1. Competitive Safety Analysis: Robust Decision-Making in Multi- Agent Systems Moshe Tennenholtz Kyle Rokos

2. Introduction: Nash Equilibrium In game theory, a set of strategies for each player so that neither player will benefit from changing their strategy while the other player keeps theirs Example: Prisoner’s Dilemma

3. Introduction: Nash Equilibrium Confess Don’t talk ConfessDon’t talk 5,5 10,0 0,10 1,1

4. Introduction: Nash Equilibrium Confess Don’t talk ConfessDon’t talk 5,5 10,0 0,10 1,1

5. Introduction: Nash Equilibrium Confess Don’t talk ConfessDon’t talk 5,5 10,0 0,10 1,1

6. Competitive Safety Analysis Nash equilibria assume that the other agents also choose the best strategy If a competitive agent does not choose an optimal strategy, the outcome could potentially be very bad Safety strategies picks a strategy which guarantees a certain payoff, regardless of the competitor's strategy

7. Competitive Safety Analysis We can think of Nash Equilibria as the ‘best’ strategy Goal: Find safety strategies which the expected payoff is equal or close to the expected payoff for the Nash Equilibrium strategy Example: Decentralized Load Balancing

8. Competitive Safety Analysis e2e2 e1e1 e2e2 e1e1 aX/2, aX/2 X,aX aX,X X/2,X/2 0.5<a<1 X>0

9. Competitive Safety Analysis e2e2 e1e1 e2e2 e1e1 aX/2, aX/2 X,aX aX,X X/2,X/2 Nash Equilibrium: P(e 1 )X/2+P(e 2 )X = P(e 1 )aX+P(e 2 )aX/2 P(e 1 )=(2-a)/(1+a) P(e 2 )=1-(2-a)/(1+a) Expected Payoff: (3aX)/(2a+2)

10. Competitive Safety Analysis e2e2 e1e1 e2e2 e1e1 aX/2, aX/2 X,aX aX,X X/2,X/2 Safety level strategy: P(e 1 )X/2+P(e 2 )aX = P(e 1 )X+P(e 2 )aX/2 P(e 1 )=a/(1+a) P(e 2 )=1-a/(1+a) Expected Payoff: (3aX)/(2a+2)

11. Competitive Safety Analysis Not true in general that safety level and Nash Equilibrium have the same expected payoff In a non-reducible, generic, 2 person game, with strictly mixed strategies, the two payoffs will always coincide If the strategies are pure instead of mixed, the safety level will have a lower expected payoff than the Nash Equilibrium

12. Competitive Safety Analysis 1-p p 1-qq d,h b,f c,g a,e Nash Equilibrium safety level

13. Competitive Safety Analysis This idea can be extended to n-person games, with more than two options per player As the complexity increases, it becomes less likely that the Nash Equilibrium and safety level strategies will produce the same expected payoff

14. Competitive Safety Analysis Desirable to find cases where the expected payoff with safety level strategies is close to the expected payoff of the Nash Equilibrium A C-competitive strategy is one where the expected payoff is 1/C of the Nash expected payoff

15. Competitive Safety Analysis Thus a 1-competitive strategy is ideal, and a 2-competitive strategy is better than a 3-competitive strategy, but not as good as a 3/2-competitive strategy The load balancing problem is a 1- competitive strategy The extended load balancing problem is a 9/8-competitive strategy

16. Competitive Safety Analysis e2e2 e1e1 e2e2 e1e1 aX/3, aX/3, aX/3 X, aX/2, aX/2 aX/2, X, aX/2 X/2, X/2, aX e2e2 e1e1 e2e2 e1e1 aX/2, aX/2, X X/2, aX, X/2 aX, X/2, X/2 X/3, X/3, X/3.. 1 2: 3:

17. Conclusion There are many problems with Nash Equilibria. (e.g. Prisoner’s Dilemma) Safety level strategies attempt to fix one of the problems, the assumption that your opponent will use an optimal strategy, while retaining the expected payoff of the Nash equilibrium Success depends on the specific problem

18. Conclusion Please ask some questions now.

19. References Moshe Tennenholtz - Competitive Safety Analysis: Robust Decision-Making in Multi-Agent Systems. Lee Erlebach - An Introduction to the Mathematics of Game Theory