CMSC 100 Multi-Agent Game Day Professor Marie desJardins Tuesday, November 20, 2012 Tue 11/20/12 1 Multi-Agent Game Day

CMSC 100CMSC 100Multi-Agent Game DayMulti-Agent Game Day

Professor Marie desJardins

Tuesday, November 20, 2012Tue 11/20/121Multi-Agent Game Day

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Multi-Agent Game DayMulti-Agent Game Day Game Equilibria: Iterated Prisoner’s Dilemma

Voting Strategies: Candy Selection Game

Distributed Problem Solving: Map Coloring

Tue 11/20/12Multi-Agent Game Day

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Distributed RationalityDistributed Rationality Techniques to encourage/coax/force

self-interested agents to play fairly in the sandbox

Voting: Everybody’s opinion counts (but how much?) Auctions: Everybody gets a chance to earn value (but how

to do it fairly?) Issues:

Global utility Fairness Stability Cheating and lying


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Pareto optimalityPareto optimality S is a Pareto-optimal solution iff

S’ (x Ux(S’) > Ux(S) → y Uy(S’) < Uy(S)) i.e., if X is better off in S’, then some Y must be worse off

Social welfare, or global utility, is the sum of all agents’ utility If S maximizes social welfare, it is also Pareto-optimal (but not vice versa)

X’s utility

Y’s utility

Which solutionsare Pareto-optimal?

Which solutionsmaximize global utility(social welfare)?


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StabilityStability If an agent can always maximize its utility with a

particular strategy (regardless of other agents’ behavior) then that strategy is dominant

A set of agent strategies is in Nash equilibrium if each agent’s strategy Si is locally optimal, given the other agents’ strategies No agent has an incentive to change strategies Hence this set of strategies is locally stable


Iterated Prisoner’s Dilemma

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Prisoner’s DilemmaPrisoner’s Dilemma

Cooperate Defect

Cooperate 3, 3 0, 5

Defect 5, 0 1, 1

AB


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Prisoner’s Dilemma: Prisoner’s Dilemma: AnalysisAnalysis

Pareto-optimal and social welfare maximizing solution: Both agents cooperate

Dominant strategy and Nash equilibrium: Both agents defect

Cooperate Defect

Cooperate 3, 3 0, 5

Defect 5, 0 1, 1

Why?

AB


Voting Strategies


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VotingVoting How should we rank the possible outcomes, given

individual agents’ preferences (votes)? Six desirable properties (which can’t all simultaneously

be satisfied): Every combination of votes should lead to a ranking Every pair of outcomes should have a relative ranking The ranking should be asymmetric and transitive The ranking should be Pareto-optimal Irrelevant alternatives shouldn’t influence the outcome Share the wealth: No agent should always get their way


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Voting ProtocolsVoting Protocols Plurality voting: the outcome with the highest number of votes

wins Irrelevant alternatives can change the outcome: The Ross Perot factor

Borda voting: Agents’ rankings are used as weights, which are summed across all agents Agents can “spend” high rankings on losing choices, making their

remaining votes less influential Range voting: Agents score each choice Binary voting: Agents rank sequential pairs of choices

(“elimination voting”) Irrelevant alternatives can still change the outcome Very order-dependent


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Voting GameVoting Game Why do you care? The winners may appear at the final exam...

The first two rounds will use plurality (1/0) voting: The naive strategy is to vote for your top choice. But is it the best

strategy? The next two rounds will use Borda (1..k) voting:

Your top choice receives k votes; your second choice, k-1, etc. The next two rounds will use range (0..10) voting

Discuss... did we achieve global social welfare? Fairness? Were there interesting dynamics?


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Let’s Vote...Let’s Vote...


Distributed Problem Solving

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Distributed Problem Distributed Problem SolvingSolving

Many problems can be represented as a set of constraints that have to be satisfied Routing problem (GPS navigation) Logistics problem (FedEx trucks) VLSI circuit layout optimization Factory job-shop scheduling (making widgets) Academic scheduling (from student and classroom perspectives)

Distributed constraint satisfaction: Individual agents have “responsibility” for different aspects of the

constraints Advantage: Parallel solving, local knowledge reduces bandwidth Disadvantage: Communication failures can lead to thrashing


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Distributed Map GameDistributed Map Game You’ll have to stand up now...

Two sets of cards – congregate with your shared color Each card has an “agent number” that identifies you Each card also has a list of “neighbors” that you have to coordinate with You have to choose one of four colors: red, yellow, green, blue Your color has to be different from any of your neighbors’ colors You can only exchange agent numbers and colors – no other information or discussion is

permitted! You can change your color (but remember this may cause problems for your neighbors...)

In five minutes, we’ll reconvene and see which group is the most internally consistent...


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Documents

CMSC 100 Multi-Agent Game Day Professor Marie desJardins Tuesday, November 20, 2012 Tue 11/20/12 1 Multi-Agent Game Day