Action Graph Games
(Albert Xin Jiang, Kevin Leyton-Brown, Navin A.R. Bhat)
Presented By:
Xuan Choo
Cheriton School of Computer Science
University of Waterloo
Sept 22, 2008
Xuan Choo
CS 886
Outline
• Game Representations
• Action Graph Games
• Action Graph Games with Function Nodes
• Computing Equilibria
• Experimental Results
• Conclusion and Final Thoughts
Xuan Choo
CS 886
Game Representations
Action Graph Games
AGG’s with Function Nodes
Computing Equilibria
Experimental Results
Game Representations
• Normal Form Game
• Extensive Form Game
• Multi-Agent Influence Diagrams
• Graphical Games
• Congestion Games
Xuan Choo
CS 886
Conclusion and Final Thoughts
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
Xuan Choo
CS 886
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.
Xuan Choo
CS 886
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
Xuan Choo
CS 886
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
Xuan Choo
CS 886
Game Representations
Action 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.
Xuan Choo
CS 886
Game Representations
Action 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 utility
function for action α
Xuan Choo
CS 886
Game Representations
Action 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
Xuan Choo
CS 886
Game Representations
Action Graph Games
AGG’s with Function Nodes
Computing Equilibria
Experimental Results
Conclusion and Final Thoughts
Example – Ice Cream Vendors
C1
C2
C3
C4
V1
V2
V3
V4
Ac
AW
Xuan Choo
CS 886
AV
Game Representations
Action Graph Games
AGG’s with Function Nodes
Computing Equilibria
Experimental Results
Conclusion and Final Thoughts
Context-Specific Independencies
C1
C2
C3
V1
V2
V3
Ac
AW
Xuan Choo
CS 886
AV
Game Representations
Action 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
Xuan Choo
CS 886
Game Representations
Action 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
Xuan Choo
CS 886
Game Representations
Action 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
Xuan Choo
CS 886
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?
Xuan Choo
CS 886
Game Representations
AGG’s with Function Nodes
Action Graph Games
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
$0.10/s delay
Xuan Choo
DEST
$1
CS 886
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
How to represent $0
delay due to$1
path usage?
$1.00/s delay
Xuan Choo
CS 886
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 utility
function for action α
Xuan Choo
CS 886
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
Xuan Choo
CS 886
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
Function Node:
Used to
represent the
number of
agents using the
route.
Xuan Choo
$0
$1
$1.00/s delay
CS 886
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
Xuan Choo
CS 886
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
Xuan Choo
CS 886
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
Xuan Choo
CS 886
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 nplayer normal-form game is PPAD-complete for
n≥2
Xuan Choo
CS 886
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 PPADcomplete
Xuan Choo
CS 886
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.
Xuan Choo
CS 886
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
Xuan Choo
CS 886
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
Xuan Choo
CS 886
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
Xuan Choo
CS 886
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
Xuan Choo
CS 886
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)
Xuan Choo
CS 886
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)
Xuan Choo
CS 886
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
Xuan Choo
• 4 player game with r x
5 grid (r  3 to 15)
CS 886
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
Xuan Choo
• 4 x 4 grid with 3 to 12
players (AGG only)
CS 886
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)
Xuan Choo
• 4 player game with r x
5 grid (r  3 to 12)
(AGG only)
CS 886
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.
Xuan Choo
CS 886
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
Xuan Choo
CS 886
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, ...
Xuan Choo
CS 886
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
Xuan Choo
CS 886
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?
Xuan Choo
CS 886