A Computational Characterization of Multiagent Games with Fallacious Rewards

1. A Computational Characterization of Multiagent Games with Fallacious Rewards Ariel D. Procaccia and Jeffrey S. Rosenschein

2. Lecture Outline • Introduction • Modeling mistakes • Robust equilibria. • Persistent equilibrium pairs. • Errors due to lies • Closing Remarks Introduction Mistakes Lies Conclusions

3. Introduction • Interactions between selfish agents are often modeled as noncooperative games. • Nash equilibrium is the central solution concept. • Computationally hard to find equilibrium. Additionally, agents lack complete knowledge. • Much attention has been devoted to learning a Nash equilibrium in games. Introduction Mistakes Lies Conclusions

4. Sources of Fallacious Rewards • It is usually assumed that agents can observe others’ rewards. • What are the sources of these observations? • Direct observation. • Modeling. • Preference revelation. • In the second and third cases, fallacious rewards might be obtained. Introduction Mistakes Lies Conclusions

5. Terminology • The Explicit Gameis the real game, whereas the implicit game is a game with same players and actions but different rewards (Bowling & Veloso 2004). • Definition: Let G be an explicit game and G’ be an implicit game. G,G’ is an -perturbed system if for all players i and actions a, |Ri(a)-R’i(a)|  . Introduction Mistakes Lies Conclusions

6. -perturbed system: example L R (-10,-10) (9,-9) (9,-11) U (-9,9) (-11,9) (10,10) (8,8) D Introduction Mistakes Lies Conclusions

7. Robust Equilibria • Definition:  is an -robust eq. of G if it is a Nash eq. in any -perturbed G’. • Example: (U,L) is 1-robust, (D,R) isn’t -robust. (2,2) (1,1) (0,0) (1,1) (0,0) (1,1) (0,0) (-1,-1) Introduction Mistakes Lies Conclusions

8. -Robust equilibrium: Results • Lemma: Robust equilibria consist of pure strategies. • Theorem: It is possible to find an -robust equilibrium in polynomial time. • Robust equilibria rarely exist. Introduction Mistakes Lies Conclusions

9. Persistent Equilibrium pairs • Definition: Let <G,G’> be a perturbed system,  an eq. in G and  in G’. <,> is an -persistent eq. pair iff for all i, |Ri()-Ri’()|  . • If  is -robust, <,> is -persistent for any -perturbed G’. (1,1) (¼,¼) (0,0) (½,½) (0,0) (-1,-1) (-1,-1) (0,0) Introduction Mistakes Lies Conclusions

10. Persistent Equilibrium Pairs: Results • Theorem: Determining whether a perturbed system has an -persistent equilibrium pair is NP-complete, even for two players. • Proposition: G and G’ are -perturbed and zero-sum  Optimal strategies are -persistent. • Proposition: G and G’ are -perturbed  coordination equilibria are -persistent. Introduction Mistakes Lies Conclusions

11. Game Manipulation • Setting: Nash equilibrium is chosen based on agents’ valuations of outcomes. • Agents may lie to improve their utility. (0,0) (1,1) (1,1) (2,0) (-1,1) (-1,0) Introduction Mistakes Lies Conclusions

12. Game Manipulation: Results • Game-Manipulation: can player i manipulate and get reward > k? • Theorem: Game-Manipulation with at least 3 players is NP-complete, with 2 players is in P. Introduction Mistakes Lies Conclusions

13. Closing Remarks • Problems of finding robust or persistent equilibria is a problem of the system designer. • We consider other manipulations: • Coalitional. • Benevolent. • Strong. • Incentive-compatible mechanisms? • Results are independent of specific algorithms; also applicable to repeated games. Introduction Mistakes Lies Conclusions