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Outline

Outline. MDP (brief) Background Learning MDP Q learning Game theory (brief) Background Markov games (2-player) Background Learning Markov games Littman’s Minimax Q learning (zero-sum) Hu & Wellman’s Nash Q learning (general-sum). Stochastic games (SG). Partially observable SG (POSG).

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Outline

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  1. Outline • MDP (brief) • Background • Learning MDP • Q learning • Game theory (brief) • Background • Markov games (2-player) • Background • Learning Markov games • Littman’s Minimax Q learning (zero-sum) • Hu & Wellman’s Nash Q learning (general-sum)

  2. Stochastic games (SG) Partially observable SG (POSG) / SG / POSG

  3. Expectation over next states Immediate reward Value of next state

  4. Model-based reinforcement learning: • Learn the reward function and the state transition function • Solve for the optimal policy • Model-free reinforcement learning: • Directly learn the optimal policy without knowing the reward function or the state transition function

  5. #times action a causes state transition s  s’ #times action a has been executed in state s Total reward accrued when applying a in s

  6. v(s’)

  7. Start with arbitrary initial values of Q(s,a), for all sS, aA • At each time t the agent chooses an action and observes its reward rt • The agent then updates its Q-values based on the Q-learning rule • The learning rate t needs to decay over time in order for the learning algorithm to converge

  8. Famous game theory example

  9. A co-operative game

  10. Generalization of MDP Mixed strategy

  11. Stationary: the agent’s policy does not change over time Deterministic: the same action is always chosen whenever the agent is in state s

  12. Example State 2 State 1

  13. v(s,*)  v(s,) for all s  S,  

  14. Max V Such that: rock + paper + scissors = 1

  15. Worst case Expectation over all actions Best response

  16. Quality of a state-action pair Discounted value of all succeeding states weighted by their likelihood This learning rule converges to the correct values of Q and v Discounted value of all succeeding states

  17. Expected reward for taking action a when opponent chooses o from state s eplor controls how often the agent will deviate from its current policy

  18. Hu and Wellman general-sum Markov games as a framework for RL Theorem (Nash, 1951) There exists a mixed strategy Nash equilibrium for any finite bimatrix game

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