Distributed Games: From Mechanisms to Protocols

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Distributed Games From Mechanisms to Protocols

• Dov Monderer and Moshe Tennenholtz
• Presented By Namrata Rastogi

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Outline

• Motivation and Goal
• Distributed System and Game
• Mechanism to Protocol
• Distributed Protocols
• Implementation by distributed protocol (uniform
  distribution case)
• Implementation by distributed protocol (general
  case)
• Strong Implementation

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Motivation and Goal

• Problems arise when mechanisms transformed into
  protocols to be used in computational
  environments (parallel interactions)
• Assumption in mechanism design that every agent
  is directly connected to center.
• Problems dealt communication structure and
  representation of messages

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Distributed System

• Internet setup is a distributed system.
• A distributed system consists of a large number
  of loosely coupled computing devices (processors)
  which together perform some computation.
• Data available locally to individual processors
  whereas solution for computational problem
  reflects a global condition.

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Software Agents

• Programs that are designed to serve the goals
  of a particular user.
• navigate in a computerized network
  transmitting messages among themselves and also
  interacting with other agents in the network.

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Distributed Games

• A distributed game is defined by following
  elements
• A set of players
• A set of locations
• A set of agents for each player, one agent for
  each location
• A set of games in strategic form with the given
  set of players, one for each location.
• A set of messages for each player.
• A probability distribution over the set of
  permutations of locations.

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Issues in a Distributed Game

• reliability
• data link ceases passing data
• introduce errors in messages
• processor malfunctions
• Reliable or good players follow a predetermined
  pattern of behavior which bad players may
  deviate from the planned procedure in arbitrary
  fashion.Despite computational process is expected
  to run its correct course.

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Mechanism
Example Auctions
Agent2
Agent1
Agent n
t2
t1
t3
Agents directly connected to Center. Center
designs such a mechanism that its objective can
be achieved with information received from agents.
C
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Mechanism to Protocol-I

• Difference evident in non-cooperative
  computational environments
• Agents resource bounded
• Players are part of distributed mechanism
• Distributed Game is a model for dealing with the
  most general multi-agent interactions in
  distributed systems.

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Mechanisms to Protocols-II

• Two aspects of Distributed Games
• communication network
• (not directly connected to center,
• Each node in a network is an agent of a
  different
• player which may have ability to modify the
  messages
• sent by other agents)
• representation of agent information
• (bit structure of messages)
• Distributed Protocols to make malicious
  behavior of agents irrational .

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Distributed Protocols

• Model and Assumptions

N Agents
vi ? W
Outcome a ? A
Utility u(a,vi)
f(x)
No action ?
Environment E (N,A,u,W,f)
Bayesian Game B(E,H)
Mechanism H (M,h)
h Mn --gt A h(m) h(m1,m2,,mn)
M set of messages
bi W --gt M
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Bayesian Games

• Strategic form game with incomplete information
  (ex- players have private information type before
  the game begins)
• set of palyers N
• an action set Ai
• a type set
• a probability function (what player i believes
  about other players types)
• a payoff function
• A pure strategy for player i in a Bayesian Game
  is a function which maps player is type into her
  action set.

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Definition of Rationality

• Utility of agent i u(h(b(v)),vi)
• (depends on messages sent by others as well
  as own message)
• Vector of strategies chosen by rational agents
  is in equilibrium.
• Rationality (utility maximizer)
• Assumption Center recommends a particular
  behavior to the agents, which is an equilibrium
  vector of strategies b.
• revelation principle

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Correlated Equilibrium

• A group of n agents play a normal form game.
• Prior to the game , they can negotiate, publicly
  ,about how the game should be played.
• They do have access to an impartial mediator who
  can make private recommendations according to any
  agreed (random) pattern.
• When a strategy bi is recommended to agent I, on
  average agent I does no better by deviating to
  ti.

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Distributed Protocols

• Model and Assumptions
• Communication network L (V,E,?)
• (E,L) distributed environment
• Assume no loops,there is a path to center from
  each agent,g can be implemented in E,agent does
  not change the contents of header,standard
  synchronous system.
• To find out if the center succeeds to implement g
  in the new environment (E,L)?
• Mechanism in distributed environment centers
  protocol.
• Strategy of an agent in a distributed game is
  called agents protocol.

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Implementation by Distributed Protocols Uniform
Distribution Case-I

• Uniform Distribution f The probability of
  occurrence is same for all values of x.
• f(x) 1/2k

k length of message I 1-1 interpretation
function
vi
?i1
y1
Ai
-Each Agent sees a set of histories
corresponding to the messages received by it so
far (stage t) and sent to the neighbors. -Agent
sends a maximum of Q messages in a round.
y2y1 ? vi
?i2
Ai1
C
Bi-connected
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Implementation by Distributed Protocols Uniform
Distribution Case-II

• Protocol of Agent i with type vi
• Use the uniform distribution on W to generate a
  random bit string ,y1 of k bits.
• Let y2 be the bit-by-bit XOR of y1 and vi.
• Send y1 and y2 to the center through your
  neighbors determined by ?i1 and ?i2,
  respectively.
• If you receive a message with a header in which
  the original sender is j, send it without any
  change to the next node in the designated path of
  j.

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Implementation by Distributed Protocols Uniform
Distribution Case-III

• Centers protocol
• 1. The center receives the message and execute
  CONTINUE until stage T.
• 2. If the sequence of messages received by the
  center up to stage T can be generated by the
  agents protocols, then it does as follows
• - XOR of y1 and y2 treated as agents type.
  Let v (v1,v2,,vn) be the vector of types
  obtained in this way.
• - It runs truth revealing mechanism that
  implements g.(h(v) ? A and halts)
• 3. If sequence of messages received by the center
  is not consistent with any vector of types, then
  it executes

(What is T?)
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Theorem 1

• Let E be an environment, in which the type of
  each agent is selected according to the uniform
  probability function on the set of types, and let
  L be a 2-connected graph.If an outcome function
  is implementable by a mechanism in E, then it is
  implementable by a distributed protocol executed
  in (E,L).
• Idea Agents protocols are in equilibrium and no
  agent gains by deviating from its protocol
  assuming others stick to their protocol.

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Inference Uniform Distribution case
A1

• What can the agent do ?

m11
A2
m20, m10/1
C
Deviates and generates inconsistent history.
Deviates to produce consistent history
Does not deviate.
Utility0
Utilityltai
Utilityai
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Implementation by distributed protocols the
general case

• Arbitrary probability function
• W x0,x1 where 0 lt x0 lt x1
• f(x0) 1/4 and f(x1) 3/4
• Outcome function Second price auction

I M1---gt W
I(1) x1 I(0) x0
A1
m11
A2 can harm A1 by decreasing the
probability that the bid of the other agent is
high.
A2
m20,m10/1
C
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Solution general case

• Using another language
• set of messages with bigger cardinality
• appropriate interpretation function
• I M2 W
• W x0,x1
• I(11) I(10) I(01) x1 and I(00) x0
• Observations
• An agent can not decrease the probability of
  other agent getting a higher bid easily.
• Agents types viewed as if they are selected from
  a uniform distribution.

f(x0) 1/4 f(x1) 3/4
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Theorem 2

• Let E be an environment, and let L be a
  2-connected graph.If an outcome function is
  implementable by a mechanism in E, then it is
  implementable by a distributed protocol executed
  in (E,L).

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Strong Implementation-I

• Strong Equilibrium No coalition can deviate
  in a way that gives each deviator a higher
  expected utility.
• (clearly highly stable against conspiracies)
• center punishes all in the group that shows
  deviation from equilibrium path.
• agent observes deviation not observed by
  center, deviates to strongly hint the center.
• Theorem 3 Let E be an environment, and let L be
  a ring.If an outcome function is strongly
  implementable by a mechanism in E, then it is
  strongly implementable by a distributed protocol
  executed in (E,L).

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Strong Implementation-II

• Communication network Ring

1
0
Stage 0 n independent uniformly distributed
draws for agents. z1,z2,z3zn of k bits W Mk
2
n
3
4
5
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Strong Implementation-III

Agents 1 lt j lt n Stage 3j-2
(j2,stage4) Center sends key zj to agent
j through the path starting at n.
1
0
2
n
3
zj
4
5
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Strong Implementation-IV

Agents 1 lt j lt n Stage 3j-1
(j2,stage5) Agent j sends yj zj XOR vj to
the center through path going through 1.
1
0
2
yj
n
3
4
5
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Strong Implementation-V

Agents 1 lt j lt n Stage 3j
(j2,stage6) Agent j sends zj the center
through path going through 1.
1
0
2
zj
n
3
4
5
Agent sees others messages when both of them
have submitted their bids ! center receives key
back at the end
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Conclusion

• Deviation of an agent from distributed protocol
  is not beneficial for it.
• It is shown that given any 2-connected
  communication network L, any desired behavior
  which is implementable by standard mechanism is
  also implementable by a distributed protocol
  executed in L.
• Any desired behavior that is strongly
  implementable by a standard mechanism is also
  strongly implementable by a communication network
  with a ring topology.