CMU 15-896 Noncooperative games 3: Price of anarchy Teacher: Ariel Procaccia
Back to prison • The only Nash equilibrium in Prisoner’s Cooperate Defect dilemma is bad; but how bad is it? -1,-1 -9,0 Cooperate • Objective function: social cost sum of costs 0,-9 -6,-6 Defect • NE is six times worse than the optimum 15896 Spring 2016: Lecture 19 2
Anarchy and stability • Fix a class of games, an objective function, and an equilibrium concept • The price of anarchy (stability) is the worst-case ratio between the worst (best) objective function value of an equilibrium of the game, and that of the optimal solution • In this lecture: Objective function � social cost o Equilibrium concept � Nash equilibrium o 15896 Spring 2016: Lecture 19 3
Example: Cost sharing players in weighted directed • graph � � � � � � � � 1 1 • Player wants to get from � to � ; strategy space is � � paths • Each edge has cost � 10 10 10 • Cost of edge is split between all 1 1 players using edge � � � � � � � � • Cost of player is sum of costs over edges on path 15896 Spring 2016: Lecture 19 4
Example: Cost sharing • With players, the example � on the right has an NE with social cost • Optimal social cost is � 1 Price of anarchy • Prove that the price of � anarchy is at most 15896 Spring 2016: Lecture 19 5
Example: Cost sharing • Think of the edges as cars, and the edge as mass transit • Bad Nash equilibrium with 0 0 0 cost … � � � � � � � � � � � � � • Good Nash equilibrium with cost 1 1 1 • Now let’s modify the example… � 15896 Spring 2016: Lecture 19 6
Example: Cost sharing • OPT • Only equilibrium has cost 0 0 0 price of stability is at • … � � � � � � � � � � � � � � 1 least • We will show that the price � � � of stability is 1 2 � � 15896 Spring 2016: Lecture 19 7
Potential games • A game is an exact potential game if there � exists a function such that � ��� � for all , for all , and for all � ��� � � , � � � � �� � �� � � Why does the existence of an exact potential function imply the existence of a pure Nash equilibrium? 15896 Spring 2016: Lecture 19 8
Potential games • Theorem: the cost sharing game is an exact potential game • Proof: Let � � � be the number of players using � under � o Define the potential function o � � ��� Φ � � � � � � � � ��� � � If player changes paths, pays � � � �� for each new o � � � � � for each old edge, so Δcost � � ΔΦ ∎ edge, gets 15896 Spring 2016: Lecture 19 9
Potential games • Theorem: The cost of stability of cost sharing games is • Proof: It holds that o cost Take a strategy profile that minimizes o is an NE o cost OPT o 15896 Spring 2016: Lecture 19 10
Cost sharing summary • In every cost sharing game NE , o NE such that o • There exist cost sharing games s.t. NE such that o NE , o 15896 Spring 2016: Lecture 19 11
Congestion games • Generalization of cost sharing games players and resources • • Each player chooses a set of resources (e.g., a path) from collection � of allowable sets of resources (e.g., paths from � to � ) • Cost of resource is a function � of the � number � of players using it • Cost of player is the sum over used resources 15896 Spring 2016: Lecture 19 12
Congestion games • Theorem [Rosenthal 1973]: Every congestion game is an exact potential game • Proof: The exact potential function is � � ��� Φ � � � � � � � � ��� • Theorem [Monderer and Shapley 1996]: Every potential game is isomorphic to a congestion game 15896 Spring 2016: Lecture 19 13
Network formation games • Each player is a vertex • Strategy of : set of undirected edges to build that touch • Strategy profile induces undirected graph • Cost of building any edge is , where � • � � � edges bought by , is shortest path in edges • ��� 15896 Spring 2016: Lecture 19 14
Example: Network formation • NE with Suboptimal Optimal 15896 Spring 2016: Lecture 19 15
Example: Network formation • Lemma: If then any star is optimal, and if then a complete graph is optimal • Proof: Suppose � � 2 , and consider any graph that is not o complete Adding an edge will decrease the sum of distances by o at least 2 , and costs only � Suppose � � 2 and the graph contains a star, so the o diameter is at most 2 ; deleting a non-star edge increases the sum of distances by at most 2 , and saves � ∎ 15896 Spring 2016: Lecture 19 16
Example: Network formation Poll: For which values of is any star a NE, and for which is the complete graph a NE? none none 15896 Spring 2016: Lecture 19 17
Example: Network formation • Theorem: If � � 2 or � � 1 , PoS � 1 1. For 1 � � � 2 , PoS � 4/3 2. • Proof: Part 1 is immediate from the lemma and poll o For 1 � � � 2 , the star is a NE, while OPT is a o complete graph Worst case ratio when � → 1 : o � � � 1 � ��� � 1�/2 � 4� � � 6� � 2 2� � � 1 � �� � 1� � 4 3 ∎ 3� � � 3� 15896 Spring 2016: Lecture 19 18
Example: Network creation • Theorem [Fabrikant et al. 2003]: The price of anarcy of network creation games is • Lemma: If is a Nash equilibrium that induces a graph of diameter , then 15896 Spring 2016: Lecture 19 19
Proof of lemma � • Buying a connected graph costs at least o � distances There are o � • Distance costs focus on edge costs • There are at most cut edges focus on noncut edges 15896 Spring 2016: Lecture 19 20
Proof of lemma • Claim: Let � � ��, �� be a noncut edge, then the distance ���, �� with � deleted � 2� � � � set of nodes s.t. the shortest path from � uses � o Figure shows shortest path avoiding � , � � � �� � , � � � is o the edge on the path entering � � � is the shortest path from � to � � ⇒ � � � � � o � ∪ � is shortest path from � to � � ∎ � � � � � 1 as � o � � � �′ �′ � � � �′ � � � �′ �′ 15896 Spring 2016: Lecture 19 21
Proof of lemma • Claim: There are noncut edges paid for by any vertex Let be an edge paid for by o By previous claim, deleting increases o distances from by at most � is an equilibrium � � o vertices overall can’t be more than o sets � 15896 Spring 2016: Lecture 19 22
Proof of lemma noncut edges per vertex • total payment for these per vertex • � overall • 15896 Spring 2016: Lecture 19 23
Proof of theorem • By lemma, it is enough to show that the diameter at a NE • Suppose for some • By adding the edge , pays and improves distance to second half of the shortest path by � • If , it is beneficial to add edge 15896 Spring 2016: Lecture 19 24
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