Communication and Cooperation General Game PlayingLecture 8 Michael Genesereth / Nat Love Spring 2006

2 Bughouse Chess

3 Diplomacy

4 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

5 Cooperation

6 p {1,2} : {a,b}  {a,b}  1..4 p 1 (a,a)=4 p 2 (a,a)=1 p 1 (a,b)=3 p 2 (a,b)=2 p 1 (b,a)=2 p 2 (b,a)=3 p 1 (b,b)=1 p 2 (b,b)=4 Hereafter, P refers to the set of all payoff matrices. Payoff Matrix ab a b

7 An agent's decision procedure is a function that maps payoff matrices into actions. W 1 : S  M The goal of design is to answer this question for all s: W 1 (s)=? This task is complicated by lack of knowledge about W 2 (s). Decision Procedures

8 An acceptability relation A i is a relation on joint actions. A i  M  N A i represents the maximal relation on M  N compatible with a given set of assumptions made by agent i about its environment. Acceptability Relation

9 P 1 : M  2 R P 1 (m)={p 1 (mn)|A 1 (mn)} A 1 ={aa, ab, ba, bb} P 1 (a)={4,3} Payoff Set Function ab a b

10 An action is irrational if and only if there is another action that yields a higher payoff for all acceptable joint actions. P i (m)<P i (m')   R i (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

11 Our agent is basically rational. R 1 (W 1 (s)) Basic Rationality

12 Row Dominance Example ab a b

13 The other agent is basically rational. R 2 (W 2 (s)) Equivalently,  R 2 (n)   m.  A 1 (mn) Mutual Rationality

14 Iterated Row Dominance Example ab a b

15 Our agent believes the other agent's actions are independent of our agent's actions. A 1 (mn)  A 1 (m'n) A 1 (mn)  n  n'   A 1 (mn') Independence

16 Case Analysis Example ab a b

17 Best Plan ab a b

18 Prisoner’s Dilemma ab a b

19 Communication

20 An offer group is a set of joint actions. Example: {aa, ab, ba} Offer Groups

21 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

22 The offer procedure u i for an agent i is a function that maps payoff matrices into offer groups. u i : S  2 M  N The action procedure v i for agent i is a function that maps payoff matrices and offer groups into actions. v i : S  2 M  N  M Offer and Action Procedures

23 The goal of design is to answer the questions: u 1 (s)=? v 1 (s,O)=? The design of v 1 is determined by the Single Offer Binding Protocol and techniques for handling cooperation without communication. As before, the task of designing u 1 is complicated by lack of knowledge about u 2 and is analogous to the task of designing w 1. New Questions

24 Action Assumptions: Basic Rationality: R 1 (v 1 (s,O)) Mutual Rationality: R 2 (v 2 (s,O)) Independence: A 1 (mn)  A 1 (m'n) Deal Assumptions: Basic Deal Rationality: R’ 1 (u 1 (s)) Mutual Deal Rationality: R’ 2 (u 2 (s)) Deal Independence: A’ 1 (OU)  A’ 1 (O'U) Assumptions

25 Theorem: If an offer group O is rational and there is a joint 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

26 Theorem: It is always rational for an agent to offer the joint action that gives it the highest return. Non-Null Offer Group Theorem

27 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

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

29 Best Plan ab a b 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.

30 Prisoner’s Dilemma ab a b 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}.

31 Concepts: Offer Groups Rationality Assumptions Theorems Lesson: Cooperation with communication requires fewer assumptions than cooperation without communication. Summary

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