Post on 22-Dec-2015
Game DynamicsOut of Sync
Michael Schapira(Yale University and UC Berkeley)
Joint work withAaron D. Jaggard
and Rebecca N. Wright
Motivation: Internet Routing
Establish routes between Autonomous Systems (ASes).
Currently handled by the Border Gateway Protocol (BGP).
AT&T
Qwest
Comcast
Sprint
Internet Routing as a Game[Levin-S-Zohar]
• Internet routing is a game!– players = ASes – players’ types = preferences over routes– strategies = outgoing edges
• BGP = Best-Response Dynamics– each AS constantly selects its best
available route to each destination– … until a “stable state” (= PNE) is reached.
But…
• Challenge I: No synchronization ofplayers’ actions– players can best-reply simultaneously.– players can best-reply based on outdated
information.
• Challenge II: Are players incentivized to follow best-response dynamics?– Can a player benefit from not best-replying?
this talk
[Nisan-S-Valiant-Zohar]
Game Dynamics and Asynchrony
• Dynamic environments– Internet protocols– large-scale markets– social networks– multi-processor computer architectures
• Game theory provides useful tools to analyze these interactions, but….
• … has so far primarily concentrated on synchronous environments (simultaneous, sequential).
•Model for asynchronous game dynamics
• Impossibility result
•Circumventing our impossibility result
•Complexity of asynchronous game dynamics
•Directions for future research
Agenda
• n nodes 1,…,n
• Node i has action space Ai
– A=A1•…•An
– A-i=A1•…•Ai-1•Ai+1•…•An
• Node i has reaction function fi:A→Ai
– f=(f1,…,fn)
Simple Model: Nodes Interacting
• Infinite sequence of discrete time steps t=1,…
• Initial state a0, Schedule :{1,…} →2[n]
– fair schedule
• The (a0,)-dynamics– Start at the initial state a0
– In each time step t let the nodes in (t) react.
Simple Model: Dynamics
•Defn: an action profile a=(a1,…,an) is a stable state if fi(a)=ai for all i.– that is, a is a fixed point of f.
•Defn: The system is convergent if the (a0,)-dynamics converges to a stable state for all choices of a0 and (fair) .
Simple Model: Convergence
• Defn: f is node independent if, for each node i, fi:A-
i→Ai
• Thm: If f is node independent, and there exist multiple stable states, then the system is not convergent.
• Can be generalized to reaction functions that are– randomized– bounded-recall– non-stationary
Guaranteed Convergence?
• Internet protocols– Internet routing [Sami-S-Zohar]– congestion control [Godfrey-S-Zohar-Shenker]
• Best-response dynamics– with consistent tie-breaking– orthogonal to the results of Hart and Mas-Colell
• Diffusion of technologies in social networks– 2 technologies {A,B}. Each node wants to be consistent with the majority of
its neighbours.
• Circuit design
Applications
• Example 1: (node-dependent reactions)Each fi is such that for every a=(a1,…,an) it holds that fi(a)=ai.
“Tightness” of Our Result
• Example 1: (node dependent reactions)Each fi is such that for every a=(a1,…,an) it holds that fi(a)=ai.
• Example 2: (unbounded recall)– 2 nodes, 1 and 2, each with action space {a,b}. – Node 2 wants to match node 1’s action.– Node 1 selects b if node 2 changed its action from a
to b in the past, and a otherwise.– What happens at the initial state (b,a)?
“Tightness” of Our Result
• Thm: If f is node independent, andthere exist multiple stable states, thenthe system is not convergent.
• Interesting connections to fundamental results in distributed computing theory.– the Fischer-Lynch-Patterson impossibility result for
consensus protocols (1983)
• But, neither result is a special case of the other.
Proving Our Result
The Dynamics Graph
action vector aS=(aS1,… aS
n)knowledge vector bS=(bS
1,…
bSn)
StateR
knowledge transition
i-transition
StateT
StateS
1. aT:=aS
2. bT:=aS
1. aR:=aS except aR
i:=fi(bS)
2. bR:=bS
• The dynamics graph captures all dynamics.
• The scenario where– the initial state is a0.– nodes 1 and 3 react simultaneously.– then nodes 2 and 3 react simultaneously.
is captured as follows:
Visualising Dynamics
• The dynamics graph captures all dynamics.
• The scenario where– the initial state is a0.– nodes 1 and 3 react simultaneously.– then nodes 2 and 3 react simultaneously.
is captured as follows:
Visualising Dynamics
State SaS=bS=a0
• The dynamics graph captures all dynamics.
• The scenario where– the initial state is a0.– nodes 1 and 3 react simultaneously.– then nodes 2 and 3 react simultaneously.
is captured as follows:
Visualising Dynamics
State SaS=bS=a0
1-transition 3-transition k-transition
• The dynamics graph captures all dynamics.
• The scenario where– the initial state is a0.– nodes 1 and 3 react simultaneously.– then nodes 2 and 3 react simultaneously.
is captured as follows:
Visualising Dynamics
State SaS=bS=a0
1-transition 3-transition k-transition 2-transition 3-transition k-transition
• Defn: A state S in the dynamics graph is stable if every outgoing edge from S leads to S.
• Defn: A fair path in the dynamics graph is a path that (1) for each i, contains an i-transition; and (2) also contains a knowledge transition.
Stability and Fairness
• Defn: The attractor region of a stable state S are all states from which any (long enough) fair path reaches S.
Attractor Regions
• Claim: A fair cycle in the dynamics graph implies an oscillation in our model.
• Proposition: If there are multiple stable states then there are states in the dynamics graph that are not in any attractor region (“neutral states”).
Proof Sketch (Cont.)
• Colour each
attractor region in a different colour – red, blue, etc.
• Colour the neutral states in purple.
Colouring States
•Key idea: We show that from every purple state there is a fair path that leads to another purple state.
•The number of purple states is finite and so this implies a fair cycle.
Creating Oscillations
• Lemma: There cannot be two edges leading from a purple state to two non-purple states that do not have the same colour.
• Intuition: We can swap the order of activations without affecting the outcome.
Proof Sketch (Cont.)
?
: different transitions
• Fix a purple state p.
• Let R be a “maximal” fair path from p to another purple state.
Proof Sketch (Cont.)
p ……
q
R
• Let be a transition that is not on R.
• Observe that at q takes us to a non-purple state.
p ……
q
R
Proof Sketch (Cont.)
• Because q is purple it must have a fair path to a non-purple non-red state.
p ……
q
R
……
u
Proof Sketch (Cont.)
• Now, we prove that at u must take us to a red state --- a contradiction!
p ……
q
R
……
u
Proof Sketch (Cont.)
• Our result holds for randomized reaction functions.– adversarially-chosen schedule
• What if the schedule is randomized?– our impossibility result breaks …– … but no general possibility result either
Circumventing Our Impossibility Result: Randomness
• Defn: A schedule is r-fair if each node is activated at least once within every r consecutive time steps.
• Can we prove our impossibility result for schedules that are r-fair? If so, for what values of r?
• We present positive and negative results.
Circumventing Our Impossibility Result: r-Fair Schedules
•Thm: Determining if a system with n nodes, each with two actions, is convergent requires exponential communication (in n).
• The proof requires reaction functions to be of exponential size.
• Combinatorial proof: a “Snake in the Box” construction
Complexity Results
• What if the reaction functions can be succinctly described?
•Thm: Determining if a system with n nodes is convergent is PSPACE-Complete.
• Hence, there is no “short” characterization of asynchronous convergence!
Complexity Results
•Other notions of asynchrony
•Other reaction functions– fictitious play, regret minimization– Observation: regret minimization is much more
resilient to asynchrony (different framework…).
•Other restrictions on schedules– random schedules– r-fair schedules– more
Directions for Future Research