is independent of F; and all i E [N].
Finally, we conclude the proof by pointing out that the proof of Lemma still applies under condition E]exp(-r"22 Zt())) s 1 (see Remark
B POLICY COMPARISON UNDER A RUIN CONDITION
Our survival condition, Assumption T defined a parameter condition in which there exists a policy that can 'lift' every individual unboundedly, as time grows. We show that different versions of the Rawlsian policy achieve this property (in addition, the constant proportions policy from Radner & Rothschild (1755) will also achieve this property). Under the survival condition, our main result shows that the Rawlsian policy will achieve better long-term social welfare.
In this section, we introduce complementary condition to the survival condition, called a ruin condition. Intuitively, under this condition, even Rawlsian policy will not be able to ensure that every individual will have positive welfare, asymptotically. As such, the lowest welfare will decay indefinitely almost surely.
Note: we borrow the 'ruin' terminology from ruin theory, but the definition of a 'ruin condition' is specific to our setting, as defined below. Our proofs make use of ruin theory in applying Lundberg's inequality, as seen in Appendix[A] A
Assumption 4 (Ruin condition).
We assume č((st (Ilftt...,,tt..... (gT ..... (55)) < Ohheee Ç RRN 3 and J = =sup fi(x) S = inf f(()) 0, of gu))
R is defined in equation
0, 9; = inf (x) 0.
Theorem 6 (Theorem 2 formal). the ruin condition is met, under regularity, modeling conditions,
and as long as fa(-) + ()))) is increasing for all E (N), the result in Theorem 7 is reversed:
Rutilitarian 2 RRawksan a.s.
where the Rawlsian and utilitarian policies are defined in the same informational contexts, i.e. (min-U, max- U), (max-g, g, max-f), (max-g, max-fg).
Proof of Theorem We prove this result similarly as in Theorem I by computing in closed-form the individual rates of growth under every policies, and then computing the long-term social welfare as an average of these rates. First, we note that we can compute the individual rates of growth under utilitarian policies just like in Theorem by noting that the proof does not make use of the survival condition (the survival condition is only necessary to compute the individual rates of growth under Rawisian policies). Next, compute the individual rates of growth for Rawlsian policies
20

---

Published as a conference paper at ICLR 2025
under the ruin condition in Theorem 7 (Corollary 4). We then uniform boundedness to compute the long-term social welfare for Raw and utilitarian policies, noting that a utilitarian policy is better than Rawlsian policy long f 2 which is true by the "rich-get-richer" modeling condition.
We present visualization of Theorem in Figure where we can observe the utilitarian policies imax-U, max-f, max-fg) converge to higher growth rate while Rawlsian policies (min-U, max-g) converges to suboptimal average growth rate. The experiment setting here is as same as Section Under uniform boundedness, the parameters for the shape of (Ji(-))s (ge (() are randomly sampled within the same interval, which is weaker than the assumption Theorem
max-g (Rawisian)
min-U (Rawisian)
Social welfare
(utilitarian)
-20
max-f (utiiitrian]
max-U (utilitarian)
-30
Ren u (Rawisian)
Rmar (utlitarrin))
-50
2000
4000
6000
8000
10000
Time step
Figure 2: We show social welfare as the finite-time growth rate averaged over all individuals, for all policies (solid lines), as well as theoretical expected growth rate (dashed lines) under the ruin condition.
Theorem 7. Under modeling, regularity, and ruin conditions (Assumption (a), policy leads to the following closed form solution of the individual rates of growth:
3 the min-U
Rg = E(If : fN).
(g.t.... 1=1...-N, a.s.
Corollary 4. With the addition of the uniform boundedness condition from Assumption (c), we can simplify the individual rates of growth, obtaining the long-term social welfare value for the Rawlsian policy under the ruin condition
M f
N M 9
Rmin -Rmax-R
<0
0.s.
N
N
Proof of Theorem obtain
Under the ruin condition, consider the upper bound in incquality 10 and
B + - ((t) [ "(. ,N). (o ........)) < <. # E(U .....
Applying Lemma with Y; =
(oT ......0 .. )), we obtain
Ü(1 + -00 and hence minggIN U.(() -o. Here we use the same intuition as Rothschild 1975a), proving there exists 7* such that
max; U.(T*) miny U,(T)) 0.
lim
(29)
We prove equation 29 by applying Proposition and Lemma in Rothschild (195aa). Now with equation 29 we know everyone has the same growth rate and hence U,(t) -0o for all i E [N]. Then by the uniform boundedness from our modeling condition and the monotonicity of (fr()).. (g)())), we have
EN (U,(t + -U())
M N
N M
lim E 1-400
Nl
N
21

---

Published as a conference paper at ICLR 2025
X = / = N J N- MMM Apply Lemmal with Y Σ (V
N gt.
we conclude
N
M R R- N
N M g
N.
Then since every individual has the same growth rate, we conclude our proof and corollary follows immediately. We note that the same result easily follows for the max-g policy, given the tie-breaking rule that favors the individual with the lowest welfare.
To show convergence of the individual rales of growth under the ruin condition, uniform boundedness is needed to obtain the same growth rate for every individual. In contrast, Theorem 3 does not require uniform boundedness for obtaining the same individual growth rate. asymptotically.
Remark 2.
The following proposition is a counterpart for Proposition in Rothschild (1975a) under the ruin conditions. In the proof below, emphasize the differences while keeping the other steps concise. Proposition 1. Under the conditions of Theorem 7 let D(t) = maxig[N] Ut(t) - minjell] (t) there exists a constant G such that if D(s) 2 G and T is the first integer such that D(s+1T < G, then there exist H and K such that PP n) S Henn
Proof of Proposition[] 7 Suppose K. is any proper subset of [N]. and K'' is the complement of I. in [V]. We take the case where the min-U policy only considers individuals in K. while ignoring individuals in K", We prove the following inequality by induction: there exists constant Ty: such
that
JEK' E max Ut(1) min U+({) | Fol S max U,(0) min (0) 2, VI 2 Tk- EEK'
(30)
iEK
=EK
The inequality in equation 30 allows us satisfy the conditions Proposition from Rothschild (1975a) and easily adapt the proof of Theorem The base case N = 2 is trivial since we know everything about the behavior of an individual with a; (t) for all 0. Now assume that for
N = No, Proposition holdd, and consider N = No + 1. Consider the set K., since [K]] S No and by induction, we have
Ui(1) = -((ff (9. liex), Vi IC, a.s.
lim 1+00
t
- As for the set K", apply the monotonicity of ())(k))se)) and the uniform boundedness condition, obtaining
where (x)(EE)) (U))) = (M- - -xex mutt
(Eiex z.-ي!)
U) - Vi ViE
lim +00
t
Hence we know that
lim maxjex U,(1) - mingex Ui(l) - -((U Jiek, (ot )oex)) < 0, a.s.
t
By applying the Latou-Lebesque theorem, have
lim max U,( min U:(() Fo = = - -((( (0. )iek)) < O,
10000
t
Then there exists Tk > such that for all > Tk and
E
max (t - min Ut(t) Fo
I max (0) JEK'
min U,( - 2.
JEK'
FEK
At this point, we may apply Lemma 4 and Lemma 5 from Rothschild (1975a) and conclude our proof.
O
Remark 3. The intuition for Proposition D is the following: when the ruin condition holds, indi-
viduals receiving an allocation will still decay in welfare, due to the strong decay functions effects that the ruin condition models. However, this happens whose welfare decays absent any intervention. As such, the welfare gap between the individuals with the maxinum welfare level and those with minimum welfare will be bounded, asymptotically, and therefore everyone will decay, yet at a slower rate given the intervention of the social planner than without any intervention.
slower rate compared to individuals
22

---

Published as a conference paper at ICLR 2025
Remark 4. The survival and ruin conditions characterize two model states in which a definite comparison between Rawlsian policies and utilitarian policies in terms of the cong-term social welfare they achieve. There is middle ground, in which neither survival nor ruin may hold, in which the direct comparison between policies becomes much more difficult. We leave this direction for future studies.
make
C EXPERIMENTAL DETAILS
This section contains detailed simulation notes for Section 5 and Appendix figures. For the simula- tions in which the return and decay functiin bounds are uniform, we choose threshold parameters FT GT G. s.t. fx) = )x) 9 for S FTT. I resppecttivll, and fi(z) ft, (((x) g for 2 F
G respectively, and we linearly interpolate between these thresh- olds. Choosing FT F.T and GT- > GT ensures that fi is increasing and 9i is decreasing the non-constant segments. We generate FT, P. G G randomly in the interval (0, A] for some A > 0. For Figures !
2 and 4 we filter to ensure that fi(.) + 9:(-) ) is increasing. For Figure 3 filter to ensure that (+(+) + g-(-) is increasing under some threshold
and decreasing above threshold r. For all figures, we average over 50 iterations and report the social welfare obtained at every timestep. Our results are qualitatively the same for other functional forms of fi, 9i: such as sigmoid functions.
All code and data used in our simulations is available in this repository
D BEYOND A MATTHEW EFFECT: MODELING VARIATIONS OF THE TREATMENT EFFECT FUNCTION
In the main text, we modeled a Matthew effect through the "rich-get sechh-et-rricher and "poor-gct-poorer" behaviors induced by increasing intervention return function fi(.) and decreasing decay function (-),), under the assumption that the treatment effect f.)-) + gr(-) is also increasing. This assumption suggests that interventions higher level of welfare have higher impact. We explore a variation this assumption in this section, assuming that there exists a threshold r above which the treatment effect is in fact decreasing. In doing so, we capture diminishing return effect, where individuals with the highest or lowest levels of welfare benefit less from an intervention than individuals with moderate levels of welfare. This is motivated by recent policies that target people with moderate welfare values: the algorithmic profiling policy introduced by Austria in 2020 (Allhutter al.) 2020) predicts a probability of an individual based on an intervention (in a sense, a prediction of the treatment effect). The policy allocates an intervention to those ii-the-middle
re-enter the job market
suggesting that moderate welfare values are predictive of the highest treatment effect. In addition, optimal taxation policy and redistributive taxation (Mankiw) et al.) 2009 Benabou 2000) often argue for an increasing tax scale or a decreasing benefit scheme as function of income. We provide theoretical extension from our results in Theorem I under stricter homogeneity assumptions, capturing a diminishing return on interventions.
Corollary 5 (Diminishing returns). Assume the conditions of Theorem 7 but with a threshold T > 0 s.t. fa(x), Ja(x) + (((x) are decreasing for x 2 T. Furthermore, assume that the functions f(, 9: are uniform for all individuals (f((x) = fz(2), gz(z) 9j (z) for Vi $ E [N]. Finally, Then with positive probability, a Rawlsian policy achieves a higher long-term social welfare than a utilitarian policy:
lim (z) + (x)) < lime-++oo +
PRRRaawhinnn 2 Rutilitarin)) > O,
where the Rawlsian and utilitarian policies are defined in the same informational contexts, i.e. (min-U, U, max-U), UU, {max-g, max-f), ff, (max-g, max-fg). Note here that all policies break the tie by choosing the individual with the lowest welfare level.
Remark 5. Corollary Shows that our analysis under the simple assumptions on the monotonicies of intervention return function and the decay function can be applied to more complicated cases. When f((-) + (.(.) is increasing, the choices of max-fg (max-f) policy and min-U policy diverge. cause long-te term loss by accumulating the decay of the ignored population. Furthermore, the ignored population enters low-welfare trap, since they will likely not be targeted again. See Figure for an illustration
The tendency of max-fg (max-f) policy of focusing on the better-off population
23

---

Published as a conference paper at ICLR 2025
of Corollary 5 where observe utilitarian policies (max-U, max-f, max-fg) show lower growth rate over the finite time horizon as compared to the Rawlsian policies (min- U, max-g).
Proof. First of all, since the asymptotic behavior of individuals under the min-U policy (i.e. being lifted unboundedly) does not depend on the simple monotonicity property of the return functions, have Ui(t) + +o for vi [M] under a Rawlsian policy with survival, regularity conditions + gi(z)) from Theorem 3 Hence we conclude RRawlsian N g- However, hefore the turning point of the monotonicity of fe(.) + gi(:), the max- ig policy tends to focus the better-off individuals by applying the rule of max-fg policy. Hence with positive probability, some individuals will be left behind (with ft(U((t) + g. (.(U((t) less than lim (x) + (x()))) then if these individuals will receive budget for all t afterwards, the probability of them never crossing the turning point (i.e., where the mononicity of the treatment effect function changes) is lowerbounded by a positive constant independent of Fe by applying
and existence of -im M - N-M f
Lemma
And after the turning point, the max-fg policy coincides with the min-U policy and lifts every dividual to infinity, asymptotically, with positive probability (not almost surely anymore since the individuals can drop below the turning point of f((-) 9-).). Hence we conclude that with positive
probability, Rmax-fg E
|M M 1ig (N M k)g
,,..., N -...,,NMM, M...
-.M...
With positive probabil-
ity, we have Rmax-tr fs S. R.
A similar argument applies to the max-f policy by substituting fi(-) + ge(-) with fi(:) and hence omitted here. For the max-U policy, Theorem 4 still applies and we have Rmaxx-f -g I Rmin-U with positive probability.
max-g (Rawisian)
min-U (Rawlsian)
max-fg fg (utilitarian)
max-f (utilitariaa)
Social welfare
max-U (utilitarian)
350
Random
Amn-u (Rawisian)
340
Rnx y (utilitarian)
310
1000
2000
3000
4000
5000
6000
Time step
Figure 3: We show social welfare the finite-time growth rate averaged over all individuals, for all policies (solid lines), as well as theoretical growth rate for min-U and max-U policies (dashed lines) under diminishing returns.
E DIFFERENT TIE- RE-BRRAKING RULE FOR THE MAX-G POLICY
In Section 2 we introduced a rule that breaks the tie favor individuals with the smallest index, when they have the same welfare values. Additionally, when the decay function values are the same, the max-g policy chooses the individual with the lowest welfare. We explore a variation where the max-g policy breaks the tie by also choosing the individual with the lowest welfare in Figure noting a slightly convergence rate than the min-U policy. All simulations details are the same Section 5
24

---

Published as a conference paper at ICLR 2025
(Rawisian)
200
min-U (Rawisian)
max-fg (utilitarian)
max-f (utilitarian)
Social welfare
max-U (utilitarian)
Random
Ros y (Rawisian)
240
Roxr (utilitariaa)
220
1000
2000
3000
4000 5000
6000
Time step t
Figure 4: Social welfare as the finite-time growth rate averaged over all individuals, for all policies (solid lines), as well as theoretical expected growth rate, asymptotically (dashed lines). The tic- breaking rule favors the individual with the lowest index for all policies.
F PROPORTIONAL RESOURCE ALLOCATION
In the main text of the paper, restrict our attention integer resources each time step and use budget 2 1 to intervene in several individuals. Here, we consider interventions that allow proportional allocation, as follows:
proportional max-U: a+())
Σ==1e0,00 e0,(0)
proportional m:i-UU: (t()) = 6 -U(())
Σ/=1 e
Theorem 8. Under regularity (Assumption and modeling conditions (Assumption 2 and assume that fi(x) f(x), 9;(x) = g(r), the proportional max- U policy leads to the following closed form solution of the individual rates of growth:
[+
Ra
-gt,i a.s.
where J is random variable with values [N] whose exact value depends on U0), f()), and g()).
Proof of Theorem 8 For Vi,j E [N] s.t U.(() 2 U,()), we have az(l) 2 az(1), then under the
assumption that [(xr = (z) g()). we further obtain
E + 1)] = at) ((t))((((((t)) f(U((t)) (1 - (t)))) ((U(())
2
- at))) g(U,(t)
2 at(()) f(U)(()) - -(1-a((t) ((U,(t)) Etz
where the last inequality holds because of modeling conditions (Assumption (a), (b)). Consider
i E M, where M, = arg waw.maxxx.uu... (U) and i MM,j E [N] such that Ui(1) (1)) 1, we have
ElU 1) ()) FR
(U,(t) - Ut))))
E[ + -1)-
= as(l) ((U((()-((1-)))) ((U((()) - (a,(1) ((U(()) - -11 - (((())
a:(t) ((-(()))(11(())-((((000
-1
- f(U((t) -(1 (t)) a,(t))
(a:(1) - (fU))) (a(()) -a)()) ((U(())
M(1) M(1-1
MD) M(0-1
ΣJENN (0) -f+ ΣJENN (0)(1) 9
1 11
(5 +
N
25

---

Published as a conference paper at ICLR 2025
Now treat (t()) -0-- (t) as the welfare process and apply adapted Lundberg's 's inequality (Lemmaa) we claim that with positive probability that Ut(t) - (t) 2 1 for vt 2 0 when U,(0) - (0 where i MM. Then combine with the regularity condition (Assumption B (c),, we have that with positive probability (lowerbounded by a constant) that (i(t) - (t >
for Vi > 0 where e Mo- Then we apply the same reasoning for E i and conclude that with probability 1, the proportional max-U policy will fixate on one single individual asymptotically.
Theorem 9. Under regularity (Assumption B) and modeling conditions (Assumption ((a)) the survival condition [Assumption 77, the proportional min-U policy leads to the following closed form solution of the individual rates of growth:
R = =((Utt "(U.................... (T.....T.....N.... i N, 0.S.
Proof of Theorem 9 The result can be proved by induction, and the proof of Theorem 3 applies here with minor modifications. We assume for N individuals the conclusion holds, and consider M == arg max, m (U) and .M = [M]MM. For VI EM,
at - 1+ (N (N 1)e- D(1)
e D())
Σ
((((t)
1
1+ (N D(o)
icM-
where ((t) maxje(M] (t minjeli] U.(t). Hence there exists constant C' such that when
D(t) 2 C, the survival condition for M is satisfied and we have
UM-(L + 1) UM-(( Σ urM a.(() (((U(()) (1 aa(()) 91 (U)(
KEMO
Σ
uzM a.(t) ST -(1 -11-a 9,
REM
Σ
ai(1) Σ
9
(
Σ
fj +
KEME fiz + g
iEME
JEM
-1
9
Σ
1 -N 1)e
Jj +9
NEM: Jk + th
JEM-
where (M), w are defined as in equation for for set MS. Hence we apply the conclusion
for M and claim that there exists constant TMM such that when ΣiEMa: a:(()= S 14NN-1ee-e for
vt 0, we have
min (t(t) JEM-
2
min +1,
Vt TM
JEM-
max Ut(t)
max-
1. VA TM.
JEM-
JEM
As for i E M,
B[ZZ(L 1)) Fi) S a(())
aa(())
( 2 e- -D(A)
NN
N - 1 1 e D(O) fit
N 1+
-e e- -D(A)
9;
and when D(t)= 2 C' for constant
C' 0, we have
1
N - 2 e- D(t)
1
N-1+c D(1) ft
N 1+e -(()) 9:
min 9
(31)
2
iE[N]
Hence for the whole population [N], if EieM. (t) min there exists constant T such that
I+(N-1)e-o y min
+, }
2T
E
min (t)
2 min U,(0) + V2T, J
JEM
E
max U (t)
< max max U,(0) 1, V JEM
JEM
The rest of the proof goes through with minor modifications given the above facts.
n
26

---

Published as a conference paper at ICLR 2025
G MODELING CHOICE ON MONOTONICITY
Use the assumption the Matthew effect, model (fi(-)); increasing and (g:(-);; de- creasing. For other combinations of monotonicities (see Table T. one could develop theoretical foundation using similar tools. We provide finite-horizon simulations for different combinations of monotonicities (as in Table I in Figure
f((-)
Decreasing
Increasing
Constant
ge(-)
Decreasing
Mixed (Rawlsian)
Rawlsian
Mixed (Rawlsian)
Increasing
Mixed (utilitarian)
Utilitarian
Utilitarian
Constant
Tie
Tie
Tie
Table I: Here "increasing" denotes non-constant increasing functions and "decreasing" denotes non- constant decreasing functions. Each cell represent the comparison between the utilitarian policies and the Rawlsian policies under the monotonicicy combination of fi(-), 9:(+) being "column-row" using the measure of long-term social welfare. We use "Mixed (utilitarian)" to represent regions where only utilitarian policies achieve the better long-term social welfare, "Mixed (Rawlsian)" to represent regions where all of Rawlsian policies and only part of utilitarian policies achieve the better long-term social welfare, "Utilitarian" to represent the region (discussed in this paper) where all of utilitarian policies show superiority over all of Rawlsian policies, "Rawlsian" to represent the region (discussed in this paper) where all of Rawlsian policies show better welfare than all utilitarian policies, "Tie" to represent regions where utilitarian policies and Rawlsian policies perform equally well. The max-g policy breaks the by choosing the individual with lowest welfare and the other policies break the tie by choosing the individual with the smallest index. Our discussion in Section 3 constrained to cell on the first row, second column because of our assumption about "rich-ger get- richer" and -poor-get-pe get-poor". All the simulations are conducted under the survival condition and the uniform boundedness assumption.
we observe the trend shifts as the monotonicity of decay functions g;(-) change. As the monotonicity of decay functions describe where the instability of the society located: decreas- ing (-)-) stands for individuals are more fragile as their welfare level are lower while increasing (-)) implies more fragility of individuals with higher welfare level. Hence Rawlsian policies, which aim to leave no one behind, would perform better under cases where g:(-) are decreasing. As when (-).) are constant (no inequality in decay functions), both types of policies will perform well show no difference in long-term welfare.
From Figure 5
H POLICY COMPARISON FOR HETEROGENEOUS POPULATIONS
In this section, we discuss the comparisons among policies when the uniform boundedness assump- tion violated and provide evidence that a Rawlsian policy still prevails over a utilitarian policy when the heterogeneity of the population is bounded below some threshold.
When the uniform boundedness condition may not hold, a direct comparison between Rawlsian and utilitarian policies becomes more intricate. We explore this case by simulating the long-term social welfare values when the limits the intervention return and decay functions fi. gi are different. We the same dataset and pre-processing procedure Section exceett generate heterogeneous bounds {fr ft+ 9 9. and conduct comparison using finite-horizon simulation.
We draw the bounds 9; .g. ST ftt from normal distributions in the following way:
The variance of the normal distribution modeled by a parameter o² that controls the heterogeneity of the bounds: larger means more heterogeneous bounds.
The mean of the normal distribution chosen differently for fa and g:: first of all, we choose means for the fo and 9; functions that makes survival condition possible. Second, we introduce a parameter b that models the strength of the decay functions: larger means that 9, and a, are closer, and therefore, the decay effect is bounded. A smaller means
27

---

Published as a conference paper at ICLR 2025
370
Social Welfare
Social welfare
welfare
250
370
mav (honsian)
moxts (olilattr)
272
700
mexd Rariaar
203
maxil (rlla
120
Aun (tawbdar)
Aw lattel
240
130
2002
2002
4300
5020
1200
2020
3010
4000
5200
0002
1200
2300
3002
4002
5000
6026
Time step
Time step
step
200
Social welfare
Soccial welfare
Social welfare
250
140
120
-92
240
1000
332
Time step
Time step
Time step
300
Social welfare
Social welfare
-000
Social welfare
370
370
200
340
1002
1800
2000
4000
5200
2002
2000
1003
Time stept
Time step
Time step t
Figure 5: Social welfare as the finite-time growth rate averaged over all individuals, for all policies (solid lines), as well as theoretical growth rate for min-U and max-U policies (dashed lines) under different combination of monotonicities of {f((-)}. {9:(-)}. The monotonicities of (f((-)) {9)(-) corresponds to Table e.g., top left plot generated for decreasing (fi(-) and (.)(-))
that the (relative) decay effect get quite large for individuals with lower welfare, i.e., we expect stronger get-poorer" effect.
Based on this set-up, we simulate the following model: w.l.o.g. we set M 1, for each in- dividual E [N], of N (bg*. ,b?o"), of Ng (gt.a?), fi N (5)())))0 fit N (rt, where gt, ff are constant parameters, b (0, 10, 02 > 0. For each pair of (b,0²) parameters, we generate 50 sets of heterogeneous bounds (fg . ff Under each set of heterogeneous bounds (f9, ff f )imI:)), we average the social welfare obtained over all individuals, averaging over 50 iterations of the generation process of the intervention and decay function bounds. present the finite-time social welfare of individuals under different poli- cies using set of randomly generated bounds {f, ft 9, 9. in Figure 6 Figure 7 illustrates a heatmap of the percentage of times when the min-U Rawlsian policy has better long-term welfare than the max-U utilitarian policy, for each value of b and (1 meaning that the min-U Rawlsian policy better in all iterations). From Figure 7 we observe the min-U Rawlsian policy maintains the tendency perform worse the max-U policy in short-term while surpasses the max-U utili- tarian policy in the long-term for range of a values (in sense, for bounded heterogeneity). As b decreases (and therefore the decay functions 9i have stronger effect), Rawlsian policy starts performing better by preventing stronger loss caused by the decay of low-welfar individuals.
28

---

Published as a conference paper at ICLR 2025
max-g (Rawisian)
min-U (Rawisian) max-fg (utilitarian)
max (utilitarian)
Social welfare
280
max-U (utilitarian)
Random
260
220
1000
2000
3000
4000
5000
6000
Time step t
Figure 6: Social welfare the finite-time growth rate average over all individuals, for all policies (solid lines) under one set of heterogeneous bounds.
1,0000
1.0000
1.0
0.7625
Coefficient b
Coefficient
0.5250
0.5250
0.4
0.2875
0.2875
0.0500
0.0500
0.0
25.19
Coefficient o²
Coefficient o²
(a)
(b)
Figure 7: Percentage of iterations where min-U obtains better social welfare than max-U the
bounds {ff ft ". 1=11 vary according to parameters b and o², at time step f = 6000 (a)
and 10 (b).
29


Reviews:
[{"review_id": "arOkMjMnLp", "reviewer": "Reviewer_DG4s", "summary": "This paper studies the behavior of Rawlsian and utilitarian policies in long-run welfare dynamics. They show that under a survival condition, Rawlsian policies are better than the idealized utilitarian approach almost surely in the long run. Under a ruin condition, a utilitarian policy will achieve better long-term social welfare. Simulation results are provided to validate the theories.", "questions": "- In line 195, the author motivates max-U policy with the setting where $f_i + g_i$ are increasing functions. However, this is not enough, right? Say $f_1 + g_1$ and $f_2 + g_2$ are such that $u_1 < u_2$ but  $f_1(u_1) + g_1(u_1) \\geq f_2(u_2) + g_2(u_2)  $ (although $f_1 + g_1$ and $f_2 + g_2$ themselves are increasing functions).\n- In lines 218 and 219, why the max-g policy is considered a variant of Rawlsian policy?  I mean, $i=\\underset{j}{\\arg \\max }\\{g_j(U_j(t))\\} $ is still maximizing, rather than minimizing the welfare.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "This paper is in general well-written and smooth to follow. The theoretical results are strong and the technical proofs seem to be rigorous and well-organized. The simulation results validates the theories clearly.", "weaknesses": "The authors basically leave the survival and ruin conditions unjustified, and the simulation settings are ad-hoc. I would appreciate it if the authors could provide discussion and common example settings where the conditions hold.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "DRb0WHHx8T", "reviewer": "Reviewer_C3iz", "summary": "The paper compares the long-term social welfare resulting from two popular policy frameworks, a Rawlsian policy that focuses on the minimizing the smallest welfare loss and a utalitarian policy that focuses on maximizing immediate welfare gain. The comparison is conducted under a model on welfare decay and intervention return, assuming a survival condition and the Matthew effect. The conclusions are that 1) when the survival condition is satisfied, the Rawlsian policy achieves better long-term social welfare than the utalitarian policy; and 2) when the ruin condition is satisfied, the utalitarian policy becomes better.", "questions": "## Major comments:\n\n1. I am curious about whether the two essential assumptions for the theoretical results in the paper are realistic. These assumption include 1) the return and decay functions are monotone, 2) the bounds on the return and decay functions are uniform across all individuals. For assumption 1), the paper mentions that the social planner does not need knowledge of $f_i$ or $g_i$. In this case, how does the social planner know whether $f_i$ and $g_i$ are monotone in practice? For assumption 2), the \"rich-get-richer\" modeling condition also needs to be satisfied. However in reality, the rich gets richer at a much faster rate than other individuals. Is assuming the same upper bound for all indivuals realistic?\n\n2. What is the motivation and advantage for considering max-f and max-g policies, as opposed to the max-fg policy? The paper mentions that the max-d policy only requires partial information about the interventions, which is less costly to measure compared to measuring both return and decay. However, the max-g policy still requires information about the decays, which then offsets the previously mentioned advantage.\n\n3. As for the survival condition, how should a social planner assess whether this condition is satisfied in practice?\n\n\n## Minor comments:\n\n1. Equation (1): The definition of $\\mathcal{F}_t$ appears much later.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The paper puts the welfare policy evaluation in a sequential decision making framework. All models and assumptions are written clearly. The theoretical results are backed with numerical experiments and mathematical proof.", "weaknesses": "The theoretical results rely on several strong assumptions, and need more clarification.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "43GLU5pWtD", "reviewer": "Reviewer_Zhdw", "summary": "The paper studies sequential policy design with the goal of maximising long-term welfare. In particular the authors compare two policy classes—Rawlsian and utilitarian—and identify conditions under which each is optimal. They model the population as N agents, where each agent’s welfare naturally decays without intervention but improves with intervention. Sections 3 and 4 compare the policy classes in terms of long-term population welfare and individual welfare, respectively. Under regularity and survival conditions, the Rawlsian policy outperforms the utilitarian; however, under the ruin condition, this implication is reversed. Section 5 contains experiments with initial welfares drawn from SIPP data and the social welfare evolution supports the theory.", "questions": "- Please address the questions above.\n- A minor question: your theory seems to extend to interventions that allocate resources to more than one agent at each time step $\\sum a_i(t) = M$. Is there a reason not to include this in the main body, with \\(M=1\\) as a special case?", "limitations": "", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "- The paper is well-structured and theoretically rigorous. Theorems 1 and 2 in Section 3 are particularly insightful, establishing necessary and sufficient conditions under which the Rawlsian and utilitarian policies are optimal.\n- The individual welfare evolution and policy optimality for Rawlsian, utilitarian, and random policies in Section 4 is also novel, requiring new sub-martingale techniques in the proofs.\n- Section 5 includes experimental results that support the theoretical finding and we see that Rawlsian policies though initially leading to a drop, are optimal for long-term social welfare.", "weaknesses": "- The modelling conditions, such as \"rich-get-richer\" and \"poor-get-poorer,\" are well-explained, but the survival condition (line 255) is difficult to interpret. Is this condition standard from Radner and Rothschild, or is it specific to your framework?\n- Clarifying how common the survival condition is would help, as knowing whether the system is in a survival or ruin state is essential to determining whether a Rawlsian or utilitarian policy is optimal.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ozW9j9WdY8", "reviewer": "Reviewer_PGAD", "summary": "In this paper, the authors study how policies can affect long-term welfare in a population. Specifically, they prove that under the survival condition, the Rawlsian policy has greater long-run utility than the utilitarian policy, and under the ruin condition, the utilitarian policy has greater long-run utility than the Rawlsian policy. They demonstrate their findings through numerical simulations.", "questions": "* In the welfare model it is assumed that everyone's welfare will decay over time without interventions. But why is this the case? In the paper [Roy Radner and Michael Rothschild, 1975], such a model is used to describe how a manager would assign his efforts to several tasks. In this circumstance, it is reasonable to have a \"decay with time\" assumption since if a task is ignored then things may get worse over time. However, I cannot see why such an assumption can be generalized to the model of welfare. Also, the \"rich-get-richer\" assumption seems at odds with the bounded $f$, $g$ assumption. I think the authors should add more explanations about the modeling choice to make the paper more accessible to readers outside the field. I also recommend the authors use real-world data or existing literature to rationalize such modeling choices. (In the current simulations only the initial state of the model is based on true data, and all other components are hand-crafted.)\n\n* In the paper the author only considers the policy that focuses on one individual at a time. Can the authors comment on what the main technical difficulty is in considering a more general policy that forms a probability distribution on the population?\n\n* In the paper, the authors do not touch on the problem of optimality of policies. It seems the current welfare model can be conceived as an average-reward continuous-state MDP. Since the functions $f_i$ and $g_i$ are assumed to be known, finding the optimal policy is equivalent to solving a planning problem in such an MDP, which is well-studied in the literature. Can the authors give a concise discussion about that?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* The paper is very well-written and easy to follow.\n\n* The paper studies the problem of how policies can affect the long-run welfare of the population, which is important and interesting.\n\n* The paper presents theoretical results that have clear implications. The proof is technically sound.\n\nHowever, since I have no background in economics, I find it challenging to evaluate the significance of the paper's contributions or the reasonableness of its welfare model. So I choose to assign a positive rating to the paper with a low confidence score.", "weaknesses": "See the \"Questions\" part.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)