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 IntroductionMistakesLiesConclusions

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. IntroductionMistakesLiesConclusions

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. IntroductionMistakesLiesConclusions

5. Terminology The Explicit Game is 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, |R i (a)-R’ i (a)|  . IntroductionMistakesLiesConclusions

6.  -perturbed system: example (-10,-10) (9,-9) (-9,9)(10,10) U D LR (9,-11) (-11,9)(8,8) IntroductionMistakesLiesConclusions

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. (0,0) (0,0)(0,0) (2,2)(1,1)(1,1) (1,1)(-1,-1) IntroductionMistakesLiesConclusions

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. IntroductionMistakesLiesConclusions

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

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. IntroductionMistakesLiesConclusions

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

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. IntroductionMistakesLiesConclusions

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. IntroductionMistakesLiesConclusions