Download presentation

Presentation is loading. Please wait.

Published byDarin Chafee Modified over 2 years ago

1
Winning concurrent reachability games requires doubly-exponential patience Michal Koucký IM AS CR, Prague Kristoffer Arnsfelt Hansen, Peter Bro Miltersen Aarhus U., Denmark

2
2 Example Player 1 chooses A {t,h} Player 1 chooses A {t,h} Player 2 chooses B {t,h} Player 2 chooses B {t,h}If A = B then move one level up, A = B then move one level up, A B = t then move to 1 st level, A B = t then move to 1 st level, A B = h then Player 1 loses. A B = h then Player 1 loses. Entrance fee: $15 Win: $20 W 7 6 5 4 3 2 1

3
3 Entrance fee: $15 Win: $20 Observation: To break even, you need at least ¾ probability to win. Good news: you can win with probability arbitrary close to 1. Bad news: the expected time to win the game with probability at least ¾ is 10 25 years (one move per day). … the age of universe: 10 11 years

4
4 Concurrent reachability games [de Alfaro, Henzinger, Kupferman ’98, Everett ’57] Two players play on a graph of states. At each step they simultaneously (independently) pick one of possible actions each and based on a transition table move to the next state. … … … …

5
5 Goals:Player 1 wants to reach a specific state or states. Player 2 wants to prevent Player 1 from reaching these states. Strategy of a player: Memory-less (non-adaptive) – π : states actions. Memory-less (non-adaptive) – π : states actions. Adaptive – π : history actions. Adaptive – π : history actions. Probabilistic strategy: π gives a probability distribution of possible actions. Patience of a memory-less strategy π = 1/min non-zero prob. in π … [Everett ’57]

6
6 Winning starting states: Sure – Player 1 has a winning strategy that never fails. Sure – Player 1 has a winning strategy that never fails. Almost-Sure – Player 1 has a randomized strategy that reaches goal with probability 1. Almost-Sure – Player 1 has a randomized strategy that reaches goal with probability 1. Limit-Sure – For every > 0, Player 1 has a strategy that reaches goal with probability at least 1 – . Limit-Sure – For every > 0, Player 1 has a strategy that reaches goal with probability at least 1 – .

7
7 Purgatory n Player 1 chooses A {t,h} Player 1 chooses A {t,h} Player 2 chooses B {t,h} Player 2 chooses B {t,h}If A = B then move one level up, A = B then move one level up, A B = t then move to 1 st level, A B = t then move to 1 st level, A B = h then move to state H. A B = h then move to state H. P n n-1 3 2 1 … H

8
8 Our results Thm:1) For every 0 1/ 2 n-2. 2) For every l 2 2 n-l-2. Thm:For every 0 61 actions in total, both players have -optimal strategies with patience 61 actions in total, both players have -optimal strategies with patience < 1/ 2 42m.

9
9 Thm:1) For every 0 t then the expected time to win the game by any ’-optimal strategy of Player 1 can be forced to be Ω( t ). patience ~ expected time to win patience ~ expected time to win All the results essentially hold also for adaptive strategies All the results essentially hold also for adaptive strategies Recall: the expected time to win Purgatory 7 with probability at least ¾ is 10 25 years (one move per day).

10
10 Algorithmic consequences Three algorithmic questions: 1. What are *-SURE states? PTIME [dAHK] 2. What are the winning probabilities of different states? PSPACE [EY] 3. What is the ( -)optimal strategy? EXP-EXP-TIME upper-bound [CdAH,…] EXP-SPACE lower-bound [our results] Cor: Any algorithm that manipulates winning strategies in explicit representation must use exponential space. … explicit representation: integer fractions

11
11 Purgatory n p i – probability of playing t in state i in -optimal strategy of Player 1. p i – probability of playing t in state i in -optimal strategy of Player 1. Claim: 1) 0< p i < 1, for all i. 2) p i < , for all i. 3) p 1 ≤ p 2. p 3 … p n 4) p i ≤ p i+1. p i+2 … p n P n n-1 3 2 1 … 1\2th tlevel+1loss hlevel=1level+1 p n p n-1 p3p3p2p2p1p1p3p3p2p2p1p1 Player 2 plays h Player 2 plays t Player 2 plays h t t t t t

12
12 Open problems Generic algorithm for - optimal strategy with symbolic representation? Generic algorithm for - optimal strategy with symbolic representation? How to redefine the game to be more realistic? How to redefine the game to be more realistic?

13
13 Goals:Player 1 wants to reach a specific state or states. Player 2 wants to prevent Player 1 from reaching these states. Winning starting states: Sure – Player 1 has a winning strategy that never fails. Sure – Player 1 has a winning strategy that never fails. Almost-Sure – Player 1 has a randomized strategy that reaches goal with probability 1. Almost-Sure – Player 1 has a randomized strategy that reaches goal with probability 1. Limit-Sure – For every > 0, Player 1 has a strategy that reaches goal with probability at least 1 – . Limit-Sure – For every > 0, Player 1 has a strategy that reaches goal with probability at least 1 – .

Similar presentations

OK

Energy and Mean-Payoff Parity Markov Decision Processes Laurent Doyen LSV, ENS Cachan & CNRS Krishnendu Chatterjee IST Austria MFCS 2011.

Energy and Mean-Payoff Parity Markov Decision Processes Laurent Doyen LSV, ENS Cachan & CNRS Krishnendu Chatterjee IST Austria MFCS 2011.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on bionics Ppt on mars orbiter mission Ppt on iso 9000 and 14000 Make a ppt on unity in diversity and organic farming Ppt on railway reservation system Jit ppt on manufacturing processes Ppt on swine flu 2015 Ppt on ibm company profile Ppt on 9-11 conspiracy theories video Ppt on mesoamerica and south america