- By Eva Tardos
- Repeated games
- player’s value/cost additive over periods, while playing
- players try to learn what is the best from past data
- what can we say about the outcome? how long do they have to stay to ensure OK social welfare?

- Result: routing, limit for very small users
- Theorem:
- In any network with continuous, non-decreasing cost functions and small users
- cost of Nash with rates ri for all i <= cost of opt with rate 2ri for all i

- Nash equilibrium: stable solution where no player had incentive to deviate.
- Price of Anarchy = cost of worst Nash equilibrium / social optimum cost;

- Theorem:
- Examples of price of anarchy bounds
- Price of anarchy in auctions
- First price is auction
- All pay auction…
- Other applications include:
- public goods
- fair sharing
- Walrasian Mechanism

- Repeated game that is slowly changing
- Dynamic population model
- at each step t each player I is replaced with an arbitrary new player with probability p
- in a population of N players, each step, Np players replaced in expectation
- population changes all the time: need to adjust
- players stay long enough…

- Dynamic population model
- Learning in repeated game
- what is learning?
- Does learning lead to finding Nash equilibrium?
- fictitious play = best respond to past history of other players goal: “pre-play” as a way to learn to play Nash

- Find a better idea when the game is playing?
- Change of focus: outcome of learning in playing

- Nash equilibrium of the one shot game?
- Nash equilibrium of the one-shot game: stable actions a with no regret for any alternate strategy x.
- cost_i(x, a_-i) >= cost_i(a)

- Behavior is far from stable
- no regret without stability: learning
- no regret: for any fixed action x (cost \in [0,1]):
- sum_t(cost_i(a^t)) <= sum_t(cost_i(x, a_-i^t)) + error
- error <= √T (if o(T) called no-regret)

- no regret: for any fixed action x (cost \in [0,1]):
- Outcome of no-regret learning in a fixed game
- limit distribution sigma of play (action vectors a = (a1, a2,…,an))

- No regret leanring as a behavior model:
- Pro:
- no need for common prior or rationality assumption on opponents
- behavioral assumption: if there is a consistently good strategy: please notice!
- algorithm: many simple rules ensure regret approx.
- Behavior model ….

- Pro:
- Distribution of smallest rationalizable multiplicative regret
- strictly positive regret: learning phase maybe better than no-regret

- Today (with d options):
- sum_t(cost_i(a^t)) <= sum_t(cost_i(x, a_-i^t)) + √Tlogd
- sum_t(cost_i(a^t)) <= (1 + epsilon)sum_t(cost_i(x, a_-i^t)) + log(d)/epsilon

- Quality of learning outcome
- Proof technique: Smoothness
- Learning and price of anarchy
- Learning in dynamic games
- Dynamic population model
- at each step t each player I is replaced with an arbitrary new player with probability p

- how should they learn from data?

- Dynamic population model
- Need for adaptive learning
- Adapting result to dynamic populations
- inequality we wish to have

- Change in optimum solution
- Use differential privacy -> stable solution