Communication and Cooperation

General Game Playing Lecture 8

Michael Genesereth / Nat Love Spring 2006

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Bughouse Chess

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Diplomacy

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Language Communication Syntax Relationship between Communication and Legality

Architecture Anonymity Enforcement

Factual Information (in incomplete info games) Does it pay to lie? Does it pay to keep secrets?

Intentions How does a player make deals/contracts?

Issues

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Cooperation

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p{1,2} : {a,b} {a,b} 1..4

p1(a,a)=4 p2(a,a)=1 p1(a,b)=3 p2(a,b)=2 p1(b,a)=2 p2(b,a)=3 p1(b,b)=1 p2(b,b)=4

Hereafter, P refers to the set of all payoff matrices.

Payoff Matrix

a b

a

1

4

2

3

b

3

2

4

1

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An agent's decision procedure is a function that maps payoff matrices into actions.

W1: S M

The goal of design is to answer this question for all s:

W1(s)=?

This task is complicated by lack of knowledge about W2(s).

Decision Procedures

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An acceptability relation Ai is a relation on joint actions.

Ai MN

Ai represents the maximal relation on MN compatible with a given set of assumptions made by agent i about its environment.

Acceptability Relation

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P1: M 2R

P1(m)={p1(mn)|A1(mn)}

A1={aa, ab, ba, bb}P1(a)={4,3}

Payoff Set Function

a b

a

1

4

2

3

b

3

2

4

1

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An action is irrational if and only if there is another action that yields a higher payoff for all acceptable joint actions.

Pi(m)<Pi(m') Ri(m)

FYI: A set of numbers is less than another set if and only if every number in the first set is less than every number in the second set.

{2,1}{4,3}{2,3}{1,4}

Action Rationality

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Our agent is basically rational.

R1(W1(s))

Basic Rationality

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Row Dominance Example

a b

a

4

4

3

3

b

2

2

1

1

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The other agent is basically rational.

R2(W2(s))

Equivalently,

R2(n) m.A1(mn)

Mutual Rationality

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Iterated Row Dominance Example

a b

a

4

4

1

1

b

3

3

2

2

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Our agent believes the other agent's actions are independent of our agent's actions.

A1(mn) A1(m'n)

A1(mn) nn' A1(mn')

Independence

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Case Analysis Example

a b

a

4

4

1

2

b

2

3

3

1

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Best Plan

a b

a

4

4

2

2

b

1

1

3

3

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Prisoner’s Dilemma

a b

a

3

3

4

1

b

1

4

2

2

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Communication

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An offer group is a set of joint actions.

Example: {aa, ab, ba}

Offer Groups

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Every agent communicates a single offer group. If there is a non-null intersection of these offer groups, then the agents are bound to execute one of the joint actions in the intersection (determined by fair arbitration). If the intersection is null, the agents are not constrained by this protocol in any way.

Example: Agent 1 offers {aa, ba} Agent 2 offers {aa, ab} Agents 1 and 2 both execute action a

Single Offer Binding Protocol

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The offer procedure ui for an agent i is a function that maps payoff matrices into offer groups.

ui: S2MN

The action procedure vi for agent i is a function that maps payoff matrices and offer groups into actions.

vi: S 2MN M

Offer and Action Procedures

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The goal of design is to answer the questions:

u1(s)=?v1(s,O)=?

The design of v1 is determined by the Single Offer Binding Protocol and techniques for handling cooperation without communication.

As before, the task of designing u1 is complicated by lack of knowledge about u2 and is analogous to the task of designing w1.

New Questions

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Action Assumptions: Basic Rationality: R1(v1(s,O)) Mutual Rationality: R2(v2(s,O)) Independence: A1(mn)A1(m'n)

Deal Assumptions: Basic Deal Rationality: R’1(u1(s)) Mutual Deal Rationality: R’2(u2(s)) Deal Independence: A’1(OU)A’1(O'U)

Assumptions

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Theorem: If an offer group O is rational and there is ajoint action mn that dominates some joint action in O, then there is a rational offer group O' containing mn and all of the elements of O.

Note: It is always rational to restrict one's attention to maximal offer groups.

Monotonicity Theorem

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Theorem: It is always rational for an agent to offer the joint action that gives it the highest return.

Non-Null Offer Group Theorem

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Theorem: An agent should never offer a joint action resulting in a payoff less that it can get without making a deal.

Corollary: An agent need never offer the joint action that gives it its lowest payoff.

Lower Bound Theorem

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Theorem: An agent should never offer an action that is dominated for all agents.

Dominated Case Theorem

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Best Plana b

a

4

4

2

2

b

1

1

3

3

Explanation: Both agents include best action aa by the Non-null Offer Group Theorem. Neither agent includes any of the other three possibilities due to Dominated Case Elimination.

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Prisoner’s Dilemmaa b

a

3

3

4

1

b

1

4

2

2

Explanation: Neither agent will include bb due to Dominated Case Elimination. Each agent will include the joint action that gives it the highest utility; agent 1 will include ba and agent 2 will include ab. Each agent knows that the other agent will not accept this best joint action due to mutual rationality. Thus, the payoff of the offer group that includes aa is greater than either singleton offer group. Hence, agent 1 will offer {aa, ba}, and agent 2 will offer {aa, ab}. The intersection is {aa}.

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Concepts: Offer Groups Rationality Assumptions Theorems

Lesson: Cooperation with communication requires fewer assumptions than cooperation without communication.

Summary

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