action graph games ( albert xin jiang, kevin leyton-brown, navin a.r. bhat)

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

Presented By:Xuan Choo

Cheriton School of Computer ScienceUniversity of Waterloo

Sept 22, 2008

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Outline

• Game Representations

• Action Graph Games

• Action Graph Games with Function Nodes

• Computing Equilibria

• Experimental Results

• Conclusion and Final Thoughts

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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

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Normal Form & Extensive Form

• General representations

• But, the representation size grows exponentially with the number of agents


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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:

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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:

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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


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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.


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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)αA , where each uα is utility

function for action α


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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


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Example – Ice Cream Vendors

C1 C2 C3 C4

V1 V2 V3 V4

Ac

AV

AW


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Context-Specific Independencies

C1 C2 C3

V1 V2 V3

Ac

AV

AW


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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


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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


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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


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That’s It?

• What? That’s it? That’s all you need to represent ALL games?


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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

SRC DEST$0

$1

$1.00/s delay

$0.10/s delay


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Simple Network Routing Game

$1$0

$0.10/s delay

$1.00/s delay

How to represent delay due to path usage?


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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)pP , where each fp is an arbitrary

mapping from neighbours of p to real numbers– u is a tuple (uα)αA , where each uα is utility

function for action α


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Example – Network Routing Game

$1$0

$0.10/s delay

$1.00/s delay


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Example – Network Routing Game

$1$0

$0.10/s delay

$1.00/s delay

Function Node:Used to represent the number of agents using the route.


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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


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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


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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


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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


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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


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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.


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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


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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


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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


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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


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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

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Representation Size

• 5 x 5 grid with 3 to 16 players

• 4 player game with r x 5 grid (r 3 to 15)


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Computing Nash Equilibrium

• 4 x 4 grid with 3 to 5 players

• 4 x 4 grid with 3 to 12 players (AGG only)

(Govindan-Wilson Algorithm)


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Computing Nash Equilibrium

• 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)

(Govindan-Wilson Algorithm)


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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.


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Conclusion

• The complexity of finding a Nash equilibrium is PPAD-complete but still exponentially smaller than that of the equivalent normal form representation


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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, ...


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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


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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)

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