Policy Design for Two-sided Platforms with Participation Dynamics
Authors: Haruka Kiyohara
Fan Yao
Sarah Dean
Venue: ICML 2025 poster (2025)
Keywords: two-sided platforms
provider-fairness
bandits
population dynamics
Primary area: social_aspects->fairness
TL;DR: This paper studies how population growth affects the allocation design in two-sided platforms where multiple stakeholders participate.

Abstract: In two-sided platforms (e.g., video streaming or e-commerce), viewers and providers engage in interactive dynamics: viewers benefit from increases in provider populations, while providers benefit from increases in viewer population. Despite the importance of such “population effects” on long-term platform health, recommendation policies do not generally take the participation dynamics into account. This paper thus studies the dynamics and recommender policy design on two-sided platforms under the population effects for the first time. Our control- and game-theoretic findings warn against the use of the standard “myopic-greedy” policy and shed light on the importance of provider-side considerations (i.e., effectively distributing exposure among provider groups) to improve social welfare via population growth. We also present a simple algorithm to optimize long-term social welfare by taking the population effects into account, and demonstrate its effectiveness in synthetic and real-data experiments. Our experiment code is available at https://github.com/sdean-group/dynamics-two-sided-market.

Policy Design for Two-sided Platforms with Participation Dynamics
Haruka Kiyohara'
Fan Yao?
Sarah Dean
Abstract
Considering such non-stationarity and two-sided dynamics is crucial, the viewers and providers are affected by each others' population in self-reinforcing feedback loops.
In two-sided platforms (e.g., video streaming e-commerce), viewers and providers engage in interactive dynamics: viewers benefit from in- creases in provider populations, while providers benefit from increases in viewer population. De- spite the importance of such "population effects" long-term platform health, recommendation policies do not generally take the participation dynamics into account. This paper thus stud- ies the dynamics and recommender policy design on two-sided platforms under the population ef- feets for the first time. Our control- and game- theoretic findings warn against the use of the stan- dard "myopic-greedy" policy and shed light the importance of provider-side considerations (i.c.. effectively distributing exposure among provider groups) to improve social welfare via popula- tion growth. We also present a simple algorithm to optimize long-term social welfare by taking the population effects into account, and demon- strate its effectiveness in synthetic and real-data experiments. Our experiment code is available at https://gituub..om/seee wtw.ghtww....mtaaaagoognnnnnnnia group/dynamicsttoo- mided-market.
Example (Video recommendation). When a platform has many videos about sports, viewers can expect that top sports videos have high quality (e.g., production and intellect). Meanwhile, if a platform is popular among sports lovers, creators will produce more sports videos to gain more views.
Example 2 (Job matching). When a platform has many applicants from target category, companies looking to fill specific role can identify more highly skilled applicants. On the other hand, if platform has more openings for specific job type, more applicants from target categories will register for the service.
The "population effects" in the aforementioned examples strongly affect viewer utility and their long-term satisfac- tion. However, their implications for recommendation pol- icy design have been under-explored. The conventional formulation of recommendation follows (contextual) ban- dits (Li et al., 2010) and assumes that viewers and providers are static across timesteps. Some recent work studies con- tent provider departures (Mladenov et al., 2020; Hutten- locher al., 2023) and the (negative) impacts on viewer welfare (Yao et al., 2024b; 2023a). However, the viewer population is modeled as static. In contrast, existing works which consider dynamic viewer populations assume that provider population is fixed (Hashimoto et al., 2018; Dean et al., 2022). Therefore, we cannot tell how we should optimize a policy, particularly in the initial launch of the platform when two-sided dynamics exist. Finally, exist- ing works considering strategic content providers (Hron et al., 2022; Jagadeesan et al., 2022; Yao et al., 2023b; 2024a; Prasad et al.. 2023) model the (strategic) evolution of provider features, assuming that the total number of view- ers and providers are fixed. These works cannot tell if a platform can "grow the pie" (i.e., viewer and provider popu- lations) improve long-term welfare.
1. Introduction
Two-sided platforms, where some individuals view content or information provided by other individuals, are ubiquitous in real-world decisions, e.g., video streaming, job matching, and online ads (Boutilier et al., 2023). In such applications, viewers and providers may co-evolve and mutually influence each other: providers increase their content production if they receive more attention from viewers (i.e.. exposure), and the platform gains more viewers if viewers receive high- quality and favored content (i.e., satisfaction). These effects are mediated by the platform's recommendation algorithm.
'Comell University University of Virginia. Correspon- Sarah Dean
In response to this gap, we study the dynamics of "popu- lation effects" on two-sided platforms. Specifically, we consider viewer and provider participation dynamics which operate as follows: (1) the population of providers increases as their exposure increases, (2) the population of viewers increases as their satisfaction increases, and (3) the potential
dence to: Haruka Kiyohara <sdean cod.
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Policy Design for Two-sided Platforms with Participation Dynamics
total welfare
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myopic long-term
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Figure 1. Comparing the myopic-greedy policy, the uniform random policy, and the long-term policy in synthetic simulation. As shown, the myopic-greedy policy loses the provider population due to concentrated exposure allocation, resulting in the negative impact on the viewer welfare in the long-run. The "long-term" policy is based on the algorithm proposed in Section 5 (Eq. (10)), and the experiment setting follows Section 6 (with a small initial population).
for viewer satisfaction increases as provider populations increase. We assume that these effects follow an arbitrary monotonically increasing function, and the immediate util- ity (in the form of exposure or satisfaction) is observed. The key consequence of setting is that the default approach to recommendation can perform much worse in the long term than even a uniform random policy. Figure illustrates this shortcoming of the default approach, a myopic-greedy policy that recommends providers to viewers on the basis of immediate utility.
iey and population regrets by projecting the long term pop- ulation that will result from the current viewer satisfaction and provider exposure. The proposed "look-ahead" policy optimizes the utility at the projected long term popula- tion instead of the immediate population. The synthetic and real-data experiments using the KuaiRec dataset (Gao et al., 2022) demonstrate that the proposed algorithm works better than both myopic-greedy and uniform random policies in multiple configurations by better trading off the long and short term goals accounting for the population growth.
We examine success and failure cases of the myopic-greedy policy through control- and game-theoretic analyses. Our primary results are the following three points. First, by analyzing the convergence conditions of the dynamics, we argue that concentrated exposure allocation among provider groups can easily cause polarization of viewer and provider populations, potentially resulting in smaller pie (i.e., pop- ulations) and long-term social welfare compared to an axposure-distributing policy. These findings highlight the importance of provider-side awareness such as exposure fair- ness (Singh & Joachims, 2018) for the long-term success of two-sided platforms under population dynamics. Second, we analyze the linear case show that the myopic-gree greedy policy is guaranteed to be optimal only if the population effects (i.e., utility gain by the population growth) are ho- mogeneous across provider groups. Third, we explain the shortcomings of the myopic-greedy policy by decomposing the welfare sub-optimality into two terms: the "policy regret" and "population regret". The former comes from the differ- ence between the policy and the myopic-optimal policy at each timestep given the current population, while the latter comes from the difference between the current population and the population under the optimal policy. By definition, the myopic-greddy policy minimizes only the policy regret (i.e., short-term objective). Because the myopic-polics ig- nores the population regret (i.e., long-term impact on the dynamics), the myopic-greedy policy fails when the scale of the long-term utility gain from the population growth is large.
Our contributions are summarized as follows:
We formulate the "population effects" in two-sided plat- forms where viewer and provider populations evolve. We find that the myopic-greedy policy can fall short when the population effects are heterogeneous.
We also find that an exposure-guarane guaranteeaggggggggggea policy can be useful for growing populations and minimizing the population regret.
We propose a simple algorithm that considers the long term population and demonstrate its effectiveness in the synthetic and real-data experiments.
2. Viewer-provider two-sided systems
This section models the dynamics of viewer and provider populations on a recommendation platform. Specifically, we consider sub-group dynamics where viewers and providers are categorized into K and subgroups'. Then, we model the populations, recommendation policy, payoffs, and social welfare follows.
1. (Viewer/provider population) Let A E Rzo be the pop- ulation of the viewer group E [K] and x be that of the provider group E [L]. We also let À :=
(A1, A1, AAK, λ1, X2, ... ..,..,,, .Ax) be the joint popula- tion vector of viewers and providers.
We can consider a "subgroup" size 1. In such cases, the viewer "population" "populato"" corresponds to the time spent by an individual viewer, while the provider "population" can be the amount of content produced by individual provider.
Finally, we propose simple algorithm that balances the pol-
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Policy Design for Two-sided Platforms with Participation Dynamics
2. (Platform's recommendation policy) The platform matches each viewer group k to provider group 1 with recommendation policy denoted by K ---y-L matrix æ. Specifically. its (k, 1)-th element TH, represents the prob- ability of allocating the provider group to the viewer group k. Thus Σi= AA.I = 1, vk E [K]. For example, the uniform random policy, which assigns equal exposure probability across all provider groups represented as given by 1 -LLK--
where i is the base utility, which indicates the matching between the preference of viewer and provider groups (e.g.. this viewer group likes sports articles). In contrast, ft,(()) represents the quality of the provider which improves as the provider population increases. fk. might be heterogeneous among different viewer and provider groups because quality might be multi-dimensional ge.g. visuals, intellects, nov- clty), viewers may have different preferences, and providers may have different abilities. We take fk. to be monotoni- cally increasing function.
3. (Viewer/provider payoffs) After viewer and provider groups are matched by the policy æ,, their perceived pay- off's can be quantified by the following metries:
Next, consider the impact of viewer and provider payoff's on the population. The number of viewers that a platform can maintain is related to the level of satisfaction; similarly, the number of providers is related to the exposure. We assume that viewer and provider subgroups have some "reference" population d (5k) and (e()) given the level of viewer sat- isfaction 8 and provider exposure eq. We assume that À is a monotonically increasing function, so higher viewer satisfaction and provider exposure result in increased pop- ulations. Based on this, we model the viewer and provider population dynamics as follows:
Viewer Satisfaction:
5k Ei= T
(1)
Provider Exposure:
el = E AR
(2)
where q is the (expected) utility that viewers l receive from the provider group 1. Eqs. (1) and (2) define viewer satisfaction as determined by the total utility they receive from recommendations, while providers care about the total amount of exposure they receive by recommenda- tion. This model is prevalent is prior works including (Singh& & Joachims, 2018; Mladenov et al., 2020).
Viewer: A4+1,A (1 ms)dee + (eLL)))
(4)
4. (Social welfare) Finally, we consider the following total viewer welfare as the global metric of the platform:
(5) where 17 E [0, 1] are the reactiveness hyperparams, deter- mining how fast the population changes. Note that similar models are widely adopted in performative predictions (Per- domo et al., 2020; Brown al., 2022). We thus have that the viewer satisfaction Sy depends on the provider population via "population effects" fa.,, while the provider exposure di- rectly depends on the viewer population. The two-sided plat- form has complex dynamics between viewers and providers. Our goal will be to consider long-term objectives under such co-evolving and two-sided dynamics.
Content: A = (1- m) Me, + (e(())))
R(T; = Σ k= A
Note that here we consider the sum of viewer-side satis- faction the social welfare, a formulation prevalent in related works (Mladenov et al., 2020; Huttenlocher et al., 2023). The sum of content-side exposure simplifies to the total size of the viewer population.
2.1. Interaction dynamics and "population effects"
We have so far seen a typical formulation in two-sided plat- forms. However, a key limitation of such formulation is to ignore potential non-stationarity in the viewer and provider populations, which is common in many real-world two- sided systems (Boutilier et al., 2023; Deffayet et al., 2024).
2.2. Game-theoretic interpretation
Next, we provide a further justification of and insight into the dynamics model by introducing a game-theoretic formu- lation that is equivalent to Eqs. (4) and (5).
First, consider the impact of provider population growth on the utility experience by viewers, which we refer to "population effects". ", An increase in provider population nat- urally leads to more high-quality content. For examplc, consider a two-stage recommendation policy, where our higher-level policy π decides the matching between viewer and provider groups, and a second-stage policy selects indi- vidual providers among the selected group. Any reasonable second stage policy should be able to select better provider from larger provider pool (Su et al., 2023; Evnine et al., 2024). To model such "population effects", we introduce the following utility decomposition:
Consider (K (K+ T.)--ayerrr game involving K viewer groups and L provider groups. Each viewer group selects a pure strategy A E R2 and each provider group chooses a pure strategy xy E Rzo. The utility functions for the viewer and provider groups, denoted by {ux K and (viH/ are defined follows:
X 2 (6)
un (A) >A -
Σ The (ba. + Smi(di))
X?
by (A) = N A I
2
We denote this game as G(T,, B, f, A), where B is a K- by-T. matrix whose (k; 1) eleemennt is b Observation 1
Tb.1 = bu.1 + (2x)
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Policy Design for Two-sided Platforms with Participation Dynamics
establishes a connection between the game instance and the dynamical formulation presented in Section 2.1.
some fixed point corresponding to an NE of G. This result is surprising in two aspects. First, it stands in stark contrast to prior work one-sided markets, where NEs can fail to exist (Hron et al., 2022; Jagadeesan et al., 2022; Yao et al., 2023a). In comparison, Theorem demonstrates that when the previously passive resource (e.g., viewer atten- tion) becomes an active participant-aas in our two-sided market model-the resulting market dynamics always ad- mit a stable equilibrium. Second, it is well-documented that the existence of NEs does not guarantee convergence to them under gradient-based dynamics, as such dynam- ics often get stuck in local equilibria (Yao et al., 2024b) or even converge to non-Nash stationary points (Mazum- dar et al., 1901). In contrast, we establish that gradient- based dynamics
Observation 1. all players in 9 apply gradient ascent to optimize their utility functions with learning rates [n=Jk= and {m Ma the resulting joint strategy evolving dynamics are exactly given by Eqs. (4) and (5).
Observation provides first-prinippes perspective for un- derstanding the dynamical formulation in Eqs. (4) and (5).2 That is, we can interpret X(-) as the marginal gain from increasing the size of a viewer or provider group by one unit. Consequently, the first terms A (-) and X xx(-) repre- sent the collective payoft' for viewer and provider groups of sizes A and A₁. The quadratic terms- - X and - Af cap- 2.
in our setting, the two-sided dynamics- converge an NE under mild con- ditions. This highlights nice structural property of the two-sided market.
ture the congestion costs associated with maintaining larger populations (e.g., if provider group becomes too large, providers within the group may face intensified competi- tion and thus reduce their productivity due to diminished marginal gains). This suggests that Eqs. (4) and (5) are quite reasonable formulation capture real-world interactions.
Moreover, our result accommodates a wide range of mod- cling choices, including reference functions X, population effects f. and recommendation policies æ, without requiring restrictive assumptions. The following sufficient condition indicates an interesting relationship between the policy de- sign and the stability of fixed points.
3. Stability and sub-optimality
This section provides theoretical analyses* of the stability and sub-optimality under the two-sided dynamics.
Proposition 1 (Sufficient condition for stability). Sup- pose that the first-order derivative of dynamics functions are bounded as (Ve,dr) (V S C, Vl E [L] and (Ve S C2. Vk E [K] at some fixed point deg- Also, suppose T7s S 17, Wk E [K]. Then, deg is stable when
3.1. Stability
An important question to ask about the two-sided dynamics is on stability: Under what conditions do the dynamics con- verge to a fixed point? The following Theorem 4 provides an affirmative answer, demonstrating that the two-sided dy- namies always converge to a stable fixed point, which is also a Nash Equilibrium (NE) (Nash Jr, 1950) of the corre- sponding game instance.'
EK TIA
4n C,C2
(8)
Consider an exposure-fair policy, which distributes the ex- posure equally among provider subgroups. When f and À are monotonically increasing concave functions, Proposi- tion 1 suggests that such an exposure-fai policy guarantees a balanced equilibrium, where both viewer and provider sub- groups maintain a moderate population and payoffs without polarization. This due to the following reasons. First, the upper bound (i.e., RHS of the inequality) becomes more re- strictive when the first order derivative of the dynamics (i.e., Ci and C2) is large, which is true when viewer satisfaction (sx), provider exposure (ei), and provider population (Ar) are small. Then, an exposure-fair policy can exclude such equilibria due to the violation of Ineq. (8). In contrast, using an exposure-concentruute
Theorem 1. For any continuous functions S with bounded first-order derivatives, consider the environment B,J,X) X). We have:
defined by the game instance J,
1. The NE of G always exists, but is not necessarily unique; 2. The two-sided dynamics (Eqs. (4) and (5)) always con- verge to one of G's 's NE, provided that Tie- qi are smaller than a constant that depends on the game parameters.
Theorem establishes a general stability result for the two- sided dynamics, showing that as long as the reactiveness hyperparams are sufficiently small, it always converges to
policy, which does not distribute exposure to some provider subgroups, can lead to polar- ized equilibrium with winners and losers, such equilibria are not excluded by Ineq. (8).
'The game 9 resembles the Cournot Duopoly competition (Cournot, 1838). When K = L = 1 and A(1) = a- - by and ((A)== = a hλ for some positive constants a and b, the game G corresponds exactly to the Cournot Duopoly model. The key distinction in ours is that Jı and À are generic increasing functions.
Consequently, the reduced subgroup population may nega- tively impact the long-term viewer satisfaction, as we have seen in Figure 1. We formally discuss such impacts through the regret analysis in the next subsection.
'll proofs are provided in Appendix B.
DDeinitionss 1-3 in the Appendix formally define the concepts of fixed point, stability, and Nash equilibrium.
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Policy Design for Two-sided Platforms with Participation Dynamics
3.2. Sub-optimality
the family of e-greedy policies as follows:
Our next question is: How does the "population effect" affect the policy design when the dynamics converge? To answer the question, we introduce the following notion of sub-optimality, called regret, to measure the performance difference between the optimal (static) policy" π and a given (possibly time-varying) policy Te:
(c) Tki = (1- 1) arg max + L/L,
where I(-} is the indicator function and e E [o, [] is the degree of exploration (i.e., random choice). Notably, T< corresponds to the myopic-greedy policy, while T is the uniform random. The subsequent results establish that π (()) optimal in the homogeneous linear setting.
1 Regret( RT T (R(T'
R(T)A)))
Theorem 3 (Optimality of the myopic-greedys Let X be the population at the NE under policy π. For any base utility B and linear increasing and homogeneous functions À and f, the social welfare R(T((); X(2) under the e-greedy policy is decreasing in E [0,, 1]. In particular, we have
where X; is the population at timestep t under the policy "'' and X₂ is that of π. T is the total horizon of the timesteps. Assuming that the policy n. converges to within d of a static policy n, the above regret can be decomposed into two factors as shown in the following Proposition 2.
When K 1, R(T((); x12 ) is strictly decreasing c. 2. When K >>> 1, we can identify functions g, such that
Theorem 2 (Regret decomposition). The (total) regret is decomposed into two main factors:
g(t) S (1) S g(c))((),
(9)
1 Regret(m (T) T
1 T
AR(A;
xT)
AR(T, mp)
and both g. h are decreasing in e. In addition, when (V (e) Ax) is sufficiently small, the function h(()) + 1 and Eq. (9) is tight.
(1)
+ (6/T) + const.
We call (1) as "population regret" and (2) as "policy regret". Each component is defined as follows.
Theorem suggests that when the population effect f is linear and homogeneous across different provider groups, the myopic-greedy policy will be always optimal. This also holds in the case when there is no population effect, i.e. fa. (A.) always equals to constant, Aλ, E R. V(k, () [K] x [T.]- In such cases, the use of the myopic-greedy policy is recommended.
A AR(A, Ac) = R(T,"" 1.* A R( At)
AR(T) x.) = R(TT); R(TT; Xe),
where t. is the one-step myopic-greedy policy timestep given population Xx, and n.." is that of under Xi-
However, when the population effect becomes heteroge- neous across different provider groups, the myopic policy ceases to be optimal, as illustrated by Proposition
The population regret refers to the sub-optimality caused by the difference of population (A, and A;) at timestep f, while the policy regret refers to the sub-optimality caused by the difference of the policy (TT, and ) This suggests that the myopic policy makes the policy regret small, but completely ignores the population regret. This presents the reason why we have observed that the uniform random policy outperformed the myopic policy in the toy example presented in Figure 1 in the Introduction.
Proposition 2. The myopic-greedy policy can be sub- optimal when {fKa] are heterogeneous across provider groups, even when X and f remain linear.
We provide a detailed example in Appendix B.5 to sup- port Proposition 2. Intuitively, the heterogeneity matters because it results in cross-over behaviors (e.g., provider group A starts low utility but becomes high utility, while provider group B has medium utility regardless of the pop- ulation), which matter for policy design. Aside from the linear case, saturation behaviors (e.g., no further increases in utility once the population becomes adequately large) also matter. When we encounter such heterogeneous or concave population effects among multiple provider groups, the ayoopicggreddp policy may not be optimal, as it ignores the impact of policy to future population changes. These results demonstrate that the myopic-grecd anteed optimal only under highly restrictive conditions and emphasize the need for practical solutions to account for the long-term effects.
4. When is "myopic-greedy"" optimal?
We have seen that the myopic-greedy policy is not always optimal. Then, the next question will be as follows: When does the myopic-greedy policy succeed? This section an- swers the question with a game-theoretic analysis in the case that f and Ā are linear functions.
Our main finding is that a ayyopccgreedy policy is nearly optimal when the provider population effects f are homoge- neous across provider groups. To formalize this, define
policy is guar-
" : arg max
Σr, R(TTAA)
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Policy Design for Two-sided Platforms with Participation Dynamics
5. Optimizing the long-term social welfare
[K] [L]. Then, update the dataset as Dr = Di-1UD, where Dn Φ (empty set). Ta is burn-in period. 2. For t > To. update the dynamics models (# (Ar), ((s)) X(er)) via supervised learning. Note that the true "reference population" is obtained from data as Xe- = " (Ar+1 - Xv)) + Ae from Eqs. (4) and (5). Then, optimize the policy with Eqs. (10) and (11) using the estimated functions À and q. collect the interaction data D,, and add them to the dataset D..
The key observation from the previous sections that myopic-greedy policy fails by ignoring the population re-
gret, which comes from the difference between the pop- ulation of the current policy and that of the optimal one. Therefore, we first establish policy learning method that optimizes for the population regret and later consider bal- ancing this with the policy regret. However, one difficulty in
minimizing the population regret A AR(A;, A (Theorem 2) that the population X, depends on the past choices of policy л. This means that when optimizing the policy, we should take into account its future influence the pop- ulation. Because we know that the population gradually changes towards the reference population X(-), we consider the following Look-ahead policy:
6. Synthetic Experiment
We first study the dynamies and the performance of the proposed method in synthetic experiment. In this task, we use K = L = 20 subgroups. To define the base utility, we first sample 20-dimensional binary feature vectors (ug, et) from a Bernoulli distribution for cach viewer and provider group and let their inner products be the base utility by.1 = "g ar (kk E [K] x [L]. Then we simulate the following concave dynamies:
7 == arg FII R
(T()))
(10)
Above, (TT) is the reference population at timestep t given the viewer satisfaction and provider exposure realized by the policy т at population Ae i..., du (8t,k) and (e(.)). # (T) is the myopic-greedi policy at the reference population (T). Thus, the lookahead policy focuses on reaching reference populations which enable high user satisfaction.
Xe(2) = X (maaz)
(a(2/TH - 0.5),
(12)
where (2) := 1/(1 + exp(-2)) is the sigmoid function, and X(-) follows the upper half of the sigmoid function.
The look-ahead policy's ' optimization problem is potentially nonconvex. To make it differential, one can consider the following softmax policy as the approximation of T.:
Next, to simulate a heterogeneous population effect, we further take inner products between viewer embeddings up and the vector of population-dependeee quality as follows.
exp(r- (bx,1 + (x(e()(()s)) Et Evecpy exp(7- - (bx, + Su, (Av(ee))
fa,(Ar) = uu (A)) (( ... A ((a) (Ai). (13)
where its i-th quality element follows the upper half of the sigmoid. i.c.. (
(z) = F(33, ( (o(z/T) (((=/)())()))
())((F)
where y > 0 the inverse temperature parameter. Then, we can optimize the objective function in Eq. (10) via gradient ascent, where present the exact gradient in Appendix A.
0.5). We use d = 20. With this model, each provider group has different improvements in quality of content, e.g., visuals, humor, and technical depth, and each viewer group has dif- ferent preferences on these aspects of quality. We visualize the population effect in Figure 6 in the Appendix.
Once we obtain the book-ahead policy, we can interpolate be- tween the look-ahead policy and the myopic-greedy policy to balance the population and policy regrets as follows:
We initialized the subgroups populations by sampling values from the normal distribution, so we have the majority and minority subgroups at = 0. Specifically, we use two initializations: (1) a small population (A ~ N22, 102)) and (2) large one (A ~ (1000, 30²)) to see how policies perform in both increasing and decreasing dynamics.
n = 3 (d) + (1 101-310 ( (m)
(11)
where B E [0, 1] is the interpolation hyperparameter and (m) is the myopic-greedy policy. ,3 can be determined by the platform's 's desire focus on short vs. long term goals,
while we later show that / 1.0 can be reasonable choice from the experiment results.
Compared methods. We compare the proposed look- ahead policy with varying interpolation hyperparameter 3 E 00.0, : 1.0). We also compare with the uni- form random policy as a reference. When computing the look-ahead policy, we assume access the dynamics and population effect functions. The lookahead policy is com- puted with gradient ascent on the objective (Eq. (10)) for 100 iterations.
5.1. Estimation the dynamics
In practice, there may be situations in which need to estimate the dynamics function (A and /) using some func- tion approximation. In such case, we can use the following Explore-then-Commin style estimation:
1. For A Ty, deploy some epsilon- policy and col- lect the data of D, := (sk, et, gr,, Ae, At), (k: 1) E
We run the compared methods for 200 timesteps and report the results in Figure 2. The results demonstrate
Results.
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Policy Design for Two-sided Platforms with Participation Dynamics
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Figure 2. Comparing the total welfare, and the viewer and provider populations with varying values of interpolation hyperparam,
Le., 3. (Top) small initial population and (Bottom) large initial population. "uniform" represents the uniform random policy.
initial utility
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5.0
Figure 3. Comparing the utility matrix of the myopic (3 = 0.0), long-term (3 = 1.0), and uniform random policies at the final timestep and the initial utility matrix. For the initial utility matrix, we use the one with small initial population.
that the long-term (look-ahead) policy performs better than the myopic-greedy policy, as the reward gain from the pop- ulation effects is large in this setting. Specifically, we ob- serve that the pure look-ahead policy (8 1.0) increases the provider populations while the myopic-greedy policy de- creases the provider populations. These population changes immediately affect the total welfare, suggesting that guar- anteeing high population among multiple subgroups via balanced exposure allocation crucial when population ef- fects matter. Indeed, we also observe different distribution of the utility matrix at the final timestep across compared methods in Figure 3. Interestingly, while the uniform policy has the largest provider population at the final timestep in Figure 2, the look-ahead policy (33 = 1.0) achieves better total welfare (and population regret). This is because the look ahead policy allocates exposure more efficiently than the uniform policy to ensure both high viewer satisfaction and high provider exposure among multiple subgroups, em- pirically demonstrating the effectiveness of our approach.
7. Real-data Experiment
This section studies the empirical behavior of the proposed method using the KuaiRec (Gao et al., 2022) dataset.
KuaiRec (dense) (Gao et al., 2022) is a viewer-provider interaction dataset consisting of 4,676,570 data samples with 1,411 viewers and 3,326 videos (i.e., providers). The data contains "watch ratio" (i.e., play dura- tion divided by the video duration) as the viewers feedback signal. We clip the maximum watch ratio by 10 and learn the viewer- viewer-provider base utility b(u, c) using a ncural col- laborative filtering (CF) model (He et al., 2017). The base utility is calculated as individual level ((u, E u x C). where and viewer and provider embeddings.
Datasets.
Simulation. We simulate the subgroup of viewers and providers, following the procedure presented in Bose et al. (2023). Specifically, we cluster viewers and providers into = = 20 subgroups respectively, based on the viewer and provider embeddings learned by the neural CF model (He al., 2017). We use the same initialization and dynamics of the population described in Section 6. Then,
7

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Policy Design for Two-sided Platforms with Participation Dynamics
population effect for viewer
population effect for viewer
population effect for viewer
provider
Ubility
utility
provider 2
provider
3-
provider 4
provider 5
0-
2-
50
100
150
200
50
100
150
200
50
100
150
200
provider population
provider population
provider population
Figure 4. Visualization of the (true) population effects in the real-world experiment. The population effects are based on the spline function (Reinsch, 1967) fitted on the empirical population effect (dotted points) observed in the KuaiRec (Gao et al., 2022) dataset. Figures and 8 in the Appendix also report the population effects and dynamics learned by the long-term policy, following Section 5.
total welfare
total viewer population
total provider population
10000
1500
total welfare
3250
total gopulation
8-044
3-2
4000
750-
1000
8-08
1-10 0
2000
500
10
10
12
10:
10
time
time
time steps
cumulaaivve total regret
total regret
population regret
policy regret
30083
0-1
50099
Bime
Figure 5. Comparing the total welfare, viewer and provider populations, and regrets in the real-data experiment. Cumulative regret is the sum of total regret by the timestep t, and the total regret is decomposed into the population and policy regrets. Note that the true optimal policies that minimize the total regret and population regret are not accessible. Thus, we report the empirical regrets by letting one of the compared policies as the optimal baseline.
we simulate the utility and population effects as follows.
tions (Reinsch, 1967) implemented in SciPy (Virtamen et al., 2020). The resulting population effects are in Figure 4.
1. Let us == E|u u E Us) be the mean embeddings of viewer group kt We define the group-wise base utility as bi, = E(b(ug, <) |c E C,] (i.e., mean utility that viewer group receives from providers in group l). 2. Next, to simulate population effect, generate random permutation of providers within each provider group. Given the current provider population At,, we let first d samples in the permutation as the the set of providers in the subgroup used at timestep t. We denote this subset as C..-- Then, we define the utility from the provider group / q(ux, := arg max blu,,c, c). Therefore, the population ef- fects are defined as
Estimation of the dynamics. In this experiment, we esti- mate the dynamics functions À and f using regression. We use the model F(2) ao co(1-ex((-a its concavity and flexibility, and fit the params (ap..... ag) from interaction data as described in Section 5. Note that we add perturbations in the population dynamics & sampled from a normal distribution as & ~ (O.00, 0.01²) x X (i.e., the scale of perturbation is proportional to the population) to account for the difficulty in learning the real-world dy- namics. During the burn-in period (10 steps), we deploy epsilon-greedy with the corresponding value of B 3(=c), e).
(1--a2)))+o due
Results. Figure 5 report the population dynamics, to- tal welfare, and the regret. Unlike synthetic experiment, the myopic-greedt policy performs better than the uniform random policy, while the proposed look-ahead policy is competitive to the myopic-policy in the total welfare after converging to the NE. However, we observe some tradeoff between the myopic-greedy and look-ahear policy. Specif- ically, Figure 5 (Bottom) suggests that the syopiccgreedy
(A,) = arg max (EC,, b(ug, e) bytts
which increases monotonically as provider population (Ae,) increases.
To obtain a smooth population effect function, we gener- ate 10 different random permutation in Step 2. Then, we take the average of 10 population effects and fit spline fune-
8

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Policy Design for Two-sided Platforms with Participation Dynamics
policy (3 = 0.0) has some population regret compared to the look-ahead policy (3- 1.0), while the look-ahead pol- icy retains some policy regret. As a result, an interpolated policy with 3 = 0.6 is the best among the compared meth- ods, while all interpolated policies with various values of perform quite well. It is also worth mentioning that the look-ahead policy (3 = 1.0) maintains high total welfare even while achieving a similar level of the provider popula- tion the uniform policy. This suggests that the proposed look-ahead policy is able to allocate exposure efficiently by considering the long-term population effects.
(i.e., non-convex optimization) can work in complex and interactive dynamics of two-sided platforms.
Strategic content providers. Another related liter- ature is the policy optimization under strategic content providers (Hron et al., 2022; Jagadeesan et al., 2022; Yao et al., 2023b). These works often formulate content providers as "selfish" agents who maximize only their own utility defined by the amount of exposure minus the cost of content generation. As described in Section 2.2, our problem setting can also be seen as variant of policy op- timization under strategic viewers and content providers. However, our formulation is distinctive in modeling the in- crease and decrease of the total population, while existing works assume that the total number of viewers and providers are fixed. This difference results in novel findings: while Hron et al. (2022) find that more exploration (randomness) can reduce incentives for producing high-quality "niche" content, we find that a more random provider-fai fair policy can be beneficial when taking the population growth of multiple groups into account.
Together with the synthetic experiment results, we observe that the proposed look-ahead policy adaptively behaves (near-Joptimally in terms of total welfare, while also guar- anteeing a high provider population through provider-fair exposure allocation. This minimizes the effort to tune the hyperparam 3 as the look-ahead policy (3 = 1.0) works reasonably well in practical situations.
8. Related work
Fairness, diversity, and welfare. Fairness and diversity among providers have been considered as necessary met- ries or constraints when optimizing policies in two-sided platforms (Singh & Joachims, 2018; Wang & Joachims, 2021; Boutilier et al., 2023). While frovider-fairness is initially considered important from provider-side perspec- tives (Singh & Joachims, 2018), recent works consider the impacts of provider fairness on viewer welfare. Specifically, provider fairness turned out important to maintain provider diversity (Yao et al., 2023b; Hron et al., 2022), and provider diversity helps maintain viewer welfare in the long-run (Su et al., 2023; Mladenov et al., 2020). Our findings align with these works in pointing out that provider- is impor- tant for long-term viewer satisfaction, but from a different viewpoint based on the growth of populations.
Population shifts. The most relevant existing works to ours are Mladenov et al. (2020) and Huttenlocher al. (2023), which consider the population dynamies by mod- eling the departure of viewers and providers. Specifically, Mladenov et al. (2020) assume that a provider will leave the platform if the provider cannot receive adequate expo- sure (i.e., exposure is below some given threshold). Then, Mladenov et al. (2020) solve the constrained optimization problem linear integer program and demonstrate that provider fairness is crucial to maintain high viewer wel- fare. To extend, Huttenlocher et al. (2023) additionally consider the departure of viewers who receive less utility than given thresholds. Huttenlocher et al. (2023) also for- mulate a matching problem to determine which viewers and providers to keep in the platform to achieve high long-term social welfare. However, both works ignore the possible growth of the platform, and how policy design affects the "growing-the-ppie behavior has remained underexplored (also, these existing works are not directly applicable to our problem setting). Our work complements these works by finding that provider faimess is important to ensure high "population effects" in a generalized setting.
9. Conclusion
This paper studies recommender policy design in two-sided platforms where viewer and provider populations matter. Through the control- and game-theoretic analyses, we found that the myopic-greed policy is guaranteed to be optimal only when the population effects are línear and homoge- neous among provider groups, and otherwise may fall short by ignoring the "population effects".. To take such long- term effects into account, we proposed a simple algorithm and empirically demonstrate that it provides both viewer satisfaction and provider exposure for future population growth. We believe our work provides a comnrrtone to build dynamics-aware allocation policies in two-sided plat- forms where multiple stakeholders engage.
Note that our work is also closely related the performa- tive prediction literature (Perdomo et al., 2020; Brown et al., 2022; Narang et al., 2023). This line of work studies the policy optimization under a stationary point of dynamics, when the algorithm affects the states (e.g., population) of the environment, and the optimal algorithm can change depend- ing on the states. While these works (theoretically) discuss the model optimization and its convergence under convex loss function (Brown et al., 2020), our work empirically demonstrates that the look-ahead policy gradient approach
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Policy Design for Two-sided Platforms with Participation Dynamics
Acknowledgements
models are few-shot learners. Advances in Neural Infor- mation Processing Systems, 33:1877-1901, 2020.
This work was partly funded by NSF CCF 2312774, NSF
OAC-2311521, and gift to the Linkedln-Cone Bowers CIS Strategic Partnership. Haruka Kiyohara is supported by the Funai Overseas Scholarship.
Bunch, R., Nielsen, C. P., and Sorensen, D. C. Rank-one modification of the symmetric eigenproblem. Numerische Mathematik, 31(1):31 48, 1978.
We thank Thorsten Joachims and Richa Rastogi for their thoughtful feedback on the early-stage discussion about the dynamics in two-sided platforms. We also thank anonymous reviewers for the valuable discussion during the review.
Cournot, A. A. Recherches sur les principes mathématiques de la théorie des richesses, volume 48. L. Hachette, 1838.
Dean, S., Curmei, M., Ratliff, L. J., Morgenstern, J., and Fazel, M. Emergent segmentation from participation dynamics and multi-learner retraining. arXiv preprint arXiv:2206.0266 02666, 2022.
Impact Statement
The real-world experiments are run public recommen- dation dataset, and we do not consider there to be any ethical concerns in our experiments.
Deffayet, R., Thonet, T., Hwang, D., Lehoux, V., Renders, J.-M., and de Rijke, M. Sardine: Simulator for auto- mated recommendation in dynamic and interactive envi- ronments. ACM Transactions on Recommender Systems, 2(3):1-34, 2024.
For the broader social impact, this paper has shed light on the population effects in two-sided platforms for the first time. Our findings suggest that ensuring the provider expo- sure among multiple subgroups, instead of concentrating single subgroup, is important to maintain a high population growth and utility. This is also relevant to provider-side fair- ness (Singh & Joachims, 2018) and diversity (Su et al., 2023) in many decision-making scenarios (Extended related work in Appendix 8 provides a further discussion). Moreover, the population effects potentially matter in broad applications including recommender systems, social networking services, and even school admission or hiring processes. Working such application-specifi configurations would also be an impactful future direction of the research community.
Evnine, A., Ioannidis, S., Kalimeris, D., Kalyanaraman, S., Li, W., Nir, I., Sun, W., and Weinsberg, U. Achieving a better tradeoff in multi-stage recommender systems through personalization. In Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 4939-4950, 2024.
Fan, K. On theorem of Weyl concerning eigenvalues of linear transformations Proceedings of the National Academy of Sciences of the United States of America, 35 (11):652, 1949.
Gao, C.. Li. S., Lei, W., Chen, J., Li, B., Jiang, P., He, X., Mao, J., and Chua, T.-S. Kuairec: A fully-observed dataset and insights for evaluating recommender systems. In Proceedings of the 31st ACM International Conference on Information & Knowledge Management, pp. 540-550, 2022.
References
Bose, A., Curmei, M., Jiang, D. L., Morgenstern, J., Dean, S., Ratliff, L. J., and Fazel, M. Initializing services in interactive ml systems for diverse users. arXiv preprint arXiv:2312.11846, 2023.
Hashimoto, T., Srivastava, M., Namkoong, H., and Liang, P. Fairness without demographics in repeated loss minimiza- tion. In Proceedings of the 35th International Conference on Machine Learning, pp. 1929-1993. PMLR, 2018.
Boutilier, C., Mladenov, M., and Tennenholtz, G. Modeling recommender ecosystems: Research challenges at the intersection of mechanism design, reinforcement learnngg and generative models. arXiv preprint arXiv:2309.06375, 2023.
He, X., Liao, L., Zhang, H., Nie. L., Hu, X., and Chua, T.-S. Neural collaborative filtering. In Proceedings of the 26th International Conference on World Wide Web, pp- 173-182, 2017.
Brown, G., Hod, S., and Kalemaj, I. Performative prediction in stateful world. In International Conference Artifi- cial Intelligence and Statistics, pp. 6045-6061, 2022.
Hron, J., Krauth, K., Jordan, M., Kilbertus, N., and Dean, Modeling content creator incentives on algorithm-curated platforms. In The Eleventh International Conference on Learning Representations, 2022.
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Dhariwal, P., Neelakantan, A., Shyam, P., Sastry, G.,
Askell, A., Agarwal, S., Herbert-Voss, A., Krueger, G., Henighan, T., Child, R., Ramesh, A., Ziegler, D. M., Wu,
J., Winter, C., Hesse, C., Chen, M., Sigler, E., Litwin, M., Gray, S., Chess, B., Clark, J., Berner, C., McCandlish, S., Radford, A., Sutskever, I., and Amodei, D. Language
Huttenlocher, D., Li, H., Lyu, L.., Ozdaglar, A., and Siderius, J. Matching of users and creators in two-sided markets with departures. arXiv preprint arXiv:2401 1.300100333.. 2023.
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Jagadeesan, M., Garg, N., and Steinhardt, J. Supply- side equilibria in recommender systems. arXiv preprint arXiv:2206. 2et.:2226..14999 13489, 2022.
Su, Y., Wang, X., Le, E. Y., Liu, L.,, Li, Y., Lu, H., Lipshitz,
B., Badam, S., Heldt, L.., Bi, S., et al. Value of exploration: Measurements, findings and algorithms. arXiv preprint arXi::2305.0776. 2023.
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Virtanen, P., Gommers, R., Oliphant, T. E., Haberland, M.. Reddy, T., Cournapeau, D., Burovski, E., Peterson, P., Weckesser, W., Bright, J., al. Scipy 1.0: fundamental algorithms for scientific computing in python. Nature methods, 17(3):261-27) 2020.
Mazumdar, E., Sastry, S. S., and Jordan, M. I. On finding local nash equilibria (and only local nash equilibria) in zero-sum games. ACM/MMS Journal of Data Science, 1901.
Wang, L. and Joachims, T. User fairness, item fairness, and diversity for rankings in two-sided markets. In Pro- ceedings of the ACM SIGIR International Conference on Theory of Information Retrieval, pp. 23-41, 2021.
Mladenov, M., Creager, E., Ben-Porat, O., Swersky, K., Zemel, R., and Boutilier, C. Optimizing long-term so- cial welfare in recommender systems: A constrained matching approach. In Proceedings of the 37th Interna- tional Conference on Machine Learning, pp. 1987-6998.
Yao, F., Li, C., Nekipelov, D., Wang, H., and Xu, How bad is top-k recommendation under competing content creators? In International Conference on Machine Learn- ing, pp. 39674-39701. PMLR, 2023a.
PMLR, 2020.
Yao, F., Li, C., Sankararaman, K. A., Liao, Y., Zhu, Y., Wang, Q., Wang, H., and Xu, H. Rethinking incentives in recommender systems: are monotone rewards always beneficial? Advances in Neural Information Processing Systems, 36, 2023b.
Narang, A., Faulkner, E., Drusvvatskiy, D., Fazel, M., and Ratliff, L. J. Multiplayer performative prediction: Learn- ing in decision-dependent games. Journal of Machine Learning Research, 24(202):1-566 2023.
Yao, F., Liao, Y., Liu, J., Nie, S., Wang, Q. Xu, H., and Wang, H. Unveiling user satisfaction and creator produc- tivity trade-offs in recommendation platforms. Advances in Neural Information Processing Systems, 2024a.
Nash Jr, J. F. Equilibrium points in n-person games. Pro- ceedings of the national academy of sciences, 36(1):48- 49, 1950.
Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et Pytorch: An imperative style, high-performance deep learning library. Advances in neural information processing systems, 32, 2019.
Yao, F., Liao, Y., Wu, M., Li, C., Zhu, Y., Yang. J., Liu, J., Wang, Q., Xu, H., and Wang, H. User welfare optimiza- tion in recommender systems with competing content creators. Proceedings of the 30th ACM SIGKDD Con- ference on Knowledge Discovery and Data Mining, pp. 3874-3885, 2024b.
Perdomo, J., Zrncc, T., Mendler-Dünner, C., and Hardt, M. Performative prediction. International Conference on Machine Learning, pp. 7599-7609, 2020.
Prasad, S., Mladenov, M., and Boutilier, C. Content prompt- ing: Modeling content provider dynamics to improve user welfare in recommender ecosystems. arXiv preprinl arXiv:2309.00944) 2023.
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Singh, A. and Joachims, T. Fairness of exposure in rankings. In Proceedings of the 24th ACM SIGKDD international conference on knowledge discovery & data mining. pp. 2219-2228, 2018.

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Policy Design for Two-sided Platforms with Participation Dynamics
A. Derivation of the gradient of Eq. (10)
derive the gradient the look-ahead policy using the chain-rule as follows:
V (R((()))TT::
V
Σ X Σ
a (T) + (xa(((((T)))
K
L
V (TH.))
K
V
(Ax)
1=1
( TT
where the (')tt element of the gradient matrix is defined as follows:
V V = ... (x = V., (Ax) I[K' = k) gerr. (X,
V (M)) V (Z)) Ve, II(' = -}
Note that (fu,), (Ax), and V (Ar) can be the gradient of the estimated dynamics and population effect functions, when the true dynamics are not accessible (i.e., we can use the estimation process described in Section 5.1).
When implementing the algorithm, one can also use autograd implemented in PyTorch (Paszke et al., 2019) to calculate the gradient directly from the look-ahead objective.
B. Omitted Proofs
This section provides proofs for the Theorems and Propositions presented in the main text. Note that define the fixed point, stability, and Nash equilibrium follows.
Definition 1 (Fixed point). Let S(.) be the dynamics function that maps the population from the previous timestep to the next timestep, i.e., Ae+1 = S(x)). Then, Xeg is a fixed point under the policy π when Xeq satisfies,
Xeg
S(Deg)>
where S is static function under static policy π.
Definition 2 (Stability). A fixed point deg is stable under the policy ι when Aeg satisfies,
> O, 38 > 0, This condition is satisfied when det S))) holds. Definition 3 (Nash equilibrium). X is Nash equilibrium of game g(T,, B, B,X) under the policy when X satisfies,
| - degll < = || - deg <E.
x = arg max up(AA. A = x-
A
Aj arg max O Xi1))
A:
where A A_ denotes the vectors that contain all elements of X except > and X and uk Un are determined by Eqs. (6) and (7).
Based on the definition of fixed point, we prove the properties of the policy and the corresponding fixed point below.
B.I. Proof Proposition 1
To prove Proposition I, we first introduce the following Theorem We also use the fixed point described the following in the proof of Theorem 1.
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Policy Design for Two-sided Platforms with Participation Dynamics
Theorem 4 (Conditions for fixed point). Xeg is stable equilibrium when the following is satisfied for all [L].
K Σ ne Vv A <1
(i - aV AV
k=1
where (V-) is the first-order derivative at Aeq.
Proof. We prove the condition for the stable equilibrium. Let 5()) be the dynamics function that maps the population from the previous timestep to the next timestep, i.c., X S(Aa). Then we have
A411 Aeg (Vx5)(A, deg)-
This matrix is decomposed into four sub-matrices
A1,1
A1,2
(VxS)
A2,1
A2.2
where AI.1 is a (K, K) dimmennsioaal matrix, A1.2 is a (K, L)-dimensional matrix, A is (L, -i--dmeniiiii matrix, and 22 is a (L, dimmensiio matrix. From the dynamics equations and 5, the element of each matrix is derived as
(A1,) = (VAp Ax) = (1 (1-m))]f [T = k} {A22hrr = (V,,, = (1 m)][(=11']
(A1,2)R. (Vx,de) aZ
(A2,1)L (V) vV AI)
When the spectrum radius (i.e., the maximum cigenvalues) of (Vx9) is less than 1, Aeg stable equilibrium of the dynamics. Here, because A1 and 22 invertible matrices,
the Schur complement
de det((1,,) det (2, A1.
Therefore, when the eigenvalues of AI,1 μn and that of A' is μ2, the eigenvalues of (VxS) is [11, Thus, the eigenvalues of (VxS) are
(nt =11 (mmm
{u2)1 = mm(1 - (e.
Σ The g An)TaN
When 7 S 17, Ak e [K] hold and the gradient norm is bounded as (V (Ve.Ai)(Vx.i) S C1 and (V S C2 at deg. we
have Proposition
Tkl S 4nj C₁C²²
where we use Vi E [0, 1), m( m)) < 0.25.
B.2. Proof of Theorem
Proof. Existence of NE; First we show the Nash equilibrium must exist. Note that ug, 2/ take negative values as A A become sufficiently large, we can without loss of generality assume each player's strategy upper bounded by a finite constant. As result, for each player in the game, its strategy set is a convex, closed, and bounded region, and its utility
13

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Policy Design for Two-sided Platforms with Participation Dynamics
function is clearly concave in its own strategies. According to Theorem in (Rosen, 1965), such game is concave n-person game and its Nash equilibrium must exist.
Non-uniqueness of NE: Next, we give an example showing that the Nash equilibrium is not necessarily unique, if we do not impose any assumption on λR. At, q. Consider the case when K = L = 1 and the following configurations
() = (4(2 - 1)), \((aa) = a (3(2 - 1)),
where (x) 1/(1 + exp(-x)) is the sigmoid function. In this case, the two players have the following utility functions
u (I(A()) (w) ((()) = (()) (4((2 - 1)) 2 (2(1))
(2(1(,) (1)) (3(2A 1)) 20-2200 (110))
(14)
Any fixed point of system (14) should satisfy the following first-order condition
( ((( 1)), A (3((2)()) - 1)),
(15)
and we can easily verify that Eq. (15) has three solutions
[a { [0.0278, 0.0555), [0.5, 0.5], [0.9722, 0.9445].
(16)
On the other hand, since each player's utility function strictly concave its own strategy, any fixed point of system (14) must correspond Nash equilibrium of the game. Hence, the game has three distinct Nash equilibria, which are given by Eq. (16).
Convergence of two-sided dynamics: According to Theorem 4, we know that two-sided dynamics converge to some stable fixed point Aeg, as long as for each reactiveness hyperparam 17. it holds that
4np
Σ
Tk. C.C.²
(17)
where Cj and C2 are upper bounds of (V (V..A)V(A..)) and (V Ax) at Xeg. A sufficient condition for Eq. (17) to hold
7) S KC105'
Next we argue that Aey must also correspond to the Nash equilibrium of if S KCIC5 This is because Xey being stable point means that for each viewer k and provider under Aeg. they cannot alter their strategies unilaterally to improve their payoffs un or U) in a small region around Aeg. That is, Ax₁ A given by Xeg are the local maximum points of ug and i under Xeq. Since both ua and 1/ are strictly concave quadratic functions in X and A their local maximum points must also be the global maximum points. Hence, they also cannot unilaterally improve their payoffs their entire strategy sets. This demonstrates that Xrg satisfies the definition of Nash equilibrium.
B.3. Proof of Theorem 2
Proof. Here, we provide proof of the regret decomposition. First of all, we have
;( R(TT; Xx)
= R(TT' A;) R( ;A;) R(T? (;)- Ae) + R(T/:A)) R(TrA)))
AR A₂) + A AR(T) T.) + const.
where
AR A₂) == R(m x;) - RiT!; A₂) is the regret arises from the population difference of stationary optimal
policy (Tr" and the policy of interest (TT).
AR(T,,TT) := R(m,; >x) R( (T;;; Ax) the one-step regret of the policy.
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Policy Design for Two-sided Platforms with Participation Dynamics
const.
R( x is the one-step regret of the stationally optimal policy. This term does depend and only depends n
Then, let XE to be the population dynamics of a stationary policy л. From the assumption about the policy convergence,
v6,8 > O, FO. E 2, s.t., AL > To, D(T, mT) < 8' and D Xx) < 6
holds true. Thus, we have + Σt AR(A. X (6)/T). Therefore,
1 Regret( ④
(R((T' x;) ((T(A)))))))
T
Σ
(AR(A) A,) + AR(T), TT) + const.)
T
Σ
(AR(X) X5) + AR(A, A.) + AR(T) TT)) + const.
T
6 T
AR(X, AT) T t=1
AR(T) T +0
+ const.
B.4. Proof of Theorem
Proof. When f, are linear functions, from Theorem we know that the NE of G(T, B,J, A) exists and is unique. For any fixed æ, let doc = =(x A²) denote the NE obtained under æ. By Proposition 1, (X A²) also the unique stable fixed point of system (5).()) and therefore satisfies
L
miK (LL + s((l0))) and 1/00 A Σ Em (1))
(w)+ Av
Σ
AA VIsIsIIkSS (18)
1=1
Next, we derive the closed-form of (Xi, (.). Suppose
f(x) = apx + bu,
(x) = aιι b
{x(x) = a2 + b2, ap, a1 a2 > 0.
From Eq. (18) we know (Ai, > the unique solution to the following linear system
T
Tk
-agl1 π
Ft, (br, bo) bj.
M (b), + bo) by, b2,
b2
(19)
T
l=1
1=1
where Ix, Ir. denote the identity matrices sizes K, L. Since Ete MI, = 1, we have
a1 TIR (br, + bb) + bj = a1 Σ Mli (b,x + bo + bj
viskkk.
Without loss of generality, we let bo, b, hereafter, since we can always absorb the term bo + b. into B by letting
by = bi + bo + l,, As a result, from Eq. (19) we can obtain the closed-form solution for (A >g) as follows:
x
(Ix u2 TT) -1 (Ix a0a -T) π a2
A:
T2 T)
1t. aga (Ix ana -1-1
@1
Tt,
b1,1,
bi b2,
(20)
15

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Policy Design for Two-sided Platforms with Participation Dynamics
where - u2 is positive definite matrix.
On the other hand, the user-side social welfare can be rewritten into
R(TT; (Ni X3))
Σ
(bibb + )(
==1
1=1
MT, Ir,, Σ nl,bbI,KK
+ tao x X
A (V 40arT T (²) + ao x Tx.
by Eq. (19)
x T X
(21)
From Eq. (20) we also have
N = (IK a2a102 - .1 T
1=1
(IK aga ag (x) " agaan .b
aa(lk aga .2 F-- [diag(T= B) + aobglx]
by n 11 1x.
(22)
Plug Eq. (22) into Eq. (21), we obtain the following explicit expression of R for any π and B:
RiT; Axo) R(;;; (Xi, x;) ai ||(I agajagg T)- [diag( ( B) agb2 1 11
(23)
where 1K = (I, 1) is column vector length K. Let mmax (-) and omin (-) denotes the largest and the smallest eigenvalue matrix. Then Eq. (23) implies
a1 diig((T (T B) + agb2 1x 12 < R(T);>o)) I a1 diag( [ BB + aob2lx 12 R4
R_ RT:) &
Aoo) (24)
22 (IK 000112m π)
2 (Ix a0a1a2 π))
Next, we consider any e-greedy policy (<) w.r.t and show that both R_ (T 6 are monotonically decreasing in e [0, 1]. Without loss of generality, we may assume the greedy recommendation policy To has the following form:
( and R2 (<); ( as functions of
1 Ki
0
0
1 K2
0
0
I[bb, = arg max be
bLN
1815
0
1
0
ie., all user groups are clustered into m sub- groups and each has size Km. Each user within a sub-group prefers the same content group and users from different sub- sub-groups prefer different content groups. The total number of user sub-groups m satisfies S m < L, K K1 + Km. and K1 2 2 Km-
Denote by diag( ग. B) ba. and b 14, Ele bLnkk By plugging n. (1 (1-E)TO + f11xK into Eq. (23), obtain
diag(T=_ B) abb2lx (1 e)bo + ebi agb21k.
Since by 2 by elementary-wise we conclude that []dia(mи] B) + anbr 1 x as a function of is decreasing [0, 1).
16

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Policy Design for Two-sided Platforms with Participation Dynamics
On the other hand, direct calculation shows
1K a2 n. =Ix- 200002 [LL ()) + E [ (1--)) (1- + i _
andn a2e 2 1KxK.
=Ik - a00112(1 TT
L
Given the explicit form of the block matrix mg. we can directly compute the smallest and the largest eigenvalues of matrix Ig a0a102(1 T no as followings:
ng
maxx (Ix ag01 (2 rT no 1,
Omin (IK - aga To TO) = - aga 02 (1 - - K
²)²
In addition, from Weyl's inequality (Fan, 1949; Bunch al., 1978), we conclude that mmax (Jx 1000 A2 n.)
a0a10ge(2- - c)
minn (IK a2TT ne)) 2 minu (IK aga102 -)² тo) max
L 1 xx
1 agaiaa(1 e)) agalaze(2 - e)
L
000102K1 + aga1agc(2 c) Ki (Ki-1) K )
L
(25)
Note that by the definition of K1, it holds that K₁ 2 K 2 function in 6 [0, 1].
K L. Hence, the lower bound Jiin in Eq. (25) an increasing
Hence, have
R_ (T (6); x(2) a1 [1(1 e)bu + cb1 agby
(26)
(1 c)bu + cb1 mgb21K 11
R4 (T (9; (0) ag - a0a <) (Ki - K)) 4))
(27)
and the RHS of both Eqs. (26) and (27) are decreasing functions of e in [0. 1].
Take g(e) = a1 [1 - bu + ebi + anb2l l12, and h(e) = (1 agaja,K1 agarazel2 - (Ki x))= -2 that
we conclude
g(t) < R(T (t); ( s g(t))((e),
and h(c) = (1 a1 (2 <) (K1 k) -2
(1 200010zK)~2
Since by definition
Ax) aga a2, our claim holds.
B.5. Proof of Proposition 2
Proof. Consider two-sided system with K = 1, T, 2, B = [1.0.9] and
x = agæ, (z( = ai {5 a2 f((x) = b1æ, f2(x) = b2æ.
According to Theorem 1, the NE of the system exists and is unique when ap 41, 02 > O are sufficiently small. Moreover, (X(a), (u) (e) x(()) at the NE satisfies
(()) X(m1 (21 + T2) + mn + 0.9m2))
(() (()
( (T((( (1)
17

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Policy Design for Two-sided Platforms with Participation Dynamics
which is equivalent
1 an (1
22 + M11 + 0.9m2))
1 a1 1 (u)
(28)
a2 A (a)
Plugin the last two equations into the first one in Eq. (28), we obtain that
1 colaabiiTii 11 -11 - -0.9721),
and therefore
a0(T11 + 0.9
X (w)
1 aga b1 m agazby
ao (0.9 + 0.1
agabbimi agozby mT1)²²
Now we can write R(T;; X²) as a function of Ti1 as the following:
R(T; X*) x(m) (TT1ii1) + =21 + Mn1 + 0.9T21) 1 (2(11)22
e (0.9 0.1
agalbbbbgo bب - agagb2 (1 - T11) 2|2
Take b = 0, agagby = 0.4, we have
R(T, x-)
0.9 + 0.1
& R(mT),
(29)
ao
1 0.4(1 =1
and it is easy verify that the RHS Eq. (29) not achieved Ti = 1. In fact, for any w11 < 0.7, holds that R(T(1) > R(1) = 1. This means the greedy policy [Mı1, T21l [1, 0] is not optimal in this example.
C. Additional experiment settings and observations
Here, we report additional details of the experiment settings and results.
Difference of population effects in the synthetic and real-world experiments. Figure 6 shows the population effect used in the synthetic experiment, defined by Eq. (13). The biggest difference between the synthetic and real-world experiment setting is that we observe saturation of the population effects as an early stage of the provider population growth, i.e., around A = 100, which is also reasonable phenomenon in real-world situations. Therefore, in the synthetic experiment, is important to distribute the content exposure among multiple subgroups to receive high population effects in many different provider groups. Thus, even the uniform policy outperforms the myopic-policy in this setting. 6 contrast, when using KuaiRec dataset (Gao et al., 2022), the situation is milder than the synthetic experiment, and therefore the myopic policy works well in the real-world experiment. Together, our synthetic and real-world experiments show that the proposed look-ahead policy performs reasonably well in two different configurations. This is because seeking for both (immediate) viewer utility (sR) and provider exposure (et) is important to maximize the look-ahead objective, which depends reference populations {(s)) and A(ez).
'Throughout the experiments, found that there tradeoffis in concentrating and distributing exposures. and which is better often depends on the problem instance. For example, suppose the special case where viewers and the total viewer population is fixed to 100. Then, consider scenario with 100 provider groups. In this situation, distributing exposure among different subgroups can result in expected exposure of for each provider group. In such cases, concentrating the exposure to one provider group be better strategy than distributing allocation for the total population growth. This is why myopic policy performs well in some scenarios, and our argument that the proposed method work adaptively well (i.c., least hetter competitive than both myopic and uniform) regardless if the myopic policy succeeds or falls short.
change their population (174 = 0, Ak [K)
18

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Policy Design for Two-sided Platforms with Participation Dynamics
population effect for viewer 1
population effect for viewer
population effect for viewer
30
content
Utility
content 2
20
content 4
content
5
101
10P
10
10
10P
107
16a
10?
10
viewer population
viewer population
viewer population
Figure 6. Visualization of the population effects the synthetic experiment. We randomly sample the scaler and temperature parameter of the sigmoid function from normal distribution for cach content- quallty feature pair as described in Section 6. The resulting quality vector is provider-dependent, and the population effects are heterogeneous across viever-provider pairs.
population effect for viewer 1
population effect for viewer 2 population effect for viewer 3
provider
provider 2
utility
utility
provider
provider
provider 5
0-
10"
10:
10
10
107
10a
101
10?
provider population
provider population
provider population
est. population effect for viewer
est. population effect for viewer
est. population effect for viewer 3
6-
provider 1 provider
utility
utility
provider 3
provider 4
provider
0
10a
10a
10a
109
10
102
102
10t
10a
provider population
provider population
provider population
Figure 7. Comparing true and estimated population effeet in the real-world experiment. (Top) True population effect used in the real-world experiment (the same figure as Figure 4 in the main text). (Bottom) Population effect learned by the long-term optimal policy at the final timestep.
Estimation results of the dynamics and population effect functions. We also report how the dynamics estimation works in the real-world experiment in Figures and 8. While we initialize the population effect and dynamics estimation with homogeneous function across viewer-provider pairs, the results demonstrate that estimation scheme provides an accurate estimation of heterogeneous functions by using the dynamics logs in the rollout process. We also observe that the policy optimization results in the main text (w/ population effect and dynamics estimation) are quite similar to those without dynamics estimation (i.c., using the true dynamics) in the experiment.
19

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Policy Design for Two-sided Platforms with Participation Dynamics
population dynamics (viewer)
population dynamics (provider)
100
viewer
provider
grovider
refenrence
viewer
refenrence
50
provider
25
viewer
provider
10
100
viewer satisfaction
provider exposure
refenrence population
est. population dynamics (viewer)
refenrence population
est. population dynamics (provider)
100
viewer
provider
grovider
viewer
provider
25
viewer viewer
provider
provider
0
0-
10
80
100
viewer satisfaction
provider exposure
Figure 8. Comparing the true and estimated population dynamics in the real-world experiment. (Top) True population dynamics simulated in the real-world experiment. (Bottom) Population dynamics learned by the long-term optimal policy at the final timestep.
20


Reviews:
[{"review_id": "qG3sn4rLnN", "reviewer": "Reviewer_E67u", "summary": "This paper proposes and analyzes a theoretical model of population effects of the (e.g. recommendation) policies for two-sided platforms, where there are consumers and producers. The basic intuition is that a recommendation system influences both the short-term satisfaction of the content consumer but also the success of the producers, which may influence the size of the producer pool, creating long-term feedback loops. The model consists of finite sets of consumer and producer types, for which consumers derive utility heterogeneously across types of producers. They compare a “myopic” policy that only optimizes for consumer satisfaction with a “look-ahead” policy that also estimates feedback loops.", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank the reviewer for their time and effort on the review. We respond to the key comments and questions below.\n\n---\n\n**(factual discussion)**\n\n---\n\n> **there is a lack of theoretical results in support of their proposed look-ahead policy**\n\nThank you for the valuable feedback. We would like to kindly argue that providing a theoretical guarantee to the look-ahead policy is challenging due to two reasons: (1) interactive dynamics between viewers and providers, and (2) non-linearity and non-convexity of the systems.\n\nFirst, one of the related works, [1] considers the departure of viewers and providers in two-sided platforms. While [1] considers a more restrictive setting than ours in modeling linear dynamics and not modeling the increase in viewer and provider participation, identifying the optimal policy is proven to be an NP-hard problem. Second, guaranteeing sub-optimality in non-linear systems is in general considered a hard problem, and without assuming convexity, we often cannot get a theoretical guarantee [2].\n\nThe most related theoretical analysis to our method is [3]. This paper studies minimizing loss function for predictions when the population gradually reacts to the model in a single-sided platform. This paper provides some theoretical guarantees when replacing the reference point to be the (estimated) fixed point and under the convexity assumption. \n\nNote that many existing papers empirically demonstrate the performance of the algorithm when some assumption (e.g., convexity) is violated. Our paper also provides empirical evidence that leveraging the reference point (instead of the fixed point) performs reasonably well (i.e., at least competitive or better than both myopic and uniform baselines) across multiple configurations. We will clarify these points and provide additional discussion about the theoretical connection to existing work in the revision.\n\n[1] Matching of Users and Creators in Two-Sided Markets with Departures.\n\n[2] Performative Prediction in a Stateful World.\n\n[3] How to Learn when Data Gradually Reacts to Your Model.\n\n---\n\n> **What additional insight do we get from multiple simulation experiments?**\n\nThis is to show that the look-ahead policy performs adaptively well in the case where the myopic succeeds and fails. As we can see in the experiment result, both uniform and myopic can be a good choice in one of the scenarios, however, fail catastrophically in the opposite scenario. In contrast, because the proposed approach can care about both myopic outcome (i.e., viewer satisfaction) and investment for provider population growth (i.e., exposure) by design, ours works adaptively well across two opposite situations. Our real-data experiment also provides an example of how an actual population effects can arise in real data. Reviewer 54YD acknowledges our contribution on this point as “it's good that the authors evaluate their method on both synthetic and real-world data.”\n\n\n**(contributions)**\n\n---\n\n> **that platforms should attend to producer welfare as well as consumer welfare, is reasonably intuitive and the paper fails to explore more implications of their model beyond this basic point**\n\nThank you for sharing your concerns, and we would like to kindly emphasize that, even if the outcome seems reasonably intuitive, highlighting the overlooked problem is crucial for scientific research. \n\nRegarding this point, we share a similar contribution to existing papers: e.g., [4], which points out the importance of guaranteeing provider exposure under the modeling of viewer/provider departure. We generalize the discussion in the setting where the viewer/provider both increases and decreases their population (i.e., a more complex situation), and demonstrate that caring about producer welfare is indeed beneficial. \n\nOur paper provides the theoretical implications and evidence for the platform designers that caring for provider welfare is indeed crucial for the long-term success of the platform, not only for the ethical perspectives. Reviewer 54YD also acknowledges this point “I believe this work contributes to the broad literature on two-sided platform. It touches on important subjects including balancing the interests of strategic agents and balancing the tradeoff between short-run and long-run welfare.”\n\n[4] Optimizing Long-term Social Welfare in Recommender Systems: A Constrained Matching Approach.", "claims_and_evidence": "The claim that the myopic policy can be sub-optimal is a mild one, since all that is necessary is for there to exist some problem instance where it performs poorly. \n\nThere is no theoretical evidence in support of the look-ahead policy. Instead, there are simulations showing that it performs well. \n\nMy main criticisms are two-fold: (1) the main conclusion, that platforms should attend to producer welfare as well as consumer welfare, is reasonably intuitive and the paper fails to explore more implications of their model beyond this basic point. (2) there is a lack of theoretical results in support of their proposed look-ahead policy. Given the intuitive and reasonably simple model, I might have expected this kind of result to be tractable.", "methods_and_evaluation_criteria": "The lack of theoretical results in support of their main conclusion is a weakness.", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "N/A", "relation_to_broader_scientific_literature": "The focus on population effects in platforms is interesting, well-motivated and deserving of more study. Their model is similar to others in the prior literature.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "N/A", "questions_for_authors": "What additional insight do we get from multiple simulation experiments? I would have preferred to just have one and then have expanded theoretical analysis. (Or an actual implementation of the policies in a lab/crowdsourced setting to see how they each do.)", "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": "4VjfyVjKzo", "reviewer": "Reviewer_KNAq", "summary": "The paper models the dynamics of a two-sided platform with viewers and providers. Viewers receive satisfaction from watching content assigned to them from providers that they like, and providers receive exposure from having their content assigned to viewers. The population of both the viewers and providers are dynamic, and a higher satisfaction for one group of viewers leads to more viewers of that group joining the platform and a higher exposure to one group of providers leads to more providers from that group joining the platform. The latter also improves the quality of the content of that platform.\n\nThey argue that myopic policies that only seek to optimize welfare according to the current population can fail to be optimal long-term. They provide an alternative \"look-ahead\" policy to do this that is solvable (approximately) via gradient ascent. They show via experiments that their policy performs as good (or better) than the myopic policy in certain scenarios.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank the reviewer for valuable feedback and the acknowledgment of the contributions. We respond to the key comments and questions below.\n\n---\n\n> **I am a bit wary about some of their claims**, though. For example, they state, after Theorem 3 and Proposition 3, (overstatement)\n\nThank you for the thoughtful feedback. We appreciate your point, and we plan to rephrase the pointed-out sentence in the revision as follows.\n\n- (Current) “.. the myopic-greedy policy **is optimal** only under highly restrictive ..\".\n- (Revision) “.. the myopic-greedy policy **is guaranteed optimal** only under highly restrictive ..”\n\n---\n\n> It would be good to have some explanation as to why the greedy policy worked so well in the real-world experiments. In fact, the greedy policy seems to have the highest (or almost the highest) welfare throughout all the timesteps.\n\nThank you for the great point. Throughout the experiments (including the ones not presented in the paper), we found that there are tradeoffs in concentrating and distributing exposures. Which is better often depends on the problem instance.\n\nFor example, suppose the special case where viewers do not change their population ($\\eta_k = 0, \\forall k \\in [K]$) and the total viewer population is fixed to 100. Then, consider a scenario with 100 provider groups. In this situation, distributing exposure among different subgroups can result in expected exposure of 1 for each provider group. In such cases, concentrating the exposure to one provider group can be a better strategy than distributing allocation for the total population growth. This is why myopic policy performs well in some scenarios, and our argument is that the proposed method can work adaptively well (i.e., at least better or competitive than both myopic and uniform) regardless if the myopic policy succeeds or falls short.\n\nNote that the reason for having a small gap between the myopic policy and the proposal in the real-data experiment is that our method involves an optimization process (e.g., estimating dynamics and optimizing policy) and thus has small modeling errors. We hope this answer could resolve your question.", "claims_and_evidence": "Their primary claim is that their look-ahead policy performs better than the myopic policy, hence justifying their model and results. This is convincing and believable. Their first set of experiments appear promising, but their second experiment shows that the myopic policy works just as well, if not slightly better. I am not sure why they haven't discussed more about why this is the case.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "I only briefly skimmed some of the proofs in the appendix.", "experimental_designs_or_analyses": "The experiments appear to be make sense within their framework.", "relation_to_broader_scientific_literature": "The paper mentions several related works such as Mladenov et al (2020), Huttenlocher et al (2023), Hashimoto et al (2018) that consider dynamics under population change of viewers or providers, but not at the same time.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "The paper is well-written and the model is interesting, as are the results. I am a bit wary about some of their claims, though. For example, they state, after Theorem 3 and Proposition 3, \n\n>\"these results demonstrate that the myopic-greedy policy is optimal only under highly restrictive conditions and emphasize the need for practical solutions accounting for the long-term effect.\". \n\nTheir results in fact, do not show this. I agree that the situation in Theorem 3 is restrictive. But that is only one scenario where it is optimal. The real-world experiments in fact also show a scenario where the myopic policy is near-optimal. Proposition 3 also says nothing about whether greedy is optimal only in such settings. I do however, believe that a stronger statement can be shown that would suggest something more substantial.", "other_comments_or_suggestions": "It would be good to have some explanation as to why the greedy policy worked so well in the real-world experiments. In fact, the greedy policy seems to have the highest (or almost the highest) welfare throughout all the timesteps.", "questions_for_authors": "N/A", "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": "iSNYVVGWSk", "reviewer": "Reviewer_54YD", "summary": "In this paper, the authors formulated and studied the dynamics of population effects on two-sided platforms, where viewer and provider populations evolve based on certain rules. Theoretically, the authors show that the myopic-greedy policy can fail to perform well when the population effects are heterogeneous across providers. They further investigated the shortcomings of the myopic policy by decomposing the overall welfare regret into both the policy regret and the population regret, where the latter captures the long-term welfare loss that is not captured by the myopic policy. In response to this, the authors proposed an algorithm that balances policy and population regrets, and show the effectiveness of this approach via synthetic and real-world numerical experiments.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank the reviewer for valuable feedback and the acknowledgment of the contributions. We respond to the key comments and questions below.\n\n---\n\n> **Could some related works that model the population departure be included as baselines?**\n\nThank you for the great point. Unfortunately, the related works, which model the population departure, cannot be directly applied to our setting. There are two reasons: (1) The existing papers assume that the system dynamics are linear, but we consider the non-linear dynamics. Therefore, their linear programming approach is not solvable in our setting. (2) The existing papers do not model the population increase, and even if the dynamics are linear, it is not trivial how to apply the existing work taking the population increase into account (because we do not have a binary indicator of the population). Our paper proposes the first reasonable baseline for optimizing long-term outcomes under flexible and non-linear dynamics of the population.\n\n---\n\n> That being said, I think the paper should have a **more in-depth discussion on the related works** which appear to be missing from the main body as of now.\n\nThank you for the thoughtful suggestion, and we appreciate your point. We will have an additional page in the camera-ready version, and we plan to move the discussion of related work that is currently in the Appendix to the main text upon publication. \n\n---\n\n> **The authors proposed to use the softmax policy as the approximation of $\\pi_t^1$; would this impact the performance of the policy?**\n\nThis is a great point. By using the softmax policy as the approximation of $\\pi_t^1$, we expect some underestimation of the objective function ($R(\\cdot)$ in Eq. (10)). However, as long as the maximizer of this objective function ($\\pi$) is the same (which is often the case), the performance of the policy should not change by a large amount.\n\n---\n\n> **It appears that one needs to know the viewer satisfaction between each viewer/provider pair in order to solve the optimization problem for the look-ahead policy. What if the platform lacks such knowledge in a realistic setting?**\n\nThis is also a great question. We consider the scenario where we work on sub-group level, and thus the (expected) satisfaction is usually accessible as a subgroup level, and this is also a standard assumption in existing work [1]. However, when this information is unavailable, we need some additional estimation process of viewer satisfaction using past interaction data. We may then need additional consideration of exploration-exploitation tradeoffs, which should be an interesting future direction.\n\n[1] Optimizing Long-term Social Welfare in Recommender Systems: A Constrained Matching Approach.\n\n---\n\n> I also think the authors should offer more discussions **how to best select $\\beta$ in real-world applications and whether any theoretical results can be obtained**.\n\n> **Could you establish any guarantees on short-term outcomes under the fully look-ahead policy?**\n\nThese are indeed important points. While we cannot provide guarantee on the short-term outcomes, it is possible to validate the short-term outcome before deploying the look-ahead policy. Throughout the paper, we consider the situation where the immediate outcome is accessible for both baseline and the proposed methods (following existing work). Thus, once we identify $\\pi_t^{(d)}$ using Eq. (10), we can compare the short-term (i.e., myopic) outcome of the proposed method and myopic policies. If the platform aims to guarantee some certain short-term outcome, the platform can adjust the value of $\\beta$ to satisfy such constraints.", "claims_and_evidence": "Overall, I think the claims made in this paper are quite convincing. I think the key assumptions used in the model quite realistic and could contribute to understanding how to best balance short-term and long-term welfare in two-sided platforms. The authors also justified their results using both solid theoretical and numerical evidence.", "methods_and_evaluation_criteria": "I think it's good that the authors evaluate their method on both synthetic and real-world data. However, I do think they can consider more baselines methods in addition to the myopic and uniform random policy. For example, could some related works that model the population departure be included as baselines?", "theoretical_claims": "The theoretical results appear sound, though I did not check the proofs in the appendix.", "experimental_designs_or_analyses": "- Overall, the experiment results appear sound. I also appreciate that the authors conduct both synthetic and real-world experiments to validate the effectiveness of their approaches.\n- I have some questions related to the practicality of the algorithm, which I detailed below.", "relation_to_broader_scientific_literature": "- I believe this work contributes to the broad literature on two-sided platform. It touches on important subjects including balancing the interests of strategic agents and balancing the tradeoff between short-run and long-run welfare. In particular, the topic of how to best evaluate and achieve long-term fairness has not been studied a lot.\n- That being said, I think the paper should have a more in-depth discussion on the related works which appear to be missing from the main body as of now.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Strength: \n- As I mentioned above, I think this paper studied an interesting problem that formulate the population dynamics in two-sided platforms. The theoretical and numerical results both appear sound and provide interesting insights.\n\nWeakness:\n- It'd be good if the authors can provide more details for the look-ahead policy in Section 5. For example, \n    (1) The authors proposed to use the softmax policy as the approximation of $\\pi_t^1$; would this impact the performance of the policy? \n    (2) It appears that one needs to know the viewer satisfaction between each viewer/provider pair in order to solve the optimization problem for the look-ahead policy. What if the platform lacks such knowledge in a realistic setting?\n- I also think the authors should offer more discussions how to best select $\\beta$ in real-world applications and whether any theoretical results can be obtained. While the authors claimed that setting $\\beta = 1.0$ already appears to work well and thus minimizes the effort to tune this parameter, it's somewhat unrealistic to adopt a fully look-ahead policy in real-world applications, especially when platforms might want to lay more emphasis on short-term outcomes. I wonder if the authors can provide more comments on this aspect. Could you establish any guarantees on short-term outcomes under the fully look-ahead policy? Or, if a platform would like to understand how to select the best $\\beta$, is there a reasonable way to do so? I suspect that simple A/B tests won't be helpful because they only capture short-term impacts.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "See my comments above.", "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": "XQ9tOwLVFa", "reviewer": "Reviewer_g8MP", "summary": "This paper studies the effect of matching policies in two-sided platforms, taking the evolution of both viewer and provider sides into consideration.\nThe authors show that myopic matching policies are only optimal in strong assumptions and appeared to be suboptimal at other cases.\nThe authors propose a new matching policy, called \"look-ahead\", that appears to perform better and more stable than myopic and uniform policies in experiments.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank the reviewer for their time and efforts on the review. We respond to the key comments and questions below.\n\n---\n\n\n**(factual discussion)**\n\n---\n\n> **W3 [polarized equilibrium]**\n\nWe clarify the reasoning behind the paragraph below Proposition 2 step by step. First, consider the case where $f$ and $\\bar{\\lambda}$ are monotonically increasing concave functions. This means that the upper bound of the first order derivative, $C_1$ and $C_2$ are monotonically decreasing to the viewer/provider populations.\n\nThen, let us assess if a polarized equilibrium can be allowed (i.e., satisfies Ineq. (8)) given some specific policies. When the equilibrium population is polarized, this means that some population diminishes, and thus $C_1$ and $C_2$ can be large. Therefore, RHS of Ineq. (8) becomes small (e.g., near-zero). While the exposure-fair policy may exclude this point due to the violation of Ineq. (8), the exposure-concentrated policy can allow this point (Ineq. (8) is satisfied) by letting the LHS of Ineq. (8) to be (near-)zero or smaller than RHS. \n\n---\n\n\n> **Theoretical guarantees of look-ahead policy**.\n\nPlease also refer to the rebuttals for Reviewer E67u.\n\n---\n\n> **Baselines in experiments**.\n\nPlease refer to the rebuttals for Reviewer 54YD.\n\n---\n\n**(contributions)**\n\n---\n\n> **W1 [why our results are not trivial]**.\nWhile our results may appear straightforward, we respectfully disagree that they are *trivial* and would like to clarify why they are novel.\n\n**Theorem 1**, a notable insight of our result is the guaranteed existence of pure NE in a two-sided market setting. This contrasts with findings in relevant prior works, such as [1], [2], and [3], which study one-sided markets involving only strategic content creators and demonstrate that pure NEs may fail to exist or only exist under restrictive conditions. Our result highlights a meaningful and somewhat surprising message: when the passive resource (e.g., user attention) in previous models becomes an active participant (as in our two-sided market), the game structure always ensures the existence of pure NE. We recognize this as an important conceptual contribution and will highlight it more clearly in the revised version.\n\n\n**Proposition 2**, we would like to emphasize two non-trivial messages. First, it is well-established (e.g., [4], [5]) that the existence of pure NE does NOT automatically imply convergence under multi-agent gradient dynamics; in fact, such dynamics may get stuck in local NEs [5] or even non-Nash stationary points [4]. In contrast, our model establishes that gradient dynamics provably converge under certain conditions, which is an important theoretical contribution. Second, the sufficient conditions we identify for convergence—namely, weak population effects and more equitable exposure allocation—are not only intuitive but also offer actionable insights for policy design. We briefly discussed these points in the paragraph following Eq. (8) and will make them more explicit in the revision.\n\n\n**Theorem 3**, we agree that fully characterizing the optimality of myopic-greedy policies is technically challenging. However, our synthetic experiments in Section 6 do well support our theoretical insight in Section 4. In fact, the main purpose of Section 4 is to conceptually illustrate the scenarios in which myopic-greedy policies perform well versus when they fall short. As a result, Section 4 serves as a motivating example that leads naturally into our exploration of optimal policy design in subsequent sections. \n\n\n[1] Modeling content creator incentives on algorithm-curated platforms. \n\n[2] Supplyside equilibria in recommender systems. \n\n[3] How Bad is Top-$ K $ Recommendation under Competing Content Creators? \n\n[4] On finding local nash equilibria (and only local nash equilibria) in zero-sum games.\n\n[5] User welfare optimization in recommender systems with competing content creators.\n\n---\n\n> **Additional clarifications about our contributions**.\n\nWe would like to kindly argue that these contributions are actually acknowledged by reviewers KANq and 54YD as “this paper studied an interesting problem ... The theoretical and numerical results both appear sound and provide interesting insights.” “the model is interesting, as are the results.”\n\nIf the proposed method appears to be straightforward, it means that our method is well-motivated by the theoretical analysis provided in advance in the paper (especially Theorem 2). Moreover, as we have discussed about Theorem 1 and Proposition 2, non-uniqueness and stability of NEs do NOT actually hold in many other similar settings. Providing theoretical evidence for a seemingly reasonable hypothesis, highlighting the overlooked problem, and giving a reasonable explanation is crucial for scientific research. We would appreciate it if the reviewer could acknowledge our contribution regarding this point.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "I find some steps confusing in the proof of Proposition 2.\n\n* In line 645-646, what do you mean by saying $\\nabla _\\lambda S$ and why can you say $\\lambda _{t+1} - \\lambda _{eq} = (\\nabla _\\lambda S) (\\lambda _t - \\lambda _{eq})$? I can only derive $\\lambda _{t+1} - \\lambda _{eq} = S(\\lambda _t ) - S(\\lambda _{eq})$ from the information given up to there. I also note that $S(\\lambda)$ is a vector instead of a real number, so one can not directly use the statement $S(\\lambda _t ) - S(\\lambda _{eq}) = (\\nabla _\\lambda S(\\lambda^*)) (\\lambda _t - \\lambda _{eq})$ for some $\\lambda^*$.\n\n* In line 654-658, I'm confused about the derivation, it seems that $\\{A _{1,1} \\} _{k,k'} = -1 \\mathbb{I} \\{k = k'\\}$, same for $\\{A _{2,2}\\}$. For $\\{A _{1,2}\\} _{k,l}$, it should be $\\{A _{1,2}\\} _{k,l} = \\nabla _{\\lambda _l} \\bar{\\lambda _k}(\\sum _{l'} \\pi _{k, l'} (b _{k,l'} + f _{k,l'}(\\lambda _{l'})))$, and no $\\eta _k$ ($\\eta _l$) term should appear in this term.", "experimental_designs_or_analyses": "Yes. The experiment results support the main findings of this paper (myopic matching policies are suboptimal).", "relation_to_broader_scientific_literature": "The author mentioned that this paper is the first paper that studies matching policies in two-side platforms, taking the evolution of both viewer and provider sides into consideration.\nUnfortunately, I'm unfamiliar with the literature and thus unqualified to evaluate this statement.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "* Strengths:\n\nS1: The presentation is good and easy to follow.  \nS2: The dynamic provided in Equation (4), (5), and the game-theoretic interpretation in Equation (6), are novel and interesting.\n\n* Weaknesses:\n\nW1: Many results stated in this paper are straightforward and not surprising. For examples,\n\n1) Proposition 1 is a straightforward statement of fact.\n2) In Theorem 1, the existence and non-uniqueness of NE are well-established throughout game theory and not surprising. The second statement of Theorem 1 is standard in optimization theory.\n3) The stability in Proposition 2 is also straightforward (same intuitions with those in Theorem 1).\n4) The derivation of Theorem 2 seems to be straight-forward. The theorem actually states some intuition of \"decomposition\".\n5) In Theorem 3, the assumption in first statement is strong (K=1). For second statement of Theorem 3, the result can say nothing about whether $R$ is monotone decreasing over $\\varepsilon$, even if $h(\\varepsilon)$ can be arbitrarily close to 1. In addition, Theorem 3 only considers the interpolation between greedy policy and uniform policy, and ignores the high-dimensional policy space.\n6) In Proposition 3, the suboptimality of myopic policy is standard in almost every field.\n\nW2: In RHS of line 176, there is no definition about the concept used in \"$\\lambda_{eq}$ is stable\".  \nW3: In the paragraph below Proposition 2, the explanation about the polarized equilibrium of exposure-concentrated policy is hard-to-believe, with no theoretical and empirical evidence.  \nW4: The proposed policy, \"look-ahead policy\", is straight-forward to design. Besides, no theoretical guarantees are provided for this policy.  \nW5: In experiments, the only considered baselines are uniform and myopic policies, both of which are weak.\n\nIn other hands, the only contribution the authors claim seems to be that this paper is the first to consider two-side dynamics in this literature, and I believe it indeed is. However, the results derived in this paper (e.g., non-uniqueness and stability of Nash Equilibriums, suboptimality of myopic policy) are likely to hold and be discovered in many other settings. So it seems that the only contribution is to extend these straight-forward results to a new setting, and such contribution I think does not warrant an ICML acceptance.", "other_comments_or_suggestions": "C1: In line 194, it seems that \"Theorem 4\" should be \"Theorem 1\".  \nC2: In Equation (3), it seems that the introduction of $b_{k,l}$ is actually unnecessary because it can be incorporated into $f_{k,l}(\\lambda_l)$.", "questions_for_authors": "I have no questions.", "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:

Meta-review: This paper studies population dynamics of two-sided matching platforms. For the most part, reviewers found the model and message of the paper (that myopic greedy policies can lead to sub-optimal welfare while the simple look-ahead policy empirically performs much better) interesting, and the experiments generally convincing. However at the same time, some reviewers felt that the theoretical contributions of the paper were marginal and would have appreciated a deeper theoretical analysis of e.g. the performance of the look-ahead policy, making this a borderline case. We encourage the authors to take the chance and further strengthen their paper regardless of the outcome.
Decision: Accept (poster)