Transcript Intelligent Agents for E

Cooperative Games, Mechanism
Design, and Auctions
Onn Shehory
March 9-13 2009
Politecnico di Milano
1
The Glove Game
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Players set: {1,2,3}
1,2 have right-had gloves
3 has left-had gloves
Players may stay alone and profit 0
Or join together: {1,3}, {2,3}, {1,2,3} to gain 1
 What mechanism should they follow?
 How do they behave given such mechanism?
 How is profit divided?
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The English Auction
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An item is placed for sale
Players free to bid
New bid must be higher than current
Winner: highest bid
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Outline
 Protocols and strategies
 Attitudes and rationality
 Stability and equilibrium
– Nash revisited
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Cooperative games
Solution concepts
Coalition Formation
RFP coalitions
Mechanism design
Auctions
Auctions – field results
4
What is a Mechanism/Protocol?
 A protocol (aka mechanism):
– provides a set of rules and behaviors to be
followed by its participants
– following the rules of a protocol is to a
player’s discretion, though deviation may
leave her “out of the game”
– examples: auctions, negotiation, voting
 Desired properties:
– maximize payoffs
– not manipulable/enforceable
– simple to implement and execute
5
What are Strategies?
 A strategy:
– is one of the possible actions a player can select given
the protocol
– is not dictated (or provided) by the protocol
– is usually the result of the player’s reasoning and
decisions, based on local algorithms and information
– examples:
 in an auction – bid as low as possible
 in elections – vote for your faction
 A good strategy:
– should maximize the player’s payoff given the protocol
and the behavior of other players
– should be difficult or impossible to manipulate
– should be computationally feasible
– may depend on the strategies of other players
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Players Attitudes
 Self-interest: a self-interested player is
attempting to maximize its own personal
payoff
 Benevolence/altruism: a benevolent player
is attempting to increase others’ payoffs
and the cumulative payoff of the society
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Risk Attitudes
 Risk prone: a player that has a preference for risk
 Risk averse: a player that has a preference for
avoiding risk
 Risk neutral: a player that has no risk preference
 Players do not need to be strictly prone, neutral or
averse – they may mix these
 Human players tend to have alternating and
context dependent risk attitudes
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Example Risk Attitudes
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You are offered two options:
1. Get 1000 Euro, cash
2. Get a lottery certificate, with a prize value of
10,000 Euro, and a 10% chance of winning
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What should you choose?
– Select 2, for a chance for getting 10,000?
– Select 1 to avoid risk?
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What should you choose if chance is 20%?
– Select 2, to maximize expected utility (2000)?
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Rationality
 A rational behavior is such that prefers a
greater payoff over a smaller one
 A rational player should always behave
rationally. That is, from among several
options available, he should select the one
that results in maximum payoff
 The problem:
– in may cases the number of options is
overwhelming
– there may be no algorithm for finding the best
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Bounded Rationality
 To overcome problems of rationality,
bounded rationality:
– limits the time/computation for option
consideration
– prunes the search space
– imposes restrictions on the types of options
 Results in fewer possibilities, hence
– computationally feasible
– may be too restrictive, far from optimal
– strategically inferior to rational
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“Good Enough” Behavior
 Make the bounded rationality rational:
– modify linear payoff functions to incorporate
computational costs
– put a cap on payoff
– add a small-amounts’ indifference
 The payoff of an option is good enough if
– too much additional computation to find other
good options, or
– other options do not provide a significant payoff
increase, or
– the player is indifferent w.r.t. the increase
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Stability and Equilibria
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Protocol Evaluation
 Payoff maximization: can refer to
individual payoffs, group payoffs, or
social welfare - the sum of individual
payoffs
 Pareto-optimality: a payoff vector
p(x1,x2,…,xn) is Pareto-optimal if there is
no other feasible payoff vector p' such
that at least one payoff is better in p'
and no payoff is worse in p
 Stability: a protocol is stable if once the
players arrived at a solution they do not
deviate from it
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Stability and Equilibria
 There are multiple stability concepts. In game
theory, the notion of equilibrium is used:
– dominant strategies: the agents have some strategies
that, regardless of what others do, maximize payoff
– Nash equilibrium: the agents have strategies that, as
long as other stick to theirs, maximize payoff
– Mixed Nash: the agents each have a set of strategies
from among which they select one with some probability
– Bayes-Nash: adds types (e.g. history) to the previous
one
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The Prisoner’s Dilemma
Payoff table for a 2- Player 2’s strategies
player, no-repetition
Prisoner’s Dilemma
Cooperate Defect
game
Player 1’s Cooperate
2, 2
-2, 4
strategies
Defect
4,-2
-1,-1
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No Pure Nash Equilibrium
Payoff table for a 2- Player 2’s strategies
player game with no
pure Nash equilibrium Cooperate Defect
Player 1’s Cooperate
strategies
Defect
4, 0
0, 2
2, 0
3,-2
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No Pure Nash Equilibrium
Payoff table for a 2- Player 2’s strategies
player game with no
pure Nash equilibrium Cooperate Defect
Player 1’s Cooperate
strategies
Defect
6, 2
0, 3
0, 1
5, 0
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Mixed Nash
 Player 1 will cooperate with probability pc
and defect with probability pd
 Player 2 will cooperate with probability qc
and defect with probability qd
 Expected utility of an agent is the utility
from a strategy times the probability of this
strategy being selected
 When there are multiple possibilities, the
expected utility is a sum over these
possibilities
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Computing the Probabilities
 The expected utility of an player x when the other
player y follows strategy s is denoted by Ux(s)
 In the case of equilibrium (mixed Nash), the
expected utility of x should be the same for all of
the possible strategies of y
 In our case we have players 1,2 and strategies
c,d
 We require that Ux(c) = Ux(d), which means that
for each of the two players, the expected utility
from the other cooperating should be equal to the
expected utility from the other defecting
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Computation Details
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For player 1 we have:
U1(c)= 6 pc+ 0 pd, U1(d)= 0 pc+ 5 pd
For player 2 we have:
U2(c)= 2 qc+ 3 qd, U2(d)= 1 qc+ 0 qd
The requirement that Ux(c) = Ux(d) results in:
qc= 0.642, qd = 0.358
pc= 0.317, pd = 0.683
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Tragedy of Common Goods
 Information on the web is (mostly) free
 Web agents that seek up to date information may
query web site as frequently as desired
 If all agents will do so, the network will be overly
congested, and some servers will crash
 So, is it undesirable to behave this way?
 If all (or most) of the agents prevent congestion,
it is in the best interest of each individual agent
to increase network use ...
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Cooperative Games
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Cooperative Games
 A cooperative game (aka coalitional game):
– Cooperation within groups is enforceable
– Groups compete, and not individuals
– Each group (=coalition) has a value v
– Characteristic function:
v : 2N, from coalitions to payments
– Players decide which coalitions to form (to
maximize payoff)
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Coalitional Games
 Coalitional game (N,v)
– A set of players N
– A coalition S is a group of players, subset of N, which cooperate
– Value (or utility) of a coalition v
 v(S) is a real, represents the gain of coalition S in the game (N,v)
 v(N) is the value of forming the grand coalition, coalition of all players
– Player payoff xi
 The portion of v(S) received by a player i in coalition S
 Characteristic function form implies:
– v depends only on the internal structure of the coalition
 Transferable utility
– The value of a coalition can be distributed arbitrarily among its
players
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Coalitional Games: Example
 Example: Majority Vote
– Prime minister is elected by majority vote
– A coalition consisting of a majority of players has a
worth of 1 since it is a decision maker
– Value of a coalition does not depend on the
external strategies of the users => characteristic
function form
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Super-additive Games
 Super-additive game
– (N,v) is super-additive if
– Here, cooperation is always beneficial
– Unification of two coalitions increases overall payoff
 Monotonicity: larger coalitions gain more
 Not all games are super-additive!
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Coalitional Stability
 Stability of a coalition
– Depends on how the value v is distributed
among the players
– How to divide v ?
 Improper payoff division => players may leave
the coalition, unhappy with their share
– Multiple solution concepts address this
point
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Coordination Game
• For 1, A>C, D>B
• For 2, a>c, d>b
• Red circles are pure Nash
• For driving side, all benefit if
all adopt the same side, but
two equilibria points exist
Strategy
Player 2’s strategies
table
Player 1’s
strategies
A, a
B,
C, b
c
D,d
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Solution Concepts
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Solution Concepts
 Assumption: the grand coalition will form
– Even when the solution includes multiple coalitions
 Solve for sub-games
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– Each players gets a payoff xi
Major issue: payoff distribution
A solution concept is a payoff allocation vector x  N
Efficiency: ∑ xi = v(N)
Individual rationality: xi ≥ v(i)
Group rationality: v(S) ≤ ∑ xi  S
Imputation: a payoff allocation vector which is efficient
and individually rational (and group rational for N)
 Many solution concepts are imputations
 Players prefer coalitions based on their respective
payoffs
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Example
 Socks selling
– Sold in pairs for 3euro a pair
– 2 sellers, each holds 5 socks
– Each can get 6euro, but together they get
15euro.
– Imputations: (6, 9), (7, 8), (7.5, 7.5)
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Some Properties
 Null player: when added to a coalition, contributes
nothing to its value
 Existence (of a solution concept): determines
whether the solution concept exists for every
game
– Examples: Kernel exists, Core does not
 Symmetry: symmetric players receive equal
payoffs
 Uniqueness: the solution concept is unique
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The Shapley Value (1953)
 Unique, efficient, symmetric, allocates zero to
null players
 Individually rational for super-additive games
 The payoff allocated to player i is
φi(v) = 1/n! ΣSN\i (s!-(n-s-1)!) (v(S∪i)-v(S))
 s = |S|
 This is considered a fair allocation
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The Glove Game Example
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N={1,2,3}
1,2 have right-had gloves
3 has left-had gloves
v(S) = 1 for {1,3}, {2,3}, {1,2,3}, otherwise 0
φ1(v) = 1/6
φ2(v) = 1/6
φ3(v) = 4/6
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The Stable Set
von Neumann & Morgenstern (1944)
 Given a game v and imputations x,y
 x dominates y if
– there is a nonempty coalition S, such that
– the members of S prefer payoff from x over
payoff from y
– v(S) ≥ ∑ xi  S
 S players can threaten to quit the grand
coalition if x not implemented
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Stable Set Example
 x=(1,2,3), y=(3,2,1)
 For players 1,2, y is better than x
 Does y dominate x?
– Depends on v({1,2})
 If v({1,2}) ≥ 5, y dominates x via {1,2}
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Stable Set Example 2
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x=(50,50,0), y=(0,60,40), z=(15,0,85)
v({1,2}) = v({1,3}) = v({2,3}) =100
y dominates x via {2,3}
z dominates y via {1,3}
x dominates z via {1,2}
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Some Properties
 Existence: the stable set may or may not exist
 Uniqueness: when exists, it is usually not unique
 Internal stability: no imputation within the stable
set dominate one another
 External stability: all payoff vectors outside the
stable set are dominated by at least on member of
the set
 Interpretation: the stable set represents conflicts,
but excludes inferior behaviors
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The Core
 The core is a set of imputations (x1, . . ., xN) satisfying
two conditions
 No coalition has a value greater than the sum of its
members’ payoffs (coalition rationality)
 The core can be empty
 A non-empty core in a super-additive game => stable
grand coalition
– No coalition has an incentive to leave (and receive a greater
payoff)
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Properties of the Core
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May be empty (rather common)
Not unique
Subset of the stable set
Has several variants
– E.g., epsilon-core
– Least-core
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Shoes Example
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A pair: a left and a right shoe. Sold for €50
21 players: 10 have 1 left shoe, 11 have 1 right shoe
The core: a single imputation, gives 10 to left shoe owners, 0 to right
shoe owners
Any left-right pair can form a coalition and sell for €50
Any such pair getting less than that will block the imputation
For imputation in the core, any of these pairs gets exactly 50 (we can
only sell 10 pairs, totaling to 500)
One right-shoe owner gets 0 payment
Examine the pairs: if any left-shoe owner gets less than 50, say 40, it
can join this player, sell their shoes, give her 5, and keep 45 to herself.
Both are better off
But such a left-shoe owner cannot exist: all left shoe owners get
already 50.
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This holds for any unequal partition
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The core is very sensitive to oversupply.
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The Kernel
Davis & Maschler (1965)
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We do not assume imputations
That’s it – no group rationality
Efficiency still required
Result: we are interested no only in payoff
distribution, but in the coalitions that form
 Implicitly assumes bargaining (but in practice
computes its result)
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Surplus
 Maximum surplus
– Given an efficient payoff vector x and players i,j, the
maximum surplus of i over j is:
sij(x)= max(v(S)- ∑kS xk : iS, jS)
– the maximal amount i can gain without the cooperation
of j by withdrawing from the grand coalition N (where x
applies), other players in i's withdrawing coalition are
satisfied with their x payoffs
 The maximum surplus measures a player's
bargaining power over another
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The Kernel Defined
 The kernel is the set of payoff vectors x that
satisfy, for all pairs i,j:
(sij(x)-sji(x))·(xj - v(j)) ≤ 0
and
(sji(x)-sij(x))·(xi - v(i)) ≤ 0
 If sij(x) > sji(x) then i outweighs j – it has more
bargaining power
 But if xj = v(j), j can obtain xj on his own, thus i’s
threat is invalidated
 For vectors in the kernel, no player can outweigh
another – no valid threat  stability
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Kernel Properties
 Equilibrium of agents‘ surpluses:
In each coalition no player can outweigh another,
thus getting a better payoff (surplus) in an alternative
coalition excluding the opponent
„I can get more without you, than you can without
me.“
 Exists, Pareto-optimal, not unique
 A subset of the bargaining set
 Exponentially hard to compute
 Computational solution: Stearns (1968)
– May converge very slowly
– A single point out of many
 Polynomial variants (Shehory/Kraus 1996;
Klusch/Shehory 1996)
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Example
 Three players
 v(i)=0, v(12)=2, v(13)=3, v(23)=4, v(123)=8
– Kernel: 2, 2.5, 3.5
– Shapley: 13/6, 16/6, 19/6
 v(i)=0, v(12)=2, v(13)=3, v(23)=4, v(123)=5
– Kernel: 2/3, 5/3, 8/3
– Shapley: 7/6, 10/6, 13/6
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More Concepts
 The Bargaining set
 The Nucleolus
 Both based on excess:
v(S)- ∑kS xk
 Possible gain of players when quitting a
coalition
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Coalition Formation
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Coalitions and Complexity
 Given N players, there are 2N-1 different
possible coalitions
– If there are k tasks, may need to multiply by
exp(k)
 The number of configurations is O(N(N/2))
 Hence, exhaustive search is infeasible
 Additionally, players may have conflicting
preferences over the possible
configurations
 Nevertheless, coalitions are beneficial
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Issues in Coalition Formation
 Given a set of tasks and a set of players, which
coalitions should a player attempt to form?
 What mechanism can agents use for coalition
formation?
 What guarantees regarding efficiency and quality
can the mechanism provide?
 Once a coalition has formed, how should its
members go about distribution of work/payoff?
 When, and how, does a coalition dissolve?
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Agent5
Agent1
T3
Agent6
Agent2
T2
T1
Agent3
Agent4
Agent7
Agent8
Agent9
Agent11
Agent10
T4
Agent12
Agent13
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Factors Affecting Solution
 Self-interest vs. benevolence:
– Simpler mechanisms for benevolent players
– Do not need means for individual payoff maximization
 Centralization vs. distribution:
– Central design of coalitions is usually much simpler to
execute and enforce
 Super-additivity:
– In super-additive environments any unification of two
coalitions increases overall payoff
– Strongly influences the mechanism - simplifies
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Example Application Domain:
E-Commerce
 B2C: Wholesale markets
– Purchasing more of the same product reduces price per
unit
– Sellers benefit from selling more and spending less on
marketing/distribution
– Usually, buyers do not need large quantities
– Can form coalitions of buyers
 B2B: Request for Proposal (RFP) markets
– A requested product/service can be provided only by
groups of suppliers, hence coalitions are a must
– Difficulties:
 Valuations of tasks vary across suppliers, and are private
information
 Time for submitting proposals is limited, and value may be
discounted
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Game Theoretic Solutions?
 Computation usually hyper-exponential
 Centralized
 Stable, but
– Have multiple equilibria points in a solution
– Are sensitive to small changes
 Not in strategic form. Transformation is
complex
 No formation mechanism provided
 No dynamics: agents cannot join or leave
existing coalitions
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Classical Solutions: Kernel Revisited
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Attaches payoff vector(s) to a configuration
The solution is Pareto-optimal, non-empty
Based on bargaining: proposals/objections
Bargaining/proposals are bilateral
Convergence to an equilibrium is guaranteed
2n coalitions, O(nn/2) configurations
No distribution, no algorithms, etc.
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Practical Solutions
 Prune the solution search space
 Design a mechanism to motivate players to follow
 Provide strategies, (and algorithms) for a player to
maximize utility given the mechanism
 Make sure that the mechanism + strategies arrive at
stability and near optimum
 Outstanding issues for most of the solutions:
– Complexity/scalability: near-optimal solutions are only good
for dozens of agents
– Dynamics: joining/leaving coalitions usually not resolved
– Unrealistic, restrictive assumptions:
 Complete information
 No uncertainty and subjectivity with regards to information
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Example: the Modified Kernel
 Use a small subset of coalitions,
configurations
 Method to compute the Kernel (even with
pruned space) is exponential (Stearns 68):
allow Kernel+D
 Modification violates equilibrium: correct
by applying a computation cost function
 Dynamism, subjective valuations?
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Coalition Formation
Mechanism Example
59
Example Mechanism
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RFP coalitions
Protocol and strategies
Use of heuristics
Experimental equilibrium
60
RFP Coalitions
 Problem properties:
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–
Tasks can only be performed by groups
A task is comprised from subtasks
A task has a deadline and a value (discounted over time)
Agents have private, subjective valuations of subtasks
Agents are self-interested utility maximizers
 Solution approach:
–
–
–
–
Agents negotiate under time pressure to form coalitions
Decisions during negotiation are derived via strategies
Complete search of the problem space is infeasible
Consequently – a simulation-based solution
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The Protocol
 General structure:
– Auction-like mechanism for task allocation to coalitions
– Negotiation for coalition formation
 Participants:
– Businesses in pressing need for complex products/services
issue RFPs (with a price and a discount rate)
– Suppliers that can address parts of the RFPs negotiate and
join potential coalitions, then submit joint proposals
– A neutral third party manager serves two roles:
 An auctioneer
 A coalition negotiation manager
 Suppliers are represented by autonomous agents,
these negotiate and submit proposals on their behalf
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The Manager Role
 Auctioneer:
– Publishes available RFPs
– Collects proposals and determines winning
coalition
– Monitors completion of task and distributes
payment
 Negotiation manager:
– Brokers proposals among agents
– Verifies joint capabilities and net costs
– Sets payments according to costs and a profit
distribution scheme (equal)
– Determines time ordering among proposals
– Monitors adherence with protocol
63
Coalition Negotiation
Iterative – one proposal per agent at each iteration
Agents either propose or wait, committed
More beneficial proposals are preferred
Time is an issue because of discount
Agents must follow protocol, can use any strategy
for proposal preparation/decision
 Strategy space is intractable – we propose some
strategies based on heuristics
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64
Strategies
 Goal: decide which coalitions to propose to which agents and
accept/reject proposals
 Strategies are based on heuristics for ranking coalitions
according to desirability
 General guidelines:
– Inspect RFP tasks and subtasks
– Inspect capabilities and capacities of other agents
– Compute candidate coalitions, then rank them
 Ranking heuristics:
– Marginal: prefer coalitions where the estimated marginal profit of
the coalition is maximal
– Expert: prefer coalitions where only a few others have the right
capabilities. Consequence - a better chance of winning (being an
expert)
– Mixture of these
 These were examined experimentally, shown beneficial with
respect to a centralized solution
65
Experimental Design
 Settings:
– The number of agents
– The number of tasks
– The number of subtasks per task (bounded)
 Parameters:
– The value of each task
– The capabilities of each agent
– The cost of a given agent to perform tasks it is capable of
 For each settings we randomly generated between 100
and 1,500 configurations making sure there are “experts”
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Experimental Design (contd)
 Each subtask was randomly assigned a mean cost
with a uniform distribution.
 The actual cost of a given subtask for a specific
agent was determined using a normal distribution,
with the mean of the subtask, and a certain deviation
(2 in the basic settings)
 We consider:
– “Complete information” case, each agent knows the costs of
the other agents
– “Incomplete information” case, they know only the mean
values of the costs
– In both cases each agent knows the capabilities of all the
agents
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Expert vs. Marginal
Incomplete Information Complete Information
Simulation profits / Optimal Profits
Complete Information
Simulation profits / Optimal Profits
Incomplete Information
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68
Experimental Equilibrium
 Given a set of agents A and a set strategies Σ;
a profile F of strategies is an experimental
equilibrium iff any agent i in A, by using
another strategy in Σ does not increase its
estimated expected utility, given that the other
agents follow their F’s strategies.
 Two main differences from Bayesian
equilibrium:
– We limit the deviation to strategies in Σ
– We use experimental estimation of the expected
utility
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Expert vs. Marginal: Deviation
Simulation profits / Optimal Profits
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
One Marginal All but one
Incomplete
Expert
Incomplete
One Expert
All but one
Marginal
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Strategies for Payment
Distribution
•Equal Distribution: estimated net divided “equally”
•Proportional: proportional to the estimated cost.
•Kernel: based on the game-theory concept of
stability (adjustments for negotiation)
•Compromise: propose/agree to an offer that is α<1 times the
“deserved” share”.
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Deviation to Compromising Strategy
utility ratio for com prom is
- hete
e
rogenous
2
1
ratio
1.5
0.5
0
1.02
1
0.9
0.85
0.8
0.7
0.5
0.45
0.4
0.3
0
alpha
I nc omplet e I nf ormat ion
C omplet e I nf ormation
72
It is not beneficial to deviate from “equal, 0.8”
0.86
0.8
0.84
0.4
0.2
0.82
0.8
0.78
0.76
0
C omplet e inf ormat ion
0.8
0.74
I nc omplet e inf ormation
C om plet e Inf orm ation
no compromise
Major 0.8
Deviation fr om e qual to k e rnel
- com prom is e0.8
0.4
0.2
ratio to optimal
0.6
One 0.5
Equal vs Proportional- hom oge nous
w ith adaptation
1
0.8
I nc om plet e I nf orm ation
1
0.8
0.6
0.4
0
C omplet e inf ormat ion
major equal
I nc omplet e inf ormat ion
0.2
one k ernel
ratio to optimal
0.6
ratio to optimal
1
ratio to optimal
Deviation fr om0.8 to 0.5 - no adaptation
Com prom is e- Hom oge ne ous
0
C omplete Inf o
Incomplete Inf o
Equal Proportional
73
It is beneficial to deviate from
“Kernel”
Deviation fr om ke rnel to equal
- no com prom is e
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
C om plet e inf orm at ion
m ajor k ernel
I nc om plet e inf orm ation
one equal
The protocol does not allow to leave a coalition that has
been formed.
74
Restrictive vs. Non-Restrictive
Protocols
Sim ulation V alue/ Optim al V alue
1.00
0.80
0.60
0.40
0.20
0.00
I nc om plet e I nf orm at ion
C om plet e I nf orm at ion
N on R es t ric tiv e R es trict iv e
Sim ulatio n Co ntr acts
/ Op tim al Co ntr acts
1.00
0.80
0.60
0.40
0.20
0.00
I nc om plet e I nf orm ation
C om plet e Inf orm ation
N on R es trict iv e R es trict iv e
75
RFP Coalitions Summary
 Subjective valuations of tasks and subtasks
impose difficulty to use traditional mechanisms
 Surprisingly, simple heuristic strategies result in
beneficial coalitions
 Adaptation and compromise further improves
results
 Stability is also arrived at, and complexity is low –
hundreds of agents and tasks can be handled
76
Recent Studies




Fuzzy utilities
Ordinal utilities
Effects of search costs
Coalitions with risk attitudes
77
Conclusion
 Coalitions are an important means for agent
collaboration
 Mechanisms are usually
–
–
–
–
too complex, or
too rigid, or
do not scale, or
hold too restrictive, unrealistic assumptions
 We have seen a few improvements on these
shortcomings
 Yet, all solutions require further improvements to
become applicable for practical use
78
Mechanism Design
79
What is Mechanism Design?
 Designing rules of a game, a protocol
– Achieve a specific outcome
– Account for agent self-interested behavior
 Incentivize
– Set rules which players prefer to follow
80
Preferred Properties







Individual rationality
Budget balance
Social welfare
Truthfulness
Resistance to attacks/collusions
Fair distributions
Computational efficiency
 Unfortunately, not all can be met simultaneously
 Trade-offs are studied widely
81
Mechanism Defined
 Given N players with types ti  Ti
 E.g. in an auction, the type of a player may be her
valuation of the good
 Depending ti, player i will select an action ai(ti)  Ai, Ai the
set of possible actions of i offered by the mechanism
 E.g. an action in an auction may be a bid of a certain value
 A player has utility ui, based on the outcome generated by
the mechanism
 E.g., in an auction the outcome may be the final allocation
of goods
 A mechanism M is a pair (A,g), where A= Πi Ai the set of
actions and g a mapping function from actions to outcomes
82
Algorithmic Mechanism Design
 Considers computational issues
– Complexity: poly-time is sought
– Optimality and worst-case analysis employed
– An algorithm is developed
– If exponential – out of scope:
 E.g., VCG (Vickrey-Clarke-Groves) acution
83
Application
 In recent years, mostly applied to
– Auctions
– Markets
– Negotiation
84
Auctions
85
Auctions
 A centralized protocol, includes one auctioneer
and multiple bidders
 The auctioneer puts a good for sale. In some
cases, the good may be a combination of other
goods, or a good with multiple attributes
 The bidders make offers. This may be repeated for
several times, depending on the auction type
 The auctioneer determines the winner
86
Auctions: pros and cons






Usually easier to prevent bidder lying
Simple protocols
Centralized: a single point of failure
Multi-attribute exponentially complex
Allows collusion “behind the scenes”
May favor the auctioneer
87
Auction Types
 Private value: the value of a good to a
bidder agent depends only on its private
preferences. Assumed to be known exactly
 Common value: the good’s value depends
entirely on other agents’ valuation
 Correlated value: the good’s value depends
on internal and external valuations
88
Auction Protocols
 English auction (aka first-price open-cry):
– bidders free to raise their bid
– end: no more raises, winner: highest bidder at bid
– agent strategy: a series of bids, based on private value,
estimates of others’ valuations, their past bids
– dominant strategy: bid a small amount more than
current highest bid, stop when private value reached
 For correlated value:
– auctioneer increases price by constant or other rate
– open-exit allows to quit without re-entry
89
More Protocols
 First-price sealed-bid auction:
– each bidder submits one bid, not knowing others’
– highest wins, pays his bid
– agent strategy: function of private value and beliefs
about others’ valuations
– no dominant strategy. Best: bid less than true value
– how much less? Nash is computable if probability
distribution of agents’ values is known
 Example: n agents, uniform value distribution,
agent i has value vi, there is Nash if each agent i
bids vi(n-1)/n
90
Yet More Auctions
 Dutch auction (decending):
– the seller lower the price until a bidder takes it
– strategically, equivalent to first-price sealed-bid
– advantage: auctioneer can accelerate auction
91
… and More
 All-pay auction:
– each bidder pays its bid to the auctioneer
– several types of such auctions are used for
resource (re-)allocation
– E.g., olympic games, political lobbying, R&D races
 Equilibrium bidding strategy must be a mixed
strategy
– Consider a common-value all-pay auction with
prize worth 1
92
Vickrey (second-price sealed-bid)
 Each bidder submits one bid, not knowing others’
 The highest bid wins, but bidder pays second-highest bid
 Agent strategy: base bid on private value and beliefs about
others’ values
 Dominant strategy: bid true valuation
– if it bids more and this increment made him win, the agent ends up
with a loss, since it may pay more that its true value
– if it bids less, there is a smaller chance of winning (but winning
price is not affected)
 Meaning: bid true value regardless of others
93
So, Which Auction is Better?
 Computation: auctions with dominant strategies
(Vickrey and English) are more efficient - no need
to speculate regarding other bidders
 Auctioneer’s revenue:
– second-price is less than the true price, however firstprice bidders under-bid. Which effect is stronger?
– for risk-neutral bidders with private independent values,
the effects are equivalent
– for risk-averse bidders, Dutch and first-price sealed-bid
auctions maximize auctioneer’s revenue
 So, are revenues equivalent?
94
Real Auctions
 In real auctions, values are not private
 As a result, for 3 or more bidders, English auctions
provides auctioneer revenue higher than Vickrey
does
 Explanation: when it observes other bidders
increasing their bid, a bidder increases its own
valuation of the good
 Both English and Vickrey are better for the
auctioneer than Dutch and first-price sealed-bid
95
Collusion
 Bidders can coordinate their bids to lower them
 In English and Vickrey auctions, collusion is a
dominant Strategy!
 Example:
– agents a,b,c values of the good are 10,10,12,
respectively
– they can agree to bid 5,5,6 respectively
– if one defects, all observe that, and can increase to real
value, so there is no benefit from defection
96
Avoiding Collusion
 In the first-price sealed-bid and Dutch auctions,
bidder collusion is not dominant, but possible:
– in the previous example, after a,b,c decided on bidding
5,5,6, it is beneficial for a,b to bid more than 5. For any
bid of c below 10 they can bid and win
 In first-price sealed-bid, Vickrey and Dutch
auctions, all bidders must identify each other and
collude jointly. External bidder can win
 In the English auction identifying is through
bidding. Computerized anonymization can prevent
identification and collusion
97
Insincere Auctioneer
 Private value auctions:
– Vickrey: auctioneer can overstate the
second highest bid to the winner
– Solution: electronic signature
– Other auctions do not motivate auctioneer
lying, since the winner pays its bid
 Non-private value:
– English: auctioneer can use shills that bid
in the auction to increase real bidders
valuation
– Any auction: auctioneer may bid, to
guarantee a minimum price
98
Example: Auctioneer Bid
 In the Vickrey auction, auctioneer is
motivated to bid over its true reservation
price
 In case his bid is second, it determines the
good’s price higher than the reservation
price
 On the other hand, auctioneer may win
although others value the good at more than
reservation price
99
Insincere Bidders
 Non-private value:
– winner’s curse: an agent that bids its true value and
wins knows that it was too high
– this means that a win is a loss (of money)
– hence, agents should bid less than true value
– this is the best strategy even in Vickrey (unlike private
value Vickrey)
 Private value, Vickrey:
– dominant truthful bidding reveals true valuations
– this may be disadvantageous:
 when subcontracting, subcontractors may re-negotiate
100
Auctions of Interrelated Goods
 Multiple homogeneous goods: truth revelation
of Vickrey holds
 Heterogeneous goods, one at a time,
interdependent values:
– for optimal bidding, agents need full lookahead
– but then agents don’t bid true values per good
 Protocol modifications to overcome that:
– pool of goods at a single auction
– allow decommitemt, with penalties
 Note: lookahead requires speculation
101
Continuous double auctions
(CDA)
 At any time during the trading period of
a good
– buyers may submit bids
– sellers may submit asks
 If open buy and sell orders are
compatible, a trade is executed
immediately
 Typically, an announcement is made to
all participants
102
CDA properties
 Widely used for securities, derivative, commodities
 Highly efficient: can respond rapidly to changing
market conditions, despite limited info. Available to
participants:
– Private utility function
– Stream of bids, asks, trades
 Prices converge fast (close) to theoretical
competitive equilibrium (for human subjects)
103
Limitations





Centralized
Requires a neutral broker
Vulnerable to manipulations
Inefficient for small volumes
Requires a matching mechanism
104
Auctions –Field Results
105
Thank You!
106
Backup Slides
107