Signaling in Bayesian Network Congestion Games: the Subtle Power of Symmetry

02/12/2020 ∙ by Matteo Castiglioni, et al. ∙ Politecnico di Milano 0

Network congestion games are a well-understood model of multi-agent strategic interactions. Despite their ubiquitous applications, it is not clear whether it is possible to design information structures to ameliorate the overall experience of the network users. We focus on Bayesian games with atomic players, where network vagaries are modeled via a (random) state of nature which determines the costs incurred by the players. A third-party entity—the sender—can observe the realized state of the network and exploit this additional information to send a signal to each player. A natural question is the following: is it possible for an informed sender to reduce the overall social cost via the strategic provision of information to players who update their beliefs rationally? The paper focuses on the problem of computing optimal ex ante persuasive signaling schemes, showing that symmetry is a crucial property for its solution. Indeed, we show that an optimal ex ante persuasive signaling scheme can be computed in polynomial time when players are symmetric and have affine cost functions. Moreover, the problem becomes NP-hard when players are asymmetric, even in non-Bayesian settings.

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1 Introduction

Network congestion games, where players seek to minimize their own costs selfishly, are a canonical example of a setting where externalities may induce socially inefficient outcomes (Roughgarden, 2005). In real-world problems, the state of the network may be uncertain, and not known to its users (e.g., drivers may not be aware of road works and accidents in a road network). This setting is modeled via Bayesian network congestion games (BNCGs). We investigate whether providing players with partial information about the state of the network may mitigate inefficiencies.

We model this information-structure design problem through the Bayesian persuasion framework by Kamenica and Gentzkow (2011). At its core, this framework involves an informed sender trying to influence the behavior of a set of self-interested players—the receivers—via the provision of payoff-relevant information. We focus on the notion of ex ante persuasiveness, as introduced by Xu (2019) and Celli et al. (2019), where players commit to following the sender’s recommendations having observed only the information structure. This assumes credible receivers’ commitments, which is reasonable in practice. In our setting, where signaling schemes are usually implemented as software (e.g., real-time traffic apps), it is natural to assume that each player decides to either follow the signaling scheme (i.e., adopting the software) or act based on his prior belief. Moreover, on a general level, the receivers will uphold their ex ante commitment every time they reason with a long-term horizon where a reputation for credibility positively affects their utility (Rayo and Segal, 2010). In some cases, the receivers could also be forced to stick to their ex ante commitment by contractual agreements or penalties.

Related Works

Arnott et al. (1991) and Acemoglu et al. (2018) study the impact of information on traffic congestion. Several recent works focus on non-atomic games (Das et al., 2017; Massicot and Langbort, 2019; Wu et al., 2018; Vasserman et al., 2015). Bhaskar et al. (2016) study the inapproximability of finding optimal ex interim persuasive signaling schemes in non-atomic games. Liu and Whinston (2019) focus on atomic games with costs uncertainties and study ex interim persuasion by placing stringent constraints on the network structure. To the best of our knowledge, the present work is the first studying ex ante persuasion in general atomic BNCGs. Other works study the problem of correlation in non-Bayesian congestion games (Christodoulou and Koutsoupias, 2005; Papadimitriou and Roughgarden, 2008). The closest to our work is that of Jiang and Leyton-Brown (2011), who provide a polynomial-time algorithm to find an optimal coarse correlated equilibrium (i.e., an ex ante persuasive signaling scheme in the non-Bayesian setting) in simple games with symmetric players selecting a single resource (a.k.a. singleton congestion games).

Original Contributions

We investigate whether it is possible to efficiently compute optimal (i.e., minimizing the social cost) ex ante persuasive signaling schemes in BNCGs. First, we show that an optimal ex ante persuasive signaling scheme can be compute in poly-time in symmetric BNCGs with affine costs. To prove this result, we exploit the ellipsoid algorithm by designing a sophisticated polynomial-time separation oracle based on a suitably defined min-cost flow problem. Then, we show that symmetry is a crucial property for efficient signaling by proving that it is -hard to compute an optimal ex ante persuasive signaling scheme in asymmetric BNCGs. Our reduction proves an even stronger hardness result, as it works for non-Bayesian singleton congestion games with affine costs, which is arguably the simplest class of asymmetric congestion games. Furthermore, in such setting, a solution to our problem is an optimal coarse correlated equilibrium and, thus, its computation is -hard.

2 Signaling in Network Congestion Games

We study atomic network congestion games where edge costs depend on a stochastic state of nature, defined as follows.

Network Congestion Game (NCG)

A network congestion game (Fabrikant et al., 2004) is defined as a tuple , where: denotes the set of players; is the directed graph underlying the game, with being its set of nodes and each representing a directed edge from to ; are the edge costs, with each defining the cost of edge as a function of the number of players traveling through ; finally, , with , denote the source-destination pairs for all the players. As usual, we assume that for all . In an NCG, the set of actions available to a player is implicitly defined by the graph , the source , and the destination . Formally, is the set of all directed paths from to in the graph . In this work, we use to denote a player ’s path and we write whenever the path contains the edge . An action profile , where , is a tuple of - directed paths , one per player . For the ease of notation, given an action profile , we let be the congestion of edge in , i.e., the number of players selecting a path passing thorough in ; formally, . Thus, denotes the cost of edge in . Finally, the cost incurred by player in an action profile is denoted by .

Bayesian Network Congestion Game (BNCG)

We define a Bayesian network congestion game as a tuple , where, differently from the basic setting, the edge cost functions also depend on a state of nature drawn from a finite set of states . Moreover, encodes the prior beliefs that the players have over the states of nature, i.e.,

is a fully-supported probability distribution over the set

, with

denoting the prior probability that the state of nature is

. All the other components are defined as in non-Bayesian NCGs. Notice that, in BNCGs, the cost experienced by player in an action profile also depends on the state of nature , and, thus, it is defined as . A BNCG is symmetric if all the players share the same pair, i.e., whenever they all have the same set of actions (paths). For the ease of notation, in such settings we let be the common source and destination. Moreover, we focus on BNCGs with affine costs, i.e., for all and , there exist constants such that the edge cost function is 111We focus on affine costs since: (i) the assumption is reasonable in many applications (Vasserman et al., 2015), and (ii) the problem is trivially -hard when generic costs are allowed (see Section 4).

Signaling in BNCGs

Suppose that a BNCG is employed to model a road network subject to vagaries. It is reasonable to assume that third-party entities (e.g., the road management company) may have access to the realized state of nature. We call one such entity the sender. We focus on the following natural question: is it possible for an informed sender to mitigate the overall costs through the strategic provision of information to players who update their beliefs rationally? The sender can publicly commit to a signaling scheme which maps the realized state of nature to a signal for each player. The sender can exploit general private signaling schemes, sending different signals to each player through private communication channels. In this setting, a simple revelation-principle-style argument shows that it is enough to employ players’ actions as signals (Arieli and Babichenko, 2016; Kamenica and Gentzkow, 2011). Therefore, a private signaling scheme is a function which maps any state of nature to a probability distribution over action profiles (signals), and recommends action to player . The probability of recommending an action profile having observed the state of nature is denoted by . Then, it has to hold , for each . A signaling scheme is persuasive if following recommendations is an equilibrium of the underlying Bayesian game (Bergemann and Morris, 2016a, b). We focus on the notion of ex ante persuasiveness as defined by Xu (2019) and Celli et al. (2019).

Definition 1.

A signaling scheme is ex ante persuasive if, for each and , it holds:

Then, a coarse correlated equilibrium (CCE) (Moulin and Vial, 1978) may be seen as an ex ante persuasive signaling scheme when . Finally, a sender’s optimal ex ante persuasive signaling scheme is such that it minimizes the expected social cost of the solution, i.e.:

The following example illustrates the interaction flow between the sender and the players (receivers).

Example 1.

Signals (B,B,B) (A,A,B) (A,B,A) (B,A,A) 1 0 0 0 0
Figure 1: Left: BNCG for Example 1. Right: An ex ante persuasive signaling scheme for the case with . The table displays only those such that for some .

Figure 1 (Left) describes a simple BNCG modeling the road network between the Tokyo Haneda airport (node ), and Yokohama (node ). It is late at night and three lone researchers have to reach the IJCAI venue. They are following navigation instructions from the same application, whose provider (the sender) has access to the current state of the roads (called A and B, respectively). Roads costs (i.e., travel times) are depicted in Figure 1 (Left). In normal conditions (state ), road B is extremely fast ( and ). However, it requires frequent road works for maintenance (state ), which increase the travel time. Moreover, it holds . The interaction between the sender and the three players goes as follows: (i) the sender commits to a signaling scheme ; (ii) the players observe and decide whether to adhere to the navigation system or not; (iii) the sender observes the realized state of nature and exploits this knowledge to compute recommendations. Figure 1 (Right) describes an ex ante persuasive signaling scheme. In this case, when the state of nature is , one of the players is randomly selected to take road B, even if it is undergoing maintenance. In expectation, following the sender’s recommendations is strictly better than congesting road A.

A simple variation of Example 1 is enough to show that the introduction of signaling allows the sender to reach solutions with arbitrarily better expected social cost than what can be achieved via the optimal Bayes-Nash equilibrium in absence of signaling. Specifically, consider the BNCG in Figure 1 (Left) with the following modifications: , coefficients always equal to zero, , , , and . Without signaling, the optimal choice yields an expected social cost of . However, a perfectly informative signal allows the players to avoid any cost.

3 The Power of Symmetry

We design a polynomial-time algorithm to compute an optimal ex ante persuasive signaling scheme in symmetric BNCGs with affine cost functions. Our algorithm exploits the ellipsoid method. We first formulate the problem as an LP (Problem (1)) with polynomially many constraints and exponentially many variables. Then, we show how to find an optimal solution to the LP in polynomial time by applying the ellipsoid algorithm to its dual (Problem (2)), which features polynomially many variables and exponentially many constraints. This calls for a polynomial-time separation oracle for Problem (2), which is not readily available since the problem has an exponential number of constraints. We prove that, in our setting, a polynomial-time separation oracle can be implemented by solving a suitably defined min-cost flow problem. The proof of this result crucially relies on the symmetric nature of the problem and the assumption that the costs are affine functions of the edge congestion.

The following lemma shows how to formulate the problem as an LP. 222LPs analogous to Problem (1) and Problem (2) can also be derived for the asymmetric setting. However, the separation problem for the latter is solvable in poly-time only in the symmetric case. For the ease of presentation, we use to denote the indicator function for the event , i.e., it holds if , while otherwise.

Lemma 1.

Given a symmetric BNCG, an optimal ex ante persuasive signaling scheme can be found with the LP:

(1a)
(1b)
(1c)
(1d)
(1e)
Proof.

Clearly, Objective (1a) is equivalent to minimizing the social cost, while Constraints (1e) imply that is well formed. Constraints (1b) enforce ex ante persuasiveness for every player : the expression on the left-hand side represents player ’s expected cost, while is the cost of her best deviation (i.e., a cost-minimizing path given and ). This is ensured by Constraints (1c) and (1d). In particular, for every player and node , the former guarantee that is the minimum cost of a path from to . This is shown by noticing that (given that ) such cost can be inductively defined as follows:

where accounts for the fact that the congestion of edge must be incremented by one if player does not select a path containing in the action profile . ∎

Lemma 2.

The dual of Problem (1) reads as follows:

(2a)
(2b)
(2c)
(2d)
(2e)
(2f)
(2g)
Proof.

It directly follows from duality, by letting (for ), (for and ), (for ), and (for ) be the dual variables associated to, respectively, Constraints (1b), (1c), (1d), and (1e). ∎

Since is exponential in the size of the game, Problem (1) features exponentially many variables, while its number of constraints is polynomial. Conversely, Problem (2) has polynomially many variables and exponentially many constraints, which enables the use of the ellipsoid algorithm to find an optimal solution to Problem (2) in polynomial time. 333For additional details on how the ellipsoid algorithm can be adopted to solve optimization problems see (Grötschel et al., 1981). This requires a polynomial-time separation oracle for Problem (2), i.e.

, a procedure that, given a vector

of dual variables, it either establishes that is feasible for Problem (2

) or, if not, it outputs an hyperplane separating

from the feasible region. In the following, we focus on a particular type of separation oracles: those generating violated constraints.

Given that Problem (2) has an exponential number of constraints, a polynomial-time separation oracle is not readily available. It turns out that, in our setting, we can design one by leveraging the symmetry of the players and the fact that the cost functions are affine, as described in the following.

First, we prove that Problem (2) always admits an optimal player-symmetric solution, i.e., a vector such that, for each pair of players , it holds , for all , and . This result allows us to restrict the attention to player-symmetric vectors .

Lemma 3.

Problem (2) always admits an optimal player-symmetric solution.

Proof.

Given any optimal solution to Problem (2), we can always recover, in polynomial time, a player-symmetric optimal solution . Specifically, for every , let , for all , and , while for every . First, notice that and provide the same objective value, as for all . Thus, we only need to prove that satisfies all the constraints of Problem (2). For and , let us denote with an action profile such that , i.e., a permutation of in which each player takes on the role of player . Moreover, let . Constraints (2b) are satisfied by , since, for every and , it holds:

Similar arguments show that satisfies all the other constraints, concluding the proof. ∎

Notice that any polynomial-time separation oracle for Problem (2) can explicitly check whether each member of the polynomially many Constraints (2c), (2d), and (2e) is satisfied for the given . Thus, we focus on the separation problem restricted to the exponentially many Constraints (2b), which, using Lemma  3, can be formulated as stated in the following.

Lemma 4.

Given a player-symmetric , solving the separation problem for Constraints (2b) amounts to finding and that are optimal for the following problem:

(3)

where we let and for all .

Next, we show how Problem (3) can be equivalently formulated avoiding the minimization over the exponentially-sized set . Intuitively, we rely on the fact that, for a fixed , we can exploit the symmetry of the players to equivalently represent action profiles as integer vectors of edge congestions , for all .

Lemma 5.

Problem (3) can be formulated as , where is the optimal value of the following problem:

(4a)
(4b)
(4c)
(4d)
Proof.

First, given a state , Problem (3) reduces to computing , where the function to be minimized only depends on the number of players selecting each edge in , rather than the identity of the players who are choosing (since they are symmetric). Letting be the congestion level of edge and using (affine costs), it holds , and, for every , . This gives Objective (4a). Moreover, Constraints (4b), (4c), and (4d) ensure that is well defined. ∎

Let us remark that computing an optimal integer solution to Problem (4) is necessary in order to (possibly) find a violated constraint for the given ; otherwise, we would not be able to easily recover an action profile from a vector .

Now, we show that an optimal integer solution to Problem (4) can be found in polynomial time by reducing it to an instance of integer min-cost flow problem. Intuitively, it is sufficient to consider a modified version of the original graph in which each edge is replaced with parallel edges with unit capacity and increasing unit costs. This is possible given that the Objective (4a) is a convex function of , which is guaranteed by the assumption that the costs are affine.

Lemma 6.

An optimal integer solution to Problem (4) can be found in polynomial time by solving a suitably defined instance of integer min-cost flow problem.

Proof.

First, notice that Objective (4a) is a sum edge costs, in which the cost of each edge is a convex function of the edge congestion , as the only quadratic term is , where the multiplying coefficient is always positive, given and . This allows us to formulate Problem (4) as an instance of integer min-cost flow problem. We build a new graph where each is replaced with parallel edges, say for . For and , let us define . Each (new) edge has unit capacity and a per-unit cost equal to . Clearly, finding an integer min-cost flow is equivalent to minimizing Objective (4a). Notice that, since the original edge costs are convex, it holds for all . Thus, an edge is used (i.e., it carries a unit of flow) only if all the edges , for , are already used. This allows us to recover an integer vector from a solution to the min-cost flow problem. Finally, let us recall that we can find an optimal solution to the integer min-cost flow problem in polynomial time by solving its LP relaxation. ∎

The last lemma allows us to prove our main result:

Theorem 1.

Given a symmetric BNCG, an optimal ex-ante persuasive signaling scheme can be computed in poly-time.

Proof.

The algorithm applies the ellipsoid algorithm to Problem (2). At each iteration, we require that the vector of dual variables given to the separation oracle be player-symmetric, which can be easily obtained by applying the symmetrization technique introduced in the proof of Lemma 3. The separation oracle needs to solve an instance of integer min-cost flow problem for every (see Lemmas 5 and 6). Notice that an integer solution is required in order to be able to identify a violated constraint. Finally, the polynomially many violated constraints generated by the ellipsoid algorithm can be used to compute an optimal . ∎

4 The Curse of Asymmetry

In this section, we provide our hardness result on asymmetric BNCGs. Our proof is split into two intermediate steps: (i) we prove the hardness for a simple class of asymmetric non-Bayesian congestion games in which each player selects only one resource (Lemma 7), and (ii) we show that such games can be represented as NCGs with only a polynomial blow-up in the representation size (Lemma 8). Our main result reads:

Theorem 2.

The problem of computing an optimal ex ante persuasive signaling scheme in BNCGs with asymmetric players is -hard, even with affine costs. 444Without affine costs, computing an optimal ex ante persuasive signaling scheme is trivially -hard even in symmetric BNCGs. This directly follows from Meyers and Schulz (2012), which show that even finding an optimal action profile (that is also an optimal Nash equilibrium) is -hard in symmetric (non-Bayesian) NCGs.

The proof of Theorem 2 is based on a reduction which maps an instance of 3SAT (a well-known -hard problem, see (Garey and Johnson, 1979)) to a game in the class of singleton congestion games (SCGs) (Ieong et al., 2005). A (non-Bayesian) SCG is described by a tuple , where is a finite set of resources, each player selects a single resource from the set of available resources, and resource has a cost . Another way of interpreting SCGs is as games played on parallel-link graphs, where each player can select only a subset of the edges.

First, let us provide the following definition and notation.

Definition 2 (3sat).

Given a finite set of three-literal clauses defined over a finite set of variables, is there a truth assignment to the variables satisfying all the clauses?

We denote with a literal (i.e., a variable or its negation) appearing in a clause . Moreover, we let and be, respectively, the number of clauses and variables, i.e., and . W.l.o.g., we assume that .

Lemma 7 introduces our main reduction, proving that finding a social-cost-minimizing CCE is -hard in SCGs with asymmetric players, i.e., whenever the resource sets are different among each other. 555The reduction in Lemma 7 does not rely on standard constructions, as most of the reductions for congestion games only work with action profiles, while ours needs randomization. Indeed, in asymmetric SCGs, a social-cost-minimizing action profile can be computed in poly-time by solving a suitable instance of min-cost flow. This also prevents the use of other techniques for proving the hardness of CCEs, such as, e.g., those by Barman and Ligett (2015). Notice that the games used in the reduction are not Bayesian; this shows that the hardness fundamentally resides in the asymmetry of the players.

Lemma 7.

The problem of computing a social-cost-minimizing CCE in SCGs with asymmetric players is -hard, even with affine costs.

Proof.

Our 3SAT reduction shows that the existence of a polynomial-time algorithm for computing a social-cost-minimizing CCE in SCGs would allow us to solve any 3SAT instance in polynomial time. Given , let , , and . We build an SCG admitting a CCE with social cost smaller than or equal to iff is satisfiable.

Mapping. is defined as follows (for every , the cost is an affine function with coefficients and ).

  • [nolistsep,itemsep=1mm]

  • ;

  • ;

  • ;

  • ;

  • ;

  • ;

  • ;

  • and ;

  • ;

  • ;

  • and ;

If. Suppose is satisfiable, and let be a truth assignment satisfying all the clauses in . Using , we recover a CCE with social cost smaller than or equal to , having in its support the action profiles for , defined as follows. First, let us consider a congestion game restricted to the players in , with action spaces limited to the resources with (since satisfies all clauses, each player has at least one action). Clearly, admits a pure NE (Rosenthal, 1973). Moreover, we show that, in any pure NE, each resource is selected by at least one player. By contradiction, suppose that there exists a resource such that no player chooses it. Then, there must be at least two players (with ) selecting a resource different from . It is easy to check that, then, there must be one player with an incentive to deviate, contradicting the NE assumption. For every and , we let (respectively, and ) be equal to the resource played by the corresponding player in a pure NE of . For every literal and , we let , , and if , while if . Similarly, for , we let , , and if , while otherwise. For every , we let and if , while if not. Finally, we let , while . We show that players have no incentive to defect from . Given that player ’s action (for and ) is determined by a pure NE of , she does not have any incentive to select a resource with (as it is not selected by other players). Moreover, in , player ’s expected cost is at most , while she would pay at least by selecting a resource with . Each player (for ) does not defect from , since her expected cost is , while she would pay: (i) the same by switching to , (ii) at least by playing with and (as there is at least one player on ), or (iii) at least by selecting with and . Each player (for and ) with does not deviate, since her cost is , while she would pay: (i) at least by switching to either or , or (ii) at least by selecting resource . Moreover, each player with does not deviate either, as her cost is , while she would pay: (i) at least by playing , or (ii) at least by switching to either or . Finally, the CCE provides a social cost smaller than or equal to , where the last inequality comes from .

Only if. Suppose there exists a CCE with social cost smaller than or equal to . First, we prove that, with probability at most , at least one player plays . By contradiction, assume that this is not the case. Then, the social cost would be at least . This implies that each player is playing either or with probability at least . We prove that is the only player on that resource with probability at least . Otherwise, by contradiction, her cost would be at least , while by playing she would pay at most . By a union bound, there exists an action profile played with probability at least in which all the players are alone on their resources (either or ). Let be a truth assignment such that if and if . Then, satisfies all the clauses, since all the players play with and, thus, all the clauses have at least one true literal. ∎

The following lemma concludes the proof of Theorem 2.

Lemma 8.

Any SCG can be represented as an NCG of size polynomial in the size of the original SCG.

Proof.

Given an SCG we build an NCG , as follows. The graph has two nodes for each resource , and, additionally, for every player , there is a source node and a destination one . Moreover, there is an edge for every and, for every and , there two edges and . Finally, for the edges , we let , while for all the other edges. ∎

5 Discussion and Future Works

The paper studies information-structure design problems in atomic BNCGs, where an informed sender can observe the actual state of the network and commit to a signaling scheme. We focus on the problem of computing optimal ex ante persuasive signaling schemes in such setting. We show that, with affine costs, simmetry is the property marking the transition from polynomial-time tractability to NP-hardness.

In the future, we are interested in studying the problem of approximating optimal ex ante persuasive signaling schemes, and in the design of practical algorithms for real-world network signaling problems. Moreover, in order to make the framework even more applicable, it would be interesting to explore how the sender can handle uncertainty about receivers’ payoffs, and to be robust to mismatching priors.

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