Resource Allocation Game on Social Networks: Best Response Dynamics and Convergence

08/24/2018 ∙ by Wei-Chun Lee, et al. ∙ Google University of Illinois at Urbana-Champaign 0

The decisions that human beings make to allocate time has significant bearing on economic output and to the sustenance of social networks. The time allocation problem motivates our formal analysis of the resource allocation game, where agents on a social network, who have asymmetric, private interaction preferences, make decisions on how to allocate time, a bounded endowment, over their neighbors. Unlike the well-known opinion formation game on a social network, our game appears not to be a potential game, and the Best-Response dynamics is non-differentiable making the analysis of Best-Response dynamics non-trivial. In our game, we consider two types of player behavior, namely optimistic or pessimistic, based on how they use their time endowment over their neighbors. To analyze Best-Response dynamics, we circumvent the problem of the game not being a potential game, through the lens of a novel two-level potential function approach. We show that the Best-Response dynamics converges point-wise to a Nash Equilibrium when players are all: optimistic; pessimistic; a mix of both types. Finally, we show that the Nash Equilibrium set is non-convex but connected, and Price of Anarchy is unbounded while Price of Stability is one. Extensive simulations over a stylized grid reveals that the distribution of quality of the convergence points is unimodal-we conjecture that presence of unimodality is tied to the connectedness of Nash Equilibrium.

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1 Introduction

Social networks play a central role in our lives, both for personal and professional growth. Information diffusion over a social network is one extensively analyzed class of problems (Feldman et al., 2014; Bakshy et al., 2012; Myers and Leskovec, 2012; Chierichetti et al., 2011; Karp et al., 2000). Within the Theoretical CS community, the information diffusion problem is often modeled as a network game where a player’s utility depends on her own and her neighbor’s actions, e.g., (Chierichetti et al., 2013; Feldman et al., 2014; Ferraioli et al., 2016). A common underlying assumption: a player has adequate resources (e.g. time) to interact with each of her neighbors, and that her payoff is independent of the amount of time spent with each of its neighbors. In the real world, agents have a finite endowment of resources. The dynamics of network games where agents have bounded resources is less well studied and is the focus of this paper.

Research on how individuals spend time, has a rich history in Economics (Juster and Stafford, 1991; Becker, 1965; Gronau, 1977). As Juster and Stafford (1991) state in the introduction to their article, “… the fundamental scarce resource in the economy is the availability of human time, and that the allocation of time to various activities will ultimately determine the relative prices of goods and services, the growth path of real output, and the distribution of income.” The work of Becker (1965) spurred the analysis of “non-market” time (i.e. outside of work), in particular how individuals utilized their time at home and consequences of time use at home on the market. Gronau (1977) further sharpened the analysis of “non-market time” by distinguishing between time spent on “work at home” and “leisure.” Juster and Stafford (1991) further identified “socializing” as a component of “leisure.”

Despite having a limited amount of time (an inelastic resource) each day, we spend it on sustaining and expanding our social capital. While there is no formal definition of “social capital”, there is a growing consensus that “social capital stands for the ability of actors to secure benefits by virtue of membership in social networks or other social structures.” (Portes, 1998). To develop social capital, Miritello et al. (2013) suggests that we spend time on communication to maintain friendships and (Roberts and Dunbar, 2011) finds that expending time is crucial to the sustenance of social networks. This connection between expenditure of time and its effect on the social network motivates the following question:

How would social interactions evolve on a social network when individuals are resource constrained?

Consider for example, an academic who receives requests on her time to meet for the coming week: from her PhD students, undergrads who are taking her class, colleagues who want to go to lunch, the occasional meeting request from her department chair. The academic may have private preferences with whom she should spend time and will respond with counter proposals: a PhD student who wants to meet for an hour may get thirty minutes; an undergrad fifteen minutes; agree to spend an hour with her peers; agree to meet for the time that her chair asks. Each person asking her for time, is simultaneously responding to requests on their time from their social network. In each case, when someone asks for time, the participants will typically agree to meet for the smaller of the two proposed times. As individuals in a social network make decisions on allocating time, we would like to know: does the social network converge to an equilibrium? If it does, is it stable? In other words, how sensitive is the network to a small change in proposals by any one person? What is the social welfare at equilibria?

Motivated by these scenarios, we define a resource allocation game on a social network, and analyze the game for convergence of the best-response dynamics to Nash equilibria. Next, we summarize our technical contributions.

1.1 Our Technical Contributions

We define the resource allocation game on a (social) network where each node is a rational player (agent) with finite time endowment, say . A player obtains utility by spending time with her network neighbors which may differ from neighbor to neighbor. First, she has a private interaction preferences over her neighbors, namely for each such that . Second, to capture decreasing marginal returns, a non-negative, concave, and increasing function captures ’s utility from interaction with . In general, we consider the asymmetric case with respect to (w.r.t) the interaction preferences , i.e., , and we consider both symmetric and asymmetric cases w.r.t the interaction utility . We consider two types of player behavior, namely optimistic and pessimistic, based on their aggressiveness in proposing time to neighbors.

A pair of players who are neighbors i.e. , can interact only when both agree to do so. Thus, if is the (time) interaction frequency “proposed” by to , and by to , then the pair will agree to . The total utility of agent is:

The main contribution of this paper is the analysis of best-response (BR) dynamics of the resource allocation game. In each round, a chosen player (in some arbitrary sequence) plays her utility maximizing (best-response) strategy given the proposals of her network neighbors. This game exhibits non-continuous dynamics, and is non-trivial to analyze. Except for weighted potential games (Monderer and Shapley, 1996), BR dynamics do not converge for most games.

Most known results on best response dynamics in (social) network games are under following settings, symmetry of weights along edges, discrete choices, and no ’s, e.g., (Ferraioli et al., 2016; Ferraioli and Ventre, 2017; Feldman et al., 2014). The symmetric games are known to be potential games (Ferraioli et al., 2016) and thus the BR dynamics converge to Nash. The general form of our game exhibits asymmetry in both and , and thereby our game does not seem to be a (weighted) potential game. Despite this, we obtain convergence for a special case and the general case, described next. These results are independent of the player behavior; optimistic, pessimistic, or a mix of two.

We first consider the case of global ranking weight system (Section 3.1): there is an intrinsic social order (e.g., an academic hierarchy, comprising say professors, Ph.D. students and undergrads, continuing our stylized example) among players, and the ’s are proportional to the global social rank, i.e., if is the rank of player then . Note that, still . Furthermore, we assume that the utility functions are symmetric on every edge — for all .

Theorem 1.1 (Informal).

The resource allocation game with global ranking weight system is a weighted potential game. As a consequence, the best-response dynamics in these games converges to a Nash equilibrium.

Next we consider the general resource allocation game when both weights and utilities are asymmetric and show the convergence of BR dynamics (in Section 3.2), despite its seemingly non-potential nature.

Theorem 1.2 (Informal).

Best response dynamics converges to a Nash equilibrium in general resource allocation games with asymmetric preferences.

Our method of proof is akin to the construction of a “two-level” potential-like function using the unused total time of a specific class of players. We show that the outer function is monotonically decreasing, and when the outer function is fixed, the inner function decreases. We note that, when the outer function is decreasing, there are no guarantees for the inner function.

What if players move simultaneously? For the simultaneous-play variant of the best-response dynamics in our game, we show the existence of a cycle through a stylized example. Note that potential games (Monderer and Shapley, 1996) e.g., battle of the sexes may also exhibit cyclic behavior under the simultaneous-play.

We show that, the Price-of-Anarchy (PoA) is unbounded while Price-of-Stability (PoS) is one (Section 4.1). Interestingly, we can show that the set of Nash Equilibrium is convex under certain conditions (when all players follow a particular strategy) and is connected in general.

Experiments on a stylized grid show that best response dynamics converge to equilibria with high social-welfare (see Section 5

). Experiments also demonstrate that the social-welfare distribution of points where BR dynamics converge is unimodal—probably a consequence of connectedness of the equilibrium set. We leave open a formal analysis of this unimodality, and the qualitative analysis of the social welfare of convergence points through notion of

average-price-of-anarchy (Panageas and Piliouras, 2016a).

1.2 Related Work

The work most related to ours is of Anshelevich and Hoefer (2012) studying contribution games on networks with symmetric utility functions on each edge. This translates to unweighted case with for each edge in our model. They analyze 2-strong equilibria, where no pair of two players can deviate and both gain, for efficiency (PoA), and shows convergence of best response dynamics. While the notion of 2-strong equilibria is stronger than the notion of a Nash equilibrium, we observe that the complexity of analyzing best response dynamics in our proposed work comes from non-symmetric weights on edges.

There has been extensive work on information diffusion in social networks, where decisions/opinions are discrete, typically binary, and graphs are weighted (but no functions). For example, (Ferraioli et al., 2016) study a discrete preference game, and show that it is a potential game and therefore the best-response dynamics converges to a Nash equilibrium. In addition they show a polynomial time convergence rate for unweighted graphs, and pseudo-polynomial time convergence rate for the weighted graphs. (Ferraioli and Ventre, 2017) studies this game under social pressure and obtain fast convergence in special cases. (Feldman et al., 2014) studies the consensus game (DeGroot, 1974) under asynchronous updates and the majority rule – a special case of the linear threshold model. (Chierichetti et al., 2013)

studies the price-of-stability of the game when the edge functions interpolates between symmetric (coordination) and non-symmetric (unilateral decision-making). Furthermore, a number of works have explored learning

(Bala and Goyal, 1998; Acemoglu et al., 2011; Narasimhan et al., 2015) and herd behavior (Banerjee, 1992).

Our work focuses on best-response dynamics. The best-response dynamics does not converge generally due to its discontinuous nature, while in coordination, or more generally in congestion games, the dynamics converges to a pure Nash equilibrium (Montanari and Saberi, 2009). Apart from BR dynamics, there is extensive literature on the analysis of no-regret dynamics (Cesa-Bianchi and Lugosi, 2006; Shalev-Shwartz, 2012)

within algorithmic game theory, see 

(Roughgarden, 2016a). For general games, the average of the points visited by dynamics converges, but to correlated equilibria, a weaker notion than Nash equilibrium (Blum et al., 2008). While in case of coordination games, the dynamics converges point-wise to a pure Nash equilibrium (Losert and Akin, 1983; Mehta et al., 2015).  Panageas and Piliouras (2016b) show that, while the social welfare at the limit-points of the dynamics may not be near optimal, the expected welfare (average price-of-anarchy) is almost optimal in a few special cases. Fast convergence of the average is known for various special cases, e.g., see (Syrgkanis et al., 2015; Daskalakis et al., 2014; Chien and Sinclair, 2011).

Work in Ecological games on foraging and more generally on predator-prey dynamics (Brown et al., 1999; Rosenzweig and MacArthur, 1963) analyzes the outcomes of agents who make decisions with resource constraints. The broad idea is that species forage for food, under limited energy constraints. However, most works focus on population dynamics (growth and depletion of species) which is not a focus of our paper.

To summarize: our main technical contribution lies in the analysis of best response dynamics of the resource allocation game where agents have bounded endowment and private, asymmetric interaction preferences. The main challenge: our game is in general not a potential game, but we are able to show, through a novel two-level potential-like function approach, convergence to Nash Equilibria. We analyze PoA, PoS, and characterize the quality of Nash Equilibria.

The rest of this paper is organized as follows. In the next section, we formally introduce the game model. Then, in Section 3, we present convergence results for best-response dynamics. In Section 4, we prove guarantees for several key properties, including the Price of Anarchy and the Price of Stability. We conclude by presenting experimental results showing the distribution of the quality of Nash Equilibrium in Section 5, and summarize and discuss future directions in Section 6.

2 Preliminaries

In this section we first formalize the game played on a social network by the resource constrained agents, its dynamics, and its Nash equilibria under two different types of player behavior.

2.1 Game Model

Consider a social network with agents (players) represented by an undirected graph , where represents the players and the links between them. The players are numbered through , and we denote the set of all players as . An edge (link) represents the interaction between player and player . The set of neighbors of player is denoted by – note that can be in .

Players gain utility from communicating/interacting with each other, however the amount of resources to communicate, such as time, is available in limited quantity to each player. In particular, player has amount of communication resource that she can distribute among her neighbors. Let denote the frequency proposal made by player to her neighbor . It follows that

If we denote the vector of frequency proposals of

to all players in by , the set of strategies (all possible allocations) of player is

where it is understood that if . For communication to happen between players and , naturally both have to agree to do so. Therefore, the realized allocation of resource, also called interaction frequency from now on, on edge is

In other words, denotes the agreed upon interaction frequency between and . To differentiate we will call , proposed interaction frequency.

To capture asymmetric liking of a player and her neighbor, we consider a weighted network with asymmetric weights. The weight assigned by player to her neighbor is denoted by . Note here that and may be different. Once every player decides her allocation, let denote the allocation profile of all the players, and let denote the agreed upon allocation. The utility of player at profile is

(1)

where is a non-negative increasing concave function of and therefore captures decreasing marginal returns. We are now ready to formally define our game.

Definition 1 (Game).

A game consists of a weighted graph where the nodes are the players , links represent the underlying social structure, and weights and on link represent player preferences. For each link we are given functions and capturing respectively utility of player from interaction with and vice-versa. Vector represents amount of resources of all the players, where the resource constraint of player is . We denote such a game by .

Interestingly, our convergence results hold for any arbitrary non-negative, increasing and concave function of ; again and need not be the same. This leads us to the definition of the social welfare of our game.

Definition 2 (Social Welfare).

Let be a game and a frequency profile of this game. Then, the social welfare of at is

2.1.1 The Global Ranking Model.

Many a times there is an inherent hierarchy among the players in a social network, e.g. the social network of a company, a network of tennis players, etc. Taking this as motivation, we define a special case of our model, in which there exists a global ranking of players capturing their social status within the network, and the weights reflect this global ranking.

Definition 3 (Global-Ranking Weight System).

A global ranking is a function . By imposing this function on a game , we can associate each player with a number . The corresponding global ranking weight system is a weighting scheme in which the weight that player places on player is defined as

(2)

2.2 Optimistic/Pessimistic Agents and Nash Equilibria

In this section we discuss the Nash equilibria of our game under two types of players, namely optimistic and pessimistic. For a given profile , to denote proposals of all player but ’s we use .

Definition 4 (Nash Equilibrium).

A strategy profile is said to be a Nash equilibrium if no player gains utility by unilateral deviation (Nash, 1951), i.e.,

At Nash equilibrium every player is playing her utility maximizing strategy given everyone else’s strategy. Given everyone’s frequency proposals, the utility maximizing proposal of a player is called her best-response. Thus, at NE every player is playing a best-response to the strategies of the other players. The best-response of player with respect to a proposal profile can be computed using the following convex program.

(3)

where note that , the proposal of to is a constant. For players and we have , if the proposals are not exactly equal, then one of the players will have some “leftover” frequency because they had to settle for a lower frequency than they would prefer. For a specific player , we call the sum of this “leftover” frequency from the interactions with all the slack of player , which is equal to

and in the case where is non-zero, we assign the weight of edge to be . Furthermore, we call the sum of all the players’ frequency slack the total slack of the game and we denote it by

If player proposes lower interaction frequency to than what proposes to , we say that “wins” over , or is in her “win” set.

Similarly, if ’s proposal to is at least what proposes to then “loses” to , or is in her “lose” set.

Note that it is possible for a player to have either or .

The best-response formulation of (3) raises an interesting question: What should a player do with her frequency slack, if she has any left even after she matches the frequency proposals of all her neighbors? Specifically, what should player do when for all but ? Since no strategy in this case strictly increases ’s utility, all possible ways of distributing will give a best-response. To answer this question, we consider two types of players, keeping in mind the dynamical nature of the system. Consider a player who has positive slack at a given profile .

We call pessimistic if she decides not to spend any slack frequency on her neighbors, therefore setting and . Similarly, we call optimistic, if she decides to spend any portion of her slack frequency on , even though this strategy does not increase her utility right now, with the hope that maybe, at some future time, some player will have slack frequency and will be willing to agree to interacting with at a higher frequency than before.

This distinction is important, as different strategy profiles for the players may lead to different results in the network in terms of dynamics as well as fixed-points. More specifically, we can define two different types of equilibria for our game.

Definition 5 (Pessimistic/Optimistic Equilibrium).

A Nash equilibrium is called a pessimistic equilibrium if for all such that , we have . It is called an optimistic equilibrium if such that and .

In words, any profile where proposals are matched on every link is a pessimistic equilibrium. On the other hand under optimistic equilibria even though players may be proposing higher frequency to a neighbor, the neighbor does not want to respond by increasing the frequency on the link to her. This is clearly a stricter condition to achieve compared to pessimistic equilibria. However, as we will see in the following sections, pessimistic equilibria also turn out to be of interest, since they possess nice convexity properties.

2.3 Best-Response Dynamics: Sequential or Simultaneous

We analyze dynamics of the interaction in our social network for its convergence properties. Whenever a player is given an opportunity to update her strategy, it is natural for her to play a best-response against the current strategy profile of the other players (solution of (3)). Therefore, we consider the best-response (BR) dynamics under it’s two natural variants, simultaneous move, and sequential move. Rounds are indexed by and the frequency proposal profile in round is denoted by . In both cases players start with certain initial proposal at which may be arbitrary or random.

In the sequential BR dynamics, in every round exactly one player updates: In round , if there exists a player not playing best-response against (in other words is not a Nash equilibrium), then an arbitrary such player plays a best-response. That is, a player such that is not a best-response against is chosen, and then is a BR of player against , while for all , .

In the simultaneous move setting, all players simultaneously update their proposal and play best-response to the strategy profile of the previous round, and inform their neighbors. Note that, there is a unique best-response for a pessimistic player, but an optimistic player may have multiple best-responses due to many possible ways of distributing her slack on the neighbors in . In the latter case, we let the best-response be arbitrary. By the definition of Nash equilibrium (Definition 4), it follows that under both sequential and simultaneous move, the convergence points of BR dynamics are Nash equilibria, i.e., where every player is playing a best-response to other player’s strategies.

3 Convergence Analysis and Results

This section presents our main convergence results. We look at how the best-response (BR) dynamics behave, both in simultaneous and sequential play. The convergence points of the best-response dynamics are states where no player wants to unilaterally deviate from her strategy profile, i.e., they are Nash equilibria.

For the sequential move case, we show convergence of BR dynamics to a Nash equilibrium in Sections 3.1 and 3.2. In Section 3.1 we show that the global ranking model gives a weighted potential game, and thus the convergence of BR dynamics follows relatively easily. To prove the convergence in general model however, we need to analyze through a different, indirect manner. This proof is presented in Section 3.2. Finally, we show in Section 3.3 that in the case of simultaneous play, the best-response dynamics need not converge, through a simple counterexample.

Before we continue, we make a quick remark regarding the notation used in this section. Since frequency proposals depend on time, i.e. the current round of our game, and all other quantities depend on the current strategy profile, we clarify the notation used below. We use to represent the frequency proposal that player made to player , at round of the dynamics. Similarly, the frequency that and end up interacting at time is denoted by , i.e. . For brevity, by abuse of notation, we will denote , and any other quantity that depends on the frequency profile by , etc, respectively. Finally, due to space constraint we discuss the main ideas here, while all the missing proofs are presented in Appendix 0.A.

Convergence in Sequential Play. First we consider the best-response dynamics under sequential-play and show our two main convergence results in Sections 3.2 and 3.2. We show that the best-response dynamics converges to a Nash equilibrium when players change their strategies one at a time. Our proof holds for any general non-negative, increasing and concave utility function , which underlines the generality and importance of our results. Furthermore, our result is independent of the order in which the players take turns to change strategies and relies only on the fact that each player is playing their best-response strategy that maximizes their utility at each time step and that eventually all players get to play their turn at some point. It is also independent of whether the players are optimistic, pessimistic or a mix of the two.

Since only one player can change their strategy at each turn, we have to clarify the time notation that will be used below. Consider a player , that makes a proposal to at time and the next proposal of to happens at time . Then, we consider for all times , and we imagine a “jump” in from to at time . This same logic applies not only to the players’ proposals but to all quantities defined so far.

3.1 Convergence of the Global Ranking Model

We start by providing strong convergence results of our global ranking model. In this section, we show that the global ranking model is a weighted potential game, as we show the game admits a weighted potential function when for any players we have . This condition does not imply symmetry between players’ interaction, since it may be the case that .

Theorem 3.1 (Weighted Potential Game).

Given a game , a global ranking weighting system and a strategy profile , if for all strategy profiles and for all players where , then admits a weighted potential function

(4)

and it is a weighted potential game.

The intuition behind our method is twofold. First of all, the players compute their utility based on , instead of , meaning that their utility only depends on the frequency they end up communicating at instead of their frequency proposals. Thus, if player changes their proposal to , changes accordingly and the difference is the same for both and . Furthermore, recall that . By scaling player’s utility in by , we obtain a symmetric expression for both players, which allows us to connect ’s effect on with ’s effect on .

Since the global ranking model is a weighted potential game, it is well-known that the best-response dynamics converge to a Nash equilibrium (Monderer and Shapley, 1996).

Corollary 1.

Consider a game that admits a global ranking weighting system and for all strategy profiles and for all players where . Then, the best-response dynamics of converge to a Nash equilibrium.

3.2 Convergence of the General Model

In the previous section, we saw that the best-response dynamics in the special case of the global ranking model converges to a Nash equilibrium, by constructing a potential function. In contrast, for the general model, with complete asymmetry in weights, utility functions and behavior of agents, existence of any such potential function seems unlikely. We obtain the convergence result for the general model in this section by an in depth analysis of the best-response dynamics. We show that the best-response dynamics converge to a Nash equilibrium for the general model as well, for any non-negative, increasing and concave utility function. As such, our result fully characterizes the best-response dynamics for this game.

Our argument for the model’s convergence is akin to using a two-stage potential-like function which decreases at each time step and reaches a minimum which is equivalent to a Nash equilibrium in our game. In order to avoid infinitesimal changes, first, we impose a reasonable constraints on proposals of the players. Let be the minimum denominator of the resource under consideration, i.e., one second, if the resource is time. Now on we assume that any pair of players interact only at multiples of a fixed constant . In other words, for all pairs of players and all times , , where is a pair-specific function and is a fixed constant. In this case, it is without loss of generality to consider s’ as well, as multiples of . Before proceeding with our main result, we will describe a sufficient condition for a frequency profile to be a Nash equilibrium.

Lemma 1.

Consider a frequency profile such that, for every player , . Then, no player can strictly increase her utility, and is a Nash equilibrium.

The first stage of our proof is to show that the total slack of the game is monotonically decreasing.

Lemma 2.

Under best-response dynamics, the total slack is monotonically decreasing.

The basic intuition behind the proof of the above lemma is that whenever a player plays her best-response at each turn, the total slack of the game decreases if she decreases her slack, or stays the same if she increases her utility without decreasing her slack.

Next, we show that, as the best-response dynamics progress, will decrease and, after some time , it will remain constant.

Lemma 3.

Under best-response dynamics, if there exist a fixed constant such that for all players and all times , there exists a for all neighbors such that , then there exists a time such that for all times .

The basic idea behind the proof of the above lemma is that whenever the total slack of the game decreases, it decreases by a constant amount, and it is also lower bounded by zero by definition. Thus, it can only decrease a finite number of times.

Finally, in the following lemma, we argue that the stabilization of the total slack of the game is sufficient to prove that the best-response dynamics converges to a Nash equilibrium within a finite number of rounds.

Lemma 4.

If there exists some time such that for all times , then the best-response dynamics converge to a Nash equilibrium within a finite number of rounds after .

The basic intuition behind the proof of the above lemma is that the total slack does not decrease in a round only when the chosen player redistributes her frequency, and this redistribution cannot cycle forever. Theorem 3.2 now follows from Lemmas 2, 3 and 4.

Theorem 3.2.

Consider a game where utility function ’s are arbitrary non-negative, increasing, and concave. The sequential best-response dynamics of converge to a Nash equilibrium.

3.3 No Convergence in Simultaneous Play

In the previous sections, we showed that when players change their strategies one at a time, their frequency proposals always converge under the best-response dynamics. Interestingly, the simultaneous setting is inherently different. In other words, if at each round all the players play their best-response strategy to the strategies of their neighbors at the previous round simultaneously, the best-response dynamics need not converge to an equilibrium. The initial starting point of the dynamics, i.e. the starting strategy is arbitrary. To illustrate this point, we present a simple counterexample that exhibits cyclic behavior of the dynamics. This cyclic behavior is well known, even in potential games. One such famous example is the simple game battle of the sexes (Roughgarden, 2016b).

Example 1.

Consider the following game with optimistic players, , where the resource constraint is uniform for all players , the utility function of each player for a neighbor is , for , and is , i.e. the complete graph with nodes. The weights between the players are represented in the following matrix, where element is equal to

for some . We want to force a cyclic behavior of the best-response dynamics for all players, where the players mismatch the frequency proposals they make to each other. Specifically, we make each player propose a slightly higher frequency to two players and a slightly lower frequency to the other two, and the weights guarantee that, at each round, each player proposes a higher frequency to the players that propose a lower frequency to her. Therefore, the players never match the proposals they make to each other.

Suppose that at time the players calculate their best-response. We can easily see that the solution to (3) for this game is . Thus, at time , each player makes the following proposals to each other, represented in the following matrix, where element is equal to

Notice that the agreement frequencies at are going to be

Now, all players compute their best-response strategies simultaneously. We observe that for player , their outcome sets are and . Furthermore, each player has exactly slack frequency. Recall that each player is optimistic and the weights for both players in are equal. Since has slack, their best-response, as calculated by (3), is to allocate of their slack to each player in .

Therefore, at time , the frequency proposals of every player are

which again yields . However, note that for all players, and . We apply the previous argument now for and get , which shows the existence of a cycle of proposals between all players, implying our model need not converge in the simultaneous setting.

4 Properties of the Game

In this section we provide several properties of our game’s best-response dynamics and equilibria. First, we show that the global optimum of our game is also a Nash equilibrium, which implies that the Price of Stability (PoS) is . We also provide an example with unbounded Price of Anarchy (PoA), which demonstrates that the social welfare of the Nash equilibria can vary significantly. Next, we show that the set of Nash equilibria for our game is connected and, more importantly, the set of pessimistic Nash equilibria is convex. Finally, we fully characterize each player’s best-response through the well-known Karush-Kuhn-Tucker (KKT) conditions for local optimality (Boyd and Vandenberghe, 2004), which provides better intuition as to how each player calculates her best-response. All the missing proofs of this section are presented in Appendix 0.B.

4.1 Price of Anarchy and Stability

In this section, we focus on the quality of Nash equilibria. We show that the optimal frequency distribution for all players is also a Nash equilibrium. On the other hand, we also show that there exist arbitrarily low quality Nash equilibria, even for the simpler case of uniform resource constraints among all players. We quantify these observations through the well-known concepts of the Price of Anarchy (PoA) (Koutsoupias and Papadimitriou, 1999) and Price of Stability (PoS) (Anshelevich et al., 2004) respectively. These results show that there is a significant difference between the Price of Anarchy and the Price of Stability in our game.

We first provide a definition of the Price of Anarchy and the Price of Stability for our game.

Definition 6 (Price of Anarchy and Price of Stability).

Consider a game , where is the frequency profile that maximizes the social welfare, and is the set of all frequency profiles that are Nash equilibria. Then, the Price of Anarchy (PoA) is defined as

while the Price of Stability (PoA) is defined as

We first want to analyze how good a Nash equilibrium can be. Theorem 4.1 shows that the optimal solution profile of our game is also a Nash equilibrium. The optimal solution profile of our game can be seen as the solution to the following (global) convex program

(5)

Note that, unlike (3) where was a constant for ’s best-response, it is now a variable in this program. It is clear that the solution to the above program is the optimal solution profile of our game, i.e. the frequency profile that maximizes the social welfare. Recall that Lemma 1 describes a sufficient condition for a frequency profile to be a Nash equilibrium. We now look into how we can transform any frequency profile into a pessimistic Nash equilibrium, with equal social welfare, through a simple process of making each player match the proposals of her neighbors.

Lemma 5.

Consider a frequency profile . Then, we can construct a frequency profile such that and is a pessimistic Nash equilibrium.

Lemma 5 is enough to guarantee that the optimal solution of our game is also a Nash equilibrium.

Theorem 4.1.

Let be a game. Then, there exists an optimal strategy profile that maximizes the social welfare of and is also a Nash equilibrium.

Proof.

Let be the solution of (5). From Lemma 5, we can construct a new frequency profile from such that and is a pessimistic Nash equilibrium. Since has the same social welfare as , we understand that is also a solution to the global convex program (5) that maximizes the social welfare. Thus, both maximizes the social welfare of and is also a Nash equilibrium. ∎

Theorem 4.1 leads us to the following corollary.

Corollary 2.

The Price of Stability of a game is .

Next, we look at how bad can equilibria be for our game. Unfortunately, the next theorem demonstrates that there exist Nash equilibria with arbitrarily bad social welfare, even for the simple case where all players have the same resource constraint.

Theorem 4.2.

The Price of Anarchy of a game is unbounded.

4.2 Properties of the set of Nash Equilibria

In this section we look at the convexity and connectedness of the set of Nash equilibria for our game. Specifically, we show that the set of pessimistic Nash equilibria is convex, which implies that the set of all Nash equilibria of our game is connected. We start with the following lemma.

Lemma 6.

Every optimistic Nash equilibrium can be transformed into a pessimistic Nash equilibrium with the same social welfare. Furthermore, every frequency profile that is a convex combination of and is also an optimistic Nash equilibrium.

We now show that the set of pessimistic Nash equilibria is convex.

Theorem 4.3.

Consider a game with two pessimistic Nash equilibria and . Any convex combination of and is also a pessimistic Nash equilibrium.

Proof.

The proof is almost identical to the proof of Lemma 6. We want to show that every convex combination of and is also a pessimistic Nash equilibrium. Let , and . Consider a pair of players . Since and are pessimistic Nash equilibria, we have and . Thus

and make matching frequency proposals to each other at . Since this holds for every such pair , we understand that is a pessimistic Nash equilibrium. ∎

Corollary 3.

The set of pessimistic Nash equilibria of a game is convex.

The following corollary now follows from Lemma 6 and Theorem 4.3.

Corollary 4.

The set of Nash equilibria of a game is connected.

Finally, we provide a complete characterization of a player’s best-response in Appendix 0.C.

5 Experimental Results

Figure 1: We show the difference between the quality of Nash Equilibria, when players adopt either pessimistic (i.e. do not redistribute slack), or optimistic (distribute slack over winning set) strategies (see  Section 2.2). We conducted 10,000 simulations on a stylized 4-regular graph with random weights, uniform endowment () and random initial proposal frequencies. The histogram (-axis shows the number of simulations that converge to a particular bin;

-axis is ratio of NE quality to OPT, appropriately binned) shows that the optimistic strategies outperform pessimistic strategies, with a higher mean quality (dashed lines) and slightly lower variance.

We present results of simulations of best-response dynamics on a stylized graph. We would like to compute the distribution of Nash Equilibria when players adopt either the pessimistic or the optimistic strategies to distribute their slack (see  Section 2.2). The pessimistic strategy for a player involves not distributing slack, while the optimistic strategy involves re-distributing slack over the winning set (i.e. set of neighbors whose proposals are higher than ’s proposals to her neighbors).

For the simulation, we use a grid (i.e. a 4-regular graph), with uniform resource constraint of , random edge weights. The simulation involves using random frequency proposals to start and then compute best-response dynamics using a random sequence of player updates. We ran 10,000 simulations and noted the quality of Nash Equilibrium for each run for each strategy. Note that, in our experiments, either all players use the pessimistic strategy or all players use the optimistic strategy. We show the results in Figure 1, by creating a histogram of the ratio of the Nash Equilibrium quality to the global optimum, for each strategy. The results show that on average the quality of Equilibria for the optimistic strategy () is significantly better than the pessimistic strategy (), and the quality of equilibria for the optimistic strategy has slightly lower variance (). It is gratifying that the optimistic strategy does so well. Notice that the distribution is unimodal in for each strategy—we conjecture that this may be an outcome of the connectedness of the Nash Equilibria (see Corollary 4).

6 Conclusion

The problem of time allocation is one of longstanding interest to Economics and to Sociology given its importance to economic output and to the sustenance of social networks. In this paper, we formally studied the resource allocation game, where agents have private, asymmetric interaction preferences and make decisions on time allocation over their social network. The game was challenging to analyze since it is not in general, a weighted potential game, and its best response dynamics are not differentiable. First we showed that a restricted subclass of games where the interaction preferences are related to the social rank is a weighted potential game. Then, for the general case, we used a novel two-level potential function approach to show that the best response dynamics converge to Nash Equilibrium. Our proof is general, and makes no assumptions on the form of the utility function beyond that it is concave, increasing and non-negative, which are reasonable and standard assumptions. We showed that the Price of Anarchy is unbounded, and that the Price of Stability is unity. Furthermore, we showed that the Nash Equilibria form a connected set. Towards understanding the quality Nash equilibrium where best response converges, through extensive simulation of a stylized graph, we showed that the distribution of quality of Nash Equilibria are unimodal, which we conjecture is related to the connectedness of Nash Equilibria.

We identify two assumptions that limit the generalizability of our results. Our analysis of time focuses on costly communication (e.g. a conversation over a phone, or meeting in person). In online social networks, communication may be asymmetric—agent may send more messages to agent than does agent send to . Second we assume that strength of the tie doesn’t change over time— remains the same. In real-world networks, tie strengths improve and degrade over time (Miritello, 2013). We can incorporate weight changes by allowing weight update of depending on how the neighbor of agent reciprocates to her proposals.

Finally, it would be interesting to further understand the quality of Nash equilibrium to which the resource allocation game converges, through the lens of average price of anarchy (Panageas and Piliouras, 2016a).

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Appendix 0.A Missing Proofs of Section 3

0.a.1 Proof of Theorem 3.1

Because the game adopts global-ranking weighting scheme, every weight can be written as

Recall the potential function we proposed in (4). If player deviates from , which is the vector of frequency proposals to all , to , then

Notice that only ’s neighbors are affected by this deviation, therefore

Since , we have

where denotes the total utility of player at strategy profile .

The difference of in after the move of player is the difference of ’s utility, scaled by a constant. Therefore is a weighted potential game and always admits a Nash equilibrium (Monderer and Shapley, 1996).

0.a.2 Proof of Lemma 1

We know that , . This implies that , . Therefore, is matching the frequency proposals of all her neighbors, meaning that when calculates her best-response, she satisfies all constraints of the form in (3) for some with equality. It follows that cannot increase her utility by unilaterally deviating from . Since this condition holds for every , we understand that is a Nash equilibrium.

0.a.3 Proof of Lemma 2

Consider that at time it is ’s turn in our model. Obviously, we have for all , since ’s turn did not affect them at all. We first prove the following claims which relate the slack of at time , before she plays her best-response, and at time , after has made new frequency proposals according to her best-response.

Claim.

If at time it is ’s turn to play her best-response strategy and , then . Moreover, .

Proof.

Since , is matching the proposal of all of her neighbors. Thus, every strategy where for all is a best-response strategy. This implies that in any best-response strategy, , for all . We know that , and also

where the second equality follows from the fact that the utility function of is increasing.

Finally, since ’s proposals did not force any change in the interaction frequency with any player , we have