The document discusses bandit problems with strategic choices and small budgets. It defines bandit problems, strategic bandit problems, and compares the two. It presents algorithms for exploring options and making recommendations in both one-player and two-player settings. Experimental results on a Go positioning problem and an online card game show that TEXP3 outperforms other algorithms in two-player settings. The document concludes with discussions on extensions to structured bandits and using strategic bandits to model investment choices.

1. METAGAMING:
Bandits with simple regret and small
budget
Chen-Wei Chou, Ping-Chiang Chou,
Chang-Shing Lee, David Lupien St-Pierre,
Olivier Teytaud, Mei-Hui Wang, Li-Wen Wu
and Shi-Jim Yen

2. Outline:
- what is a bandit problem ?
- what is a strategic bandit problem ?
- is a strategic bandit different from a bandit ?
- algorithms
- results

3. What is a bandit problem ?
A finite number of time steps
A (finite) number of options,
each of them equipped with a (unknown) proba distribution
At each time step:
- you choose one option
- you get a reward, distributed according to its proba distribution
At the end:
- you choose one option (you can not change anymore...)
- your reward is the expected reward associated to this option

4. What is a bandit problem ?
A finite number of time steps
A (finite) number of options,
each of them equipped with a (unknown) proba distribution
At each time step:
- you choose one option
- you get a reward, distributed according to its proba distribution
At the end:
- you choose one option (you can not change anymore...)
- your reward is the expected reward associated to this option
Here we collect
information

5. What is a bandit problem ?
A finite number of time steps
A (finite) number of options,
each of them equipped with a (unknown) proba distribution
At each time step:
- you choose one option
- you get a reward, distributed according to its proba distribution
At the end:
- you choose one option (you can not change anymore...)
- your reward is the expected reward associated to this option
Here we use
information for
the final choice

6. What is a bandit problem ?
A finite number of time steps
A (finite) number of options,
each of them equipped with a (unknown) proba distribution
At each time step:
- you choose one option
- you get a reward, distributed according to its proba distribution
At the end:
- you choose one option (you can not change anymore...)
- your reward is the expected reward associated to this option
Here, we
explore

7. What is a bandit problem ?
A finite number of time steps
A (finite) number of options,
each of them equipped with a (unknown) proba distribution
At each time step:
- you choose one option
- you get a reward, distributed according to its proba distribution
At the end:
- you choose one option (you can not change anymore...)
- your reward is the expected reward associated to this option
Here, we take no risk

8. What is a bandit problem ?
A finite number of time steps
A (finite) number of options,
each of them equipped with a (unknown) proba distribution
At each time step (exploration):
- you choose one option
- you get a reward, distributed according to its proba distribution
At the end (recommendation):
- you choose one option (you can not change anymore...)
- your reward is the expected reward associated to this option

9. Which kind of bandit ?
- in the bandit literature, options are
also termed “arms”
- here the criterion is the expected reward
of the option chosen at the end
(sometimes it is the sum
of the rewards during exploration)
- we presented here stochastic bandits
(a probability distribution
per option) ==> next slide is different

10. And adversarial bandit ?
A finite number of time steps
A (finite) number of options for player 1,
and a finite number of options for player 2.
An unknown probability distribution for each pair of options
At each time step:
- you choose one option for P1 and one option for P2
- you get a reward, distributed according to the
corresponding proba distribution
At the end:
- you choose one **probabilistic** option for P1
(you can not change anymore...)
- your reward is the expected reward associated to this option,
for the worst choice by P2

11. What is meta-gaming ?
What is “strategic choice” ?
Strategic choices:
- decisions once and for all, at a high level
- ≠ from tactical level
Meta-gaming: choice at a strategic level, in games:
- choosings cards, in card games
- choosing handicap positioning, in Go
==> once and for all, at the beginning of the game

12. Example of stochastic bandit
(i.e. 1P strategic choice)
Game of Go handicap bandit problem, at each time step:
- you choose one handicap positioning
- then you simulate one game from this position
==> only one player has a strategic choice
==> stochastic bandit

13. Example of adversarial bandit
(i.e. 2P strategic choice)
Urban Rivals bandit problem, at each time step:
- you choose
- one set of cards for you (P1)
- one set of cards for P2
- then you simulate one Urban Rivals game from this position
PLAYER 1:
PLAYER 2:
==> two players have a strategic choice
==> adversarial bandit

14. Is a strategic bandit problem
different from
a classical bandit problem ?
No difference in nature
Just a much
smaller budget

15. Algorithms
Reminder:
- two algorithms needed:
- one for choosing during N exploration steps
- one for choosing during 1 recommendation step
- two settings
- one-player case
- two-player case

16. Algorithms for exploration
Uniform: test all options uniformly
Bernstein races:
- uniformly among non discarded options,
- discard options with statistical tests
Successive reject:
- uniformly among non discarded options,
- discard periodically the worst option
UCB: choose option with best average result + bonus
for options weakly sampled,
Adaptive-UCB-E: a variant of UCB aimed at removing
hyper-parameters
EXP3: empirically best option + random perturbation

17. Algorithms for recommendation
Empirically Best Arm: choose empirically best option
Most Played Arm: choose most simulated option
Successive reject:: the only non discarded option
UCB: choose option with best average result + bonus
for options weakly sampled.
LCB: choose option with best average result + malus for
options weakly sampled.
Empirical distribution of play: an option has its
frequency (during exploration) as probability (for
recommendation)
TEXP3: idem, but discard low probability options

18. Experimental results
Big boring tables of results
are in the paper.
Only a sample of most clear
results here.

19. One player case
Killall Go stones positionning

20. One player case
Killall Go stones positionning
Uncertainty
should
have
malus in
recommend.

21. One player case
Killall Go stones positionning
EXP3 for
2player
case

22. Experimental results: TEXP3
outperforms EXP3 by far
2-player case, game =
Urban Rivals (free online card game)

23. Do you know killall-Go ?
Black has stones in advance (e.g. 8 in 13x13).
If white makes life, white wins.
If black kills everything, black wins.
Black choose stones
positioning
(strategic decisions).

24. Left: human is Black and chooses E3 C4.
Right: computer is Black and chooses D3 D5.
White won both.
Human said that the computer choice D3 D5 is good.

25. Killall Go, H8 (left) H9 (right)
Left: Human Pro Player (5P) as black has 8 handicap stones.
White (computer) makes life and wins.
Right: Human Pro Player (5P) as black has 9 handicap stones
and kills everything and wins.

26. CONCLUSIONS
1 player case:
UCB for exploration,
LCB or MPA for recommendation
2 player case:
TEXP3 performs best.
Killall-Go
Win against pro with H2 in 7x7 Killall-Go as white.
Loss against pro with H2 in 7x7 Killall-Go as black.
13x13: Computer won as white with H8, lost with H9.
13x13: Computer lost as black with H8 and with H9.
Further work:
Structured bandit: some options are close to each other.
Batoo: Go with strategic choice for both players; nice test case.
Industry: choosing investments for power grid simulations – in progress.