Action Graph Games ( Albert Xin Jiang, Kevin Leyton-Brown, Navin A.R. Bhat)

Action Graph Games(Albert Xin Jiang, Kevin Leyton-Brown, Navin A.R. Bhat) Presented By:Xuan ChooCheriton School of Computer ScienceUniversity of WaterlooSept 22, 2008

Outline • Game Representations • Action Graph Games • Action Graph Games with Function Nodes • Computing Equilibria • Experimental Results • Conclusion and Final Thoughts

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Game Representations • Normal Form Game • Extensive Form Game • Multi-Agent Influence Diagrams • Graphical Games • Congestion Games

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Normal Form & Extensive Form • General representations • But, the representation size grows exponentially with the number of agents

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Graphical Games • Able to any game that has a normal form representation • Compact • Computation can be done that depends on the size of the representation rather than the size of the induced normal form • But, does not take advantage of anonymity • Agent’s utility depends only on the number of agents who took each action, rather than the identity of these agents. If there is time:

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Congestion Games • Able to take advantage of anonymity, symmetry, and context-specific independencies • They always have a pure-strategy equilibria • But, it cannot represent all games • Some games do not have a pure-strategy equilibria If there is time:

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Action Graph Games • Combines advantages of graphical games and congestion games • Able to represent any game • Compact • Takes advantage of anonymity, symmetry, and context-specific independencies • It can also compactly represent many games that are neither compact as graphical games or congestion games

Game RepresentationsAction Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Action Graphs • What is an Action Graph? Definition: An action graph G = (A, E) is a directed graph where: • A is a set of nodes, and each node is a distinct action • E is a set of directed edges, which represents the relationship between the actions.

Game RepresentationsAction Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Action Graph Games • What is an Action Graph Game? Definition: An action graph game is a tuple (N, A, G, u) where: • N is the set of agents • A is a set of action profiles (a set of actions for each agent) • G is an action graph • u is a tuple (uα)αA, where each uα is utilityfunction for action α

Game RepresentationsAction Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Example – Ice Cream Vendors • There are 4 locations at the beach • There are n ice cream vendors • 3 kinds of vendors: • Sells only Vanilla ice cream • Sells only Chocolate ice cream • Sells both but only on the west side • Vendors are negatively affected by other vendors selling the same flavours in neighbouring or same locations • Vendors are positively affected by other vendors selling different flavours in neighbouring or same locations

Game RepresentationsAction Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Example – Ice Cream Vendors C1 C2 C3 C4 Ac AW V1 V2 V3 V4 AV

Game RepresentationsAction Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Context-Specific Independencies C1 C2 C3 Ac AW V1 V2 V3 AV

Game RepresentationsAction Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Representation Size • From the definition, to completely specify an AGG, you need to specify the set of agents, each agent’s set of actions, the action graph, the utility functions • Set of agents: • N = {1, ... ,n}  can be specified by the integer n

Game RepresentationsAction Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Representation Size • Each agent’s set of actions: • The set of all actions A can be specified by |A| • Therefore, each agent’s set can be specified in O(|A|) space. • The action graph: • Can be represented by a list of neighbours • Space required is bounded by:|A|I where I is the maximum number of neighbours any action can have

Game RepresentationsAction Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Representation Size • The utility function • Theorem: If I is bounded as n increases, then the number of payoff values stored by the utility functions is in O(|A|nI) • Theorem: The number of payoff values stored in an AGG is always less than or equal to the number of payoff values in the induced normal form representation • The size of an AGG representation is determined by the size of the payoff values

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts That’s It? • What? That’s it? That’s all you need to represent ALL games?

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Consider This Game: • Simple network routing game • There are two types of agents • One is charged $0.10 / sec of delay • The other is charged $1.00 / sec of delay • There are two paths to take • One route costs $0 • The other costs $1 • Paths are affected by number of agents using it $1.00/s delay $0 SRC DEST $0.10/s delay $1

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Simple Network Routing Game $0.10/s delay $0 $1 How to represent delay due to path usage? $1.00/s delay

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Definition • An action graph game with function nodes is a tuple (N, A, P, G, f, u) where: • N is the set of agents • A is a set of action profiles • P is a finite set of function nodes • G = (A U P, E)is an action graph • f is a tuple (fp)pP, where each fp is an arbitrary mapping from neighbours of p to real numbers • u is a tuple (uα)αA, where each uα is utilityfunction for action α

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Example – Network Routing Game $0.10/s delay $0 $1 $1.00/s delay

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Example – Network Routing Game $0.10/s delay $0 $1 Function Node:Used to represent the number of agents using the route. $1.00/s delay

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Representation Size • We have seen the sizes for N, and A • We can apply the arguments for A for P as well • The action graph: • The graph now contains extra function nodes, so the space complexity becomes: O((|A| + |P|)2) • The utility function: • The size representation remains the same as the induced AGG

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Representation Size • The functions fp: • In the worst case: same order as the utility function • However, the functions can often be defined such that the representations take up a negligible amount of space

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Representation Size • So this means that the representation size of an AGGFN is the same as the representation size of the induced AGG • In fact, the use of function nodes can reduce the representation size! • See the coffee shop game example in the paper

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Nash Equilibrium • Complexity for finding the Nash equilibrium for an AGG? • PPAD-Complete! • Theorem: Finding a Nash equilibrium in an n-player normal-form game is PPAD-complete for n ≥ 2

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Nash Equilibrium • Theorem: The problem of finding a Nash equilibrium for an AGG can be reduced to finding a Nash equilibrium in a two-player normal form game with the size polynomial in the size of the AGG •  This follows that the problem of finding a Nash equilibrium for an AGG is also PPAD-complete

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Nash Equilibrium • What’s the significance? • Consider this: • Instead of finding a Nash equilibrium for an n-player game, we are instead finding a Nash equilibrium for a 2-player game in the size of the AGG.

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Representation Size • Theorem: If I is bounded as n increases, then the number of payoff values stored by the utility functions is in O(|A|nI) • Theorem: The number of payoff values stored in an AGG is always less than or equal to the number of payoff values in the induced normal form representation

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Nash Equilibrium • What’s the significance? • Consider this: • Instead of finding a Nash equilibrium for an n-player game, we are instead finding a Nash equilibrium for a 2-player game in the size of the AGG. • This means that the complexity is be PPAD-complete, but may be exponentially smaller than finding a Nash equilibrium of the equivalent normal-form game

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Practical Algorithms • The Govindan-Wilson Algorithm • Start with random values • Do something similar to gradient descent search • The Simplicial Subdivision Algorithm • Divide and conquer algorithm • Start with a rough approximation and refine it

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Coffee Shop Game • Set in a downtown area – which is represented by an r x k grid of blocks • Any player can choose to • Set up their coffee shop in any one of those blocks • Decide not to enter the market • Their utility depends on • The number of players that choose the same block • The number of players that choose neighbouring blocks • The number of players that choose any other block

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Coffee Shop Game (3 x 4 Grid)

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Representation Size • 5 x 5 grid with 3 to 16 players • 4 player game with r x 5 grid (r  3 to 15)

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Computing Nash Equilibrium (Govindan-Wilson Algorithm) • 4 x 4 grid with 3 to 5 players • 4 x 4 grid with 3 to 12 players (AGG only)

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Computing Nash Equilibrium (Govindan-Wilson Algorithm) • 4 player game with r x 5 grid (r  3 to 12) • 4 player game with r x 5 grid (r  3 to 12) (AGG only)

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Conclusion • AGG’s present a compact way of representing all games • Compact – takes advantage of structures like anonymity, and context-specific independencies • Representation size is determined by the size of the payoff values • AGG representations can be extended by introducing function nodes.

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Conclusion • The complexity of finding a Nash equilibrium is PPAD-complete but still exponentially smaller than that of the equivalent normal form representation

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Final Thoughts – What I Liked • The paper was very well written and structured • Although for a person with basic game theory knowledge, it does present a lot of information to digest. • Lots of examples explaining how to represent different game representations as AGG’s • Graphical Games, Congestion Games, Symmetric Games, Polymatrix Games, Local Effect Games, ...

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Final Thoughts – What I Didn’t Liked • The experimental data presented only compared the AGG to the normal-form representation • Would have liked to see comparisons to other game representations as well

Game Representations Action Graph Games AGG’s with Function Nodes Computing Equilibria Experimental Results Conclusion and Final Thoughts Questions? • Is the AGG the “ultimate” representation? • Are there any disadvantages to using the AGG over another representation? • Can the AGG truly represent ALL games?

Action Graph Games ( Albert Xin Jiang, Kevin Leyton-Brown, Navin A.R. Bhat)