Network Congestion Games Evdokia Nikolova Austin Houston Dallas College Station TX Assistant Professor Texas A&M University College Station, TX

Best route depends on others Evdokia Nikolova Network Congestion Games

Travel time increases with congestion Highway congestion costs were $115 billion in 2009. Avg. commuter travels 100 minutes a day. Evdokia Nikolova Network Congestion Games

Example: Inefficiency of equilibria Delay is 1.5 hours for everybody at the unique Nash equilibrium 1/2 x hours 1 hour Town A Town B 1/2 Suppose drivers (total 1 unit of flow) leave from town A towards town B. (Ex): Verify that the 50-50 split is the unique Nash equilibrium of the system shown above. Every driver wants to minimize her own travel time. What is the traffic on the network? In any unbalanced traffic pattern, all drivers on the most loaded path have incentive to switch their path. Evdokia Nikolova Network Congestion Games

Example: Inefficiency of equilibria Delay is 2 hours for everybody at the unique Nash equilibrium 1 x hours 1 hour Town A Town B 0 hours A benevolent mayor builds a superhighway connecting the fast highways of the network. What is now the traffic on the network? No matter what the other drivers are doing it is always better for me to follow the zig-zag path. Evdokia Nikolova Network Congestion Games

Example: Inefficiency of equilibria 1/2 A B 1 x hours 1 hour x hours 1 hour vs Adding a fast road on a road-network is not always a good idea! Braess’s paradox In the RHS network there exists a traffic pattern where all players have delay 1.5 hours. Price of Anarchy: measures the loss in system performance due to free-will Evdokia Nikolova Network Congestion Games

Network Congestion Games Game model Directed graph G = (V,E) Multiple source-dest. pairs (sk,tk), demand dk Players (users): nonatomic (infinitesimally small) Strategy set: paths Pk between (sk,tk) for all k Players’ decisions: flow vector Sometimes will use for path flow. Edge delay (latency) functions: typically assumed continuous and nondecreasing. Evdokia Nikolova Network Congestion Games

Network Congestion Games Outline Wardrop Equilibrium Social Optimum Price of Anarchy Evdokia Nikolova Network Congestion Games

Network Congestion Games Outline Wardrop Equilibrium Social Optimum Price of Anarchy Evdokia Nikolova Network Congestion Games

Wardrop’s First Principle “Travel times on used routes are equal and no greater than travel times on unused routes.” Definition: A flow x is a Wardrop Equilibrium (WE) if for every source-dest. pair k and for every path with positive flow between this pair, where Also called User Equilibrium or Nash Equilibrium. Equilibrium flow is called Nash flow. All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems Evdokia Nikolova Network Congestion Games

Network Congestion Games Outline Wardrop Equilibrium Social Optimum Price of Anarchy Evdokia Nikolova Network Congestion Games

Wardrop’s Second Principle “The average [total] journey time is minimum.” The cost of flow x is defined as the “total journey time”: Denote , assumed convex. All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems Evdokia Nikolova Network Congestion Games

Wardrop’s Second Principle “The average [total] journey time is minimum.” Definition: A flow x is a Social Optimum if it minimizes total delay: flow constraints All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems Evdokia Nikolova Network Congestion Games

Network Congestion Games Social Optimum “The average [total] journey time is minimum.” Definition: A flow x is a Social Optimum if it minimizes total delay: All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems Evdokia Nikolova Network Congestion Games

Network Congestion Games Social Optimum Definition: A flow x is a Social Optimum if it solves Lemma: A flow vector x is locally optimal if for each path p with positive flow and each path p’, where All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems Proof sketch: marginal benefit of marginal cost of reducing traffic on p increasing traffic on p’ Evdokia Nikolova Network Congestion Games

Network Congestion Games Social Optimum (SO) Definition: A flow x is a Social Optimum if it solves Lemma: A flow vector x is locally optimal if for each path p with positive flow and each path p’, Corollary 1: If costs are convex, local opt is a global opt, and lemma gives equivalent defn of Social Optimum. All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems Evdokia Nikolova Network Congestion Games

Network Congestion Games Social Optimum (SO) Definition: A flow x is a Social Optimum if it solves Lemma: A flow vector x is locally optimal if for each path p with positive flow and each path p’, Corollary 2: If costs are convex, SO is an equilibrium with respect to modified latencies All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems Evdokia Nikolova Network Congestion Games

Network Congestion Games Social Optimum (SO) Mechanism Design Interpretation: If users value time and money equally, imposing tolls per unit flow on each edge will cause selfish players to reach the Social Optimum! Definition: A flow x is a Social Optimum if it solves Lemma: A flow vector x is locally optimal if for each path p with positive flow and each path p’, Corollary 2: If costs are convex, SO is an equilibrium with respect to modified latencies All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems Evdokia Nikolova Network Congestion Games

Computing Social Optimum (SO) Definition: A flow x is a Social Optimum if it solves Corollary 3: If costs are convex, SO exists and can be found efficiently by solving convex program above. All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems Evdokia Nikolova Network Congestion Games

Network Congestion Games Outline Revisit Wardrop Equilibrium Social Optimum Price of Anarchy Evdokia Nikolova Network Congestion Games

Equilibrium existence WE Definition: A flow x is a Wardrop Equilibrium if for every source-dest. pair k and for every path with positive flow between this pair, compare with: SO Definition: A flow vector x is a Social Optimum for every source-dest. pair k and for every path with positive flow between this pair, All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems Evdokia Nikolova Network Congestion Games

Equilibrium existence WE Definition: A flow x is a Wardrop Equilibrium if for every source-dest. pair k and for every path with positive flow between this pair, where SO Definition: A flow vector x is a Social Optimum for every source-dest. pair k and for every path with positive flow between this pair, All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems Evdokia Nikolova Network Congestion Games

Equilibrium existence WE Definition: A flow x is a Wardrop Equilibrium if for every source-dest. pair k and for every path with positive flow between this pair, where Alternative SO Definition: A flow vector x is a Social Optimum if it solves: All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems Evdokia Nikolova Network Congestion Games

Equilibrium existence Alternative WE Definition: A flow vector x is a Wardrop Equilibrium if it solves: where Alternative SO Definition: A flow vector x is a Social Optimum if it solves: All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems Evdokia Nikolova Network Congestion Games

Equilibrium existence Alternative WE Definition: A flow vector x is a Wardrop Equilibrium if it solves: where Alternative SO Definition: A flow vector x is a Social Optimum if it solves: All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems Evdokia Nikolova Network Congestion Games

Equilibrium existence Alternative WE Definition: A flow vector x is a Wardrop Equilibrium if it solves: Theorem: A Wardrop Equilibrium exists and can be computed in polynomial time. Also, if program above is strictly convex, equilibrium is unique, up to same flow cost. All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems Evdokia Nikolova Network Congestion Games

Network Congestion Games Outline Wardrop Equilibrium Social Optimum Price of Anarchy Evdokia Nikolova Network Congestion Games

Example: Inefficiency of equilibria Delay is 1.5 hours for everybody at the unique Nash equilibrium 1/2 x hours 1 hour Town A Town B 1/2 Suppose drivers (total 1 unit of flow) leave from town A towards town B. (Ex): Verify that the 50-50 split is the unique Nash equilibrium of the system shown above. Every driver wants to minimize her own travel time. What is the traffic on the network? In any unbalanced traffic pattern, all drivers on the most loaded path have incentive to switch their path. Evdokia Nikolova Network Congestion Games

Example: Inefficiency of equilibria Delay is 2 hours for everybody at the unique Nash equilibrium 1 x hours 1 hour Town A Town B 0 hours A benevolent mayor builds a superhighway connecting the fast highways of the network. What is now the traffic on the network? No matter what the other drivers are doing it is always better for me to follow the zig-zag path. Evdokia Nikolova Network Congestion Games

Example: Inefficiency of equilibria 1/2 A B 1 x hours 1 hour x hours 1 hour vs Adding a fast road on a road-network is not always a good idea! Braess’s paradox In the RHS network there exists a traffic pattern where all players have delay 1.5 hours. Price of Anarchy: measures the loss in system performance due to free-will Evdokia Nikolova Network Congestion Games

Network Congestion Games Price of Anarchy Cost of Flow: total user cost Social optimum: flow minimizing total user cost Price of anarchy: (Koutsoupias, Papadimitriou ’99) One remarkable new use of game theory: equilibrium is best known approximation for a purely optimization problem All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems Max Prob(both players ontime) inapproximable since NP-hard to decide if there are 2 paths < t each. Evdokia Nikolova Network Congestion Games

Variational Inequality representation of equilibria Theorem: Equilibria in nonatomic games are solutions to the Variational Inequality (VI) where VI Solution exists over compact convex set with ℓ(x) continuous [Hartman, Stampacchia ‘66]. ∎ VI Solution unique if ℓ(x) is monotone: (ℓ(x)-ℓ (x’))(x-x’) ≥ 0. [Exercise: verify] Proof: Flow x is an equilibrium if and only if ℓ(x).x <= ℓ(x).x’ . Proof: (=>) Equilibrium flow routes along minimum-cost paths ℓ(x). Fixing path costs at ℓ(x), any other flow x’ that assigns flow to higher-cost paths will result in higher overall cost ℓ(x).x’. (<=) Suppose x is not an eq. Then there is a flow-carrying path p with ℓp(x) > ℓp’(x). Shifting flow from p to p’ will obtain flow x’ with ℓ(x).x’ < ℓ(x).x, contradiction. One remarkable new use of game theory: equilibrium is best known approximation for a purely optimization problem All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems Max Prob(both players ontime) inapproximable since NP-hard to decide if there are 2 paths < t each. Evdokia Nikolova Network Congestion Games

Price of Anarchy with linear latencies Theorem**: The price of anarchy (PoA) is 4/3 in general graphs and latencies i.e. where x is WE and x* is SO flow. Pf: One remarkable new use of game theory: equilibrium is best known approximation for a purely optimization problem All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems Max Prob(both players ontime) inapproximable since NP-hard to decide if there are 2 paths < t each. Evdokia Nikolova Network Congestion Games **References: Roughgarden, Tardos ’02; Correa, Schulz, Stier-Moses ‘04, ‘08

Network Congestion Games Take-away points Equilibrium and Social Optimum in nonatomic routing games exist and can be found efficiently via convex programs. Social optimum is an equilibrium with respect to modified latencies = original latencies plus toll. Price of anarchy: 4/3 for linear latencies, can be found similarly for more general classes of latency functions. One remarkable new use of game theory: equilibrium is best known approximation for a purely optimization problem All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems Max Prob(both players ontime) inapproximable since NP-hard to decide if there are 2 paths < t each. Evdokia Nikolova Network Congestion Games

Network Congestion Games References Wardrop ‘52, Beckmann et al. ’56, … A lot of work in AGT community and others. Surveys of recent work: AGT Book Nisan et al. ‘07 Correa, Stier-Moses ’11 One remarkable new use of game theory: equilibrium is best known approximation for a purely optimization problem All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems Max Prob(both players ontime) inapproximable since NP-hard to decide if there are 2 paths < t each. Evdokia Nikolova Network Congestion Games

Network Congestion Games Some open questions What is the price of anarchy with respect to other Social Cost functions? Dynamic (time-changing) latency functions? Uncertain delays? One remarkable new use of game theory: equilibrium is best known approximation for a purely optimization problem All these are hard computational problems that would require new methods, insights for understanding the properties and structure of equilibria and the associated global optimization problems Max Prob(both players ontime) inapproximable since NP-hard to decide if there are 2 paths < t each. Evdokia Nikolova Network Congestion Games