# Finding and Certifying (Near-)Optimal Strategies in Black-Box Extensive-Form Games

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We developed an Monte Carlo CFR-like equilibrium-finding algorithm that converges at rate O( log(t)/t), and does not require a lower-bounded sampling vector

Abstract:

Often---for example in war games, strategy video games, and financial simulations---the game is given to us only as a black-box simulator in which we can play it. In these settings, since the game may have unknown nature action distributions (from which we can only obtain samples) and/or be too large to expand fully, it can be difficult...More

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Introduction
• Computational equilibrium finding has led to many recent breakthroughs in AI in games such as poker (Bowling et al 2015; Brown and Sandholm 2017, 2019b) where the game is fully known.
• In many applications, the game is not fully known; instead, it is given only via a simulator that permits an algorithm to play through the game repeatedly (e.g., Wellman 2006; Lanctot et al 2017).
• This, assumes the whole game to be known exactly
Highlights
• Computational equilibrium finding has led to many recent breakthroughs in AI in games such as poker (Bowling et al 2015; Brown and Sandholm 2017, 2019b) where the game is fully known
• We develop an algorithm for extensiveform game solving that enjoys many of the same properties of outcome-sampling Monte Carlo CFR (MCCFR) but works without the problematic assumption of having an a-priori uniformly-lowerbounded “sampling vector” that is required by MCCFR
• Our goal in this paper is to develop equilibrium-finding algorithms that give anytime, high-probability, instance-specific exploitability guarantees that can be computed without expanding the rest of the game tree, and are better than the generic guarantees given by the worst-case runtime bounds of algorithms like MCCFR
• Assuming that the confidence sequence is correct at time t, the pessimistic equilibrium computed by Algorithm 6.1 is an εt-equilibrium of Gt. This allows us to know when we have found an ε-equilibrium, without expanding the remainder of the game tree, even in the case when chance’s strategy is not directly observable
• We developed an MCCFR-like equilibrium-finding algorithm that converges at rate O( log(t)/t), and does not require a lower-bounded sampling vector
Methods
• The authors conducted experiments on two common benchmarks:

(1) k-rank Goofspiel. At each time t = 1, . . . , k, both players simultaneously place a bid for a prize.
• The authors conducted experiments on two common benchmarks:.
• (1) k-rank Goofspiel.
• At each time t = 1, .
• K, both players simultaneously place a bid for a prize.
• The prizes have values 1, .
• K, and are randomly shuffled.
• The valid bids are 1, .
• The higher bid wins the prize; in case of a tie, the prize is split.
• The winner of each round is made public, but the bids are not.
• The authors' experiments use k = 4
Conclusion
• The authors developed algorithms that construct high-probability certificates in games with only black-box access.
• The authors' method can be used with either an exact game solver (e.g., LP solver) as a subroutine or a regret minimizer such as MCCFR.
• Table 1 shows which algorithm the authors recommend based on the use case.
• The authors developed an MCCFR-like equilibrium-finding algorithm that converges at rate O( log(t)/t), and does not require a lower-bounded sampling vector.
• The authors' experiments show that the algorithms produce nontrivial certificates with very few samples.
Summary
• ## Introduction:

Computational equilibrium finding has led to many recent breakthroughs in AI in games such as poker (Bowling et al 2015; Brown and Sandholm 2017, 2019b) where the game is fully known.
• In many applications, the game is not fully known; instead, it is given only via a simulator that permits an algorithm to play through the game repeatedly (e.g., Wellman 2006; Lanctot et al 2017).
• This, assumes the whole game to be known exactly
• ## Objectives:

After t playthroughs, to efficiently maintain a strategy profile σt and bounds εi,t on the equilibrium gap of each player’s strategy that are correct with probability 1 − 1/ poly(t).
• ## Methods:

The authors conducted experiments on two common benchmarks:

(1) k-rank Goofspiel. At each time t = 1, . . . , k, both players simultaneously place a bid for a prize.
• The authors conducted experiments on two common benchmarks:.
• (1) k-rank Goofspiel.
• At each time t = 1, .
• K, both players simultaneously place a bid for a prize.
• The prizes have values 1, .
• K, and are randomly shuffled.
• The valid bids are 1, .
• The higher bid wins the prize; in case of a tie, the prize is split.
• The winner of each round is made public, but the bids are not.
• The authors' experiments use k = 4
• ## Conclusion:

The authors developed algorithms that construct high-probability certificates in games with only black-box access.
• The authors' method can be used with either an exact game solver (e.g., LP solver) as a subroutine or a regret minimizer such as MCCFR.
• Table 1 shows which algorithm the authors recommend based on the use case.
• The authors developed an MCCFR-like equilibrium-finding algorithm that converges at rate O( log(t)/t), and does not require a lower-bounded sampling vector.
• The authors' experiments show that the algorithms produce nontrivial certificates with very few samples.
Tables
• Table1: Algorithms we suggest by use case in two-player zero-sum games. Sampling-limited means that the black-box game simulator is relatively slow or expensive compared to solving the pseudogames. Compute-limited means that the simulator is fast or cheap compared to solving the pseudogames. In general-sum games, only Algorithm 7.2 is usable
Reference
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• A.6 Proposition 7.5 Identical to Theorem 1 of Farina, Kroer, and Sandholm (2020).
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