1. 1 Coinduction Principle for Games Michel de Rougemont Université Paris II & LRI

2. 2 Coinduction (inspired by D. Kozen) Naive notion Induction: F monotone, F has a unique least-fixed point. Coinduction: F is antimonotone, unique smallest greatest fixed-point. Example: Even numbers over N General Principle Function F defined by an equation, admits a fixed-point F*. We want to show that F* satisfies a property: it suffices to show that the property is preserved by the equation. Foundation: Let B a Banach Space, R is a linear Operator, of spectral radius <1. It admits a fixed point. Applications to Computer Science: 1.Evolutionary Games 2. Combinatorics 3.Stochastic processes

3. 3 1. Evolutionary Dynamics Example: Rock-Scissor-Paper: Mixed strategy= density of agents playing pure strategies Replicator Strategy:

4. 4 2. Enumerative Combinatorics Coinductive Counting (J. Rutten), Stream Calculus Male Bees=Drones D (Q,D)  Q Female Bees=Queens Q Q  D How many Q ancestors at level k? Q D Q Q D D Q Q D Q D Q k q0q0 q1q1

5. 5 3. Probabilistic Processes Equivalence of Markov Chains ? Metric Analogue of Bisimulation (Desharnais and al.) D-bar measure in Statistics Approximate Equivalence Property of Markov Chains: …. p 1-p

6. 6 Coinduction to compare 2 processes Property of Markov Chains: Examples from D. Kozen, Lics 2006 Generation of a biased coin (q) given a biased coin (p). Consider 3 different processes.

7. 7 Biased coin simulation Algorithm : qflip(q): If q >p( if pflip=head) return head else return qflip(q-p/1-p) ) else ( if pflip=head) return qflip(q/p) else return tail ) p q 0 1 q’ 0 1 Given: pflip a biased coin (head, tail) with probability (p,1-p). Task(q) : Generate a biased coin with probability (q,1-q). q p 0 1 q’ 0

8. 8 Strategy : qflip(q) Convergence: output with probability Min(p,1-p) at each step. Halts with probability 1. H(q)=Probability qflip(q)=head p q 0 1 q’ 0 1

9. 9 Time estimation of qflip(q) Estimated Time: p q 0 1 Algorithm : qflip(q): If q >p( if pflip=head) return head else return qflip(q-p/1-p) ) else ( if pflip=head) return qflip(q/p) else return tail ) q’ 0 1

10. 10 Strategy 1: qflip(q) Estimated Time: p q 01 Algorithm 1 : qflip(q): If ( q>1-p) ( if pflip=head) return qflip((q-1+p)/p) else return head Else if q >p( if pflip=head) return head else return qflip(q-p/1-p) ) else ( if pflip=head) return qflip(q/p) else return tail ) 1-p

11. 11 Comparison between Strategies 0 and 1

12. 12 Bounded Linear Operators B, Banach Space, R is a linear Operator, of spectral radius <1. Affine Operator τ(e) =a+Re Φ closed non empty region preserved by τ. Conclusion: there exists a fixed point e* (τ(e*) =a+Re*) s.t e* is in Φ. Example: E(q) is bounded. B Φ E*

13. 13 Coinduction principle Co induction principle:

14. 14 Application Co induction principle

15. 15 Case 1 Co induction setting

16. 16 Strategy 2: qflip(q) Estimated Time: p q 01 Algorithm 2 : qflip(q): If ( q>1-p) ( if pflip=head) return qflip((q-1+p)/p) else return head Else If ( q>0.5) ( if pflip=head) return tail else qflip(q/1-p) Else if q >p( if pflip=head) return head else return qflip(q-p/1-p) ) else ( if pflip=head) return qflip(q/p) else return tail ) 1-p 1/2

17. 17 New Application Co induction principle on pairs (E,E’)

18. 18 Probabilistic Processes Equivalence of Markov Chains ? Are M1,M2 ε-close ? Metric Analogue of Bisimulation (Desharnais and al.) Distance d between distributions obtained by iterations, also the maximum fixed-point of a Functional F. Property of Markov Chains: …. p-ε 1-p+ ε …. p 1-p M1M1 M2M2

19. 19 Conclusion General Principle: Given τ linear bounded operator of spectral radius <1, Φ closed non empty region preserved by τ, we conclude that there exists a fixed point e* in Φ. Applications: 1.Stochastic processes. Compare Expected time between two fractal processes. 2.Evolutionary Games. Compare convergence time. 3.Analysis of Streams.