CMSC 100
Multi-Agent Game Day
Professor Marie desJardins
Tuesday, November 20, 2012
Multi-Agent Game Day
1
Tue 11/20/12
Multi-Agent Game Day

Game Equilibria: Iterated Prisoner’s Dilemma

Voting Strategies: Candy Selection Game

Distributed Problem Solving: Map Coloring
Multi-Agent Game Day
2
Tue 11/20/12
Distributed 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
Multi-Agent Game Day
3
Tue 11/20/12
Pareto 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)
Which solutions
are Pareto-optimal?
Y’s utility
Which solutions
maximize global utility
(social welfare)?
X’s utility
Multi-Agent Game Day
4
Tue 11/20/12
Stability

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
Multi-Agent Game Day
5
Tue 11/20/12
Iterated Prisoner’s
Dilemma
Multi-Agent Game Day
6
Tue 11/20/12
Prisoner’s Dilemma
Cooperate
Defect
Cooperate
3, 3
0, 5
Defect
5, 0
1, 1
A
Multi-Agent Game Day
B
7
Tue 11/20/12
Prisoner’s Dilemma:
Analysis

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
A
B
 Why?
Multi-Agent Game Day
8
Tue 11/20/128
Voting Strategies
Multi-Agent Game Day
9
Tue 11/20/12
Voting

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 
Multi-Agent Game Day
10
Tue 11/20/12
Voting 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
Multi-Agent Game Day
11
Tue 11/20/12
Voting 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?
Multi-Agent Game Day
12
Tue 11/20/12
Let’s Vote...
Multi-Agent Game Day
13
Tue 11/20/12
Distributed Problem
Solving
Multi-Agent Game Day
14
Tue 11/20/12
Distributed Problem
Solving


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
Multi-Agent Game Day
15
Tue 11/20/12
Distributed 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...)
Multi-Agent Game Day
16
Tue 11/20/12
2
1
5
3
6
4
12
11
16
Multi-Agent Game Day
9
8
7
15
14
13
18
17
17
10
19
20
Tue 11/20/12