
Submitted by
Assigned_Reviewer_1
Q1: Comments to author(s).
First provide a summary of the paper, and then address the following
criteria: Quality, clarity, originality and significance. (For detailed
reviewing guidelines, see
http://nips.cc/PaperInformation/ReviewerInstructions)
This paper studies Monte Carlo tree search in zerosum
extensive form games with perfect information and simultaneous moves. It
is proved that the MCTS algorithm converges to an approximate Nash
equilibrium under certain conditions. Empirical study confirms the formal
result.
The detailed comments are as follows. 1. Overall I
think it is a good paper which provides a sufficient condition for the
convergence of MCTS algorithms. The result is useful and the presentation
is clear. However, as the main contribution of the paper should be the
theoretical part, I have concern on the novelty of the proof technique.
2. UCB algorithms are widely used algorithms in the literature but
they are not analyzed in this paper. Given there have been some analysis
on the UCB for Trees, it will be better that the authors also consider UCB
algorithms and compares them with other algorithms in the paper. 3. It
is only proved that the propagation of the mean has good convergence.
However, in the experiments the propagation of current sample value also
seems good under different criteria. Can you give some explanation?
Q2: Please summarize your review in 12
sentences
This paper studies Monte Carlo tree search in zerosum
extensive form games with perfect information and simultaneous moves. The
findings in the paper are interesting, however, technically speaking, the
paper is not very outstanding. Submitted by
Assigned_Reviewer_3
Q1: Comments to author(s).
First provide a summary of the paper, and then address the following
criteria: Quality, clarity, originality and significance. (For detailed
reviewing guidelines, see
http://nips.cc/PaperInformation/ReviewerInstructions)
This paper analyzes the class of zerosum
perfectinformation simultaneous move games. There are some interesting
games in this class e.g. goofspiel and Tron. The authors analyze MCTS,
implemented with regretmatching for this class of games and prove
convergence to an approximate Nash equilibrium.
The paper is clear
and wellwritten, and the results are wellpresented. The main
contribution is new results in the formal analysis. Previous work, that
the authors cited [20] Lanctot et al, 2013, looks at the setup but has
more preliminary analysis. The proof is inductive  with optimal
strategies at the leaf nodes.
I think the biggest weakness in this
type of approach is that not very interesting things can be proved about
the inner nodes of the game tree. This misses out on some key ideas in
game theory: Subgame perfect nash equilibria. I think the paper can be
improved by tying the analysis to those concepts. Recommended citation:
"Existence of subgame perfect equilibria in games with simultaneous
moves". C. Harris 1990. Q2: Please summarize your review
in 12 sentences
This paper presents novel analyses of regretmatching
strategy with MTCS in zerosum perfect information simultaneous move
games. While interesting from a computational standpoint, the analyses
misses out on tying to powerful tools of subgame perfect Nash equilibria
for these class of games; something that has been wellstudied by Game
Theorists. Submitted by
Assigned_Reviewer_7
Q1: Comments to author(s).
First provide a summary of the paper, and then address the following
criteria: Quality, clarity, originality and significance. (For detailed
reviewing guidelines, see
http://nips.cc/PaperInformation/ReviewerInstructions)
The authors present an analysis for a class of Monte
Carlo tree search algorithms using essentially (with some mild
restrictions), any noregret selection method. Their primary theoretical
contribution is a proof of convergence to epsilonNash in the case of
simultaneous moves.
One weakness is that the convergence is
asymptotic (no rates are provided) which is a necessary consequence of the
authors' analysis since the selection method is assumed only to be
epsilonHannan consistent. This is not necessarily detrimental since, as
far as I can tell, there don't seem to exist any proofs of convergence for
MCTS methods for extensiveform games with simultaneous moves.
However, I think the question of rates is important, since we know
that backwards induction will give us an exact equilibrium anyway. Do the
authors know what rates are attained when a specific regret rate is
assumed for the selection method, for example, by using the \sqrt{T}
guaranteed by exp?
The authors conclude with some experiments.
They compare the approach of propagating mean reward values used in their
analysis to the standard approach of propagating the current sample value.
They find that mean values slightly underperform experimentally. Finally,
they attempt to experimentally derive some insight on the convergence
rate.
To conclude: This is a wellwritten paper that is
technically correct. I have a mild reservation regarding the ultimate
significance of the work given the lack of convergence rates. On the one
hand, MCTS methods are knows to work well in practice, so even if rates
don't follow from the authors' analysis, establishing asymptotic
convergence is a good first step. On the other hand, convergence results
are already known for MCTS with sequential moves, so the result feels a
little incremental.
Nitpicks Figure 1: leafs > leaves
Line 144: negated, not "inverted," right? Line 10 of Algorithm 1:
I think (a_1,a_2) should be (i,j), or possibly (\sigma_1,\sigma_2). This
is especially confusing since a_{i,j} was introduced as notation for the
payoffs in a singlestage matrix game. Line 172: It seems like
notation hiccups from (i,j) to a's might continue throughout the paper.
I'll stop pointing them out, but these should be edited.
Q2: Please summarize your review in 12 sentences
This was a wellwritten paper, that I enjoyed reading.
I have some reservations about its impact, given the lack of convergence
rates in the analysis.
Q1:Author
rebuttal: Please respond to any concerns raised in the reviews. There are
no constraints on how you want to argue your case, except for the fact
that your text should be limited to a maximum of 6000 characters. Note
however that reviewers and area chairs are very busy and may not read long
vague rebuttals. It is in your own interest to be concise and to the
point.
We thank all the reviewers for the thoughtful and
valuable feedback.
Response to Reviewer 1:
Concerning the
novelty of the proof technique: we provide the first convergence proof for
MCTS in simultaneous move games. In order for the proof to cover the
general class of epsilonHannan Consistent selection policies, we define
the algorithms with guaranteed exploration and repeated games with bounded
error. These notions, their properties, lemmas, and theorems are novel and
reusable in other analyses.
We agree that UCB is an important
algorithm and it is often used in practice. However, it has been shown not
to converge to a Nash equilibrium of a simple (onestate) game [17], as we
note in the second paragraph of Section 3. In fact, it can converge to a
strategy with large regret, which occurs in practice even in small
subgames of Goofspiel [20]. It is for this reason that we chose not to
include it in the empirical analysis. However, we agree that it should be
discussed and will add more explicit discussion of this point to the
paper.
Propagation of values also performs very well in
experiments. However, in the proof of Theorem 4.11, if we propagate the
current value, the nodes above the leaves will not become a matrix game
with epsilon bounded error and the proof is not applicable. A deeper
analysis is required and we hope to generalize this in future work.
Response to Reviewer 2:
We agree that subgame perfect
equilibria are important in this setting. However, we disagree with the
main objection of the reviewer that the biggest weakness in this type of
approach is that not very interesting things can be proved about the inner
nodes of the game tree. In fact, we can take an intuitive definition of
the epsilonsubgameperfect equilibrium, which would say that playing the
computed strategy from any inner node of the game ensures payoff no worse
than epsilon far from the actual value of the subgame rooted in the node.
With this definition of subgame perfection, our proof shows that the
algorithm converges to an epsilonsubgameperfect equilibrium. The proof
is by induction from leaves to the root. In order to prove that the
strategy executed from the root is an approximate equilibrium, it proves
that executing the strategy from the inner nodes forms an even tighter
approximate equilibrium. Furthermore, as reported also by [Harris 90, page
2], there are no issues with existence of the subgame perfect equilibrium
as we deal only with finite games. We did not include this discussion as
we could not find a definition of epsilonsubgameperfect equilibrium for
this class of games in literature. After this discussion, we admit it was
a mistake and that discussion about subgame perfect equilibria and
references to relevant literature for this class of games is important.
Therefore, we will add it to the final version if accepted for
publication.
Response to Reviewer 3:
The question of a
finite time regret bound is indeed very interesting and we are currently
working on it. However, unless we count in the a positive influence of the
random samples executed in MCTS before the tree is completely constructed,
at the moment we do not expect to have bounds that would support using
this approach rather than full backward induction (in theory). Taking the
random samples into account has not been done even in the sequential moves
setting and it most likely requires a completely different proof
technique. In this paper we focus rather on a clear and intuitive proof,
but we are actively investigating a finite time bound for future work.
 