Equilibrium and Potential in Coalitional Congestion Games
aa r X i v : . [ c s . G T ] N ov Equilibrium and Potential in Coalitional CongestionGames by Sergey Kuniavsky and Rann Smorodinsky Abstract
The model of congestion games is widely used to analyze games related to traffic andcommunication. A central property of these games is that they are potential games andhence posses a pure Nash equilibrium. In reality it is often the case that some playerscooperatively decide on their joint action in order to maximize the coalition’s total utility.This is by modeled by Coalitional Congestion Games. Typical settings include truckdrivers who work for the same shipping company, or routers that belong to the same ISP.The formation of coalitions will typically imply that the resulting coalitional congestiongame will no longer posses a pure Nash equilibrium. In this paper we provide conditionsunder which such games are potential games and posses a pure Nash equilibrium.JEL classification: C72Key words: Congestion games, Equilibrium, Potential, Coalitions This work is based on Sergey Kuniavsky’s M.Sc thesis done under the supervision of Rann Smorodin-sky. Financial support by the Technion’s fund for the promotion of research and the Gordon Centerfor System Engineering is gratefully acknowledged. Valuable comments by an anonymous referee aregratefully acknowledged. Munich Graduate School of Economics, Kaulbachstr. 45, Munich 80539, Germany. Financialsupport from the Deutsche Forschungsgemeinschaft through GRK 801 is gratefully acknowledged. < [email protected]. > . Corresponding author: Faculty of Industrial Engineering and Management, Technion, Haifa 32000,Israel. < [email protected] > Introduction
Congestion games, introduced by Rosenthal [5], form a very natural model for studyingmany real-life strategic settings: Traffic problems, load balancing, routing, network plan-ning, facility locations and more. In a congestion game players must choose some subsetof resources from a given set of resources (e.g., a subset of edges leading from the Sourceto the Target on a graph). The congestion of a resource is a function of the number ofplayers choosing it and each player seeks to minimize his total congestion accross all cho-sen resources. In many modeling instances players and the decision making entity havebeen thought of as one and the same. However, in a variety of settings this may not bethe case.Consider, for example, a traffic routing game where each driver chooses his route inorder to minimize his travel time, while accounting for congestion along the route causedby other drivers. Now, in many cases drivers are actually employees in shipping firms,and in fact it is in the interest of the shipping firm to minimize the total travel timeof its fleet. Similarly, routers in a communication network participate in a congestiongame. However, as various routers may belong to the same ISP we are again in a settingwhere coalitions naturally form. This motivated Fotakis et al. [2] and Hayrapetyanet al. [3] to introduce the notion of Coalitional Congestion Games (CCG). In a CCGwe think about the coalitions as players and each coalition maximizes its total utility.The coalitional congestion game inherits its structure from the original game, once thecoalitions of players from the original congestion games (now, becoming the players of thecoalitional congestion game) have been identified.The most notable property of congestion games is that they have posses pure Nash equilib-rium. This has been shown by Rosenthal in [5]. Later, Monderer and Shapley [4] formallyintroduce potential games and show the equivalence of these two classes. The fact thatpotential games posses a pure Nash equilibrium is straightforward. Unfortunately, thestatement that a CCG is a potential game or that it possesses a pure Nash equilibriumis generally false. In this paper we investigate conditions under which this statement istrue. We focus on a subset of congestion games called simple congestion games, whereeach player is restricted to choose a single resource.Our main contributions are:1. Whenever each coalition contains at most two players the CCG induced from asimple congestion game possesses a pure-strategy Nash equilibrium (Theorem 1).2. If some coalition contains three players, then there may not exist a pure-strategyNash equilibrium (Example 1). 1. If a the congestion game is not simple then there may not exist a pure-strategy Nashequilibrium (Example 2); and4. Suppose there exists at least one singleton coalition and at least one coalition com-posed of two players, then a coalitional congestion game induced from a simplecongestion game is a potential game if and only if cost functions are linear (Theo-rem 2).Our results extend and complement the results in Fotakis et al. [2] and Hayrapetyan et al.[3]. For example, Fotakis et al. [2] show that if the resource cost functions are linear thenthe coalitional congestion game is a potential game. We show that, with some additionalmild conditions on the partition structure, this is also a necessary condition. Hayrapetyan[3] shows that if the underlying congestion game is simple and costs are weakly convexthen the game possesses a pure Nash equilibrium. We demonstrate additional settingswhere this holds.Section 2 provides a model of a coalitional congestions games, section 3 discusses theconditions for the existence of a pure Nash equilibrium in such games and section 4discusses the conditions for the existence of a potential function. All proofs are relegatedto an appendix.
Let G = { N, S, U } be a non-cooperative game in strategic form. Let C = { C , . . . , C n c } be a Partition of N into n c nonempty sets. Hence: ∪ n c k =1 C k = N and C k ∩ C l = ∅ ∀ k = l ∈ [1 , . . . n c ].The game G and the partition C form a Coalitional Non-Cooperative (CNC) Game G C = { N C , S c , U c } defined as follows: • N C is the set of agents which are the elements of C . • The strategy space is S c = { S ck } k ∈ C where S ck = × i ∈ C k S i .Note that × S k = × k ∈ C × i ∈ C k S k,i is isomorphic to S = × ni =1 S i , since we only changed theorder of the coordinates. Thus, we can look on s c as a vector in S . • The utility function is defined as follows: ∀ s c ∈ S c U ck ( s c ) = P i ∈ C k U i ( s c ) and U c = { U ck } k ∈ C . 2n the context of G C , G is the Underlying Game and a player in G is referred to as a subagent . As always, a Pure Nash Equilibrium of the game G C is a strategy profile s ∈ S C such that ∀ k ∈ N C , U ck ( s ) ≥ U ck ( s − k , t k ) ∀ t k ∈ S ck . N E ( G C ) the (possibly empty) set ofPure Nash equilibrium strategy profiles in G C .A congestion game is a game G = { N, R, Σ , P } where N is the finite set agents, R is thefinite set resources, P = { P r } r ∈ R are the resource costs functions, where P r : [1 , . . . , n ] → R and Σ = × i ∈ N Σ i , where Σ i ⊆ R , is i ’s strategy space. Agent i selects s i ∈ Σ i and pays P r ∈ s i P r ( c ( s ) r ), where c ( s ) r = P j ∈ N I { r ∈ s j } is the number of agents who select r in s . Inutility terms, the utility of agent i is U i ( s ) = − P r ∈ s i P r ( c ( s ) r ). If Σ i = R ∀ i ∈ N then G = { N, R, Σ , P } is called a Simple Congestion Game .We will assume that the P r functions are non-negative and increasing ( P r (1) can be zero).Fix a Simple Congestion Game or Coalitional Congestion Game (CCG) G with R resourcesand n (sub-)agents. A congestion vector is an element of N R whose elements sum up to n .A Simple Congestion Game (CCG) G and a strategy profile s induce a congestion vector c ( s ): { c ( s ) r } r ∈ R .A strategy profile s of a Coalitional Congestion Game G C induces a private congestionvector for each of the agents in G C . Such vector for agent k will be c k : c k ( s k ) r = |{ i ∈ C k : s k,i = r }| which is an element if N R whose elements sum up to | C k | .Let X be a subset of the strategy profiles space. We denote c ( X ) as the correspondingset of congestion vectors: ∀ X ⊆ S c ( X ) = { c ( s ) s.t : s ∈ X } The following preliminary result asserts that if the a Nash equilibrium the underlyinggame is composed of strategies such that all sub-agents of any agent choose differentresources, then it is also an equilibrium of the coalitional game.
Proposition 1
Let G be a Simple Congestion Game , C a partition of N and G C be theinduced CCG. Let s be a strategy profile of G C where s k,i = s k,j ∀ k ∈ N C , ∀ i, j ∈ C k . If c ( s ) ∈ c ( N E ( G )) then s ∈ N E ( G C ) . This is key to proving our central result about the existence of a Nash equilibrium inCCGs: 3 heorem 1
Let G be a Simple Congestion Game , C a partition where the largest elementis of size 2, and G C a CCG with the underlying game G and the partition C . Then N E ( G C ) = ∅ . That is, if the largest coalition is a Pair, a Pure Nash equilibrium alwaysexists. Does this existence result extend to other partition forms, where the maximal elementhas more than two sub-agents? The following example demonstrates that this is not truein general:
Example 1
Consider a game with two identical resources A and B and four sub agents,with the following payment functions:Resource / Agents C = [ { , , } , { } ] the matrix form of the resulting 2-player CCG is: G C A BA,A,A -54, -18 -48, 0A,A,B -32, -16 -36, -12A,B,B -36, -12 -32, -16B,B,B -48, 0 -54, -18Whereas the underlying Simple Congestion Game has a pure Nash equilibrium this CCGhas none. To verify this note for the compound agent (made up of 3 sub-agents) thestrategies
AAA and
BBB are dominated. Following their deletion the remaining game isone of matching pennies and hence has no Pure Nash equilibrium .
Can the result of Theorem 1 be extended to CCGs with small coalition size, but with anunderlying congestion game that is not simple? Again, the answer is negative:
Example 2
Let G be a Congestion Game with three identical resources and three agents.Each agent of G chooses two of the three resources. The cost of each resource is P ( n ) =6 − n .Let C = [ { , }{ } ] , and G C is the CCG with underlying game G and partition C . Afteromitting identical strategies due to sub agents symmetry G C looks as follows: C AB AC BCAB,AB -16,-8 -14,8 -14,-4AC,AC -14,-4 -16,4 -14,-4BC,BC -14,-4 -14,-4 -16,-8AB,AC -11,-7 -11,-7 -12,-6AB,BC -11,-7 -12,-6 -11,-7AC,BC -12,-6 -11,-7 -11,-7Note that compound agent’s strategies (AB,AB), (AC,AC) and (BC,BC) are dominated.Note that the remaining game has no pure Nash equilibrium. An Exact Potential is a function P : S → R satisfying: P ( s ) − P ( s − i , t i ) = U i ( s ) − U i ( s − i , t i ) (1) ∀ i ∈ N, ∀ t i ∈ S i , ∀ s ∈ S × S . . . × S n Games with a potential function are called
Potential Games . It is well known that poten-tial games have a pure Nash equilibrium (see Monderer and Shapley [4]). In particularCongestion Games are potential games. Fotakis et al [2] prove that a CCG, where thecost functions of the resources of the underlying game are linear, is a potential game.Removing the linearity assumption is problematic. In fact, even in the case of a CCG witha maximal coalition of size 2, which guarantees the existence of a pure Nash equilibrium(Theorem 1), the existence of Exact Potential is not guaranteed. In the following examplewe show that the existence of a potential function implies linearity of the cost functions:
Example 3
Consider a congestion games with 2 resources, A and B , with costs a , a , a and b , b , b , correspondingly. Assume there are 3 players and set the coalition structureto C = [ { , }{ } ] . This induces the following two player CCG, given in matrix form: G C A BA,A a , a a , b A,B a + b , a a + b , b B,B b , a b , b Assume this game has an exact potential with the following values: C A BA,A P P A,B P P B,B P P From the definition of exact potential the following must hold (see, in addition, Theorem2.9. in Monderer and Shapley [4]): ( a + b − a ) + ( a − b ) + (2 a − a − b ) + ( b − a ) =( P − P ) + ( P − P ) + ( P − P ) + ( P − P ) = 0 . Similarly: a + b − b + a − b + 2 b − a − b + b − a = 02 a − b + a − b + 2 b − a + b − a = 0 . Manipulating these equalities leads to: a = a + a b = b + b , which implies that the cost functions are linear. Using this example we can now prove our final result:
Theorem 2
Let G be a Simple Congestion Game and let C be a partition that has atleast one element of size 1 and at least one element of size 2. Let G C be a CCG with theunderlying game G and partition C . G C will posses an Exact Potential iff the CCG islinear. Proof:
Sufficiency - This has been obtained Fotakis et al. [2] (Theorem 6).Necessity - Recall that example 3 provides a 3 player congestion game and a coalitionstructure that yields a CCG for which linear cost function are necessary for the existenceof a potential. The reason that the the linearity extends beyond the example to all6ituations implied in the theorem is that for any general CCG we can fix the strategyfor all but 2 agents, of which one has 2 sub agents and one has a single sub agent. Wecan now look at the induced 2 player game. If the original game was a potential gameso must be the induced game. The example then implies linearity in the induced game.However, as we can arbitrarily fix the strategy for all but the relevant 3 sub agents theresult follows. (cid:4)
A Appendix - Omitted Proofs
We begin with the definition of an auxiliary game. Let G be a Simple Congestion Gameand let C be a partition. The Restricted Coalitional Congestion Game , denoted G C ,is a CCG where coalitions are restricted strategies such that distinct sub-agents choosedistinct resources. More formally: Definition 1
Let G be a Simple Congestion Game and let C be a partition. The Re-stricted Coalitional Congestion Game , denoted G C , is the game G C = { N c , S c , U c } , where N c and U c are as before and S c = { S ck } k ∈ C where S ck = {× i ∈ C k S k,i : s k,i = s k,j ∀ i, j ∈ C k } . The following result about pure Nash equilibria in restricted coalitional congestion gameswill be useful for proving our main result:
Lemma A.1
Let G be a Simple Congestion Game and C a partition of N . Let G C be aRestricted CCG with the underlying game G and the partition C . Let s be a strategy profileof G C (where s k,i = s k,j ∀ k ∈ N C , ∀ i, j ∈ C k ). If c ( s ) ∈ c ( N E ( G )) ⇒ s ∈ N E ( G C ) Proof:
For two congestion vectors u, v , let d ( u, v ) = P r ∈ R | v r − u r | , denote the distance betweenthese two vectors.Let s be a strategy profile satisfying the condition of the lemma, let k be an arbitraryagent in the game G C and denote by s − k be the strategy profile of all players except k . We denote by BR ( s − k ) the set of k ’s best reply strategies to s − k . Assume , by wayof contradiction, that s k BR ( s − k ) and let t k ∈ BR ( s − k ) be a best reply to s − k whichcorresponding congestion vector has a minimal distance to c ( s k ). Namely, d ( c ( t k ) , c ( s k )) ≤ d ( c ( t ′ k ) , c ( s k )) ∀ t ′ k ∈ BR ( s − k ).In G C each agent selects each resource at most once. As c k ( t k ) = c k ( s k ) this impliesthat there are two resources, say r and x , such that: ( c k ( t k ) r , c k ( t k ) x ) = (1 ,
0) and7 c k ( s k ) r , c k ( s k ) x ) = (0 ,
1) and, in addition, that c ( s ) r = c ( t k , s − k ) r − c ( s ) x = c ( t k , s − k ) x + 1.Let i be the sub-agent of k that chooses r in t k but chooses a resource different than r in s k . Let t ′ k be a strategy for agent k , derived from t k by moving sub agent i from r to x .This results in c ( t ′ k , s − k ) x = c ( s ) x and c ( t ′ k , s − k ) r = c ( s ) r . By assumption, c ( s ) ∈ N E ( G ).Therefore P x ( c ( s ) x ) ≤ P r ( c ( s ) r + 1) and so: P x ( c ( t ′ k , s − k ) x ) = P x ( c ( s ) x ) ≤ P r ( c ( s ) r + 1) = P r ( c ( t k , s − k ) r ) (2)Thus, the contribution of sub agent i to agent k ’s payment in ( t ′ k , s − k ) is less or equal itscontribution to k ’s payment in ( t k , s − k ). As the only change in k ’s strategy between t k and t ′ k is i ’s choice, we conclude that k ’s payment in ( t ′ k , s − k ) is less or equal its payment in( t k , s − k ), and so t ′ k ∈ BR ( s − k ). However, by construction d ( c ( t ′ k ) , c ( s k )) ≤ d ( c ( t k ) , c ( s k )),contradicting the way t k was chosen. ✷ . Proof of Proposition 1 :Let s be a profile as described in the Proposition. Let t k be the best reply strategy foragent k to s − k . We show that c k ( t k ) r ≤ ∀ k ∈ N C and ∀ r ∈ R .Assume this is not true and that some resource r , t k,i = t k,j = r . Since c k ( s k ) r ≤ c ( s − k , t k ) r > c ( s ) r . Therefore there must exist some resource x such that c ( s − k , t k ) x < c ( s ) x .Let t ′ k be a strategy profile derived from t k by moving sub agent i from r to x . In thestrategy profile ( s − k , t ′ k ) agent i pays P x ( c ( s − k , t ′ k ) x ). By construction: c ( s − k , t ′ k ) x = c ( s − k , t k ) x + 1 ≤ c ( s ) x (3)From Equation 3 and the fact that c ( s ) ∈ N E ( G ) we get that: P x ( c ( s − k , t ′ k ) x ) = P x ( c ( s − k , t k ) x + 1) ≤ P x ( c ( s x ) x ) ≤ P r ( c ( s ) r + 1) (4)Using monotonicity of the cost functions and the fact that c ( s ) r + 1 ≤ c ( s − k , t k ) r we get: P r ( c ( s ) r + 1) ≤ P r ( c ( s − k , t k ) r ) (5)Combining Equations 4 and 5 we get that P x ( c ( s − k , t ′ k ) x ) ≤ P r ( c ( s − k , t k ) r ) (6)8e will now show that k is better off in the strategy profile ( s − k , t ′ k ) than in ( s − k , t k ),thus contradicting the fact that t k is a best response to s − k . We do this but analyzingeach of k ’s sub-agents: • Sub agent i pays in ( s − k , t ′ k ), where he chose x , no more than than in ( s − k , t k ), wherehe chose r (equation 6). • From definition of x , c k ( t k ) x < c k ( s k ) x . Since c k ( s k ) x = 1 and c k ( s k ) x > c k ( t k ) x , weget that c k ( t k ) x = 0. Thus, apart from agent i no other sub agent of k chose x in t k . • Sub agent j , who selects r both in ( s − k , t ′ k ) and ( s − k , t k ), pays strictly less in ( s − k , t ′ k )than in ( s − k , t k ), because c ( s − k , t ′ k ) r < c ( s − k , t k ) r . This inequality holds for any othersub agent of k who chose r in t k • All sub agents who choose a resource in the set R \ { r, x } pay the same in ( s − k , t k )and ( s − k , t ′ k ).To conclude, agent k pays strictly less in ( s − k , t ′ k ) than in ( s − k , t k ). This contradicts thefact that t k is a best reply to s − k . Therefore, any best reply of k to s − k must be such thatall the sub-agents choose different resources.Thus, agent k ’s best reply strategy to s − k is a strategy that is allowed also in G C . Wecouple this observation with the observation that s is a Nash equilibrium of G C (followsfrom Lemma A.1) and the fact that k is arbitrary to conclude that s ∈ N E ( G C ).QED Proof of Theorem 1:
Let s be an arbitrary Nash equilibrium of G Case 1 - Assume that c ( s ) r ≤ | N C | for all resources r ∈ R . In this case we can re-arrange the players over the resources so that the result will be a strategy profile with thesame congestion vector, and furthermore, for any k ∈ N C its two sub agents, i, j , choosedifferent resources. The resulting vector is also in N E ( G ) and complies with the conditionsin Proposition 1. Therefore that proposition suggests that s is a Nash equilibrium of G C .Case 2 - Let us denote by r the resource with the highest congestion in s and assume c ( s ) r > | N C | . We argue that without loss of generality (by rearranging the players) s hasthe following two properties: (a) if agent k has its 2 sub agents on the same resource thenit must be the case that the corresponding resource is r , that is, ∀ k ∈ N C c ( s k ) r ′ > r ′ = r ; and (b) all agents have at least one sub-agent choose r .Note some properties of the strategy tuple s :9. Let k be an agent with a single sub agent. Then this sub agent must be on r and ithas no profitable deviation.2. Let k be an agent with a two sub agents, i and j . Assume i is on r and j is on some r ′ = r . Moving a single subagent cannot be profitable.3. Let k be an agent with two sub-agents, i and j . Assume i is on r and j is on some r ′ = r . Moving both sub agents simultaneously cannot be profitable as at least oneof these moves makes k worse off, while the other cannot improve k payoff.4. Let k be an agent with two sub-agents, i and j , both on r . Moving both sub agentscannot be profitable.Thus, if s is not a NE of G c , the only profitable deviation possible is for an agent k withtwo sub-agents, i and j , both on r , to move one sub-agent, say j to another resource, say r ′ . Furthermore, let us assume that this is the most profitable deviation for k . That is P r ′ ( c ( s r ′ ) + 1) ≤ P r ′′ ( c ( s r ′′ ) + 1) for all r ′′ = r . We denote the resulting strategy profileby s ′ . Note the properties of s ′ :1. All agents with a single sub-agent choose the resource r .2. All agents with two sub agents, have at lease one sub agent in r .3. The payment of all sub agents in r is lower compared with s , while the payment ofall subagents in R \ { r } is at least as large compared with the payment in s .4. Any agent with two sub agents, one on r and one on some r ′′ = r pays (weakly) lessthan what k paid in s Assume s ′ is not a Nash equilibrium, then there must be some profitable deviation. Whatare the possible profitable deviations?1. Let k ′ be an agent with a single sub agent. Then this sub agent must be on r andit has no profitable deviation. Recall that the payment of k is s ′ is lower than thepayment of k ′ is s .2. Let k ′ be an agent with a two sub agents, i and j . Assume i is on r and j ison some r ′ = r . Moving i cannot be profitable for the same argument as above.Moving j to another resource in R \ { r } cannot be profitable, so the only profitabledeviation might be moving j back to r . However, this would result k ′ paying thesame payment that k paid in s , which by construction of s ′ is higher than what k pays in s ′ . Implying that k ′ had a profitable deviation in s , thus contradicting whatwe already know. 10. Let k be an agent with two sub-agents, i and j . Assume i is on r and j is on some r ′ = r . Moving both sub agents simultaneously cannot be profitable as at least oneof these moves makes k worse off, while the other cannot improve k payoff.4. Let k be an agent with two sub-agents, i and j , both on r . Moving both sub agentscannot be profitable.Once again, the only profitable deviation possible is for an agent k ′ with two sub-agentsboth on r , to move one sub-agent. The resulting strategy profile s ′′ , once more, only allowsfor profitable deviations of the same form. Namely, for an agent k ′ with two sub-agentsboth on r , to move one sub-agent. We continue iteratively in the same manner. As theprocess is bounded by the number of agents selecting r with both sub agents in s , it mustend in finitely many steps. The final strategy vector has no profitable deviations and is,therefore, a Nash equilibrium of the game.QED References [1] Fotakis D., Kontogiannis S., Spiraklis P. “Symmetry in Network CongestionGames: Pure equilibria and Anarchy Cost,”
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